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\documentclass{article}
\usepackage{amssymb}
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{#1}\hyperlink{Our general topics:}{\hfil$\leftarrow$}}\end{center}}}
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\begin{document}
\raggedright
%%tth:\special{html: <A NAME="Top of file"></A>}
\pagestyle{myfooters}
%\pagestyle{plain}
\thispagestyle{empty}
%%tth:\special{html:<title> Assessing Risks</title>}
%Slide 1
\title{\ \newline \newline{\LARGE\bf Assessing Risks}\newline \newline \newline}
\author{Tom Carter
\newline
\newline
\newline
\tthdump{\href{http://cogs.csustan.edu/\~tom/SFI-CSSS}{http://cogs.csustan.edu/\~\ tom/SFI-CSSS}}
%%tth:\href{http://cogs.csustan.edu/~tom/SFI-CSSS}{http://cogs.csustan.edu/\~tom/SFI-CSSS}
\vfill
Complex Systems
\newline
}
\date{July, 2002}
\maketitle
%Slide 2
\sectionhead{Our general topics:}
%%tth:\begin{itemize}
%%tth:\item
\tthdump{\hyperlink{Assessing risks}
{\ $\circledcirc$ Assessing risks\newline}}
%%tth:\makehyperlink{Assessing risks}
%%tth:\item
\tthdump{\hyperlink{Using Bayes' Theorem}
{$\circledcirc$ Using Bayes' Theorem\newline}}
%%tth:\makehyperlink{Using Bayes' Theorem}
%%tth:\item
\tthdump{\hyperlink{The `doomsday argument'}
{$\circledcirc$ The `doomsday argument'\newline}}
%%tth:\makehyperlink{The `doomsday argument'}
%%tth:\item
\tthdump{\hyperlink{References}
{$\circledcirc$ References\newline}}
%%tth:\makehyperlink{References}
%%tth:\end{itemize}
\pagedone
\quotesection{The quotes}
%%tth:\begin{itemize}
%%tth:\item
\tthdump{\hyperlink{Science and wisdom}
{\ $\circledcirc$ Science and wisdom\newline}}
%%tth:\makehyperlink{Science and wisdom}
%%tth:\item
\tthdump{\hyperlink{Miracles}
{$\circledcirc$ Miracles\newline}}
%%tth:\makehyperlink{Miracles}
%%tth:\end{itemize}
%\thepage
\tthdump{\hyperlink{Our general topics:}{\hfil To topics $\leftarrow$}}
%%tth:{\special{html: <a href="\#Top of file"> Back to top of file</a>}}
\pagedone
%Slide 3
\quotesection{Science and wisdom}
%%tth:\begin{quote}
``Science is organized knowledge. Wisdom is organized life.''
- Immanuel Kant
``My own suspicion is that the universe is not only stranger than we suppose,
but stranger than we can suppose.''
- John Haldane
``Not everything that can be counted counts, and not everything that counts can be counted.''
- Albert Einstein (1879-1955)
``The laws of probability, so true in general, so fallacious in particular .''
- Edward Gibbon
%%tth:\end{quote}
\pagedone
\sectionhead{Assessing risks}
\begin{itemize}
\item The task of assessing risks in our lives is notoriously difficult.
One thing we can try to do is calculate the probabilities of various
events happening.
Unfortunately, humans (even well trained scientists, mathematicians,
and probabilists) often do a poor job of estimating probabilities.
There is an apocryphal story of a statistician who always packed a bomb
in his luggage when flying in a plane. When asked why, he explained that
he knew that if the probability of there being one bomb on a plane was
$\frac{1}{1000}$, then the probability of there being two
bombs would be $$\frac{1}{1000} * \frac{1}{1000} = \frac{1}{1,000,000},$$
and he felt much safer with the $\frac{1}{1,000,000}$ chance \ldots
\pagedone
\item Obviously there is something wrong with this statistician's calculation
of probabilities.
I sometimes wonder whether we in the US, holding on to our nuclear
weapons, have fallen into a similar sort of confusion about calculating
risks and ``feeling safer.''
\item I won't go through much, but some probability basics, where $a$ and $b$ are events: \newline
$ P(not\ a) = 1 - P(a).$\newline
$ P(a\ or\ b) = P(a) + P(b) - P(a\ \mathrm{and}\ b).$
The probability of two events happening, $P(a\ \mathrm{and}\ b)$ (often denoted
by $P(a, b)$), can be quite difficult to calculate, since we often do not
know how the events $a$ and $b$ are related to each other.
\pagedone
\item Conditional probability: \newline\newline
$ P(a \vert b) $ is the probability of $a$, given that we know $b$.
The joint probability of both $a$ and $b$ is given by:
$$P(a, b) = P(a \vert b) P(b).$$
Since $P(a, b) = P(b, a)$, we have Bayes' Theorem:
$$P(a \vert b)P(b) = P(b \vert a) P(a),$$
or
$$P(a \vert b) = \frac{P(b \vert a) P(a)}{P(b)}.$$
\item If two events $a$ and $b$ are such that
$$P(a \vert b) = P(a),$$
we say that the events $a$ and $b$ are {\em independent}.
\pagedone
\item Note that if $a$ and $b$ are {\em independent},
$$P(a \vert b) = P(a),$$
then from
Bayes' Theorem, we will also have that
$$P(b \vert a) = P(b),$$
and therefore,
$$P(a, b) = P(a \vert b)P(b) = P(a)P(b).$$
This last equation is often taken as the definition of {\em independence}.
\item We have in essence begun here the development of a mathematized methodology for drawing
inferences about the world from uncertain knowledge.
\end{itemize}
\pagedone
\quotesection{Miracles}
%%tth:\begin{quote}
``The opposite of a correct statement is a false statement. The opposite of a profound truth
may well be another profound truth.''\newline
- Niels Bohr (1885-1962)
``Groundless hope, like unconditional love, is the only kind worth having.''\newline
- John Perry Barlow
``There are only two ways to live your life. One is as though nothing is a miracle.
The other is as though everything is a miracle.''\newline
- Albert Einstein (1879-1955)
``The Universe is full of magical things patiently waiting for our wits to
grow sharper.''\newline
-Eden Phillpotts
\pagedone
\
\newline \newline \newline
``Nature uses only the longest threads to weave her patterns, so that each
small piece of her fabric reveals the organization of the entire tapestry.''
- Richard Feynman
%%tth:\end{quote}
\pagedone
\sectionhead{Using Bayes' Theorem}
\begin{itemize}
\item A quick example: \newline
Suppose that you are asked by a friend to help them understand the results of
a genetic screening test they have taken. They have been told that they
have tested positive, and that the test is 99\% accurate. What is the
probability that they actually have the anomaly?
%% (Hint: We don't yet have enough information \ldots)
%% \pagedone
You do some research, and find out that the test screens for a genetic anomaly
that is believed to occur in one person out of 100,000 on average. The lab that
does the tests guarantees that the test is 99\% accurate. You push the question,
and find that the lab says that one percent of the time, the test falsely reports
the absence of the anomaly when it is there, and one percent of the time the test
falsely reports the presence of the anomaly when it is not there.
The test has come
back positive for your friend. How worried should they be? Given this much
information, what can you calculate as the probability they actually have the anomaly?
In general, there are four possible situations for an individual being tested:
\begin{enumerate}
\item Test positive (Tp), and have the anomaly (Ha).
\item Test negative (Tn), and don't have the anomaly (Na).
\item Test positive (Tp), and don't have the anomaly (Na).
\item Test negative (Tn), and have the anomaly (Ha).
\end{enumerate}
\pagedone
We would like to calculate for our friend the probability they actually have
the anomaly (Ha), given that they have tested positive (Tp):
$$P(Ha \vert Tp).$$
We can do this using Bayes' Theorem.
We can calculate:
\begin{eqnarray*}
P(Ha \vert Tp) & = & \frac{P(Tp \vert Ha) * P(Ha)}{P(Tp)}.
\end{eqnarray*}
We need to figure out the three items on the right side of the equation. We can
do this by using the information given.
\pagedone
Suppose the screening test was done on
10,000,000 people. Out of these $10^7$ people, we expect there to be
$10^7/10^5 = 100$ people with the anomaly, and 9,999,900 people without the
anomaly. According to the lab, we would expect the test results to be:
\begin{itemize}
\item Test positive (Tp), and have the anomaly (Ha):
$$ 0.99 * 100 = 99\ \mathrm{people}.$$
\item Test negative (Tn), and don't have the anomaly (Na):
$$ 0.99 * 9,999,900 = 9,899,901\ \mathrm{people}.$$
\item Test positive (Tp), and don't have the anomaly (Na):
$$ 0.01 * 9,999,900 = 99,999\ \mathrm{people}.$$
\item Test negative (Tn), and have the anomaly (Ha):
$$ 0.01 * 100 = 1\ \mathrm{person}.$$
\end{itemize}
\pagedone
Now let's put the the pieces together:
\begin{eqnarray*}
P(Ha) & = & \frac{1}{100,000}\\
\\
& = & 10^{-5}\\
\\
P(Tp) & = & \frac{99 + 99,999}{10^7}\\
\\
& = & \frac{100,098}{10^7}\\
\\
& = & 0.0100098\\
\\
P(Tp \vert Ha) & = & 0.99
\end{eqnarray*}
\pagedone
Thus, our calculated probability that our friend actually has the anomaly is:
\begin{eqnarray*}
P(Ha \vert Tp) & = & \frac{P(Tp \vert Ha) * P(Ha)}{P(Tp)}\\
\\
& = & \frac{0.99 * 10^{-5}}{0.0100098}\\
\\
& = & \frac{9.9 * 10^{-6}}{1.00098 * 10^{-2}}\\
\\
& = & 9.890307 * 10^{-4}\\
\\
& < & 10^{-3}
\end{eqnarray*}
In other words, our friend, who has tested {\em positive}, with a test that
is 99\% correct, has less that one chance in 1000 of actually having the anomaly!
%% \pagedone
%% \item There are a variety of questions we could ask now, such as, ``For this anomaly,
%% how accurate would the test have to be for there to be a greater than 50\%
%% probability that someone who tests positive actually has the anomaly?''
%%
%% For this, we need fewer false positives than true positives. Thus, in
%% the example, we would need fewer than 100 false positives out of the
%% 9,999,900 people who do not have the anomaly. In other words, the proportion
%% of those without the anomaly for whom the test would have to be correct
%% would need to be greater than:
%% $$\frac{9,999,800}{9,999,900} = 99.999\%$$
%% \pagedone
%% \item Another question we could ask is, ``How prevalent would an anomaly have to be
%% in order for a 99\% accurate test (1\% false positive and 1\% false negative)
%% to give a greater than 50\% probability of actually having the anomaly when
%% testing positive?''
%%
%% Again, we need fewer false positives than true positives. We would therefore
%% need the actual occurrence to be greater than 1 in 100 (each false positive
%% would be matched by at least one true positive, on average).
%% \pagedone
%% \item Note that the current population of the US is about 280,000,000 and the
%% current population of the world is about 6,200,000,000. Thus, we could
%% expect an anomaly that affects 1 person in 100,000 to affect about 2,800
%% people in the US, and about 62,000 people worldwide, and one affecting
%% one person in 100 would affect 2,800,000 people in the US, and 62,000,000
%% people worldwide \ldots
%% %%\pagedone
%% \item Another example: suppose the test were not so accurate? Suppose the test
%% were 80\% accurate (20\% false positive and 20\% false negative). Suppose
%% that we are testing for a condition expected to affect 1 person in 100.
%% What would be the probability that a person testing positive actually has
%% the condition?
%% \pagedone
%% We can do the same sort of calculations.
%%
%% Let's use 1000
%% people this time. Out of this sample, we would expect 10 to have the
%% condition.
%% \begin{itemize}
%% \item Test positive (Tp), and have the condition (Ha):
%% $$ 0.80 * 10 = 8\ \mathrm{people}.$$
%% \item Test negative (Tn), and don't have the condition (Na):
%% $$ 0.80 * 990 = 792\ \mathrm{people}.$$
%% \item Test positive (Tp), and don't have the condition (Na):
%% $$ 0.20 * 990 = 198\ \mathrm{people}.$$
%% \item Test negative (Tn), and have the condition (Ha):
%% $$ 0.20 * 10 = 2\ \mathrm{people}.$$
%% \end{itemize}
%% \pagedone
%% Now let's put the the pieces together:
%% \begin{eqnarray*}
%% P(Ha) & = & \frac{1}{100}\\
%% \\
%% & = & 10^{-2}\\
%% \\
%% P(Tp) & = & \frac{8 + 198}{10^3}\\
%% \\
%% & = & \frac{206}{10^3}\\
%% \\
%% & = & 0.206\\
%% \\
%% P(Tp \vert Ha) & = & 0.80
%% \end{eqnarray*}
%% \pagedone
%% Thus, our calculated probability that our friend actually has the anomaly is:
%% \begin{eqnarray*}
%% P(Ha \vert Tp) & = & \frac{P(Tp \vert Ha) * P(Ha)}{P(Tp)}\\
%% \\
%% & = & \frac{0.80 * 10^{-2}}{0.206}\\
%% \\
%% & = & \frac{8 * 10^{-3}}{2.06 * 10^{-1}}\\
%% \\
%% & = & 3.883495 * 10^{-2}\\
%% \\
%% & < & .04
%% \end{eqnarray*}
%%
%% In other words, one who has tested {\em positive}, with a test that
%% is 80\% correct, has less that one chance in 25 of actually having this
%% condition. (Imagine for a moment, for example, that this is a drug test
%% being used on employees of some corporation \ldots)
%% \pagedone
%% \item We could ask the same kinds of questions we asked before:
%% \begin{enumerate}
%% \item How accurate would the test have to be to get a better than 50\%
%% chance of actually having the condition when testing positive?
%%
%% (99\%)
%% \item For an 80\% accurate test, how frequent would the condition
%% have to be to get a better than 50\% chance?
%%
%% (1 in 5)
%% \end{enumerate}
\pagedone
\item Some questions:
\begin{enumerate}
\item Are examples like this realistic? If not, why not?
\item What sorts of things could we do to improve our results?
\item Would it help to repeat the test? For example, if the
probability of a false positive is 1 in 100, would that mean
that the probability of two false positives on the same
person would be 1 in 10,000 ($\frac{1}{100} * \frac{1}{100}$)?
If not, why not?
\item In the case of a medical condition such as a genetic anomaly,
it is likely that the test would not be applied randomly, but would
only be ordered if there were other symptoms suggesting the anomaly.
How would this affect the results?
\end{enumerate}
\end{itemize}
\pagedone
\sectionhead{The `doomsday argument'}
\begin{itemize}
\item There is a line of reasoning called the ``doomsday argument'' (attributed
to Brandon Carter in the 1980's) suggesting that we consistently
underestimate the likelihood that the human race will end soon.
What follows is a brief summary of the general theme of the argument.
\item Suppose you have before you two urns, and are told that one contains
ten balls labeled 1 through 10, and the other contains one thousand
balls labeled 1 through 1000. You choose one of the urns at random.
A ball is drawn at random from the urn you chose, and the ball drawn
has on it the label `7'. What is the probability that the urn you chose
is the one with ten balls in it?
In the beginning, before drawing the ball labeled `7', we have
$$P(ten) = P(thousand) = \frac{1}{2}.$$
After drawing the ball, however, we can use Bayes' theorem to calculate:
\begin{eqnarray*}
P(ten\ \vert\ \mathrm{draw}\ `7`) & = &
\frac{P(\mathrm{draw}\ `7`\ \vert\ ten) * P(ten)}{P(\mathrm{draw}\ `7`)}\\
& = & \frac{\frac{1}{10} * \frac{1}{2}}
{\frac{1}{2}*\frac{1}{10} + \frac{1}{2}*\frac{1}{1000}}\\
& = & \frac{\frac{1}{20}}
{\frac{1}{20} + \frac{1}{2000}}\\
& = & \frac{\frac{1}{20}}
{\frac{101}{2000}}\\
& & \\
& = & \frac{2000}{2020}\\
& & \\
& = & 0.990099
\end{eqnarray*}
\pagedone
\item Now the `doomsday argument' \ldots Consider the two possibilities:
D: We humans destroy ourselves before we leave earth and colonize the
universe.
U: We colonize the universe.
We now estimate: In the case `D', no more than $100,000,000,000 = 10^{11}$
humans will ever live, and I am one of those $10^{11}$.
In the case `U', many more humans will live, say $10^{15}$.
I know that I am among the first $10,000,000,000$ people to live, so in
terms of human birth order, we can say that my `label' is say $8,000,000,000$
(call this `L8B').
\pagedone
We now use Bayes' theorem:
\begin{eqnarray*}
P(D\ \vert\ \mathrm{L8B}) & = &
\frac{P(\mathrm{L8B}\ \vert\ D) * P(D)}{P(\mathrm{L8B})}\\
& = & \frac{\frac{1}{10^{11}} * P(D)}
{P(D)*10^{-11} + (1 - P(D))*10^{-15}}\\
& = & \frac{\frac{1}{10^{11}} * P(D)}
{\frac{10^4*P(D)+(1 - P(D))}{10^{15}}}\\
& = & \frac{10^4*P(D)}
{10^4*P(D)+(1-P(D))}
\end{eqnarray*}
Now put in a value for $P(D)$, and see what happens:
If we start with an estimate $P(D) = \frac{1}{100}$, then
\begin{eqnarray*}
P(D\ \vert\ \mathrm{L8B}) & = & \frac{10^4*10^{-2}}
{10^4*10^{-2}+(1-10^{-2})}\\
& = & \frac{10^2}
{10^2 + 0.99}\\
& = & 0.990197
\end{eqnarray*}
\pagedone
\item In other words, if our original estimate for the likelihood of `doomsday'
was $P(D) = \frac{1}{100}$, we should revise that estimate upward to
0.990197!
You can try other values for the various pieces on your own \ldots
\end{itemize}
\pagedone
\pagedone
\footnotesize
\bibliographystyle{plain}
\tthdump{\hypertarget{References}{}\hyperlink{Our general topics:}{\hfil \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ To top $\leftarrow$}}
%%tth:{\special{html: <A NAME="References"></A><a href="\#Top of file"> Top</a>}}
\vspace{-1.0 in}
\begin{thebibliography}{12}
%%tth:{\special{html: <font size="+0">}}
\bibitem{bostrom1}
Bostrom, Nick,
{\em Existential Risks -
Analyzing Human Extinction Scenarios and Related Hazards},
http://www.nickbostrom.com/existential/risks.html.
\bibitem{bostrom2}
Bostrom, Nick,
{\em The Doomsday Argument: a Literature Review},
http://anthropic-principle.com/preprints/lit/index.html.
\bibitem{feller}
Feller, W.,
{\em An Introduction to Probability Theory and Its Applications},
Wiley, New York,1957.
\bibitem{hamming2}
Hamming, R. W.,
{\em Coding and information theory}, 2nd ed,
Prentice-Hall, Englewood Cliffs, 1986.
\bibitem{neumann}
von Neumann, John,
Probabilistic logic and the synthesis of reliable organisms
from unreliable components,
in {\em automata studies( Shanon,McCarthy eds)}, 1956 .
\bibitem{pierce}
Pierce, John R.,
{\em An Introduction to Information Theory -- Symbols, Signals and Noise},
(second revised edition),
Dover Publications, New York, 1980.
\end{thebibliography}
\tthdump{\hyperlink{Our general topics:}{\hfil To top $\leftarrow$}}
%%tth:{\special{html: <a href="\#Top of file"> Back to top of file</a>}}
%
% \sectionhead{On-line references}
%
%
% Some of the references listed above are available on line. They are listed again here for easy access:
%
% %\bibitem{abrams2}
% Abrams D S and Lloyd S,
% Non-Linear Quantum Mechanics implies Polynomial Time
% solution for NP-complete and $\#$P problems,
% %in {\it LANL e-print} quant-ph/9801041, http://xxx.lanl.gov (1998)
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9801041}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9801041}
% {http://xxx.lanl.gov/abs/quant-ph/9801041}
%
%
% %\bibitem{aharonov5}
% Aharonov D, Beckman D, Chuang I and Nielsen M,
% What Makes Quantum Computers Powerful?
% \hyperref{http://wwwcas.phys.unm.edu/\~mnielsen/science.html}{}{}
% %\hyperURL{http}{wwwcas.phys.unm.edu/~mnielsen}{science.html}
% {http://wwwcas.phys.unm.edu/\~mnielsen/science.html}
% %
% %
%
% %\bibitem{decoherence2}
% Chuang I L, Laflamme R and Paz J P,
% Effects of Loss and Decoherence on a Simple Quantum Computer,
% %in {\it LANL e-print} quant-ph/9602018, http://xxx.lanl.gov (1996)
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9602018}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9602018}
% {http://xxx.lanl.gov/abs/quant-ph/9602018}
%
% %\bibitem{grover2}
% Grover L K,
% A framework for fast quantum mechanical algorithms,
% %in {\it LANL e-print} quant-ph/9711043, http://xxx.lanl.gov (1997)
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9711043}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9711043}
% {http://xxx.lanl.gov/abs/quant-ph/9711043}
%
% %\bibitem{grover4}
% Grover L K,
% A fast quantum mechanical algorithm for estimating the median,
% %in {\it LANL e-print} quant-ph/9607024, http://xxx.lanl.gov (1997)
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9607024}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9607024}
% {http://xxx.lanl.gov/abs/quant-ph/9607024}
%
%
% %\bibitem{knill4}
% Knill E, Laflamme R and Zurek W H 1997
% Resilient quantum computation: error models and thresholds
% %in {\it LANL e-print} quant-ph/9702058, http://xxx.lanl.gov (1997)
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9702058}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9702058}
% {http://xxx.lanl.gov/abs/quant-ph/9702058}
%
% \pagedone
%
%
% %\bibitem{preskill2}
% Preskill J 1997
% Fault tolerant quantum computation,
% %in {\it LANL e-print} quant-ph/9712048, http://xxx.lanl.gov (1997),
% to appear in {\it Introduction to Quantum
% Computation}, edited by H.-K. Lo, S. Popescu, and T. P. Spiller
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9712048}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9712048}
% {http://xxx.lanl.gov/abs/quant-ph/9712048}
%
% %\bibitem{preskill3}
% Preskill J, Kitaev A, Course notes for Physics 229, Fall 1998, Caltech Univ.,
% \hyperref{http://www.theory.caltech.edu/people/preskill/ph229}{}{}
% %\hyperURL{http}{www.theory.caltech.edu/people/preskill}{ph229}
% {http://www.theory.caltech.edu/people/preskill/ph229}
%
%
% %\bibitem{rieffel}
% Rieffel E, Polak W
% An Introduction to Quantum Computing for Non-Physicists
% %{\it LANL e-print} quant-ph/9809016, http://xxx.lanl.gov (1998),
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9809016}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9809016}
% {http://xxx.lanl.gov/abs/quant-ph/9809016}
%
% %\bibitem{Steane-97}
% Steane A,
% Quantum Computation, Reports on Progress in Physics 61 (1998) 117,
% %(preprint in {\it LANL e-print} quant-ph/9708022, http://xxx.lanl.gov)
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9708022}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9708022}
% {http://xxx.lanl.gov/abs/quant-ph/9708022}
%
% %\bibitem{zalka2}
% Zalka C,
% Grover's quantum searching algorithm is optimal,
% %in {\it LANL e-print} quant-ph/9711070http://xxx.lanl.gov (1997)
% \hyperref{http://xxx.lanl.gov/abs/quant-ph/9711070}{}{}
% %\hyperURL{http}{xxx.lanl.gov/abs/quant-ph}{9711070}
% {http://xxx.lanl.gov/abs/quant-ph/9711070}
%
\end{document}
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https://ctan.math.washington.edu/tex-archive/info/examples/PSTricks_en/29-01-8.ltx
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washington.edu
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CC-MAIN-2022-27
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| 244,030,103 | 1,132 |
%%
%% A DANTE-Edition example
%%
%% Example 29-01-8 on page 556.
%%
%% Copyright (C) 2011 Herbert Voss
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
%%
%%
%% ====
% Show page(s) 1
%%
\documentclass[]{article}
\pagestyle{empty}
\setlength\textwidth{177.70511pt}
\setlength\parindent{0pt}
\usepackage{pst-eucl}
\begin{document}
\begin{pspicture}(-1.5,-1.5)(1.5,1.5)
\psset{linecolor=black,CodeFig=true}
\pstGeonode[PosAngle={-90,90,90,-90}](-1.25,-1.25){A}%
(-1.25,1.25){B}(1.25,1.25){C}(1.25,-1.25){D}
\pstLineAB{A}{B} \pstLineAB{B}{C}
\pstLineAB{C}{D} \pstLineAB{D}{A}
\pstRightAngle{A}{B}{C}
\pstRightAngle[RightAngleSize=1]{B}{C}{D}
\pstRightAngle[RightAngleType=german]{C}{D}{A}
\pstRightAngle[RightAngleType=german,
RightAngleSize=.75]{D}{A}{B}
\end{pspicture}
\end{document}
|
https://texample.net/media/tikz/examples/TEX/spiderweb-diagram.tex
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texample.net
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CC-MAIN-2023-06
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text/x-tex
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text/x-matlab
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crawl-data/CC-MAIN-2023-06/segments/1674764500392.45/warc/CC-MAIN-20230207071302-20230207101302-00598.warc.gz
| 567,375,653 | 2,026 |
% Spiderweb Diagram
%
% Author: Dominik Renzel
% Date; 2009-11-11
\documentclass{article}
%%%<
\usepackage{verbatim}
\usepackage[active,tightpage,floats]{preview}
\setlength\PreviewBorder{5pt}%
%%%>
\begin{comment}
:Title: Spiderweb diagram
Such a diagram defines a set of dimensions D = {D1,...,Dn} and a common scale unit
range from 0 to a maximal value U. Each individual sample contains a sequence of
pairs (Dx,Vx) with 0 <= Vx <= U for all Dx in D.
The diagram is rendered as a spiderweb, where the D dimension axes meet each
other in the origin in an angle of 360/D and are each divided in U units. Each
sample is rendered as half-opaque colored path along the particular value unit nodes
on each dimension. Overlapping parts will be rendered in a composite color.
\end{comment}
\usepackage{tikz}
\usetikzlibrary{shapes}
\begin{document}
\newcommand{\D}{7} % number of dimensions (config option)
\newcommand{\U}{7} % number of scale units (config option)
\newdimen\R % maximal diagram radius (config option)
\R=3.5cm
\newdimen\L % radius to put dimension labels (config option)
\L=4cm
\newcommand{\A}{360/\D} % calculated angle between dimension axes
\begin{figure}[htbp]
\centering
\begin{tikzpicture}[scale=1]
\path (0:0cm) coordinate (O); % define coordinate for origin
% draw the spiderweb
\foreach \X in {1,...,\D}{
\draw (\X*\A:0) -- (\X*\A:\R);
}
\foreach \Y in {0,...,\U}{
\foreach \X in {1,...,\D}{
\path (\X*\A:\Y*\R/\U) coordinate (D\X-\Y);
\fill (D\X-\Y) circle (1pt);
}
\draw [opacity=0.3] (0:\Y*\R/\U) \foreach \X in {1,...,\D}{
-- (\X*\A:\Y*\R/\U)
} -- cycle;
}
% define labels for each dimension axis (names config option)
\path (1*\A:\L) node (L1) {\tiny Security};
\path (2*\A:\L) node (L2) {\tiny Content Quality};
\path (3*\A:\L) node (L3) {\tiny Performance};
\path (4*\A:\L) node (L4) {\tiny Stability};
\path (5*\A:\L) node (L5) {\tiny Usability};
\path (6*\A:\L) node (L6) {\tiny Generality};
\path (7*\A:\L) node (L7) {\tiny Popularity};
% for each sample case draw a path around the web along concrete values
% for the individual dimensions. Each node along the path is labeled
% with an identifier using the following scheme:
%
% D<d>-<v>, dimension <d> a number between 1 and \D (#dimensions) and
% value <v> a number between 0 and \U (#scale units)
%
% The paths will be drawn half-opaque, so that overlapping parts will be
% rendered in a composite color.
% Example Case 1 (red)
%
% D1 (Security): 0/7; D2 (Content Quality): 5/7; D3 (Performance): 0/7;
% D4 (Stability): 6/7; D5 (Usability): 0/7; D6 (Generality): 5/7;
% D7 (Popularity): 0/7
\draw [color=red,line width=1.5pt,opacity=0.5]
(D1-0) --
(D2-5) --
(D3-0) --
(D4-6) --
(D5-0) --
(D6-5) --
(D7-0) -- cycle;
% Example Case 2 (green)
%
% D1 (Security): 2/7; D2 (Content Quality): 2/7; D3 (Performance): 5/7;
% D4 (Stability): 1/7; D5 (Usability): 4/7; D6 (Generality): 1/7;
% D7 (Popularity): 7/7
\draw [color=green,line width=1.5pt,opacity=0.5]
(D1-2) --
(D2-2) --
(D3-5) --
(D4-1) --
(D5-4) --
(D6-1) --
(D7-7) -- cycle;
% Example Case 3 (blue)
%
% D1 (Security): 1/7; D2 (Content Quality): 7/7; D3 (Performance): 4/7;
% D4 (Stability): 4/7; D5 (Usability): 3/7; D6 (Generality): 5/7;
% D7 (Popularity): 2/7
\draw [color=blue,line width=1.5pt,opacity=0.5]
(D1-1) --
(D2-7) --
(D3-4) --
(D4-4) --
(D5-3) --
(D6-5) --
(D7-2) -- cycle;
\end{tikzpicture}
\caption{Spiderweb Diagram (\D~Dimensions, \U-Notch Scale, 3 Samples)}
\label{fig:spiderweb}
\end{figure}
\end{document}
|
https://mis.hiroshima-u.ac.jp/English/application.tex
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hiroshima-u.ac.jp
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application/x-tex
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crawl-data/CC-MAIN-2022-33/segments/1659882570741.21/warc/CC-MAIN-20220808001418-20220808031418-00042.warc.gz
| 382,633,989 | 1,156 |
\documentclass[a4paper,12pt]{jarticle}
\pagestyle{empty} % ページの有無
%\setlength{\textheight}{200mm}
\setlength{\textwidth}{140mm}
\setlength{\footskip}{0mm} % 下マージン
\setlength{\topmargin}{0mm} % 上マージン
\setlength{\oddsidemargin}{5mm} % 奇数ページの左マージン
\setlength{\evensidemargin}{-15mm} % 偶数ページの左マージン
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%namelist environment
%from: \begin{namelist}{width}
\newcommand{\namelistlabel}[1]{\mbox{#1}\hfil}
\newenvironment{namelist}[1]{%
\begin{list}{}
{\let\maketable\namelistlabel
\settowidth{\labelwidth}{#1}
\setlength{\leftmargin}{1.1\labelwidth}}
% \setlength{\itemindent}{-5mm}}
}{%
\end{list}}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{document}
\hspace{4cm}
Technical Report 発行申請書 \\
\vspace{5mm}
\hspace{10cm}
2000 年 1 月 1 日 % 申請日
\vspace{1cm}
Technical Report No. \hspace{3cm} 希望部数 33部 % No.は、記入不要。最低部数は33部。
\vspace{5mm}
Author: % 著者
\vspace{5mm}
Title: % 論文タイトル
\vspace{5mm}
Abstract: % 論文の概要（英文が望ましい。）
\end{document}
|
https://ctan.math.washington.edu/tex-archive/info/examples/PSTricks_7_de/14-01-31.ltx
|
washington.edu
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text/x-tex
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| 232,116,865 | 1,043 |
%%
%% Ein Beispiel der DANTE-Edition
%%
%% 7. Auflage
%%
%% Beispiel 14-01-31 auf Seite 197.
%%
%% Copyright (C) 2016 Herbert Voss
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
%%
%%
%% ====
% Show page(s) 1
%%
%%
\documentclass[]{exaarticle}
\pagestyle{empty}
\setlength\textwidth{193.16928pt}
\setlength\parindent{0pt}
\usepackage{pst-plot}
%StartShownPreambleCommands
\usepackage{pst-plot}
%StopShownPreambleCommands
\begin{document}
\begin{pspicture}(-0.5,-0.5)(3,2)
\psaxes[showorigin=false]{->}(3,2)
\end{pspicture}
\end{document}
|
https://sandbox.amusewiki.org/library/max-sartin-raffaele-schiavina-esperienza-it.tex
|
amusewiki.org
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CC-MAIN-2019-18
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application/x-tex
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crawl-data/CC-MAIN-2019-18/segments/1555578527566.44/warc/CC-MAIN-20190419101239-20190419123239-00549.warc.gz
| 545,394,380 | 3,018 |
\documentclass[DIV=12,%
BCOR=10mm,%
headinclude=false,%
footinclude=false,%
fontsize=11pt,%
twoside,%
paper=a4]%
{scrartcl}
\usepackage{fontspec}
\setmainfont[Script=Latin]{Linux Libertine O}
\setsansfont[Script=Latin,Scale=MatchLowercase]{CMU Sans Serif}
\setmonofont[Script=Latin,Scale=MatchLowercase]{CMU Typewriter Text}
\let\chapter\section
% global style
\pagestyle{plain}
\usepackage{microtype} % you need an *updated* texlive 2012, but harmless
\usepackage{graphicx}
\usepackage{alltt}
\usepackage{verbatim}
% http://tex.stackexchange.com/questions/3033/forcing-linebreaks-in-url
\PassOptionsToPackage{hyphens}{url}\usepackage[hyperfootnotes=false,hidelinks,breaklinks=true]{hyperref}
\usepackage{bookmark}
\usepackage[shortlabels]{enumitem}
\usepackage{tabularx}
\usepackage[normalem]{ulem}
\def\hsout{\bgroup \ULdepth=-.55ex \ULset}
% https://tex.stackexchange.com/questions/22410/strikethrough-in-section-title
% Unclear if \protect \hsout is needed. Doesn't looks so
\DeclareRobustCommand{\sout}[1]{\texorpdfstring{\hsout{#1}}{#1}}
\usepackage{wrapfig}
\usepackage{indentfirst}
% remove the numbering
\setcounter{secnumdepth}{-2}
% remove labels from the captions
\renewcommand*{\captionformat}{}
\renewcommand*{\figureformat}{}
\renewcommand*{\tableformat}{}
\KOMAoption{captions}{belowfigure,nooneline}
\addtokomafont{caption}{\centering}
\usepackage{polyglossia}
\setmainlanguage{italian}
% footnote handling
\usepackage[fragile]{bigfoot}
\usepackage{perpage}
\DeclareNewFootnote{default}
\DeclareNewFootnote{B}
\MakeSorted{footnoteB}
\renewcommand*\thefootnoteB{(\arabic{footnoteB})}
\deffootnote[3em]{0em}{4em}{\textsuperscript{\thefootnotemark}~}
% avoid breakage on multiple <br><br> and avoid the next [] to be eaten
\newcommand*{\forcelinebreak}{\strut\\*{}}
\newcommand*{\hairline}{%
\bigskip%
\noindent \hrulefill%
\bigskip%
}
% reverse indentation for biblio and play
\newenvironment*{amusebiblio}{
\leftskip=\parindent
\parindent=-\parindent
\smallskip
\indent
}{\smallskip}
\newenvironment*{amuseplay}{
\leftskip=\parindent
\parindent=-\parindent
\smallskip
\indent
}{\smallskip}
\newcommand*{\Slash}{\slash\hspace{0pt}}
\addtokomafont{disposition}{\rmfamily}
\addtokomafont{descriptionlabel}{\rmfamily}
% forbid widows/orphans
\frenchspacing
\sloppy
\clubpenalty=10000
\widowpenalty=10000
% http://tex.stackexchange.com/questions/304802/how-not-to-hyphenate-the-last-word-of-a-paragraph
\finalhyphendemerits=10000
% given that we said footinclude=false, this should be safe
\setlength{\footskip}{2\baselineskip}
\title{Esperienza}
\date{}
\author{Max Sartin [Raffaele Schiavina]}
\subtitle{}
% https://groups.google.com/d/topic/comp.text.tex/6fYmcVMbSbQ/discussion
\hypersetup{%
pdfencoding=auto,
pdftitle={Esperienza},%
pdfauthor={Max Sartin [Raffaele Schiavina]},%
pdfsubject={},%
pdfkeywords={}%
}
\begin{document}
\thispagestyle{empty}
\strut\vskip 2em
\begin{center}
{\usekomafont{title}{\huge Esperienza\par}}%
\vskip 1em
\vskip 2em
{\usekomafont{author}{Max Sartin [Raffaele Schiavina]\par}}%
\vskip 1.5em
\strut\par
\end{center}
\vskip 3em
\par
Noi abbiamo una esperienza che i nostri figli non avranno che per sentito dire. Dovremmo farne tesoro.
\emph{L'esperienza dei partiti e delle organizzazioni colossali, svaniti per così dire in una notte di tempesta e rimasti esclusivamente nella coscienza e nelle convinzioni di pochi individui}, quelli che veramente avevano una coscienza e delle convinzioni.
Queste sono le cose che contano e noi dovremmo concentrare il nostro pensiero sulle cose che contano dando poca importanza alle cose accessorie. Perciò non vi affannate troppo se vi sono quelli che vorrebbero mettersi in gara con gli arrivisti per far numero. Non è roba nostra. Se sono sinceri se ne accorgeranno o prima o poi; se non lo sono è tempo perso. In mezzo alla rovina totale delle cose e alla disperazione degli uomini, bisogna cercare di rimettere in onore l'ideale che può essere raggiunto ed è, in ogni modo, la sola guida sicura alla risurrezione. Senza fede nella vita, senza l'assegnamento di scopi elevati che la nobilitino e la rendano degna d'essere vissuta, si va nel buio di disastro in disastro ripetendo all’infinito gli errori e le tragedie del passato.
\bigskip
\bigskip
[da una lettera datata 31 gennaio 1945]
% begin final page
\clearpage
% if we are on an odd page, add another one, otherwise when imposing
% the page would be odd on an even one.
\ifthispageodd{\strut\thispagestyle{empty}\clearpage}{}
% new page for the colophon
\thispagestyle{empty}
\begin{center}
A·Muse·Wiki Sandbox
\smallskip
Anti-Copyright
\bigskip
\includegraphics[width=0.25\textwidth]{logo-en}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
Max Sartin [Raffaele Schiavina]
Esperienza
\bigskip
Volontà, 1945
\bigskip
\textbf{sandbox.amusewiki.org}
\end{center}
% end final page with colophon
\end{document}
|
http://dlmf.nist.gov/15.4.E5.tex
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nist.gov
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CC-MAIN-2014-10
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application/x-tex
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crawl-data/CC-MAIN-2014-10/segments/1394010683244/warc/CC-MAIN-20140305091123-00006-ip-10-183-142-35.ec2.internal.warc.gz
| 52,255,845 | 619 |
\[\mathop{F\/}\nolimits\!\left(\tfrac{1}{2},\tfrac{1}{2};\tfrac{3}{2};-z^{2}%
\right)=z^{{-1}}\mathop{\ln\/}\nolimits\!\left(z+\sqrt{1+z^{2}}\right).\]
|
https://eml.berkeley.edu/~saez/course/introPE/intro-slides.tex
|
berkeley.edu
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CC-MAIN-2019-43
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application/octet-stream
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application/x-tex
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crawl-data/CC-MAIN-2019-43/segments/1570987813307.73/warc/CC-MAIN-20191022081307-20191022104807-00378.warc.gz
| 471,391,834 | 9,230 |
\documentclass[landscape]{slides}
\usepackage[landscape]{geometry}
\usepackage{pdfpages}
\usepackage{hyperref}
\def\mathbi#1{\textbf{\em #1}}
\topmargin=-1.8cm \textheight=17cm \oddsidemargin=0cm
\evensidemargin=0cm \textwidth=22cm
\author{Emmanuel Saez}
\date{}
\title{Graduate Public Economics \\
Introduction and Road Map } \onlyslides{1-300}
\newenvironment{outline}{\renewcommand{\itemsep}{}}
\begin{document}
\begin{slide}
\maketitle
\end{slide}
\begin{slide}
\begin{center}
{\bf PUBLIC ECONOMICS DEFINITION}
\end{center}
Public economics = Study
of the role of the government in the economy
Government is instrumental in most aspects of economic life:
1) Government in charge of huge regulatory structure
2) Taxes: governments in advanced economies collect 30-50\% of National Income in taxes
3) Expenditures: tax revenue funds traditional \textbf{public goods} (infrastructure, public order and safety, defense),
and \textbf{welfare state} (education, retirement benefits, health care, income support)
4) Macro-economic stabilization through central bank (interest rate, inflation control), fiscal stimulus, bailout policies
\end{slide}
\begin{slide}
\includepdf[pages={8}]{introPE_attach.pdf}
\end{slide}
\begin{slide}
\begin{center}
{\bf Bigger view on government}
\end{center}
Economists have a narrow minded view of individual behavior: selfish, rational, and utility based on own consumption only
But social interactions critical for humans: we cooperate at many levels: families, communities, nation states, global treaties; Beyond subsistence, value of income is always relative
Governments are a formal way to organize cooperation
Archaic human societies depended on social cooperation for protection and taking care of the young, sick, and old
$\Rightarrow$ Explains best why our modern nation states have defense and provide education, health care, and retirement benefits
Replacing social institutions by markets does not always work
\small E.g., Retirement benefits: Saving for your own retirement is economically rational but in practice most people unable to do so unless institutions (employers/government) help them
\end{slide}
\begin{slide}
\begin{center}
{\bf For Economists: \\ Two General Rules for Government Intervention}
\end{center}
1) Failure of 1st Welfare Theorem: Government intervention can help
if there are market or individual failures
2) Fallacy of the 2nd Welfare Theorem: Distortionary Government
intervention is required to reduce economic inequality
\end{slide}
\begin{slide}
\begin{center}
{\bf Role 1: 1st Welfare Theorem Failure}
\end{center}
{\bf 1st Welfare Theorem:} If (1) no externalities, (2) perfect
competition, (3) perfect information, (4) agents are rational,
then private market equilibrium is Pareto efficient
Government intervention may be desirable if:
1) Externalities require government interventions (Pigouvian
taxes/subsidies, public good provision)
2) Imperfect competition requires regulation (typically studied in
Industrial Organization)
3) Imperfect or Asymmetric Information (e.g., adverse
selection may call for mandatory insurance)
4) Agents are not rational ({\bf = individual failures}
analyzed in behavioral economics, field in huge expansion): e.g.,
myopic or hyperbolic agents may not save enough for retirement
\end{slide}
\begin{slide}
\begin{center}
{\bf Role 2: 2nd Welfare Theorem Fallacy}
\end{center}
Even with no market failures, free market might generate
substantial inequality.
Inequality is an issue because of people care
about their relative situation.
{\bf 2nd Welfare Theorem:} Any Pareto Efficient outcome can be
reached by (1) Suitable redistribution of initial endowments
[individualized {\bf lump-sum} taxes based on indiv.
characteristics and not behavior], (2) Then letting markets work
freely
$\Rightarrow$ No conflict between efficiency and equity [1st best taxation]
Redistribution of initial endowments is not feasible
(information pb) $\Rightarrow$ govt needs to use {\bf distortionary} taxes
and transfers $\Rightarrow$ Trade-off between
efficiency and equity [2nd best taxation]
This class will focus primarily but not exclusively on role 2
\end{slide}
\begin{slide}
\begin{center}
{\bf Illustration of 2nd Welfare Theorem Fallacy}
\end{center}
Suppose economy is populated 50\% with disabled people unable to work (hence they earn \$0) and 50\% with able people
who can work and earn \$100
\textbf{Free market outcome:} disabled have \$0, able have \$100
\textbf{2nd welfare theorem:} govt is able to tell apart the disabled from the able [even if the able do not work]
\small
$\Rightarrow$
can tax the able by \$50 [regardless of whether they work or not] to give \$50 to each disabled person $\Rightarrow$ the able keep working [otherwise they'd have zero income and still have to pay \$50]
\normalsize
\textbf{Real world:} govt can't tell apart disabled from non working able
\small
$\Rightarrow$ \$50 tax on workers + \$50 transfer on non workers destroys all incentives to work $\Rightarrow$ govt can no longer do full redistribution $\Rightarrow$ Trade-off between equity and size of the pie
\small
\end{slide}
\begin{slide}
\begin{center}
{\bf Normative vs. Positive Public Economics}
\end{center}
{\bf Normative Public Economics:} Analysis of How Things Should be
(e.g., should the government intervene in health insurance market?
how high should taxes be?, etc.)
{\bf Positive Public Economics:} Analysis of How Things Really Are
(e.g., Does govt provided health care crowd out private health
care insurance? Do higher taxes reduce labor supply?)
Positive Public Economics is a required 1st step before we can
complete Normative Public Economics
Positive analysis is primarily empirical and Normative analysis is
primarily theoretical
Positive Public Economics overlaps with Labor Economics
{\bf Political Economy} is a positive analysis of govt outcomes
[public choice is political economy from a libertarian view]
%focusing on \textbf{government failures}]
\end{slide}
\begin{slide}
\begin{center}
{\bf Individual Failures vs. Paternalism}
\end{center}
In many situations, individuals may not or do not seem to act in
their best interests [e.g., many individuals are not able to save
for retirement]
Two Polar Views on such situations:
1) {\bf Individual Failures [Behavioral Economics View]}
Individual Failures exist: Self-control problems, Cognitive
Limitations
2) {\bf Paternalism [Libertarian Chicago View]} Individual
failures do not exist and govt wants to impose on individuals its
own preferences against individuals' will
Key way to distinguish those 2 views: Under Paternalism,
individuals should be opposed to govt programs such as Social
Security. If individuals understand they have failures, they will
tend to support govt programs such as Social Security.
\end{slide}
\begin{slide}
\begin{center}
{\bf Plan for 230B Lectures}
\end{center}
1) {\bf Labor Income Taxation and Redistribution (SAEZ):} (a) Normative Aspects: Optimal Income Taxes and Transfers,
(b) Empirical Aspects: Labor Supply and Taxes and Transfers, (c) Social security retirement and disability benefits
2) {\bf Wealth inequality and taxing capital income (ZUCMAN):} (a) Wealth inequality, (b) Taxation of capital income, (c) International tax and tax enforcement issues
3) {\bf Environmental Economics (JOE SHAPIRO):} Guest lecture, last day of class
%3) {\bf Local public finance (YAGAN):} (a) Property taxation, the Tiebout model, and education finance, (b) Place-based policies and capitalization
%2) {\bf Capital Income Taxation and Redistribution} (a) Empirical Aspects: Wealth Accumulation, Savings, and Taxation, (b) Normative Aspects: Optimal Capital Income Taxation
%3) {\bf Social Insurance:} (a) Social Security and Retirement and Savings Decisions, (b) Disability Insurance
\end{slide}
\begin{slide}
\begin{center}
{\bf Income Inequality: Labor vs. Capital Income}
\end{center}
Individuals derive market income (before tax) from {\bf labor} and
{\bf capital}: $z=wl+rk$ where $w$ is wage, $l$ is labor supply,
$k$ is wealth, $r$ is rate of return on wealth
1) {\bf Labor income inequality} is due to differences in working abilities
(education, talent, physical ability, etc.), work effort (hours of
work, effort on the job, etc.), and luck (labor effort might
succeed or not)
2) {\bf Capital income inequality} is due to differences in
wealth $k$ (due to past saving behavior and inheritances
received), and in rates of return $r$ (varies dramatically
overtime and across assets)
Entrepreneurs start with labor which then transmutes into
wealth (e.g., Zuckerberg with Facebook)
\end{slide}
\begin{slide}
\begin{center}
{\bf Macro-aggregates: Labor vs. Capital Income}
\end{center}
Labor income $wl \simeq$ 70-75\% of national income $z$
Capital income $ rk \simeq$ 25-30\% of national income $z$ (has increased
in recent decades)
Wealth stock $k \simeq 400-500\%$ of national income $z$ (is increasing)
Rate of return on capital $ r \simeq 6\%$
$\alpha = \beta \cdot r$ where $\alpha= r k/z$ share of capital income and
$\beta=k/z$ wealth to income ratio
In GDP, gross capital share is higher (35-40\%) because it includes
depreciation of capital ($\simeq 10\%$ of GDP)
National Income = GDP - depreciation of capital + net foreign income
\end{slide}
%\begin{slide}
%\includepdf[pages={19}, scale=.9]{introPE_attach.pdf}
%\end{slide}
%
%\begin{slide}
%\includepdf[pages={9,10}]{introPE_attach.pdf}
%\end{slide}
\begin{slide}
\begin{center}
{\bf Income Inequality: Labor vs. Capital Income}
\end{center}
Capital Income (or wealth) is more concentrated than Labor
Income. In the US:
Top 1\% wealth holders have 40\% of total wealth (Saez-Zucman 2016). Bottom 50\% wealth holders
hold almost no wealth.
Top 1\% incomes earn about 20\% of total national income on a pre-tax basis (Piketty-Saez-Zucman, 2018)
Top 1\% labor income earners have about 15\% of total labor income
\end{slide}
\begin{slide}
\begin{center}
{\bf Income Inequality Measurement}
\end{center}
Inequality can be measured by indexes such as Gini, log-variance,
quantile income shares which are functions of the income
distribution $F(z)$
Gini = 2 * area between 45 degree line and Lorenz curve
Lorenz curve $L(p)$ at percentile $p$ is fraction of total income
earned by individuals below percentile $p$
$0 \leq L(p) \leq p$
Gini=0 means perfect equality
Gini=1 means complete inequality (top person has all the income)
\end{slide}
\begin{slide}
\includepdf[pages={29}, scale=1]{introPE_attach.pdf}
\end{slide}
\begin{slide}
\begin{center}
{\bf Key Empirical Facts on Income/Wealth Inequality}
\end{center}
1) In the US, labor income inequality has increased substantially
since 1970: due to skilled biased technological progress
vs. institutions (min wage and Unions) [Autor-Katz'99]
2) US top income shares dropped dramatically from 1929 to
1950 and increased dramatically since 1980. Bottom 50\% incomes
have stagnated in real terms since 1980 [Piketty-Saez-Zucman '18 distribute
full National Income]
%3) Top US incomes used to be primarily capital income. Now, top
%incomes are divided 50/50 between labor and capital income (explosion of top labor incomes with stock-options, etc.)
3) Fall in top income shares from 1900-1950 happened in most
OECD countries. Surge in top income shares has happened primarily
in English speaking countries, and not as much in Continental Europe
and Japan [Atkinson, Piketty, Saez JEL'11]
\end{slide}
\begin{slide}
\includepdf[pages={1}, scale=.9]{introPE_attach.pdf}
\end{slide}
\begin{slide}
\includepdf[pages={30}, scale=1]{introPE_attach.pdf}
\end{slide}
\begin{slide}
\includepdf[pages={23}, scale=.9]{introPE_attach.pdf}
\end{slide}
\begin{slide}
\includepdf[pages={24-25}, scale=.9]{introPE_attach.pdf}
\end{slide}
%\begin{slide}
%\includepdf[pages={17-18}, scale=.9]{introPE_attach.pdf}
%\end{slide}
\begin{slide}
\begin{center}
{\bf Measuring Intergenerational Income Mobility}
\end{center}
Strong consensus that children's success should not depend too much on parental income [Equality of Opportunity]
Studies linking adult children to their parents can measure link between children and parents income
Simple measure: average income rank of children by income rank of parents [Chetty et al. 2014]
1) US has less mobility than European countries (especially Scandinavian countries such as Denmark)
2) Substantial heterogeneity in mobility across cities in the US
3) Places with low race/income segregation, low income inequality, good K-12 schools, high social capital, high family stability tend to have high mobility [these are correlations and do not imply causality]
\end{slide}
\begin{slide}
\includepdf[pages={11-16}]{introPE_attach.pdf}
\end{slide}
\begin{slide}
\begin{center}
{\bf Govt Redistribution with Taxes and Transfers}
\end{center}
Government taxes individuals based on income and consumption and
provides transfers: $z$ is pre-tax income, $y=z-T(z)+B(z)$ is
post-tax income
1) If inequality in $y$ is less than inequality in $z$
$\Leftrightarrow$ tax and transfer system is redistributive (or
progressive)
2) If inequality in $y$ is more than inequality in $z$
$\Leftrightarrow$ tax and transfer system is regressive
\small
a) If $y=z \cdot (1-t)$ with constant $t$, tax/transfer system is
neutral
b) If $y=z \cdot (1-t)+G$ where $G$ is a universal (lumpsum)
allowance, then tax/transfer system is progressive
c) If $y=z-T$ where $T$ is a uniform tax (poll tax), then
tax/transfer system is regressive
Current tax/transfer systems in rich countries look roughly like b)
\end{slide}
\begin{slide}
\begin{center}
{\bf US Distributional National Accounts}
\end{center}
Piketty-Saez-Zucman (2018) distribute both pre-tax and post-tax US national income across adult individuals
Pre-tax income is income before taxes and transfers
Post-tax income is income net of all taxes and adding all transfers and public good spending
Both concepts add up to national income and provide a comprehensive view of the mechanical impact
of government redistribution
\end{slide}
\begin{slide}
\includepdf[pages={28}, scale=.9]{introPE_attach.pdf}
\end{slide}
\begin{slide}
\includepdf[pages={26}, scale=.9]{introPE_attach.pdf}
\end{slide}
\begin{slide}
\includepdf[pages={27}, scale=.9]{introPE_attach.pdf}
\end{slide}
\begin{slide}
\begin{center}
{\bf US tax/transfer System: Progressivity and Evolution}
\end{center}
{\bf 0) US Tax/Transfer system is progressive overall:} pre-tax national income is less
equally distributed than post-tax/post-transfer national income
{\bf 1) Medium Term Changes:} Federal Tax Progressivity has declined
since 1970 but govt redistribution through transfers has increased (Medicaid, Social Security retirement, DI, UI
various income support programs)
{\bf 2) Long Term Changes:} Before 1913, US taxes were primarily
tariffs, excises, and real estate property taxes [slightly
regressive], minimal welfare state (and hence small govt)
http://www.treasury.gov/education/fact-sheets/taxes/ustax.shtml
\end{slide}
\begin{slide}
\includepdf[pages={21-22}]{introPE_attach.pdf}
\end{slide}
\begin{slide}
\begin{center}
{\bf Federal US Tax System (2/3 of total taxes)}
\end{center}
1) Individual income tax (on both labor+capital income)
[progressive](40\% of fed tax revenue)
2) Payroll taxes (on labor income) financing social security programs [about
neutral] (40\% of revenue)
3) Corporate income tax (on capital income) [progressive if incidence on
capital income] (15\% of revenue)
4) Estate taxes (on capital income) [very progressive] (1\% of
revenue)
5) Minor excise taxes (on consumption) [regressive] (3\%
of revenue)
Fed agencies (CBO, Treasury, Joint Committee on Taxation) and think-tanks (Tax Policy Center)
provide distributional Fed tax tables
\end{slide}
\begin{slide}
\begin{center}
{\bf State+Local Tax System (1/3 of total taxes)}
\end{center}
Decentralized governments can experiment, be tailored to local views, create tax competition and make redistribution harder (famous Tiebout 1956 model)
hence favored by conservatives
1) Individual + Corporate income taxes
[progressive] (1/3 of state+local tax revenue)
2) Sales taxes + Excise taxes (tax on consumption) [regressive] (1/3 of
revenue)
3) Real estate property taxes (on capital income)
[slightly progressive] (1/3 of revenue)
See ITEP (2018) ``Who Pays'' for systematic state level distributional tax tables
US Census provides Census of Government data
%http://www.census.gov/govs/www/qtax.html
\end{slide}
%\begin{slide}
%\begin{center}
%{\bf Why has govt grown so much over 20th century?}
%\end{center}
%
%1) {\bf Demand Side Argument: Wagner law} Govt provided goods
%(education, health, social insurance) are luxury goods
%
%2) {\bf Supply Side Argument: Tax Enforcement Ability} Ability of govt to tax
%increases dramatically over the course of economic development
%[easy to tax large companies which need careful records for their
%operations]
%
%
%\end{slide}
\begin{slide}
\begin{center}
{\bf REFERENCES CITED}
\end{center}
{\small
Alvaredo, F., Atkinson, A., T. Piketty and E. Saez ``The Top 1 Percent in International and Historical Perspective.'' \emph{Journal of Economic Perspectives} 27(3), 2013, 3-20. \href{http://eml.berkeley.edu/~saez/alvaredo-atkinson-piketty-saezJEP13top1percent.pdf} {(web)}
Alvaredo, F., Atkinson, A., T. Piketty, E. Saez, and G. Zucman \emph{World Inequality Database},
\href{http://www.wid.world/} {(web)}
Alvaredo, F., Atkinson, A., T. Piketty, E. Saez, and G. Zucman. 2018 \emph{World Inequality Report},
\href{https://wir2018.wid.world/} {(web)}
Atkinson, A., T. Piketty and E. Saez ``Top Incomes in the Long Run of History'', Journal of Economic Literature
49(1), 2011, 30--71. \href{http://elsa.berkeley.edu/~saez/atkinson-piketty-saezJEL10.pdf} {(web)}
Chetty, Raj, Nathan Hendren, Patrick Kline, and Emmanuel Saez, ``Where is the Land of Opportunity? The Geography of Intergenerational Mobility in the United States,'' \emph{Quarterly Journal of Economics}, 129(4), 2014, 1553-1623.
\href{http://eml.berkeley.edu/~saez/chetty-friedman-kline-saezQJE14mobility.pdf}{(web)}
ITEP (Institute on Taxation and Economic Policy). 2018. ``Who Pays: A Distributional Analysis of the Tax Systems in All 50 States'', 6th edition.
\href{https://itep.org/whopays/}{(web)}
%Kleven, Henrik, Claus Kreiner, and Emmanuel Saez ``Why Can Modern Governments Tax So Much? An Agency Model of Firms as Fiscal Intermediaries,'' NBER Working Paper No. 15218, August 2009.
%\href{http://www.nber.org/papers/w15218} {(web)}
Kopczuk, Wojciech, Emmanuel Saez, and Jae Song ``Earnings Inequality and Mobility in the United States: Evidence from Social Security Data since 1937,'' Quarterly Journal of Economics 125(1), 2010, 91-128. \href{http://www.econ.berkeley.edu/~saez/kopczuk-saez-songQJE10mobility.pdf} {(web)}
Piketty, Thomas, \emph{Capital in the 21st Century}, Cambridge: Harvard University Press, 2014,
\href{http://piketty.pse.ens.fr/en/capital21c2}{(web)}
Piketty, Thomas and Emmanuel Saez ``Income Inequality in the United States, 1913-1998'', Quarterly Journal of Economics, 118(1), 2003, 1-39. \href{http://links.jstor.org/stable/pdfplus/25053897.pdf} {(web)}
Piketty, Thomas and Emmanuel Saez ``How Progressive is the U.S. Federal Tax System? A Historical and International Perspective,'' Journal of Economic Perspectives, 21(1), Winter 2007, 3-24.
\href{http://www.econ.berkeley.edu/~saez/piketty-saezJEP07taxprog.pdf} {(web)}
\textbf{Piketty, Thomas, Emmanuel Saez, and Gabriel Zucman, ``Distributional National Accounts:
Methods and Estimates for the United States'', Quarterly Journal of Economics, 133(2), 553-609, 2018
\href{https://eml.berkeley.edu/~saez/PSZ2018QJE.pdf} {(web)} }
Piketty, Thomas and Gabriel Zucman, ``Capital is Back: Wealth-Income Ratios in Rich Countries, 1700-2010'', \emph{Quarterly Journal of Economics} 129(3), 2014, 1255-1310
\href{http://www.econ.berkeley.edu/~saez/PikettyZucman2014QJE.pdf} {(web)}
Saez, Emmanuel and Gabriel Zucman, ``Wealth Inequality in the United States since 1913: Evidence from Capitalized Income Tax Data'', \emph{Quarterly Journal of Economics} 131(2), 2016, 519-578
\href{http://eml.berkeley.edu/~saez/SaezZucman2016QJE.pdf}{(web)}
Tiebout, Charles M. ``A Pure Theory of Local Expenditures''
Journal of Political Economy, 64(5), 1956, 416-424
\href{https://www.jstor.org/stable/pdf/1826343.pdf}{(web)}
}
\end{slide}
\begin{slide}
\begin{center}
{\bf GENERAL BOOK REFERENCES}
\end{center}
{\small
\textbf{Graduate Level}
Atkinson, A.B. and J. Stiglitz, Lectures on Public Economics, New York: McGraw Hill, 1980.
Auerbach, A. and M. Feldstein, eds., Handbook of Public Economics, 4 Volumes, Amsterdam: North Holland, 1985, 1987, 2002, and 2002.
\href{http://www.sciencedirect.com/science/handbooks/15734420/} {(web)}
Auerbach, A., Chetty, R., M. Feldstein, and E. Saez, eds., Handbook of Public Economics, Volume 5,
Amsterdam: North Holland, 2013
\href{http://www.sciencedirect.com/science/handbooks/15734420/} {(web)}
Kaplow, L. The Theory of Taxation and Public Economics. Princeton University Press, 2008.
Mirrlees, J. Reforming the Tax System for the 21st Century The Mirrlees Review, Oxford University Press, (2 volumes) 2009 and 2010.
\href{https://www.ifs.org.uk/publications/mirrleesreview} {(web)}
Piketty, Thomas, \emph{Capital in the 21st Century}, Cambridge: Harvard University Press, 2014,
\href{http://piketty.pse.ens.fr/en/capital21c2}{(web)}
Salani\'e, B. The Economics of Taxation, Cambridge: MIT Press, 2nd Edition 2010.
Slemrod, Joel and Christian Gillitzer. Tax Systems, Cambridge: MIT Press, 2014.
\pagebreak
\textbf{Under-Graduate Level}
Gruber, J. Public Finance and Public Policies, 6th edition, Worth Publishers, 2019.
Rosen, H. and T. Gayer Public Finance, 10th edition, McGraw Hill, 2014.
Stiglitz, J. and J. Rosengard. Economics of the Public Sector, 4th edition, Norton, 2015.
Slemrod, J. and J. Bakija. Taxing Ourselves: A Citizen's Guide to the Debate over Taxes. 5th edition, MIT Press, 2017.
}
\end{slide}
\begin{slide}
\begin{center}
{\bf REFERENCES ON EMPIRICAL METHODS:}
\end{center}
{\small
Angrist, J. and A. Krueger, ``Instrumental Variables and the Search for Identification: From Supply and Demand to Natural Experiments,'' Journal of Economic Perspectives, 15 (4), 2001, 69-87 \href{http://www.jstor.org/stable/pdfplus/2696517.pdf} {(web)}
Angrist, J. and Steve Pischke. Mostly Harmless Econometrics: An Empiricist's Companion, Princeton University Press, 2009. \href{http://www.mostlyharmlesseconometrics.com/} {(web)}
Bertrand, M. E. Duflo et S. Mullainhatan, ``How Much Should we Trust Differences-in-Differences Estimates?,'' Quarterly Journal of Economics, Vol. 119, No. 1, 2004, pp. 249-275. \href{http://www.jstor.org/stable/pdfplus/25098683.pdf} {(web)}
Imbens, Guido and Jeffrey Wooldridge (2007) What's New in Econometrics? NBER SUMMER INSTITUTE MINI COURSE 2007. \href{http://www.nber.org/minicourse3.html} {(web)}
Meyer, B. ``Natural and Quasi-Experiments in Economics,'' Journal of Business and Economic Statistics, 13(2), April 1995, 151-161. \href{http://www.jstor.org/stable/pdfplus/1392369.pdf} {(web)}
}
\end{slide}
\end{document}
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\title{Dust envelopes around RV Tauri stars}
\author[A. V. Raveendran]
{A. V. Raveendran \\
Indian Institute of Astrophysics, Bangalore 560034, India}
\date{Accepted 1988 December 15.
Received 1988 December 14;
in original form 1988 October 11}
\pagerange{\pageref{firstpage}--\pageref{lastpage}}
\pubyear{1994}
\begin{document}
\maketitle
\label{firstpage}
\begin{abstract}
In the {\it IRAS\/} [12]--[25], [25]--[60] colour--colour diagram, RV
Tauri stars are found to populate cooler temperature regions
$(T<600\,\rmn{K})$, distinctly different from those occupied by the
oxygen and carbon Miras. The {\it IRAS\/} fluxes are consistent with the
dust density in the envelope varying as the inverse square of the
radial distance, implying that the grain formation processes in these
objects are most probably continuous and not sporadic. It is found that
the spectroscopic subgroups A and B are well separated in the
far-infrared two-colour diagram, with group B objects having
systematically cooler dust envelopes. We interpret this as being due to
a difference in the nature of grains, including the chemical
composition, in the two cases.
\end{abstract}
\begin{keywords}
circumstellar matter -- infrared: stars.
\end{keywords}
\section{Introduction}
It has been well established that RV Tauri variables possess infrared
emission far in excess of their expected blackbody continuum, arising
from their extended cool dust envelopes (Gehrz \& Woolf 1970; Gehrz
1972; Gehrz \& Ney 1972). Recently, Lloyd Evans (1985) and Goldsmith
et al.\ (1987) have given detailed descriptions of the
near-infrared properties of RV Tauri stars. In this paper we present an
analysis of the {\it IRAS\/} data of RV Tauri stars with the help of the
far-infrared two-colour diagram and a grid computed using a simple
model of the dust envelope. Such two-colour plots have already been
employed extensively by several investigators to study the
circumstellar envelopes around oxygen-rich and carbon-rich objects
which are in the late stages of stellar evolution (Hacking et al.\
1985; Zuckerman \& Dyck 1986; van der Veen \& Habing 1988; Willems \&
de Jong 1988).
Table 1 summarizes the basic data on the 17 objects detected at
\hbox{60\,$\umu$m}. Apart from the {\it IRAS\/} identification and the flux
densities at 12-, 25-, 60- and 100-$\umu$m wavebands, it gives the
spectroscopic groups of Preston et al.\ (1963), the light-curve classes
of Kukarkin et al.\ (1969) and the periods of light variation. The
list, which contains about 20 per cent of all the known RV Tauri
stars, is essentially the same as that given by Jura (1986). The
spectroscopic subgroups are from either Preston et al.\ (1963) or Lloyd
Evans (1985).
\begin{table*}
\centering
\begin{minipage}{140mm}
\caption{Data on the RV Tauri stars detected by {\it IRAS}.}
\begin{tabular}{@{}llrrrrlrlr@{}}
Name & & \multicolumn{4}{c}{Flux density (Jy)%
\footnote{Observed by {\em IRAS}.}}\\
Variable & {\it IRAS} & 12$\,\umu$m & 25$\,\umu$m & 60$\,\umu$m
& 100$\,\umu$m & Sp. & Period & Light- & $T_0\,(\rmn{K})$ \\
& & & & & & group & (d) & curve \\
& & & & & & & & type \\[10pt]
TW Cam & 04166$+$5719 & 8.27 & 5.62 & 1.82 & $<$1.73 & A & 85.6 & a & 555 \\
RV Tau & 04440$+$2605 & 22.53 & 18.08 & 6.40 & 2.52 & A & 78.9 & b & 460 \\
DY Ori & 06034$+$1354 & 12.44 & 14.93 & 4.12 & $<$11.22 & B & 60.3 & & 295 \\
CT Ori & 06072$+$0953 & 6.16 & 5.57 & 1.22 & $<$1.54 & B & 135.6 & & 330 \\
SU Gem & 06108$+$2734 & 7.90 & 5.69 & 2.16 & $<$11.66 & A & 50.1 & b & 575 \\
UY CMa & 06160$-$1701 & 3.51 & 2.48 & 0.57 & $<$1.00 & B & 113.9 & a & 420 \\
U Mon & 07284$-$0940 & 124.30 & 88.43 & 26.28 & 9.24 & A & 92.3 & b & 480 \\
AR Pup & 08011$-$3627 & 131.33 & 94.32 & 25.81 & 11.65 & B & 75.0 & b & 450 \\
IW Car & 09256$-$6324 & 101/06 & 96.24 & 34.19 & 13.07 & B & 67.5 & b & 395 \\
GK Car & 11118$-$5726 & 2.87 & 2.48 & 0.78 & $<$12.13 & B & 55.6 & & 405 \\
RU Cen & 12067$-$4508 & 5.36 & 11.02 & 5.57 & 2.01 & B & 64.7 & & 255 \\
SX Cen & 12185$-$4856 & 5.95 & 3.62 & 1.09 & $<$1.50 & B & 32.9 & b & 590 \\
AI Sco & 17530$-$3348 & 17.68 & 11.46 & 2.88 & $<$45.62 & A & 71.0 & b & 480 \\
AC Her & 18281$+$2149 & 41.47 & 65.33 & 21.12 & 7.79 & B & 75.5 & a & 260 \\
R Sct & 18448$-$0545 & 20.88 & 9.30 & 8.10 & $<$138.78 & A & 140.2 & a \\
R Sge & 20117$+$1634 & 10.63 & 7.57 & 2.10 & $<$1.66 & A & 70.6 & b & 455 \\
V Vul & 20343$+$2625 & 12.39 & 5.72 & 1.29 & $<$6.96 & A & 75.7 & a & 690
\end{tabular}
\end{minipage}
\end{table*}
\section[]{Description of the Envelope\\* Model}
If we assume that the dust grains in the envelope are predominantly of
the same kind and are in thermal equilibrium, the luminosity at
frequency $\nu$ in the infrared is given by
\begin{equation}
L(\nu)=\mskip-12mu\int\limits_{\rmn{envelope}}\mskip-12mu
\rho(r)Q_{\rmn{abs}}(\nu)B[\nu,T_{\rmn{g}}(r)]\exp [-\tau(\nu,r)]\>
\rmn{d}V,
\end{equation}
where
$Q_{\rmn{abs}}(\nu)$ is the absorption efficiency at frequency $\nu$,
$\rho(r)$ is the dust grain density,
$T_{\rmn{g}}(\nu)$ is the grain temperature,
$B[\nu,T_{\rmn{g}}(r)]$ is the Planck function, and
$\tau(\nu,r)$ is the optical depth at distance {\it r\/} from the
centre of the star.
The temperature $T_{\rmn{g}}(r)$ is determined by the condition of energy
balance: amount of energy radiated = amount of energy absorbed. The
amount of energy absorbed at any point is proportional to the total
available energy at that point, which consists of:
\begin{enumerate}
\item the attenuated and diluted stellar radiation;
\item scattered radiation, and
\item reradiation from other grains.
\end{enumerate}
Detailed solutions of radiative transfer in circumstellar dust shells
by Rowan-Robinson \& Harris (1983a,b) indicate that the effect of
heating by other grains becomes significant only at large optical
depths at the absorbing frequencies $[\tau(\rmn{UV})\gg 10]$, and at
optical depths $\tau(\rmn{UV})<1$ the grains have approximately the
same temperature that they would have if they were seeing the starlight
unattenuated and no other radiation.
The Planck mean optical depths of circumstellar envelopes around
several RV Tauri stars, derived from the ratios of the luminosities of
the dust shell (at infrared wavelengths) and the star, range from 0.07
to 0.63 (Goldsmith et al.\ 1987). There is much uncertainty in the
nature of the optical properties of dust grains in the envelope. The
carbon-rich RV Tauri stars are also reported to show the 10-$\umu$m
silicate emission feature typical of oxygen-rich objects (Gehrz \& Ney
1972; Olnon \& Raimond 1986). The pure terrestrial silicates or lunar
silicates are found to be completely unsuitable to account for the
infrared emission from circumstellar dust shells around M-type stars
(Rowan-Robinson \& Harris 1983a). We assume that the absorption
efficiency $Q_{\rmn{abs}} (\nu)$ in the infrared varies as
$\nu^{\gamma}$. ${\gamma}=1$ appears to provide a reasonable fit in a
variety of sources (Harvey, Thronson \& Gatley 1979; Jura 1986). Under
these circumstances the condition of energy balance implies that the
dust temperature $T_{\rmn{g}}$ will vary as $r^{\beta}$.
In view of the low value of the observed Planck mean optical depth for
the stellar radiation and the nature of the assumed frequency
dependence of the absorption efficiency, the extinction of the infrared
radiation by the dust envelope can be neglected. If we consider the
envelope to be spherically symmetric, equation (1) reduces to
\begin{equation}
L(\nu)=\!\!\int_{r_{1}}^{r_{2}}\!\!4\upi r^2\rho(r)\> Q_{\rmn{abs}}(\nu)B[\nu,T_{\rmn{g}}(r)]\> {\rmn{d}}r,
\end{equation}
where $r_1$ and $r_2$ are the inner and outer radii of the shell. For
a dusty density distribution $\rho(r)\propto r^{\alpha}$ and $r_2\gg
r_1$, equation (2) reduces to
\begin{equation}
L(\nu)\propto \nu^{2+\gamma-Q}\int_{X_0}^{\infty}{{x^Q}\over
{\rmn{e}^x-1}}\rmn{d}x ,
\end{equation}
where $Q=-(\alpha+\beta+3)/\beta$ and $X_0=(h\nu /kT_0)$. $T_0$
represents the temperature at the inner boundary of the dust shell
where grains start condensing. In a steady radiation pressure driven
mass outflow in the optically thin case, values of $\alpha$ lie near $-2$
(Gilman 1972). $\gamma$ and $\beta$ are related by
$\beta=-2/(\gamma+4)$.
In the {\it IRAS\/} Point Source Catalog (PSC, Beichman et al.\ 1985a),
the flux densities have been quoted at the effective wavelengths 12,
25, 60 and \hbox{100\,$\umu$m}, assuming a flat energy spectrum $[\nu F(\nu)=1]$
for the observed sources. For each model given by equation (3), using
the relative system response, the colour-correction factors (Beichman
et al.\ 1985b) in each of the {\it IRAS\/} passbands were
calculated and the fluxes were converted into flux densities expected
for a flat energy distribution, as assumed in the {\it IRAS\/} PSC, so
that the computed colours can be directly compared with the colours
determined from the catalogue quantities. Such a procedure is
more appropriate than correcting the {\it IRAS\/} colours for the energy
distribution given by a particular model and then comparing them with
those computed by the model.
\subsection{Colour--colour diagram}
The IR colour is defined as
\[
[\nu_1]-[\nu_2]=-2.5\log [f(\nu_1)/f(\nu_2)],
\]
where $\nu_1$ and $\nu_2$ are any two wavebands and $f(\nu_1)$
and $f(\nu_2)$ are the corresponding flux densities assuming a flat
energy spectrum for the source.
In Fig.~1, we have plotted the [25]--[60] colours of RV Tauri stars
against their corresponding [12]--[25] colours derived from the {\it
IRAS\/} data. Filled circles represent stars of group A and open circles
stars of group B. The two sets of near-parallel lines represent the
loci of constant inner shell temperature $T_0$ and the quantity $Q$
defined above. The models correspond to the case of absorption
efficiency $Q_{\rmn{abs}}(\nu)$ varying as $\nu$ (with $\gamma=1$ and
hence $\beta=-0.4$). We have omitted R Sct in Fig.~1 because it shows a
large deviation from the average relation shown by all the other
objects. R Sct has a comparatively large excess at 60$\,\umu$m, but the
extent of a possible contamination by the infrared cirrus (Low et
al. 1984) is unknown. Goldsmith et al.\ (1987) found no evidence
of the presence of a dust envelope at near-IR wavelengths and the
spectrum was consistent with a stellar continuum. This explains why R
Sct lies well below the mean relation shown by stars of groups A and C
between the [3.6]--[11.3] colour excess and the photometrically
determined (Fe/H) (Dawson 1979). R Sct has the longest period of
140$\,$d among the RV Tauri stars detected at far-infrared wavelengths
and does not have the 10-$\umu$m emission feature seen in other objects
(Gehrz 1972; Olnon \& Raimond 1986). R Sct is probably the most
irregular RV Tauri star known (McLaughlin 1932).
\begin{figure}
\vspace{302pt}
\caption{Plot of [25]--[60] colours of RV Tauri stars against their
[12]--[25] colours after normalizing as indicated in Beichman et al.\
(1985b). Some of the objects are identified by their variable-star
names. Typical error bars are shown in the bottom right-hand corner.
The lines represent the loci for constant inner shell temperature and
the quantity $Q$. Note the separation of group A and B stars at $T_0
\sim$ 460$\,$\,K. Positions occupied by a sample of carbon and oxygen
Miras are also shown. The $Q=1.0$ line differs from the blackbody line
by a maximum of $\sim 0.05$.}
\end{figure}
The inner shell temperatures $(T_0)$ derived for the various objects
are also given in Table~1 and we find the majority of them to have
temperatures in the narrow range 400--600$\,$K. If the dependences of
$Q_{\rmn{abs}}(\nu)$ on $\nu$ and $\rho(r)$ on $r$ are similar in all
the objects considered, then in the colour--colour diagram they all
should lie along a line corresponding to different values of $T_0$ and
in Fig.~1 we find that this is essentially the case. In view of the
quoted uncertainties in the flux measurements, we cannot attach much
significance to the scatter in Fig.~1.
At \hbox{100\,$\umu$m} the infrared sky is characterized by emission, called
infrared cirrus, from interstellar dust on all spatial scales (Low et
al. 1984), thereby impairing the measurements at far-infrared
wavelengths. In Fig.~2, we have plotted the [60]--[100] colours of the
six RV Tauri stars detected at \hbox{100\,$\umu$m} against their [25]--[60]
colours, along with the grid showing the regions of different values
for inner shell temperature $T_0$ and the quantity $Q$, as in Fig.~1.
The results indicated by Fig.~2 are consistent with those derived from
Fig.~1. AR Pup shows a large excess at \hbox{100\,$\umu$m} but, in view of the
large values for the cirrus flags given in the catalogue, the intrinsic
flux at \hbox{100\,$\umu$m} is uncertain.
\subsection{Radial distribution of dust}
\begin{figure*}
\vspace*{174pt}
\caption{Plot of the [60]--[100] colours of RV Tauri stars against
their [25]--[60] colours after normalizing as indicated in Beichman
et al.\ (1985b). The solid lines represent the loci for constant
inner shell temperature and the quantity $Q$. The dashed line shows
the locus for a blackbody distribution.}
\end{figure*}
From Fig.~1, it is evident that all RV Tauri stars lie between the
lines corresponding to $Q=1.5$ and 0.5. With
\[
\alpha=-(1+Q)\beta-3,
\]
these values suggest limits of $r^{-2.0}$ and $r^{-2.4}$ for the
dust density variation, indicating a near-constant mass-loss rate. Jura
(1986) has suggested that the density in the circumstellar envelope
around RV Tauri stars varies as $r^{-1}$, implying a mass-loss rate
that was greater in the past than it is currently. By fitting a power
law to the observed fluxes, such that $f(\nu)$ varies as $\nu^q$,
values of $q$ determined by him for the various objects given in Table~1
lie in the range 0.6--1.2, with a mean $\skew5\bar q=0.98$. The
assumption of a power law corresponds to the case of $X_0=0$ in
equation (3) and hence we get
\[
q=2+\gamma -Q.
\]
Since we assume that $Q_{\rmn{abs}}(\nu)$ varies as $\nu$, the
resulting value for $Q$=2.0. None of the objects is found to lie in the
corresponding region in the colour--colour diagram. Even this extreme
value for $Q$ implies a density which varies as $r^{-1.8}$.
Goldsmith et al.\ (1987) have reported that the simultaneous
optical and near-IR data of AC Her can be fitted by a combination of
two blackbodies at 5680 and 1800\,K, representing, respectively, the
stellar and dust shell temperatures, and suggested that in RV Tauri
stars the grain formation is a sporadic phenomenon and not a continuous
process. Apparently, they have been influenced by the remark by Gehrz
\& Woolf (1970) that their data in the 3.5--11$\,\umu$m region of AC Her
indicated a dust temperature of $\sim$300\,K. We find that the {\it K--L\/}
colours given by Gehrz (1972), Lloyd Evans (1985) and Goldsmith et
al. (1987) are all consistent with each other. Surely, hot dust ($\sim
1800\,$K), if present at the time of observations by Goldsmith et
al. (1987), would have affected the {\it K--L\/} colour significantly.
AC Her, like other members of its class, is found to execute elongated
loops in the ({\it U--B\/}), ({\it B--V\/}) plane (Preston et al.\
1963), indicating that significant departure of the stellar continuum
from the blackbody is to be expected. Further, their data show only a
marginal excess at the near-IR wavelengths. We feel that the case for
the existence of hot dust around AC Her and hence for the sporadic
grain formation around RV Tauri stars is not strong. In Fig.~3 we find that AC Her and RU Cen lie very close to R Sct
which, according to Goldsmith et al.\ (1987), shows no evidence
for the presence of a hot dust envelope.
\subsubsection{Comparison with oxygen and carbon Miras}
In Fig.~1 we have also shown the positions of a sample of oxygen-rich
and carbon-rich Miras. At the low temperatures characteristic of the
Miras, a part of the emission at 12$\,\umu$m comes from the photosphere.
For a blackbody at 2000$\,$K, the ratio of fluxes at wavelengths of 12
and 2$\,\umu$m $(f_{12}/f_{2})\sim 0.18$. The Miras shown in Fig.~1 have
$(f_{12}/f_{2})$ ratios larger than twice the above value. It is
clear that the three groups of objects populate three different regions
of the diagram. Hacking et al.\ (1985) have already noticed that
there are distinct differences between the {\it IRAS\/} colours of
oxygen-rich and carbon-rich objects. On the basis of an analysis, using
a bigger sample of bright giant stars in the {\it IRAS\/} catalogue, this
has been interpreted by Zuckerman \& Dyck (1986) as being due to a
systematic difference in the dust grain emissivity index. U Mon shows
the 10-$\umu$m silicate emission convincingly and, in most of the other
objects for which low-resolution spectra in the near-infrared have been
reported (Gehrz 1972; Olnon \& Raimond 1986), the 10-$\umu$m emission
may be partly attributed to silicates. Hence it is reasonable to expect
that, in the envelopes around at least some of the RV Tauri stars, the
dust grains are predominantly of silicates, as in the case of oxygen
Miras (Rowan-Robinson \& Harris 1983a). The fact that none of the RV
Tauri stars is found in the region of the two-colour diagram occupied
by the oxygen Miras indicates that the emissivity indices of the
silicate grains in the two cases are different. Because of the higher
temperatures and luminosities, the environment of grain formation will
be different in RV Tauri stars.
\subsubsection{Correlation with subgroups}
Preston et al.\ (1963) have identified three spectroscopic
subgroups, which are designated as groups A, B and C. Objects of group
A are metal-rich; group C are metal-poor; group B objects are also
metal-poor, but show carbon enhancements (Preston et al.\ 1963; Lloyd
Evans 1974; Dawson 1979; Baird 1981). It is interesting to see that
Table~1 contains no group C objects and that in Fig.~1 there is a clear
separation of the two spectroscopic subgroups A and B, with the
demarcation occurring at an inner shell temperature of about 450$\,$K,
group B stars having lower temperatures than group A. SX Cen is the
only exception. Lloyd Evans (1974) has reported that metal lines are
stronger in SX Cen than in other group B objects. It may be worth
noting that SX Cen has the shortest period among the 100 or so objects
with the RV Tauri classification. RU Cen has the coolest inner shell
temperature, as already suggested by the near-infrared spectrum (Gehrz
\& Ney 1972).
\begin{figure}
\vspace*{174pt}
\caption{Plot of ({\it K--L\/}) colours of RV Tauri stars detected by
{\it IRAS\/} against their corresponding ({\it J--K\/}) colours. The
position of AR Pup is indicated. The three objects lying close to the
blackbody line are AC Her, RU Cen and R Sct.}
\end{figure}
Group B objects follow a different mean relationship from those of group
A, having systematically larger 11-$\umu$m excess for a given excess at
3$\,\umu$m (Lloyd Evans 1985). For a general sample of RV Tauri stars,
the distinction between the oxygen-rich and carbon-rich objects is not
that apparent in the {\it JHKL\/} bands. In Fig.~3 we have plotted the
near-IR magnitudes of the objects given in Table~1 (except V Vul which
has no available measurements) in the {\it J--K, K--L\/} plane. The
colours, taken from Lloyd Evans (1985) and Goldsmith et al.\
(1987), are averaged if more than one observation exists, because the
internal agreements are found to be often of the order of observational
uncertainties, in accordance with the earlier finding by Gehrz (1972)
that variability has relatively little effect on colours. Barring RU
Cen and AC Her, it is evident that stars belonging to group B show
systematically larger excesses at {\it L\/} band for a given excess at
{\it K}. The low excesses at near-IR wavelengths for AC Her and RU Cen
are consistent with the very low dust temperatures indicated by the
far-infrared colours.
%
\begin{figure*}
\vbox to 220mm{\vfil
Landscape figure to go here. This figure was not part of the original
paper and is inserted here for illustrative purposes.\\ See the author
guide for details on how to handle landscape figures or tables, and
\verb|mnland.tex|.
\caption{}
\vfil}
\label{landfig}
\end{figure*}
It is already well established that from {\it UBV\/} photometry one can
distinguish between groups A and B, members of group A being
significantly redder than those of group B (Preston et al.\ 1963).
Similarly, Dawson (1979) has found that the two spectroscopic groups
are well separated in the DDO colour--colour diagrams when mean colours
are used for the individual objects.
The clear separation of the spectroscopic subgroups A and B in the IR
two-colour diagram suggests that the natures of dust grains in the
envelopes in the two cases are not identical. This is to be expected
because of the differences in the physical properties of the stars
themselves. The average colours of group B stars are bluer than group
A, but the envelope dust temperatures of B are cooler than those of A.
The near-IR spectra of AC Her and RU Cen are extremely similar (Gehrz
\& Ney 1972). The striking similarities in the optical spectra of AC
Her and RU Cen have been pointed out by Bidelman (O'Connell 1961). We
feel that the physical properties, including the chemical composition,
of the grains formed in the circumstellar envelope strongly depend on
those of the embedded star. This, probably, explains the diversity of
the energy distributions of RV Tauri stars in the near-infrared found
by Gehrz \& Ney (1972). On the basis of the observed differences in
chemical abundances and space distribution of RV Tauri stars, Lloyd
Evans (1985) has already pointed out that there is no direct
evolutionary connection between group A and group B objects, thus
ruling out the possibility that group B objects are the evolutionary
successors of group A, in which grain formation has stopped and the
cooler temperatures for the former are caused by an envelope expansion.
Kukarkin et al.\ (1969) have subdivided RV Tauri stars
into two classes, RVa and RVb, on the basis of their light curves; the
former shows a constant mean brightness, whereas the latter shows a
cyclically varying mean brightness. Extensive observations in the
near-infrared show that, on average, RVb stars are redder than RVa
stars, and Lloyd Evans (1985) has suggested that in RVb stars dust
shells are denser in the inner regions and hence radiate strongly in
the 1--3$\,\umu$m region. Fig.~3 confirms this; RVb objects show
systematically larger ({\it J--K\/}) and ({\it K--L\/}) colours than RVa
objects. Apparently, there is no distinction between objects of the two
light-curve types at far-infrared wavelengths (Fig.~1).
\section{Conclusions}
In the [12]--[25], [25]--[60] colour diagram, RV Tauri stars populate
cooler temperature regions $(T<600 \,\rmn{K})$, distinctly different from
those occupied by the oxygen and carbon Miras. Using a simple model
in which
\begin{enumerate}
\item the envelope is spherically symmetric,
\item the IR-emitting grains are predominantly of the same kind, and
\item in the infrared the absorption efficiency $Q_{\rmn{abs}}
(\nu)\propto\nu$,
\end{enumerate}
we find that the {\it IRAS\/} fluxes are
consistent with the density in the envelope $\rho(r)\propto r^{-2}$,
where {\it r\/} is the radial distance. Such a dependence for the dust
density implies that the mass-loss rates in RV Tauri stars have not
reduced considerably during the recent past, contrary to the suggestion
by Jura (1986). In the two-colour diagram, the blackbody line and the
line corresponding to $\rho(r)\propto r^{-2.2}$ nearly overlap and the
present data are insufficient to resolve between the two cases. The
latter case is more physically reasonable, however.
The spectroscopic subgroups A and B are well separated in the {\it
IRAS\/} two-colour diagram, with group B objects having systematically
cooler dust envelopes. If we consider only the objects detected by {\it
IRAS}, we find that stars belonging to group B show systematically
larger excess at {\it L\/} band for a given excess at {\it K}.
Apparently, there is no correlation between the light-curve types (RVa
and RVb) and the far-infrared behaviour of these objects. It is fairly
certain that the physical properties, including the chemical
composition, of the embedded stars are directly reflected by those of
the dust grains. Most probably, the grain formation process in RV Tauri
stars is continuous and not sporadic as suggested by Goldsmith et
al.\ (1987).
\section*{Acknowledgments}
I thank Professor N. Kameswara Rao for some helpful suggestions,
Dr H. C. Bhatt for a critical reading of the original version of the
paper and an anonymous referee for very useful comments that improved
the presentation of the paper.
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\appendix
\section[]{Large gaps in L\lowercase{y}${\balpha}$ forests\\* due to fluctuations in line distribution}
(This appendix was not part of the original paper by A.V.~Raveendran
and is included here just for illustrative purposes.)
Spectroscopic observations of bright quasars show that the mean number
density of Ly$\alpha$ forest lines, which satisfy certain criteria,
evolves like $\rmn{d}N/\rmn{d}z=A(1+z)^\gamma$, where $A$ and~$\gamma$ are two
constants. Given the above intrinsic line distribution we examine the
probability of finding large gaps in the Ly$\alpha$ forests. We
concentrate here only on the statistics and neglect all observational
complications such as the line blending effect (see Ostriker, Bajtlik
\&~Duncan 1988).
Suppose we have observed a Ly$\alpha$ forest between redshifts $z_1$
and~$z_2$ and found $N-1$ lines. For high-redshift quasars $z_2$~is
usually the emission redshift $z_{\rmn{em}}$ and $z_1$ is set to
$(\lambda_{\rmn{Ly}\beta}/\lambda_{\rmn{Ly}\alpha})(1+z_{\rmn{em}})=0.844
(1+z_{\rmn{em}})$ to avoid contamination by Ly$\beta$ lines. We
want to know whether the largest gaps observed in the forest are
significantly inconsistent with the above line distribution. To do
this we introduce a new variable~$x$:
%
\begin{equation}
x={(1+z)^{\gamma+1}-(1+z_1)^{\gamma+1} \over
(1+z_2)^{\gamma+1}-(1+z_1)^{\gamma+1}}.
\end{equation}
%
$x$ varies from 0 to 1. We then have $\rmn{d}N/\rmn{d}x=\lambda$, where $\lambda$
is the mean number of lines between $z_1$ and $z_2$ and is given by
%
\begin{equation}
\lambda\equiv{A[(1+z_2)^{\gamma+1}-(1+z_1)^{\gamma+1}]\over\gamma+1}.
\end{equation}
%
This means that the Ly$\alpha$ forest lines are uniformly
distributed in~$x$. The probability of finding $N-1$ lines between $z_1$
and~$z_2$, $P_{N-1}$, is assumed to be the Poisson distribution.
%
\newpage
%
\begin{figure}
\vspace{11pc}
\caption{$P(>x_{\rmn{gap}})$ as a function of $x_{\rmn{gap}}$ for,
from left to right, $N=160$, 150, 140, 110, 100, 90, 50, 45 and~40.}
\label{appenfig}
\end{figure}
\subsection{Subsection title}
We plot in Fig.~\ref{appenfig} $P(>x_{\rmn{gap}})$ for several $N$ values.
We see that, for $N=100$ and $x_{\rmn{gap}}=0.06$, $P(>0.06)\approx20$
per cent. This means that the probability of finding a gap with a size
larger than six times the mean separation is not significantly small.
When the mean number of lines is large, $\lambda\sim N>>1$, our
$P(>x_{\rmn{gap}})$ approaches the result obtained by Ostriker et al.\
(1988) for small (but still very large if measured in units of the mean
separation) $x_{\rmn{gap}}$, i.e., $P(>x_{\rmn{gap}})\sim N(1-
x_{\rmn{gap}})^{N-1}\sim N {\rmn{exp}}(-\lambda x_{\rmn{gap}})$.
\bsp
\label{lastpage}
\end{document}
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http://www.ipm.ac.ir/papers/pdf/abs15414.tex
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ipm.ac.ir
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\documentclass[12pt]{article}
\usepackage{amsmath,amssymb,amsfonts}
\begin{document}
Relying on multifractal behavior of pulsar timing residuals ({\it PTR}s), we examine the capability of Multifractal Detrended Fluctuation Analysis (MF-DFA) and Multifractal Detrending Moving Average Analysis (MF-DMA) modified by Singular Value Decomposition (SVD) and Adaptive Detrending (AD), to detect footprint of gravitational waves (GWs) superimposed on {\it PTR}s. Mentioned methods enable us to clarify the type of GWs which is related to the value of Hurst exponent.
%We compare scaling exponents determined by multifractal analyses of synthetic {\it PTR}s induced by a robust model for GWs with that of computed for pure pulsar timing residuals.
%Combining the mentioned methods enables us to clarify the type of GWs which is related to the value of Hurst exponent.
We introduce three strategies based on generalized Hurst exponent and width of singularity spectrum, to determine the dimensionless amplitude of GWs. For a stochastic gravitational wave background with characteristic strain spectrum as $\mathcal{H}_c(f)\sim \mathcal{A}f^{\zeta}$, the dimensionless amplitude greater than $\mathcal{A}\gtrsim 10^{-17}$ can be recognized irrespective to value of $\zeta$.
%}
%\txg{we applied our simulation in predicted interval for $\mathcal{A}$ . Maybe our strategies detect smaller values of $\mathcal{A}$ but we dont need them because they are not predicted for GWs. So} \txb{our relations are reliable for $\mathcal{A}\gtrsim 10^{-17}$}
We also utilize MF-DFA and MF-DMA to explore 20 millisecond pulsars observed by Parkes Pulsar Timing Array (PPTA). Our analysis demonstrates that there exists a cross-over in fluctuation function versus time scale for observed timing residuals representing a universal property and equates to $s_{\times}\sim60$ days. To asses multifractal nature of observed timing residuals, we apply AD and SVD algorithms on time series as pre-processes to remove superimposed trends as much as possible. The scaling exponents determined by MF-DFA and MF-DMA confirm that, all data are classified in non-stationary class elucidating second universality feature. The value of corresponding Hurst exponent is in interval $H \in [0.35,0.85]$. The $q$-dependency of generalized Hurst exponent demonstrates observed {\it PTR}s have multifractal behavior and the source of this multifractality is mainly devoted to correlation of data which is another universality of observed data sets.
\end{document}
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https://melusine.eu.org/syracuse/B/BaseBrevet/dta/a2000/geodivers/2000exo01.tex?enregistrement=ok
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eu.org
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%@metapost:orleans2000.mp
%@Titre: Orléans -- 2000
\textit{Attention : le barème de cet exercice sur trois points est le
suivant : 1 point pour une réponse exacte, -0,5 point pour une
réponse inexacte, 0 point s'il n'y a pas de réponse. La note globale
de l'exercice ne pourra être négative.}
Pour chaque ligne du tableau ci-après, trois réponses sont proposées,
mais une seule est exacte dont vous devez indiquer le numéro dans le
second tableau.
$$
\begin{tabular}{|c|m{6cm}|*{3}{c|}}
\hline
Question & Enoncé & Réponse 1 & Réponse 2 & Réponse 3 \\
\hline
Q & Sachant que les coordonnées de deux points dans un repère du plan
dont données par $A(1;-4)$ et $B(3;6)$, alors les coordonnées du
milieu de $[AB]$ sont : & $(4;2)$ & $(2;1)$ & $(1;5)$ \\
\hline
R & Si on multiplie par 1/5 les dimensions d'un trapèze, son périmètre
est multiplié par : & 4/5 & 1/25 & 1/5 \\
\hline
S & Soit $FGHI$ un rectangle de centre $E$, alors :
$\vecteur{FG}+\vecteur{FI}=$ $$\includegraphics{orleans2000.2}$$ &
$\vecteur{FH}$ & $\vecteur{FE}$ & $\vecteur{GI}$ \\
\hline
\end{tabular}
$$
$$
\begin{tabular}{|c|m{8cm}|}
\hline
Question & Indication du numéro (1, 2 ou 3) de la réponse choisie \\
\hline
Q & Réponse : \\
\hline
R & Réponse : \\
\hline
S & Réponse : \\
\hline
\end{tabular}
$$
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https://theanarchistlibrary.org/library/wildcat-the-deep-state.tex
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theanarchistlibrary.org
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\title{The Deep State}
\date{September 11, 2014}
\author{Wildcat}
\subtitle{Germany, Immigration, and the National Socialist Underground}
% https://groups.google.com/d/topic/comp.text.tex/6fYmcVMbSbQ/discussion
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\begin{document}
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{\usekomafont{title}{\huge The Deep State\par}}%
\vskip 1em
{\usekomafont{subtitle}{Germany, Immigration, and the National Socialist Underground\par}}%
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{\usekomafont{author}{Wildcat\par}}%
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% start a new right-handed page
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Nearly three years ago, in November 2011, news of a double suicide after a failed bank robbery developed into one of the biggest scandals in postwar German history.\footnote{What follows is based on four articles previously published in \href{http://www.wildcat-www.de/index.htm}{\emph{Wildcat}}. These in turn were based on the research of antifascist groups, on newspaper articles, on the reports from parliamentary investigation committees and on books. We use a lot of names of German Nazis, German towns, German cops and politicians. Most do not have any meaning outside of Germany. But we hope that in the “Age of Google” they can help you if you want to check the facts or go deeper.} Even now, it remains unresolved. For thirteen years the two dead men, Uwe Mundlos and Uwe Böhnhardt, had lived underground, together with a woman, Beate Zschäpe. The three were part of the National-Sozialistischer Untergrund (NSU), a fascist terror organization which is supposed to have murdered nine migrant small entrepreneurs in various German towns and a female police officer, and to have been responsible for three bomb attacks and around fifteen bank hold-ups. Although the NSU did not issue a public declaration, the connection between the nine murders committed between 2000 and 2006 as obvious: the same weapon was used each time, a Ceska gun.
At the time they were called “doner murders” (as in doner kebab) and the police called their special investigation team “Bosphorus.”\footnote{We will refer to some of the German security agencies. There are three intelligence agencies in Germany. The Bundesnachrichtendienst (BND; Federal Intelligence Service) is the foreign intelligence agency of Germany, directly subordinated to the Chancellor’s Office. The Militärischer Abschirmdienst (Military Counterintelligence Service, MAD) is a federal intelligence agency and is responsible for military counterintelligence. The third agency, Germany’s domestic intelligence service, is called “Verfassungsschutz” and has a federated structure. Aside from the federal “Bundesamt für Verfassungsschutz”(BfV; Federal Office for the Protection of the Constitution) there are also 16 so called Landesämter für Verfassungsschutz (LfV; State Authorities for the Protection of the Constitution) – one for each state – which are independent of the BfV. They are tasked with intelligence-gathering on threats against the state order and with counter-intelligence.} Nearly all the police departments working on the murders focused mainly on the victims and their alleged involvement in “organized crime,” the drug trade, etc. Not only was it eventually revealed that the murderers were organized Nazis, but that the killers had been supported by some branches of the state apparatus and the search for the murderers had been systematically obstructed. As one famous public television news presenter said: “One fact is established: the perpetrators could have been stopped and the murders could have been prevented.” She also voiced “the outrageous suspicion that perhaps they were not supposed to be stopped.” The final report of the parliamentary investigation committee of the Thuringia state parliament, published in August 2014, stated a “suspicion of targeted sabotage or conscious obstruction” of the police search. The Verfassungsschutz (VS, the German domestic secret service) had “at least in an indirect fashion protected the culprits from being arrested.”
Since the supposed double suicide on the November 4, 2011, the intelligence services, the interior ministries of the federal and central state, and the BKA collaborated to cover tracks, just as they had collaborated before to keep the existence of the NSU from becoming publicly known. One day before the connection between the NSU and the last bank robbery was publicly announced, a consultation in the chancellery took place. Since then, the investigation has been systematically obstructed by the destruction of files, lies, and the refusal to surrender evidence. In the current criminal case against the alleged sole survivor of the NSU (Beate Zschäpe) and five supporters at the higher regional court in Munich, the public prosecutor wants it to be believed that the series of terror acts were the work of three people (“the Trio”) and a small circle of sympathizers. “The investigations have found no indication of the participation of local third parties in the attacks or any of organizational integration with other groups.” But it is clear that the NSU was much larger and had a network all over Germany. And it is highly unlikely that the two dead men were the only perpetrators.
Research on the NSU has shown that the VS had the organized fascists under surveillance the whole time, without passing its information on to the police. It had many Confidential Informants (CIs)\footnote{In the language of the German police and intelligence, confidential informants are called “V-Leute” or “V-Männer.” The V stands for Vertrauen, which means confidence. \hyperref{}{amuse}{rf3-3262}{↩}} in leading positions in the fascist structures – or rather, the CIs even built up large parts of these structures. It is very unlikely that the secret services acted without consultation with the government – but it is certain that we will never find any written order. Sometimes public prosecutors and leading police officials were included in the cover-up. For example, the current President – at the time Vice-President – of the Landeskriminalamt (LKA) or Criminal Police Offices of Thuringia ordered his police in 2003 to “go out there, but don’t find anything!” after receiving a tip about Böhnhardt’s whereabouts.
Obviously the German state apparatus has erected a (new?) parallel structure that operates in accordance with government policies and out of the reach of parliamentary or legal control. The National-Sozialistischer Untergrund was a flagship project of this “deep state,” supporting the new policy towards migrants that started in 1998 at the instigation of Otto Schily, then Interior Minister. Since the NSU became known to the public, this apparatus has even been financially and operationally strengthened.
The NSU complex gives us a glimpse of the way the German state functions, and can therefore sharpen our criticism of the capitalist state. This is of international relevance for two reasons. First, many countries, such as Hungary, the Czech Republic, Morocco, and Russia, have recently seen mobilization, pogroms, and violence against migrants. In a weaker form this has also happened in Germany, and as usual one can see a pattern: the government stirs up hatred, fascists take action (there have been at least five arson attacks in the first half of 2014). Second, many states are preparing militarily for mass strikes and social unrest. In accordance with an operational scheme that has shaped interior policies in many Western countries since the Second World War, state institutions make use of paramilitary fascist structures. A recent example is the relation between the Greek security apparatus and the fascist \emph{Golden Dawn}.\footnote{Wildcat has published an article about the \emph{Golden Dawn} in Greece, “\href{http://wildcat-www.de/en/wildcat/95/e\_w95\_greece.html}{Fascists in Greece: From the streets into parliament and back}.”}
\section{The Background: The State Lays the Ground for Racism}
In October 1982 the new German Chancellor Helmut Kohl told Margaret Thatcher in a confidential conversation that he wanted to reduce the number of Turks in Germany by half within four years. They were “impossible to assimilate in their present number.” A few months before this conversation his predecessor Schmidt blared: “I won’t let any more Turks cross the border.” In October 1983, the government passed a repatriation grant. In the following years, the Christian Democrats (CDU) began a debate about the alleged rampant abuse of the asylum law. Although hate was stirred against “gypsies,” “negroes,” and others, in its core this racism was always aimed against “the Turks,” the largest group of immigrants. Kohl made this clear in his conversation with Thatcher: “Germany does not have a problem with the Portuguese, the Italians, not even the Southeast Asians, because all these communities are well integrated. But the Turks, they come from a very different culture.”\footnote{Claus Hecking, “\href{http://www.spiegel.de/politik/deutschland/kohl-wollte-jeden-zweiten-tuerken-in-deutschland-loswerden-a-914318.html}{Britische Geheimprotokolle: Kohl wollte offenbar jeden zweiten Türken loswerden},” \emph{Spiegel Online}, August 1, 2013.}
Already in the second half of the 1980s, this government policy was accompanied by Nazi attacks on foreigners. After German reunification this process culminated in the racist pogroms of Rostock-Lichtenhagen in August 1992.\footnote{There were many racist pogroms in Germany at the beginning of the 90s. The first peak was in September 1991 in Hoyerswerda, a town in northeastern Saxony. On four nights there were attacks against a hostel mainly used by Mozambican contract workers. The second peak was the pogroms in Rostock-Lichtenhagen in Mecklenburg-Vorpommern: Between August 22 and 24, 1992, violent xenophobic riots took place; these were the worst mob attacks against migrants in postwar Germany. There were also arson attacks against Turkish houses in which eight people died.There are two Wildcat articles in English about these pogroms and their consequences, “\href{http://wildcat-www.de/en/wildcat/60/w60e\_ros.htm}{Rostock, or: How the New Germany is being governed},” from Wildcat 60, 1992; and “\href{http://libcom.org/library/critique-autonomous-anti-fascism-wildcat-germany}{Critique of autonomous anti-fascism},” from Wildcat 57, 1991.} Less than four months later, the SPD (Social Democratic Party of Germany) and the CDU (Christian Democratic Union of Germany) agreed to abolish the right of asylum almost completely.
The state racism was bloody, but it was not quantitatively successful in deporting large numbers or discouraging immigration. At the beginning of Kohl’s Chancellery there were 4.6 million foreigners in Germany; when it ended in 1998 there were 7.3 million. Consequently, interior policy focused on “police penetration” of “parallel societies” after the Rostock pogroms and especially under the Schröder government. Interior minister Kanther and his successor Schily imposed the definition of immigration as “criminally organized” throughout Europe. This policy, too, was primarily directed not against “newcomers” but against the “Turks” who already live here. Small businesses owned by migrants are generally suspected of involvement in organized crime. Even before 9\Slash{}11, the financial transactions and phone calls of whole communities were screened and analyzed on suspicion of organized crime and trafficking. In particular, the investigations targeted small businesses frequented by large numbers of people: coffee shops, internet cafes, kiosks, and so forth. From these places migrants can transfer money to another country without the involvement of banks, using the \emph{Hawala} system.\footnote{The \emph{Hawala} system is an informal value-transfer system based on a huge network of money brokers. This network makes it possible to send money to an acquaintance in a cheap and confidential way. There are no promissory instruments exchanged between the \emph{hawala} brokers: the system is solely based on trust between the brokers.} “Police penetration” reached its climax with the search for the Ceska killers: the “BOA Bosphorus” organized the largest dragnet among migrant communities in the history of Germany: massive surveillance of phone calls, mobile phones, money transfers, hotel bookings, rental car use, etc.
\section{The Nazis}
Although the global economic crisis of the early 1990s reached Germany a bit later than elsewhere because of the “reunification boom,” it was relatively more severe. Unemployment doubled, “floodgates opened wide” in the factories. The unions supported the crisis policy of employers with new collective agreements to ensure “job security” and company agreements implementing “working time accounts” over a full year. The workers were left alone in their defensive struggles, even though some were quite militant and creative. The (radical) Left was preoccupied with the struggle against fascism and racism. They no longer analysed racism as a governmental policy, but as a “popular passion.” Anyone who tries to fight against ethnic racism in all its shades but omits the dimension of social racism remains toothless at best: in the worst case s\Slash{}he becomes an agent of state racism.\footnote{By social racism we mean racism against people from lower social strata, people who don’t integrate well in society, people living from benefits, etc. Étienne Balibar uses a similar concept in Étienne Balibar and Immanuel Wallerstein, \emph{Race, Nation, Class}: Ambiguous Identities (London: Verso, 1991).} Jacques Rancière described it this way: “The racism we have today is a cold racism, an intellectual construction. It is primarily a creation of the state\dots{} [It is] a logic of the state and not a popular passion. And this state logic is primarily supported not by, who knows what, backward social groups, but by a substantial part of the intellectual elite.” Rancière concludes that the “‘Leftist’ critique” has adopted the “same conceit” as the right wing (“racism is a popular passion” which the state has to fight with increasingly tougher laws). They “build the legitimacy of a new form of racism: state racism and ‘Leftist’ intellectual racism.”\footnote{All quotes from a lecture by \href{http://www.akweb.de/ak\_s/ak555/17.htm}{Jacques Rancière} in 2010, printed in German translation in \emph{ak 555}, November 19, 2010. The English translation is available at: http:\Slash{}\Slash{}wrongarithmetic.wordpress.com\Slash{}2010\Slash{}09\Slash{}21\Slash{}ranciere-racism\Slash{}} After that shift, there was a strong tendency for antifascist activities to focus on the socially deprived and their primitive racism, and the state became increasingly attractive as an ally. From the mid-90s onward, it funded most of these anti-racist initiatives. All these changes were completed by the self-disarming of most of the radical Left, which started adopting the aim of “strengthening civil society” at the same time as it removed all references to class struggle.
The most important NSU members were born in the mid-1970s in East Germany and were politically socialized in the “asylum debate” in the early 90s. It was a phase of massive de-industrialization and high unemployment in the East of Germany. The young Nazis learned that they could use violence against migrants and leftist youth without being prosecuted by the state. They realized that they could change society through militant action.
In West Germany a new youth culture grew in the ’80s as well: right-wing skinheads. The skinhead scene in the East and in the West was held together by alcohol, excessive violence, concerts, and the distribution of illegal videos and CDs. This music business allowed them to set up their own financing. Still, a large part of their money was organized through petty crime. From the beginning, many Nazis were involved in prostitution, and arms and drug trafficking. Later they became heavily involved with biker gangs and security firms, which are booming due to the the privatization of state functions.
In the mid-90s various militant groups and other groups from the rightwing music scene united under the banner of the Blood \& Honour network (B\&H).\footnote{\emph{Blood \& Honour} is a neo-Nazi music promotion network and political group founded in the United Kingdom in 1987. \emph{Combat 18} was founded in 1992 as its militant arm.} Soon after the German Nazi scene organized internationally, making contacts worldwide and building an infrastructure that stretched from CD production to arms dealing and shooting ranges. At that time the police could no longer countenance Nazi violence, and the Nazis had to hide their actions. In that context, the B\&H\Slash{}\emph{Combat 18} concept of clandestine struggle and small, independent terrorist groups (“leaderless resistance”) helped them reorganize.
In the former East German state of Thuringia, the Nazi scene was built up by “Freie Kameradschaften,”\footnote{In the early 90s the militant neo-Nazi scene began to organize in groups called \emph{Freie Kameradschaften} (free associations, free camaraderie). These have no formal membership and no centralized national structure, but keep in close contact. Over 150 such \emph{Kameradschaften} exist in Germany.} the Thüringer Heimatschutz (THS),\footnote{The \emph{Thüringer Heimatschutz} (THS) was a coordinating network of the \emph{Freie Kameradschaften} in Thuringia with up to 170 members. Its head Tino Brandt was a paid CI for VS in Thuringia.} Blood \& Honour, and the Ku Klux Klan. This is the environment that gave birth to the National-Sozialistischer Untergrund. The “Kameradschaft Jena” consisted of Ralf Wohlleben, Holger Gerlach, André Kapke, Böhnhardt, Mundlos, and Zschäpe. From 1995 onwards they were filed as “rightwing extremists” in the VS Information System. Organized in the THS, they practised the use of explosives and firearms, and committed their first attacks. The other members of the “Kameradschaft Jena” remained active in the scene after the Trio went underground in 1998. And they supported their comrades: Holger Gerlach gave them his driver’s licence, passport, and birth certificate, and he rented motorhomes for them. Kapke and Wohlleben organized weapons and passports. Those two organized the largest right-wing rock festival in Germany and maintained international contacts. In 1998 Wohlleben became a member of the NPD, the largest neo-Nazi party at the time. Over time he became its deputy chairman in Thuringia. With the help of this network, Böhnhart, Mundlos and Zschäpe could move underground and commit their attacks, probably with local support.
\section{The Informants System}
The German State is directly involved in organized fascist structures. But the direct and extensive involvement in the Thüringer Heimatschutz and the National-Sozialistischer Untergrund stands out. In and around these groups the VS positioned more than two dozen Confidential Informants, or CIs. These CIs were not used to catch violent Nazis like the Trio, instead they organized the militant Nazi scene in Germany, developing it ideologically and militarily. The VS recruited mostly very young fascists and made them into leaders of the scene. In an internal document of 1997, the Bundeskriminalamt (Federal Criminal Police Office, or the BKA) called these CIs “incendiaries” in the Nazi scene.\footnote{\emph{Von Baumgärtner, Maik; Röbel, Sven; Stark, Holger,} “\emph{Innere Sicherheit: Der Brandstifter-Effekt},” Der Spiegel 45, November 5, 2012; “\href{https://www.antifainfoblatt.de/artikel/der-\%C2\%BBbrandstifter-effekt\%C2\%AB-des-verfassungsschutzes}{Der »Brandstifter-Effekt« des Verfassungsschutzes},” \emph{Antifaschistisches Infoblatt}, March 8, 2014.} It saw “the danger that the CIs egged each other on to bigger actions” and found it questionable “whether some actions would have happened without the innovative activities of the CIs.” There are many statements by former CIs descriving how they discussed their political actions with their handlers. In some of those cases the handlers prevented their CIs from leaving the scene or told them to appear more aggressive. For the German intelligence agencies, maintaining CIs is more important than law enforcement. They protected them from the police in multiple cases so that they could operate undisturbed. In the mid-90s there was a brief debate about this problem, because it became known that CIs of the German intelligence agencies fought and killed as mercenaries in the Yugoslavian civil war.
In 1996 the Federal Interior Ministry began \emph{Operation Rennsteig}: the Bundesamt für Verfassungschutz (BfV, federal domestic secret service), Militärischer Abschirmdienst (MAD, German military intelligence agency), and local VS agencies of Thuringia and Bavaria coordinated their intelligence activities relating to the THS and the NSU, at least until 2003. They discussed the recruitment of informants but also how they could achieve discursive hegemony within “civil society.” \emph{Operation Rennsteig} marks a turning point in German interior policy, which really took hold when Otto Schily, a former ’60s student radical and defense lawyer of the Red Army Faction, became interior minister in 1998. There was an unseen extension of the security apparatus and an adjustment of the focus of the intelligence agencies. To adapt themselves to the new international constellation (Yugoslavian wars, the first attack on the World Trade Center in 1993), they centralized the German intelligence structure and unified the handling of the Nazi scene. In this process they also expanded intelligence activities within the Nazi scene. All this happened at the same time as the shift in “foreigners policy” from the attempt at “reduction” under Kohl to the “fight against parallel societies in our midst” under Schily.
Everyone involved in \emph{Operation Rennsteig} knew that it was an explosive and not entirely legal operation. Most of the files concerning recruitment and handling were incomplete, some CIs were not even registered. Between November 12, 2011 and the summer of 2012, 310 case files were destroyed in the BfV alone. They tried to destroy everything connected with \emph{Operation Rennsteig,} CI “Tarif,” and other important CIs around the NSU. Again, the commands were coming from the top of the hierarchy. A few days after the first destruction of files, the Federal Interior Ministry gave the order to continue the destruction. Not only did they destroy physical files, they also manipulated computer files and deleted the phone data of CIs in contact with the NSU.
\section{Who Was in Control?}
When more and more high-level CIs in the NSU’s immediate environment were exposed, they began to tell the fairy tale of “CIs out of control.” This was just the secret service’s next smokescreen, after such cover stories as “we didn’t know anything” and “we were badly coordinated” collapsed when \emph{Operation Rennsteig} became publicly known. It is a lie, but many on the Left believe it because it fits into their picture that “the Nazis can do what they want with the state.” It is therefore worth taking a closer look at this point.
Who are CIs? The services usually try to recruit people with problems: prison, debts, and personal crises. These people then receive an allowance that can amount to a normal monthly income for important CIs. CIs get support for their political actions and warnings before a house search. On the other hand, there is a lot of control: surveillance of all telephones, tracking of movements, sometimes direct shadowing. In order to crosscheck the reports, the VS runs more CIs than it would otherwise need. Time and again there are meetings of Nazi cadres with four or five CIs sitting around the table. There were several CIs within the NSU structure who did not know about each other. The great majority of them did what the VS wanted them to do — passing on information, betraying everything and everyone, while also directly supporting armed struggle by providing passports, logistics, propaganda and weapons.
Some examples of CIs in the NSU structure:
\begin{itemize}
\item\relax
Tino Brandt, the chief of the THS, was the best paid CI of the Thuringia VS from 1994 to 2001; he helped the Trio go underground, and afterward provided passports and money.
\item\relax
Thomas Starke (LKA CI in Berlin from 2000 to 2011) organized weapons and the Trio’s first hideout, and he delivered explosives before they went underground. He gave clues as to where they could be found in 2002, but these were “not investigated.”
\item\relax
Thomas Richter was CI “Corelli” for the BfV from 1994 to 2012; after this became public he was kept hidden by the agency and was found dead in April 2014. He had “immediate contact” with Mundlos as early as 1995, and was the link between the NSU and the KKK and co-founder of the anti-antifa.
\item\relax
Andreas Rachhausen – “GP Alex” – brought back the getaway car the three had used for going underground in January 1998, when Rachhausen was already a CI.
\item\relax
Ralf Marschner was CI “Primus” for the BfV from 1992 until about 2001. He rented motorhomes through his building company at exactly the time when two of the murders occurred.
\item\relax
Carsten Szczepanski tried to build up a German branch of the KKK in the early 1990s, while monitored by the VS. Between 1993 and 2000, he was imprisoned for a brutal attempted murder. In prison he co-edited the Nazi magazine “Weißer Wolf” (White Wolf), which propagated the concept of \emph{leaderless resistance} and sent greetings “to the NSU” even then. He became a CI in prison. For his work he received many prison privileges (besides lots of money). He supplied much information, for example that Jan Werner had organized the Trio’s weapons. Immediately after his release he tried to set up a terror cell like \emph{Combat 18}. When his cover blew in 2000, the VS got him a new identity and sent him abroad.
\item\relax
Michael von Dolsperg, (formerly See), a member of \emph{Combat 18}, close to the THS. From 1995 to 2001 he was a BfV CI with the code name “Tarif.” He was rewarded with at least 66,000 D-Mark. After 1994 he was editor of the magazine “Sonnebanner,” which proposed “going underground” and “forming independent cells.” We know that some of its articles were discussed by Mundlos, Böhnhardt, Zschäpe and their close contacts. Dolsperg produced a total of 19 issues. In an interview he claimed that “the BfV got all issues in advance.”\footnote{\emph{Der Spiegel}, September 2014.} This is not the only case where the VS partly financed and “fine-tuned” the contents of a Nazi magazine. In Thuringia, the VS was consulted for anti-antifascist leaflets and did the proofreading.\footnote{“\href{https://www.antifainfoblatt.de/artikel/der-th\%C3\%BCringer-nsu-untersuchungsausschuss}{Der Thüringer NSU-Untersuchungsausschuss},” \emph{Antifaschistisches Infoblatt} 101 \Slash{} 4.2013, 28.01.2014.} In 1998 Kapke asked Dolsperg if he could provide housing for the Trio in hiding. Dolpsberg refused after his handler advised him to do so.
\end{itemize}
Parallel to the story about “CIs out of control,” the intelligence agencies created another one: “too much chaos in the intelligence apparatus.” To support this legend they put on display all the internal conflicts between the various law enforcement and intelligence agencies, cases of “conflicting authorities” and the competition between different agencies. One highpoint was the scandal around Roewer, the former President of the local VS agency in Thuringia.\footnote{From 1994 to 2000 Helmut Roewer was president of the Thuringia Verfassungsschutz. He is famous for his excessive leadership of the VS, involving prostitutes and spiked helmets. In summer 2000 he had to resign because it came to light that he financed important militant Nazis not only with help of the ‘normal’ VS structures but also with a system of front companies. Exactly who got the money remains unclear. Roewer himself said some time ago that the Thuringia Verfassungsschutz funded the neo-Nazi scene with 1.5 million DM. Today Roewer publishes with the right wing Ares-Verlag.} All this show of confusion was used to make the NSU a pretext for the enhancement of the security apparatus.
\section{1998: The So-called Disappearance of the Underground}
In January 1998, the LKA found pipe bombs and explosives in a garage rented by Zschäpe in Jena. The VS had known about these explosives all along. Nonetheless, Böhnhardt was able to leave undisturbed in his car during the raid. It took days until the police issued a warrant for the Trio because all those responsible were on sick leave, on vacation, or otherwise unavailable. Obviously they wanted the Trio to go underground. Already in November 2011, the famous German feuilletonist Nils Minkmar described the nature of the “underground” as follows: “They didn’t have to hide very deep, it was more like snorkeling in a bathtub: They used to have a social life in \emph{Zwickau}, kept in contact with a wide circle of supporters and attended demonstrations, concerts and other events. Many did know where the three were hiding. And if the right wing scene in Germany has a problem, it is certainly not that it is extremely sealed off, but that it is heavily interspersed with CIs.” In fact, today we know that the three operated in an environment that was structured and monitored by the VS; most of their main supporters were CIs. After searching the garage, the police even found two address lists belonging to Mundlos containing 50 names, including at least five CIs.\footnote{Von Maik Baumgärtner, Hubert Gude und Sven Röbel, “\href{http://www.spiegel.de/panorama/nsu-behoerdenversagen-bei-der-fahndung-nach-dem-trio-a-883431.html}{Ermittlungspanne: Fahnder werteten NSU-“Garagenliste” nicht richtig aus},” \emph{Spiegel Online}, February 14, 2014; Wolf Wetzel, “\href{http://wolfwetzel.wordpress.com/2013/11/16/die-garagenliste-die-gold-card-des-nationalsozialistischen-untergrundesnsu/}{Die Garagenliste – die Gold Card des Nationalsozialistischen Untergrundes\Slash{}NSU},” Eyes Wide Shut, November 16, 2011.} The lists displayed the national network of the NSU, with contacts in Chemnitz, Jena, Halle, Rostock, Nuremberg, Straubing, Regensburg, Ludwigsburg. Officially, the police never analyzed the lists or used them for investigation purposes!
\section{2000: The Extremism Doctrine and the Beginning of the Murders}
Two and a half years later, on September 9, 2000, the Ceska murders began with the death of Enver Simsek. In early summer the BfV had informed the interior ministry that “a few groups” were trying to get the “structure and the equipment” to “attack certain targets.” These groups were especially active in the states of Berlin and Brandenburg, Saxony, Saxony-Anhalt and Lower Saxony. The BfV also kept an eye on the Trio — after they went underground they were closely watched by the unit for right-wing terrorism (!). Nevertheless the BfV claimed that these small Nazi groups had “no political concept for armed struggle,” although they actively propagated such concepts by supporting newspapers such as the “Sonnenbanner.” Federal Interior Minister Schily used this information to make a press statement in which he warned of the “danger of Antifa actions radicalizing individual right-wing extremists. These militant right-wing extremists or small groups could decide to retaliate.”
The strategy was to build up fascist structures and to blame the radical left for their existence in the public discourse, employing the extremism doctrine.\footnote{The “extremism doctrine” is the state doctrine in the Federal Republic of Germany, which says that the democracy of the Weimar republic (1918-1933) was destroyed by the violent extremism of the right and the left. The term was coined in the 1970s by the VS. Before the 1970s it was called “radicalism,” but had to be changed because in the 60s “radical” became a positive term.} The film \emph{Youth Extremism in the Heart of Germany}, made by the Thuringian VS in May 2000, is a clear example. At the beginning it states that fascist and antifascist “scenes need each other, they cannot live without each other” and that “violence as a means to an end is accepted in the left-wing scene.” It describes the fascists with the usual clichés: unemployed, uneducated, disorganized, committing crimes when drunk. Roewer, the president of the VS, explains the high number of right offenses “solely with the fact that scrawling swastikas, roaring \emph{Sieg Heil} \dots{} are offenses in Germany \dots{} because of that the statistics appear very high with over 1,000 crimes per year, but nearly all are propaganda offences.” The THS is mentioned positively, Kapke and Tino Brandt are allowed to speak: “the Anti-Antifa Ostthüringen was formed in response to violence from the left, to bring those perpetrators to light,” and “We are representatives of the \emph{National Democratic Party} of Germany in Jena \dots{} We are fundamentally opposed to violence.”
\section{2003-2005: The Manhunt is Discontinued; Bomb Attack in Cologne}
In 2003 four immigrants from Turkey had already been killed. Evidence piled up that the murders could have a right-wing extremist background. In March 2003 the Italian secret service gave the VS evidence of a network of European Nazis that prepared murders of immigrants. The FBI had analysed the murders and regarded “hatred of Turks” as a motive for the murders. In Baden-Württemberg CI “Erbse” revealed that there was a Nazi group called NSU and one member was called “Mundlos:” the handler was advised to destroy this information. It was decided to let the Trio disappear.
In June 2004, a nail bomb exploded in the Keupstraße in Cologne. The attack resembled other right-wing attacks, for example the London nail bombings by the Nazi David Copeland five years earlier. But the Federal Interior Minister Schily announced two days later: “The findings of our law enforcement agencies do not indicate a terrorist background, but a criminal one.” He definitely knew better!
The shops and restaurants in the Keupstraße are almost exclusively run by immigrants. Many of these shops are very successful; some businesspeople even joined in an initiative to become active in local politics with their own demands. The attack ended these attempts. The uncertainty as to who was behind the attack and the crackdown by the police on the victims directly after created great distrust in the Keupstraße, which is still felt to this day.
The Keupstraße bombing and its aftermath exemplify the structural interaction of state institutions with the fascist terror: first the attack terrorizes the immigrants, then they are harassed by the police and the media. This harassment makes the intentions of the NSU a reality: “foreign profiteers” and “foreign mafias” were marked and cut off from the German “Volkskörper” (“German people’s body”).
\section{2006-2007: Murders of Migrants Stop, Police Officer Kiesewetter is Murdered}
In April 2006 two people were killed within three days: kiosk owner Mehmet Kubasik in Dortmund and Halit Yozgat in his internet café in Kassel. The body count of the Ceska murders went up to nine. The victims’ relatives organized joint demonstrations in Kassel and Dortmund, shouting the slogan “No tenth victim!” After the demonstrations the series of murders stopped.
The murder in Kassel showed clearly that the VS wanted to sabotage all investigations – and that this was a decision from the top of the hierarchy: at the time of the murder the Hessian VS officer Andreas Temme was present in Yozgat’s internet café. Temme was known as a gun fanatic and collected fascist literature. He was the only person present at the murder scene and did not come forward to the police. At that time he was the handler of a fascist CI with whom he had a long phone call an hour before the murder. The police saw Temme as a suspect for the entire Ceska series. Nevertheless, the Hessian VS refused to give the police any information; otherwise someone “would just have to put a dead body near a CIs or a handler” to “paralyze the whole VS.” The dispute between the police and the VS was taken up to the Hessian interior minister Bouffier, who stopped the investigations after consultation with the BfV.
Just over a year later, on April 25, 2007, the police officer Michèle Kiesewetter was shot in her police car. Her colleague Martin Arnold, sitting next to her, survived a headshot. After four and a half years the investigations still had not gotten anywhere. After the NSU became publicly known, politicians and the public prosecutor insisted obstinately that Kiesewetter had been murdered by chance and that Böhnhardt and Mundlos had been the sole perpetrators. But that story does not add up!\footnote{Why would Böhnhardt and Mundlos go all the way to Heilbronn to kill at random a police officer who was also from Thuringia? A police officer whose immediate superior was a member of the KKK? Kiesewetter’s uncle is a police officer involved in fascist structures himself; he said to the police in 2007 that the murder of his niece was connected to the Ceska murders. The police officers investigating Heilbronn concluded from eyewitness accounts that there were six perpetrators and made composite sketches, but those were not used in the investigation, etc.} In the case of Kiesewetter, the poor performance of the investigation teams cannot be explained by “racism.” The murder victim was part of the police. Why the need for a cover-up?
After the murder in Heilbronn, it became quiet around the NSU. Four and a half years later, suddenly there were two bank robberies that were attributed to the NSU. After the second of these failed, Böhnhard and Mundlos allegedly committed suicide and the NSU became a matter of public knowledge.
\section{Germany’s “Security Structure” and the Nazis}
\emph{One has to make use of the far right, no matter how reactionary they are\dots{} Afterwards it is always possible to get rid of them elegantly\dots{} One must not be squeamish with auxiliary forces.}
– Franz Joseph Strauß\footnote{Franz Josef Strauß was a German politician. He was the chairman of the CSU (independent party in Bavaria, but in an electoral union with the CDU), a member of the federal cabinet in various positions and for a long time minister-president of Bavaria. During his political career Strauss was a controversial figure, a law-and-order politician, well connected to the intelligence agencies and often leaning to the far right. He was involvement in several large-scale scandals.}
Since at least the disclosures starting in Italy in the second half of 1990, it has been known that NATO keeps armed fascist troops as a reserve intervention force. Only states with such a “stay-behind” structure could become NATO members after the Second World War. In case of a Soviet occupation this reserve was supposed to fight as a guerrilla force behind the front (hence the name \emph{stay-behind}). But it also had to prevent Communist Party election victories and other forms of radical social change. In West Germany the stay-behind troops were called \emph{Technischer Dienst} (technical services) and were built up by Nazi war criminals such as Klaus Barbie under US leadership. This became publicly known for the first time in 1952.\footnote{See Daniele Ganser, \emph{Nato’s Secret Armies: Operation Gladio and Terrorism in Western Europe} (Cass: New York, 2004).}
According to a German government report of December 1990, in which the existence of stay-behind structures was admitted, “preparations for the defence of the state” were made in cooperation with the Bundesnachrichtendienst (BND, German foreign intelligence agency) from 1956 onwards. Heinz Lembke was part of these structures. He delivered weapons to the \emph{Wehrsportgruppe Hoffmann}\footnote{The Wehrsportgruppe Hoffmann was one of the largest paramilitary groups in Germany. It was founded by Karl-Heinz Hoffmann in 1973 and prohibited in 1980. Part of the group subsequently went to Lebanon to receive military training. In September 1980 a bomb exploded at the Oktoberfest in Munich, killing 13 people. The alleged individual perpetrator Gundolf Köhler, who died in the explosion, was a member of the Wehrsportgruppe Hoffmann.} in the ’70s. Lembke’s huge arsenal was discovered incidentally by forestry workers in 1981. The night after Lembke agreed to disclose who had pulled the strings, he was found hanged in his cell.
The stay-behind structures obviously changed their character in the 70s and 80s (in Italy they were called \emph{Gladio} and took part in something they must have understood as a civil war from 1969 to 1989.) In the 1990s they changed their direction again: now Islamism was the main enemy – it was perhaps at this point that new personnel were recruited. The thread connecting them: fascist groups as reserve intervention forces.
Christian Menhorn’s testimony at the penultimate session of the BUA\footnote{Before that, the BUA had not paid attention to the BfV. The delegates had not even known about its department for right-wing terrorism.} is typical of the secret services’ self-confidence. Menhorn was responsible for the THS at the time. He appeared as the best-informed VS analyst. He gave the BUA members the impression that he knew a lot more about the Nazi scene than they did and reprimanded them repeatedly. The questions put to him centered on why the VS prevented any mention of the Trio in a joint internal paper by the VS and BKA. Menhorn said that the VS, in opposition to the BKA, knew that the Trio was “irrelevant.” That was after the first murders had already happened. When he was asked for the reasons for this fatal denial, his immediate reply was very brief but still revealed what the VS did at that time: “We adjusted our information.”\footnote{Hajo Funke, \href{http://hajofunke.wordpress.com/2013/05/28/hajo-funke-abbruch-der-untersuchung-auf-halber-strecke-das-vorzeitige-ende-der-offentlichen-ermittlung-des-nsu-untersuchungsausschusses-des-bundestags/}{\emph{Abbruch der Untersuchung auf halber Strecke. Das vorzeitige Ende der öffentlichen Ermittlung des NSU Untersuchungsausschusses des Bundestags}}.}
Menhorn, Richard Kaldrack (alias; Marschner’s handler), Thomas Richter, Mirko Hesse, Martin Thein (Dolsperg’s handler) and Gordian Meyer-Plath, Scepanski’s handler and head of the Saxony VS, are all part of a new generation, born in 1966 or later, who came straight from school or university and started working for the VS. They all stand for the extremism doctrine; some of them have used it for an academic career. Thein for example has published books on Ultras and “fan culture” with leftwing publishers. It is very unlikely that those agents\Slash{}handlers, who were very young at the time, could have taken important decisions (not stopping the Trio, giving them arms, keeping information from the police \dots{} ) without consultation with the hierarchy. They were instructed by old hands like Norbert Wießner, Peter Nocken and Lothar Lingen (alias), who won their wings fighting the Red Army Faction. Lingen set up a department in the BfV exclusively for “right terror” at the beginning of the 90s. He could be called the highest-ranking agent\Slash{}handler: it was he who coordinated the destruction of files after the existence of the NSU became public knowledge.
Behind them there was a strategic level of a very few high officials whose careers swung between the interior ministry, the chancellery and the top levels of the services (e.g. Hanning and Fritsche).
\section{Intelligence, Nazis, and the War}
Since the mid-90s Germany has almost always been at war. The biggest missions were those in Yugoslavia since 1995 and in Afghanistan since 2002. The role of intelligence became far more important, playing a greater role in securing German territory, holding down the domestic opposition to the war, and monitoring the Bundeswehr (German army) soldiers. To these ends it uses intelligence operations against opponents of the war, it infiltrates Islamist groups, and it cooperates with neo-fascist soldiers and mercenaries.
Many German and Austrian Nazis fought in the Yugoslavian civil wars, especially on the Croatian side. This involvement was organized by contacts in the “Freien Kameradschaften” and was known to the German government all along. At the same time, the German government ignored the embargo and sent military instructors to Croatia. August Hanning (see below) told the BUA that they were fighting against the Islamists since the mid-90s – and could not be bothered with the Nazis. They were focused on the “presence of al-Qaeda terrorist groups” and not on the extreme right-wing terrorists in Bosnia. What this statement obscures, of course, is that they had previously had strongly supported the Islamist militias, when these were not yet called “al-Qaeda.”
These wars were quite lucrative for some Nazis. Normally they got no pay but they were allowed to loot. They took part in “ethnic cleansing.” The regular Croatian army and the professional mercenaries\footnote{U.S. companies heavily involved in the conquest of Krajna.} conquered a town and marked the houses of “Serbs.” Then the Nazis were allowed to plunder and murder. After their return to Germany some Nazis could build up companies (and get leading positions in the NPD and other organizations).
The Bundeswehr has been called an “expeditionary force” since 2006. It became an all-volunteer military in July 2011 and can also be used inside Germany. So far, Germany has had little direct experience of the “privatization of warfare,” but the Bundeswehr is actively trying to eliminate this “shortcoming,” seeking to create its own private shadow armies with the support of the Federal Employment Agency. (This agency finances the training and “certification of safety personnel for international assignments”).
\section{Operational Cores and Control from Above}
You can see that the structure that led the NSU is still intact by looking at the systematic action to destroy important files. The heads of the agencies were immediately operationally active. On a strategic level they set the course for the further upgrade of the law enforcement agencies with targeted public relations work. In total five presidents of VS agencies were forced to resign. These resignations were intended “to provide breathing space for the Minister of the Interior,” as one of these directors put it. But above all the resignations were supposed to allow the the operational work to continue undisturbed. The “deep state” – this dense web of intelligence agencies, military, and police that supports government actions and implements its regulations with extra-legal means, “freelance” employees and “auxiliary forces” – must not become visible.
August Hanning is certainly one of the strategic coordinators of this structure. From 1986 to 1990 he was Security Officer in the embassy in East Berlin, among other things responsible for prisoner ransom. In 1990 he moved to the German chancellery and in 1998 he became president of the BND. Under his leadership the BND assisted in abductions and torture by the CIA. Among other things, Hanning argued against the return of Guantanamo prisoner Murat Kurnaz, although he knew of his innocence.\footnote{Murat Kurnaz is a Turkish citizen and resident of Germany. He was arrested was arrested in Pakistan late in 2001 then imprisoned at Guantanamo Bay for five years. From 2002 onwards the USA was ready to return Kurnaz to Germany, but the German government declined that offer. According to the German government Kurnaz had lost his residency permit because he had left Germany for more than 6 months without notice. Kurnaz couldn’t return to Germany until a court ruled that he still had his residency permit because in Guantanamo he was unable to apply for an extension of his “leave to remain.”} He became secretary of state in the interior ministry late in 2005. During his examination before the BUA he said in relation to the NSU complex that “the security structure of Germany has proved itself.”
Another important figure is Klaus-Dieter Fritsche (CSU). Since the beginning of this year he has been federal government commissioner for the federal intelligence services, a newly created post. He is at the height of his career now. In 2009 he succeeded Hanning as Interior Ministry Secretary of State: in this capacity he was known as “Germany’s most powerful official” and “the secret interior minister.” Previously he was intelligence coordinator at the federal chancellery and before that, from 1996 to 2005, he was vice-president of the BfV with responsibility for the management of CIs like Corelli, Tarif and Primus. At the BUA he expressed the self-image of the “deep state” clearly: “secrets that could affect the government’s ability to act if revealed, must not be revealed\dots{} the interests of the state are more important than a parliamentary investigation.”
This “deep state” has a long tradition in Germany: it survived both 1933 and 1945. In 1933 the Nazis could smash the (Communist) opposition quickly, because the political police had previously created files about them which they immediately made available to the Nazi government. After 1945 the secret services, police agencies, and the administrative apparatus continued with essentially the same personnel. The BND, the VS and the \emph{stay-behind} structures were made up of old Nazis. But today this complex runs across party lines. In the case of the NSU, both CDU and SPD Interior Ministers of the states played a role. BKA chief Ziercke is member of the SPD, while the public prosecutor is from the FDP [Free Democratic Party, a liberal party]. In Thuringia, interior ministers openly fought antifascist activities in cooperation with the VS whether they were from the SPD or the CDU, and so on.
The VS was an important tool in the domestic policy of all previous governments. In the mid-50s it helped to ban the Communist Party; in the 60s it worked with intelligence operations and agent-provocateurs against the youth movement. At the beginning of the 70s it helped the Brandt government to implement “professional bans”: 3.5 million applicants for civil service were audited, 11,000 applicants were banned from work as civil servants. There were unofficial disciplinary procedures and dismissals, too.
These structures survived the collapse of the Eastern Bloc: the security services were even able to use them to expand their sphere of influence. This was reinforced by 9\Slash{}11: During the \emph{war on terror} intelligence agencies worldwide had a massive boost, similar to that of the Cold War. The United States enhanced its security apparatus with the Patriot Acts to ensure “homeland security.” In Germany, the \emph{Joint Counter-Terrorism Centre} was founded in 2004 to coordinate BKA, BND, VS and the LKAs. The BKA Act of 2009 provides the BKA with means “to respond to threats of international terrorism,” which were previously only available to the police authorities of the states (computer and network surveillance, dragnets, use of undercover investigators, audio and video surveillance of housing and telecommunications). In addition, the BKA can now investigate without concrete suspicion on its own initiative, without the approval of an prosecutor.
The development of the scandals surrounding the NSU and the surveillance of the NSA and its western partners (which include the German agencies) has made clear that the power of the ‘deep state’ in Germany is stronger than was expected. It was never touched and has survived all scandals. Across party lines, parliamentary investigation is conducted with special consideration for \emph{raison d}‘\emph{état}. This ensures that the deep state is not affected and that police and intelligence agencies continue to be empowered, provided with additional rights and encouraged to cooperate more closely. Because of this, the Bundestag investigation committee arrived at the non-factual conclusions that there is “no evidence to show that any authority was involved in the crimes (of the NSU) in any manner, or supported or approved them” and that there was no evidence “that before November 4, 2011 any authority had knowledge” of the NSU or its deeds or “helped it to escape the grasp of the investigating authorities.”\footnote{From the final report of the parliamentary investigation committee. Available at: http:\Slash{}\Slash{}dipbt.bundestag.de\Slash{}dip21\Slash{}btd\Slash{}17\Slash{}146\Slash{}1714600.pdf.}
This \emph{raison d}‘\emph{état} also includes the PdL (Partei die Linke – Left Party), which participated “constructively” in the BUA and supported its final report. The PdL is the left-wing opposition party in Germany. It was formed in 2007 through a merger of the successor of the SED (state party of former East Germany) and the Left opposition in the SPD. It is increasingly supported by sections of the radical left. So far, the VS had spied on the PdL. As part of the final declaration of the BUA the PdL has been assured that it will be no longer monitored by the secret services.
The Nazi scene is hardly affected: the unmasking of the NSU has not weakened it, instead many are encouraged to pursue their goals at gunpoint. They are arming themselves. In 2012, there were 350 cases of gun use registered. That was a peak, but in 2013, the use of firearms by Nazis increased further. Refugee shelters are attacked much more frequently again.
There is no reason to believe that we could take action against the brown plague via the state. At the trial in Munich, the public prosecutor is doing a political job, trying to deal with the case according to the ruling doctrine.
A weakness of large parts of the “left” opposition and the radical Left becomes apparent: after the pogroms of the early ’90s many abandoned the working class as a revolutionary force. They could therefore only turn to “civil society” and thus ultimately the state as an ally against the Nazis. This ally supported fascist structures and helped to establish them, while at the same time it gave the left-wing opposition the opportunity to turn itself into a force supportive of the state. This fact paralyses many Antifa and other leftwing groups. Instead of naming the state’s role in the NSU complex, they focus on the investigation committees and the trial, they lose themselves in the details which are produced there. There were no significant movements on the streets when the NSU became public. All this allows the state apparatus to minimize the NSU – but many people still feel the horror.
\bigskip
\section{NSU timeline:}
\bigskip
\noindent
\begin{minipage}[t]{\textwidth}
\begin{tabularx}{\textwidth}{|X|X|}
\hline
\relax 1993\Slash{}1994 & Foundation of the “Kameradschaft Jena” \\
\relax 1996 & Foundation of the Thüringer Heimatschutz (THS) \\
\relax 1996-1998 & Small actions with dummy bombs and deactivated bombs \\
\relax 1\Slash{}26\Slash{}1998 & Raid on the garage of Zschäpe; the Trio (Zschäpe, Mundlos, Böhnhardt) goes underground \\
\relax 1998-2011 & Numerous bank robberies \\
\relax 07\Slash{}27\Slash{}2000 & Bomb attack on eastern European, mostly Jewish migrants in Düsseldorf \\
\relax 09\Slash{}09\Slash{}2000 & Murder of Enver Şimşek in Nürnberg \\
\relax 01\Slash{}19\Slash{}2001 & Bomb attack in the Probsteigasse in Cologne \\
\relax 6\Slash{}13\Slash{}2001 & Murder of Abdurrahim Özüdoğru in Nürnberg \\
\relax 6\Slash{}27\Slash{}2001 & Murder of Süleyman Taşköprü in Hamburg \\
\relax 8\Slash{}29\Slash{}2001 & Murder of Habil Kılıç in Munich \\
\relax 2\Slash{}25\Slash{}2004 & Murder of Mehmet Turgut in Rostock \\
\relax 06\Slash{}09\Slash{}2004 & Bomb attack on the Keupstraße in Cologne \\
\relax 06\Slash{}09\Slash{}2005 & Murder of İsmail Yaşar in Nürnberg \\
\relax 6\Slash{}15\Slash{}2005 & Murder of Theodoros Boulgarides in Munich \\
\hline
\end{tabularx}
\end{minipage}
\bigskip
\bigskip
\noindent
\begin{minipage}[t]{\textwidth}
\begin{tabularx}{\textwidth}{|X|X|}
\hline
\relax 04\Slash{}04\Slash{}2006 & Murder of Mehmet Kubaşık in Dortmund \\
\relax 04\Slash{}06\Slash{}2006 & Murder of Halit Yozgat in Kassel \\
\relax 4\Slash{}25\Slash{}2007 & Murder of the police officer Michèle Kiesewetter \\
\relax 11\Slash{}04\Slash{}2011 & The NSU becomes publicly known \\
\hline
\end{tabularx}
\end{minipage}
\bigskip
\bigskip
\section{List of Abbreviations}
VS = German domestic secret service
BfV = Federal office of the domestic secret service
MAD = German military intelligence agency
BND = German foreign intelligence agency
BUA = Parliamentary investigation committee
NSU = National Socialist Underground
B\&H = Blood \& Honour
Trio = Böhnhardt, Mundlos, Zschäpe
BAW = German public prosecutor
BOA = Special investigation team
LKA = The “Criminal Police Offices” of Germany’s 16 federal states (\emph{Länder}). Each incorporates a ‘state security’ division.
BKA = Federal equivalent of the LKA, with reponsibility for “national security,” “counter-terrorism,” etc.
THS = \emph{Thüringer Heimatschutz} (Thuringia Homeland Protection): coordinating network of the neo-Nazi \emph{Freie Kameradschaften} groups in Thuringia, eastern Germany. See also footnotes 9 and 10.
% begin final page
\clearpage
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% the page would be odd on an even one.
\ifthispageodd{\strut\thispagestyle{empty}\clearpage}{}
% new page for the colophon
\thispagestyle{empty}
\begin{center}
The Anarchist Library
\smallskip
Anti-Copyright
\bigskip
\includegraphics[width=0.25\textwidth]{logo-en}
\bigskip
\end{center}
\strut
\vfill
\begin{center}
Wildcat
The Deep State
Germany, Immigration, and the National Socialist Underground
September 11, 2014
\bigskip
Retrieved on June 3, 2016 from https:\Slash{}\Slash{}viewpointmag.com\Slash{}2014\Slash{}09\Slash{}11\Slash{}the-deep-state-germany-immigration-and-the-national-socialist-underground\Slash{}
Also available at Wildcat website: http:\Slash{}\Slash{}www.wildcat-www.de\Slash{}en\Slash{}actual\Slash{}e075\_nsu.html
\bigskip
\textbf{theanarchistlibrary.org}
\end{center}
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\end{document}
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\input zb-basic
\input zb-matheduc
\iteman{ZMATH 2003a.00063}
\itemau{Reynolds, Eric; Treahy, Diana; Chao, Chin-chi; Barab, Sasha}
\itemti{The Internet learning forum: Developing a community prototype for teachers of the 21st century.}
\itemso{Comput. Sch. 17, No. 3-4, 107-125 (2001).}
\itemab
This paper reports an effort to create a community of practice for teachers' professional development via the World Wide Web. Beginning with a discussion of our theoretical foundations and current online models of professional development, we address the problem of how developing Web and video technologies may provide innovative and effective professional development for teachers. We describe the initial conceptions of the Internet Learning Forum (ILF), a Web site developed to support mathematics and science teachers sharing and evolving their pedagogical practices. This site includes exemplary instructional units, teachers' reflections, and peer discussion. Starting with video, ILF participants examine assumptions, reflect on practices, and share within the ILF community. The goal of this unique community of practice is to create quality professional development. While this paper provides an overview of our initial design work, the site has evolved into a nationally funded project. However, the work described here, which guided the development of the prototype, has important implications for other Web-based efforts to support teacher professional development. (Authors' abstract)
\itemrv{~}
\itemcc{B50}
\itemut{}
\itemli{doi:10.1300/J025v17n03\_07}
\end
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https://www.mathematik.tu-dortmund.de/lsiii/cms/bibtex/41223916.tex
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tu-dortmund.de
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@TECHREPORT{GevelerRibbrockGoeddekeTurek2009,
author = {Geveler, M. and Ribbrock, D. and G\"{o}ddeke, D. and Turek, S.},
title = {{Lattice}--{Boltzmann} Simulation of the Shallow--Water Equations with Fluid--Structure Interaction on Multi-- and Manycore Processors},
year = {2009},
month = dec ,
institution = {Fakult\"{a}t f\"{u}r Mathematik, TU Dortmund},
note = {Ergebnisberichte des Instituts f\"{u}r Angewandte Mathematik, Nummer 400},
}
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https://unilab.gbb60166.jp/math_ia/tex/nia22x2ansc.tex
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%\vskip-2ex
{\large\gtfamily\bfseries ■ 条件付き確率}
\toi 52枚のトランプ（ジョーカーは含まない）からAさん、Bさんが
この順にカードを引くとき、2人とも絵札を引く確率を求めなさい。
ただし、引いたカードは戻さないとする。\kotae{\frac{11}{221}}
\vfill
\toi 袋の中に赤玉6個と白玉3個が入っている。1個ずつ玉を取り出すとき
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{\subtoi 2個目が赤\kotae{\frac{2}{3}} }
\vfill
\toi 8本のくじの中に2本のあたりが入っている。Aさん、Bさんが
この順にくじを引くとき次の確率を求めなさい。
ただし、引いたくじは戻さないとする。
\nidangumi{\subtoi Aさんが当たりを引く\kotae{\frac{1}{4}} }
{\subtoi Bさんが当たりを引く\kotae{\frac{1}{4}} }
\vfill
\ifnum\make_pdf_with_insatsu>0
\hfill\rotatebox{0}{{\normalsize\gtfamily\bfseries 数学プリント\#22その2}\scriptsize
{\normalsize\ajKaku{1}}~\( \frac{11}{221} \)\
{\normalsize\ajKaku{2}}~
\ajKakko{1}\( \frac{5}{12} \)\
\ajKakko{2}\( \frac{2}{3} \)\
{\normalsize\ajKaku{3}}~
\ajKakko{1}\( \frac{1}{4} \)\
\ajKakko{2}\( \frac{1}{4} \)\
{\normalsize\ajKaku{4}}~
\ajKakko{1}\( \frac{1}{6} \)\
\ajKakko{2}\( \frac{1}{3} \)\
{\normalsize\ajKaku{5}}~
\ajKakko{1}\( \frac{2}{5} \)\
\ajKakko{2}\( \frac{4}{9} \)\
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\toi 1個のさいころを2回投げる。出た2つの目の和が奇数であるを$A\,,$
大きいほうの目が奇数を$B$とするとき、次の確率を求めなさい。
\nidangumi{\subtoi \( P(A\cap B) \)\kotae{\frac{1}{6}} }
{\subtoi \( P_A (B) \)\kotae{\frac{1}{3}} }
\vfill
\toi 箱の中に
\quad\ajKakkoroman{1}\ 1と書かれた赤玉が2個\
\quad\ajKakkoroman{2}\ 2と書かれた赤玉が3個\
\quad\ajKakkoroman{3}\ 1と書かれた白玉が4個\
\quad\ajKakkoroman{4}\ 2と書かれた白玉が5個入っている。
玉を1個取り出すとき、赤玉である事象を$A\,,$\quad 1が
書かれている事象を$B$とする。
このとき次の確率を求めなさい。
\nidangumi{\subtoi \( P_A (B) \)\kotae{\frac{2}{5}} }
{\subtoi \( P_{\ \overline{A}\>} (B) \)\kotae{\frac{4}{9}} }
\vfill
\end{document}
■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■■
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\startcomponent ma-cb-vn-enumerations
\project ma-cb
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%\chapter{Numbered definitions}
\chapter{Đánh số các diễn giải}
%\index{numbered definition}
\index{lời bình}
\Command{\tex{defineenumeration}}
\Command{\tex{setupenumerations}}
%With \type{\defineenumeration} you can number text elements
%like remarks or questions. If you want to make numbered
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Với lệnh \type{defineenumeration}, bạn có thể đánh số các phần văn bản giống
như các lời phê hay các câu hỏi. Nếu bạn muốn đánh số các lời phê trong tài
liệu, bạn dùng:
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%For example:
Ví dụ:
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% inbetween=\blank,
% after=\blank]
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inbetween=\blank,
after=\blank]
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%\type{\resetremark} and \type{\nextremark} are available and
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Với các lệnh \type{\remark}, \type{\subremark}, \type{resetremark} và
\type{\nextremark}, bạn có thể tạo các lời phê như thế này:
% \startbuffer[b]
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%
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% \stopbuffer
\startbuffer[b]
\remark Vào đầu thời kì trung cổ, Hasselt là nơi hành hương. {\em Heilige Stede}
(Thánh Địa) bị đóng cửa trong suốt thời kì Cải cách. Vào năm 1930, sau 300 năm
{\em Heilige Stede} được mở lại.
\subremark Ngày nay, {\em Heilige Stede} bị đóng cửa lần nữa nhưng mỗi năm một
lần, một triễn lãm hàng không được tổ chức tại cùng nơi đó. \par
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Bạn có thể đặt lại cách đánh số với lệnh \type{\resetremark} hay
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%You can set up the layout of \type{\defineenumeration} with:
Bạn có thể thiết đặt khung nền với lệnh:
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%You can also vary the layout of {\bf Remark} and {\bf
%Subremark} in the example above by:
Bạn cũng có thể biến đổi khung nền của {\bf Lời phê} trong ví dụ trên với:
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%If a number becomes obsolete you can type:
Nếu không dùng số, bạn có thể dùng
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\remark[-]
\stoptyping
%If the remark contains more than one paragraph you will
%have to use the command pair
%\type{\startremark} $\cdots$ \type{\stopremark} that becomes
%available after defining {\bf Remark} with
%\type{\defineenumeration[remark]}.
Nếu lời phê chứa nhiều hơn một đoạn văn, bạn sẽ phải dùng cặp lệnh
\type{startremark} $\cdots$ \type{stopremark} sau khi đã định nghĩa với
\type{\defineenumeration[remark]}.
%So the example above would look like this:
Vì vậy, ví dụ ở trên sẽ như thế này:
% \startbuffer[c]
% \startremark
% In the early medieval times Hasselt was a place of pilgrimage. The
% {\em Heilige Stede} (Holy Place) was torn down during the
% Reformation.
% After 300 years in 1930 the {\em Heilige Stede} was reopened.
% Nowadays the {\em Heilige Stede} is closed again but once a year an
% open air service is held on the same spot.
% \stopremark
% \stopbuffer
\startbuffer[c]
\startremark
Vào đầu thời kì trung cổ, Hasselt là nơi hành hương. {\em Heilige Stede}
(Thánh Địa) bị đóng cửa trong suốt thời kì Cải cách.
Vào năm 1930, sau 300 năm {\em Heilige Stede} được mở lại. Ngày nay,
{\em Heilige Stede} bị đóng cửa lần nữa nhưng mỗi năm một lần, một triễn lãm
hàng không được tổ chức tại cùng nơi đó.
\stopremark
\stopbuffer
\typebuffer[c]
\start
\getbuffer[a]\getbuffer[c] \par
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http://dlmf.nist.gov/10.67.E1.tex
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nist.gov
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\[\mathop{\mathrm{ker}_{{\nu}}\/}\nolimits x\sim e^{{-x/\sqrt{2}}}\left(\frac{%
\pi}{2x}\right)^{{\frac{1}{2}}}\*\sum_{{k=0}}^{\infty}\frac{a_{k}(\nu)}{x^{k}}%
\mathop{\cos\/}\nolimits\!\left(\frac{x}{\sqrt{2}}+\left(\frac{\nu}{2}+\frac{k%
}{4}+\frac{1}{8}\right)\pi\right),\]
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https://dlmf.nist.gov/29.10.E2.tex
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nist.gov
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CC-MAIN-2021-43
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\[z^{\prime}=\mathrm{i}(z-K-\mathrm{i}{K^{\prime}}),\]
|
http://edshare.soton.ac.uk/2274/10/MA17xexam00A18.tex
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soton.ac.uk
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CC-MAIN-2017-43
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\documentclass[a4paper,12pt]{article}
\newcommand\ds{\displaystyle}
\begin{document}
\parindent=0pt
QUESTION
Using de Moivre's theorem, or otherwise, calculate
$\ds\left(\cos\left(\frac{\pi}{4}\right)+j\sin\left(\frac{\pi}{4}\right)\right)^4$.
ANSWER
$\ds\left(\cos\left(\frac{\pi}{4}\right)+j\sin\left(\frac{\pi}{4}\right)\right)^4
=\cos(\pi)+j\sin(\pi)=-1+j(0)=-1$
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The Project Gutenberg EBook of The Romance of Lust, by Anonymous
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.net
Title: The Romance of Lust
A classic Victorian erotic novel
Author: Anonymous
Release Date: October 14, 2009 [EBook \#30254]
[Last updated: February 18, 2012]
Language: English
*** START OF THIS PROJECT GUTENBERG EBOOK THE ROMANCE OF LUST ***
Produced by Daniel Fromont
The Romance of Lust (1873)
A classic Victorian erotic novel
by Anonymous
1892 edition
VOLUME I.
This text was converted to LaTeX by means of \textbf{GutenMark} software (version Jul 12 2014).
The text has been further processed by software
in the iTeX project, by Bill Cheswick.
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\tableofcontents
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\pagenumbering{arabic}
\gutchapter{CONTENTS}
The Novice---Mrs. Benson---Mary---Mrs. Benson's Correspondence with Mrs. Egerton---Miss Evelyn---Eliza
There were three of us---Mary, Eliza, and myself. I was approaching fifteen, Mary was about a year younger, and Eliza between twelve and thirteen years of age. Mamma treated us all as children, and was blind to the fact that I was no longer what I had been. Although not tall for my age, nor outwardly presenting a manly appearance, my passions were awakening, and the distinctive feature of my sex, although in repose it looked magnificent enough, was very sufficiently developed when under the influence of feminine excitement.
As yet, I had absolutely no knowledge of the uses of the different organs of sex. My sisters and I all slept in the same room. They together in one bed, I alone in another. When no one was present, we had often mutually examined the different formations of our sexes.
We had discovered that mutual handlings gave a certain amount of pleasing sensation; and, latterly, my eldest sister had discovered that the hooding and unhooding of my doodle, as she called it, instantly caused it to swell up and stiffen as hard as a piece of wood. My feeling of her little pinky slit gave rise in her to nice sensations, but on the slightest attempt to insert even my finger, the pain was too great. We had made so little progress in the \textit{attouchements} that not the slightest inkling of what could be done in that way dawned upon us. I had begun to develope a slight growth of moss-like curls round the root of my cock; and then, to our surprise, Mary began to show a similar tendency. As yet, Eliza was as bald as her hand, but both were prettily formed, with wonderfully full and fat mounts of Venus. We were perfectly innocent of guile and quite habituated to let each other look at all our naked bodies without the slightest hesitation; and when playing in the garden, if one wanted to relieve the pressure on the bladder, we all squatted down together, and crossed waters, each trying who could piddle fastest. Notwithstanding these symptoms of passion when excited, in a state of calm I might have passed for a boy of ten or eleven.
My father had left us but moderately provided for, and mamma, wishing to live comfortably, preferred giving me lessons along with my sisters at home to sending me to school; but her health beginning to fail, she inserted an advertisement in the \textit{Times} for a governess. Out of a large number of applicants, a young lady, of the name of Evelyn, was selected. Some ten days afterwards she arrived, and became one of the family.
We did not see much of her the first evening, but after breakfast the following morning, mamma accompanied her to what was considered our school-room, and said, ``Now, my dears, I place you under Miss Evelyn's care; you must obey her in all things; she will teach you your lessons, as I am unable to do so any longer.'' Then, turning to our new governess, ``I fear you will find them somewhat spoiled, and unruly; but there is a horse, and Susan will make you excellent birch rods whenever you require them. If you spare their bottoms when they deserve whipping, you will seriously offend me.'' As mamma said this, I observed Miss Evelyn's eyes appeared to dilate with a sort of joy, and I felt certain that, severely as mamma had often whipped us, if we should now deserve it, Miss Evelyn would administer it much more severely. She looked amiability itself, and was truly beautiful in face and person, twenty-two years of age, full and finely formed, and dressed always with the most studied neatness. She was, in truth, a seductive creature. She made an instantaneous impression on my senses. There was, however, somewhat of a sternness of expression, and a dignity of carriage, which caused at once to fear and respect her. Of course, at first, all went smoothly enough, and seeing that mamma treated me precisely as she did my sisters, I came to be regarded as quite a child by Miss Evelyn. She found that she had to sleep in the same room with my sisters and myself. I fancied that on the first night Miss Evelyn did not approve of this arrangement, but gradually became familiarized with it, and seemed to think no more about it.
When bedtime came, we all kissed mamma and retired early, as usual. Miss Evelyn followed some hours later. When she came in, she carefully locked the door, then looked at me to see if I was asleep. Why, I know not, but I was instinctively prompted to feign sleep. I did so successfully, notwithstanding the passing of the candle before my eyes. So she at once commenced undressing. When her back was turned, I opened my eyes, and greedily devoured her naked charms as they were gradually exhibited before me. The moment she turned round, I was again as if asleep. I have said that my passions had begun to develope themselves, but as yet I did not understand their force or direction. I well remember this first night, when a fine ripe woman gradually removed every particle of dress within a couple of yards of me---the effect of each succeeding charm, from her lovely and beautifully formed bubbies to the taking off her shoes and stockings from her well-formed legs and small feet and ankles, caused my prick to swell and stiffen to a painful extent. When all but her chemise was removed, she stopped to pick up her petticoats that she had allowed to fall to her feet, and in lifting them, raised also her chemise, and exposed to my view a most glorious bottom---dazzlingly white and shining like satin. As the light was full upon it, and she was still in a stooping position, I could see that below her slit she was well covered with dark hair. Turning round, to put her petticoats on a chair, and to take up her night-gown, she slipped her chemise from her arm, and letting it fall to the ground while she lifted the night-gown over her head, I had for some seconds a view of her beautiful belly, thickly covered with dark curly hair over the mount of Venus. So voluptuous was the sight, I almost shuddered, so intense was my excitement. She now sat down on the bed to take off her shoes and stockings. Oh! what beautiful thighs, legs, ankles, and feet she had!
I am now advanced in life, and have had many handsome and well-formed women, but I never saw limbs more voluptuously formed.
In a few minutes the light was extinguished, and a rushing rill flowed into the night vase; very different from the gentle tricklings from myself and sisters as we often squatted down opposite each other and crossed water, laughing at the different sources from which they flowed. My sisters often envied me the power of directing the spurt where I pleased, so little were we from dreaming of the real intent of that projecting little instrument.
I heard the charming creature get into bed, and shortly breath hard. As for me, I could not sleep. I lay awake the greater part of the night, afraid to be restless, lest I should disturb Miss Evelyn and give her reason to think I had been observant of her undressing. When at last I dozed off, it was but to dream of all the charms I has seen.
About a month passed thus. Every night Miss Evelyn became more and more at her ease, and confident of my mere childishness, often gave me glorious and lengthened glimpses of her beautifully developed charms: although it was only about every other night that I could enjoy them, for, as they always produced sleeplessness afterwards, the following night nature assured her rights, and I usually slept profoundly when I would have preferred continued gazing on the charms of my lovely governess. But, doubtless, those exhausting sleeps helped to throw her off her guard, and gave me better opportunities than I should otherwise have had. Once or twice she used the night ware before putting on her night-gown, and I could see the rosy-lipped opening embosomed in exquisite dark curls, pouring out its full measure of water; showing a fine force of nature, and driving me wild with excitement. Yet it is singular that I never once thought of applying to my fingers for relief from the painful stiffness that nearly burst my prick asunder.
Whether mamma had observed my very frequent projection of my trousers, or began to think it better I should not sleep in the same room as Miss Evelyn, I cannot say, but she had my bed removed into her own. However, I was so thoroughly treated as a mere boy by every one in the house, that Miss Evelyn seemed to forget my sex; and there was at all times a freedom of carriage and an \textit{abandon} in her attitudes that she certainly would not have indulged in if she had felt any restraint from considering herself in the presence of a youth of the age of puberty.
In cold weather I used to sit on a low stool by the fire---Miss Evelyn was seated in front, I had my lesson book on my knee, and she herself would place her beautiful feet on the high school fender, with her work in her lap, while she heard my sisters repeat their lesson, totally unconscious that for half an hour at a time she was exposing her beautiful legs and thighs to my ardent gaze; for sitting much below her, and bending my head as if intent on my lesson, my eyes were below her raised petticoats. Her close and tight-fitting white stockings displayed her well-formed legs, for while confined to the house during our morning lessons she did not wear drawers; so that in the position she sat in, with her knees higher than her feet on the already high fender, and her legs somewhat apart to hold her work in her lap more easily, the whole glorious underswell of both thighs, and the lower part of her fine large bottom, with the pinky slit quite visible, nestled in a rich profusion of dark curls, were fully exposed to my view. The light from the fire glancing under her raised petticoats tinged the whole with a glow, and set me equally in a blaze of desire until I was almost ready to faint. I could have rushed headlong under her petticoats, and kissed and fondled that delicious opening and all its surroundings. Oh, how little she thought of the passion she was raising. Oh! dear Miss Evelyn, how I did love you from the dainty kid slipper and tight glossy silk stocking, up to the glorious swell of the beautiful bubbies, that were so fully exposed to me nearly every night, and the lovely lips of all that I longed to lovingly embrace.
Thus day after day passed away, and Miss Evelyn became to me a goddess, a creature whom, in my heart of hearts, I literally worshiped. When she left the school-room, and I was alone, I kissed that part of the fender her feet had pressed, and the seat on which she sat, and even the air an inch above, imagination placing there her lovely cunt. I craved for something beyond this without knowing exactly what I wanted; for, as yet, I really was utterly ignorant of anything appertaining to the conjunction of the sexes.
One day I had gone up to my sisters' bedroom where the governess slept, that I might throw myself on her bed, and in imagination embrace her beautiful body. I heard some one approaching, and knowing that I had no business there, I hid myself under the bed. The next moment Miss Evelyn herself entered, and locked the door. It was about an hour before dinner. Taking off her dress, and hanging it on the wardrobe, she drew out a piece of furniture, which had been bought for her, the use of which had often puzzled me; she took off the lid, poured water into its basin, and placed a sponge near it. She then took off her gown, drew her petticoats and chemise up to her waist and fastened them there, straddled across it, and seated herself upon it.
I thus had the intoxicating delight of gazing on all her beautiful charms, for when she tucked up her clothes she stood before her glass, presenting to my devouring glance her glorious white bottom in all its fullness, turning to approach the bidet, she equally exposed her lower belly and beautiful mount, with all its wealth of hair. While straddling over the bidet before she sat down, the whole of her pinky-lipped cunt broke on my enraptured sight. Never shall I forget the wild excitement of the moment. It was almost too much for my excited senses; fortunately, when seated, the immediate cause of my almost madness vanished. She sponged herself well between the thighs for about five minutes. She then raised herself off the bidet, and for a moment again displayed the pouting lips of her cunt---then stood fronting me for two or three minutes while she removed, with the rinsed sponge, the trickling drops of water which still gathered on the rich bush of curls around her quim. Thus her belly, mount and thighs, whose massy-fleshed and most voluptuous shape were more fully seen by me than they had heretofore been, and it may easily be conceived into what a state such a deliberate view threw me.
Oh, Miss Evelyn, dear, delicious Miss Evelyn! what would you have thought had you known that I was gazing on all your angelic charms, and that my eager eyes had been straining themselves to penetrate the richness of those charming pouting lips which lay so snugly in that rich mass of dark curling hair. Oh! how I do long to kiss them; for at that time I had no other idea of embracing and still less of penetrating them.
When her ablutions were completed, she sat down and drew off her stockings, displaying her beautiful white calves and charming little feet. I believe it was this first admiration of really exquisitely formed legs, ankles and feet, which were extraordinarily perfect in make, that first awakened my passion for those objects, which have since always exercised a peculiar charm over me. She was also so particularly neat in her shoes---little dark ones---that were \textit{bijoux} to look at, I often took them up and kissed them, when left in the room. Then her silk stockings, always drawn up tight and fitting like a glove, set off to the greatest advantage the remarkable fine shape of her legs.
Putting on silk for cotton stockings, she took down a low-bodiced dress, finished her toilet, and left the room. I crawled out from under the bed, washed my face and hands in the water of the bidet, and even drank some in my excitement.
Some six weeks had now elapsed since the arrival of Miss Evelyn. The passion that had seized me for her had so far kept me most obedient to her slightest command, or even wish, and, from the same cause, attentive to my lessons, when not distracted by the circumstances already detailed. My example had also had the effect of keeping my sisters much in the same groove, but it was impossible this could last ---it was not nature. As long as all went smoothly, Miss Evelyn seemed to be all amiability. We fancied we could do as we liked, and we grew more careless.
Miss Evelyn became more reserved, and cautioned us at first, and then threatened us with the rod. We did not think she would make use of it. Mary grew impertinent, and one afternoon turned sulky over her lessons, and set our teacher at defiance. Miss Evelyn, who had been growing more and more angry, had her rise from her seat. She obeyed with an impudent leer. Seizing her by the arm, Miss Evelyn dragged the struggling girl to the horse. My sister was strong and fought hard, using both teeth and nails, but it was to no purpose. The anger of our governess was fully roused, and raising her in her arms, she carried her forcibly to the horse, placed her on it, held her firmly with one hand while she put the noose round her with the other, which, when drawn, secured her body; other nooses secured each ankle to rings in the floor, keeping her legs apart by the projection of the horse, and also forcing the knees to bend a little, by which the most complete exposure of the bottom, and, in fact, of all her private parts too, was obtained.
Miss Evelyn then left her, and went to mamma for a rod. In a few minutes she returned, evidently flushed with passion, and proceded to tie Mary's petticoats well up to her waist, leaving her bottom and her pinky slit quite bare and exposed directly before my eyes. It was quite two months since I had seen her private parts, and I was well surprised to observe the lips more pouting and swelled out, as well as the symptoms of a mossy covering of the mount much more developed. Indeed, it was in itself more exciting than I had expected, for my thoughts had so long dwelt only on the riper beauties of Miss Evelyn that I had quite ceased to have any toying with Mary.
This full view of all her private parts reawakened former sensations and strengthened them. Miss Evelyn first removed her own scarf, laying bare her plump ivory shoulders, and showing the upper halves of her beautiful bubbies, which were heaving with the excitement of her anger. She bared her fine right arm, and grasping the rod, stepped back and raised her arm; her eyes glistened in a peculiar way. She was indeed beautiful to see.
I shall never forget that moment---it was but a moment. The rod whistled through the air and fell with a cruel cut on poor Mary's plump little bottom. The flesh quivered again, and Mary, who had resolved not to cry, flushed in her face, and bit the damask with which the horse was covered.
Again the arm was raised, and again, with a sharp whistle, it fell on the palpating buttocks below it. Still her stubborn temper bore her up, and although we saw how she winced, not a sound escaped her lips. Drawing back a step, Miss Evelyn again raised her hand and arm, and this time her aim was so true that the longer points of the rod doubled between the buttocks and concentrated themselves between the lips of Mary's privates. So agonising was the pain that she screamed out dreadfully. Again the rod fell precisely on the same spot.
``Oh! oh! oh! Dear Miss Evelyn. I will never, no, never, do so again.''
Her shrieks were of no avail. Cut succeeded cut, yell succeeded yell ---until the rod was worn to a stump, and poor Mary's bottom was one mass of weals and red as raw beef. It was fearful to see, and yet such is our nature that to see it was, at the same time, exciting. I could not keep my eyes from her pouting quim, the swelling lips of which, under the severity of the punishment it was undergoing, not only seemed to thicken, but actually opened and shut, and evidently throbbed with agony. But all this was highly exciting for me to witness. I then and there resolved to have a closer inspection at a more convenient opportunity, which did not fail me in the end.
Meanwhile, her spirit was completely cowed, or rather, crushed. Indeed, we were all fully frightened, and now knew what we had to expect, if we did not behave ourselves. There was now no fear of any manifestation of temper, and we felt we must indeed obey implicitly whatever our governess chose to order. We instinctively learned to fear her.
A very few days after this memorable whipping, some visitors arrived ---a gentleman and lady. The gentleman was an old friend of mamma's, who had lately married, and mamma had asked them to visit her on their wedding tour and spent a short time with us.
The gentleman was a fine-looking man, tall and powerfully built; the lady rather delicate looking, but well shaped, with good breasts and shoulders, small waist, and spreading haunches, well-formed arms, small hands and feet, and very brilliant eyes.
I think it was about three days after their arrival that one afternoon I went into the spare room, which was occupied by these visitors; while there, I heard them coming upstairs. The lady entered first, and I had just time to slip into a closet and draw the door to; it was not quite closed, but nearly so. In a minute the gentleman followed, and gently shutting the door, locked it. Mrs. Benson smiled, and said---
``Well, my love, you are a sad teaser; you let me have no rest. Surely, you had enough last night and this morning without wanting it again so soon?''
``Indeed, I had not,'' he said, ``I never can have enough of your delicious person. So come, we must not be long about it, or our absence will be observed.''
He seized her round the waist, and drew her lips to his, and gave her a long, long kiss; squeezing her to him, and moving himself against her. Then seating himself, he pulled her on his knee, and thrust his hand up her petticoats, their mouths being glued together for some time.
``We must be quick, dear,'' she murmured.
He got up, and lifted her on the edge of the bed, threw her back, and taking her legs under his arms, exposed everything to my view. She had not so much hair on her mount of Venus as Miss Evelyn, but her slit showed more pouting lips, and appeared more open. Judge of my excitement when I saw Mr. Benson unbutton his trousers and pull out an immense cock. Oh, dear, how large it looked; it almost frightened me. With his fingers he placed the head between the lips of Mrs. Benson's sheath, and then letting go his hold, and placing both arms so as to support her legs, he pushed it all right into her to the hilt at once. I was thunderstruck that Mrs. Benson did not shriek with agony, it did seem such a large thing to thrust right into her belly. However, far from screaming with pain, she appeared to enjoy it. Her eyes glistened, her face flushed, and she smiled most graciously on Mr. B. The two appeared very happy. His large cock slipped in and out quite smoothly, and his hands pressed the large glossy buttocks and pulled them to him at each home thrust. This lasted nearly five minutes, when all at once Mr. B. stopped short, and then followed one or two convulsive shoves---he grinning in a very absurd way at her. He remained quiet for a few minutes, and the drew out his cock, all soft, with slimy drops falling from it onto the carpet. Taking a towel, he wiped up the carpet, and wrapping it round his cock, went to the basin and washed it.
Mrs. Benson lay for a few minutes longer all exposed, her quim more open than before, and I could see a white slime oozing from it.
You can hardly imagine the wild excitement this scene occasioned me. First, the grand mystery was at once explained to me, and my ignorant longings now knew to what they tended. After giving me plenty of time to realise all the beauties of her private parts, she slipped down on the floor, adjusted her petticoats, and smoothed the disordered counterpane, and then went to the glass to arrange her hair. This done, she quietly unlocked the door, and Mr. Benson went out. The door was then relocked, and Mrs. B. went to the basin, emptied and filled it, then raised up her petticoats, and bathed the parts between her legs with a sponge, and then rubbed all dry with a towel; all this time exposing everything to my ardent gaze. But, horror of horrors! she after this came straight to the closet and gave a slight scream on discovering me there. I blushed up to the ears, and tried to stammer out an excuse. She stared at me at first in silent amasement; but at last said---
``How came you here, sir, tell me?''
``I was here when you came up; I wanted my football, which was in this closet, and when I heard you coming, I hid myself, I don't know why.''
For some minutes she seemed to consider and examine me attentively. She then said---
``Can you be discreet?''
``Oh, yes, ma'am.''
``You will never tell any one what you have seen?''
``No ma'am.''
``Well, keep this promise, and I shall try what I can do to reward you. Now, go downstairs.''
I went to the school-room, but I was greatly agitated, I scarcely knew what I was doing. The scene I had witnessed had complete possession of my thoughts. In years but a boy, the mystery now practically explained to me had awakened all the passions of a man. Instead of studying my lessons, my thoughts wandered to Mrs. B., thrown back on the bed with her fine legs and thighs fully exposed; above all, the sight of the pinky gash, with its fleecy hair at the bottom of her belly, which I had seen for some minutes all open and oozing out the slimy juice that followed the amorous encounter they had been indulging in. It seemed so much more developed than Miss Evelyn's. I felt sure that Miss Evelyn could never take in such a thick long thing as Mr. B. had thrust into his wife, and yet it appeared to go in so easily, and moved about so smoothly, and so evidently to the satisfaction and utmost delight of both, as was proved by their ardent embracings, fond murmurs, and voluptuous movements, especially just before they both ceased together all movement whatever.
Then I thought, how delicious it would be to treat Miss Evelyn in the same way, and to revel with my stiff-standing prick in her delicious quim, which in my mind's eye I saw before me as I had viewed in on her rising from the bidet, when I lay hid under the bed. Then I thought of my sister Mary's smaller, although attractive little quim, and I resolved, as that was the easiest to get hold of, to initiate her in all the newly discovered mysteries. I fully determined that my own first lesson, as well as hers, should be taken on her little fat chubby cunt. Then the recollection of its pouting and throbbing lips under the fearful flagellation she had undergone, began to excite me, and made my cock stand stiff and throb again. All the weeks of excitement I had now constantly been under had produced a wonderful effect on my pego, which had become considerably more developed when in a state of erection. As you may suppose, with such distracting thoughts, I did not get on with my lessons. Miss Evelyn, for some reason or other, was out of humour that morning, and more than once spoke crossly to me for my evident inattention. At length she called me to her, and finding that I had scarcely done anything, she said---
``Now, Charles, I give you ten minutes longer to finish that sum, if not done in that time I shall whip you; you are exhibiting the mere spirit of idleness. I do not know what has come over you, but if persisted in, you shall certainly be punished.''
The idea of the beautiful Miss Evelyn whipping my bare bottom did not tend to calm my excitement, on the contrary, it turned my lewd thoughts upon the beauties of her person, which I had so often furtively gazed upon.
It was close upon four o'clock, at which hour we always broke up for a run in the garden for an hour, and during this period I had resolved to begin instructing Mary in the secret mysteries I had so lately been a witness to. But fate had ordered it otherwise, and I was to receive my first practical lesson and be initiated on the person of a riper and more beautiful woman; but of this hereafter. At four o'clock I had done nothing with my task---Miss Evelyn looked grave:
``Mary and Eliza, you may go out, Charles will remain here.''
My sisters, simply imagining that I was kept to finish my lessons, ran into the garden. Miss Evelyn turned the key in the door, opened a cupboard, and withdrew a birch rod neatly tied up with blue ribbons. Now my blood coursed through my veins, and my fingers trembled so that I could hardly hold my pencil.
``Put down your slate, Charles, and come to me.''
I obeyed, and stood before my beautiful governess, with a strange commixture of fear and desire.
``Unfasten your braces, and pull down your trousers.''
I commenced doing this, though but very slowly. Angry at my delay her delicate fingers speedily accomplished the work. My trousers fell to my feet.
``Place yourself across my knees.''
Tremblingly, with the same commixture of feeling, I obeyed. Her silk dress was drawn up to prevent its being creased---my naked flesh pressed against her snowy white petticoats. A delicate perfume of violet and vervain assailed my nerves. As I felt her soft and delicate fingers drawing up my shirt, and passing over my bare posteriors, while the warmth of her pulpy form beneath me penetrated my flesh, nature exerted her power, and my prick began to swell out to a most painful extent. I had but little time, however, to notice this before a rapid succession of the most cruel cuts lacerated my bottom.
``Oh, dear! Oh, dear! Oh, dear! Oh, Miss Evelyn. I will do the sum if you will only forgive me. Oh, oh, oh, \&c.''
Holding me firmly with her left arm, Miss Evelyn used the rod most unmercifully. At first, the pain was excruciating, and I roared out as loud as I could, but gradually the pain ceased to be so acute, and was succeeded by the most delicious tickling sensation. My struggles at first had been so violent as to greatly disorder Miss Evelyn's petticoats, and to raise them up so as to expose to my delighted eyes her beautifully formed silk-clad legs up to the knees, and even an inch or two of naked thigh above.
This, together with the intense tickling irritation communicated to my bottom, as well as to the friction of my cock against the person of Miss Evelyn in my struggles, rendered me almost delirious, and I tossed and pushed myself about on her knees in a state of perfect frenzy as the blows continued to be showered down upon my poor bottom. At last the rod was worn to a stump, and I was pushed off her knees. As I rose before her, with my cheeks streaming with tears, my shirt was jutting out considerably in front in an unmistakeable and most prominent manner, and my prick was at the same time throbbing beneath it with convulsive jerks, which I could by no means restrain.
Miss Evelyn glared at the projection in marked astonishment, and her open eyes were fixed upon it as I stood rubbing my bottom and crying, without attempting to move or button up my trousers. She continued for a minute or two to stare at the object of attraction, flushing scarlet up to the forehead, and then she suddenly seemed to recollect herself, drew a heavy breath, and rapidly left room. She did not return until after my sisters came back from the garden, and seemed still confused, and avoided fixing her eye upon me.
In two days afterwards, all disagreeable marks of this very severe whipping had disappeared. On the following day we were invited to pass the afternoon at the grange, a beautiful place about two miles from us. The afternoon was fine and warm; we walked there, and arrived about four o'clock. Mr. and Mrs. Robinson were in the drawing-room, but at once desired us to go in the garden and amuse ourselves with their three daughters, whom we would find there. We went at once, and found them amusing themselves on a swing. Sophia, the eldest, about nineteen, was swinging a sister about two years younger, a very fine, fully developed young woman. Indeed, all three sisters were finer women and more beautiful than the average of young ladies.
Another sister, Agnes, was not seated, but standing on the board between the ropes. Sophia was making both mount as high as possible. They were laughing loudly, when we found them, at the exposure each made---one in advancing, the other retiring. Agnes's light dress of muslin and single petticoat, as she retired and the wind came up from behind, was bulged out in front, and exposed her limbs up to her belly, so that one could see that her mount was already well furnished. The other, in advancing, threw her legs up, and exposed all the underside of her thighs and a part of her bottom, and you could just discern that there was dark hair between the lower thighs and bottom.
As they considered me but a child, I was no check to their mirth and sport. On the contrary, the gave me a long rope to pull down the swing when at its highest, and I sat down on the grass in front for greater convenience. The fine limbs and hairy quims exposed freely before me from moment to moment excited my passions. None of them wore more than one petticoat, and they had no drawers, so that when they mounted to the highest point from me, I had the fullest possible view of all. My cock soon rose to a painful extent, which I really believe was noticed and enjoyed by them, I observed, too, that I was an object of attention to Miss Evelyn, who shortly seated herself in the swing, and allowed me to swing her with the end of the rope. I even fancied that she threw up her legs more than was at all necessary; at all events, she naturally, with the strong feelings I had towards her, excited me more than all the rest.
We were as merry as could be, and we passed a delightful evening until eight o'clock, when it began to rain. As it continued, and became very heavy, Mr. Robinson ordered out the closed carriage to take us home. It was a brougham, only seated for two. Mary took Eliza on her knee, Miss Evelyn took me upon hers. I know not how it happened, but her lovely arm soom passed round my body as if to hold me on her knee, and her hand fell, apparently by accident, exactly on my cock---the touch was electric. In an instant, my member stood stiff and strong beneath her hand. Still Miss Evelyn, who must have felt the movement going on beneath her fingers, did not remove her hand, but rather seemed to press more upon it. In my boyish ignorance, I imagined she was not aware of what was happening. The motion and jolting of the carriage over rough road caused her hand to rub up and down upon my erected and throbbing member. I was almost beside myself, and to conceal my condition I feigned sleep. I let my head fall on Miss Evelyn's shoulder and neck---she allowed this.
Whether she thought I had really fallen asleep I know not, but I was quite sensible that her fingers pressed my swollen and throbbing cock, and I fancied she was measuring its size.
The tight grasp she managed to gain, and the continued jolting of the carriage, brought me up at last to such a pitch state that a greater jolt than usual, repeated two or three times in succession, each followed by a firmer pressure of her charming fingers, caused me such an excess of excitement that I actually swooned away with the most delicious sensation I had ever experienced in my life. I was some time before I knew where I was, or what I was about, and was only made conscious of our arrival at home by Miss Evelyn shaking me to rouse me up. I stumbled up, but though partially stupefied, I fancied Miss Evelyn's eyes shone with a brilliancy I had never before observed, and that there was a bright hectic flush on her cheek. She refused to go into the parlour, but hurried to bed on pretence of a headache.
When I retired to bed, and took off my shirt, I found it all sticky and wet in front.
It was thus I paid down my first tribute to Venus. I thought long over this evident approach to familiarity on the part of Miss Evelyn, and went to sleep with a lively hope of a more private interview with her, when I trusted that her evident passion would initiate me in the pleasures to be derived from her beauteous body.
But again fate intervened, and another, not less beautiful, more experienced, and more inclined for the sport, was to be my charming mistress in love's revels.
Two days after this, Mr. Benson was unexpectedly called away on pressing affairs, which he feared might detain him three weeks. He left Mrs. B. with us. As he had to be driven about nine miles to the town where the coach passed, mamma took the opportunity of going to the town with him. Mrs. B. complained of not being equal to the fatigue, and mamma told Miss Evelyn she would like her company, and as the two girls wanted new shoes, they could go also; I was to remain at home, and mamma desired me to be quiet and attentive to Mrs. Benson, who, observing no one, said to me, with a peculiar look:
``I shall want you to hold my skeins, Charlie, so don't go out of the way, but be ready for me as soon as they are gone.''
She then went up to her bedroom, where Mr. B. immediately joined her, no doubt to re-enact the scene I had already witnessed from the closet on a previous day. They were fully half an hour occupied together. At length, all was ready, and off they went, leaving me to a fate I had little dreamt of.
Mrs. B. proposed we should go up to the drawing-room, which looked out to the garden, and was nowhere overlooked. I followed her, and could not help admiring her fine figure as she preceded me in going upstairs. Although pale in complexion, she was well made, and very elegant in her carriage, and sat down on a low easy chair, throwing herself completely back, and crossing one leg over the other, apparently without being aware that she carried her petticoats up with the action, and exhibited the beautiful underleg up to the garter.
I had never forgotten the day, when secreted in the closet, I had seen them completely exposed, and how charming they were. Her present negligent attitude, although far from the same exposure I speak of, was still, with the former recollection running in my head, enough to set my whole blood on fire. I have before remarked what a power beautiful and well-stockinged legs, and ankles and small feet, had upon my nervous system, and so it was now. As I gazed upon her handsome legs, ankles, and feet, I felt my prick swell and throb in a manner that could not fail to be perceptible to Mrs. B, especially as her head lay on a level with that part of my person as I stood before her.
Although she continued knitting, I could see that her eyes were directed to that part of my person, and fixed upon the increasing distention of my trousers. In a few minutes she gave me a skein of worsted to hold, and desired me to kneel in front of her, so as to bring my hands down to the level of the low chair on which she was seated.
I knelt close to the footstool on which her foot rested; it was raised up, and a very slight movement brought it against my person, at first rather below where my throbbing prick was distending my trousers. As she commenced to wind her ball, she gradually pushed her foot further forward, until the toe actually touched the knob of my cock, and occasionally moved it right and left, exciting me beyond measure.
I flushed up to the very ears, and trembled so violently that I thought I should have dropped the skein.
``My dear boy, what is the matter with you, that you blush and tremble so, are you unwell?''
I could not answer, blushed more than ever. The skein at length was finished.
``Charles,'' she said, ``get up, and come here.''
I rose and stood by her side.
``What have you got in your trousers that is moving?''
And here her busy fingers commenced unbuttoning them. Released from confinement, out started my prick---stiff as iron, and as large as that of a youth of eighteen. Indeed, I was better hung than one boy selected out of five hundred of that age. Mrs. B., who had pretended to be perfectly astonished, exclaimed---
``Good gracious, what a pego! Why Charles, my darling, you are a man not a boy. What a size to be sure!'' and she gently handled it. ``Is it often in this state?''
``Yes, ma'am.''
``For how long?''
``Ever since Miss Evelyn came.''
``And pray, sir, what has Miss Evelyn's coming had to do with it?''
``I---I---I---I---''
``Come now, Charles, be candid with me; what is it you mean where you say Miss Evelyn has caused you to be in such a state, have you shown her this, and has she handled it?''
``Oh! dear no; never, never!''
``Is it her face, her bosom, or her legs that have captivated you?''
``It was her feet and ankles, ma'am, with her beautiful legs, which she sometimes exhibited without knowing.''
``And do all ladies' legs and ankles produce this effect upon you?''
``Oh, yes, ma'am, if they are neat and pretty!''
``And what make you so excited now?''
``It was the sight of your beautiful legs just now, and the recollection of what I saw the other day, ma'am,'' I stammered out, blushing more than ever.
While this conversation was going on, her soft hand grasped my distended prick, and had commenced slowly slipping the loose skin over the swollen head, and allowing it to slip back again.
``I suppose, Charles, after what you saw in the closet, you know what this is meant to do.''
I muttered out an indistinct reply that I did, and I hung down my blushing face.
``You have never put it into a lady, have you?''
``Oh! dear no, ma'am.''
``Would you like to do so?''
I did not answer, but sheepishly held down my head.
``Did you see what I had in the same place, when you were in the closet?''
I muttered, ``Yes, ma'am.''
``Would it afford you any pleasure to see it again?''
``Oh, yes; so much!''
Mrs. B. rose, went to the window, drew down the blind, then gently turned the key in the door. Returning to the chair, and drawing well up her dress, petticoats and chemise, she exposed all her person up to the middle of her belly; and sat down stretching herself backwards, and opening her thighs well.
``Well, my dear boy, look at it if you wish.''
I was no longer shy. Nature prompted me to an act of gallantry that gratified the lady immensely. Falling on my knees, I glued my lips to the delicious spot, pushing my tongue in as far as I could, and sucked it. It was quite spunky; I had no doubt but that Mr. B. had fucked her two or three times just before leaving. This, however, made no difference to me. The attack was as unexpected as it was delightful to the lady. She placed both hands on my head and pressed my face against her throbbing cunt. She was evidently hotly excited, not only by what I was then doing, but by the scene, the conversation, and the handling of my prick, which she had been indulging in. She wriggled her bottom nervously below me, I continued to greedily lick her moist and juicy cunt.
``Oh! oh! dear Charles, what exquisite delight you are giving me. Oh! oh!''
And she pressed my face more fully into the gaping sheath, and thrusting her bottom up at the same time, spent right into my mouth, over my cheeks, chin, and neck. Her thighs closed convulsively round my head, and for some moments she remained still. I continued to lick away, and swallowed the delicious spunk that still flowed from her. At last she spoke again---
``Oh! you darling Charles, I love you for ever; but get up, it is now my turn to give you a taste of the exquisite pleasure you have given me.''
I raised myself, and she drew me to her, and gave me a long kiss, licking her own sperm from off my lips and cheek; and desiring me to thrust my tongue into her mouth, she sucked it deliciously, while her soft hand and gentle fingers had again sought, found, and caressed my stiff-standing prick. She then desired me to lay myself on the floor, with three pillows to raise my head, and lifting up all her petticoats, and striding across me, with her back to my face, she knelt down, then stooping forward, she took my standing prick in her mouth, and at the same time lowering her buttocks, brought her beautiful cunt right over and down upon my mouth, the pillows exactly supporting my head at the proper level, to command a thorough enjoyment of the whole, which now I had completely before my eyes.
In the former sucking my own position hid everything from view beyond the rich mass of hair adorning her splendid mount of Venus, which I found to be much more abundant than it had appeared to me when I had seen it from the closet. When I applied my lips to the delicious gap, I found that she had the most beautiful silky light curls running up to and around her charming pink bottom-hole, and losing themselves in the chink between the buttocks. I applied myself furiously to the delicious gash, and sucked and thrust my tongue in alternately. I could see by the nervous twitching of her buttocks, and the bearing down of her whole bottom on my face, how much she was enjoying it. I, to, was in an extasy of delight. One hand gently frigged the lower portion of my prick, while the other played with my balls, and her beautiful mouth, lips, and tongue sucked, pressed, and tickled the head of my excited prick. The more furiously I sucked her cunt, the more her lips compressed the head of my pego, and her tongue sought to enter the urethra, giving me almost overpowering delight. Such reciprocal efforts soon brought on the extatic crisis, I cried out:
``Oh, lady! oh, dear lady! let me go; I am dying!''
She knew well enough what was coming, but she had her own way, and at the instant that she again poured down upon my mouth and face a plenteous discharge, her own rosy mouth received a torrent of my sperm.
For some minutes we lay mutually breathless and exhausted. Then Mrs. B. rose, shook down her clothes, assisted me to rise, and taking me in her arms, and pressing me lovingly to her bosom, told me I was a dear charming fellow, and had enraptured her beyond measure. She then embraced me fondly, kissing my mouth and eyes, and desiring me to give her my tongue, sucked it so sweetly.
``Now, fasten up your trousers, my darling boy.''
When I had done so, the blind was drawn up, and the door unlocked.
We sat down, I by her side with one arm round her lovely neck, and the other clasped in her hand.
``I am sure I can depend upon your prudence, my dear Charles, to keep all this a profound secret from every one. Your mamma thinks you a child, and will suspect nothing. I shall take an opportunity of suggesting that you shall sleep in the small room adjoining my bedroom, and with which there is a door of communication. When every one is gone to bed, I shall open the door, and you shall come and sleep with me, and I will let you enjoy me as you saw Mr. B. do the other day. Will you like that?''
``Oh! above all things, oh, yes. But you must also allow me to kiss that delicious spot again that has just given me such pleasure. Will you not, ma'am?''
``Oh, yes, my darling boy, whenever we can do so safely, and unobserved; but I must impress upon you never to seem very familiar with me before any one, or to take the slightest liberty unless I invite you to do so. Anything of the sort would certainly draw attention, and lead to our detection, and at once put an end to what I mean shall be a delightful connection for you as well as myself.''
I, of course, promised the most perfect obedience to her very prudent directions. The ice was broken, and we allowed no ceremony to stand between us. I grew again very excited, and would fain have proceeded at once to try again to fuck her as well as suck her, but she was inexorable, and told me I should only spoil the pleasure we should afterwards have in bed. The day passed like an hour in her charming society.
The carriage brought mamma and party to dinner. Mamma hoped I had behaved well, and been attentive to Mrs. B. in her absence. She answered nothing could be better, and that I was quite a model youth---so gentle and so obedient.
My mother found that she had caught cold, and had febrile symptoms after dinner. Mrs. B. persuaded her to retire to bed, and accompanied her. When in her room, she apparently noticed, for the first time, my little bed. She took the opportunity of suggesting that it would be much better to remove it to the small room, so as to leave my mother in perfect quiet, which my coming to bed might disturb.
This was said in such an innocent natural manner, that no suspicion was excited on the part of mamma or anybody else. Mamma only making the objection that my early rising might by my noise disturb Mrs. B. in the next room.
``Oh, no; I am not so easily disturbed, besides he has been so well behaved all day, that I am sure, if I tell him to be quiet in the morning, he will not fail to do so.''
So it was settled, and my bed was at once removed to the little room.
I know not what Miss Evelyn thought of this; at any rate, she made no remark, and I went to bed early. It will easily be conceived that I did not go to sleep. The hours struck one after the other, and no appearance of my amiable instructress. The remembrance of all her charms was ever present to my mind's eye, and I longed once more to dart my tongue into her moist and juicy cunt, as well as to try the new method that was to initiate me into the real secrets of Venus.
The long delay of her coming put me in a perfect fever. I tossed and tumbled in bed; my prick throbbed almost to bursting. Fortunately, I had never frigged myself, and that resource never occured to me, or I might have rendered myself quite incapable of enjoying the raptures my beautiful benefactress afterwards entranced me with. At last I heard voices and footsteps on the stairs. Mrs. B. bid Miss Evelyn good night, and the next minute her door was opened, closed again, and the key turned in the lock. I had taken the precaution to do so with my door. I heard her use the night vase, and then she opened my door, at once coming to my bed side. Seeing me awake and quite flushed, she kissed me, and whispered---
``Have you not been to sleep, Charles?''
``No, ma'am,'' I answered, in the same subdued tone, ``I could not sleep.''
``Why, dear boy?''
``Because I was going to sleep with you.''
Her lips pressed mine, and her soft hand, thrust under the clothes; sought for and caressed my stiff-standing prick---it was as hard as iron.
``Poor boy, I am afraid you have been suffering. How long has it been in this state?''
``All the evening, ma'am, and I did think you were such a long time in coming.''
``Well, Charles, I could not come sooner without causing suspicion---I thought Miss Evelyn was suspicious, so I pretended to have no desire to go to bed; and even when she showed evident symptoms of drowsiness after her long ride, I rallied her upon it, and begged her to sit up with me yet a little; until at last she could hold out no longer, and begged me to let her retire. I grumblingly complied, and she is thrown completely off any scent on our account, as she could never suppose I was impatient as you to come here. I shall undress as fast as possible, and then do my best to relieve you of this painful stiffness. Get up, shut this door, and come to my bed. My room has an inner baize door, and we shall there be certain of not being overheard.''
I instantly complied, and she commenced undressing. Every detail of her charming toilet was devoured by my greedy eyes. Her smooth, glossy, and abundant hair, arranged in braids, was neatly fastened in under a coquettish lace cap with pretty blue ribbons. Her \textit{chemise de nuit} of the finest, almost transparent cambric was edged with fine openwork. She looked devine. The drawers of the commode contained scent bags of that peculiar odour which is generally found to perfume the persons of the most seductive women. In another moment she was in bed, clasping me in her arms.
``Now, Charles, you must be a good boy, and make no noise, and allow me to teach you your first love lesson, see I will lay myself down on my back, thus---do you place yourself on your knees between my out-spread thighs---there, that is a darling---now let me lay hold of your dear instrument. Now lay yourself down on me.''
I placed myself on her beautiful smooth and white belly and pressed against the hair of her mount. With her long taper fingers she guided my prick---I trembled in every limb and almost felt sick with excitement---but when I felt the delicious sensation caused by the insertion of my skinned pintle between the smooth warm oily folds of the lady's cunt---I gave but one shove which carried me up so that I swooned away on her belly and milk-white bosom.
When I came to myself I still lay on her belly, enfolded in her lovely arms, my prick sheathed up to the cods in her delicious cunt, which was throbbing in the most extatic way and pressing and closing with every fold on my prick---which had hardly lost any of its pristine stiffness; as my eyes began to discern her features, an exquisite smile played upon my darling companion's lips.
``You sad rogue,'' she whispered, ``you have given me a baby; what have you been doing to make you spend so soon, and in such a quantity, Did you like it?''
``Oh, dearest madam, I have been in heaven---surely no joy can be greater than you have given me.''
``But you do not know as yet everything that is to be done, and to how much greater an extent the pleasure may be enhanced by mutual efforts; move your instrument gently in and out---there, that is delicious, but not so fast. Good, is it not nice!''
And she moved in unison with me, meeting each slow thrust down by an equal movement upwards, and squeezing my prick in the most delicious manner internally, as she retired again to meet succeeding thrusts in the same way.
Oh! it was extatic---my prick, swollen to its utmost size, seemed to fill her exquisite vagina, which although capable of easily accommodating the larger prick of Mr. B., appeared to be sufficiently contracted to embrace tightly with its smooth and slippery folds my stiff throbbing prick. So we continued, I shoving myself into her, and she upheaving her beautiful bottom to meet me. My hands removed everywhere, and my mouth sucked her lips and tongue, or wandered over her pulpy breasts sucking their tiny nipples. It was a long bout indeed, prolonged by Mrs. Benson's instructions, and she enjoyed it thoroughly, encouraged me by every endearing epithet, and by the most voluptuous manoeuvres. I was quite beside myself. The consciousness that I was thrusting my most private part into that part of a lady's person which is regarded with such sacred delicacy caused me to experience the most enraptured pleasure. Maddened by the intensity of my feeling I at length quickened my pace. My charming companion did the same, and we together yielded down a most copious and delicious discharge.
Although I retained sufficient rigidity to keep him in his place, Mrs. B. would not allow any further connection with her, and she made me withdraw, and bade me go to sleep like a good boy, and she would give me a further lesson in the morning.
Finding that she was determined on this point, and that she disposed herself to slumber, I felt I was obliged to follow her example, and at last fell fast asleep. It might be about five in the morning, quite light at that time of year, when I awoke, and instead of finding myself, as usual, in my own little bed---I found my arms round the person of a charming woman, whose large plump smooth bottom lay in my lap, pressing against my belly and thigh. I found my prick already in a rampant state, and it at once began throbbing and forcing its way between the delicious cheeks of her immense bottom, seeking the delightful sheath it had so enjoyed the previous part of the night. Whether Mrs. B, was asleep or not, I do not know, but am inclined to think she really was so, from the muttered mistake she made in waking. She was probably dreaming, for she mechanically raised her thighs. I pressed my prick stoutly forward against her luxurious body, knowing that the entrance to the temple of pleasure which had so entranced me the night before lay in that direction. I found more difficulties than I expected, but at length began to penetrate, although the orifice appeared much tighter than on the previous evening. Excited by the difficulties of entrance, I clasped the lady firmly round the waist and pushed forcibly and steadily forward. I felt the folds give way to the iron stiffness of my prick, and one-half of it was fairly embedded in my extremely tight sheath. I put down my hand to press my prick a little downwards to facilitate the further entrance; you may imagine my astonishment when on so doing I found myself in the lady's bottom-hole, instead of her cunt. This at once explained the difficulty of entrance. I was about to withdraw and place it in the proper orifice when a convulsive pressure of the sphincter caused me such exquisite satisfaction by the pressure of the folds on the more sensitive upper half of my prick, which was so delicious, and so much tighter, and more exciting than my previous experience of the cunt that I could not resist the temptation of carrying the experiment to the end. Therefore, thrusting my two fingers into her cunt, I pressed my belly forwards with all my might, and sheathed my prick in her bottom-hole to its full extent. Mrs. B at this awoke, and exclaimed, ``Good Heavens! Fred, you hurt me cruelly. I wish you would be content with my cunt, I shall be unable to walk tomorrow. You know it always has that effect. It is downright cruel of you---but since you are in, stay quiet a little, and then continue to frig me with your fingers, as you know that eventually gives me great pleasure.''
She calls me Fred, what can she mean? I was, however, too agreeably situated to speculate on anything, but as I was now buried within her bottom-hole, I lay quiet for a few minutes as she had requested; and as her complaints subsided, and I felt a slight reciprocating movement, I, too, moved within her, working at the same time my two fingers in her cunt. By this time she was wide awake, and became conscious of who was her bed-fellow.
``What are you about, Charles?'' she exclaimed, ``do you know where you are?''
``I did not know I was doing anything wrong.''
``Doing wrong, indeed! My, a lady's bottom-hole was never intended for a pego. How came you to put it in there?''
``I cannot tell; I did not do it on purpose. I thought I was going into the same delightful place I was in last night.''
All this time I was moving my prick in and out of one aperture, and my fingers were working away in the other. The tightness of the sheath round my prick was delicious beyond anything I could conceive, and I think, from the way the lady conducted herself, she liked it as much as I did. At any rate, she permitted me to go on until I had a delicious discharge; and she herself spent all over my hand.
When the bout was over, she jumped out of bed, went to the basin, and with a sponge purified herself. After which, she said---
``My dear boy, you had better come and wash yourself, too; and take care not to make a mistake of this kind again, as it is sometimes attended with disagreeable consequences.''
It was now perfect sunny daylight, and my enchanting mistress looked so lovely in her almost transparent cambric night-shirt that I was emboldened to ask her to let me see her perfectly naked in all her glorious beauty of form. She gratified me at once; but laughingly, pulled off my night-shirt, and said---
``I, too, must have the pleasure not only of contemplating your promising youthful charms, but of embracing your dear form disencumbered of all the superfluities of dress.''
We clasped each other in a most enrapturing embrace, and then my lovely and engaging companion allowed me to turn her in every direction so as to see, admire, and devour every charm of her exquisitely formed body. Oh! she was indeed beautiful---shoulders broad, bosom, or rather upper neck, flat, not showing any projection of the collar bone; bubbies firm, well separated and round, with most exquisite rosy nipples not much developed; a perfect waist, small naturally, with charming swelling hips, and an immense bottom---it was almost out of proportion, large, but oh, how beautiful. Then her belly, undulating so enticingly, and swelling out, the lowest part into a very fine and prominent mons Veneris, covered with a thick crop of silky and curly light hair; then the entrance to the grotto of Venus had such delicious pouting lips, rosy, but with hair still thick on each side, which is often not the case even with women who have a sufficient tuft above, how beautiful where it exists as it did in this charming and perfect woman, continuing in beautiful little curls not only down to but around her lovely pinky and puckered little bottom-hole, the delights of which I had already, in this infancy of my love education, tasted and enjoyed. Her two alabaster thighs, worthily supporting by their large well-rounded fleshy forms, the exquisite perfections of the upper body, I have already described. How beautiful, elegant, and elongated her legs were, rising from well-turned ankles and most tiny beautiful feet. Her skin was white as milk, and dazzlingly fair and smooth. To my young eyes she was a perfect goddess of beauty. Even now, in advanced life, I can remember nothing that, as a whole surpassed her, although I have met many with points unsurpassingly beautiful---some carry it in the bosom, some in the general carriage, some in the mount of Venus and bottom together, and some in legs and thighs; but this devine creature, without having the appearance of it when dressed, was, when stripped, perfect in all her parts as well as beautiful in face---caressing and voluptuous by nature, and lending herself, with the most enchanting graces to instruct me in all the mysteries of love, and let me say, of \textit{lust} also.
We caressed each other with such mutual satisfaction that nature soon drove us to a closer and more active union of the bodies. Fondly embracing one another, we approached the bed, and being equally excited threw ourselves upon it, and, in the exquisite contact of our naked flesh, enjoyed a long, long bout of love, in which my most charming companion exhibited all the resources of amorous enjoyment. Never shall I forget the luxury of that embrace. She checked my natural tendency to rush at once to a completion. I think we must have enjoyed the raptures of that embrace fully half an hour before bringing on the grand finale, in which my active companion showed the extraordinary suppleness of her delicious body by throwing her legs over my back, pushing my bottom forward with her heels, and raising and sinking her bottom in unison with each thrust of my terribly stiff prick, which seemed to swell and become thicker and harder than ever. In retiring from each thrust, her cunt seemed to close upon my prick with the force of a pair of pincers. We both came to the extatic moment at the same time, and both actually screamed with delight; my ardent mistress in her fury of excitement actually bit my shoulder and drew blood; but I felt it not---I was in the seventh heaven of delight, and lay for long almost insensible on her beauteous body, clasped in her loving arms. On coming to our senses:
``Oh, my beloved boy,'' she said, ``never, never, have I experienced such pleasure. You are a perfect angel. I only fear I shall come to love you too much.''
We turned on our sides without dislodging the dear instrument of our enjoyment, and my lovely friend prattled on and delighted me with her toying, embracing, and gaiety. My prick had once more swelled up, and I wished to quietly enjoy a fuck in the luxurious position in which we lay; but my lovely friend said---
``That must not be, my dear Charles, I must consider your health. You have already done more than your age warrants, and you must rise and go to your bed to recover, by a sound sleep, your strength.''
``But feel how strong I am,'' and I gave a forcible thrust into her glowing and well-moistened sheath. But, though she certainly was greatly excited, she suddenly turned round and unseated me, and drew away from me, refusing to take it again. As she was quite naked, the movements of her beauteous form were most graceful and enchanting, and one leg being thrown backwards left her lovely cunt full in view, and actually gaping open before me. Seized with the strongest desire to suck and kiss it, as I had done the night before, I begged that at least she would grant me that last favour, as it could not in any way do me harm. To this she readily consented, and lay down on her back, opening her glorious thighs, and with a pillow under her bottom so as to raise up her cunt into a better position for me to gamahuche her, as she called it. Before letting me begin, she said---
``My dear Charles, do you see that little projection at the upper part of my quim, that is my clitoris, and is the site of the most exquisite sensation; you see it is rather hard, even now, but you will find as you titillate it with your tongue or suck it, that it will become harder and more projecting, so apply your lips there.''
I did as my lovely mistress desired, and soon found it stiffen and stand up nearly an inch into my mouth.
The convulsive twitches of her buttocks, the pressure forward of her hand on my head, all proved the exquisite felicity my lovely friend was enjoying. I slipped my hand under my chin---the position was awkward, but I managed to thrust my thumb into her cunt. My forefinger was somewhat in the way---but finding it exactly opposite the rosy hole of her bottom, and all being very moist there, I pushed it forward and it easily entered. I could not move my hand very actively, but I continued to gently draw my finger and thumb a little back together, and then thrust forward again. It seemed to add immensely to the pleasure I was giving her; her whole body quivered with excessive excitement. My head was pressed so firmly against her cunt that I had difficulty in breathing, but I managed to keep up the action of tongue and fingers until I brought on the exquisite crisis---her buttocks rose, her hand pressed hard on my head and her two powerful and fleshy thighs closed on my cheeks on each side and fixed me as if in a vice, while she poured down into my mouth and all over my chin, neck, and hand a perfect rush of sperm, and then lay in convulsive movements of enjoyment, hardly knowing what she was doing. As she held me so fast in every way, I continued to lick up the delicious discharge, and continued at the same time to pass my tongue over her clitoris. This, by producing a new excitement, brought her senses round. So relaxing her hold of me with her thighs she said---
``Oh, my darling Charles, come up to my arms that I may kiss you for the exquisite delight you have given me.'' I did so, but took care, in drawing myself up, to engroove my stiff-standing prick in the well-moistened open cunt that lay raised on a pillow so conveniently in the way.
``Oh, you sad traitor,'' cried my sweet companion. ``No, I cannot, I must not allow it,'' but I held her tight round the waist, and her position was too favourable for me to be easily unhorsed.
``Ah! you must not, my dear boy. If you will not consider yourself, consider me. I shall be quite exhausted.'' I shut her mouth with my kisses and tongue, and soon the active movements I was making within her charming vagina exercised their usual influence on her lubricity, so as to make her as eager for the fray as myself.
``Stop, my dear Charles, and you shall have it in a new position, which will give you as much more pleasure as it will me.''
``You are not going to cheat me, are you?''
``Oh, no! my darling, I am now as much on fire as you are---withdraw.''
I obeyed, half in fear. My fair mistress turned herself round, and getting on her hands and knees, presented to my ardent gaze her magnificent bottom. I thought she meant me to once more put it into the rosy little orifice, and said so.
``Oh! no,'' she replied, ``not there''; but putting her hand under her belly, and projecting it backwards between her thighs, she said---
``Give it me and I will guide it into the proper place.''
Before doing so I stooped forward and pushing my face between the glorious cheeks of her bottom, sought and found the lovely little orifice, kissed it, and thrust my tongue in.
``Oh! don't Charles, dear, you tickle me so,'' then flinching, and squeezing her buttocks together, I had nothing for it but to put my prick in her hand. She immediately guided it to and engulphed it in her burning cunt up to the very hair. I found I apparently got in fully an inch further this way---the position also gave my beautiful instructress more power of pressure on my prick---then her glorious buttocks, heaving under my movements, and exposed in all their immensity, was most exciting and beautiful. I seized her below the waist with a hand upon each hip, pressing her magnificent backside against me each time that I thrust forward. Oh! it was indeed glorious to see. I was beside myself, and furious with the excitement the view of all these charms produced upon me. My charming mistress seemed equally to enjoy it, as was evinced by the splendid movements of her body; till at last overcome by the grand finale, she sank forward on her belly, and I followed on her back, without losing the position of my throbbing prick within her. We both lay for some time incapable of movement, but the internal squeezing and convulsive pressure of her cunt on my softened, but still enlarged prick, were exquisite beyond imagining. At last she begged me to relieve her. Getting out of bed, she sighed deeply, kissed me tenderly, and said, ``My dear Charles, we must not be so extravagant in future, it will destroy us both---come, let me see you to your bed.'' The sight of my lovely mistress standing naked in all the glory of her beauty and perfection of form began to have its usual effect upon my prick, which showed symptoms of raising his head again; she gave it a pat, stooped down, and for a moment plunged its head into her beautiful mouth, then seizing my night-shirt, she threw it over my head and conducted me to my own bed, put me in, tucked me up, and tenderly kissing me, left the room, first unlocking my door and then locking the door of communication between the two rooms. Thus passed the first glorious night of my initiation into all the rites of Venus, and at the hands of a lovely, fresh and beautiful woman, who had only been married long enough to make her a perfect adept in the art. Never, oh never! have I passed such a night. Many and many a fine woman, perfect too in the art of fucking, have I enjoyed, but the novelty and the charm, the variety and the superiority of the teacher, all combined to make this night the \textit{ne plus ultra} of erotic pleasure.
It need not be said that, exhausted by the numerous encounters I had in love's battlefield, I fell into a deep and sound sleep, until aroused by being rudely shaked up. I opened my eyes in astonishment. It was my sister Mary. She threw her arms round my neck, and kissing me, said---
``You lazy boy, do you know they are \textit{all} down at breakfast, and you still asleep. What has come over you?''
``Oh!'' I said, ``I got frightened with a horrible dream, and lay awake so long afterwards that when I did sleep, I overslept myself.''
``Well, get up at once,'' and pulling the clothes quite off me, she laid bare my whole private parts, with my cock, as usual in youth on waking, at full stand.
``Oh! Charlie,'' said Mary, fixing her eyes upon it in astonishment at its thickness and length. ``How your doodle has grown,'' and she laid hold of it. ``Why it is as hard as wood, and see how red its head is.'' Without her knowing why, it evidently had its natural effect on her sense, and she flushed as she squeezed it.
``Ah! my dear Mary, I have learnt a great secret about that thing, which I will tell you the first time we can be quite alone and secure from interruption. Just now there is no time, but before you go downstairs, let me see how your poor little Fanny is.''
We had been used to these infantile expressions when in our ignorance and innocence we had mutual examinations of the difference of our sexes, and my sister was still as ignorant and innocent as ever. So when I said that I had not seen it since it was so ill-treated in the terrible whipping she had received from Miss Evelyn, she at once pulled up all her petticoats for me to look at it.
``Lie back for a moment on the bed.''
She complied. I was delighted. The prominence her mons Veneris had assumed, the increased growth of moss-like little curls, and the pouting lips of her tiny slit---all was most promising and charming. I stooped and kissed it, licking her little prominent clitoris with my tongue; it instantly hardened, and she gave a convulsive twitch of her loins.
``Oh! Charlie, how nice it is! What is it you are doing? Oh, how nice! Oh, pray go on.''
But I stopped, and said---
``Not at present, my darling sister, but when we can get away together I will do that and something much better, all connected with the great secret I have got to tell you. So run downstairs, and tell them why I had overslept myself, but not a word to anyone about what I have told you. I will be down in a trice.''
She went away, saying---
``Oh, Charlie, dear, what you did just now was so nice, and has made me feel so queer; do find an early opportunity of telling me all about it.''
Very few minutes sufficed to finish my toilet and bring me to the breakfast table.
``Why, Charlie,'' broke out my mother, ``what is this horrid dream?''
``I can hardly tell you, my dear mother, it was so confused; but I was threatened to be murdered by horrid-looking men, and at last taken to high rocks and thrown down. The agony and fright awoke me, screaming, and all over perspiration. I could not sleep for hours after, even though I hid my head under the clothes.''
``Poor child,'' said Mrs. Benson, who was quietly eating her breakfast. ``What a fright you must have had.''
``Yes, ma'am, and at the same time, as I awoke with a scream, I was afraid I might have disturbed you, for all at once I remembered I was no longer in mamma's room, but next door to you. I hope I did not wake you?''
``Oh, no, my dear boy; I never heard you, or I should have got up to see what was the matter.''
So it passed off, and no further observation was made about it, but I once caught Mrs. Benson's eye, and the expression and a slight nod was a sign of approval of my story. After breakfast we went as usual to the school-room. I thought Miss Evelyn was kinder in her manner to me than usual. She made me stand close to her when saying my lessons, occasionally letting her left arm fall round my neck, while she pointed to my book with the finger of the right, and there was always a certain pressure before raising her arm again. These little caresses were frequently repeated, as if she were wishing either to accustom me or herself to a habit of it, so as, doubtless, gradually to increase them to something more definite. I could not help feeling what a different effect these endearments would have had twenty-four hours earlier; but now, momentarily satisfied passions, and the new love that had seized me for Mrs. B., prevented at first the inevitable cockstand that would otherwise have been produced by these approaches of Miss Evelyn. Not that I had given up all desire to possess her. On the contrary, my last night's instruction only made me more anxious to have Miss Evelyn too. Therefore, I by no means repulsed her present caresses, but looked up innocently in her face, and smiled affectionately. In the afternoon she was more expansive, and drew me to her by her arm round my waist, and pressed me gently to her person, saying how well I was attending to my lessons, and how sorry she was to have been obliged to punish me so severely the week before.
``You will be a good boy in future, will you not, dear Charlie?''
``Oh, yes; as long as you are so kind to me. I love you so much, and you are so beautiful when you speak so kindly to me.''
``Oh, you little flatterer.''
And she drew me to her lips and gave me a sweet kiss, which I returned with eagerness. I felt my prick had raised itself up to its full extent as these caresses were exchanged, and as Miss Evelyn held me tight pressed against her thigh, she must have felt it throbbing against her. That she did so, I have no doubt, as her face flushed, and she said---
``There, now, that will do, go to your seat.''
I obeyed; she rose in an agitated manner, left the room, and was absent for a quarter of an hour. I had no doubt but that she was overcome by her feelings, and I thought to myself she will manage to have me some of these days. I could afford to leave it to her own discretion, as my charming mistress of last night was there to keep me in exercise and cool the effervescence of passion under which I should otherwise have laboured. Nothing particular occured during the day; Mrs. B. was apparently indifferent about me, and never sought to approach or be in any way familiar; I studied her looks and followed her example. Mamma sent me early to bed, as she feared I had not had sleep enough the previous night by reason of my bad dream, and hoped I should have no more of the kind. This time my beautiful mistress found me sound asleep when she came to bed. She did not awake me until she had completed her night toilet, and was all ready to receive me in her arms. I sprung up, and in an instant, without a word being said, had her on her back, and was into her delicious cunt as far as I could drive my stiff-standing prick. My energy and fury seemed to please and stimulate the lady, for she replied to every eager thrust with as eager a spring forward. In such haste matters were very speedily brought to a crisis---with mutual sighs, and ``oh's'' and ``ah's,'' we sank exhausted, and lay for a very short time, when charming Mrs. B. said---
``Why, Charles, you are quite wild to me; what a hurry you have been in, but it was very nice, and I forgive you, but you must be more rational in future.''
``Oh, my beloved mistress, how can I help it; you are so beautiful, and so good to me; I quite adore and love every part of your charming body. I know I was too impetuous, but I must make it up by kissing and fondling the dear source of all my joys.''
She did not resist, but let me do as I liked. Pushing myself down the bed, I applied my lips and tongue to her lovely cunt, all wet with our mutual discharge, which was so sweet to the taste that I first began licking between the lips, and the applied myself to her excited clitoris, and with my finger and thumb working as on the previous morning I threw her into an extasy of delight, until again she had a delicious discharge. Then creeping up, I thrust my prick into her well-moistened and velvety cunt---as you may imagine it was rampant as ever after my mouth contact with the exquisite quim I had been sucking.
``Stop, Charles, darling, I will show you another position, where you can lie easily with your dear delightful prick up to the hilt in the sheath you have so charmingly excited. Here, lie down by my right side ---on your side.''
She lay down on her back, and throwing her right leg over my hips, told me to bend my knees forward and open my legs, or rather lift up my right leg. She placed her left thigh between my thighs, then slightly twisting her bottom up towards me brought the lips of her cunt directly before my prick, which she seized with her delicate fingers, and guided safely into Venus's grotto. I gave one or two shoves, and she a heave or two, to house him comfortably.
``And now,'' she said, ``we will take it reasonably in this way; we can go on, or stay occasionally; embrace, cuddle, or talk, just as we please. Are you quite comfortable?''
``Oh! deliciously so!'' I replied, as my hand wandered all over her beautiful belly and bubbies, and then my mouth sucked the last.
``There, darling, that will do for the moment; I want to have some talk with you. First, let me thank you for your very discreet behaviour this day, it quite justifies the confidence I had in you. Your story of the dream was capital, and just suited the purpose. I hope, my dear Charlie, that under my auspices you will become a model lover---your aptitude has already proved in several ways. First and best, with all the appearance of a boy, you are quite a man, and even superior to many. You have already shown great discretion and ready wit, and there is no reason to fear that you will become a general favourite with our sex, who soon find out who is discreet and who is otherwise---discretion is the trump card of success with us. Alas! few of your sex understand this. Let me impress one lesson on you, my dear Charles. You and I cannot continue long on our present footing. My husband will return and carry me away, and although circumstances will throw us at intervals into each other's arms---for you may be sure you will be always welcome to me---yet my very absence will force you to seek other outlets to the passions I have awakened and taught their power. I have one piece of advice to give you as to your conduct to newer lovers---for have them you must, my dear Charles, however much you may fancy yourself now attached to me; with these, let them all for some time imagine that each possesses you for the first time. First of all, it doubles their satisfaction, and so increases your pleasure. Your early discretion causes me to think that you will see all the advantages of this conduct. I may add that if they suppose you have had previous instruction, they, if they are women, will never rest until they have drawn from you the secret of your first instructress. You might, of course, tell some tale of a `cock and a bull,' but in searching for the truth and cross-questioning you when you are least aware of it, they will lead you into contradictions, and the truth will at last be ferreted out. Now this would be unjust to me, who have risked a good deal to give you the delightful instructions of last night, and, as I hope, of many more. So you see, my dear Charles, in all early cases you must enact the part of an ignoramus \textit{seeking} for instruction, with vague ideas of how to set about it. I hope, while I am near you,'' she added, ``no such occasion will arise, but I feel certain, with your passions and your power, dear, darling fellow---push away---I!---I!---I feel for cer---certain they will ar---arise.''
Thus ended the very wise and excellent advice this charming woman was giving me. Do not imagine that I did not pay great attention, and, indeed, her very reasonable maxims became the guide of my after-life, and I owe to them a success with women rarely otherwise obtained. Her sensible remark had been drawn out to such a length, that my prick had so far rebelled that he had throbbed inside of her delicious cunt so forcibly as to produce a happy movement of her body that interrupted and cut short her words.
``Charlie, my darling, pass your middle finger down and rub it on my clitoris, and then suck the nipple of my bubby next you, and work away with your glorious prick.''
I did as desired. She seconded me with an art quite peculiar to herself, and at last we both died away in that love's death which is so overpowering and so delicious. The glorious position we were in rendered it almost impossible to lose ground, spend as often as you please; but if my prick had been one that would have shrunk to nothing, the wonderful power of retaining it within her possessed by my delicious mistress would have prevented the possibility of exit.
In after-nights I have often fallen sound asleep with it entirely engulphed within her, and awoke hours afterwards to find her extraordinary power of retention had held him firm, notwithstanding his having shrunk up to a mere piece of doughy flesh. In this instance, after recovering our senses, I still retained my place, and we recommenced our conversation; my lovely instructress giving me many and most useful hints for my after-life. I have often since dwelt on the wisdom of all she so charmingly taught me, and wondered how so young a woman could have so thorough a knowledge of her sex and the world. I suppose love is a great master and inspired her on this occasion. I may here remark that for forty years afterwards this charming woman and I remained the fastest of friends after being the most ardent of lovers. She was the depository of all my erotic extravagancies, and never showed any jealousy, but really enjoyed the recital of my wildest love combats with others.
Alas! death at last took her from me, and I lost the mainstay of my existence. Forgive this digression, but I am writing long after these events, and sorrows will have their vent. Woe is me!
To return to present joys. We continued talking and toying, until I was again anxious to commence love's combat. My prudent mistress wished me to finish for the time, and to sleep and refresh ourselves for renewed efforts; but youth and strength nerved me for the fight, and being securely fixed, I held her as in a vice, with my thighs around only one of hers that could have allowed her to escape. Passing my finger down on her stiffened clitoris I so excited her that she had no wish but to bring matters to a crisis.
``Stop, my dear,'' she said, ``and we will renew our pleasure in another attitude.''
So withdrawing her leg off my loins, she turned on her side, so as to present her glorious buttocks before me, and pressed them into my belly and against my thighs, which seemed to introduce my prick even further than he was within before. Besides, in all these positions, where a woman presents her splendid backside to you, it is always more exciting, and has a greater hold of you than any other way. We did most thoroughly enjoy this splendid fuck, and without withdrawing, both fell into the sweetest imaginable slumber. This was one of those occasions in which, having fallen asleep engulphed, I awoke some five hours later, to find my prick still lightly held within the velvety folds of one of the most delicious cunts ever created for the felicity of man, or, I may say, woman either. You may easily imagine how soon my prick swelled to his wonted size on finding himself still in such charming quarters. I let him lay quite still, barring the involuntary throbs he could not avoid making, and bending my body away from my lovely mistress, I admired her breadth of shoulders, the beauty of her upper arm, the exquisite \textit{chute} of her loins, the swell of her hips, and the glorious projection and rotundity of her immense buttocks. I slowly and gently pushed in and out of her juicy sheath, until, awakened by the exquisite sensations of my slow movements, all her lubricity was excited, and we ended one of our most delicious encounters, finishing, as usual, with a death-like exhaustion. She declared I had done enough for one night, and jumping out of bed, compelled me to betake myself to my own room, where, I must confess, I very shortly slept as sound as could be, without at the same time oversleeping myself.
Thus passed several successive nights, until the fall of the moon, when one day Mrs. B. complained of headache and feeling unwell. I was very much alarmed, but she took occasion to tell me it was quite natural, and she would explain to me how it was so at night. I was obliged to be content with this. At night, she came and sat on my bed, and told me all the mysteries of the case. How women, not with child, had these bleedings monthly, which, so far from being hurtful, were a relief to the system, and that they happened at the full or the new moon, generally at the former. Further, that all connection with men must cease at such a time. I was in despair, for my prick was stiff enough to burst. However, my kind and darling mistress, to relieve me from the pain of distention, took my prick in her mouth, and performed a new manoeuvre. Wetting her middle finger with her saliva, she thrust it up my bottom-hole, and worked in unison with the suction of the knob, and the frigging of the root of my prick with the other hand. I had a most exquisite and copious discharge, the pleasure being greatly enhanced by the action of the finger up my fundament. My charming mistress swallowed all I could give her, and did not cease sucking until the last drop had exuded from my throbbing prick.
I was obliged to be satisfied with this, and my mistress informed me I could have no more enjoyment for four or five days; which, to my impatience, was like condemning me to as many ages of hope deferred. I observed, while she was kissing me, that her breath had a peculiar odour, and I asked her what she had been eating.
``Why do you ask, my dear boy?''
``Because of the difference of your breath, generally so sweet and fragrant.''
She smiled and said it was all from the same cause she had just been explaining to me, and was very generally so with women at that period. I mention this because it was the means of my discovering that Miss Evelyn was exactly in the same state. She had continued her endearing caresses without proceeding much further than I have already described, except more frequently kissing me. She now always did so on first entering the school-room, and also when we were dismissed. I suppose to prevent an observation or inference, she had adopted the same habit with my sisters. On this day, having drawn me with her arm round my waist close to her, when she kissed me I felt the very same odour of breath that I had observed in Mrs. Benson. She too was languid that day and complained of headache. I also observed a dark line under her eyes, and on afterwards observing Mrs. B., saw precisely the same---so I became convinced they were unwell from the same cause. Mrs. B. had told me that most women were so at the full of the moon---which was then the case.
The next day my mother proposed to drive to town, and probably knowing the state of the case, asked Mrs. B. and Miss Evelyn to accompany her, as she thought the airing would be beneficial. They at once accepted---my younger sister cried out, ``Oh, mamma, let me go with you also.'' Mary interposed, and thought she had the best right---but Lizzie said she had spoken first. I managed to give Mary a wink and a shake of the head, which she instantly comprehended, so gracefully giving way, although with apparent reluctance, it was arranged that Eliza should accompany the ladies. I now felt my opportunity was at hand to initiate my darling sister into the delightful mysteries that I had just been myself instructed in.
At eleven o'clock the carriage drove up, and we stood looking after them until they were lost to sight. Then returning into the parlour, Mary threw her arms round my neck, and kissing me, said---
``Oh! I am glad, Charlie, you winked to me, for now you know we can do as we like, and you can tell me all about this secret, and you must kiss my little Fanny as you did before, it was so nice. I have thought of nothing else, but how to have it done again.''
``Well, my darling, I shall do all that, and more, but we cannot do so here. I tell you what we will do---we will pretend to go for a long walk in the country, but instead of that, we will pass through the shrubbery into the orchard and hazelwood, and so gain the little remote summer house, of which I have secured the key; there we shall be safe from all observation.''
This little summer house was at some distance from the house, and in a lonely corner of the orchard, raised on an artificial mount, so that its windows should command a lovely view beyond the walls of the grounds. It was about ten feet square---was beautifully sheltered, and the ladies in summer took their work there, and occupied it for hours every fine day; so it was furnished with tables and chairs, and on one side a long couch without a back. It had already entered into my idea that this was the spot I should contrive to get to with Mary---little thinking how chance would throw so glorious an opportunity in my way so soon. It was always kept locked to prevent it being used by the servants, gardeners, or others. I knew where the key was kept, and secured it when the ladies were dressing for their drive---so after staying sufficiently long to prevent any suspicion, and saying then we were going for a long walk in the country, so as to prevent them seeking for us at the summer house if any visitors should chance to call, we sallied out, but re-entered the grounds where we could not be observed, and speedily gained the spot we had in view---entered and locked the door. Then I drew down the blinds, threw off my coat and waistcoat, and told Mary to take off her shawl and bonnet, and outer gown.
``But why all this, Charlie, dear?''
``First, my darling---all those are in the way of kissing and toying with your charming little Fanny, and next, I don't want anything to appear tumbled when we go back.''
This was enough, and she did everything as I desired, indeed, more, for she took off her petticoat and little corset, saying she would be cooler thus. So, following her example, I took off my trousers, saying she would be better able to see and play with my doodle. When these preliminaries were accomplished, I drew her on my knees---first pulling up her shift and my own shirt, so that our naked flesh should be in contact. Seeing that her chemise fell off from her bosom, I first felt her little bubbies, which were beginning to develope themselves, and had the tiniest little pink nipples that even my lips could hardly get hold of. She had pulled up my shirt to look again at the great change that had occured to my prick---of course, our preliminaries had already excited it to a stiff-standing position.
``Oh, Charlie, what a size it is to be sure; and how nice to pull this skin over its head; look how it runs back again. Oh! how funny!''
It was time to stop this, or she would have soon made me discharge.
``Well, then, what is this great secret, and what has it to do with your doodle and my Fanny?''
``I will tell you, but you must never say a word to a soul---not even to Eliza, she is too young yet.''
``Well, go on.''
``I was one day seeking something in the closet in Mrs. Benson's room, when I heard them coming, and had only the time to slip into the closet. They entered, locked the door, and Mr. B. laid her on the bed, and lifted up all her petticoats so that I saw her Fanny quite surrounded with hairs, as yours will be by and by. Mr. B, stooped down, and applied his tongue as I did to you the other morning.''
``Oh, yes; and it was so nice, Charlie!''
That is exactly what Mrs. B. said when he had done. Then he pulled out his doodle, such a size, much bigger than mine, and whipped it into her Fanny. I was quite frightened, and thought he must have killed her. But no, it went in quite easy; and she hugged and kissed him while he pushed it up and down for some time, till they both stopped all at once. He then drew it out, hanging down all wet, and asked if it had not given her great pleasure. `Delightful,' she said. I have now got used to it, but you know you hurt me, and made me so sore the first time you did it.' After this they left the room, and I got away without being discovered. But I found out what our two things were made for, we will do as they did, so lie down on the couch whilst I kneel at the end, and begin in the way I kissed it the other morning.''
``Oh, Charlie, if it is all like that, I shall be so pleased with it.''
Down she squatted, drawing up her chemise. My hand wandered all over her charming belly and mount. Then kneeling down, and putting her legs over my shoulders, and my hands under her thighs and bottoms, I applied my tongue at once to her little clitoris, which I found was already stiff, and showing its head at the upper part of her pinky slit. The action of my agile tongue produced an instantaneous effect---her loins and thighs heaved up her bottom to press her little pouting cunt against my face. Mechanically she put her hand on my head, and muttered terms of endearment---
``Oh, darling Charlie, how delicious! Oh! do go on! it is so nice, \&c.''
I wanted no stimulant, but licked away until, with shortened breath; and greater heavings of her body, she began to stammer---
``Oh! oh! I feel so queer---ah, stop; I am going to faint---I, I, I, can't---can't bear it any longer---oh!---oh!'' Her limbs relaxed, and she died away in her first discharge, which was very glutinous and nice, but only scanty in quantity. I let her quiet until she came to; then looking in her face, and smiling, I asked her how she liked it.
``Oh! I was in heaven, dear Charlie, but I thought it was killing me ---it was almost too much to bear---nothing could be more delicious.''
``Oh, yes!'' I replied, ``there is something more delicious still, but, I must kiss you in this way again before we try the other; the more moist the inside is the easier I shall get in.''
``But, Charlie, you don't mean to say you will ever get in your doodle, now that it has grown so big.''
``Well, we will try, and if it hurts you too much we can stop.''
So I began again to gamahuche her; this time it took a longer effort to produce the ultimate result; but apparently with still greater effect, and a more copious discharge. Her little cunt being now relaxed, and well moistened with her own discharge and my saliva, and well inclined to receive my prick, I spat upon it and lubricated it from head to root. Then rising from my knees, I stretched myself over Mary's belly, and gently directing my prick, and rubbing it up and down first between the lips, and exciting her clitoris by the same action, I gently and gradually inserted its head between the lips of her charming little cunt. There was less difficulty than might have been expected, the gamahuching and double spending had relaxed the muscles, and her passions being excited also acted on her organs of generation; at all events, I got in the head, and about two inches of its length without her murmuring anything beyond---
``How big it feels---it seems to stretch me so.''
All this was exciting me dreadfully, and it was only by the greatest effort that I did not thrust rudely forward. I now felt I was pushing against some obstacle, I thrust hard and hurt her. She cried out, begged me to stop. I was so near the finale that I felt I must go on. So, plunging forward, I rushed, at the impediment, and made her cry out most lustily. Probably another push would have decided my position, but nature could hold out no longer, and I yielded down my erotic tribute to her virginal charms, without having actually deflowered her. So far, perhaps, it was fortunate, because I poured into her a torrent of sperm which was not only balm to her partially wounded hymen, but so relaxed and lubricated the interior of her cunt as greatly to facilitate my after-efforts.
I lay quiet still for some time, and the gradual swelling out and throbbing of my prick reawakened her young passions. She said---
``Charlie, my dear, you said that it would prove delicious in the end, and I can feel it is becoming so. I have no more pain, and you shall go on just as you like.''
As my prick stiffened at her endearing words and involuntary pressures, and as I had it completely under control, since I had taken the edge off its immediate appetite by the last discharge, I held it literally well in hand; and as I had lost no ground by withdrawing, I started with the advantage of possession. First I slipped my hand down between our two bellies and began frigging her clitoris, which immediately excited her passions to the highest pitch.
``Oh! Charlie, dear, now push it all in---I do so long for it---and I don't care how it hurts me.''
I had been giving short thrusts more to stimulate her passions than to alleviate my own; and as she was totally unaware of what was going to happen, she widened her thighs and heaved up her bottom, expanding her vagina in the act. I gathered my strength together, and as my cock was standing as stiff as iron, I suddenly drove it forward, and felt that I broke through something, and gained two inches more insertion at least. The effect on my poor sister was most painful, she shrieked out lustily; strove hard to unsheath me, wriggled her body in all directions to effect this; but I was too securely engulphed for that, and all her struggles only enabled me the more easily to sheathe him up to the very hairs. So excited was I by her tears and screams, that I was no sooner there than a torrent of sperm burst from me, and I lay like a corpse on her body, but perfectly maintaining the ground I possessed. This death-like quiet lasted some minutes, and, to a certain extent, assuaged the violence of the pain I put poor Mary to. Doubtless, also, the balmy nature of the ample quantity of sperm I had shot up to her womb helped to soothe her suffering. At all events, when we were both able again to converse, she unbraided me with the agony I had caused her, and wished me to get off her at once; but retaining the advantageous possession of her very tight and delicious sheath, I told her all was now over, and we might look forward to nothing but enrapturing pleasure.
Some minutes had elapsed in these remonstrances on one side; and coaxings on the other, when I suddenly felt her charming little cunt actually throb upon and give an involuntary squeeze to my prick, which was still throbbing her. He was far too ready to stand at any time, still more when engulphed in the exquisite young cunt he had just initiated into love's mysteries---\textit{bref}---he stood stiff as ever, and Mary, at first with a shudder of fright, then with all the energy of awakened passion, began to move her body under me. I held off from any interference, feeling certain that if the desire came naturally to her it would doubly enhance my own pleasure. My foresight did not fail me. Mary's passions became fully aroused, and when so, the trifling soreness passed out of mind, and we actually had a most delicious fuck, in which my prick appeared as if in a vice, and Mary wriggled her backside almost as well as the more artistic movements of Mrs. Benson. All things must come to an end, but this did so amid screams of delight on both sides. This single bout began and finished the education of my darling sister. She hugged and fondled me afterwards, declaring I was quite right in telling her pleasure followed pain; for nothing could exceed the enrapturing nature of the sensation my prick had produced. She thought now that it was not a bit too big, but just made to give the utmost satisfaction. We remained locked in each other's arms, my prick still engulphed in its tight and exciting sheath. We fondled and prattled, until it became again in a state of violent erection, equally stimulating her tight little cunt, so that we were forced to recommence our love encounter. I found that my dear little sister possessed naturally the power of throbbing on or nipping a prick, which the French call casse-noisette. It is a great gift and adds immensely to the man's pleasure, and I should think to the woman's too. In my sister's case it began from the very first complete insertion of my prick and the years that I afterwards continued to fuck her added nothing to this delicious accomplishment, except in the variety of positions in which it could be exercised.
The dear girl was in extasies at the pleasure she had received, and at the pain which seemed to be past. Oh! she was so sweetly caressing that I could not withdraw from her, and we fondled and toyed until again my cock rose to his first vigour, and she nothing loath, began her new and naturally taught gift of bottom upheavings and cunt pressures until again we sank exhausted in the death-like ending of love's battles. On recovering our senses, I was obliged to withdraw and relieve my sister of the dead weight of my body on her person.
It has always struck me as extraordinary how the most delicate women will support a heavy man on their persons, not only without flinching, but even with ease and pleasure---but so it is. On rising and withdrawing, we were both alarmed to see that my prick was all bloody, and that blood and semen were oozing from her cunt. We had no idea this would be the case, and at first I was as frightened as she was. A moment's reflection showed me that it was only the natural result of forcing my way in, and that the pleasure since enjoyed proved it to be of no consequence. I soon convinced and calmed my sister on the point---fortunately the sofa covering was red, and applying my handkerchief, I wiped up all the semen mixture, and, in fact, no marks remained; the same handkerchief wiped all results from Mary's dear little cunt, and as her shift had been kept well up, fortunately no stains appeared upon that.
We now ate some luncheon and drank some wine that we had prudently brought with us. We then began playing and romping together---she wanting always to get hold of my prick, and I to pull her about in every way. It was gloriously warm weather, so I proposed we should off with every thing. In a trice we were as naked as we were born, and flew into each other's arms in a frenzy of delight, then we had a mutual thorough inspection. My darling sister gave every promise of becoming a magnificent woman---her shoulders were already wide---her arms well shaped, although still thin---her waist small---the swell of the hips already well developed---as to her bottom, it stuck out well and hard behind, quite charming to see, and giving promise of very ample dimensions hereafter.
I made her kneel on the low couch, with her head well up and her thighs open; kneeling behind, I gamahuched her until she spent; then rising, shoved my prick into her cunt, in her then position, and had a downright good poke, which she, too, found was a way that gave her extra excitement. We passed thus some hours in mutual delights. I taught her the side fuck which had so charmed me with my delightful instructress, and I found dear Mary even an apter scholar than myself had proved. The afternoon advancing, we dressed, and eradicating all signs of what we had been doing, returned to the house, mutually promising to keep thoroughly secret all that had passed and agreeing that no sign of unusual familiarity should escape us. I strongly advised Mary to get some warm water and bathe her cunt well, for, as may be supposed, I had taken the opportunity of teaching her the true erotic language as applied to the organs of generation of both sexes, and the name of the connection itself, ``fucking.''
Thus delightfully ended the first lesson in love taught to my sister, and such was my first triumph over a maidenhead, double enhanced by the idea of the close ties of parentage between us. In after-life, I have always found the nearer we are related, the more this idea of incest stimulates our passions and stiffens our pricks, so that if even we be in the wane of life, fresh vigour is imparted by reason of the very fact of our evasion of conventional laws.
We had both returned to the drawing-room for more than an hour before the arrival of the ladies. Dear Mary complained of feeling sore and stiff in every limb. I had advised her to lie down on the sofa and try to sleep. I did the same, and happily we both dozed off, and never awoke until the loud rat-tat of arrival at the house door roused us up. I told Mary to hide all appearance of pain, and only to say, as an excuse for going early to bed, that we had gone further afield than we at first intended, and that she was very tired. We were both sent early to bed, for I was still treated as quite a boy, and I was sound asleep when my charming Mrs. B. woke me up by her warm caresses. I could well have spared them that night, but when did one of my years not respond to the endearments of the woman he loved, and who yielded all to him. She sucked me dry as usual, and I slept soundly till morning.
The next three days passed without anything to record. Mary did not allow her real soreness to appear, but heroically went through her sufferings, for she told me afterwards she felt very severe pains all over, doubtless her whole nervous system had been overexcited, and this was the natural reaction; it was so far fortunate that not a shadow of a chance of our having fresh connection occurred, so she had time to perfectly recover from the ill effects of her first initiation into the erotic raptures. I continued to have the relief each night of the charming mouth of my loved and beautiful instructress. At last, the abominable \textit{menses}, as she called them, were past and gone. For a full twenty-four hours after, she would not allow me to reassume all the privileges she had previously granted, and admit me to share her bed. She told me this was necessary to prevent any recurrence, and also that in some cases a virulent white discharge occasionally followed for some hours, sufficiently acrid to affect my local health, and ``that,'' she added, ``was now too precious in her estimation to risk it in any way.'' I thought it hard at the time, but it was only another proof of the thoughtful wisdom of this estimable woman. At last, I was again in full possession of her charming person. Oh! how we did revel in all the luxuries and lubricity; almost every night my enchanting friend found some new position to vary and enhance our erotic raptures. One new dose was laying me down flat on my back, then straddling over me, she sank on her knees, and with body erect, lifted up or rather bent back my stiff-standing prick, until he was fairly below her open cunt, then guiding it exactly to the proper entrance, she sank her body slowly down upon it until fully engulphed, hair crushed hair, then as slowly raising again, she drew off until all but the nut was uncovered, to again sink down. In this position we could both see the whole process. At length, becoming too excited, she sank on my bosom, then one arm and hand pressed her splendid buttocks down on my throbbing prick after every elevation of her magnificent backside while my other hand, doubling round behind her, introduced the middle finger up her charming bottom-hole, and worked in and out in unison with both our heaving movements, until stopped by the grand crisis, when death-like langour overcame us both almost at the same moment. I must not forget to mention that from time to time I paid a visit to the small and rosy orifice that lay so near to the more legitimate altar of Venus. It was a variety of enjoyment that my lovely mistress acknowledged to me she at times felt much inclined to enjoy, but only after having the front path of pleasure well fucked and lubricated with sperm, which alone caused the other mucous membrane to feel inclined that way.
I will here insert a characteristic letter from my loved mistress to her intimate and bosom school friend, with the reply thereto. It was several years before they were shown to me, and some time after I had possessed \textit{both} the charming writers, for we all three became fast friends; indeed, I may call myself or rather my prick, the pivot on which their friendship turned, yet there never was the shade of jealousy on either part, but in these remarks I am anticipating what I may, perhaps, be hereafter tempted to describe more fully. I give these letters now, because they immediately refer to the events I am at present relating. They show the secret working of my loved mistress's mind, and the voluptuous nature of her temperament, and the satisfaction that my delicious initiation had given. Her affectionate and flattering remarks, relating to myself, are greater than I deserved. The following is the first letter addressed to her friend:
\gutchapter{MRS. BENSON TO THE HON. MRS. EGERTON.}
Dear Carry,
I am about to keep my promise, and give you an account of our honeymoon. You, my dear, must be equally faithful, and reply as frankly as I am now about to write to you.
Two giddier girls than you and I never entered the bonds of matrimony, or more earnestly longed for the sights connected with it. Well, after the usual breakfast, we left by rail for Leamington, where we were to pass our first night. We had a \textit{coup\'{e}} to ourselves; and beyond seating me on his knee, and kissing me, Fred behaved with much decency and propriety. We arrived and dined. The hour between tea and bedtime was sufficiently tedious, as both of us were naturally much preoccupied. My husband wrote a letter to mamma, telling her of our safe arrival, and of his intense happiness. After which he asked me if I would go to bed, in the most matter-of-fact way imaginable. I murmured an affirmative, scarcely knowing what to say. He rang for a candle, and told me he would follow shortly. It seemed like a dream to me. The maid showed me to a room containing a large four-post bedstead, heavily hung with curtains, and provided with old-fashioned furniture.
I seated myself on the edge of the bed and began to meditate. I sat thus, for, I dare say, ten minutes, and then commenced undressing. I had put on my night-gown, and removed everything but my stockings, when I heard footsteps approach the door. I opened, and my husband entered, closed it, and turned the key. Oh! Carry, I did feel so funny. I was undressed in a bedroom with a man, and that man had a right to my person. He seated himself in an armchair, and drew me on his knee. Nothing but my thin night-gown separated my bottom from his bare knee, for he had quite undressed in an adjoining room and had nothing on but his shirt under his dressing-gown, which flew open as he sat down. He drew my lips to his, and kissing me, thrust his tongue between them, while his hand first caressed and squeezed my bosom, which, you know, is pretty full and well developed; it then wandered down upon my thigh, pressed and felt the fleshy form. Little by little he approached my belly, and for a moment pressed my mount. These preliminaries are at all times exciting, but now they made me almost ill, so great was my confusion. Seeing this, he drew up my night-gown, and placed his hand, first on my naked thigh, then on my mount, and you know, Carry dear, what a forest I have got there. He seemed delighted with it. His fingers played with the silky curls, drawing them out to their full length, so long that it appeared to surprise him, and his eyes sparkled, and his face showed much excitement.
``Open your thighs, dearest,'' he whispered.
I obeyed mechanically, and his middle finger forced itself between the lips of my cunt, and commenced rubbing my clitoris. You know, by experience, what an excitable one it is and to what a size it developes itself when excited. Again Fred seemed delighted with his discovery.
``Does that please you, my darling?''
``Yes,'' I faltered out.
He thrust his finger up my cunt, then rose up, threw off his dressing-gown, took me in his arms, and lifted me on the bed, placing a pillow under my head. Then letting my legs fall over the sides, he knelt on the floor, and separating my thighs with his arms, stooped and kissed my quim. He did more, he sucked and then licked with his tongue my already excited clitoris. It set me on fire, and I could not avoid showing it by the convulsive twitchings of my loins and buttocks.
``Do you like that, my love?''
``Oh! yes; so much!---so very much!''
I was nearly mad with the excitement he was putting me into. He again stood up, and lifting my legs, his hands pressed them again and again.
``What delicious legs,'' he exclaimed.
I could see his shirt bulging out. He leant forward, and with his arms under my legs, lifted them well up, and I felt a stiff thick thing pressing against my cunt. His left hand opened the lips, his right hand guided it between them, and a cruel push lodged its great head completely within. Neither you, or I, Carry, were strictly virgins, our fingers and other means had opened our vaginas to a certain extent. We had played too many tricks together to have left our maidenheads quite intact, so that the passage was less difficult than it might have been. Nevertheless, it had never been penetrated by the male organ, and that of my husband was of the largest. I experienced, therefore, a great deal of pain, and cried out---
``Oh, my dear Fred, you hurt me dreadfully, what are you doing?''
``Doing, my darling! why, I am getting into you. Have a little patience, and I will make you mad with pleasure.''
Another determined thrust sent him halfway, and then with another, still more violent, he lodged himself up to the hilt within. I screamed with real pain, and struggled to free myself.
``Good heavens, sir, you are killing me; I will not endure such treatment.''
He heeded me not, but holding me fast by the thighs commenced shoving in and out furiously. A sensitive woman never receives an insertion of this kind with impunity. The friction began to excite feelings that first deadened the pain of entrance, and then began to awaken the delicious sensations of lubricity. The enjoyment I began to experience was delicious, and I could not refrain from heaving up to meet his thrusts.
``That is right, my angel; was I not correct in saying it would soon turn from pain to pleasure? Do you not enjoy it now?''
``Yes; but you make me feel so funny. I don't know what---it---is.''
His increased and rapid movement filled me with delight; I bounded up and down in response to his thrusts, and felt so queer when, all of a sudden, he gasped for breath, stopped, and I felt a greater and stiffer swelling of his instrument, and then a gush of hot liquid dashed against my womb, which continued running for some seconds. This, Carry, was my first experience of what a man can do for us.
Withdrawing his huge affair---for he since admits he is larger than most men---letting go my thighs---he pressed down upon me, and tenderly embraced me, and said that I had behaved admirably; in future there would be no more pain, and from what he had already experienced he felt sure I was made for the fullest enjoyment that husband and wife could indulge in. After a little fondling, he rose, drew off my stockings, and helped me into bed, immediately following me. On throwing back the clothes to enter the bed, he said he must kiss the dear little hairy thing that had given him such pleasure. He kissed and toyed with it admiring the profusion of hair on my mount, the whiteness and beauty of my belly, and then, baring my breasts, admired, kissed, and sucked them. All this not only excited me, but I could see very well it had again caused his affair to stick out. Seeing that I was timidly glancing at it, he seized my hand, and made me lay hold of it, showed me how the skin covered and uncovered its head; then becoming rampageous, he got on my belly and between my thighs, and again introduced his cock to where it had already given such pleasure. He still rather hurt me, and made me smart for a little while, but as the interior was well lubricated by his former discharge, the penetration was easily accomplished. When up to the hilt, and the two hairs were closely joined, he paused and said---
``We will take it less impatiently this time, that my darling Bessie may enter into all the joys of fucking, for that is what we call it my dear; so I shall go slowly to work until my darling's passions awake and urgently call for more rapid movements.''
He did so, and gradually produced the most lascivious excitement in my whole body. I writhed beneath him in the utmost extasy, threw my arms round his body, and hugged him to me.
``Oh! you are an angel,'' he cried, ``and made for enjoyment. Throw your legs also over my back---there, that is it---and now I will hasten my movements, and we will die away together.''
Oh, the delight he gave me was inexpressibly delicious; his rapid and eager thrusts were as eagerly met by the upheaving of my bottom to reciprocate them. The grand crisis seized us simultaneously, and we sank momentarily exhausted in each other's arms, leaving the dear exciter of such joys soaking within. My dear husband was so pleased, he kissed and fondled me in the sweetest manner, telling me that never woman before had yielded him such intense pleasure, that nature had prompted me to as much enjoyment as if I had been already married a month.
We were locked closely in the warmest embrace; his tenderness and fondling began to have its effect on my passions, and involuntarily I made; some internal convulsive twitchings.
``I feel you, my darling, calling on my instrument for renewed efforts; he will soon respond.''
And, in fact, I felt it swelling and swelling so deliciously that I could not help continuing the interior pressures, although feeling confusedly ashamed of the notice my husband took of it.
``Don't be afraid, my sweetest love, but give way to whatever your passions dictate, and thus you will best please me, and give to yourself double enjoyment. I mean to initiate you into every secret that the rites of Venus possess, and wish that my loved wife should become a devoted votary, and I will do my best that she may revel in all the luxuries of perfect coition.''
We completed this course with even greater abandon than before, and I began to enjoy his embraces beyond anything our imaginations used to suggest. This time he withdrew and lay down by my side, and taking me within both his arms, continued his charming endearments. I never slept that night; I was in a fever of restless excitement. My husband fucked me five times before he dozed off. Towards morning I tossed and tumbled, and could not sleep. Daylight soon came, my restlessness had shaken all the bed clothes off, except a part of the sheet, and turning towards my husband, I perceived that the sheet stuck up over the lower part of his body. Curiosity seized me---I looked at him, and saw he was evidently sleeping. So gently removing the sheet, I beheld the dear instrument of all my last night's joys as well as pains. You know how we used to long to see man's cock when we were at school, and how, when we did sometimes see a boy's limp thing hanging down, we used to wonder what change would come over it, and how. Well, here was an opportunity of examining, at my ease, the wonderful curiosity that had so puzzled us. The last edge of the sheet passing over it touched its ruby head; it throbbed and pulsated to the view. I was afraid this had awakened Fred, but no, he slept as sound as ever. So I gently raised myself on my bottom, and gazed on the dear object I had so longed to see and feel. There it stood up like a pillar, rather bending towards his belly: and what surprised me much was to see a dark strongly wrinkled bag at its roots, with apparently two large balls inside; the hair on its roots spread in dark mass up to his navel, and beautifully bright and curling it was. I approached my lips, and made the action of kissing, without touching it. Whether it felt my warm breath, I know not, but it actually throbbed a response. What a great big thing it was, equally long as it was thick, I did not think I could encircle it with my hand; I longed to try, but was afraid I should waken Fred, and what would he think of me, I blushed at the very idea; but my passions became excited, and too strong to resist the temptation. So first lying gently down again, I very quietly dropped my arm over him and touched his cock, it throbbed at the touch, but Fred slept on. So raising myself again, I very gently laid bold of it. It was as much as I could grasp below the head, but was beyond my grasp at the root; I found it took three of my hands to measure its length from the root to the nut, which stood out in all its redness above. I was almost breathless with excitement, and lost some of my caution. Stooping down, I gently kissed the ruby head, when, before I knew where I was, it was pushed up into my mouth, and my husband's voice said---
``Oh, you dear darling creature! how kind of you to waken me so luxuriously!''
I was horrified at being discovered; and blushing up to the eyes, I hid my face in his bosom.
``Do not be ashamed, my angel, it is now as much yours as mine, and have you not as much right to see, kiss, and handle it? come, don't be ashamed.''
However, I could not face him, and when he tried to raise my head I turned my back. He seized me round the waist, and, before I knew where I was, passed a hand between my thighs, and guided his huge cock to the lips of my cunt, and was in me, I thought further than ever, in a moment. It is true the previous toying with his instrument had terribly excited me, and I had felt that my cunt had become very moist, but I had no idea that anything could be accomplished in that position. I was most delightfully undeceived, for not only did it feel tighter in it, but transferring his fingers from guiding his prick, he touched and played with my clitoris, and produced such excessive lubricity that I went off and spent with a scream of delight before he was ready; but continuing with finger and cock to ravish me inside and out, he soon brought me again to such a pitch of lewdness that I was quite ready to spend with him when the grand crisis arrived. Nothing could exceed the pleasure; my internal pressures, he declared, were the most exquisite he had ever experienced. My clitoris, too, he declared was quite unique. You remember how it used to stick out when excited as far as the first thumb joint, and how, when sometimes I played the husband on your belly, you declared that it actually entered between the lips of your cunt, rubbed against your smaller development, and gave you great pleasure, as indeed it gave me. My husband has often examined and sucked it, and admires it beyond measure. At present he did not withdraw, declaring that I held him so tight he did not think he could pull it out if he tried. In fact, it was involuntary on my part, and I could not help clinging to his dear instrument for the life of me.
Oh, how he fondled and embraced me, making me partially turn my body so that he might kiss and tongue me, and then suck my bubbies; his busy finger all the time tickling and frigging my clitoris. I soon felt his cock swelling so deliciously within me, and he shortly recommenced his rapturous pushings in and out. We made a long, long bout of it, and I am sure that I spent twice before joining him at the last moment, when he died away in a shout of joy that I feared must have been heard by the servants in the house, who long before this had been on the move. After this we lay soaking and enjoying it for more than half an hour, when my husband declared he felt as if a wolf was at his stomach, and that he must have some breakfast. He got up and quickly dressed, desiring me to lie still, and he would bring me some breakfast in bed, and that, while it was getting ready, he would order some warm water to bathe myself with. I felt his delicacy, and loved him for it. The water came, I was much refreshed after using it, and got into bed again, but I felt awfully stiff and done up all that and next day.
My darling husband waited on me himself at breakfast, stimulating me to eat freely as a means of restoring my lost strength; which he very soon put to the test again, for he fucked me three times during the day, and each time he gave me greater pleasure than before. He was just as active at night. And the whole three weeks we stayed at Leamington, he never fucked me less than four times a night, declaring that I had become most perfect in the exercise.
We then came here, our old friend, Mrs. Roberts, having kindly insisted upon our paying her a long visit Fred has been called away suddenly and will not return for a month. I am sure you will pity me, as you know my temperament is too hot to keep chaste so long. You remember Charlie Roberts; you would consider him a child, but he is not so. One afternoon Fred followed me into my bedroom, as was usual, and gamahuched and fucked me on the edge of the bed. I was about to leave the room after he was gone, when on opening a closet, in which my dresses were hung, who should I discover but this same Charlie. I was in a fix.
There was no doubt the lad had seen everything. I spoke kindly to him, and he promised secrecy. In order to ensure it, I determined to have his maidenhead. A few days afterwards my husband left me, and the girls with their mamma and the governess went to town with him, leaving Charlie to keep me company. I went upstairs with him to the drawing-room, and seating myself in a low chair, crossed my legs carelessly, exposing them, and letting the garter and part of the bare skin of one thigh be visible. The effect was what I expected. I saw Charlie's eyes fixed on the exposure, he blushed scarlet, and I could distinctly see his cock swell out under his trousers. In a little while I had unbuttoned them, and, oh, Carry, would you imagine it, I found he had the cock of a man. I could scarcely believe my eyes. He is not quite fifteen, and yet he is almost as large as Fred. Here was a godsend, indeed! I drew up my petticoats, and the gallant little fellow instantly fell on his knees, kissed and sucked my cunt. To reward him, I placed him on his back on the couch, and got on the top of him. I took his pego into my mouth, and pressed my cunt against his face, we devoured each other with our luxurious caresses until we both spent copiously. Nothing was lost, we both greedily swallowed all we could get.
At home he is looked upon as still a child, and I had little difficulty in arranging for him to sleep in a little dressing-room adjoining my bedroom, with which there is a door of communication. He was sent early to bed, but when I came I found him still awake, expecting me, and I had the delicious treat of initiating him into the pleasures of fucking. If you ever wish to enjoy \textit{par excellence} this pleasure, get hold of a vigorous boy who has never had a woman. My good fortune threw into my hands a wonderfully provided youth, whose aptitude, as well as size and powers, it would be very difficult to match. I had already given him several lessons in the enrapturing art when we fell asleep, and now I must mention a little episode, which it would not do to omit.
In the morning I was dreaming of Fred, when I became conscious that something was entering me. I was in that half-dreaming state when it is difficult to be quite certain what is happening, but gradually I became aware that although there was no doubt I was being entered, it was not in the usual way. My husband had frequently of late pushed his prick up my bottom-hole, and as he told me that all husbands did so, I could make no objections. I, therefore, at first took it for granted that Fred, finding my naked bottom in his lap, could not resist the temptation of entering it. I, therefore, humoured him, and so moved my bottom as to facilitate his complete entrance, and began to feel myself the excitement it occasioned, but as I became wider awake, I gradually called to mind that Fred had left me, and that Charlie was my bedfellow. The audacity of the young rogue paralysed me, but his delicious movements had become too nice for me to think of dislodging him. He insisted that he was quite unconscious of his mistake, and that he believed himself buried in the delicious grotto of the night before. It probably was so, for so perfect an ignoramus as he is, although ever so apt a scholar in Venus's rites, he could hardly have imagined there could be any entrance in the smaller orifice. I let him go on, and with his well hung cock in my bottom, and two or three fingers in my cunt, he fucked and frigged me most deliciously, until we both spent in an agony of pleasure. If, Carry, you have not tried this route I strongly recommend you to do so without delay, but you must be well fucked in the first instance, to stimulate a desire in those parts, and your lover must be up to the art of frigging you at the same time, or you can pass your hand under your belly, and rub your clitoris, which was the plan I adopted with Charlie, until I taught him the art of rubbing the clitoris properly. As there is always more excitement when this is done by a male, it is better to have them when one can, but, \textit{faute de mieux}---one can do it oneself with much additional lascivious satisfaction.
To give you an instance of the precocious aptitude of this dear little fellow, I mounted upon him one morning, keeping my body erect, that we might see the delicious instrument in its action of being engulphed and then withdrawn, a most exciting pose which I recommend you to try, if your husband has not already taught it to you. At last, overcome by the lascivious movements, I sank on his bosom. He pressed my bottom down with one hand, and with the other embracing the nearer buttock, introduced his middle finger up the rosy orifice of my bottom, and frigged me in unison with our ups and downs of fucking, giving me the most delicious additional sensations.
What do you think of that for a \textit{tyro}? His discretion, too, is extraordinary. The first night after I sent him to his own bed, he overslept himself. I had not thought of that, and had not looked into his little room before descending to breakfast. His sister was sent to call him. He at once excused himself by saying he had had a bad dream, she came down and told us. In a few minutes he followed, and in the most natural way possible, told a tale of fright, declared he had awoke screaming and afterwards had been so frightened that he could not sleep, and turning to me in the most natural way, hoped his scream had not disturbed me. He never came near me, or appeared in any way attracted by me---a discretion worthy of a man of the world. Oh! my dear Carry, I shall make a great deal of this boy. We have had several delicious nights since, and he improves wonderfully. Splendidly as my husband fucks, Charley already beats him. He is quite as often ready, indeed, oftener, and it is I that hold him back, but there is something still so charmingly infantine in his way of caressing me, and then the lascivious idea he is all my own, and that I initiated him in love's mysteries, adds an inexpressible charm to our lascivious encounters. I feel that I shall almost regret my husband's return, as it will force me to give up this delicious indulgence. Not the slightest shadow of suspicion of our doings is excited in the family, thanks to the very guarded and admirable conduct of Charlie, which is above all praise.
Write to me soon, my dear Carry, and be sure you are as candid as this long, long letter is to you, for the life of me I could not make it shorter. I only hope you will give me one as long, and have as much delicious intelligence for me. I know you too well to suppose that you have not found means as I have done, to try what other men are made of, although you can scarcely have had such wonderful luck as mine. Write then, and write without reserve. Our mutual affection is too sincere to allow of any concealment whatever between two such loving and lewd lascivious friends.
Ever your affectionate friend,
E. BENSON.
Such was the long letter my adored mistress wrote at the time to her school companion. It will be seen that their attachment had led to something more than the usual fingerings and caressings of school girls, indeed, had led them on to the lewdest and most lascivious indulgences that two girls could practise in common, and had first excited their passions and given them the delicious power of pleasing coition they were both so perfect in, for, as I before said, about two years after this time, I was the possessor of both and many and many an orgy we three had together, without the shadow of jealousy on any side. It will be seen that Mrs. Egerton, in her reply, even looks forward to the delicious indulgence, which in the end was happily effected and long continued. The following is her reply---
\gutchapter{THE HON. MRS. EGERTON TO MRS. BENSON.}
How can I ever sufficiently thank my darling Lizzie for her delicious letter, I have devoured its delightful details a dozen times already. I keep it in my bosom, and renew the pleasure of its perusal at every spare moment. \textit{Too long?} Oh! with such a charming power of description, why did you not cover fifty more pages. Never in my life have I enjoyed such an exquisite description of those dear lascivious encounters. How delighted I am at your good fortune in meeting with such a miracle of a boy as that dear Charlie Roberts. Why, he has every quality of a man, united to the charm of extreme youth. What a splendid man he will become, the very perfection of a lover, and already possessing so lewd and lascivious a lubricity. Oh! how I envy you his possession. What luck for him too, to have fallen into the hands of so delicious a teacher as my beloved Lizzie is. Am I not myself her pupil, and were you not my own delicious instructress in all that one of our sex could teach each the other.
You will remember a long-standing engagement entered into, between us made, when we were both so lewd and so longing for the real knowledge of man, and how we pledged ourselves that if either got possession of a lover, we should manage after a while to share him between us. Your description of Charlie Roberts has brought this pledge most vividly to my recollection. I am sure my dear Lizzie will not be angry or jealous when I avow that I long to participate with her in the possession of that darling boy; and if my Lizzie is as of old, I feel certain she will rather indulge and cultivate this propensity than otherwise. Think how easy it will be for us both to arrange the meeting of all three together, because I wish to possess him in common, certain that it will increase the lascivious pleasure of coition. No one will suspect us when we drive out, two women with one man. It will naturally be supposed that one fears the other, and so there will be no danger. See, here I am at once anticipating future scenes, but it is all owing to the extremely exciting and lascivious details you have so vividly given me.
I have no such delicious scenes to depict as those you have so delightfully described to me. My honeymoon passed off in a much more common-place way than yours. Our marriage, which was performed within a day of your own, went off as such events do. My husband was loving, without being very warm. I felt very much as you describe on going to bed the first night, but the discretion or delicacy of my husband, which I could well have pardoned him for dispensing with, left me time not only to get into bed, but kept me waiting there some time. He entered like yours in his dressing-gown, but immediately put out the light and found his way into bed, as best he could. He crept to my side and embraced me tenderly enough, and began to fondle and kiss me, telling me how dearly he loved me, \textit{etc}., but for some time he avoided any indecent liberties. I suppose he thought it necessary to gain my confidence and quiet any alarm I might be in. He might have saved himself the trouble, for in reality I was longing for and at the same time somewhat dreading an attack on my maiden charms. At last, little by little, he approached the object of delight, and eventually begging me not to be alarmed, he mounted upon me and effected the object of his desires. He did not hurt me much, not nearly as much as I expected, nor so much as you seem to have suffered. I deemed it politic to affect more suffering than he really inflicted. Towards the end I had slight scintillations of pleasure, but not worth mentioning; it is true my husband is not so well-armed as yours and Charlie appear to be, and he is also much colder in his passions; for instance, he did not attempt to fuck me again, although I would have been gratified if he had done so; perhaps it was considerate towards me in his idea, but, merely embracing me in his arms, he talked himself and me to sleep.
In the morning he again fucked me, this time giving me something like pleasure, but I was altogether disappointed with my night's experience. It was not such as you or I, my dear Lizzie, had pictured to ourselves, in our anticipations of the marriage night. My husband since has never exceeded twice a night, but he has become more exciting, and has generally made me spend twice to his once, first exciting my passions by feeling all my private parts, and frigging my clitoris, so that I generally have lubricated the passage by my own discharge before he attempts to make an entrance. I find he likes this, and so far it pleases me, because only one discharge would leave me in a state of excitement unbearable. He has never attempted any of those lewder and more lascivious methods, of which you have had such delicious experience. Altogether, I cannot but say I am disappointed. My husband is loving, and very anxious that I should improve my mind in every way. You know I was rather more proficient than usual at school in Italian. My husband speaks it fluently, and as we mean to spend a winter at Rome, was anxious that I should have further instruction. He asked me if my school teacher was a good one, but I did not encourage that idea. You may remember our former master was a Count Fortunio, so handsome and so enterprising that you and I had both formed the plan of having him, and had already put over some of the preliminaries when, unfortunately, he was caught with that impudent Miss Peace, with whom, doubtless, he had accomplished everything. Of course, he was instantly changed for another, and we saw no more of him, to the sad disappointment of our then libidinous hopes. My husband proposed advertising for a master, when I had the happy instinct to tell him that schoolmistresses generally applied to Rolandi, of Berner's Street, for language masters, and that, if he would write or call, he would be sure to get every information. That evening, after dinner, as we sat dozing over the fire in the library---very imperfectly lighted---my husband informed me that he had seen Rolandi, who had most strongly recommended a very gentlemanly man, moving in good society, namely, the Count Fortunio. I started in amazement; fortunately, owing to the half-light we were in, my surprise and confusion were unnoticed by my husband. He said that he had been referred to one or two gentlemen of standing as to the Count's character, that he called upon them, and felt satisfied that I could not be in better hands. You may imagine what an effect this information had upon me. All night long I could think of nothing else. What seemed most difficult to me was the hiding from my husband our previous knowledge of each other. I feared the Count would at once recognise me and claim acquaintance, which was what I most wished to avoid; to you, from whom I have no secrets, I may own it immediately occurred to me that this would be an opportunity (for which I had in heart been longing) of obtaining the services of a lover I could trust. How to manage it I knew not, but chance, that favourer of all wrongdoers, stood me in good stead.
My husband had intended to be present to receive the Count. Fortunately, a letter arrived in the morning requiring his instant attendance in the City about the sale of some stock, of which he was trustee. He begged me to see the Count, and arranged as to hours of attendance, \&c., the more frequently the better. I felt my embarrassment was at an end; the next thing was to avoid letting the servants, those domestic spies on our conduct, see the first meeting. There was a small room off our drawing-room that had no door but the opening into the drawing-room; this was fitted up as a sort of boudoir writing-room, and my husband had pointed it out as a convenient place for me to take my lessons in. Here, therefore, I posted myself, and awaited the hour of arrival, to which he was punctual. He was announced and I told the servants to show him in. I sat purposely with my back to the entrance, apparently engaged in writing, as if I did not know he had approached, until I heard the door of the drawing-room shut. I then rose, turned, and smilingly held out my hand. He started with surprise, but immediately and gallantly kissed the hand held out to him.
``I hope you are not disappointed in finding who is going to be your pupil.''
``Oh, no, certainly not; I did not know you under your married name; but I am so happy to renew an acquaintance which at one time had such charming promise.''
``Stop, signor, I am now married, and it is necessary to be very cautious. I do not wish to deny that I am much pleased to renew acquaintance with you, but it must be with great reserve. Sit down by my side, and be reasonable.''
``Reasonable! and by the side of one whom I so much loved, and from whom I had such hope. Oh! dear Mrs. Egerton, you are surely not going to treat me as a mere master. You would render me miserable if you did so. How can I help admiring one whom I so fondly loved, and with whom I hoped for such happiness long ago.''
Here, having possession of my hand, his other arm was passed round my waist, and he drew me to his lips, and I must own, I reciprocated the ardent kiss he gave me. You remember how handsome he is, and how soft and loving was the expression of his eyes. Well, my dear, to cut matters short, I was so excited that I hardly observed that he had passed his hand up to my petticoats, until I found he had got it on my mount. My passions being excited, and knowing that my husband could not return, and also that he had given strict orders that I was not to be disturbed in my Italian lessons, I gave way unreservedly to the excitement the Count raised. Before I well knew where I was, he was on his knees in front of the low chair on which I was seated. He had thrown up my petticoats, and I felt a long and extremely hard prick rush up my cunt, and begin the most lively action. In fact, he carried me (not unwillingly I must avow) by storm, and made haste to secure the fortress at once, so that I had a very quick fuck, that did not assuage the fire he had raised within me. He has since apologised for his haste, saying that he wished to secure possession of me before I could think of resistance, so as to ensure more facilities of connection hereafter. We had no lesson in language that day, but another bout of love, in which he did his utmost, and with perfect success, to give me the most delicious enjoyment.
In fact, my dear Lizzie, I may say it was the first fuck that thoroughly realised my, or rather our, anticipations of the act. We arranged the line of conduct necessary to be followed so as neither to compromise me or him either. In a short time we had again a delicious fuck. Seated, with outstretched legs, on a chair, he got me to straddle over him, and sink down on his stiff upstanding prick. I have tried this position kneeling, with my husband on his back; but it does not equal the chair fuck. One has so much better a spring from one's feet than from one's knees, besides, the man is brought more face to face, and there is more facility for mutual embracings; but both ways have their charm. I had repeatedly observed that the Count apparently lost his place, and on recovering it, partially penetrated the smaller orifice, which you so picturesquely describe. I thought it accident, and as it hurt, I always put him back, and joked him on his awkwardness. But after I read your dear delightful letter. I became convinced that he had a wish to penetrate there, without the courage to tell me so.
I must confess to you, that our stolen embraces at home had become too unsatisfactory, and the Count had arranged for a private house to be at our disposal. Of an afternoon I drove out shopping, called at Swan and Edgar's in Regent Street, leaving the carriage at the door, walked upstairs, made some trifling purchase, paid for and left it until I should call in an hour; then descending by another staircase, left by the Piccadilly entrance, and taking a cab, joined my expectant lover, where he was waiting for me. There stripping perfectly naked, we enjoyed each other most lasciviously, and practised every act of lubricity. When satiated with our efforts, a second cab conducted me to St. James's passage, in Jermyn Street, from whence I gained on foot Swan and Edgar's in Piccadilly, received my parcel, and rejoined my carriage. Thus no suspicions were excited, either in the household or otherwise.
We have met thrice since your dear delicious letter fired my imagination, and I have seized the occasion to taste the sweets of the neighbouring altar to Venus's legitimate one. After the Count had fucked me twice I turned my back as if wishing it in a way we often enjoyed it, but took care to place my bottom in such a position that the smaller orifice was nearest to his standing prick. Whether he saw my drift I know not, but finding with his finger how conveniently it lay, he plunged boldly forward, and half sheathed himself at the first push. I started with the sudden pain, and should have disengaged myself at once, notwithstanding that I purposely placed myself to receive his prick in my bottom-hole, but with his arms round my waist I was perfectly powerless, and another thrust sent him up to the hilt, but really hurting me most sensitively; I begged him to desist and withdraw, but he said---
``I will remain quite quiet for a time, and you will see that your pain will diminish, and then you will like it.''
I could not help myself, and sure enough he was right. Shortly I felt no pain; slipping one hand down, he began to frig my clitoris, and in a little time, finding by the involuntary movements of my loins that my passions were excited, he began to move very slightly and slowly. I soon found a strange excitement seize me, which increased to such a degree that I almost fainted, when my nature gave down its divinest essence. We have since repeated the new experience, but I quite agree with you in thinking that we must be well fucked first.
The Count is a master of his weapon, which, neither quite so long as you describe your husband's nor nearly so thick at the point, is very much so at the root, and as stiff and hard as iron. I assure you, the wild excess of passion he drives me into is indescribable. You shall experience the delight of his fucking, for, with you and me, there must be no difficulty, diversion, nor jealousy. Nay, I shall try to seduce your husband, with a view to cover our delinquencies. I would offer you mine, but, truly, he is not worth having to a woman who can find better, as my dear Lizzie so charmingly does. We have managed matters so prudently that my husband has taken a great fancy to the Count, and he dines frequently at our house.
We have often talked of you. I told him of your marriage, and of a probability of your eventually settling in London. I marked the sparkle of his eyes at the news, but was silent as to your letter and adventures. It is better we should manage the affair between us when you are here.
So you see, after all, I have not come off so badly, although, I must say, tamely in comparison with the delicious adventures of my dear and charming Lizzie. I think, when we meet, we shall be able to get up parties of the most delightful kind. I even hope we may induce the Count to join you and Charlie in a \textit{partie carr\'{e}e}; what fun and pleasure we should have, and then the delight of exchanging lovers at each bout. Oh! the very idea has set me on fire; fortunately, I am expecting my lover at every moment. I will close my letter with this lascivious picture, and in hopes of some day realizing it with my loved Lizzie, Whose most affectionate and attached friend,
I shall ever remain,
CARRY EGERTON.
Such were these two charming letters, and I may immediately mention now that the lascivious picture dear Carry drew of a \textit{partie carr\'{e}e}---we four the actors---was afterwards realised to the utmost extent of every salacious enjoyment that the most experienced lubricity could suggest.
The Count and I often sandwiched them between us, which they declared to be the \textit{ne plus ultra} of pleasure, while the upper operator gamahuched the unoccupied quim. Nay, these giddy delicious creatures were not satisfied until they had induced us to alternate the joys of coition with each other; but that was rarely the case. These enchanting women were so exquisitely seductive that, while we had them at our disposal, we sought no other source of delight. But I am digressing, and talking of events that occurred long after the period which I am more particularly describing.
The three weeks' absence of Mr. Benson terminated, alas, far too soon; in fact, time flew so quick that it hardly appeared three days when a letter arrived announcing his return for the next day. My heart was ready to burst, but I managed to make no show or mention when Mrs. B. told the news at breakfast Mrs. B. observed that I turned pale, but no one else remarked anything. We contrived to meet for a short time in the middle of the day, and she embraced me tenderly, with tears in her eyes, and looking so loving that my passions became overexcited, and hers too. Notwithstanding the imprudence of the risk, we there and then had a most delightful and salacious fuck; and at night this charming woman allowed me full liberty to do anything I liked; and as often as nature would support us we revelled in a sea of lubricity. How often I cannot say, although my loved mistress declared that I had spent ten times, I am certain she did oftener than that, for neither closed an eye, nor ceased from the most loving embraces. She exerted all the wonderful powers of seduction for which she was so distinguished. Never mortal man could have passed a more intoxicating night of pleasure. We heard movements in the house before we parted with mutual tears coursing down our cheeks.
It was with difficulty I tore myself from her; indeed, I could not have done so if she had not herself risen, and tenderly embracing me, told me to have courage and hope, for, some how or other, we should manage an occasional interview. Particularly cautioning me to be perfectly on my guard when her husband came, she said it would be better if I kept out of the way until after the first interview was over, as it might he too much for me to see him embrace her. I did as she desired. No one noticed me in the confusion of his arrival.
Mamma had insisted upon my returning to my bed in her room, as she was sure Mr. Benson would require the dressing-room. Mrs. B., from policy, objected, saying that there was no occasion, that I had been so quiet she had never once been conscious of my being there, \&c., but mamma had her own way, and I really believe very much to the satisfaction of Mrs. B. herself; for I doubt, if Mr. B. had been aware of my close proximity, whether he would altogether have liked it. Nevertheless, he so completely treated me as little more than a child that I am quite sure he had no suspicion of my having occupied his place so continuously during his absence.
Mr. and Mrs. B. retired shortly after his arrival, doubtless to plunge into all the joys of venery after his long absence, and his wife's supposed privation of them. The idea of that being the case did not so much annoy me as I expected; on the contrary, imagination portrayed them in all the agonies of delight, and actually excited me extremely. All at once, the idea struck me that I might be purposely hid in the closet, behold all their delicious encounters, and when he had left his wife to put herself to rights, and the key was turned upon him, I might then in my turn, fly into my enchanting mistress's arms, and revel in all the joys her well moistened and juicy cunt could give. I determined to propose this to dear Mrs. Benson the first moment I could get her apart from all observation.
I was a little \textit{distrait} in the school-room that day, but an appeal from Miss Evelyn recalled me to my senses. She asked me what I could be thinking of; I held down my head and blushed. Already an adept in dissimulation, I faltered out that it was of herself and of her endearing caresses the day before, which had made me feel so queer all over. In fact, the previous day she had hugged me rather close to her, and kissed me more lovingly than usual, which really had, at the time, inflamed my desires, and given me great hope of matters coming to a more satisfactory termination with her. She patted my check, and kissed me again, saying I was a naughty boy to have any such thoughts, and I must not indulge in them, or she would not love me any more. But there was a sparkle in her eye, and a flush on her cheek, which showed me she was anything but displeased.
At our usual break-up at four o'clock, I went to the parlour to see if, by chance, I could get a secret word with Mrs. B., but found that she and her husband had again retired. I knew what that meant; it set me too on fire, and I flew to the garden where my sisters had gone to play. I gave Mary a hint, which she readily understood, and proposed a game of hide and seek. To prevent Eliza interrupting us, I took up a stone, which I furtively dropped again, and proposed that Eliza should guess first, in which hand I had got it, and if she guessed wrong she was to be the seeker. Of course, she guessed wrong. So we bound up her eyes, and she was to stand behind a tree and count one hundred before she attempted to look for or seek us. We made a detour, and as fast as we could run reached the summer house, which, as all the ladies were in the house occupied, I knew to be untenanted. We entered and locked the door, in an instant I had Mary down on her back on the sofa, my head between her thighs, and my tongue in her cunt, and then on her clitoris. She was as eager for it as myself. A week had passed since the happy day of giving up her maidenhead to me. She had thoroughly got over all the pains and inconveniences of that day, and was as ready for a renewal of what could only be joys now as I was. She spent in my mouth almost as soon as I began to gamahuche her clitoris. Waiting an instant to lick up and swallow the soft and delicious young discharge, I rose, pulled out my bursting prick, and engulphed it in her well-moistened sheath with one rapturous shove up to the hilt, positively taking away her breath by the energy of the attack. I was almost as rapid in coming to a conclusion as she had been. Nevertheless, she died away a second time, the moment she felt the warm gush of my raging discharge. We lay some minutes rapt in the lascivious lap of lubricity. But in our young and unbroken energies, nature soon reasserted her power. I must give my sister the palm. It was the internal pressures of the inner folds of her deliciously tight cunt that first awakened my vigour. Somewhat more slowly we began another love encounter, which speedily became much more rapid and energetic, ending as usual in an extasy of delight, and closing with actual cries of intense pleasure.
It was well we had completed our second course, for we heard the footsteps of Eliza, who, after in vain searching for us near to where we had left her, had at last sought us in the summer house. I had just time to arrange my trousers and unlock the door when she arrived and burst in upon us. She said it was unfair to go so far away, but we only laughed, and proposed that Mary should now seek us. We were standing outside below the mound, tying on the handkerchief, when Miss Evelyn was seen approaching. She came up and noticed the flush still on Mary's cheeks, but we at once told her that we had been playing at hide and seek, and had had a good run, and that it was now Mary's turn to be the seeker. However, Miss Evelyn said she thought we had had enough exercise for the time, and that it would be better to walk gently about to get cool, as it only wanted a few minutes of the hour for renewing our lessons, so we all demurely returned to the house. A reflection struck me that it would be necessary to initiate my sister Eliza in our secrets, and although she might be too young for the complete insertion of my increasingly large cock, I might gamahuche her while fucking Mary, and give her intense pleasure. In this way we could retire without difficulty to spots where we should be quite in safety, and even when such was not the case, we could employ Eliza as a watch, to give us early notice of any one approaching. It will be seen that this idea was afterwards most successfully carried out to the immense increase of my pleasure.
It was a lovely summer evening. After dinner Mr. B., who, doubtless, had no longer any amorous longing, after having twice retired during the day, challenged Miss Evelyn to a game at chess, of which she was a great proficient. Mamma, Mrs. B., and the two girls stepped out into the flower garden, to enjoy the beauty of the evening. Fortunately mamma fancied she felt chilly, and shortly went back again, taking the two girls with her, and setting Mary down to the piano. I seized the happy moment, and drew Mrs. B. to a seat, far removed beyond the hearing of any listeners, but in sight of the windows. There I unfolded to her the plan I had proposed to myself; she smiled at my precocious ingenuity, but added it would not be safe to leave the closet door open, even partially, as by chance Mr. B. might open it, and that would never do; but she might lock me in---or rather I might do so from the inside.
``Ahh! but then I want to see it all---it is so exciting to see Mr. B. working into that divine body of yours.''
She laughed heartily at my remark, and said I was a lewd lascivious young rascal---adding:
``But are you not jealous to see another in possession of me?''
I admitted that that was my first impression, but on thinking over it, I had become convinced I should like her and enjoy her all the more lasciviously if I were a witness to their love contests, but I must be able to see them.
``Well! can you not bore a couple of holes an inch and a half apart, below the middle panel and cut a narrow slit from hole to hole? I will take care to place myself in a proper position, and do my best to gratify your premature lubricity. My darling boy, you progress wonderfully, and make me proud of my pupil.''
Seeing she took it thus kindly, I said---
``Do tell me, my beloved mistress, how often he has fucked you to-day?''
``Will it please you really, my dear Charlie, to know that?''
``Oh! yes, so much.''
``Well, then, six times in the morning, and four before dinner. He was bursting with desire, and could not hold. He spent twice before giving me time to come once, but then you know, my dear Charlie, how actively you had been employing your time all the previous night, you sad rogue that you are.''
``Did you enjoy it much, my dear Mrs. B.?''
``Why, if I must tell you, you little curiosity box, I did; you know how powerfully my husband is hung, and loving him as I do, it is impossible to undergo his powerful and lascivious embraces without feeling all one's libidinous passions stirred up within me, but even while in his possession, my dear boy, I thought of your young charms, and the fierce delights we had enjoyed together last night. My husband little imagined it was of you, not him, that I was thinking and stimulating myself to wild upheavings of voluptuous movements, while he was revelling in all the lubricity of his own passions, and fucking me to my heart's content.''
``Oh! how delicious! my angelic mistress,'' I cried, ``the pleasure of your vivid description almost makes me faint with desire---oh! that I could possess you at once.''
``You must not think of that, my dear darling boy. We must manage it to-morrow; I shall go into the house at once, and occupy your mother's attention, do you get a gimlet and chisel, slip up at once to my bedroom, and prepare a peep-hole for to-morrow; be careful to put it low down, below the projection of the middle panel of the door in which the lock is placed, and take care to remove the pieces of wood you take out. I shall put the key inside of the door. Your sisters always take two hours at the piano after your midday meal, our luncheon is served at the same time. Mr. B. is sure to require my attendance in my room after that, but I shall detain him by some excuse till I observe that you have disappeared, and after giving you sufficient time, we shall follow, and you shall have the extraordinary satisfaction you require; but above all remember---not a movement to betray yourself until my husband leaves and I have locked the door behind him.'' So saying, she pressed her lovely hand on my stiffly excited member, rose and joined mamma. I lost no time in following her advice, and happily executed all I wanted, and returned unconcernedly to the drawing-room, without my absence having occasioned any remark Next day I got safely to my hiding-room, and had comfortably stowed myself away in such a position that the opening I had made was on a level with my eyes, before they arrived. She, dear creature, anticipating my vista, had merely slipped on a dress, without a corset, and told her husband that he was so insatiate that she was obliged to be ready at a moment's notice to satisfy his inordinate passion, so she had only to take off her gown to be at her ease. ``Most admirable, my darling wife, but drop off every thing, and let me contemplate, at my ease, all the beauties of your exquisite body.''
No sooner said than done, and my lovely mistress stood in all the glory of her magnificent and beautiful naked form. He kissed and fondled her from head to foot, laid her on the bed and gamahuched her till she squealed again with pleasure. Then pulling out his magnificent prick, he plunged it into her delicious cunt at a single bound, evidently giving her the most exquisite delight, as was evidenced by the instantaneous clasping of him with her arms and legs, and the rapid wriggling of her backside. They soon ran a first course, but Mr. B. remained engulphed in the closely fitting sheath of his salacious wife. She evidently exerted herself more than usual, both for her own pleasure as well as to give satisfaction to me, for once when she turned her head in my direction I caught her eyes, and she smiled, giving a still more vigorous heave than usual, and showing me all her cunt at full stretch with the noble prick in it. I was ready to burst. At last their bout was over for the present; Mr. B. withdrew his prick, all slimy from its sheath, pendant, but still full of size.
Most extraordinary! I would have given a good deal to have dared to rush out, put it in my mouth and suck it dry, I can hardly describe how strongly this desire took possession of me. It was the first promptings of a passion I have since often indulged in, where I have met with companions with whom I could join in orgies of both sexes. Mrs. B. professed to be dead beaten by the constant and frequent renewals of these interviews in addition to night work and lay perfectly still, while he performed his ablutions and readjusted, his habiliments.
``Fasten the door after me.'' said he, as he ardently pressed her form in his arms and kissed her. She had continued stretched on the bed, exactly facing me, with legs widely extended, so as to show me the whole of her lovely cunt, which I could see still panted under its late excitement. My charming mistress told me it was palpitating not for what had passed, but for what it was waiting for. She rose at last and closed the door, turning the key upon her husband. She then approached the bidet to purify herself, but I bounded from the closet, seized her in my arms, dashed her back on the bed and immediately glued my lips to her glowing and foaming cunt, with all the froth and spending of her husband oozing out. I greedily devoured it, and raised her to such a frenzy of lewdness that she dragged me up and cried, frantically---
``For God's sake fuck me---fuck me!''
Of course my cock was bursting to do so; with one shove he was sheathed to the cods; my loved mistress spent with that alone, so highly was she excited, not only by the preparations, but as she herself acknowledged to me, by the idea of the instantaneous infidelity to her husband, at the moment after he had just fucked her---such is the wild imagination of women when they give way to every libidinous thought. It would have been exactly the same if some equally fortunate lover had been awaiting my retiring from the field. The idea of success in deception is a passion with them, and they would almost sacrifice any thing to obtain it. Before I could arrive at the grand crisis, she was again ready, and we died away in an agony of blissful lubricity---she held me, as usual, so tight that I never thought of withdrawing from the folds of her delicious cunt, but lay still enjoying the never ceasing compressions of its velvety folds, which sometimes really had almost the force of a vice. I was rapidly ready for a second bout, which, like the first, ended in extatic joys, beyond the power of description. My charming mistress thought I ought now to desist, but pleading my forty hours' fast (for, of course, she knew nothing of my fucking Mary), I begged her to allow me to run one more course.
``Then, my darling Charlie, you must let me turn on my side, for I am so heated with your weight and my husband's that I must have some relief, but there is no occasion for you to withdraw, leave me to manage it.''
With an art quite her own, she accomplished her object, her splendid buttocks' pressing before my eyes against my belly fired me immediately. My cock swelled and stood firm as ever. Then passing an arm round her body, I used my fingers on her excited and stiffly projecting clitoris. We had a much longer and more voluptuous fuck than before; nothing could exceed the delicious movements of my divine mistress; she twisted her body so, that I could suck one of her bubbies, while I fucked and frigged her; she spent with such a scream of delight that I am sure she must have been heard in the house, had it not been for the inner baize door to the room. She continued throbbing so deliciously on my prick that I began to flatter myself I should obtain a fourth favour, but she suddenly bolted out of my arms and out of bed. Turning round, and taking my whole prick into her mouth, and giving it a voluptuous suck, she said---
``No, my loved boy, we must be prudent if we mean to have a repetition of these most exquisite interviews. You have given me most extatic pleasure, and by moderation, and running no risk in too long indulgence of our passions, we may safely manage to enjoy similar interviews every day. Get into the dressing-room, remain there until I leave my room and pass your door. After I have seen that no one is near, I will cough twice, wait a minute longer, then quietly leave and descend by the back stairs.''
All was happily effected, and for the week longer they remained with us, I found means to repeat the charming lesson every day, without raising suspicion in any one's mind.
At last this admirable woman departed. It was with difficulty I could bear the scene, but I gulphed down my feelings as best I could. She had become a universal favourite, and all regretted her leaving, so that my distress was not noticed in the general regret. It was more than two years before fortune favoured me in again meeting with this charming woman. And then we saw very much of each other, both alone and with other congenial spirits, of which, perhaps, I may hereafter write a detail; but at present I have got events to relate that followed fast on her departure.
I have said that Miss Evelyn had been gradually growing more familiar in her manner of partially caressing me. She drew me closer to her, almost invariably placing her arm round my waist, frequently kissing and pressing me against her firm and well-formed bosom. This had frequently an evident effect on my lower person, even while I was kept less excitable by the constant relief my passions were obtaining in the arms of my adored Mrs. B. Now I no longer had that vent, for the little relief I could get at rare intervals from my sister Mary was as nothing, after the constant exercise I had been provided with for a whole month. Ever since I had practised that little deception on Miss Evelyn by attributing to her embraces the evident distraction I was in on the day of Mr. Benson's return, she had increased her pressures of my person, and could not but feel my stiff prick throbbing against her thigh, while she closely pressed my body against it with her arm. I often noted the increased sparkle of her eyes and changes of colour on her face when she kissed me, and I put up my hand and caressed her cheek. At times she would push me suddenly away, and beg me to resume my seat; frequently she would quit the room in an agitated manner, till this led me to suppose that an internal conflict was going on, and that passion urged one course, reason another. Remembering the sage advice given to me by my loved and beautiful mistress, Mrs. B., I resolved to play the part of an innocent ignoramus, and let her own passions develope and produce the result I so longed for. I doubt if I could have held out but for the relief I found in dear Mary's embraces, who, each time we could manage to meet, became more and more attractive, and more capable of giving and receiving pleasure. We had some difficulty in keeping Eliza blind to our doings. At last Mary agreed to initiate her into gamahuching, and to tell her I did so to her when we shut ourselves up together, and that if she would keep the secret, I would do the same to her; but that it was necessary that one should keep watch while the other amused herself with me, for fear Miss Evelyn should chance to come. Mary proceeded to gamahuche her, which delighted Eliza beyond measure; indeed, although a year and a half younger, she speedily showed a developement of passion superior to Mary. At first I only gamahuched her, letting her play with my prick as I did so, but not attempting to instruct her in the art of insertion into her charming little quim, which already showed symptoms of a hairy growth on her well-formed and very prominent mount. When I had done enough in this way, Mary, who had previously been fucked by me, returned, and Eliza took up the watch, while I appeased in Mary's deliciously tight cunt the thirst that gamahuching Eliza had raised.
It was thus I could more coolly await the gradual approximation that Miss Evelyn's evident passion for me was bringing about. That she struggled against it was evident, but passion was gaining the advantage, as was shown by her nervous tremblings and sudden clutches, drawing me up to her parched lips, and sometimes pushing me away with a shudder that shook her frame and paled her lovely cheeks. I fancied that nature had been too much for her on these occasions, and that in reality the sudden clutching was the approach of love's crisis, and that when she shuddered, and suddenly repulsed me, she was discharging. It was evident this could not continue. At last the happy day for which I so longed arrived. Mamma was going to go to the town, and taking my two sisters with her, to get something or other for them. She invited Miss Evelyn to accompany her, but the latter declined, on the excuse of an alleged headache. In truth, the violent nature of the conflict going on between her passions and her prudence had visibly affected her health; she had become pale and anxious-looking, and my mother was somewhat uneasy about her. She told her not to occupy herself too much with my lessons that day, and only give me work for an hour in the morning and an hour in the afternoon, and begged her to take a quiet stroll in the garden, and rest as much as possible.
On leaving us, she cautioned me to be as gentle and obedient as possible, as Miss Evelyn was poorly and out of spirits. Mamma and the girls departed. Miss Evelyn, almost as pale as death, and quite visibly trembling, falteringly begged me to go to our school-room and study the lesson she had given me the previous evening, saying she would join me shortly. I went, but no lesson could I do that day. The evident agitation and apparent illness of Miss Evelyn distressed if not alarmed me; I was still too inexperienced in her mind. It was a phase of woman's nature which I had as yet no knowledge of. I had merely a vague kind of idea that it all tended to the ultimate gratification of my libidinous hopes, and I only held off to a certain extent in obedience to the counsel my loved Mrs. Benson had so wisely impressed upon me, and was waiting in lively, hopes of the result I so ardently wished for.
At last Miss Evelyn joined me, her eyes were swollen and red as if she had been weeping; my own filled with tears when I saw her, and I approached, hesitatingly, and said---
``Oh, my dear governess, I am so grieved to see you look so poorly. Oh, do nothing to-day, and I promise to work twice as hard to-morrow.''
At the moment I really felt quite distressed at the sad expression of her features. For an instant she smiled languidly, then, by some compulsion of feeling, she seized me in both arms and drawing me to her bosom, covered me with kisses; her eyes became almost perfectly brilliant.
``Oh, you dear, dear, darling boy, I love you beyond expression. Kiss, oh, kiss me! my darling! and comfort me, because I love you all too well.''
Then, again, there was a change, she seemed to fear she had said too much, and turned away her head and tears started to her eyes, but her arms did not relax the embrace in which she held me. I was deeply moved at her evident agitation. I thought she was really ill, and suffering greatly; so I threw my arms round her neck, kissing her tenderly, and weeping myself, tried to comfort her in my inexperienced way, sobbing out---
``Oh, dear, dear Miss Evelyn, do be comforted, I so dearly love you that it makes my heart bleed to see you so unhappy. Oh, let me see you smile, and do try not to cry so. Why are you so unhappy and low spirited? Oh, that I could do anything to make you happy?'' And redoubling my endearments, she again turned her lovely face to me. Again there was the unnatural fire in her eyes, and a hectic glow flushed her cheek.
``You darling angel of a boy; it is you that makes me so unhappy.''
I started back in surprise.
``I make \textit{you} unhappy! Oh! Miss Evelyn, how can that be, when I adore the very ground you stand on, and love (\textit{sobbing})---love (\textit{sob})---love you more than anything in the world.''
She seized my head in her two lands, glued her lips to mine, gave me a long, long kiss of love; then, pressing me to her bosom---
``Oh, say that again, my loved, my darling boy; it is the love I feel for you that is breaking my heart, but I can resist it no longer. Will my Charlie love his Evelyn always as he does now?''
``Oh, how could I do otherwise? I have worshipped you from the first moment of your arrival, and have had no other idea. What can I do to prove it---try, oh, try me. I have never breathed a syllable of my love for you, even to yourself, let alone other people.''
Her eyes, sparkling with passion, were searching the depths of mine, as if to fathom my thoughts. I, too, began to feel my amorous passions excited by her warm embraces and kisses. She held me tight to her body, and could not help feeling the hard substance that jutted out against her.
``I believe you, my Charlie, and will trust you with my life---with more, with my honour! I can no longer resist my fate. But, oh! Charlie, love me always, for I run a fearful risk in loving you as I do.''
She again drew me to her lips, my hands clasped her neck in a close embrace. Her hands wandered---pressed upon my throbbing prick. With trembling and hasty fingers she unbuttoned, or rather tore open, my trousers, and her soft fingers clasped my naked instrument.
``Oh, I shall die, dear Miss Evelyn; what must I do to make you happy?''
My apparent ignorance could not but please her. She sank back on the long low chair on which she was seated, apparently accidentally drawing up her petticoats with her hand in falling back. I threw myself on my knees, and pushing her petticoats further up disclosed the rich, dark, curly beauty of her mount. She covered her burning face with her hand, while, pressing my head forward, I began pressing her beauteous cunt, sucking it without daring to lick her clitoris. She tried to push me away---``No! no! I must not.''
But I suppose my proceedings fired her passions still more, for she was quite moist and juicy, and I have no doubt had already had one discharge while embracing me so warmly. She suddenly said---
``Come then, my loved boy, and I will be all in all to you.''
Drawing me up---nothing loath---I was soon extended on her belly, with my stiff-standing cock pressing against her cunt. I had still the prudence not to show any knowledge of the act. I sighed deeply---
``Oh! my loved Miss Evelyn, do help me, I know not what to do.''
Her hand glided down between us, she guided my glowing instrument between the longing lips of her delicious cunt. I pushed, and buried the head and two inches of its body at the first thrust. The second brought it against an unexpected obstacle, for it never had struck me that Miss Evelyn was a virgin. I pushed hard at it.
``Oh, Charlie, love, be gentle, you are hurting me very much.''
Knowing that the best way would be to excite her by short shoves, without at first trying to go further, I did so, and she began to feel all the raging desires that so formidable a prick as mine must excite, when moving between the soft velvety folds of her tight and juicy quim. I held myself in, and continued my proceedings until the convulsive movements of her loins, and the increased pressure of the folds of her cunt, showed me that the crisis was approaching, and she was about to spend. She hugged me close in her arms, and at the moment of spending involuntarily heaved up her bottom. This was the very moment I was with difficulty waiting for. I retired a little and plunged forward with irresistible force. I burst my way through every barrier, up to the very roots of my prick. The attack was as painful as unexpected. Miss Evelyn gave a shriek of agony and swooned away. I at once improved the opportunity, and thrusting in and out with the utmost vigour, broke down every obstacle, and enlarged the opening by side movements as much as possible, while she was insensible to the pain. I then died away myself in an agony of delight. I lay soaking within the delicious sheath until her convulsive shudders and short sobs showed that my now fully deflowered mistress was recovering her senses. The thought of the unexpected victory I had won had already begun to make my cock stand again, although it was still comparatively soft. I could feel an involuntary pressure on it, as she came to a full consciousness of our position. She threw her arms round my neck, gave me a most impassioned kiss, and then sobbed and cried as if her heart would break.
It is a curious idiosyncracy of my nature to be most libidinously excited by a woman's tears, and although I really suffered to see her in such grief, it stiffened my prick to its utmost dimensions. I tried to comfort her with words, but she sobbed, sobbed on. I suddenly thought that a renewal of action might bring about a revulsion of feeling, and began vigorous movements. She sighed deeply, but I could tell by the nervous twitchings of her loins that her passions were being excited. They soon decided the contest. She threw her arms round my waist, and pressed me to her, devouring my mouth with her kisses. Nature prompted her movements, and in a very few minutes we both poured down a plenteous offering on Venus's altar. She shook and trembled as she felt the warm gush within her, and squeezed me with all her might to her bosom. We lay in a trance for some ten minutes, my charming governess fainting with love, and giving my delighted prick the most luscious pressure, which speedily fired him to new efforts. Miss Evelyn herself was most amorously excited, and we again dashed on love's delicious path---to end, as usual, in the death-like swoon of satiated passion. When we came to our senses, my loved mistress, embracing me tenderly, and throwing her eyes up to heaven, said---
``Oh, my dear darling boy, you made me suffer horribly at first, but I have been in heaven since. Oh, how I love and adore you. But we must rise, my Charlie, we may be discovered. We have, in fact, run great risk, as the door has not been fastened.''
I rose, and withdrew my prick from her reeking quim, which seemed by its close pressure to let me go with regret. I found it was all bloody.
``Stop, Charles, let me wipe it with my handkerchief, lest it stain your shirt.''
She did so, and folding it up and placing it in her bosom, said---
``I shall keep this precious relic as a memorial of the sacrifice I have made to you, my loved boy. Ah! Charlie, you cannot yet understand the value of that sacrifice and the risk of ruin I have run for your sake. I love you as I never loved anyone before, or can ever love again. My honour and happiness are now in your hands, and it is on your discretion they rest. Be careful never to exhibit any liberty of conduct towards me or to mention to anyone what has occurred.''
It may readily be imagined I gave her every assurance on that head, and told her I loved her too dearly, and was too grateful for the extatic happiness she had taught me how to enjoy, for any chance of betrayal to take place through my indiscretion. She embraced me tenderly, told me to go straight to the garden, that she must seek some repose after all that had happened, and we should meet again at midday meal.
I did as desired, full of sweet thoughts at the exquisite delights she had afforded me, and already longing for the afternoon school hour to renew the enrapturing union of our souls and bodies. Miss Evelyn did not come down to her luncheon, but had something sent up to her room. However, she joined me in the school-room at two o'clock, as usual. She was very pale, but embraced me tenderly, and was very endearing. Of course, I immediately became excited, and very enterprising, but she gently repulsed me, and requested that I would leave her quiet that day, as she felt not only exhausted, but in pain, and would be all the better for perfect repose. I begged hard to be allowed some slight favours, if not all, but she was inexorable. Finding that I could neither do any lessons nor be quiet, she said---
``Then we must go into the garden, I think the fresh air and a gentle walk will do me good.''
It instantly occurred to me that if I could draw her away to the summer house, I should have a better chance of succeeding in again enjoying her delicious embraces. Accordingly, when she went up to her room to put on her bonnet and shawl, I possessed myself of the key, to be prepared for my chance of success.
We walked about the flower garden for a time, Miss Evelyn taking my arm, and most lovingly conversing with me. She walked somewhat stiffly. We sat down for a rest, shortly she felt the heat of the sun too great, so I proposed a walk in the shaded shrubbery. I kept prattling on, so as not to let her see how far I was leading her away, she appeared surprised that we had got so far, when we came in sight of the summer house.
``Oh! Charlie, my dear, I am afraid it will fatigue me too much to walk all the way back without rest and we have not the key.''
``Sometimes it is left in the door, I will run and see.'' Off I bounded, slipped the key in the lock, and ran back to say it was there, she followed me in, and sank on the long backless sofa, which had already served me so often. I begged her to extend herself at length. I placed pillows for her head, and drew a chair for myself near her. She did not appear to have any suspicion of any act on my part, but lay down on her side. She took my hand in hers, and we began a conversation, very interesting, in as much as it was how we should regulate our conduct, so as not to raise any suspicion of our amorous connection, and also of how we should manage to meet from time to time.
``You, dear boy,'' she said, ``I cannot now live without the comfort of your embraces, but you must remember, in my dependent position, discovery would be my ruin. I rely on your silence and discretion, and if I am as dear to you as you, my adored Charlie, are to me, I may safely trust to you.'' I threw my arms round her neck, and told her I loved her all too dearly, and longed too much to return to her endearing and delicious embraces, for her to have any fear of my committing either her or myself. She fondly embraced and kissed me. I became fired with passion. My hand wandered, her position only enabled her to make a feeble resistance, I reached her beauteously covered mount, she murmured supplications to be left alone, and held her thighs close together.
She was not aware of my knowledge of the parts, so inserting my finger into the upper part of the lips, I reached her clitoris, and began rubbing in and out, purposely, in an awkward way, but taking care to hit the right point.
``Charlie, my Charlie, you must not do that---I---I cannot bear it.''
At the same time she threw her arm round my neck and drew me to her lips, which glued themselves to mine. I felt her thighs yield and open. I immediately improved the occasion, and began frigging her with my middle finger up her quim. Her passions became inflamed.
``Come then, my darling boy, to my arms, I cannot resist you longer.''
In an instant I was unbuttoned and had my trousers down, and was between her legs almost before she had concluded her sentence. The excitement of my caresses had moistened her juicy cunt, and the head of my prick entered without any difficulty. In my ardour I was about to rush on with a vigorous shove, when she implored me to be more gentle, as she still smarted from our morning encounter. Moderating my movements, and gently insinuating my stiff instrument, I gradually made my way up to its utmost limits, and hardly occasioned even a grimace of pain. Here I stopped, leaving it sheathed up to the root, and making it throb from instant to instant. Then seeking my loved Miss Evelyn's mouth, our lips and tongues met. Her arms round my waist became tighter in their embrace. The delicious folds of her luscious juicy quim began to throb and press on my excited member. Allowing her to become thoroughly excited, I waited until she actually quite unexpectedly yielded down her nature, and spent profusely, to the exquisite pleasure of my saturated organ. I still held all off, to give her time after the delight of that spend, which was probably the first of unalloyed extatic pleasure she enjoyed; for as I was an inactive participator, there was nothing to cause any action on the still raw edges of her broken maidenhead. Her internal pressures were most exquisite. Our embraces with tongues and lips were like the billing and cooing of doves, and very rapidly brought her again to a raging point of desire. I then began with slow and gentle movements, drawing my prick slowly nearly all the way out, and then as slowly driving it up to the hilt. Her previous very copious discharge had so oiled the delicious folds of her cunt, that no pain was felt, only the intense pleasure. At last it became overpowering; her arms were thrown round my waist, and her legs were involuntary cast over my hips. Nature prompted her to the most delicious movements of her bottom; she met my forward thrusts, and responded to them in the most libidinous manner.
``Go on, go on, dear Charlie---faster!---faster!''
I wanted no spur. Fast and furious grew our movements, until at last, with a mutual cry of delight, we sank in each other's arms in the blissful extasy of the most complete enjoyment. It was several minutes before we regained our senses, and both our organs of generation were pulsating, the one within the other, in all the luxury of satiated passion. With her beauteous legs still thrown over mine, she moved her arms to my neck, kissed me voluptuously, and mingled the sweetest accents of gratification with the most endearing caresses and flatteries. I lay, as it were, in the paphian bower of bliss, in a state of exquisite sensations quite impossible to describe. It seemed even a greater pleasure than the more active state of delight we had been to. I could have lain so for hours, but for that excitable prick of mine, whose sensibilities were far too rapidly set in motion by the luscious pressures of that most delightful cunt in which it lay engulphed. It had gradually resumed its pristine firmness, and was now at full stand, throbbing impatiently for further combats. I began to move. Miss Evelyn said---
``Oh, my Charlie, you must cease, my dear boy; we must not only be prudent, but consider your youth and health. Do, oh! do! my dear boy. Oh!---pray cease.''
Her words were cut short by the increasing passion that the vigorous movements of my prick occasioned to her whole system. She could resist no longer, but with arms and legs closely embracing me, and devouring me with kisses, she threw herself into the fight, and with body and soul so seconded me that we died away in screams of delight, and sank quite insensible in each other's arms.
It was many minutes before we recovered speech. I still lay entirely embedded in her most exquisite cunt, and would have liked to have continued in her delicious embrace. But Miss Evelyn so imploringly beseeched me to cease for this time, and pointed out how necessary prudency was, if we ever wished to meet again, that I felt compelled to raise myself from her body. But, in doing so, I slid off downwards, and before she could prevent me, I glued my lips to the open pouters below me, and greedily devoured all her delicious discharge, and did not desist until I had so licked her clitoris as to make her spend most copiously again. At first she had tried to resist, saying---
``Charlie, what on earth are you at? You must not, my dear boy, it is dreadful.''
But, as I roused her passions, her hand, instead of trying to draw away my head, held it firm and pushed it well against her throbbing and delicious quim, her thighs closed against the sides of my head, and she almost swooned away with the extasy of her discharge. I greedily swallowed it and rising completely, took her in my arms, and placing her on her bottom, sweetly kissed her.
``Oh, what a charming creature you are, my beloved Miss Evelyn, I adore you from the sole of your feet to the crown of your bead.''
``But you, my beloved Charlie, have more than justified my imprudence. You have given me a joy which I could never have dreamt of. I am yours, body and soul; do with me as you like. I, too, adore the very ground you tread on.''
We continued exchanging the sweetest vows of affection, until, seeing my prick rising to its usual stiffness, she said---
``Oh, my darling, you must put this away; it would be most imprudent to continue any longer. Now, let me button it up.''
First stooping and kissing it, she put it into my trousers with some difficulty, buttoned me up, and we strolled towards the house.
Our conversation turned on our chance of fresh encounters. She begged I would not think of attempting anything of the kind next day, and she would try and arrange for the day after, although my sisters were terribly in the way.
I suggested she should keep me in as when she flogged me, nay, indeed, she should flog me in reality if she liked.
She laughed at my idea, but said something might be done in that way as a blind. So I said---
``I will neglect my lesson on purpose to furnish an excuse.''
``We shall see---we shall see. Meanwhile, remember to be very prudent.''
We reached the house; she retired to her room until mamma returned. Very kind inquiries were made, she said she had suffered severely from headache, but, on the whole, felt better and hoped that a good night's rest would put her all to rights. We all retired early, both mamma and the girls were tired with their drive and shopping. I had resumed my bed in the little dressing-room, and went to sleep with thoughts of my delicious day's doings, to dream of re-enacting them with every amorous excess that the utmost lubricity could suggest.
The next day Miss Evelyn began to resume her former looks---the struggle was at an end. She was very gentle in her manner, and seemed even more affectionate than usual to my sisters, who, fancying she was not very well, were attentive, rather trying to anticipate her wishes than following them.
There was rather a greater appearance of reserve than previously in her manner to me, but when I went up to her to repeat my lessons, there was a warmer clasping of my waist and a suppressed manner that showed she was restraining her desire to press me to her bosom. Her face slightly flushed, and she turned her beautiful eyes upon me with such an endearing expression of affection that I could have thrown myself into her arms but for the check upon my ardour which her own reserve imposed upon me.
Nothing more took place between us that day. At our usual hour of recreation, from four till five, Miss Evelyn retired to her room to repose after the efforts of restraint that she had put upon herself all day, and left us to ourselves. I need not say an immediate resort to the summer house followed. There, first deliciously fucking Mary, and then gamahuching Eliza, with the addition of gently introducing, at the same time, a finger a short distance up her quim, I finished off with another voluptuous fuck with Mary. I thus was enabled to bear the bridle Miss Evelyn put upon the indulgence of my appetite in her person, and was apparently more reasonable than in reality. She again, on the second day, failed to give me the opportunity I so longed for. Thinking she might hesitate, from fear of discovery, and the fact of having no apparently reasonable excuse of being alone with me, I determined to play the idler next day in the afternoon. On being called up, I had done nothing. Miss Evelyn looked grave, but blushed deeply at the same time.
``What do you mean, Charlie, by this idleness? Go, do your lesson, or I shall be obliged to punish you.''
She took me by the arm, and gently pressed it as she told me to resume my seat. At four o'clock, of course, my lesson was as far as before from being done.
``Mary and Eliza, you can go into the garden. Charles will remain until he finishes his lesson, or is punished for his idleness.''
They left and Miss Evelyn locked the door after them. Then we flew into each other's arms, and indulged in the most endearing caresses for a very few seconds. I had been in a state of most violent erection for some time, so that my hand was up her petticoats immediately. I gently pushed her back on her low long easy chair, and kneeling in front, first thrust my head between her thighs, and taking a glance at her beautifully haired cunt, already all moist and juicy, showing that she was as ready as myself, I gamahuched her until she spent in my mouth, and sucked the delicious liquid most greedily. There was something peculiarly sweet in her spend, and my tongue sought the innermost lining of her delicious quim as far as its limited length would admit, that I might not lose a drop of her exquisite nectar, worthy of the gods. The excitement I occasioned her was almost too much for her to bear, she drew me up, saying---
``Oh! Charlie, my angel of a boy, come, oh, come to my arms.'' I raised myself up, threw myself into her arms, and in a moment I was engulphed up to the cods in her exquisite and throbbing cunt; she closed upon me with arms and legs, we were both too violently excited to pause for any of the more voluptuous movements of less violent desires, but rushed on in passion's wildest extasy, both far too eager to think of any restraint, and with the utmost vigour on both our parts, we ran our first course with great rapidity. My adored Miss Evelyn had quite got over every feeling of pain, and could not but be delighted with the heat and vigour of my attack. We both died away together, at the extatic moment pouring down a mutual flood of spunk to cool the inflamed members that had the instant before been in such tumultuous action. Darling Miss Evelyn hugged me close to her bosom, and threw her beautiful eyes, screaming with passion, up to the ceiling, as if to thank heaven for the joys she had felt. Our lips then met and glued themselves together in one long, long kiss of love, which quickly lighted up our lust; she was as eager as myself, and we had another vigorous encounter, ending in all the agonies of delight, as before. Then after a longer interval of the most endearing caresses and fond accents of murmured love, we ran our third course, with more abandon---lengthening out our exquisite sensations, by slower and quicker movements and pauses between---in which my beautiful governess began to develope an art in which she shortly became even superior to the more experienced Mrs. Benson, who had so charmingly initiated me into love's mysteries.
There was a peculiar charming and endearing softness in the manner of Miss Evelyn most winning and most exquisitely attractive. It was evidenced even in her mode of handling my prick; without grasping it, her hand appeared to pass over it hardly touching it, but in so exciting a manner that after any number of encounters, she could raise it by her fairy touch in a moment. Our third encounter lasted quite half an hour, and we sank in the death-like luxury of discharge, our whole souls seemed to exude with the exquisite distillation of our seed. We had long before regained our senses. I was still engulphed in her delicious cunt, but she begged me to relieve her of my weight. We rose, she shook her petticoats down, and assisted me to arrange my trousers. I then sat down and took her on my knee. Our lips met in a mutual warm kiss of gratified passion. She thanked me for the joys of paradise I had given her---and for my discretion in procuring an excuse for our meeting. She acknowledged that she had been as impatient as myself, but was obliged to take every precaution against raising the slightest suspicion in the house.
``You must always remember, my darling boy, that for me discovery would be my ruin for ever. I risk every thing to possess you, my beloved boy, I would care little for discovery, if it would not also separate us for ever. That idea, my adored Charlie, is insupportable, I can no longer exist without you.'' Here she threw her arms round my neck, and burst into tears.
I have already described the effect of tears on my unruly member, which, while I was consoling and vowing eternal attachment to my loved mistress, burst from its bonds and stood out in all its glory. I took her soft and beautiful little hand, and laid it on it. She grasped it tightly, and looking at it, while smiling through her tears, said---
``My Charlie, what a great big thing it is. I wonder how it could ever get into me, without killing me.''
``You shall soon see that,'' said I, and changing places, I laid her down, lifted her petticoats and was into her in a moment. She begged me to proceed slowly, and to lengthen out our pleasures as much as possible. We had a most glorious and truly delicious fuck; my lovely and charming mistress giving me most extatic pleasure by the exquisite pressures of the internal folds of her delicious and lascivious cunt.
We lay enraptured for long after we had spent, and then resumed our sitting position, and arranged every thing in order, as the time for the return of my sisters from their hour of recreation was close at hand.
Our conversation naturally turned upon how we should arrange for our next meeting. Miss Evelyn insisted that we must not think of meeting more than once in three or four days, as otherwise we might raise suspicions fatal to our meeting at all. However reasonable this was, I raised an outcry against such a tantalizing delay, and begged hard for a shorter period between our intervals.
``It cannot be my darling boy, remember discovery would separate us for ever. By prudence, we may long continue these delicious meetings.'' I suddenly suggested that as I slept alone in the little room, which, when the spare room was unoccupied, was far away from every one, she might steal along at night, when all were asleep, and thus I could enjoy the whole of her exquisite charms, without hindrance. She did not reply, but I could see her eye sparkled, and her cheek flushed as if already in imagination she was revelling untrammelled in all the luxury of voluptuousness such a plan opened out. However, she did not at once accept, but kissing me fondly, called me her dear and ingenious boy, and said she would think over my suggestion. We resumed our lessons on my sisters' return. Miss Evelyn was again four days before she gave me another opportunity of an amorous meeting. It was only my purposed insubordination that obtained me this interview. We again indulged in all the luxuries of carnal enjoyment, as far as could be done, incommoded as we both were by dress and locality. Reverting more strongly than ever to my plan of meeting in my lonely room, I begged so hard that at last she promised to come the night of the following day. I was obliged to put up with this, although I would fain have had her come that very night, but as her passions were evidently gaining stronger possession of her, and she was becoming more loving, and more voluptuous than ever, I felt certain she would not disappoint me on the next night. The delicious idea of revelling in charms I had so often furtively gazed on, kept me away from my sisters next day. Under a plea of headache I went early to bed, and took up some oil, to oil the hinges and lock of the door, to be prepared for my loved mistress. I lay long awake, and was almost in despair of her coming, when I heard the clock strike twelve. All at once I became aware she was at my bedside. She had entered the room with so gentle a step that though on the watch for her, I did not hear her even when she opened the door, shut, and locked it. She had come in her dark-grey cloak, and when at my bedside this was dropped on the floor, she stood in nothing but a very fine and thin chemise. She flung herself in my arms, as I rose to embrace her, and we instantly sank closely clasped in each other's arms. I was far too sharply set to practise any preliminaries. I turned her on her back, and was into her in a moment, with one vigorous thrust, which almost took away her breath, and gave her intense delight. I was too quick for her, however, as I spent in two or three shoves into that delight-giving cunt. But as this hardly allayed the fires of my too ardent desires, the convulsive internal movements of her unsatisfied orbit quickly restored my scarcely reduced member to a renewed vigour. Miss Evelyn being greatly excited by the unsatisfying nature of my first bout, was extremely warm, and throwing her arms and legs around my body, we again rushed headlong into all the fury of fucking, and as my previous spendings had somewhat reduced the power of immediate discharge, I was able to suit my movements exactly to those of my most active companion, and we sank together in all the voluptuousness of satisfied desires, lying long locked in each other's arms, before we were again in a state to renew our combats in love's delicious domain. We spent the interval in whispered vows and fond endearments and embracings of each other's naked charms, both of us admiringly passing our hands over every part of our bodies.
Miss Evelyn at last concentrated all her attention on my well-developed member, which she most endearingly embraced and fondled tenderly, very quickly putting him into an ungovernable state of erection. I was lying on my back, and she partially raised herself to kiss my formidable weapon; so gently putting her upon me, I told her it was her turn to do the work. She laughed, but at once mounted upon me, and bringing her delicious cunt right over my prick, and guiding it to the entrance of love's grotto, she gently sank down upon it and engulphed it until the two hairs pressed against each other. A few slow up and down movements followed, when becoming too libidinous for such temporizing delays, she sank on my belly, and began to show most wonderful activity of loins and bottom. I seconded her to the utmost, and finding she was so excited, I slipped my hand round behind and introduced my middle finger in the rosy and very tight orifice of her glorious backside. I continued to move in and out in unison with her up and down heavings. It seemed to spur her on to more vigorous actions, and in the midst of short gaspings and suppressed sighs, she sank almost senseless on my bosom, I, too, had quickened my action, and shot into her gaping womb a torrent of boiling sperm.
We lay entranced in the raptures of satiated desire for a long time. At last she came to her senses, and fondly kissing me, turned off, and we lay side by side closely embraced.
``Oh! my beloved Charlie, what exquisite delight you have given me; you are the most delicious and loving creature that ever could be created. You kill me with pleasure, but what was that you were doing to my bottom? What put such an idea into your head?''
``I don't know,'' I replied. ``I put my arm round to feel the beautiful globes of your bottom, and found in grasping one that my finger was against a hole, all wet with our previous encounters, and pressing it, found that my finger slipped in; you gave it such a delicious pressure when in that the idea entered into my head that, as it resembled the delicious pressure your enchanting other orifice gives my shaft when embracing you, this orifice would like a similar movement to that which my shaft exercised in your quim. So I did so, and it seemed to add to your excitement, if I may judge by the extraordinary convulsive pressures you gave my finger when you died away in all the agony of our final rapture. Tell me, my beloved Miss Evelyn, did it add to your pleasure as much as I fancied?''
``Well, my darling Charlie, I must own it did, very much to my surprise; it seemed to make the final pleasure almost too exciting to bear, and I can only account it a happy accident leading to an increase to pleasure I already thought beyond the power of nature to surpass. Naughty boy, I feel your great instrument at full stretch again, but you must moderate yourself, my darling, we have done enough for to-night No, no, no! I am not going to let him in again.''
Passing her hand down, she turned away its head from the charming entrance of her cunt, and began handling and feeling it in apparent admiration of its length, thickness, and stiffness. Her gentle touch did anything but allay the passion that was rising to fever heat; so sucking one of her bubbies, while I pressed her to me with one arm under her, and embracing her on the other side, I passed my hand between our moist and warm bodies, reached her charming clitoris, already stiff with the excitement of handling my prick. My titillations soon decided her passions, and gently prompting her with the arm under her body, I turned her once more on the top of me. She murmured an objection, but offered no resistance; on the contrary, she herself guided my throbbing and eager prick into the voluptuous sheath that was longing to engulph it. Our movements this time were less hurried and more voluptuous. For some time she kept her body upright, rising and falling from her knees. I put my finger to her clitoris, and added to the extatic pleasure she was so salaciously enjoying. She soon found she must come to more rapid and vigorous movements, and lying down on my belly embraced and kissed me. Toying with our tongues I put an arm round her waist, and held her tight, while her glorious buttocks and most supple loins kept up the most delicious thrust and pressures on my thoroughly engulphed weapon. I again stimulated her to the highest pitch of excited desires by introducing my finger behind, and we both came to the grand crisis in a tumultuous state of enraptured agony, unable to do ought, but from moment to moment convulsively throb in and on our engulphed members. We must have lain thus languidly, and deliciously enjoying all the raptures of the most complete and voluptuous gratification of our passions, for fully thirty minutes before we recovered complete consciousness. Miss Evelyn was first to remember where she was. She sprang up, embraced me tenderly, and said she must leave me at once, she was afraid she had already stayed imprudently long. In fact, it was near five o'clock in the morning. I rose from the bed to fling my arms round her lovely body, to fondle and embrace her exquisite bubbies. With difficulty she tore herself from my arms. I accompanied her to the door, and with a mutual and loving kiss we parted. I to return and rapidly sink into the sweetest slumber after such a delicious night of most voluptuous fucking.
She came again three times in the next six nights; each time we renewed our mutual joys, with ever increasing voluptuous indulgencies. On coming to me for the fifth time, she said---
``Dear Charlie, I have only come to kiss you, and say I cannot stop.''
``Cannot stop!'' I cried, ``and why not, beloved Miss Evelyn?''
``I am not well, but cannot explain more.''
I had sprung out of bed, and clasped her in my arms, then passing a hand down to her beauteous and well-covered mons Veneris, I found that she was tied up there in cloth. I immediately remembered how my loved Mrs. Benson had been exactly in the same way. I then also remarked the peculiar odour of breath, but pretending ignorance, I begged to know what had happened to my darling little grotto.
``I cannot tell you more, my dear boy, but it will keep me away from you for four or five nights.''
``But why should that be the case; cannot you let me enter that delicious cave of delight only once?''
``No, no, impossible! my dear Charlie, absolutely impossible! It would do me very great harm, and you too. Let us be quiet in that way, and I shall be the sooner well again to come and embrace you as before.''
``Oh! but darling, how can I support five nights' absence, I shall go mad with desire and burst---feel how he grows, and is longing for his loved companion.''
Her soft and gentle hand caressed it. I thought to succeed by a \textit{coup de main}, but she was too quick for me.
``No, Charles, I am serious, and you must not try to force me, or I shall never come near you again.''
I saw she was in earnest, and flung myself on the bed in a pet.
``Come, my darling Charlie, be reasonable, and I will do my best to give you some satisfaction. Lay yourself on your back---so. I will kneel on the floor at right angles to you, because you must not attempt to touch me down there. That is a dear boy.''
So taking my prick in her soft hand, she gently moved it up and down; then, suddenly stopping, took it into her mouth, sucking as much as she could get in, and titillating the knob with her tongue, while one hand frigged at the root of my prick and the other gently handled my two crisped-up cods. She prolonged the pleasure by occasional pauses, and at last, on finding the electric-like sensations coming, she hastened her movement, and I poured a torrent of sperm into her mouth. She continued her delicious sucking until not a drop more was left for her to swallow. This was the first time she ever gamahuched me, but it was not the last by scores of times. Ever after we improved upon the model, and added other endearments. When not under her courses, we mutually gamahuched each other, and she was the first to repeat upon me, with the intensest gratification, the delicious introduction of a finger behind while gamahuching me. At present, when she had thus taken the edge off my carnal appetite, she lovingly embraced me, and left me to my lovely slumbers. Of course, the four days' grace, saving two more passing visits ``to keep me cool,'' as she said, turned all to the advantage of my sisters, whom I fucked and frigged to their utmost gratification and delight.
I thus passed about four months. Miss Evelyn becoming a perfect adept in love's delicious mysteries; but, although I had attempted to enjoy the orifice of the lower temple of Venus, my member was too large, and gave too much pain, to completely succeed, so that I became the faithful worshipper at the more legitimate altar of love. My sisters were gradually developing their forms. Mary particularly so. The hair on her quim had increased to a most charming curling profusion. Her hips spread out, and her bottom, hard and prominent, promised to be very large. Eliza, too, began to show increased bubbies, and an enlarged and mossy mons Veneris.
We were approaching summer, and near the full of the moon, Mary had complained of feeling very low spirited, and very much inclined to cry. I tried to comfort her, and thought success would best attend my efforts if I fucked her. So enticing her down into the garden, we entered the summer house, and I at once proceeded to action. She was rather unwilling, she could not say why, but had an instinctive reluctance. She yielded, however, to my entreaties, and I fucked her without apparently exciting her in the usual way. I consequently withdrew as soon as I had run the first course, and at once discovered what ailed poor Mary. My member was covered with blood. For the first time her courses had come upon her. She was greatly alarmed, but I told her I had beard it was quite natural to young women when they reached a certain age, that she had better tell mamma at once, who would instruct her what to do. I carefully wiped my reddened member, and then retired to my room to purify myself. That very night, on Miss Evelyn coming to me, I found she was exactly in the same state. She gave me my usual relief with her soft hand and caressing lips, and then left me for five nights, as at that time.
I now found myself reduced to my dear little sister Eliza. Up to this time I had never actually fucked her, and her maidenhead was still intact. She was now approaching fourteen, and the down on her charming little cunt was becoming more decided; her bubbies too, under the erotic excitement of my \textit{attouchements} and gamahuching, had assumed a decided prominence. My finger had somewhat rendered the opening of her little pinky slit more easy of access. So I resolved to complete her carnal education and fuck her thoroughly. The opportunity was perfect; both Miss Evelyn and Mary retired to their rooms to lie down at our usual hour's recreation, Eliza and I at once hied to the summer house, and locked ourselves in. I immediately laid her down on the long couch, and gamahuched her until she spent in my mouth, and then continued until she was again almost mad with desire. I then told her I should initiate her into a new mystery, more delicious than any she had yet experienced, but that the first initiation was always painful.
``Oh! what is it, my dear Charlie, everything you do is so nice, I know I shall like it---what is it?''
``Then you must know, dear Eliza, that this little cunt of yours is made for the express purpose of having a prick put into it; only, as mine is so large, and you are still so small and so young, I was afraid it would give you too much pain to do it sooner; but now, I think, I may get it in, if I do it gently.''
``Oh, Charlie, dear, put it in at once, I have often felt I should like it so; but, as you never attempted to do it, I thought it was a mere fancy of mine. Have you ever put it inside Mary's quim?''
``Often; nay, always, my darling.''
``Does she like it?''
``She adores it.''
``Then put it into me directly, Charlie.''
I wanted nothing better, and told her that in order to thoroughly enjoy it, she must strip. In a minute she dropped off everything, while I took off my trousers---coat and waistcoat having been already laid aside. I had brought a towel to lay on the couch below her bottom, to prevent any telltale stains. Laying her down on her back, with her bottom close to the end, her legs gathered up, and her two feet resting on the sofa, with her knees falling outwards (in the very best position for my intended operation), I put a pillow on the floor, on which I knelt, thus bringing my cock a little above her quim to give me a good purchase. I then first gamahuched her well again, until she spent and cried out---
``Oh, put it in, my dear Charlie, I do feel to want it so!''
She was already well moistened by her previous discharges, and by my licking the lips of her cunt, and covering them with saliva, with which I also, at the same time, wetted my prick itself. I then made the point approach the charming pouting and longing lips of her sweet little cunt, and rubbing it first up and down between the lips, proceeded to insert its knob between them. Thanks to the precautions taken, and the excitement I had raised by my previous caresses with tongue and prick, the immediate entrance was effected with greater ease than might have been expected. No sooner was it in about an inch beyond the knob than the passion of excitement I had raised so stimulated the natural lubricity of Eliza's nature that she heaved up her buttocks energetically, letting her knees drop quite down sideways, thus favouring to the utmost my forward thrust made at the moment, so that my prick was sheathed in an instant more than half his length, and but for the obstacle of her maidenhead, which he then met with, would have been entirely engulphed. As it was, it gave her a very sharp pang of pain, which made her shrink back, and utter an---
``Oh! Charlie!''
``Do not fear, I will be gentle, keep still a moment and then you will find the pain pass away, and great pleasure follow.''
So we lay still for a time, until I felt those involuntary internal pressures, the true precursors and infallible indicators of rising desires; so commencing a slow and continuous in and out movement, I shortly produced such an excess of pleasure in her delicious orbit, that her movements became almost furious, and nature alone prompted her to second me with as much art as if she had already been long instructed in the delicious movements so calculated to add to the libidinous delights of true enjoyment.
But Eliza was a rare example of a truly salacious and voluptuous nature, and proved herself in that way far in advance of Mary; although she was of a very warm temperament, Eliza's passions were far more excitable, and in the end she became one of the most voluptuous fuckers possible, abandoning herself to all the wildest raptures that the most erotic nature could suggest. Of this, hereafter; at present I had worked her up to the utmost pitch of excited desire; she was in the very act of discharging, and as I withdrew for a final thrust, she heaved up her buttocks in an agony of pleasure, I felt it was now or never, and striking home with all my force, I burst with irresistible strength through every obstacle, and tore my way inwards, until sheathed to my very cods. Poor Eliza! at the very moment she thought herself in the seventh heaven of delight, she experienced the most excruciating agony. She gave a piercing cry and fainted away; her arms fell senseless from my body---her legs would have also fallen, but twining my arms round them, I continued for several successive thrusts to penetrate fully and easily into every recess, for I myself was wound up to a fearful state of excitement. I died away in an excess of joy, sending a torrent of balmy sperm to soften and mitigate the pain of her terribly torn quim. Finding that Eliza could not regain consciousness, I rose somewhat in alarm, and was horrified to see the quantity of blood that followed my withdrawal. It was fortunate my forethought of the towel, as it had not only saved the sofa, but helped to stanch her swollen and bleeding quim, and to wipe the blood from her thighs and bottom. I had effected all this before the dear girl showed the least symptoms of animation. She first sighed, then shivered, and at last opened her eyes, and looked confusedly at me, and asked---
``What has happened to me, Charlie?''
Then observing how she was lying naked, she recovered her complete consciousness of all the circumstances of the case.
``Oh! Charlie, now I know; I thought you had killed me; Charlie, oh! it was so frightfully painful. How could you hurt me so, and just as I thought it was the most heavenly pleasure I had ever experienced in all my life.''
``My darling, it is all over now, and it will never hurt again, and we shall both of us have greater pleasure than ever, but not just now; it has been greater pain to you than I thought it would be, and for the present we must not attempt any more.'' I helped her to rise, but she felt very faint, and I had great difficulty in getting her dressed. She was shocked to see the bloody state of the towel. I told her to put my handkerchief between her thighs, and partly up her slit, to prevent any marks of blood staining her shift. I then laid her down on the sofa, while I ran to get some water from the fountain in the garden. I took a glass and the towel with me. I returned with the water, which greatly refreshed Eliza. I begged her to lie still as long as she could stay. However, when she attempted to walk, she found herself very much incommoded with the smarting pain. I was terribly afraid lest this would be observed when we got to the house, so I suggested she should purposely fall down when in sight of any one, and say she could not move because she had hurt her knee by the fall.
This stratagem succeeded admirably. We were seen approaching by Miss Evelyn, my mother, and Mary. Dear Eliza acted her part admirably, was seen to fall heavily, and screamed. They all rushed out, we lifted her carefully on her legs, and supported her to the house, she complaining of the pain in the knee and ankle. My mother insisted on her going to bed at once, and having embrocations and hot towels applied. Eliza let them do as they liked, and eventually was left to quiet repose, which soon relieved the painful sensations she had undergone. Next day she complained of great stiffness, and walked lame, but thought the hot applications had prevented the swelling, so thus happily passed off all observations of suspicion of the real circumstances of the case. It was not until the third day after that I attempted to make an entrance. Of course, I excited her first to the utmost by a long continued gamahuching. She then let me, but with fear and trembling, introduce my bursting member into the delicate folds of her cunt. As I was very gentle in my movements, the pain was scarcely felt, and when once well sheathed, and the first thrusts given slowly and luxuriously, the whole lubricity of her nature was soon awakened, and by the time I was ready to spend she was as ready to second me, and we died away in a mutual flood of delighted extasy. She held me close, and would not let me withdraw.
``No, Charlie, it took some trouble to get it in, let it stop where it is so deliciously engulphed,'' and at once anticipating her natural desires, she began the most exquisite pressures upon me, which very shortly brought us both up to the point of demanding more active measures. However, I rather restrained her, and told her we must retard our movements to increase our pleasures, because mere quick repetitions would only exhaust her, without yielding the true extasies of enjoyment. I, therefore, taught her the pleasures of the slow movements, and I worked her up to spending point, without giving way myself. The dear little creature clung to me with the most close and endearing embraces, as if she should force a complete amalgamation of our two bodies, and died away in the sweetest bliss of contented desire, with such a heavenly expression of extasy on her face as made me devour it with kisses. I had great difficulty in restraining myself from precipitately following her example; her delicious movements at the moment of spending, and the close pressures on my prick, were so exciting that resisting them was quite a triumph of control. I succeeded, and lay quite quiet, embalmed in the delicious suction of those exquisite folds of her charming little cunt, which exercised the most delightful pressures as well as suction on my enraptured prick. I left it entirely to her to lie as we were as long as she pleased, or to again begin the dear delightful friction that should once more make us dash on passion's furious course, to end as usual in the ever delicious extasies of the final crisis.
This last bout had been a double one for my sister; she all but swooned away with the rapture my spending in unison with her produced. She declared it was a death of the most delicious extasy, which it was perfectly impossible to describe. She clung to me, kissing me in the most endearing manner, and telling me how happy I had at last made her by completing the insertion of my prick in her cunt. It was worth the suffering of twenty times as much agony to arrive at so exquisite a result as every fuck I now gave her conferred upon her. We adjourned to the flower garden, that we might be seen playing together, and not excite suspicion by our constant disappearance, now that we were only two together. Of course, Mary knew what we were at and probably guessed that I had completed the initiation of Eliza. She smiled, and gave me a significant pressure of the hand, when we met again in the school-room to resume our lessons. For two days more I enjoyed Eliza all to myself; at each new fuck she became more and more perfect in conferring as well as receiving pleasure.
On the third day, Miss Evelyn whispered, ``to-night,'' as she gave me a stolen pressure of the hand. She came, and we indulged in every whim of our fancy. I had further the delicious pleasure of gazing on all her naked beauties, as it was daylight before we parted; I had gamahuched her twice, and fucked her five times. She gave me credit for a long fast, and allowed so much indulgence on that account, but told me I must in future be more moderate, for her sake, if not for my own. She allowed three nights to pass before again coming to me. I cannot say I regretted it, because now that Eliza as initiated, as well as Mary, we indulged in the most delightful orgies of fucking and gamahuching at the same time. At first we used to fuck with one laid on her back to be fucked, while the other backed on her knees over the face of the one being fucked, and was gamahuched by her, while I introduced my finger into the rosy orifice of the bottom before me. But we found the most voluptuous way was for one to lie down on her back, and the other on hands and knees over her. She thus brought her mouth over the cunt of the one lying down, and presented her bottom to me, who knelt behind her. The one below guided my prick into the cunt above her face; she had thus all the satisfaction of seeing our action, while with one hand she tickled my cods, and the other felt my bottom-hole, and inserted a finger. Meanwhile, she was gamahuched and bottom-frigged at the same time by the one I was fucking, and we used all three to die away in agonies of enraptured delight, to recommence with a change of places between the two girls. Sometimes I tried to introduce my prick into the rosy little orifice of Mary's backside, but, although the finger-frigging gave her much additional pleasure while her cunt was operated upon by my virile member, she as yet could not support the insertion of my large prick. I had not even attempted little Lizzie, but one day, when Miss Evelyn and Mary were again under menstruation, and I had dear Lizzie all to myself, she was seized with such an irresistible desire to ease herself, that she had only time to get behind a bush and squat down. I remained waiting for her, when she called to me, to ask if I had any paper. I advanced to give her some. She was in a half-standing position, with her clothes held up to her waist. While giving her the paper, my eyes accidentally fell upon what she had voided. I was struck with its extraordinary thickness. I made no observation at the time, but it raised an idea that preoccupied me much. I had often thought over the pleasure that fucking Mrs. Benson's bum-hole had given me, hence I had tried to initiate both Miss Evelyn and Mary in that delightful route of pleasure, but, as before stated, had been unable to succeed with them from the great developement of my weapon. Thinking that if they could not bear the insertion, there could be no possibility of success with my younger and less developed sister, I had never attempted with Lizzie more than the insertion of one finger. It is true, with her it seemed to produce more excitement than either upon Miss Evelyn or Mary. The sight of the extraordinary dimensions of the matter she had voided now suggested the idea that if her apparently very small and rosy-lipped bottom-hole could allow so large a mass to come out, with gentle efforts my scarcely larger machine might be inserted. I determined to try the initiation into that route of delight the very next day. Remembering that dear Mrs. Benson always made it a rule that she should be first well fucked and gamahuched, and the prick well moistened, I began by exciting dear Lizzie to the utmost. I first fucked her, and made her spend twice to my once; then gamahuched her until she implored me to shove my prick into her. I had managed to introduce my two forefingers at once into her bottom, and had frigged her while sucking her cunt, without apparently giving any pain; on the contrary, from her movements I fancied she felt greater excitement. I took care to enlarge, as much as possible, or rather to stretch her bottom-hole as open as I could with two fingers. It was at the moment of her greatest excitement, when she was pressing me to fuck her at once, that I said---
``My dearest sister, there is still another mystery of sensual voluptuousness that you have as yet not experienced or been initiated into, and I am about to instruct you in it.''
``Oh, what is it? dear Charlie; but do anything you like, and as quick as possible.''
``Well, then dear, it is this sweet little orifice in your bottom that I am going to introduce my prick into. It may give you some little pain the first time, but by gentleness of movement, and halting from time to time when it hurts too much, we shall get him completely inserted, and then it will be an immense pleasure to both of us.''
``Dear, dear Charlie, do as you like, your darling prick can only give me the greatest delight; I am dying to have him into me, I don't care where, as long as I get the dear creature into me. I suppose I must be on my hands and knees.''
Upon which she turned with great agility, and presented the two hard and already promising globes of her charming backside. I lost no time in first thrusting my prick up to the hilt in her cunt, to moisten it. It made her shudder again with excess of lust, and she exercised such a pressure upon it that I had some difficulty in withdrawing it. It was so snug and nice therein that was a great temptation to run a course in her cunt at once, but having the other object in view, and knowing that I wanted all its stiffness to succeed, I did summon up courage enough to withdraw; then applying the very plenteous saliva in my mouth that gamahuching her had stimulated, I added it to the already moistened prick, and applying some to her bum-hole, and introducing a well-wetted finger, I put the knob of my formidable prick to the small and smiling orifice that lay before me. The disproportion struck me as so great that I dreaded success would be much too painful for her, but remembering the dimensions of what had come out of it, I boldly proceeded with the operation. I got in over the knob without making her flinch, but, as I proceded to push gently forward, and had got in about two inches, she cried---
``Stop a little, Charlie, it feels so queer---I can't bear it in further.''
I stopped where I was, but slipping a hand under her, I applied my finger to her clitoris, holding her bottom tight against me with the other hand round her waist, so as not to lose ground. My agile finger soon worked her passions up, and I felt her bottom give convulsive twitches on my prick. I allowed her to become still more excited, and then gently pushing forward found I was slowly, and almost imperceptibly, gaining ground. My prick was then inserted almost two-thirds of its length, when, thrusting rather too sharply, she again cried out, and, but for the arm that held her fast round the waist, would have unseated me.
``Oh, Charlie, dear, do stop; it seems to choke me, and makes me feel so queer, that I thought I was going to faint.''
``I shall lie quite still, now dear Lizzie. It is quite in''---this was a little bit of deception to calm her fears---``and when the pain of insertion passes, which will be the case in a minute, we shall have nothing but pleasure.''
So I kept my prick just where he was, but redoubled my frigging her clitoris, and very soon brought her up to spending point, resolved that I would not attempt complete insertion until I felt she was in the raptures of sensual discharge. This quickly came upon her, and it was the delicious movement of her own buttocks that sheathed my prick to the hilt without an effort on my part, and so far from giving her pain, made her positively scream with the intense voluptuousness of her sensation in spending. She could not speak for many minutes, but continued the exquisite pressures of the sphincter muscle on my enraptured prick. But for my determination not to give way, and rather to wait for another bout that would completely initiate dear Lizzie in all the luxury and abandon of this delicious mode, I must have at once vigorously finished my own course. My restraint was well rewarded. The first words my beloved sister uttered were those of almost delirious joy at the extraordinary delight I had given her. Never, never, had any fuck so enraptured her. She turned up her lovely face to me, and tears of sensuality and voluptuousness filled her eyes.
I had hardly begun my titillations on her still excited clitoris, which, by the way, had lately considerably developed itself, when she was as eager for another bout as I was. I held sufficient restraint on myself to practice every salacious movement, that I might give Lizzie such exquisite pleasure as should induce her on future occasions to grant me the use of her charming bottom-hole whenever I should desire. I worked her up to the utmost pitch of the most salacious excitement, and at the moment when she spent, in an agony of shrieking extasy, I poured a perfect flood of spunk right up into her entrails, and we both sank forward, but without unseating me, quite overpowered by the intensity of our delight. When we came to our senses I rose from off her. On withdrawing my prick I found a few traces of blood, but of no moment. I wiped my prick on a handkerchief, and also wiped between the cheeks of dear Lizzie's bottom, for fear any tell-tale marks should be made on her linen. I then helped her up, and she threw her arms round my neck, and sweetly kissing me, thanked me for a new lesson in love, which had overwhelmed her with delight.
Thus ended the first lesson that Lizzie ever received by that route of pleasure, and I may incidentally state that she was peculiarly constituted for giving and receiving the most exquisite pleasure in that way. She afterwards developed into a magnificent woman, with one of the naturally largest and finest backsides I almost ever met with; and she came to love backward fucking to the utmost extent. In after-days, when married, she told me that her husband was a muff, who had no idea of enjoying a woman but in one way. She had often deceived him, and slipped it into her bottom-hole without his ever having any suspicion of the sort of pleasure he had given her.
Three months passed with the rapidity of a dream, while we indulged in these scenes of delicious lubricity and voluptuousness, without ever attracting any observation within the house and, more curious, without Miss Evelyn either discovering or suspecting anything between my sisters and myself---thanks to my natural powers and the unfailing resources of youth. Both she and my sisters thought they each gave me as much as I could get through, and, therefore, neither ever imagined I could seek carnal delights in other arms. So it was but now there happened one or two events which had a considerable effect on the after-tenour of our loves.
A neighbour, a very nice good-looking man, about thirty-five years of age, a gentleman farmer, very well off, had for some time past always waited for us at the church door on Sundays, apparently for a chat with mamma, Miss Evelyn, and us. He treated and evidently considered us as mere children, nor did he appear to fix particular attention to any one.
One Monday my mother received a note from him, to beg she would grant him a short interview on the following day, as he wished for her advice on a subject of much interest to him. Mamma's reply begged him to come at eleven o'clock, when she would be happy to see him.
He came, and was particularly neatly dressed. My mother had been very agitated all the morning, and looked flushed and nervous as the hour drew near; I really believe the old lady fancied it was for an idle avowal to herself that he was coming. Be that however as it may, the object of his visit turned out to be a proposal to Miss Evelyn, with an offer of marriage. He was ready to make such settlements upon her as could not but be satisfactory. He told my mother that before speaking to Miss Evelyn, whom he had loved from her first appearance in the parish, and whose quiet, modest character had daily made a deeper impression, he thought it only his duty to first break the subject to her, and to ask her permission for an interview with Miss Evelyn, and next, if he was acceptable to her, for leave to visit at our house, while courting his wished-for wife. He further stated that he had never ventured to hint the state of his feelings to Miss Evelyn, and prayed my mother to be the kind intermediary in opening the subject to her, and to beg as a favour that she would grant him an interview to state his case in person on the following day, so that he might learn his fate from her own lips. My mother, although probably inwardly a little disappointed, had the interest of Miss Evelyn too much at heart not to take up the matter warmly, and urged, with all the volubility elderly ladies can so well exercise, whenever the marriage of a younger friend is in question, all the benefit that would accrue to her from so advantageous a proposal. Miss Evelyn was really taken quite by surprise, and stammered out some vague expressions of wishing for time to consider.
``Stuff and nonsense, my dear, remember your dependent position, and the advantages this match holds out to you. You must not think or talk of delay. He will be here to-morrow, and I hope his lover eloquence will soon decide the question in his favour.''
Poor Miss Evelyn burst into tears and said it was so sudden, and she was so ill-prepared to take any decision. She would, however, think over it very seriously and in the morning be better able to give an answer. My mother seeing that she was much agitated by what she had told her, very kindly said---
``Give the children a holiday this afternoon, and I advise you to keep your own room, and write to your widowed mother, to tell her of the offer, and to ask her advice how you should act.''
We thus had many hours to ourselves; I had heard all that had passed, and felt a sad pressure at my heart, when I began to realise the fact that the proposal of Mr. Vincent would, if accepted, lead to our separation, and deprive me of my loved Miss Evelyn. The idea made me very sad, and I showed no alacrity in taking advantage of our extra hours of recreation with my sisters, until Mary began to rally me about my melancholy, and asked what I meant by it. I at once said---
``Don't you see, if Miss Evelyn marries Mr. Vincent we shall get another governess, and can we ever expect to get one who is so kind and excellent a teacher, and who troubles us so little at our games.''
``Ah! that is very true, and we should be horribly annoyed if we were watched and interrupted. However, more reason that we should make the most of the present moment, so come along, Charlie, and let us have some real good fucking. We have plenty of time, mamma is not very well. No one will come near us, and there is nothing to hinder our having a jolly time of it, all three stark naked together, so come along.''
Her words had already changed the current of my ideas; before she ceased speaking my prick responded, which her quick eye immediately observed, and patting it with her hand, she said---
``Ah! my dear little dummy, I am glad to see you are of my opinion, so come along.''
Away we went, and a most glorious afternoon of orgies we spent.
Miss Evelyn came to me at night and threw herself into my arms, in an agony of sobs and tears, and pressing me to her throbbing bosom, she sobbed out---
``Oh! my dear Charlie, I love you so dearly, you have become as necessary to me as life itself. I cannot bear the thought of parting from you, my loved one. You, whom I have initiated into all the delights of mutual love. Oh! the thought of parting is bitter, and breaks my heart. Oh! love me, my own darling boy, and press me to your heart.''
I did more, for, as I have before stated, a woman's tears have a never failing effect on the erective nerves of my machine. It was but the commencement of a night of most luxurious enjoyment. Miss Evelyn put no restraint either on herself or me, but indulged in every act of lubricity and voluptuousness were drawing to a close. In fact, when eventually she left me in the morning, and I thought over all she had said, it became evident to me that she had already made up her mind to accept the very advantageous offer made to her. The instinctive intelligence of woman had at once shown to her that such an opportunity was not to be lost for the sake of a mere boy, whom circumstances must naturally soon remove far away from her. At the same time, doubtless, the idea that I was all her own making, for she never had any suspicion of my previous initiation, held a charm over her, to say nothing of the powerful weapon she had so unexpectedly found by her side, and which had so great an influence over her passions. We spent a most luxurious night, and hardly closed our eyes, notwithstanding my afternoon's debauch; such is the power and resources of nature, in a well-constituted youth of fifteen and upwards, that Miss Evelyn had rather to force our embraces, than to stimulate by any artificial excitement my ever ready prick. I won from her a promise to come next night, and let me know what fate was in store for us.
Next day Mr. Vincent was true to his appointment. Mamma received him with Miss Evelyn by her side, and after the usual compliments, rose and apologised for leaving them, as she had household duties to attend to. Miss Evelyn informed me afterwards that Mr. Vincent, on my mother leaving the room, rose from his seat, and approaching her, said, in the most frank gentlemanly manner---
``You are aware, my dear Miss Evelyn, of the object of my visit, and I augur from your kind condescension in giving me this interview that my suit is not disagreeable to you.''
Then taking her passive hand, and pressing it to his heart, he continued---
``I have loved you, Miss Evelyn, from the first moment of my seeing you. I feel that my future happiness hangs on your lips, for without your love, my life would now be a blank. I am here to-day to offer you my hand and fortune. If I have not yet your heart, I seek to be allowed to cultivate your society, that I may try to win it.''
Then seeing that she was greatly agitated, he begged her to be seated (for she had risen when he approached and took her hand), he led her to a sofa, and seated himself by her side. He pressed for an answer. She said---
``You must be fully aware, Mr. Vincent, that your generous offer has taken me greatly by surprise. I feel most grateful to you for it, but must implore you to allow me to pause, until at least I have heard from my mother, to whom I will communicate the noble offer you have made to me, a poor governess, who cannot but feel grateful to you for condescending to think of her in such a way.''
``Ah! say not so, my dear Miss Evelyn, and believe me, it is no sudden impulse that has driven me to your feet, but ardent love, and real admiration of your great beauty and admirable conduct ever since you entered this family.''
The dear creature smiled through her tears upon me when she recounted those terms of affection that Mr. Vincent poured out to her.
To be brief---before they parted he won from her that his frequent meetings at church, and elsewhere, had gained him something more than esteem, but hopeless of ever becoming his wife, she had done her utmost to suppress wanner feelings. Oh! woman, thy name is deception! So she sent him away the happiest man in existence. He rode over every day afterwards, and was with Miss Evelyn from four to five; indeed, he was often the cause of our having half an hour's longer recreation. He also frequently dined with us. Miss Evelyn's mother naturally jumped at the offer, and most delightedly gave her consent.
When Mr. Vincent heard of this, be became very urgent in claiming an early day for making him the happiest of men. Miss Evelyn wanted a delay of six weeks, but this raised such an outcry on his part, seconded by my mother, that at last she was driven from six weeks to a month, and then to a fortnight from that date; so all became extremely busy in getting ready marriage dresses, \&c. The marriage was to take place from our house, and my mother insisted that she should provide the marriage breakfast. Mrs. Evelyn was invited to our house for a week at the time of the marriage, to keep my mother company. My two sisters and a young sister of Mr. Vincent's were to be the bridesmaids, and a young man, courting Miss Vincent, to be bridegroom's man. So all was thus arranged, and eventually came off most happily. When Mrs. Evelyn arrived she occupied the spare room, where charming Mrs. Benson had so deliciously initiated me in all the pleasures of sensuality and passion.
To return to the day when Mr. Vincent had his first interview, and declared his love and admiration, and ended with the offer of marriage. Before going away, he rang for mamma, thanked her for all her kindness to him, informed her how happy Miss Evelyn had made him in granting permission to prosecute his suit for her hand, \&c. Then begging the favour of a chaste kiss, he left all radiant with hope.
The interview had naturally been very trying tor Miss Evelyn, and she was so evidently nervously agitated that my mother begged her to go to her room, and lie down to repose herself, as after so much agitation she must be quite unfit for any school work, and that she herself would hear our lessons that morning and give us an afternoon's holiday in honour of the happy event that had occurred.
We thus, my sisters and I, were thrown again into another prolonged opportunity of fully enjoying ourselves, but, notwithstanding the wonderfully regenerative power that nature had gifted me with, I felt that if I wanted to enjoy again my dear Miss Evelyn, who had promised to be with me that night. I must not only restrain myself from such excess as we had indulged in the previous day, but also manage to get some sleep, of which I had scarcely tasted the night before, so I contented myself with first gamahuching and then fucking each sister; afterwards again gamahuching them, and making them each spend five times, so as to satisfy them without exhausting myself, and then finishing off with a delicious fuck in Lizzie's bottom-hole, while each gamahuched the other. This quite satisfied them, and they allowed me to steal up to my room to sleep, Mary promising to call me in time for tea. I slept the sleep of the just for some three hours, and came to tea perfectly ready for anything that could happen that night. It was well it was so, for now that there could not be any long lapse of time before we must part, Miss Evelyn became a very glutton for pleasure, and every art and position was made use of to stimulate and lengthen out our joys. She came every night, even up to the very night before the marriage, although in the last three nights before the event came off, her mother, Mrs. Evelyn, slept in the spare bedroom with which my room communicated. Nevertheless, we met and carried on our amorous sports with bated breath and suppressed sighs.
We had of late often tried in our moments of greatest excitement to introduce my prick into her delicious tight little bottom-hole. Once, by a sudden manoeuvre, I managed to get in at the moment she was spending, and actually made an entrance as far as about two inches beyond the nut, and I think I should have fully succeeded at that time if my own excitement had not made me spend too soon. This oiled the way, and my prick, having already fucked several times, becoming too limp, the squeeze of her bottom actually forced him out, as if she were voiding herself naturally. I fancied that, at the moment, but for my too excited passion, she would have rather I had completely initiated her. However, the night preceding her marriage, I at last succeeded. We had fucked in every varied way. She was on her knees, with her head on the pillow, and I on my knees, behind her; this was a favourite way of hers, as she declared I got further in, nay, seemed to touch her heart and fill her whole body; besides the frigging her clitoris and the action of my finger in her bottom-hole added greatly to the raptures this position gave her. She had been already well fucked, and we had mutually gamahuched each other, so her whole system was in a most excited and well-moistened state. Taking care to put two fingers at once into her bottom-hole, I worked them so as to stretch it as much as possible, while exciting her with my prick in her cunt, and a finger on her clitoris. Just as she was going into the raptures of spending I dropped from my mouth a quantity of saliva onto her bottom-hole, and as she was pushing her buttocks back to me I suddenly withdrew my prick, and with one vigorous thrust housed him half his length in her delicious bum-hole. She almost cried out aloud at the suddenness of the attack, and would have flinched away but for the grasp of both my hands upon her hips; a more vigorous shove sent me up to the hilt against her beautiful buttocks. She whispered---
``For heaven's sake, dear Charles, do stop a moment, I can't bear it, and must cry out if you do not be quiet for a time at least.''
As I was safely fixed, it exactly suited me to remain still, for had I gone on, a push or two would have made me spend. Now fairly engulphed, I wished not only to fully enjoy it myself but, if possible, make her enjoy it too. So remaining quite still, as far as regarded my prick, I stole one hand down to her clitoris, and began to excite that; the other I ran up to her bubbies, and played with the nipples, a thing which I had found out excited her almost as much as playing with her clitoris. Her passions were soon reawakened, and the involuntary twistings of her loins and pressures of her sphincter convinced me that in a very short time I should work her up to the utmost; and so it was, and immensely she enjoyed both her own spend and mine when she felt my hot spunk shooting up into her very entrails. We sunk gently on our sides after this bout, but without unsheathing me; and here embracing, kissing and tonguing each other when she turned her head, and sometimes sucking the nearest nipple to me, we soon once more were in a state to renew our delicious combat; and a second course was run in the delightful callipygian recesses of Venus's second temple of lubricity. This was our last bout, for, alas, it was getting the hour when the house would be all astir. My lovely mistress embraced me most tenderly, and acknowledged that I had at last taught her a new pleasure. She wept as she tore herself from my arms, and I wept too when she left me, as I thought I had now lost her for ever as a mistress, and what a charming one she had been to me!
Morning came, and with it bridesmaids, bridegroom, and man. To church we all went, my sisters perfectly enchanted with the idea of being bridesmaids, and beautifully arrayed in new dresses. They were also still more delighted with some handsome jewelry presented by Mr. Vincent. In their eyes he became the handsomest and finest man they had ever seen. The breakfast went off as usual, and when the bride, who had changed her bridal dress for a neat travelling one, came down, pretty near all were in tears on taking leave of her. She pressed me tenderly to her bosom, and whispered---
``Courage, Charlie, dear.''
It was almost too much for me, but I managed to restrain any extreme demonstration of my grief. The carriage door was shut, and off they rattled to spend the honeymoon at Leamington. The friends assembled remained until the evening, and after the sensations of the day, and the fatigues of the previous night, I was glad to get to bed. I cried myself to sleep, thinking that another at that moment was revelling in all the delights of amorous enjoyment of those charms that had been so long in my sole possession.
Thus ended one of the most delightful episodes of my life, and although I, at some rare intervals, from time to time found an opportunity of enjoying my loved mistress, they were flying fucks, very delicious, but very unsatisfactory.
This was the first great incident that had an effect of changing the tenour of our existence for some time, but I will reserve the details of our after-adventures for a second part of these reminiscences of Early Experiences.
\gutchapter{END OF VOL. I.}
\section*{\raggedright VOLUME II}
\gutchapter{CONTENTS}
Mr. James MacCallum---Mrs. Vincent---Miss Frankland---Miss F., Mary, and Eliza---Doctor and Mrs. Brownlow---
The house was scarcely itself even the day after the marriage. Mrs. Evelyn was still with us, and did not leave until the following day. She and my mother spent most of the day in the summer house, so that our pastimes therein were interrupted. Mary complained of severe headache, which, in fact, was the premonitory symptom of her courses, which declared themselves violently in the evening. I had arranged with my sisters to steal up to their room when all were asleep, as now that we had lost our governess they had it all to themselves. I went, of course, but found only Eliza capable of entering into our sensual enjoyments. I made her come to me in Miss Evelyn's bed, and while fucking her, was thinking all the time of my darling governess; and even when I was fucking her I could only remember the complete insertion of my prick into Miss Evelyn's bum-hole the very night before her marriage, and wondered whether or not her husband had discovered her loss of maidenhead. And yet, I fancied woman's natural cunning would easily deceive him, as millions before him have been deceived. Coupling Mary's attack and Miss Evelyn's choice of the marriage day on the full moon, I could not help imagining that she intended to help her deception by the advent of her menstruation. It will be seen hereafter how far I was correct in my conjecture. I passed a delicious night in the arms of my charming Lizzie, and only stole away just in time not to be observed by the early-rising servants. Mrs. Evelyn departed the next day. My mother, feeling poorly, desired Lizzie to sleep with her, so perforce I had to pass a very quiet night, but which the agitation and excessive venery of the last week rendered very acceptable.
Another week passed without anything particular beyond Mary being able to join Lizzie and me in our orgies. The doctor had recommended my mother to go for a few weeks to the seaside, and she resolved that we should all go for six weeks before engaging a new governess. So we left town for a charming little retired village on the west Welsh coast. It was but a small place, with one street, and some straggling houses here and there, but with a beautiful stretch of sand ending in abrupt rocks. Our lodgings were but small; a sitting-room and bedroom above a shop, and two rooms over that. I slept in the small back room off the sitting-room, my mother had the front upper room, and my two sisters were in the room beside her, with only a thin partition between them, so we found ourselves obliged to seek for some outside place to enjoy the erotic pleasures that had now become necessary to us. Very few visitors ever came near the retired little village. In our explorations we found that at the far end of the sands there were some nice retired spots behind the rocks, which soon became the scenes of our sensual enjoyments. The place was more than a mile from the village, and we could see if any one was coming towards us for the whole distance; but still as we might forget how fast time flies, we prudently established either one or the other of my sisters as a sentinel to give us warning if any one was approaching. So I took them in turn, laid them down, had a mutual gamahuche, and then a fuck; after which the previous watcher took the place of the one just fucked, and the same process was followed in her case. We had done this for three days, and were congratulating ourselves upon having found out so safe a place to indulge all our propensities in. We always spent the mornings with mamma, who kept us so far to our lessons, but after our midday meal, which mamma also made her dinner hour, she retired for a siesta, and we went out for a long walk and something better. I have said we fully enjoyed the first three days without any apparent chance of discovery. On the fourth, while Lizzie was on the watch in front, and Mary and I after a delicious gamahuche had just died away in all the ecstasies of a prolonged fuck up to the moment of discharge, and I was saying to her---
``Did not that feel delicious, and was it not up to the hilt?''
``I should think so, with such a rammer as that up her cunt,'' said a strange voice close to us.
You may easily suppose how we startled with surprise.
``Oh, don't do that, I did not mean to spoil sport,'' said the same voice.
It was a very gentlemanly man, with a soft quiet voice, and charming amiable expression of countenance, who stood smiling upon us close to our side, with his breeches open, and his standing pego in his hand. So great was our surprise that we never thought of the state we were in. Mary lay with legs spread out, and belly exposed, and cunt gaping open; and I with my breeches down, and my great big cock pendent, it is true, but hardly diminished in thickness. The stranger said again---
``I am not here to spoil sport, on the contrary, to aid you in every way. I accidentally observed you two days ago. I am here, a stranger, like yourselves. I know you to be brother and sisters, and admire you all the more for being above the usual prejudices of that relationship. But you must be aware that as I know all about you, the best way is to let me be a participator in your sport; and then you not only shut my mouth, but it will be the means of vastly adding to all your pleasures, as well as giving me the most intense satisfaction. Now, for instance, your elder sister there, who was about to replace the younger on the watch, will be all the more satisfied, if I first fuck her. Don't be alarmed, my dear,'' said he, as he observed a sudden move of Mary, who all at once recollected how exposed her whole person was. ``I shall do nothing without your full consent, but I am quite sure your brother, who takes you each in turn, will rather be pleased than otherwise, to see you in my arms, or I much mistake his character.''
I could not help, internally, thinking how exactly be had hit off my very thought, for I had just been calculating, in my own mind, how much better it would be for us to make him a participator with us, rather than an enemy by a refusal. So I at once averred that as it had turned out, it was likely to add greatly to all our pleasure, and I begged Mary to let him have his way. The natural reluctance of woman to appear too easy of access made her simulate a refusal, but as she still lay on her back, I leant over her, and opening her legs, begged him to kneel between and help himself. He gallantly, on kneeling, first stooped forward, and gave a good lick-up of all her cunt's spunk-covered lips, and then proceeded to gamahuche her, which quickly made her as anxious for his prick as he was to fuck her. As soon as they were fairly at it, I whistled, and beckoned to Lizzie to come up. You may easily imagine her surprise to see Mary in the arms of a strange man; but as the sight had had its usual effect on my sensitive organ, and as it was standing, almost ready to burst, I made her kneel opposite to them, and introduced my prick into her cunt from behind, so that we could both see the delicious fuck going on before us. It redoubled our excitement, and all four of us spent together in cries of rapture. After this bout we sat down to make further acquaintance, which, you may suppose, was not difficult, after such an introduction. Our new friend gave us some hints very useful for future proceedings, meanwhile he was feeling young Lizzie's cunt with one hand, and my prick with the other, very nicely and gently frigging it. He brought me to full stand very quickly, and then made me lie on my back, while he proceeded to admire and praise the extraordinary development which he declared was the greatest for one of my age he had ever met with, and his experience was very extensive. When it was at full stand, he stooped forward, and in the most delicious manner sucked my prick. It was more exciting than when either of my sisters, Miss Evelyn, or Mrs. Benson had gamahuched me. He also inserted a finger in my bottom-hole, and eventually made me spend in his mouth, which he greedily swallowed, nor did he cease sucking until every drop was drawn out of me. This had, of course, excited him, and he said---
``Now, I must have the young one in her turn.''
Lizzie, nothing loath, lay down on the grass at once, I conducted his prick into her cunt, and frigged his bottom-hole, while their bout lasted. His prick was one of the middlings, hot very long, nor very thick, but of a uniform size throughout, without any large projection of the nut, like mine. He advised us to stop for that day, and to walk towards the village with him, and then when in full sight, but far beyond hearing, we could sit down and concert measures for future pleasures of the most delicious lubricity.
``I see,'' he said, ``that we shall just hit it. I shall greatly add to your pleasures, and you to mine; you have something yet to learn, and I am the very person to instruct you in even higher delights than any of you have yet enjoyed.''
We followed him as desired, and, seated on a sand hillock, we held a long conversation, and arranged everything for future indulgencies. We agreed to meet at the rocks next day at our usual hour, he undertaking to be there ahead of us, to see that no lurking stranger should have hidden himself as he had done that day. He would think over the matter in the meantime, and contrive some way of meeting where we could be fully at ease, and strip ourselves naked, so as to enjoy a complete orgie of the most salacious lubricity. He showed us where he was lodged, a small inn a little way out of the village with its front to the road, and behind the stables there was attached to it a small cottage, consisting of a bedroom above, with a dressing-room, or small bedroom if necessary, over the passage; the door opened upon the coast, and there was no other communication with the inn than by going round past the stable yard to the front door. The servant of the inn came round in the morning, and laid his modest breakfast of tea, eggs and toast, and when he was done, cleared away and made his bed, \&c. He took his dinner in the inn parlour at the hour the landlord and family dined. Nothing overlooked his windows, and he was sufficiently away from the village not to be easily observed, still less so from the inn; so that on approaching his lodgings from the sands he was almost as safe from observation as if he had lived in a lonely house far distant from any other. I am thus particular in describing his lodgings, as the advantages of the situation afterwards induced us to turn them to a profitable use. Our friend's name was MacCallum, James MacCallum, an offshoot of the great Scotch clan of that name, then in about his thirtieth year, fond of sporting, particularly fishing. His room was surrounded with the necessary implements, and he much frequented Wales from its advantage of possessing so many good trout streams. He it was who gave me a taste for the piscatory art, and I afterwards accompanied him on many a fishing excursion, which often led to new and singular erotic adventures, of which I may, perhaps, hereafter recount a few. His ordinary residence was London, and our present acquaintance led to some most intimate relations of true erotic extravagance, of which more anon.
Meanwhile we met at the rocks on the next day, a Saturday. We found Mr. MacCallum at his post, and all being secure, proceeded to action. It was Mary's turn to take the first watch. Our friend constituted himself master of the ceremonies. He desired me to take off my breeches, and Lizzie to take off her gown and ease her corset, for as yet she wore no stays; then telling me to lie down on my back, he made Lizzie kneel at my head, with her bottom to me, and then to press back so as to bring her charming little cunt over my mouth, her under-petticoat and chemise being well canted over her shoulders. I thus had complete command of her clitoris with my tongue, and she could sink her buttocks quite down on my face, so that I could shove my tongue well up her cunt, and lick up all her spendings when she discharged; and at the same time, while embracing the charming plump hard buttocks with one hand, the other was left free to frig her bum-hole, and stimulate her passions up to the utmost. I have already told you how naturally she had taken to posterior pleasures. While thus engaged, Mr. MacCallum proceeded to gamahuche my prick in the most delicious manner, for he had an art in this delightful accomplishment that far exceeded that of the many by whom I have been gamahuched---of course, he added the \textit{postilion}, as the French say, by frigging my bottom-hole at the same time. He made me most voluptuously discharge in his mouth at the very instant dear Lizzie was pouring into mine her delicious spendings. We lay enraptured for some time before we could stir. Then rising, I wished to return the compliment Mr. M. had paid my prick, by sucking his. But this he declined, saying---
``I shall teach you all a new pleasure before we part, and my powers are not quite so active as your youth enables you to be, so for the moment we will indulge in close observation and sweet caresses of our members until by gentle titillations I get you two more prepared for the amorous contest---''
He gamahuched Lizzie while handling my prick, and a very short period elapsed before he had us both in such a state of excitement that we were ready for anything he chose to direct. This time he also required me to lie down on my back, but he placed Lizzie on the top of me, and guided my prick himself into her delicious tight little notch. When fully inserted, which was completely accomplished before she quite lay down upon me, he desired us to go slowly to work. For a short time, with his face close to my cods, he watched the in and out movement of my prick, inserting a finger into both Lizzie's bottom and mine. Then rising, he said---
``Stop a little, my dears, but don't withdraw. I am about to give your sister a lesson in the double action of most delicious pleasure.''
Then spitting on his prick, and applying a quantity of saliva to the rosy orifice of her bottom, he proceeded to insert his prick---little thinking how fond she was of taking pleasure in this route, and how often she had already enjoyed it. He took every precaution not to hurt her, and to be as gentle as possible, telling her to push out her bottom, and to strain as if she wanted to void something, which he told her would facilitate his entrance, and give her less pain. You may imagine how secretly pleased Lizzie was; she did all he desired---and with great gentleness he succeeded in sheathing his prick up to the close junction of his belly against her buttocks.
``Capital, my dear, you have borne it admirably. I see you will make an apt scholar; now you will have nothing but the most ecstatic raptures from the action of two pricks at once. Now, Charles, it is for you to work, and for your most charming sister to continue only the exquisite pressures she is already at this moment so rapturously conferring on our excited members.''
We thus commenced the first lesson we ever had in the double fuck. Dear Lizzie was almost mad with the agonising sensations of rapturous pleasure the double thrusting produced upon her erotic nerves. I, too, felt the rubbing of Mr. M.'s prick so closely upon mine, for the slight membrane dividing the bottom passage from the vagina, by the powerful stretching of the two members between which it was sandwiched, became so thin a division that it really appeared as if there was nothing between our pricks. Such ecstatic excitement brought matters to a speedy conclusion. Lizzie screamed so loudly with her excess of pleasure that it somewhat alarmed Mary, who came running up to see what was the matter. Her surprise was great at the sight she beheld, but we were far too deliriously wrapt in the lap of most salacious luxury and lubricity to be sensible to any interruption. As for Lizzie she was in convulsions of ecstasy, which ended in quite a hysteric attack which rather alarmed us, and made us withdraw from the exquisite sheaths in which we had been engulphed with such rapture. It was some time before dear Lizzie recovered her senses, and then she burst into tears, declaring she had never before known what pleasure meant, and she had been in the seventh heaven of delight, that she could wish for no better death than to die in such agony of pleasure. She then threw herself into Mr. M.'s arms, and kissing him with the utmost fervour, said---
``Oh, you dear man, how I love you for teaching me such a delicious way of loving; you shall have me whenever and wherever you please. I shall love you as much as I do my darling brother Charlie.''
She then turned to me and warmly embraced me too. Then, putting on her gown, she proceeded to take up the watch, while Mary remained to be likewise initiated in the luxury of the double fuck. She somewhat dreaded the experiment, but having witnessed the ecstasies of pleasure it had thrown Lizzie into, she was not unwilling to try if it could be accomplished with Mr. MacCallum's somewhat less massive member. He put us through the same preliminary manoeuvres of backing Mary on her knees over my mouth, and while he sucked my prick, he feasted his eyes at the same time on Mary's really finely developed buttocks, giving him promise of great after-pleasure. He even begged me to leave her bottom-hole to his finger so that he frigged the bum-hole of the sister while he sucked the prick of the brother, a combination which afforded him the most racy delight, Mary was greatly excited, and spent most copiously in my mouth, while I quickly followed suit in the mouth of Mr. M., who did not allow a drop to be wasted. When we had reposed ourselves sufficiently, his lascivious touches and caresses and praises of our parts soon sufficiently re-excited us to let him see if we might again proceed to action. As before, I lay down on my back, and Mary, straddling across me, had my prick guided into her longing cunt by the hand of Mr. M. When I was fairly engulphed in her hot and throbbing cunt, she began her exquisite \textit{casse-noisette} pressures, which talent she possessed in the greatest perfection; then bending down to me I clasped her in my arms, and glued my lips to hers in a loving kiss and tongue embrace. Her bottom presented itself in all its beauty to our worthy master of the ceremonies, who, delighted with its more fully blown beauties than that of the younger sister, paid first due homage to it by fondly kissing it, and thrusting his tongue up the rosy orifice, titillating her excessively, then wetting his prick he applied it to the tender rosebud-like dimple at first without success, Mary telling him she did not think he could possibly succeed.
``Patience and perseverance, my dear girl,'' said he, ``will enable me to get into a mouse; we must try another way; it is that great huge monster of a prick in your cunt that is so blocking up the route as to close almost entirely the way to the more secret temple of salacious delights. Withdraw for a moment.''
I did so; upon which he plunged in an instant up to the hilt in her cunt and gave a few shoves to excite her and throw her off her guard, for he told us afterwards, the first difficulty was all owing to Mary's involuntary opposition, by squeezing in her bottom-hole, instead of pushing it out. When he thought he had sufficiently excited her, and made her suppose he was going to continue regularly fucking her, he suddenly withdrew the two fingers he had in her bum-hole, by a jerk substituted his prick, and before Mary was aware, had sheathed it more than halfway into her bottom. She gave a half scream, but his hold of her hips, and my close embrace of her waist, for I all along knew what he was at, prevented her from flinching and throwing him out, which was her first impulse. He said---
``I will keep still, and any unpleasant feeling will go off in a moment.''
He stopped for two or three minutes, which I occupied in first rubbing the end of my prick on Mary's clitoris, which was a well-developed one, and when by her nervous movements I found her passions were being roused, I slipped it into her tightened cunt without much difficulty. Mr. M. took the opportunity of finding me penetrating to glide in on his point of attack up to his utmost limit. Mary gasped again, and declared it was choking her. However, by a little more patience, and then by very gentle movements, we gradually worked her up to the utmost state of excitement, and she, as well as both of us, went off in a delirium of enraptured felicity. She lay panting and throbbing between us for nearly a quarter of an hour.
I was already in a state for renewed efforts, but Mr. M. rose, and withdrew his reeking prick from the tight recess in which it had enjoyed such ecstasies, and told us we must be content with that day's work, expressly as he had a plan in his head for the next day, that would require us to have all our erotic powers at command. Then, as before, we approached the village, so as to be seen, but not overheard, so that our going away to more distant places should create no suspicion. Mr. M. then informed us that we could come to his cottage the next afternoon, instead of the rocks; we should be able to undress ourselves in the buff, and have a perfect orgie of salacious delights. We heartily approved of this plan, and after an amusing conversation, we parted to meet the next day on the sands, but in the contrary directions to the rocks, for the purpose of afterwards approaching his cottage from the least observable site.
After dinner the next day we started at our usual hour apparently for our ordinary promenade, but after leaving the village, and allowing most of the people to be safely stowed away in church for the afternoon service, we turned on our steps and made for Mr. M.'s door. He saw us coming, and was ready to admit us, without knocking. We immediately adjourned to the bedroom upstairs, and lost no time in all of us stripping stark naked. After some preliminary admiration of the two girls, whose forms were certainly cast in beauty's mould, we lay down in bed. I and Lizzie mutually gamahuched each other, with the usual accompaniments in the charming orifices of our bottoms. Mr. MacCallum and Mary, for he had taken a great fancy to her and her splendid bottom, followed our example, After we had a happy and most delicious spend, and then mutual embraces and kisses, we put the girls into all conceivable poses, until we were once more ready to go on with something more serious than gamahuching. Mr. M., as usual, acted as master of the ceremonies, and ordered Mary to lie down on her back, then Lizzie reversed upon her, so that she could gamahuche Mary's cunt, and tickle her bum-hole, while Mary was to frig Lizzie's clitoris with one hand, and play with my cods with the other, Mr. M. himself guided my prick into the delicious bottom-hole of Lizzie, and when we were all fixed, and he had frigged my bum-hole with two fingers, he said---
``Now I am going to initiate you, Charlie, into the delight of being alike operator and receiver.''
So saying, he moistened his tool and spit in my bum-hole, and proceeded very gently to introduce his prick therein. I have described his cock as not very thick at the point, consequently the first part introduced itself very easily, but when the pillar pushed its way in, and began to stretch the parts, it produced a curious sickening feeling, very like as if I had received a kick on the bottom; so I was obliged to ask him to halt a little. He was too experienced in the art not to fully understand my feelings, and knew well it would go off in a minute or two, if I was left quiet. So pausing until I told him he might now try to get in further, he drew back a little and applying more spittle to the shaft, gently and firmly, and slowly guided his prick up to the hilt, or as far as his belly and my buttocks would allow. Again pausing a little, until feeling by the throbbing of my prick, which produced the same pressure on my bum-hole, that I was warming to the work, he began slow movements of thrusts in and out, which, together with the hot and voluptuous pressures and movements of my own little partner excited both by Mary's finger and my prick, began to fire my passions, and we soon grew very fierce in our movements. Nothing I could ever have imagined equalled the extraordinary and delicious ecstasy that the double action produced upon my erotic nerves. I gasped, I shuddered with the agony of intense pleasure, and at the moment when the grand and rapturous finale approached, I actually brayed exactly like a donkey, which, in after cooler moments, amused all of us. The action of pleasure had come upon all at once, and we sank in an inert mass on those below us. How poor Mary endured it astonished us, but the scene had so excited her that she said it never occurred to her, and she felt nothing. We eventually rose, and after a necessary purification, partook of wine and cake, which Mr. MacCallum, with great foresight, had provided. After that he would not allow us to fuck for some time; and we had a regular romp all about the room, which we enjoyed very much, and nothing was heard but slaps on our bottoms, and the wildest rollicking laughter---until our two cocks, by their stiff-standing, showed that we were again ready to enter on new combats. This time Lizzie lay down, Mary gamahuched her. Mr. M. got into her bum-hole, and I proceeded to attempt to do the same to him, but all to no purpose. I was too heavy hung for his bottom-hole, a very small one for a man. He had every wish to accommodate me, but do what I would, I could not overcome the physical difficulties. So reversing our positions, I lay on my back, Mary straddled over me, my prick was put into her cunt, and stooping down, and presenting her anus, M. succeeded more easily than the day before in getting into her bum-hole. Lizzie standing up with a leg on each side of Mary's and my body, brought her quim up to M.'s mouth, and he luxuriously gamahuched her, while his finger acted \textit{postilion} in her bottom. The erotic storm raged with great fury for a long time, and then, growing more fast and furious, brought us all standing in ecstasies of the most salacious enjoyment, for us to sink once more into the annihilation of satiated desire. We lay long wrapped in close embrace. Recovering our senses in long-drawn sighs, we again refreshed ourselves with wine and cake, and as our passions were not so quickly reawakened as those of our more excitable companions, we proceeded to gamahuche them, without their exercising a like skill upon our pricks. We then had another romp, and replacing Mary below and Lizzie above, I, this time, fucked her cunt, at her request, as she said it must not be altogether neglected. M., as previously, took me behind, and as there was a greater facility, so there was greater enjoyment, and as our previous exertions had taken off the sharper appetite, we were enabled to draw out our pleasure to a much greater length, until at last we died away in all the agony of such a glorious conjunction of parts. We had one more delicious general fuck before we parted. Lizzie was again fucked by me, and buggered by Mr. M., which she declared she preferred to any other combination, my prick so deliriously gorging her tight little cunt, and making M.'s prick, from the pressure of my larger weapon in the cunt, feel as tight in her bottom as my prick did, when nothing but Mary's finger was in her cunt. We ran our course with even greater luxury and lasciviousness than before. Lizzie actually was hysterical with the force of her enjoyment, and we all sank sideways off poor Mary, and lay long locked in each other's arms. This, for that day, ended our most delightful orgie. We purified ourselves, and then dressed. We parted with many sweet embraces, and promises of renewing the delightful scenes we had just gone through, and, in fact, we often and often repeated them, varying from time to time with a visit to the rocks, lest we should draw observation upon us by constantly going to the cottage.
Our six weeks came to an end so rapidly that we could hardly believe the time had already passed. Mamma one morning informed us we were to leave on the day after the next. You may suppose our disappointment, but there was no help for it. We met that day at the rocks, we were melancholy at the thought of parting with our charming friend, whom we now really loved. We were not near so fiery as usual, but resolved to have one thorough good orgie the next day at the cottage, as a farewell benefit to us all. We met, as agreed on, and put in force every art to augment our pleasures, and every contrivance to excite anew our powers to the utmost. Both M. and I must have spent six to seven times, but the girls being more easily excited in their finer organs of coition, went off in ecstasies some nine or ten times; until fairly exhausted, we had, from want of power, to give up the game, dress and part. We hoped to meet again. The girls wept at parting with our delightful friend, to whom we owed so many delicious orgies. We exchanged addresses, and he promised to come on a fishing excursion to our neighbourhood, where he hoped we should find means of renewing the lascivious sports we had already so much enjoyed. We tore ourselves away from him at last. It will be seen in the sequel, that unforeseen events carried me to London, or rather away from home, before we could meet again; and it was in London, at his own chambers, where we again renewed our charming intercourse, and practised every art of venery.
We returned home, and mamma again advertised for a governess, and stated that she required one of not less than thirty years of age, and with much experience in teaching. Numerous responses were made to the advertisement; but one lady desired to see mamma and her pupils before accepting the place, at the same time forwarding very satisfactory testimonials. Mamma was rather struck with the style of letter, and the unusual demand of previous acquaintance before entering into final arrangements. So she wrote to Miss Frank-land, begging her to come and spend three days with us, and if her visit should prove as agreeable to both as her letter had done to mamma, she had no doubt matters might be arranged to their mutual satisfaction. Accordingly, at the expected hour, Miss Frankland arrived. She was, to our then thinking, an elderly lady, rather above thirty years of age than under, of tall and commanding figure, somewhat large, but no superfluous fat, broad shouldered, and wide hipped, with bosoms well separated, but not too prominent. Her hair was coal black, and her eyes equally so, but with the most determined expression, rendered more so by very thick eyebrows, which met in the middle. She showed also a well marked downy moustache, and the small curly hairs below her head, at the back of her neck, literally lost themselves beneath her high-necked dress. She always wore long sleeves, and never showed bare arms. I afterwards found the reason of this was that her arms were so black with thick hair that she was ashamed to let them be seen, although, in reality, beautifully formed and plump. Her mouth was large; it showed animal passion, but at the same time determined firmness of character. You could not call her handsome, but there was altogether an appearance of face, expression, and person that might well be styled a fine woman. As for us, at the period of first seeing her, we only marked the determined character of her countenance, and at once dreaded her becoming our governess, as we felt we should not only have one who would master us, but who would also be severe in every way. Youth is often a better physiognomist than it is credited with. It will be seen in the sequel whether we had judged correctly or not. Suffice it to say that her three days' visit ended in her being perfectly satisfied with the offered position, and mamma being equally satisfied with her. We did not know at the time, but afterwards found out, that she had made it a \textit{sine qua non} that she should have carte blanche as to the use of the rod. She had observed to mamma that she thought we had been too leniently treated by our late governess, and it would be necessary to exert severe discipline, which, in her own experience, she had always found most efficacious. My mother, who had during the last two months found us rather headstrong and wilful, quite chimed in with her idea, and gave every authority to do quite as she liked, either with her girls or her son.
Terms being so arranged, Miss Frankland required a week to make all her arrangements before definitely taking up her new residence. My mother, thinking we should be well kept in on the arrival of Miss Frankland, left us in uninterrupted liberty until then, you may be sure we improved the occasion, and did our best to make up for the loss of our inestimable and amiable friend, Mr. MacCallum. Not only did we make use of the summer house by day, but every night I stole up to my loved sisters' room, where we tried to emulate the luxurious scenes of lubricity we had lately been so deliriously indulging in at the seaside in Wales. Of course, the week passed far too quickly, and on the appointed day my mother drove into the town to bring Miss Frankland home, on the arrival of the coach. My two sisters accompanied her, as something or other was always wanted for the girls; and as Miss Frankland and her luggage would quite fill the carriage on their return, I was left by myself at home, a most fortunate circumstance, as it turned out.
I was somewhat annoyed at being left alone. But how true it is that ``man proposes and God disposes.'' Had I gone with them I should have missed a most delicious and unexpected treat. I had strolled to the summer house in a sort of despair at the lost opportunity of again fucking my sisters before the arrival of the dreaded governess. I was listlessly gazing out of the window when I suddenly became aware of a lady waving her hand to me from a gig coming down the road which our summer house commanded. In an instant I recognised Mrs. Vincent. To run down the hillock, unbolt the private door, and welcome her to our house, was the work of a moment. I begged her to get out and walk to the house through the grounds, her servant could drive round to the stables and wait there. She did so at once. I never said a word of all being absent until I had her safe in the summer house. Without a word I seized her round the waist, and pressing her back on the couch, quickly unbuttoned my trousers, and pulling up her petticoats, was pushing my stiff prick against her belly before she was almost aware of my intentions.
``My dear Charlie,'' she cried, ``what are you at? We shall be discovered, and it will be my ruin.''
``Oh, no, my ever loved Mrs. Vincent; they are all away to town, and we have nothing to fear.''
She loved me too well to make further resistance; on the contrary, seconding me with all her accustomed art, we both quickly sunk in all the voluptuous raptures of satisfied desire. I would not quit my position, but kissing her rapturously, I shoved my tongue into her mouth, and stopped her remonstrances. The excitement of meeting her after a two months' separation stimulated my passions to the utmost, and with hardly bated breath I began a fresh career, but with more moderation and greater pains to make her a perfect participant in the raptures I myself was receiving. She thoroughly enjoyed it, and being relieved from any fear of surprise, after my informing her of the absence of all the family, she gave way to all the force of her ardent amorous propensities, enjoyed our delicious fuck thoroughly, and spent at the same time as myself with screams of satiated passion. After this I withdrew. She kissed me most tenderly, and said I was as bad and wild a boy as ever, that she loved me too tenderly ever to refuse me anything I desired, and begged me to sit by her side and talk of old times.
``No,'' I said, ``on the contrary, tell me all about yourself; I have not seen you since your marriage day, and I want to know how the after-part went off. I was in dread lest our embracings should have left traces that would make your husband suspect you were not all he had anticipated.''
``You are a strange boy, my dear Charlie, and more of a man in every way than many ten years older than yourself. Who would have thought such ideas would have been running through so young a head. Well, my darling boy, I was somewhat uneasy on that very point myself, and, indeed, had fixed the marriage day when I expected I should be unwell on the very night, but in that I was disappointed; nothing came, and I was driven to act in the best way I could. I kept my legs close together. I got my hand down to that part of my person, and kept squeezing my affair as close as possible. I pressed hard with my fingers on his weapon as he forced an entrance, and all at once gave way with a scream of apparent pain, as he gave an extra thrust, and let him penetrate at once. An inexperienced husband takes much on credit and imagination, I quite satisfied him that he was the first possessor of my person; but, oh! my beloved Charlie, I found I was really ready in the family-way, and you, my dear fellow, are the father of the baby now within my womb.''
``What? I! I! the father of your baby? Oh, dear, darling Mrs. Vincent; oh, say that again.''
``It is indeed true, my dear Charlie; and the knowledge that I first possessed you, and you me, reconciles me to giving my husband a child that is not his.''
``My child! my child!'' I cried, and I danced round in a paroxysm of delight at the idea of being a father. It seemed at once to elevate me to manhood, and puffed me up with pride. I rushed upon dear Mrs. V., embraced her most warmly, and pushing her back on the sofa, said---
``I must see how the little angel looks in his cell.''
I turned up her petticoats, and exposed all her beauteous belly, already by its swelling showing there was more there than ever went into her mouth. Her cunt too had become more prominent. I stooped, kissed her lovely quim, gave it a good suck, then gamahuched her till she cried out for my prick to fuck her, and a most exquisite and rapturous fuck we had. The thought that I was baptising my own babe with my sperm stimulated my lubricity, and we ran a course of the most libidinous delights until we dissolved away in the most voluptuous death-like exhaustion of satisfied desires.
``Charlie, my darling, you must get up; remember you may injure the dear little creature by too great an excess, so pray rise.''
I rose at once, but only to embrace her most tenderly. She complained of feeling somewhat faint, and said we must now go to the house to get some wine. We put ourselves in order, and all radiant at the thoughts of paternity, I strutted along as proud as a peacock, and thinking no small beer of myself. I hardly knew whether I stood on my head or my heels, and was quite extravagant in my conduct. Dear Mrs. V. was obliged seriously to caution me before I could come to a proper reserved behaviour in presence of the servants. She rested about half an hour, and was about to order the gig up to the door, but I implored her to send it round to the road below the summer house, as I should all the longer have the pleasure of being with her. She smiled, and again gave me a pat on the cheek, as much as to say, ``I understand you, you rogue,'' but did as I suggested. So we proceeded through the grounds, and were at the summer house some time before the gig could be harnessed and come round to the road below. I did not wait for that, but embracing darling Mrs. V., wanted to push her down on the sofa.
``No, no, dear Charlie, that will tumble my dress too much, and we shall have no time to put it in order; stop, I will kneel on the low couch, and you will stand behind, I can guide you from below, and you know I always thought you got further in and gave me more pleasure that way than any other.''
She knelt down, and I canted her clothes right over her shoulders, and exhibited her fine buttocks, which, now she was in the family-way, had widened out, and were fatter and rounder than ever. First gluttonously kissing them, I brought my prick right against them. Mrs. Vincent projected her hand behind, seized and guided him into her glowing and longing cunt, and he plunged at one bound up to the hilt.
``Gently, Charlie dear,'' she cried, ``remember our dear baby is there, and you must not be too violent.''
This at once reduced me to moderation. I had a hand on each hip, and as I slowly glided in, I pressed her splendid buttocks backwards to meet me. I kept my body upright so as to enjoy the lovely sight of the movement of her bottom.
``Put your arm round and feel my clitoris, Charlie, dear.''
I did so for a minute, and then whispered---
``It is such delight to gaze on your splendid bum in action, so pray apply your own finger to your clitoris, and let me enjoy the lovely sight.''
``Very well, darling.''
And I could feel her frigging away most furiously. This enabled me to introduce first one and then two fingers into her most delicious bottom-hole. When I found she was in the greatest state of excitement, I suddenly shifted my prick, and substituted it for my fingers. In her surprise and excitement, she had no time to resist, and I glided in, not too rapidly, quite up as far as I could go. She flinched a little, and called me a bad fellow, but I held her hips too tight to allow of her unseating me, even if she had wished. I begged she would let me go on, for I had never forgotten the delight of doing it this way the night before her marriage. She made no reply, but I could feel redoubled action with her finger on her clitoris; and the muscular twitchings of her loins and sphincter soon convinced me that nothing would please her better than finishing our course where I was---and most delicious it proved. We should have died away in loud cries of agonised delight but for the necessity of prudence, for doubtless the gig was then awaiting but a few yards off. My darling mistress seemed unwilling to let me withdraw; she held my prick in such close and firm embrace, throbbing on it from moment to moment, and so exciting him that she shortly felt he was again stiffening inside of her. She rose on her legs, and by that action unsheathed me. Then, turning round, she threw her arms about my neck, and most tenderly embraced me, thanking me for having given her such exquisite proofs of love.
``But I must go, my dear Charlie, and I hope we shall have occasionally some other delicious opportunity of enjoying such raptures again. Say everything kind to your mother and the girls, and tell them I shall come ever again shortly and see them all.''
I saw her into her gig, and watched her until a turn in the road hid her from my sight. I returned to the summer house, and kissed the spot she had last pressed with her lovely body. My soul was filled with love of her, and pride that I was man enough to put a babe into her belly. I strutted about the room, and if any one could have seen me I should doubtless have appeared ridiculous. Mamma, our new governess, and the girls returned to tea. I told them of Mrs. Vincent's visit, and her regret at finding them absent, also of her promise to drive over again on an early day. My mother hoped I had been attentive to her. I said I had, as well as I could, and had got some wine and biscuits, as she complained of not feeling very well, she thought the jolting road had tired her.
It may well be supposed that after the impression our new governess had made upon us, we were very attentive for some time. Indeed, her system of teaching was really excellent, far superior, in that respect, to our former governess. She had a method of interesting you in what she was teaching, and for quite two months we paid such great attention, and made such really extraordinary progress, that she could not help praising us highly to mamma while we were in the room. This was bad policy, because, with the natural thoughtlessness of youth, we fancied ourselves so clever that we became less attentive. This was patiently borne with for some time, probably in consequence of our previous good behaviour. But at last Lizzie was somewhat impudent when blamed rather harshly by Miss Frankland.
``Oh! it has come to that, has it? We shall see.'' She continued our lessons until four o'clock as usual, and then desired Lizzie to remain where she was; she dismissed Mary and me, locked the door on poor Lizzie, and went away, doubtless for a rod. She soon returned, and locking herself in, most severely whipped poor Lizzie's bottom. She sent her out when it was finished, and Lizzie joined us, weeping bitterly from the pain she was suffering. We laid her on the couch, and turned her petticoats over her head to cool her bottom, which she declared felt as if burning hot coals were spread over it. I kissed the dear red buttocks that were all covered with weals and looked like raw beef, but no blood had been drawn. We fanned her with our handkerchiefs, which she said was a delightful relief. In a very few minutes she began to wriggle her bottom in a state of excitement, and cried out---
``Dear Charlie, do shove your prick into my cunt, it has begun to long for a fuck.''
I wanted nothing but this to instantly act, for the sight of her bare bum had already made my cock stand as stiff as iron. She raised herself on her hands and knees, presenting the back entrance to her cunt, and telling me it was there she must have it instantly. I plunged up to the hilt in a moment, for she was as juicy and moist as if she had spent, which it is more than probable was the case. Very few powerful thrusts on my part, seconded by energetic action on hers, and she spent again with a scream of delight, and with a pressure on my cock that almost hurt it. She hardly paused a moment before she cried out---
``Shove on, dear Charlie, push it in further if you can, I am burning with desire.''
She wriggled her backside in every way in the most lascivious and delicious manner, and when she felt the crisis approaching, by the increased swelling and hardness of my prick as well as the peculiar electric effect at the moment, she met my flood of sperm with so copious a discharge that it literally spurted out and deluged my cods and thighs. She held me tight, and would not allow me to withdraw until I had myself spent four times and she seven at least. We then rose, her nerves calmed by the repeated doses of hot boiling sperm shot into her interior. She declared that never in all her fucking had she felt such insatiable desire, or more ravishing delight in satisfying it, that she would undergo a dozen such floggings to have the same rapturous enjoyment.
``I am sure,'' she said, ``it was all the effect of the rod, I never felt anything like it before.''
Mary all this time had been but a spectator, and a pleased one to see the erotic fury of her sister and my powerful efforts to allay it. It is true we had both had a delicious fuck during the time poor Lizzie was catching it on her backside, and I had just gamahuched her deliciously afterwards as Lizzie came in in such pain.
Miss Frankland had retired to her room, and looked still flushed and somewhat wild looking when she joined us after the usual hour's recreation. As may well be supposed, we were all as attentive as possible. There was one circumstance that evidently pleased Miss Frankland immensely. When Lizzie, in her turn, went up to repeat her lesson, she suddenly threw her arms round Miss Frankland's neck, and with tears running over her cheeks, sobbed out---
``Dear Miss Frankland, pray forgive me, and let me kiss you, for I love you dearly.''
There was a bright sparkle of delight in Miss Frankland's eyes. She clasped Lizzie round the waist, and drew her to her lips in a long sweet kiss of love, which seemed as if it would never end. We observed Miss Frankland's colour rise. She at last put Lizzie away, and said she was a dear amiable girl, whom she could not help loving.
``Go to your seat, you are too agitated, my dear, to say your lesson just now; so send Mary up.''
Lizzie came back to her seat, but I could not help fancying I saw a complete expression of erotic desire on her countenance. When afterwards we were alone together, she told us that when the governess kissed her, she felt Miss F.'s tongue glide into her mouth, and ``tip her the velvet'' in a most delicious and exciting manner, and she believed that if they had been alone they must have given each other mutual embraces of a warmer description. This led me to think that Miss Frank-land was herself rendered lecherous by the action of even wielding the rod. Lizzie during the whole of the next week did nothing but rave of the excessive excitement that her whipping had put her into, and the extreme felicity she felt in having her salacious lechery satisfied. We were not able to meet every day, for frequently Miss Frankland accompanied us, and joined in the youthful sports we then gave way to. Lizzie continuing to harp on the extraordinary enjoyment the whipping had procured her, after it was over, fired the imagination of Mary, until she was wound up to a pitch of actually longing to be whipped. In such a case it was easy to incur the penalty; she had but wilfully to neglect her studies, and she was sure to get it. This she accordingly did, and it resulted as before. When released, she rushed to the summer house, and without any preliminaries, called upon me to fuck her directly; and a very similar scene followed to that which had occurred when dear Lizzie was whipped. Mary did not, however, give way to the uncontrollable desire to throw herself into Miss Frankland's arms as Lizzie had done. Miss F., as usual, retired to her room after the punishment was over, and was late in coming down, with the same flushed face and excited eye. I became convinced that she herself was salaciously excited by the act, and I began to fancy that with such passions, if I could but excite her in any way, it might be worth my while. When once these lecherous ideas were raised in my imagination, desire soon painted her with every charm of beauty, and I became excessively lewd and anxious to possess her. The more I looked at and scanned the really beauteous proportions of her finely developed form, the more my determination to have her took root, and grew strong within me.
About this time Miss Frankland, who had become a great favourite with mamma, obtained permission to take possession of the spare bedroom, with an understanding that she was to cede it to any visitor who might come. Of course, this circumstance made my desire to get into her good graces doubly strong, inasmuch as the opportunity of sleeping with her afterwards could be so easily effected. I determined to watch her when retiring to bed, and try to get a view of her naked form. For this purpose I removed the stopping of moistened bread I had put in the hole I made to see Mr. Benson fucking his wife. I lay awake, until she came to bed. I saw her undress, but only caught sight of her naked bubbies, over her chemise. As I have said, they were not large, but widely separated, with a fine flat neck up to the throat. I mean that she showed no collar bone, which is a great beauty in woman. She had evidently been quite naked, and had used the bidet, but the extent of the slit in the door did not allow me to command the part of the room where she had used it. I remedied this defect next day, and the following night was rewarded with a most glorious sight. You may well suppose that I did not let sleep overcome me, but was at my post as soon as ever I heard her enter her room. I was on my knees in a moment, at my peephole, and saw her deliberately undress to her chemise. She then arranged all her magnificent head of hair, brushing it out as far and further than her arms would extend; and after well brushing and combing it, she plaited and rolled it up, in a great big rouleau behind, then washing her hands, she drew out the bidet, poured water into it, and then divested herself of her shift. She was standing in front of the dressing-table, with two candles shining on her, so that when she lifted her shift over her head; I had a well-lighted full view of her wonderfully covered belly. She was all over hair; it was as black as coal, and shone as if polished in all its beautiful curls. I am now an old man, but never have I seen the equal to that dear woman in a hairy belly. It was quite up to her navel, and several inches down the inside of her thighs, besides running thickly in the chinks of her bottom, and with two bunches where the beautiful back dimple is usually situated, as thick, and even thicker than ordinary women have in on their mounts. In addition to this, there was a beautiful little line of curls that ran up her belly, as far as between her bubbies, to say nothing of the very hairy thighs, legs, and arms. I never saw a more deliciously hairy woman, and she was all that such excessive growth of hair denoted---passionate and lecherous to a degree, when once she had confidence in her companion, to let her feelings have vent. Of course, I am now describing my after-experience; at the moment I was only dazzled by the extraordinary richness and quantity of that exquisite ornament---hair---not only in splendid quantity on the head, but in a profusion such as I had never then and have not since witnessed. I was struck dumb with astonishment and admiration. She laved her hairy cunt, and all the adjacent parts, then wiped herself dry, put on her night-gown, extinguished her light, and, of course, got into bed. So did I but only to toss and tumble, and at last, in troubled sleep, to dream of that most gloriously covered cunt, and to imagine myself revelling therein. So great was my excitement that I had the first wet dream I ever experienced. It is needless to say, it was under the dreaming idea that I was enjoying to the utmost that wonder cunt.
I was quite exhausted by morning with such a restless night, and was not only very \textit{distrait}, but was really so fatigued that I could not attend to my lessons. Of course Miss Frankland noticed this, and being unaware of the cause, attributed it to wilful idleness and bravado of her authority. She spoke very gravely and seriously to me, and told me if I did not improve my conduct by next day it would be her painful duty to punish me with severity.
``I expect to see you exhibit very different conduct tomorrow, otherwise you will drive me to do that which I would much rather not.''
It rained hard that afternoon, and we had to amuse ourselves within doors. On retiring for the night, I determined to watch again for Miss Frankland, but my want of rest the previous night overpowered me, and I fell fast asleep until far in the night. I rose and crept to my peep-hole, but all was dark. I could hear Miss Frankland breathing heavily. The thought at once struck me that I might safely steal up to my sisters' room, as they were now alone, since Miss F. had the previous night removed to the spare bedroom, where she was now fast asleep. So softly opening my door, and leaving it ajar, I crept along the passage, gained my sisters' room, and gently awakening them, jumped in between them, to their great joy and satisfaction. We immediately began with a gamahuche, I taking Mary's cunt, while Lizzie crossed her legs over her head, and was gamahuched by Mary, whose finger was at the same time acting \textit{postilion} to her charming bottom-hole, while I had the exquisite prospect before me of their operations. As soon as ever Mary spent I made Lizzie lie down on her back, with her head towards the bottom of the bed, Mary knelt over her in the opposite direction, presenting her very full backside, which was daily developing larger proportions. I plunged into her cunt, plugging her little rosy bum-hole at the same time with my middle finger, while Lizzie did as much for me, at the same time rubbing Mary's clitoris with the fleshy end of the thumb, while Mary, at the same time she herself was fucked and frigged in two places, was employed in gamahuching Lizzie, and frigging her bottom-hole with two fingers, Lizzie declaring that one finger felt as nothing. We lengthened out our delicious proceedings until excess of excitement compelled us to give way to all the fury of our feelings, and we managed to spend all together with such rapturous and lascivious delight as rendered us quite powerless for some time. We then had a delicious cuddle, the girls having each one hand on my prick and the other on my buttocks. When we had once more worked ourselves up to fucking heat, we reversed the previous position, and I fucked Lizzie. Mary was gamahuched and bottom-fingered by Lizzie, while she employed herself with Lizzie's clitoris and my bum-hole. Lizzie was far hotter and more salacious than any of us, and spent copiously on my delighted prick, which enjoyed excessively the warm bath of glutinous liquid that was poured down upon it. I gave a few slow-drawn thrusts in and out, to moisten well its whole shaft, and removing my two fingers from her delicious bottom-hole, and wetting it with my saliva, I withdrew my prick from the reeking sheath of her cunt, and to her great delight slowly housed it in her longing and exquisitely delicious bottom-hole, keeping it quiet there for some time, so as not to spend before Lizzie was ready. I enjoyed the delicious throbbing of her body, which at last becoming too exciting, I stooped over her, passed a hand under her belly, replaced Mary's fingers, rubbing her clitoris while Mary frigged her cunt with two fingers thrust into it. We thus quickly brought matters to an end, and died off in all the ecstasies of satiated lust. As daylight was beginning to dawn, I tore myself from their loving embraces, gained my room in safety, and slept the sleep of the just until late in the morning.
My orgie with my sisters had so far satisfied my animal passions that I rather began to dread the severity I knew Miss Frankland would use if I came under her hand. This made me so far attentive next day as to satisfy her; and as it was a fine afternoon she came out to walk in the garden, while we innocently amused ourselves. That evening I kept awake, and again enjoyed the superb display of Miss Frankland's wonderfully hairy cunt, all the lower part of her body was as black as a chimney sweeper's. The sight awakened every lustful feeling within me. I felt I must possess her, and determined to brave the severest infliction she could give me with the rod. I somehow, instinctively, arrived at the conclusion that this extraordinary profusion of hair could only grow where nature had implanted the hottest animal passions, and had but to greatly excite them to turn their lust to my advantage. I determined that to-morrow I should bring things to a crisis, and that I might be equal to every effort I went to my bed, and did not attempt to steal up to my sisters' room. Next day nothing could be made of me in the morning; Miss Frankland sternly warned me that if such conduct was pursued after dinner nothing should save my skin from a severe scourging. However, my mind was made up, and I went in ``for the whole hog,'' as our vulgar Yankee cousins say. I was more idle and insubordinate than ever. Miss F. looked thunder; at four o'clock she ordered me to stay, and the girls to go. She then locked the door, took out from the desk a formidable rod, and told me to approach her. I did so---really half in fear, for she could look dreadfully fierce and determined, in which case I came up to her side.
``Now, Charles,'' she said, ``your conduct, for two or three days past, has been such as I cannot put up with. Your mother has given me full power to punish any of you severely, if I think you deserve it; you are getting to be of an age that I hoped you would have so acted as to give me no cause of offence, but I am sorry to see my hopes are disappointed. I am now about to punish you, submit to it quietly, or it will be all the worse for you. Unbutton and put down your trousers.''
I felt I must submit, but when brought to this point I really so much dreaded her that there was not the slightest erection in poor cockey.
While I was undoing my trousers, I observed that Miss Frankland had quite lifted up her outer frock, and had sat down, evidently intending to flog me across her knee. Both being ready, she told me to put the footstool by her side and kneel upon it, then desiring me to bend forward over her knees she put one hand over my body to hold me down; then uncovering my bottom, and taking the rod, which was by her side, she raised her arm and gave me a fearful cut, which made me not only flinch, but cry out most lustily. Blow followed blow, causing at first great agony, that made me cry again in good earnest; then the very continuance of the blows seemed to deaden the parts until I hardly felt them. This was succeeded by a titillation and lascivious excitement which speedily brought my prick out in the fullest vigour. I then began to push it against Miss Frankland's thigh, and to wriggle myself nearly off her knees. Seemingly to prevent this, she passed her left arm quite round my body, bringing her hand under my belly, and, apparently by accident, against my prick, which she grasped, and I could feel her hand pass both up and down it as if she was measuring its length and thickness, continuing all the time to shower down blow after blow on my devoted backside. As she held a firm grasp on my prick, I pretended to be evading the blows, while in reality I was thrusting it in and out of her hand with the utmost energy and excitement, which speedily brought on the delightful crisis, and with a cry of rapture I gave down a copious discharge into her hand, and sank almost senseless on her lap. I pretended complete loss of consciousness, which she believing, she gently felt, and even frigged a little, my prick, pressing me the while close to her body, and then I felt a shudder run through her whole frame. I have no doubt she was in a paroxysm of lust, and had spent, I gave her time to recover a little, and then pretending to come to my senses, but in a confused state of ideas, said---
``Oh, what has happened? I have been in heaven!''
Then raising myself, and apparently only just recognising Miss Frankland, I threw my arms round her neck, and exclaimed---
``Dear Miss Frankland, do flog me again if it will produce again such ecstasies as I never before experienced.''
Her face was flushed, her eye shone with all the fire of libidinous passion. My prick had hardly lost its stiffness when I spent, and was now projecting out firmer than ever.
``Why, Charles, I thought you a mere boy, while you are quite a man with such a thing as this.''
``Oh!'' I cried, ``do continue to hold it, you give me such pleasure!''
``Has anyone else ever held it in this way?''
``No, I never felt anything like it before.''
``But don't you know what this is meant to do?''
``Oh, yes, it is what I piddle from.''
She laughed, and asked if it was often in its present state of stiffness.
``Every morning when I awake it is so, and it hurts me very much until I piddle.''
``And has no one ever taught you any other use of it?''
``No, what use can it be of?''
``You dear innocent boy, if I could trust you, I would teach you a secret that this dear thing would greatly enjoy. But can I trust you?''
``Oh, certainly, dear Miss Frankland, I know what you mean now, to repeat the delicious sensations you gave me a few minutes ago. Oh, do, do! do it again, it was far too nice for me ever to tell anybody, as long as you will do it for me.''
``Well, Charles, I will trust you. Do you know that women are differently formed from you?''
``Yes, I used to sleep in mamma's room, and I have often been surprised to see that she piddled from a long hole, and had not got a doodle like I have to piddle from.''
``My dear innocent Charlie, that long hole was made to take in this dear fellow here that is throbbing almost to bursting in my hand, and if you promise me faithfully never to tell any one, I will teach you how it is done.''
You may be sure my protestations of secrecy were most earnest.
``Look here then, my dear boy, and see what I have got between my legs.''
She laid herself back on the long chair, drew up her petticoats, and exhibited to my charmed gaze the wondrous wealth of hair she possessed. Opening her legs, I saw the wide-spread rosy lips showing themselves in beautiful contrast to the coal-black hair that grew in the greatest profusion all round the lower lips, and extended also some five or six inches down the side of each thigh. But what at the moment most astonished me, and drew all my attention, was to see a deep red clitoris standing out from the upper part of her cunt quite stiff, and as long and as thick as the middle finger of a man. I very nearly betrayed myself at the sight, but, fortunately, was able to keep up the character of apparent ignorance I had hitherto shown, and said---``You, also, have got a little doodle to piddle with?''
She laughed, and said---
``It is very different from yours. Give me yours here, that I may kiss it.''
She fondled it for a second or two, and then could not resist the impulse to take it into her mouth and suck it.
``Oh, what pleasure! I shall die!''
``Not yet, dear boy; kneel down there, and I shall instruct you in the real secret of pleasure.''
But, before she could do anything, I threw my head down, crying out---
``I must give this pretty little fellow a taste of the pleasure you have just given mine.''
And in an instant I had the delicious thing up to the root in my mouth, sucking furiously at it. Her twistings, and up and down action of her loins, showed how rapturously I was exciting her, in fact, I brought on the crisis, when she pressed my head down hard upon it, and closed her thighs on each side of my head, as she poured over my chin and breast a perfect torrent of sperm. A minute after she seized my arms, and drew me up on her belly, then slipping her hand down between us, she seized my prick and guided him, nothing loath, into her burning hot and foaming cunt. She placed her hands on my buttocks, and pressing me right up to the hilt, began a movement which she told me how to second, that in a very short time brought down an exquisite spend from me. The idea that she was giving me the first lesson in love, and of being the first possessor of my person, seemed to excite her lust to the utmost, and she immediately followed my discharge with another, so copious that it spurted all over my thighs. Her force of pressure on my prick in her agonies of enjoyment was so great as nearly to hurt me. I never knew any one but her with such strength of pressure of cunt on the prick. She has often actually brought tears into my eyes, so powerful was her grip that it made me really feel as if in a vice. She lay back with closed eyes and panting bosom in a rapturous trance of lascivious lubricity, her throbbing cunt holding me tightly pressed between its palpitating folds in the most delicious imprisonment, and from time to time grasping my prick with a pressure that very shortly restored it to its fullest vigour and stiffness. She was as hot as fire and responded immediately to the renewed life she found stirring within me. She gave way to her salacious lust with, if anything, a more passionate excess than the first time. My superb weapon seemed to stir up within her a force of lubricity that nothing could seem to satisfy. Her hands clutched my buttocks convulsively, and seemed to wish to force my whole body into her wildly excited cunt. With such vigour was the action carried on that the grand crisis soon arrived, most rapturous to both, and almost maddening to Miss Frankland. The heavings of her body and gaspings for breath were quite hysterical, while, with one of those real vice-like pressures, I felt as if she were nipping my prick in two. It was not a mere throbbing pressure, but a long continued convulsive squeeze, as if her cunt had been seized like the jaws of the mouth with lock-jaw, and could not open. It was nearly ten minutes before she recovered her senses. She seized my head between her hands, kissed me most lovingly, declared I was the dearest creature that ever lived, that she had never before had any one who had so satisfied her, and filled her with inexpressible rapture, \&c. This fondling had again brought up my prick to full stand. Miss Frankland said---
``Dear Charlie, we must be prudent, as the time is drawing near for your sisters' return.''
But there was no stopping, the exquisite pleasure of her splendid interior cunt pressures was irresistible. My movements speedily determined matters in my favour. Miss Frankland's temperament was far too warm not to quickly set her passions to the highest fucking heat; and again we had a most exquisite fuck, lengthened out more luxuriously by the more urgent fires of desire having been moderated by the three previous discharges. With more abandon we both sank in the death-like ecstasies of the delicious melting away in all the luxury of contented and voluptuous discharges. Miss Frank-land lay for some short time luxuriously closing in my delighted prick, but raising her body, she said---
``Charles, we must cease for the present.''
And, pushing me away, I was forced to withdraw; but her dear cunt seemed as reluctant as myself, and held my prick so tight that I had to pull hard to draw it out, and, at last, he left with a noise like drawing a cork from a well-corked bottle. Before I rose, or she could hinder me, I threw myself down and glued my lips to her reeking cunt, and greedily licked up the foaming sperm that had surged out of her well-gorged quim. She with difficulty drew away her body, but as I rose she clasped me to her bosom and kissed me most fervently, and licked her own sperm off my richly covered lips. Begging me to button up, and putting herself to rights, she desired me to sit down by her side. She wiped my mouth with her handkerchief, arranged my disordered neck-tie, collar, and hair. We then embraced most tenderly, and she thanked me for the immense gratification I had given her; she praised my parts as being of extraordinary development and more satisfying than any she had yet had any experience of. This was the second time she referred to other experiences. I took no notice of this all the time, as if I was supposed to be too ignorant or innocent to think any harm of it, but I determined in some excess of passion to get her to give me a recital of some of her previous experiences.
Before my sisters came in, she said---
``I shall try and arrange some means for our meeting unobserved to-morrow. Meanwhile, you must sit as if you had been severely punished, and I shall assert that you had done everything to resist my authority, for which I had punished you further by not allowing you to leave the school-room.''
I said not a word to Miss F. about the ease of meeting by merely opening the door of communication between our rooms. I was afraid to make her suspicious of a former use of it. But I determined, when she came to bed, to rap at the door and beg her to open it, and I had no doubt she would be as delighted as myself to find with what facility she could indulge to the utmost every libidinous passion which her lascivious nature could suggest. My sisters returned, and appeared disappointed that I had not been able to join them, as they had anticipated a glorious fuck or two each, after the whipping had excited me as it did them. They told me afterwards they had been obliged to content themselves with a double mutual gamahuche, but it did not make up for my absence.
While they were all engaged after tea, I slipped up to Miss Frankland's room to see that the key was in the lock of the door between our two rooms. I opened it, oiled the hinges, and locked it again from her side. I also, with a view to sometimes slipping up to my sisters' room, oiled my own and their doors, hinges, and locks, as now that the ice was broken with Miss Frankland, it would be necessary to be doubly careful not to excite suspicion of my visits to my sisters. Having finished everything to my satisfaction, I joined them in the drawing-room, and while my sisters were playing duets on the piano to mamma, I challenged Miss Frankland to a game of chess. She, of course, was a far superior player to me, but our legs meeting under the chess table, her little charming foot sought mine, rested on it, and pressed it from time to time. This distraction of her ideas enabled me to win two games successively. My mother sent the girls to bed, and told me to follow their example, but as I did not wish to lie long waiting for Miss Frankland's appearance in her bedroom, I pleaded for relaxation in the hour of retiring, to enable Miss Frankland to regain her chance of beating me, at the same time pressing her foot as a sign to her to second my request. She took the hint, though she had no idea of the object. Mamma came near us to look over our game. This induced Miss Frankland to play with more caution and thought, and she won three games in succession, making her the final winner. Mamma now said I must go to bed, as it was very late for me. She still treated me as a child. I, however, had gained my object in obtaining nearly two hours' delay in going to bed, so that I had not long to wait before I heard Miss Frankland enter her room. I determined to let her finish her toilet before I called her attention to me. I watched through my peep-hole, and could now calmly and leisurely see all the beauties of her well-developed form, and the rich wealth of hair she possessed. She went through all her ablutions as usual. I observed she also used a syringe to thoroughly purify the inside of her glorious cunt. When she had dried herself, and was about to pull on her chemise, I rapped on the door of communication, and in a loud whisper called her attention to me.
``Are you there, Charlie?''
``Yes, pray unlock the door and open it, that I may come to you.''
She actually had not yet discovered that the door, locked and bolted on her side, communicated with my bedroom, but her delight at the discovery was greater than her surprise. I flew into her arms, and was hugged to her bosom, and covered with kisses. But as my prick was in a bursting state of erection, I drew her to the bed, upon which we both threw ourselves, she on her back, and I above, and in an instant I was engulphed up to the cods in her glorious and glowing cunt, and we ran an eager course of rapturous thrustings, until nature could stand no more, and we sank in all the delights of a most delicious mutual spend. I lay soaking in bliss for some time, and after fondling each other, Miss Frankland said---,
``Get up, dear Charlie, and let us get into bed.''
For we had been in too great haste to do otherwise than tumble on the top to it. My charming bed-fellow also rose for a necessary purpose, which I had interrupted when I knocked at the door. She sat down on the \textit{pot de chambre}, and a mighty rush of water followed. I cried---
``Oh, do let me see you piddle from your beautiful fanny.''
I still kept up my character of innocence, and used none but infantine words in reference to our organs of generation.
She laughed, but pulled up her shift, and raised her thighs above the pot, so advancing the light, I had the delicious sight of her wide-stretched cunt, pouring out a stream of piddle with great force. Her position brought out all the beauties of the vast wide-spread mass of black curly hair that thickly covered all the lower part of her magnificent quim, ran down each thigh, up between her buttocks, and opening out on her back, had two bunches just below the two beautiful dimples that were so charmingly developed below her waist. There was as much hair there as most women have on their mons Veneris. Her whole body had fine straight silky hair on it, very thick on the shoulders, arms and legs, with a beautiful creamy skin showing below. She was the hairiest woman I ever saw, which, doubtless, arose from or was the cause of her extraordinary lustful and luxurious temperament. The sight I was indulging in brought out my pego in full bloom; as we both rose she saw it sticking out under my shirt.
``Off with all that, and let me gaze on your charming young perfections.''
I did as she desired, begged her to do the same, and there she stood, in all the glory of her superb form. We encircled each other's naked bodies, and then turned each other round to gaze on all the exciting charms displayed to each other.
``Come, my darling boy, and let me kiss and fondle you all over.''
She laid me on my back, reversed herself above me, and taking my prick in her mouth, after first feeling it most gently, and praising its large proportions, again declaring it was the finest she had ever seen, she began to gamahuche me with a skill such as I had never before experienced, and gave me the most exquisite and most luxurious delight. For my part, seeing her wonderful clitoris, stiff standing out of the bright red lips of her luscious cunt, I took it bodily into my mouth, sucked it, and rolled my tongue about it, to the evident delight of my salacious companion. Her buttocks rose and fell, and the lips of her cunt immediately before my eyes opened, or closely pressed the lips together, showing the delicious nature of her enjoyment. I felt her put her hand to my bottom and insert her finger, and begin frigging me there. I let her see how it pleased me. She stopped a moment, to beg me to do the same to her, anticipating my earnest desire to do so. I lost no time in following her example. The parts adjacent were well lubricated by our previous indulgence, and first inserting two fingers into her deliciously juicy cunt to moisten them, I slipped one of them into her charming bottom-hole, and finding great ease of space, slipped the second in as well. My other hand and arm embraced and caressed her magnificent backside, which rose and fell on my face with unwearied speed, as my finger frigged her bottom-hole in unison with her movements, and my mouth more closely sucked her stiffly excited clitoris.
Her whole body became convulsed with erotic movements, showing what force of lubricity our mutual embracings were most rapturously exciting. I, too, grew wild with desire, and was equally energetic in my movements, and would have thrust my prick down her throat but for her hand, which grasped the lower part of the shaft. The rapturous crisis came at last and laid us prostrate with soul-killing ecstasies. We each retained the dear object of our mutual caresses within our lips and our fingers remained within the delightful recesses that had so much contributed to the excessive raptures we had enjoyed. We lay for some time in this sweet languid enjoyment. Miss Frankland then rose from off me, saying---
``My darling boy, we must now get into bed.'' We did so, quite naked as we were, closely embracing, and covering each other with kisses and caresses, murmuring soft terms of endearment, and in whispered accents told of the ecstatic joys each had given the other. Our hands wandered over every charm. Miss Frankland had an art of gently passing her fingers over my prick that had the instant effect of raising him into the fullest vigour. It was the most exquisite method of feeling my cock I ever experienced. She seemed scarcely to touch it, but drew her fingers along its length, from foot to head, with a delicacy of touch I have never found equalled by any other woman. The effect was magical, and invariable, no matter how many times I might have fucked her before. With her hot temperament, and excessive lubricity, it was almost a necessary art. She was one of those libidinous natures that could well employ several men at once. At my happy age, she found ready to her hand one who could respond to her every desire in every way, so happily does nature second youth and health that she never found me wanting, when called on. There was no excess of lubricity we did not afterwards practise. We satisfied our passions in every way in which they could be indulged, nor did we hesitate at any thing which imagination could fancy would stimulate them. She was surprised at my aptitude, and rejoiced and congratulated herself on having found so powerful and charming a satisfier of her libidinous nature. How delighted she was to think she was the first to cull the sweets of my innocence, and how happy to find so apt a scholar, who in one sweet lesson became a master of the art. The more I gained experience of the charming sex, the more I appreciated the wisdom of the counsels of my really first and ever loved mistress, dear, charming, lovely Mrs. Benson. How truly she had foretold that all who might hereafter think that they were giving me the first lesson in love would doubly, trebly, a hundred fold enjoy the sweet intercourse from such self-deception. Here was my fiery Miss Frankland, who had had considerable experience in the amatory world, pluming herself upon instructing an innocent youth in all the mysteries of the passions for the first time. It evidently added immensely to her excitement. Indeed, in our after-conversation, she avowed that as it was the first time she had ever taken the maidenhead of a youth, so it had been the greatest degree of excitement she had ever experienced. I might fancy her delight at finding combined with such a satisfaction a wonderfully well-hung youth, and who proved so apt, and so equal to every luxurious whim that the most erotic lust could suggest. But I digress. At present, her magic touch had brought me up to bursting point, she threw a leg over me, and raising her body, said she would help herself this time. Guiding my prick to the wanton lips that were longing for him, she sank slowly down on the stiff pole on which she was so delightedly impaling herself, until our hairs were crushed beneath her weight, and nothing more could be engulphed. She again rose, until the edge of the nut showed itself at the mouth of her cunt, and then as slowly sheathed it again. She continued this exquisite movement for some time, to our delicious mutual enjoyment, then falling down on my belly, and telling me to pass my arm round her bottom and finger her as before, she glued herself to my lips, our tongues interlaced, and shot in and out of our luxurious mouths; our movement grew fast and furious, until we again sank in all the luxury of the last grand crisis. It was the very act of voluptuous rapture, and we lay lost to every sense but that of erotic ecstasy and satisfied lust. When we recovered our senses, she lay down by my side, cuddling me most closely, and toying and prattling, until she thought we had paused long enough. She slid her hand down to my prick, and very quickly, by her delicious and delicate handling of it, renewed its full vigour. Throwing her right leg over me, while lying on her back, she heaved up her body into a position half turned to my belly, I lying on my side; she then bid me embrace her other thigh between mine, then guiding cockey to the entrance, she gave a push backwards, to meet my forward thrust, when it was instantly sheathed to the hilt.
``Now, my darling boy, in this way we can lengthen out our pleasures as long as we please; you can make me spend oftener than yourself, which will satisfy my very lustful nature, and not over exhaust your young powers.''
Giving one or two delicious side wriggles to her bottom, and nestling her backside close to my belly, she told me to pass my left arm under her waist that I might embrace her left bubby and finger its nipple, a proceeding which she told me was as exciting as playing with her clitoris---then turning her head, our tongues interlaced; she put my right hand down to her stiff-projecting clitoris, which I continued to frig just as I might have done to a boy's cock. Keeping up a slow in and out movement with my prick, excited by so many points of lascivious friction, she spent most copiously before I was prepared to join her. Her head sank back in the ecstasy of her discharge, drawing away from me, and leaving my mouth free. I instantly dropped it upon her other firm and elastic bubby, at which I sucked away, pushing my prick as far as possible into her cunt, and leaving it there, without movement, to enjoy the rapture-giving pressures of her delicious cunt, slowly passing my hand up and down her still sufficiently indured clitoris. She lay for some time in the luxurious enjoyment of the position, then once more sucking my lips, she thanked me over and over again for the pleasure I had given her, heightened as it was by knowing that it had not exhausted me. I began to move slowly in and out, keeping up my movements at the other points of excitement. She was ready on the instant to second me, and as she meant this time that we should spend together she left nothing to desire. Her movements were of the most exciting and stimulating description, and we were not long before the ecstatic moment arrived, and we sank in the lap of luxury, pouring forth streams of ecstatic bliss. We lay close locked in the most delicious embrace, only conscious of unutterable joy. It was some time before we could venture to break this exquisite trance of enjoyment. It was followed by the sweetest toyings and prattlings, until again my delighted prick, stimulated by the internal pressures of the luxurious sheath in which it had remained engulphed, again awoke her scarce-slumbering passions to dash on pleasure's heavenly course. Again she spent before me with, if anything, increased rapture, and, after a pause, renewing her lascivious movements in response to my own, we sank in a perfect death-like swoon of thoroughly satiated lust, and gradually and imperceptibly fell into the deepest slumber for many hours, locked as we were in each other's arms. Her wonderfully retentive power of cunt held my happy prick a willing prisoner through our long sleep. I awoke first, to find it standing stiff within the charmed circle which even in her sleep was deliciously grasping it with its nervous folds. I passed my hand down to her clitoris, and began fucking her. She heaved her bottom up and down, and murmured some incoherent words, being evidently still under the influence of sleep, and probably dreaming of some former events, for in her half expressed murmurings, I could make out something---
``Henry---my only---ever loved one---meet again---oh! how ineffable---how exquisitely delicious. Do push it in---more faster---beloved of my soul.''
She clasped me with a hug, as if she would make but one body of us both, and spent with a scream of agonized delight, pouring down and spurting out a perfect torrent of boiling spunk all over my cods and thighs.
``Dearest, beloved Henry, it is too much,'' she uttered, and fainted away.
I lay quite still, and determined not to speak until she should come to herself. It was evident her dreams had brought back some former loved and happy man and no doubt the fact of my being in possession, in full fuck, had made her believe in the reality of her sleeping thoughts. She was quite a quarter of an hour before recovering her senses; daylight had broken, and she looked round in a sort of alarm, and exclaimed---
``Where am I?''
Then her eye catching my face---
``Oh! my darling Charlie, it is you! I have been dreaming of being far away, and, I suppose, the fact of your dear weapon throbbing within me made me think of former events. Well, the dream had its pleasures, if only in a dream.''
``It was no dream, my dear Miss Frankland, or at least, only partially so, as far as regarded your loved Henry---for that was the name you applied to me, and most deliciously did you embrace me under the idea, and die away in an excess of pleasure I quite envied; but you alarmed me by really fainting afterwards. I am so pleased to have turned a mere vision of the night into ecstatic reality, and I am not at all jealous of your former lover, because had you not had any, you would, probably, never have loved me. Oh, no! I should never be jealous of you my dear mistress. I would even like to see you in all the ecstasies of passion, in the arms of another, provided that I should share in your delights.''
She listened in an astonishment, acknowledged that she had imagined herself in the arms of one she had greatly loved, and had thought the whole affair was a dream, and was not conscious of its absolute reality as to her being fucked.
``Well, I must have mine now, feel how it is bursting for relief.''
``Yes, yes, the dear fellow, push him away, my Charlie, and you will see, I shall enjoy the real Charlie quite as much as the dreamt-of Henry---of whom I shall some day speak to you. You are worthy of him and of me---and I fear I shall love you as I do him, far too dearly.''
Then lending herself to the work we were at, she did, indeed, exert all her lascivious power, and we enjoyed such a fuck as seldom falls to mortals here below. We lay prostrate and panting with satisfied lust, until prompted by the urgency of natural wants we were both obliged to rise and relieve ourselves. My darling mistress then used her bidet and told me to lave my parts in the basin, as it was not only cool and refreshing, but also reinvigorating. After which as it was now broad daylight, she allowed me to pose her, and turn her in every position, that I might admire and handle every part of her superb form. Her bottom was larger and harder than any I had yet seen, and, indeed, excepting one, of which, dear reader you will presently hear something, it was about the finest in form and size of any I ever met with. Of course, this handling was not effected without producing erotic excitement in both parties. Miss Frankland had occupied herself as much with me as I had done with her, and her beautifully large clitoris was showing its head in full stand out from among the vast mass of bushy curls surrounding it. I proposed we should have a mutual suck on the floor, with her bottom to the light, that I might have a full view of all her glorious parts. She humoured my fancy, and pulling a couple of pillows off the bed to prop up my head, she stepped across my body, and kneeling down, took my prick in her mouth, and brought her splendid backside and lascivious cunt down to my face. I first glued my lips to the open cunt, thrust my chin in, and then my tongue, as far as I could reach, licking the luscious moisture which our previous handlings had excited; it was as sweet and delicious as cream. This stimulated her very much, and she closed the sides of her cunt upon my tongue so closely as to give it a good squeeze. I never saw a woman but her, who had such a wonderful power in that way. My nose actually felt it was reciprocating the pressures of the cunt, so I changed the venue, and slipped my tongue into her bottom-hole, evidently to her excessive delight. But things were approaching a crisis, and she cried to me to take her clitoris in my mouth, and substitute fingers in both the other orifices. This I quickly did, while she sucked and postillioned me, handling the root of my prick, and my buttocks with the delicious gentle titillations in which she had such skill, until, in an excess of joy, we both poured a tribute of sperm into each other's mouths, and both greedily swallowed it. After this we got into bed again, to have one loving cuddle before parting. Of course, it ended in raising such a storm of desire as a fuck could only allay---she said---
``My loved Charlie, this must really be the last.''
I told her it had so excited me to see her splendid bum before my eyes when we were on the floor that I should like to kneel behind and put it in that way. I really meant into her cunt, but she thought I meant her bottom-hole, and said---
``Well, you are a strange boy, what on earth made you think you could put that great big thing of yours into my bottom-hole; but, to tell you the truth, after being well fucked, I rather like it that way, so you shall try, but you must be gentle in getting in.''
I said, ``I did not know I could do it that way with my prick, I meant to put it into your cunt from behind, but now, from what you say, I should like to try what the other is like.''
You see, I was keeping up my apparent ignorance. She turned on her face, and keeping her head on the pillow, drew up her knees to her belly and exposed to the greatest advantage her glorious backside. I knelt behind, but previous to beginning, I glued my lips to the delicious orifice, and shoved my tongue in as far as I could, and deliriously excited her. Then approaching my stiff-standing prick, and thrusting it into her cunt up to the roots two or three times, so as thoroughly to lubricate it, I withdrew and placed it before the smaller temple of lust; then, by a gentle uniform pressure, I gradually and almost imperceptibly glided in to the utmost extent. She pushed her bottom out, and, I could feel, was straining as if to void something, which is the real method to accelerate the entrance of a prick in that enchanting channel with the least difficulty and pain. We then commenced a slow movement---she wanted me to stoop forward and place my arm round her body, and frig her clitoris, but I begged her to do it herself, and allow me the luxury of looking on the delightful wriggling of her superb backside, and also the sight of my own prick surging in and then withdrawing. She humoured me, and we had a most exquisite fuck. Her bottom-hole had hardly so tight a pressure as she could exercise with her cunt, but, nevertheless, it held me in very firmly, and had a peculiar heat which was most exciting. We both died off together, she so completely overcome with ecstatic delight that her body sank flat on the bed, drawing me with her, without unsheathing my weapon. We lay for a short period, she convulsively shuddering from time to time with the intense degree of excitement this delicious route had produced upon her. At last she begged me to rise and relieve her. As we must now separate, I rose. She assisted me in my ablutions, put on my night-shirt, conducted me to my bed, fondly kissed and thanked me for the exquisite night of every species of delight I had conferred upon her, promising a repetition the following night. She left me and locked the door of communication, but previously unlocked mine, in case I should oversleep myself.
Thus ended the first delightful night I ever passed with that most charming and deliciously lascivious woman---the first of many scores that followed, but in none of which were her raptures more intense, if as much. She ever after dwelt on the night when she had been the happy means of initiating me into all love's mysteries, for she never knew of my previous experiences, and always plumed and prided herself on being my first instructress.
The next day I was somewhat somnolent, of which you may be sure Miss Frankland took no notice. She retired to her own room when we went for our recreation. My sisters scolded me for not coming to them the previous night, but I told them that Miss F. had continued to move about her room for so long a time that I had fallen fast asleep, and even then had not had enough, as they might have observed how sleepy I had been all day. However, to satisfy them, I gamahuched them both, and fucked them both while each was giving the other a second gamahuche, so that then each spent three times to my twice. I thus kept in my forces for the renewed delights I anticipated at night. I went to bed early and slept soundly at once, having no anxiety about keeping awake, feeling certain that Miss F. would awaken me as soon as she was ready to take me to her arms. She came, and we passed another most delicious night of every salacious and libidinous enjoyment. A third night followed, which differed only in the lascivious proposition of Miss Frankland to deflower my bottom-hole with her wonderfully prominent and elongated clitoris, little dreaming that there, too, she had been anticipated by our loved and charming friend MacCallum. She had, however, all the imaginary pleasure of first possession. As you may well suppose, I did not attempt in any way to enlighten her ignorance thereon. We had gamahuched each other, I had fucked her twice in the cunt and once in her bottom-hole, when the fancy seized her to bugger me with her clitoris. Of course, I made no objection; on the contrary, sucking it up to a proper stiffness, I placed myself on my hands and knees in the most favourable position to satisfy her erotic fancy. She first slipped her tongue into my bottom-hole, then spit upon her clitoris, and then anointed my aperture with the delicious slime of her well-fucked cunt, and then with the utmost ease pushed the dear thing up to its utmost limits. I humoured her in every way, wriggling my bottom sideways, which she declared was a vast improvement on her back and forward movements. She passed her arm round my belly, and with that exquisitely delightful touch on my prick for which she was so distinguished, she excited me to the utmost, making my sphincter ani respond to the throbbings of my exquisitely delighted prick, and equally exciting her lascivious passions with the idea of first possession of that narrow abode of voluptuousness. She could feel by the electric excitement of my prick how near I was to spending, and quickening the action of hand and clitoris, we both died away together in all the raptures that such an extra exciting conjunction could produce.
Several nights thus passed in the indulgence of every form of the most lascivious enjoyment. We used to amuse our moments of relaxation in trying who could suggest any new position or varied manner of effecting the delicious junction of our bodies. On one occasion, recurring to the state of excitement her flogging had thrown me into, I asked her, as if I did not know the fact very well already, if the application of the rod on the bottom of a woman, or the mere act of being flogged, at all excited her sex. She told me both acted with great force on her erotic nerves. She thought, from experience, that being whipped caused the greatest excitement and produced the greatest longing to be fucked.
``Then,'' said I, ``do you think it had erotically excited my sisters?''
``Certainly, especially your sister Eliza. I do not know whether you noticed her sudden impulse to embrace and kiss me after her return to school work the day I flogged her; that was a stray erotic impulse, and had we been alone, I could not have avoided responding to it in a way that would have delighted her, and initiated her into some of the delicious mysteries of venery. Nay, I think, but for my happy discovery of your great and delightful merits, I should have sought for and found an opportunity of being alone with that dear girl, for you must know we can lasciviously embrace our own sex with immense mutual pleasure, and although not equal to that which this noble fellow''---(taking hold of my prick)---``inspires, is not without its merit, and even as a little variety from time to time is very enticing.''
``Then, I suppose, you still have some hankerings after the virgin charms of dear Lizzie?''
``I have, and what is more, I believe both Mary's and her passions have already developed themselves. I have sometimes fancied I heard suppressed sighs and gentle movements going on in their beds, and I shrewdly suspect they were practising masturbation on each other. I did not interfere, and after what has passed between you and me, I will tell you that I had a little plan in my head to let them proceed to such lengths that when I chose to make the discovery they would be at my mercy. I then could initiate them in every lascivious and voluptuous delight that woman can have with woman. The happy discovery of your excellences, and the perfect facility my change of room has given for meeting without the slightest chance of discovery, has for the present driven that idea out of my head. I am, however, indebted to it for the change of room, as I asked for it solely to leave the two girls the utmost liberty to indulge in their voluptuous mutual enjoyments, certain that it would increase and give them every desire for the further instruction I could impart to them.''
``I suppose you would have fucked them with this dear stiff little thing?'' said I.
``Oh, yes, you darling, but you have so excited me talking about it, that you must fuck me directly.''
We indulged in a most exciting fuck, and when recovered from the confusion of ideas the delightful crisis always produces, we resumed our conversation on the interesting subject of my sisters. I observed that she had not lately flogged them again.
``All your fault; I am now so satisfied with you that I no longer seek for relief to pent-up desires in that way.''
``Tell me, dear Miss Frankland, did flogging my sisters excite you much?''
``It did, even to spending; but the fear of proceeding further with them at that time rendered me ferocious. The very severity I used was as it were in revenge for stopping short of other salacious embraces, but if once I had gone so far as to make them partakers of my lubricity, I should never have flogged them again so severely, but only to such a gentle extent as would raise their passions to an uncomfortable pitch, rendering them slaves to my burning lust. Even now I have, from time to time, a desire to do so, especially with dear Eliza, as I think she had far more of venereal lust in her nature than Mary. You would not object, dear Charlie?''
``Not in the least, if you will only give me the voluptuous satisfaction of hearing all the details from your lips afterwards; it would stimulate us both to additional raptures, and spur our desires to renewed combats.''
``I don't think it wants much to do that; your glorious prick is as hard as iron.''
``It was the lascivious idea of your enjoying Lizzie that made it get up, but I must fuck you again or it will burst.''
``I, too, my dear boy, am inflamed at the idea; put it in behind this time; I have a great letch in that way at this moment.''
I did as I was directed, and so great was the agony of delight when we died away that she sank on the bed dragging me after her, and we lay almost insensible, soaking in bliss for quite half an hour. We did not again renew our conversation that night, but I determined to push her forward to carry out her idea, and also to give Lizzie a hint to second her wishes in every way, without giving her any idea of what had passed between Lizzie and me, and being equally reserved as to my nightly connection with Miss Frankland.
The following night we passed again in all the amatory delights we could imagine. After our deep midnight sleep, which always took place locked in each other's arms, and poor cockey held firm as if in a vice, I awoke her first, and found my prick stiff-standing in her cunt, which was involuntarily pressing it in the delicious interior folds. I began moving gently, until she was so excited as to quite wake up, when she joined me in all the raptures of a delicious and voluptuous fresh morning fuck. We then rose to satisfy natural wants, and cool our excited nerves by a copious ablution. As we were returning to bed, I observed that Miss Frankland took something out of her wardrobe wrapped up in a handkerchief, and placed it under her pillow with a certain air of mystery. I said nothing. After purifying ourselves we always indulged in a voluptuous gamahuche; after which Miss Frankland generally asked, as a favour, that I should finish off \textit{in culo}. I loved her delicious bottom-hole too dearly ever to refuse. She placed herself as usual on her knees, thighs well drawn up, and head down, so as to make the most of her glorious backside. After I had followed the usual preamble of thrusting in and out of her luscious and juicy cunt so as to lubricate my prick well, I then introduced it, always with the slow and gradual pressure, until it was sheathed to the hilt, when we generally paused some minutes to reciprocate mutual throbs and pressures. In this lascivious pause I saw her hand steal under the pillow, and draw out the handkerchief and put it under her belly. I shortly found a considerable substance entering her cunt, and making my quarters still more tight and narrow. I began to move, and found the substance in the other entrance keeping time to my movements. I had a tight hold of her projecting clitoris, which I had frigged up to a stiff-standing point. I slipped my hand down and found she was dildoing herself with what proved to be a very handsome dildo, in not very formidable proportions.
``That's right, my darling,'' I cried, ``why did you not do it openly, you ought to know that my greatest wish is for you to enjoy these salacious meetings in every possible voluptuous manner; frig on then, my beloved, and be sure that if it adds to your delight it adds to mine.''
``Thank you, my darling Charlie, shove away, I am in the seventh heaven of delight in having as good as two pricks working in me at once.''
She would have explained more, but her words were cut short by the ecstasies the double fuck produced, and she spent copiously before me, on finding which I held back, and was rewarded by making her spend eventually with the utmost excess of delight twice to my once. By this time it was broad daylight, and too late in the morning to enter into any conversation on the new partner in our amatory combats, which was reserved for the next meeting.
This did not occur so soon as we expected, for that day Miss Frankland's flowers declared themselves. It was a fortunate thing for me that she had them at the period of the new moon, and as Mary had them at the full, it enabled me to dedicate a night or two to my beloved sisters, who considered I had been neglecting them of late. I said I had not felt very well, and that I began to think that our excessive fucking was becoming too much for me; that they must remember I was one to two, and I felt if I continued to overexert myself I should break down and fail altogether.
``That would never do, dear Charlie, and it is very true you do twice our work and more, because we don't pour down such a torrent as you do when we spend; you must take care of yourself, we will not be so exacting in future, but cool ourselves first by a mutual gamahuche between Lizzie and me.''
I thus arranged a certain amount of cessation of fucking in that quarter that I might dedicate the more to the far more exciting powers of the delicious and salacious Miss Frankland.
I had always remained in my own bed until I heard her heavy breathing, denoting that she slept, before I dared to leave my own room to go to my sisters. The desire of racking me off, as dear charming Mrs. Benson used to call it, might have seized her, and my absence would have discovered all.
However, she had, no doubt, considered that it would be all to her advantage that I should be left perfectly quiet to recruit my system, after the heavy drain on my amatory resources which she had kept up for the previous fortnight. She never sought in any way to excite me until a day and a night after the cessation of her menses. She told me it was much better to have done with it entirely at once, rather than by erotic excitement keep up the discharge for a week or more.
``And it is not, my dear Charlie, from any want of randy lust on my part, for, especially at first, there is an extreme desire to be well laboured by the biggest prick one could find in existence; the natural irritation of the parts seem to be increased by the way in which the sensual system is affected in that quarter. Former experience has taught me that it is much better to bear this, than by seeking for erotic excitement to keep up the natural discharge for twice as long as it would otherwise endure. Besides which, there would have been a danger of affecting your dear health. Sometimes conjunctions, at such a period, produce a urethral irritation very prejudicial to a man, and such as might deprive me of the delight of your embraces for some weeks. So you see, my own beloved boy, that in every way it is prudent to avoid any amorous excitement at such a period, however hard nature may press for venereal relief. Some women hazard all this, and for a momentary gratification, run risks perfectly unwarrantable, not only for themselves, but above all for their lovers. I, too, my darling, have had my day of imprudence, and knowing the result, I should be both cruel and stupidly insensate to let you run the risk of what already occurred.''
As she recounted those sage counsels, I could not but remember my loved Mrs. Benson, whose advice had been of such service to me, and here was another loved mistress instructing me in further matters connected with the sex. It certainly was a stroke of great good fortune for me to have met at so early an age two such admirable women, not only most amorous and lascivious, but instructing me in the real knowledge of their sex, and the world, at the very time that they were indulging my every lascivious desire, as well as their own. Mistresses of their art, no mystery in love's catalogue of excitements, and of means of gratifying the same, was unknown to them. But they knew, too, how to inculcate wisdom for future conduct. I owe every amatory success of my after-life to the admirable teachings of these two charming and estimable women.
The next night, after we had sacrificed sufficiently often to Venus to enable us more calmly to resume the delightful discussion on the various ways of pampering and exciting the passions, I turned the conversation on flogging; for to take you, dear reader, into my confidence, I was seized with an uncontrollable letch to flog the superb bottom of my loved mistress. I had often seen it palpitating under the vigorous attacks of my stiff-standing pego, while belabouring either of the delicious entrances to the temples of lust. I had often given her glorious bottom good sound slaps of the hand, but I longed to apply to it in earnest a good birch rod, see it flush to a raw meat hue, and then to shove my prick with the utmost force into either or both of the delicious orifices. I thought the best way of arriving at this desired object was to recur to her own description of a less severe flogging exciting the passions with pain; and as she had also admitted that it excited her equally to be flogger or flogee, I proposed that she should exercise a gentle discipline on my bottom, to try its efficacy. She jumped at the idea, but there was no rod in her room, perforce the ceremony was put off until the next night. On that occasion, she advised me first to indulge in every excess of lubricity, and when nature should begin to flag, then the real efficacy of the rod would be experienced. She aided me with the utmost skill in every act of most voluptuous and luxurious venery, and we mutually poured down six tributes to our blessed Mother Venus, with very little cessation, for we both wished to feel somewhat exhausted, before trying the effects of the birching system. We lay quiet for a short time, and then dear Miss Frankland began exciting me, but only in an ordinary way. My prick had already been too well satiated with the previous encounters to respond at once to the calls made on it.
``Ah,'' said she, in her sweetest way, ``I see we want the rod here. Prepare yourself, sir, and take care to make no resistance, or it will be the worse for your bottom.''
Following her cue, I began to implore pity, to promise I would behave better in short time, \textit{etc}., \textit{etc}. But she was inexorable, and ordered me to lie across her knees. Then, taking me round the waist, she gave a smart cut or two, really sharp, that made me for the moment wince.
``Take care, sir, you are resisting, and you know your punishment will be severe, if you so continue.''
``Forgive me, mistress dear, and I will never do so again.''
``We shall see.''
Cut three, sharp, though not so severe. I did not flinch. ``Ah! that is something like a good boy, now we shall have no difficulty.''
She began a series of less and less severe blows, until it ended in a gentle irritable titillation which very shortly began to show its effects by the stiffness of my pego---fiercely shoving against the naked thigh of my loved castigator, who, passing a hand round my body, laid hold of it, delighted to find how efficacious her proceedings had been. Pretending to be quite exhausted, she sank back on the bed, and said she could do no more. I sprang upon her, and we had two more \textit{coups} without withdrawing, with the greatest excess of voluptuousness. It was now my turn, and as she let me slip out of her delicious cunt, I took that up as a cause of dissatisfaction.
``What! you naughty girl,'' I cried, ``is that the way you treat your master, bundling him out of his room in that manner; here, give me the rod, I must make your bottom pay for your ill conduct---here, kneel on this footstool, and lay your body over my thighs, no resistance, or it will be the worse for you.''
``Oh! pray, sir, do forgive me this time,'' and she knelt at my side, and pretended to cry. I forced her down, and she presented her glorious backside, in all its splendour of rotundity and size, before my delighted gaze. I seized her round the waist, and first gloated my sight with all the full and lascivious charms, not only displayed, but in my power, and I armed with a splendid rod. I gave her two or three sharp cuts, which made her beauteous buttocks wriggle, but called forth no remonstrances; but as I continued, in all the rage of lust the exercise excited, to flog away most severely, she begged me to be somewhat more gentle. But I flogged on with increased vigour, until she began to writhe under the severity of the punishment I was inflicting. She struggled fiercely, at last, to be free, but she was completely in my power, and I did not spare her until I saw that, changing from severe pain, her feelings were turning to a storm of lechery and lust. She became frantic with excitement, and screamed out---
``Cease, darling Charlie, and fuck me directly. I am dying for it.''
I threw down the rod, jumped on the bed, and drew up her loins, so that she was placed in a kneeling position; she herself seized my bursting prick, and carried it to the lips of her cunt, where he instantly engulphed himself to the hilt. Her movements became lascivious beyond expression, and were urged with a vigour, which brought down in a very short time a torrent of sperm from both of us. We were too much excited to stop short, and almost without a pause, a second course was run still more voluptuously. She was not even then satisfied, but making me lie on my back, she reversed herself upon me, and we commenced a mutual gamahuche. I succeeded in making her spend again, and she was able to bring my pego up to a standing point.
``Now, Charlie, dear, we must finish off behind.''
So getting again on her hands and knees, she guided my willing prick to the narrower abode of felicity. After first steeping it for a moment in the moisture of her foaming and reeking cunt, I thrust it into her bottom-hole. I seized hold of her clitoris, she had her dildo all ready, and working it herself with one hand, we ran a last course of most lustful and lecherous enjoyment, which ended in such killing raptures that we both sank all but insensible on the bed. Exhausted as we were by the wild excesses we had indulged in, we fell, without moving or regaining our senses, into a deep and profound slumber, until almost too late in the morning, so that I had to regain my room the moment we awoke, without attempting any further amorous toyings. Thus ended my first experiences as a flogger. The sensation was so new, and the temptation to lay on with a vengeance was so great, that I had gone beyond all reasonable bounds in inflicting such a severe punishment on the glorious bum of my beloved Miss Frankland. I must, however, do her the justice to say that she comprehended and excused the feelings under which I acted, only begging me, on any future occasion not to let them carry me away so far as they had done on this. We several times renewed this bum flogging, but with more moderate inflictions---sufficient to highly excite without actually punishing the patient, whichever of us it might be.
We often after this made flogging the theme of our discussions, and I gradually led on to the idea she had expressed of Lizzie's evidently amorous disposition. She still affirmed that such was her conviction. I then suggested that it would be worth her while to try and gratify it, as well for Lizzie's sake as for the satisfaction of her own letch in that way.
``I suppose you could easily find a pretext if you desired to do so?''
``Yes, easily enough, the idea excites me, and I shall indulge it.''
I do not remember what the pretext was, but Lizzie was kept in next afternoon at four o'clock---Mary and I proceeded to the summer house. I knew we should not be interrupted by Lizzie, and that I need not hold in for her satisfaction. So I gave Mary all the benefit of our being alone, and we had four most exquisite and refined indulgencies in every attitude admitted of by the legitimate entrance to love's temple. For, as yet, I had never been able to gain an entrance to the narrower orifice, which was too small for my formidable weapon to penetrate. It is odd how easily Lizzie accommodated me in her delicious bottom-hole, while Mary, older and more womanly in form, was as yet unable to make room for me in that strait path of bliss. When night came I was all curiosity to know how my dear mistress had carried on matters with Lizzie. She told me that Lizzie had been somewhat nervous at first, but she had spoken kindly to her, told her how her amiable and loving conduct after her first whipping had won her affection; that she did not mean to be so severe as on the former occasion, but that discipline must be kept up.
``So come, my dear girl, drop off your frock, as I shall mine, that the bundle of clothes may be out of the way, as well as to avoid their being creased.''
Seeing that Lizzie still trembled a little after she had dropped her gown, she took her in her arms, and kissing her lovingly, desired her not to be afraid---she would not punish her much.
``Lift up all your things, my dear, and let me see if any marks of the former punishment remain.''
Lizzie had a very prominent and very promising bottom. Miss Frankland felt it all over, and admired loudly its form and firmness, declaring it was quite beautiful to look at, and how womanly it was growing.
``Turn round, and let me see if you are as womanly in front. Upon my word, a well-formed mount with a charming mossy covering.''
Her hand wandering over her form excited Lizzie, whose face flushed and eyes glistened with rising desires. Miss Frankland herself became moved, but proceeded at once to lay her across her lap, and began with gentle switches, just sufficiently sharp to attract the blood in that direction, which, of course, acted with double force on all the already excited erotic organs, and Lizzie began to wriggle her bum in all the lasciviousness of lust under the excited gaze of Miss Frankland, who, seeing how matters were going on in her favour, increased the force of her blows, but only sufficiently to still more lecherously excite her patient---until, driven to an excess of lust, she cried out---
``Oh, my loved Miss Frankland, I am dying with pleasure, do embrace and caress me.''
Miss Frankland lifted her up, and drew her to her bosom and lips, and, while sucking her tongue, slipped her hand down and found Lizzie's quim wet with her flowing spunk, and her little clitoris stiff with the erotic passion that was consuming her. She frigged her until she spent again, while their tongues were in each other's mouth. As Lizzie spent, Miss F, shoved a finger up her cunt, which, of course, met with no resistance, but as Lizzie possessed in perfection the art of nipping, she was sufficiently tight to leave a doubt of anything but finger-fucking.
``Ah, you little puss, you have been playing with this before now, tell me the truth?''
``I will tell you everything, if you will only play with me again. Ever since you flogged Mary and myself, we have both been so often burning down there, and have found out that feeling it, and pushing fingers in, was so nice, although at first we often hurt ourselves. But you do it so much better than Mary---oh, do, do it again, dear Miss Frankland!''
``I shall do it much better, my darling, with what I have got down there---look here!''
And, lifting up her petticoat and chemise, she exposed, to the absolute astonishment of Lizzie, her extraordinary mass of hair, and her fiery red clitoris glowing and sticking out of its black mass of curls.
``How beautiful!'' cried Lizzie. ``I declare, you have got a doodle, for which I have been so longing; I must kiss it.''
Stooping down, she took it in her mouth, and sucked it.
``Stop, dear Lizzie, we shall both enjoy it.''
Taking the cushion from the chair, she lay down on her back on the floor, telling Lizzie to turn her face the other way, and to kneel down across her body, so that both their mouths could adapt themselves to each other's quim.
Lizzie told me afterwards that she took care to show no previous knowledge, but to let Miss Frankland apparently initiate her into all the ceremonies of gamahuching.
Miss Frankland glued her lips to dear Lizzie's charming quim, while Lizzie took her extraordinary clitoris into her mouth. After a few ardent caresses Miss Frankland pushed a finger up Lizzie's bottom-hole, then paused an instant to tell Lizzie not only to follow her example in that respect, but to use her other hand in her quim while sucking her clitoris. Then, both adapting themselves as prescribed, they gamahuched on, until both could no longer move from the excessive raptures produced by their profuse discharge. After this first bout Lizzie became curious to see all the wonderful hair-covered organ and limbs of Miss Frankland, who gratified her to the utmost extent of her wishes. Nor did she leave this inspection entirely to Lizzie, but reciprocated it. Undoing her dress above, she uncovered the charming budding beauties of Lizzie's bubbles, and began sucking the nipples. Their mutual caresses and handlings very quickly refired these hot and lecherous women. After a little renewed gamahuching, until both were wild with excitement, Miss Frankland proposed to put her clitoris into Lizzie's quim; told her to kneel down, and kneeling behind her, she sheathed it with ease in the hot and juicy folds of Lizzie's beautiful cunt. Passing her hand under Lizzie's belly, she frigged her clitoris until again nature gave down her delicious tribute, and they sank in all the voluptuous languor that follows. A third time they renewed their salacious and lascivious raptures, then resumed their dresses so as to be ready to receive us. Miss Frankland begged Lizzie to keep her counsel and not reveal, even to Mary, what had passed. But Lizzie urged Miss F. to admit Mary into the new mysteries she had just herself been taught, and said she could assure her that Mary had a far more beautiful body than hers, and would like it quite as well as she did.
``Well, my dear, I shall think of it, and find an occasion to flog her, as I have done you.''
``Oh, that will be jolly!'' cried Lizzie. ``She will like it just as much as I do; it is so nice, you must flog me every day, dear Miss Frankland. I loved you from the first, I adore you now.''
They embraced most lovingly, but our return put an end for the present to any further conversation.
These details were accompanied and interrupted by two or three delicious and most voluptuous fucks, without once withdrawing my burning prick from her equally heated and throbbing cunt---for her description of these proceedings was most exciting. When she had finished, I withdrew, that we might gamahuche each other, and lick up all the delicious spunk in which her juicy cunt abounded. We then renewed our combats, sacrificing to holy Mother Venus in both orifices. Then we slept as only easy-conscienced people like ourselves could sleep; and, like giants refreshed by slumber, renewed our devotions on every altar before separating in the morning.
Two days later Mary was initiated by Miss Frankland in a like manner to Lizzie, while Lizzie and I made the most of our time in the summer house. Excited by her naive description of her scene with Miss Frankland, we indulged in every salacious device that we could cram into the hour's absence, which, by the way, we lengthened out by more than a quarter of an hour, for which Miss Frankland thanked me at night. Her scene with Mary had been one of even greater lubricity, in consequence of Mary at once lending herself to everything, and acknowledging that she knew from Lizzie what she had to expect. Besides, Mary's more developed form and something about her greatly excited Miss F., and she was quite amorous upon her. She had done so much in the way of spending, that after I had gamahuched and fucked her two entrances three times, she required the stimulus of the rod to bring her up to the highest point of lascivious lubricity. And, to tell the truth, I afterwards required and received it myself. Thus our voluptuous passions acted one on the other, and we passed an exhausting night in every excess and refinement of venery, in which Miss Frankland's dildoes, for she had two, of different sizes, played no small part in both our persons.
Now that the ice was broken, I easily persuaded Miss F. to have occasionally first one and then the other of my sisters to sleep with her, alleging that an occasional early night's rest would recruit my powers, and that when she dismissed her bedfellow in the morning, I could finish her off in force; she could thus initiate them in mutual floggings, and in the use of dildo. Of course, I need not say that my ultimate object was to succeed in our making it a general orgie. In this indeed it ended, but not exactly as I had intended. That mattered not, as long as the desired object was attained. I had the delightful opportunity, too, of watching through my peep-hole many of the delicious scenes of lubricity enacted, and when driven to the fiercest excess of passion, I used to withdraw, steal up to the unoccupied sister, and vent my raging lust in every indulgence with her.
This had been carried on for about a fortnight, one or other of the girls sleeping every other night with Miss Frankland. Lizzie, it appeared, had often professed to long to see a real cock, and had managed to worm out of Miss F. that she had enjoyment of mine. The little hussey importuned Miss F. to let her see me fucking her, saying that she could easily hide behind the curtains, and I would never know. Miss F., whose passions were at the utmost tension of desire, consented, and placing Lizzie where she could see without being seen, opened my door, but found an empty bed. She at first suspected that I had gone to one of the female servants, but thought she would make sure and see if Mary was not the object. So she stole softly upstairs, and found us in the act of enjoying a double gamahuche, which as it was early morning light, she could see without difficulty. She had the kindness to let us enjoy it to the end, and then dragging me off, said---
``Oh! Charles! this is dreadful! Why could you not be content with me?---have I ever refused you? Do you know this would be the ruin of all of us if ever it should become known? You are too young to know the dreadful consequences of discovery.''
Here she burst out in a torrent of tears---it was evident from real fear of the sad results that might ensue, and not from any feeling of jealousy. I threw myself into her arms, and as she had herself acknowledged our intimacy, I had less difficulty in alluding to it. I caressed and fondled her, and told her there was no fear of discovery---less now than ever---as we would be all interested alike in keeping our secret; she would cover my intimacy with my sisters, and they would cover my intimacy with her. All at once she said---
``How long has this been going on?---tell me truly.''
I had long prepared myself for such a question, and at once replied that after the description of the libidinous scenes that had taken place between her and them, and her exquisite account of their young charms, I got so lecherous upon them that I had sought Mary out while she was engaged with Lizzie, and Lizzie when Mary was with her; they were both too much delighted to refuse me anything, and we had now enjoyed each other about a dozen times. I had previously told my sisters to support any story I might recount to Miss F. Lizzie had stolen up after she found Miss Frankland had passed through my room, and now both confirmed the tale told. We surrounded Miss Frankland, caressing her in every way. My pego got terribly excited. Drawing up my nightshirt, I said---
``Let this dear fellow make peace between us, and become equally dear to all. I know, my loved mistress, that my sisters are longing to see him exercised on your glorious person, and buried in your delicious hairy cunt, so let me offer up sacrifice to its juicy charms. Lizzie has just said you sought me for the purpose---see, the dear clitoris is raising its head---let Mary lie down under you to suck your clitoris, and see my prick close above her eyes in vigorous action filling your exquisite cunt. You can gamahuche her and Lizzie can look on behind, witness the glorious sight, and act postillion to my bottom-hole.''
``Well, my beloved children, the die is cast, it is no use crying after spilt milk, so let us make the best of it. I never could resist the eloquent look of this loved and long thick thing, that was made for giving poor woman all she could crave for.''
So arranging our relative positions as I had prescribed, we ran a course of the most luxurious and salacious enjoyment imaginable. Lizzie, who had taken possession of one of the dildoes, manipulated herself, while watching every voluptuous movement of our bodies, and we all managed to spend most rapturously together. We could not afford to do more at that moment, as time was creeping on, and the household would soon be astir. Miss Frankland regained with me my room, her own door being locked, and kissing me tenderly, said I was a bad boy, but she supposed it must eventually have come to this, so it was well it was sooner than later.
Thus passed our first general orgie, which was the precursor of many much more luxuriously and salaciously libidinous, and which I shall more minutely describe as events progress.
Miss Frankland would not allow us to have a general orgie the next night. She was now aware of our summer house doings---only of late begun, as she supposed---for my story had been too plausibly off-hand not to deceive her, especially as she had felt convinced by all that occurred on our first fucking that she had had the delightful pleasure of taking my maidenhead. She was quite satisfied on that head. But she now suspected that what I had just begun I should be too glad to repeat. She accompanied us to the garden in our recreation hour, so that nothing erotic took place. We sat down all together after a little running about, and Miss F. opened to us a rule of conduct we must in future pursue. She said---
``However delightful it would be for you all, as well as for myself, to meet every night, it would in the first instance become a dangerous habit, dangerous because of engendering carelessness in the necessary precautions against discovery; and next, and above all, because it would be the destruction of our loved and darling Charlie, who could not possibly long continue such excessive venery as three loved objects at once would constantly require of him.''
Seeing my inclination to interrupt her, and declare that I felt quite equal to it, she stopped me, and told us I was too young to know what such excessive indulgence would lead to; that we must trust to her experience and be guided by her, and we should all find the advantage of it. Three times a week was the utmost she could allow, when we should be all together. The other nights she would take care that I committed no excess. Such were the sage counsels of this admirable woman, and such in future became the programme of our proceedings. I rebelled and kicked against what I thought at the time too great a restriction, but I eventually became convinced that greater pleasure followed the enforced delays. Of course I slept with Miss Frankland on what might be called our off nights, but she soon established a custom of restraining my spendings to twice a night, allowing me to excite and make her spend as often as I pleased. I was difficult to manage at first, but eventually settled down in great regularity to the rules she dictated, and, indeed, enforced. I soon found out the wisdom of her proceeding, for often afterwards my lagging efforts required the spur of the rod to be applied in earnest for the completion of our orgies.
The second night after the discovery of my intercourse with my sisters was the first of meeting all four together, in Miss Frankland's room. We had been sent, as usual, early to bed, and Miss F. had privately recommended us to go quietly to sleep as soon as possible, and not to be under any anxiety, as she herself would go for the girls, after all the household had retired. As for me, it was the plan I had always adopted, as it enabled me to reap the greater amount of enjoyment, and its longer continuance, by the rest I had previously secured. Winter had passed away, and summer came round again. It was a lovely, warm, moonlight night. As soon as we were all assembled, stripping to the buff was the order of the night; then followed charming embraces and mutual posings, so as each should admire the beauties of all. Hands wandered everywhere over every charm, chiefly concentrating on the wonderful and finely developed form of the fascinating Frank-land, whose richness of coal-black hair was so deliciously exciting. It soon became necessary to calm the first effervescence of our passions, which we always did by a general gamahuche. Miss Frankland, who had taken an extraordinary letch for Mary, paired off with her, while Lizzie and I accommodated each other. Miss Frankland, who had provided herself with a store of dildoes, furnished us all with one, differing in size, according to the intention of their application. As Mary's bottom-hole as yet could only accommodate a moderate size, Miss F. kept the smallest for her particular use, the others were indiscriminately used. Thus armed, we proceeded to enter on all the voluptuous excesses of gamahuching in every form, lengthening out our pleasures as much and as long as possible, that we might pass the whole night in the most libidinous raptures. When the ecstatic moment overtook us, our mouths had to cease their operations to give vent to the expressions of the rapturous nature of our feelings. We lay panting for some time before being able to rise and resume our mutual caresses. Now that we had taken off the edge of our lustful appetite, we prepared more calmly for further and more voluptuous combinations. The upper coverings of the bed were entirely removed, so that it presented nearly a square field of combat for love's encounters, admirably adapted for its purpose. We held a council as to our next movements, and finally decided to begin as follows: Mary to lie down on her back, Lizzie reversed above her, Miss Frankland was to indulge in her letch for Lizzie, which was that of fucking her bottom-hole with her extraordinary clitoris, while I was to fuck Miss Frankland's cunt, and postillion her smaller orifice with two fingers, Lizzie was to postillion Mary with her finger, while gamahuching her, Mary to apply the smaller dildo to my bottom-hole, and frig Lizzie's cunt with a larger one. It was also agreed we should run two courses in this voluptuous group, varying only in the substitution of my prick in Miss Frankland's bottom-hole, instead of her cunt, in which was to be placed one of the dildoes. We were none of us to press matters to a speedy termination, but to make the most of the exquisite conjunction of our parts. We enjoyed a most salacious and voluptuous fuck, and so managed matters as all should spend together in perfect raptures of lubricity and lust. Notwithstanding the pleasure of the final discharge, we managed, as previously agreed, to hold our mutual positions, our parts palpitating with repeated throbbings on or in the delicious quarters with which they were conjoined. These soon reawakened our passions, which we as yet had done but little to calm, and when sufficiently heated, the slight change agreed upon was effected, and I plunged up to the hilt in the glorious and hairy bottom-hole of the divine Frankland, who gave almost a scream of delight as she felt my huge pego rushing up into her burning entrails. We had to pause some minutes to allow her excitement to subside to a certain extent, or she would have discharged after two or three thrusts of my potent weapon. We then proceeded more leisurely, and after drawing out our enjoyment in the most salacious and voluptuous manner, the ecstatic moment seized us all together, with such an excess of wild enjoyment that with screams of almost agonised delight, we poured into or upon each other whole torrents of hot boiling sperm, and sank almost insensible into a confused heap of naked forms. We were a long time in recovering our senses. Then disentangling ourselves, we rose and laved our parts in cold water, not only to purify ourselves, but as a stimulant to further exertions in all the wildest excesses of lubricity that any of us could fancy. But we always managed so as to make Miss F. think that she was the author of any new salacious idea or suggestion. In fact she nearly was so in every case, for her experience in every letch, and its gratification in every form of libidinous refinement, was great and we owed to her many new and delicious combinations in our salacious orgies. After partaking of wine and cake, which Miss F. had taken care to secure, we indulged in some delicious romping and pulling about of the rich curls and hairy coverings of nearly all Miss Frankland's superb form. The girls above all admired the magnitude, hardness, and beauty of her truly magnificent buttocks, and what with one now and then sucking her bubbies, and at other times toying with her already standing clitoris, we soon brought her to such a state of excitement that, seizing hold of Mary, she got her on the table and gamahuched her, while Lizzie, creeping under, sucked her clitoris, and I pushed my prick from behind into her cunt. We brought on a delicious spend, and the glorious creature died away in excess of pleasure along with Mary, while I had not yet arrived at the climax. So I contented myself with making my prick throb to her delicious squeezings, until the fatigue of the position required us to break up the pose. She was so far calmed that she could now propose and discuss after-proceedings, and what our next form of enjoyment should be. As Mary had had an extra spend with Miss Frankland, Lizzie was now placed on her knees, with her head well down. I thrust my prick into her longing cunt. Miss Frankland standing up, strode across Lizzie's body in front of me, here I introduced first a smaller dildo up her bottom-hole and then a larger one up her cunt, both up to the cod pieces. She then pushed forward her belly and put her stiff-standing clitoris into my mouth, and placed her two hands on my head. I then passed one hand under her open legs, and seizing both dildoes in one hand, proceeded to work them up and down both holes at once, in unison with my suction of her clitoris, and my fucking movements in Lizzie's cunt, who at the same time was frigging her own clitoris with her fingers. Mary, armed with two dildoes, applied one to my bottom-hole, while she fucked herself with the other. In this way we ran a most exciting and delicious course. Miss F., in the ecstatic moments, seemed as if she would have pressed my head into her belly. She was so charmed with the voluptuous delights this pose had given that she cried out we must not change until another course was run. Lizzie said she must change from front to back, and begged Mary to hand her a dildo with which she might frig herself. The women were ready directly, but my pego was longer in answering the call, so Miss Frankland told Mary to apply the birch rod skilfully.
This she did with great art, working the dildo, which was still in her cunt while so occupied. The effect was almost electrical, and my glorious rampant prick filled dear Lizzie's delicious and longing bottom-hole to her utmost delight. Miss F. begged Mary to give her a gentle stimulus with the rod. Nothing could better have pleased Mary, for she afterwards admitted she had long had the greatest letch to flog that glorious and immense backside. With such stimulants as these this course proved one of the most salacious and voluptuous we had yet had, and the ecstatic ending was accompanied with screams of delight, as we died away in the deathlike swoon of rapturous and satiated desires. We again rose to purify and refresh ourselves, and for some time after lay closely embraced on the bed. As Mary had not yet had my prick in her cunt, Miss F. proposed that I should fuck her, that Lizzie should kneel close behind us, she could fuck Lizzie's bottom-hole with her clitoris, and work one dildo up my bottom, while she worked a second in her own. No sooner said that done. Lizzie's head was shoved almost below Mary's belly, so as to bring Miss F. close enough to me to operate as she desired, and we ran another delicious course with such extreme pleasure that all sank sideways down on the bed and dropped into a sound slumber. We did not awaken until so late that we only had time to lave ourselves in cold water, finish off with a general gamahuche, and then regain our separate rooms. On this last occasion Miss Frankland said she must gamahuche me, as she delighted to break her fast on cream. The joke amused the two girls amazingly.
It was about this time Mrs. Vincent gave birth to a fine boy. I have not spoken of her since our first interview after her marriage in the summer house, when all had gone into the town to bring out Miss Frankland. We had only had two stolen interviews since that time, which I have not mentioned, because they were too hasty, and with too little comfort to have been thoroughly enjoyed; then she became too heavy with child to afford me any further opportunity. Mamma wrote a congratulating letter to Mr. Vincent, wishing him joy of the advent of a son and heir, little dreaming that her own son was the father thereof. This brought a visit from Mr. Vincent to beg that mamma would kindly become godmother to the little fellow. My mother at once assented, and asked who the godfathers were. He said an uncle, from whom they had expectations, had consented to be one, but he was at a loss to know whom to ask as second.
``Why not ask Charlie, he was always very fond of your wife as his governess, and he, too, has an uncle from whom we hope some day to receive something handsome.''
``That is a very good idea of yours, Mrs. Roberts, and if you will kindly send for Charles I shall put it to him, and if he consents, it saves me all further trouble.''
I was sent for, and, you may be sure, accepted immediately, thanking Mr. Vincent for the honour he did me, and hoping that Mrs. Vincent would be equally agreeable that I should be godfather, although so young.
``Leave that to me, my dear wife is so much attached to me that my wish is her law, so do not make yourself uneasy on that head.''
It may well be supposed I was not at all uneasy, but quite certain that it was the very thing Mrs. V. would have proposed if she had not been withheld by prudence. We heard afterwards from Mr. V. that she had simulated objections on account of my youth, but the very first moment she could say a word to me in private it was to tell me what delight it had given her that her husband should have fulfilled in the matter the very wish nearest and dearest to her heart.
The ceremony eventually came off as had been proposed, but it was at very rare intervals that I could find an opportunity of renewing our old combats in the field of Venus. Meanwhile I had no reason to regret this as far as indulgence of my erotic passions went, because, for nearly two years, that is until I had passed my eighteenth birthday, I continued to enjoy uninterrupted bliss in the arms of the luxurious and fascinating Miss Frankland, or in orgies with her and my sisters, which culminated in every excess of venery capable of being enjoyed by three women and one youth. In fact, we all indulged rather too freely, if I may judge from the fact that, at least to Miss Frankland and myself, the rod had almost become a necessity, and occasionally even my sisters admitted it gave them a fillip. Under the able tutorship of Miss Frank-land we became the most perfect adepts in every voluptuous indulgence of lubricity. But I must also give her the credit of never neglecting our education. Indeed, I may say it gained by the intimate union of our bodies. For that estimable woman impressed upon us that to keep her friendship and confidence we must do justice to her teaching. I have already said her system of instruction was very superior to anything we had previously known, and now that she had won our unbounded love and affection, there was nothing we were not ready to do in school to second her efforts for our mutual improvement. She had very superior attainments-spoke French and German like a native, had sufficient knowledge of Latin and Greek to ground me well in them, and her knowledge of music was very superior. I have hardly ever heard anyone with a more charming touch on the piano. In the two years that followed our first orgie we made really astonishing progress. We all spoke French very fairly, had a pretty good knowledge of German, especially Mary, who really spoke it well; as for myself I was well up in French, fairly so in German, and with a very good ground work of Latin and Greek.
It was about this time that an event happened which completely changed the order of my life. My mother had hinted that I had some expectations from an uncle. These were very vague. He was my father's brother, but they had never agreed, and we were almost strangers to each other. He died, and one day we were all surprised, not to say delighted, to hear from his executor, a Mr. Nixon, a rich merchant in London, that my uncle had left my mother four hundred pounds a year as long as she did not marry again, but at her death the said annuity was to be divided between my two sisters, independent of any coverture. The residue and bulk of the property was settled on me, under trust to Mr. Nixon until I was of age, with a request that I should be brought up to the law and entered as a barrister in the Inner Temple. Further, a sum of five hundred pounds was allowed for a new outfit, in every way becoming to all of us. Mr. Nixon announced that in a fortnight he would take the opportunity of being in our neighbourhood to come over and make the necessary arrangements consequent upon the altered state of affairs. He added that the residue of the property would yield about one thousand pounds a year, and that, therefore, my education must be looked to more closely than it probably had been. Here was, indeed, a change. My father had left the house and grounds, and something like six hundred pounds a year in the funds, entirely to my mother as long as she remained a widow, or until her death. Afterwards one hundred and fifty pounds per annum to each of my sisters, and the house and residue to me---a moderate income requiring other efforts to make it comfortable to one's upbringing. Here I was now the heir eventually to something like fifteen hundred pounds a year, two country houses, and a very fair house besides attached to my uncle's house. You may easily imagine the joy of the whole family when from somewhat pinched economy, we found ourselves in easy circumstances, with at once quite double our previous income. We indulged in somewhat wild dreams of what all this might produce; but mamma brought us to our senses by informing us that until I was of age Mr. Nixon would entirely control our destinies, and that it was more than probable he would insist upon sending me to a public school. This news dashed all our hopes to pieces with a vengeance, because it was precisely on our greater freedom that we had been counting, and now there was every probability our delightful intercourse and delicious orgies would come to an abrupt termination. We exchanged sad and crestfallen looks on hearing this from mamma, and met in a very disconsolate humour that night in Miss Frank-land's room; but that charming and estimable woman cheered us up with the hope that if a temporary separation did occur, it would only lead to our safer and more perfect reunion hereafter.
``And, to tell you the truth,'' she said, ``my dear Charlie, we have been of late too much for you, and your health and constitution will benefit by a forced inactivity, for I have observed some symptoms about you lately that prove we three have taxed you too hard. I have no doubt I shall be retained as governess to your sisters, and leave me alone to keep them to a point that will not disappoint you when we meet again, which must always occur at intervals of not longer than six months.''
To our loving minds six months seemed an age. At the same time Miss F.'s remarks had, to a certain extent, reassured us, and although we could not enter into our orgie with the usual fury and letch, nevertheless we managed to pass a night sufficiently rapturous in the enjoyment of our libidinous passions, which many would have thought excessive.
In due course Mr. Nixon made his appearance. He was a pleasant-looking elderly gentleman, and a complete man of the world. Finding that I had been educated entirely at home under governesses, he fancied I must be a milk-and-watery ignorant youth, and had already hinted as much to mamma---who, having told me, put me on my mettle. Mr. Nixon sent for me into the parlour alone, and began an agreeable conversation apparently leading to nothing, probably with a view not to render me nervous and timid, gradually turning the conversation upon educational subjects. He was agreeably surprised to find the progress I had made, not only in historical and geographical subjects, but in languages, and above all was surprised at my knowledge of Latin and Greek. He was particular in asking if some clergyman had not lent his aid to the governess. After dinner, during which he paid great attention to Miss Frankland, he warmly complimented her on her system of teaching and its extraordinary success. At the same time he observed that, as his dear old friend had desired that his nephew should become a barrister, it would be necessary he should be sent to some clergyman taking a few boys, and then to King's College, London, before entering a barrister's chambers. Miss Frankland at once admitted the justice of the remark, and hoped that Charles would not shame her teaching.
``Quite the contrary, I assure you, Miss Frankland. I have been struck with the admirable ground work you have established, and especially the advantages you have given him of the knowledge of modern languages. I am so much pleased that I intend to beg of Mrs. Roberts to keep you as the able governess of the girls until they are so much older as to require a little knowledge of the world which a metropolitan ladies' school is sure to impart.''
All this was said with a certain deference of manner to Miss Frankland, that I felt certain the old gentleman was greatly struck with her person, as well as her system of teaching. But of this it is probable my readers will learn more hereafter.
My mother, hearing of the intention of sending me to some clergyman, immediately suggested that her own brother-in-law, the Rev. Mr. Brownlow, rector of Leeds, in Kent, a retired village close to the castle of that name, would be a suitable person. He was a gentleman who had taken honours at Cambridge, and was in the habit of receiving one, two or even three young gentlemen, but never more, to prepare them for the universities. At that moment she knew by a letter from her sister that he had a vacancy. His name, she said, stood high as an instructor, as Mr. Nixon would find on inquiry; and as Charles had never been away from home, it would be a great satisfaction to her to know that he was under the care of her own sister. Mr. Nixon said he perfectly agreed to her suggestion, provided, as to which he had no doubt, his inquiries justified his sending me there. He left us with a promise of an early decision, and, indeed, before the week had passed we received his full concurrence to my mother's suggestion. So my aunt was written to, and it being the period of the holidays, Mr. and Mrs. Brownlow were asked to come over and spend a week, and then I could return with them to Kent. We had not seen aunt or uncle since we were little children, and only remembered her as a very tall immense person. The distance had prevented personal intercourse, and we only knew of them by interchanges of hams, Canterbury brawn, and oysters at Christmas time. As they replied by return of post, saying they would be with us in two or three days following their letter, you may be sure Miss Frankland and all of us made the most of what was to be the last of our mutual orgies for the time. No restrictions were put upon us, and every night was dedicated to the god of lust and voluptuousness.
At last the fatal day arrived. My mother and the two girls went into the town to fetch uncle and aunt out, leaving Miss Frankland and me to our studies. You may well suppose it was the prosody of love and not that of grammar that occupied us. There was a tenderness of manner, and a loving kindness and fondling, which I had not before observed in Miss Frankland, and which I should have thought alien to her character. Embracing me tenderly, and pressing me lovingly to her bosom, she burst into a flood of tears, and sobbed as if her heart would break as her head sank on my shoulder. I tried to comfort her in the best way I could, and as my kind reader knows, a woman's tears always had a most potent effect on my prick, I placed it in her hand, she hysterically laughed amidst her crying, but instantly sank her head down to the loved object, embraced, sucked, and frigged it until I poured a flood of boiling sperm into her mouth, which she greedily swallowed, and continued sucking until not a drop was left. Then rising once more to caress and embrace me, she said---
``Yes, my own beloved boy, that was indeed a means to stop my tears, I not only adore it, but have come to love you, my darling, more than I ever loved anything in my life---you are my own scholar, bodily and mentally. I shall miss you greatly, and I bitterly regret our parting; but we shall meet again, although never with such freedom and ease as we have done. You will spend your holidays at home, and we shall make the most of them. I can feel the dear object already to be made the most of again, and so it shall, dear fellow, so come to its own nest.''
These last fond words were addressed to my prick, which, already rampant again, was claiming attention. We went at it, hammer and tongs. Recruited at luncheon, we renewed the raptures of lubricity as that estimable woman alone knew how to indulge them. We were the less reasonable, as it had been decided by us the night before that I was to find out the habits of the coming couple before I should venture on leaving my room to slip up to theirs, and thus I had a night of relaxation before me.
At five o'clock the carriage drove up, and uncle and aunt were welcomed to our house. My uncle was a tall, portly, unctuous-looking clergyman, quite a gentleman in his manners, and with a very agreeable voice. My aunt, who was some fifteen years my uncle's junior, was very tall for her sex, a fine portly figure, broad shouldered, large bubbies well apart, a small waist for her size, immense hips and evidently buttocks to more than match. She was very stout, but stood firm upon her pins, and walked with great elasticity of step, showing there was a good deal in her, or rather she could take a good deal out of anybody. She had a profusion of fair hair, with thick eyebrows, that promised abundance elsewhere. Her eyes were of a deep blue that could look very far into you. She had a very pleasing expression, a small mouth, and very white teeth. Her complexion was exceedingly fair, her arms immense, but beautifully formed, hands and feet small, fat and plump. She looked thirty-five, but was nearly forty, and was altogether a most desirable woman to look at, on a large scale. She embraced me tenderly, which I did not fail to return, and complimented me and the whole family on our late good fortune. The first introduction was altogether most agreeable, and I already began to imagine I might not be so badly off after all.
We were allowed to sit up rather later than usual, and as my aunt was fatigued with her day and night's journey, they were glad to follow our example almost immediately. I had only just time to get undressed, when I heard them enter the room which Miss Frankland had vacated the previous day. This had previously been arranged, and she now slept in my sisters' room, as formerly, until we should depart. I quickly blew out my light, for fear they should observe it shining through the chinks I had made. Kneeling down, I began to watch the proceedings. The first thing my aunt did was to squat on the pot just opposite my peep-hole, and as she held up her dress well, I could see that she had a most prominent mons Veneris, thickly covered with very fair ringlets. Her power of piss was something wonderful, it was like a cataract in force and quantity, and at once made my mutinous prick stand at the mighty rush of waters that could be so plainly heard. As she rose, and before she dropped her dress, I saw her splendid proportions of limb, the like of which had never before met my eyes. Alas! it was but a passing glimpse. However, I determined to watch on, hoping to see a further display in the course of undressing. She took off all her upper clothes, until nothing but her stays and chemise remained. I could now mark the real grandeur of her proportions. The stays kept in the waist, and allowed the splendour of her hips and buttocks to stand out in all their glory. Never in my life have I seen a finer backside than my aunt had got. I am now speaking from a vast amount of after-inspection and adoration, but in its covered magnificence in which I at this moment viewed it, it appeared the finest backside I have ever met with, and was in fact the one I alluded to some time back, when I observed that Miss Frank-land's was the finest but one I ever saw. It is true, her stoutness added greatly to its prominence, but though stout, even very stout, it was not a stoutness you could call fat. For in after-intimacy, which became of the very closest and most voluptuous nature, I was never able to pinch her in any muscular part. She had the hardest, as well as the biggest, backside I ever met with. I am quite sure that when she was standing upright, a child might have stood on the immense projections of her buttocks. Her thighs were positively monstrous in their mighty proportions, as hard as iron, exquisitely moulded, and of a fairness and smoothness that rivalled ivory, which, in another respect, they much resembled, namely, in feeling cold to the touch. Her legs were worthy of the glorious frame they supported, and finished off with a pair of charming, clean-run ankles, and very small feet for her size. As her chemise was short sleeved, the grand magnificence and beauty of form of her splendid arms and neck, where the bubbles came out in all their perfection and brilliancy of skin, were fully displayed. As may be supposed, not a bone was to be traced in her upper neck, but all was dazzling in colour and flesh, which is such a beauty in woman. When a woman shows her gaunt collar bones, it is a proof of bad breeding, and a common nature. Aunt's truly grand bubbles rose magnificently over her bodice, which I thought at the time was their support, but this glorious woman required nothing of the sort, for when perfectly stripped, her bubbles stood out firm and projecting in all their grandeur, and they were of the largest, worthy of all her other fully developed charms. Her belly alone was somewhat too prominent, when standing up, but as she never had had children, it did not at all hang flabbily, and ended in one of the most prominent and largely developed montis Veneris I have ever met with, profusely covered with the fairest of curls, which did not prevent her lovely creamy skin from shining through them. She was well provided with hair on that part, but after the extraordinary hairy covering that Miss Frankland possessed, and with which I had so often toyed, all other women appeared as nothing in that way. My aunt, after donning a night-robe, sat down to her toilet, and proceeded to let down her massive bunch of tresses. Here, she was, indeed, richly gifted, her hair was all her own, in the utmost profusion, and, tall as she was, fell much below her buttocks, and was so thick that she could let it spread over both back and front, and completely cover her nakedness. Titian must have had such another magnificent head of hair for one of his models, for it exactly resembled, except in being somewhat of a fairer hue, his celebrated Magdalen, in the Pitti Palace, at Florence, where she is represented covered only with the rich profusion of her ringlets. Such was my aunt, and often and often afterwards has she indulged all my fancies, by showing herself off in every voluptuous attitude with this, the greatest ornament of woman, flowing in the utmost profusion over her glorious and mighty charms. Meanwhile, the doctor had undressed, but it may well be supposed perfectly unnoticed by me. I had better game in view. He, too, had donned a \textit{robe de chambre}, and sat down by his wife to have a chat over the occurrences of the day. Of course, their conversation very naturally turned upon myself. They began by congratulating themselves that the good fortune of the family was partly reflected on them by the circumstances of my being put under the doctor's care. The lady remarked how doubly fortunate it was, as the little scandal that had happened had, for some time, prevented their having any pupils at all. The doctor said---
``Never mind that, my love, this little fellow will soon be the decoy duck for others; he seems a nice, gentle lad, but I shall seek to have some talk with him to-morrow, and see what he is made of; boys, under women's instructions, are generally mere milk-sops.''
``I don't think you will find it so in this case,'' added my aunt. ``I am not a bad judge of character, and I feel certain that Miss Frankland is too stern and firm of purpose not to have bent any boy's will to her bidding; I fear, on the contrary, she has, if anything, been too severe with him, for my sister told me that she had full power to wield the rod, but, after one or two severe bouts, she completely mastered them, and that their progress was really very great, and most satisfactory, as Mr. Nixon, Charles's guardian, who had examined him, had reported most favourably thereon. But he appears to be insignificant, and undersized, thin as a whipping post, pale, and somewhat sickly-looking, he appears much younger than he is, and seems hardly fitted for what you and I would delight in. Eh! dear doctor?''
I did not understand at this time what her allusion meant, but it was followed by the doctor stooping forwards, kissing her, and, I have no doubt, tongueing her too. He first thrust a hand below her beauteous bubbles, and then pulling up her chemise, began foraging between her legs. She put down her hair brush; and laid hold of his cock, but quickly said---
``Don't excite me, my dear, you see this poor fellow can do nothing without a rod, and we have none here, so be quiet and go to bed, that is a good boy.''
Obeying her, he rose, threw off his robe, put on a nightcap, and tumbled into bed, and was sound asleep before his magnificent spouse had finished her toilet. When it was concluded, she took off her stays, and drew her chemise over her head, I doubt if it could have fallen over her enormous buttocks. She then walked across the room in my direction, stark naked as nature made her, and strikingly magnificent in the firmness of her tread, and the glorious uprightness of her truly superb grandeur of form. I was positively awestruck. I could imagine her to be Juno in all her glory before Jupiter, and well he might be tempted to stray to the forbidden path of love, if Juno had such a backside as the enormous and glorious one my aunt possessed. She again squatted down, naked as she was, and poured out another torrent into the pot. I felt overpowered at the sight, and staggered back to my bed, and for the first time in my life felt constrained to rack off by self-pollution the excess of lust the gazing on such superhuman beauties had engendered. I could hardly refrain from shouting out to relieve my till then suppressed excitement, especially when nature gave way, and there spurted forth a jet of sperm, actually from the bed against the door towards which I had pointed my prick while wildly frigging it, and in imagination shoving it into aunt---anywhere; for if ever the saying that ``there was plenty of good fucking about all these parts'' was applicable to any one, it was supremely so in my glorious aunt's case. Any one might shove his prick against any part of her body, and spend at once from excess of lust, at her very beauty and splendour of form and exquisite colour and fineness of skin. Never, never have I met her equal. Her power of fuck, too, was on a par with the immensity of size, and of a quality to please the most fastidious, or the most lustful. Such were the first experiences that I had of my aunt's person, and as my narrative extends, the reader will become more intimate with her person and proceedings. I sank to sleep, to dream of possessing her in every way, rivalling Jupiter with Juno, and Mars with Venus, mere visions of the night, but which were in after-days converted into sweet realisations of the most voluptuous and rapturous nature.
The next day, at our hour of recreation, Miss Frankland walked out with us, and seeking a retired part of the grounds, while the girls amused themselves, I recounted to Miss F. all I had seen and heard. She at once came to the conclusion that I was destined to fall into the arms of my aunt.
``I am so far pleased, my dear Charlie, that it will be into those of an extraordinary fine woman; you must, after your present experiences with me, have had some one to go to, and certainly you could not have a finer. There will, evidently, be every facility, for I read those hints, which have puzzled you, as intimating any thing but reserve once you are admitted into the inner arcana of their lives, or I am much mistaken. There is one point I must strongly caution you about, and your general prudence and great good sense will make you appreciate its importance. Your aunt is evidently much experienced in erotic pleasures. If at once she found in you the extraordinary adept you are, she would never cease tormenting you until she discovered who had been your instructress. Now it must be evident to you that if she thought you and I were intimate in that way, she might draw evil inferences with regard to your sisters, or if not going so far as to think we had equally corrupted them, it is probable enough she might seek to remove me from their society. So you see, my darling boy, though it may be very difficult to do, you must, for all our sakes, determine to appear quite innocent and ignorant of every thing connected with indulgence in amorous passions. You must not let yourself appear excited, but leave her to take all the initiatory steps, and I much mistake if she will not be extremely ready to do so, but all the more so if she finds you apparently innocent. However much you now know of love's proceedings, you must keep a guarded check upon your feelings, so as not to let your knowledge become apparent in the smallest degree. She will, eventually, be twice as well pleased if she fancies she has had your first fruits. Before you leave I shall give you some short hints as to how to conduct yourself.''
All this time I was getting rampageous, so begging her to stoop forward upon a stump, I tilted up her petticoats and fucked her from behind, frigging her delicious clitoris, and making her spend at the same time as myself. It was a hasty fly, but very sweet nevertheless, for we were both conscious that it was necessary to make the most of the short time I had yet to remain at home. I mentioned my aunt's remark about having no rod at hand, and it was agreed that Miss Frankland should put one on an upper shelf of her wardrobe, and accidently leave the key in the door. As this wardrobe remained in the room uncle and aunt were sleeping in, woman's curiosity was sure to induce an examination of it. This answered a double purpose, for Miss F. so arranged things that some excellent books full of little bits of paper inserted here and there, at highly moral or religious passages, led both uncle and aunt to have a very high idea of her moral character---for these were works that apparently could only be for her own private reading.
The rod was placed, and the bait laid next day. Meanwhile, that afternoon, the doctor called me aside, and put me through a conversational sort of examination. I was studiously modest, but being very fairly grounded by the admirable system of teaching pursued by Miss Frankland, I not only satisfied him, but he took occasion to compliment Miss Frankland very highly for the admirable groundwork she had laid. I fancied also, as he continued in conversation with her, that he grew more kindly and unctuous, as if the spirit of lust was infusing itself in his veins, as he continued to converse with and gaze on that most engaging and lust-creating creature.
That night I watched, as before, their preparations for sleep, and heard their conversation. This time the doctor was profuse in his praise of me, but aunt thought I was timid and lifeless; there seemed no spirit about me, as there ought to be, she added, at his age, but this education by females makes girls of boys. I thought to myself, I guess, I shall very soon undeceive you on that point, my dear aunt. The doctor went quietly to bed; aunt stripped and used the bidet, giving me a most exciting and voluptuous view of all her full-blown charms. No sooner was her light out, and she in bed, than I slipped out and crept up to my sisters' room, where three randy cunts were impatiently awaiting my advent with an equally randy and inflamed pego. We indulged in every complicated combination of lust and lubricity, and never ceased until daylight forced my unwilling retreat. Before leaving, as the rod was to be put in the wardrobe, and the key left in the door, it was arranged that the next night the girls, and Miss F., too, if she could, were to endeavour to sleep soundly before I came. For if our stratagem succeeded, I should remain to see the result which would probably occupy more than an hour or two, and I would awaken them by applying Moses's rod to their water courses as doubtless I would be in a rampageous state, if our expectations of the doctor's and aunt's tendencies that way were realised.
I kept myself awake until aunt and uncle came to bed, and then I immediately placed myself \textit{en vedette}. At first no notice was taken of the key being in the lock. Aunt continued her operations, and uncle became somewhat more tentative than usual, when aunt, finding by placing her hand on his prick that it was mere useless desire, rose and scolded him. He grew more emboldened, and followed her up, wishing to feel her splendid cunt. It so happened she had drawn back as far as the wardrobe itself, until the key actually hurt her back.
``Ah what have we here?'' she cried, and then turning round, said that as the key had been left in the lock, there could be no harm in looking in. Her husband became as curious as she. Of course, the first things they saw were the prearranged books. They were seized upon with avidity probably with the expectation of finding something smutty, but to their surprise, and especially that of the doctor, it was quite the reverse.
``Well, I should never have thought this; do you know, my dear, I had begun to suspect that, under a demure exterior, there was lurking an enormous deal of animal passion in that Miss Frankland, but if so, these works prove that it is under complete regulation. More's the pity, for she is made for the real enjoyment of the passions.''
``Oh you have been speculating in that quarter, have you, you old lecher?''
``Well, my dear, you know we have both liberty to stray now and then, and you, yourself, have not a little availed yourself on our mutual understanding.''
``Now, doctor, you are too bad; do I not quite overlook all your weakness for the younger members of your own sex, and do I not lend myself to your fantasies in that way, when chance deprives you of any opportunity of pederasty?''
``Well, well, my love, I was not upbraiding you, you are too dear and too kind to me to permit of any thing beyond a joking allusion; but what have we here? A birch rod! by all that is holy.''
Reaching up to the high shelf, he drew down the rod. At first they suspected Miss Frankland operating on herself, but the perfectly untouched state of the rod proved that it was there in reserve only, and had not yet been used.
``What a lucky chance,'' cried my aunt. ``I shall now be able to birch you into something like a fit state to fuck me---and you shall birch me afterwards, if it will only produce a second fuck, back or front, whichever you like.''
``You are an angel, my darling wife, and I shall try to content both orifices; it is an abominable shame that with such a gloriously made magnificent woman as God has given me in your noble form, I should ever require any other stimulant than a glance at your exquisitely exciting proportions; but I suppose it is age that weakens our sensibilities.''
``You are right, my dear John, for I, who used to think your dear old cock was enough for me, find I require the excitement of younger ones to give me the real excess of pleasure my constitution demands; it would be a shame if I did not humour all your little caprices, when you so readily throw opportunities in my way. I only wish this nephew of mine had been more worthy of us, we should have made him a glorious \textit{bonne bouche} between us, equally to his satisfaction as to ours.''
``Well, my dear, the air of Kent, and more manly treatment, may yet develop his somewhat stinted growth, and under your tuition, he may yet prove not so bad an object as you seem to think, at all events, he may serve as a \textit{pis aller}, until a better turns up; but you must proceed with caution, for he seems as modest as a maid.''
``My dear John, your modest ones always make the best, when once broken in. I only wish his physique had been more to my liking, but we shall see, we shall see; meanwhile let us both strip to the buff, and proceed to make the most of this happy discovery of the rod---the very thing we most wanted and wished for.''
Aunt rapidly twisted up her magnificent tresses, and as rapidly stripped to the skin; the doctor likewise. I assure you he was a well-made, muscular, portly, handsome man, with a large well-filled pair of cods. His pego still hung down his head, but had a certain amount of size, doubtless stimulated by the exciting nature of their conversation and reminiscences. His skin and his cock were beautifully white, and the ball of his prick of a tempting scarlet. I felt at the moment that, if I dared, I would have bolted into the room, and sucked it into such a stiffness as would have instantly satisfied the insatiable cunt of my glorious aunt. This was a delight to be left for a future day, when I allowed the doctor all the credit and pleasure of persuading me to do that which I was burning with desire to do. But I digress. No sooner were both fully prepared than my aunt, in a stern voice, ordered the doctor to approach.
``Come here, sir, I must whip you, you have not done your duty as you ought lately, and you are a very naughty boy.''
The doctor, putting on the air of a schoolboy, begged to be excused this time, but his inexorable mistress was not to be moved, and seizing him by the arm, pulled him over her broad and massive thighs, and with one arm round his waist, seized his cock in her hand, and began whacking away at his backside in such real earnest and, apparently, with all the force of her powerful arm, that I began to think the doctor must cry out in earnest. But he took it all without a murmur, only wriggling his fat and smooth buttocks about in a way that rather inferred satisfaction than suffering. Presently my aunt, who, doubtless, knew by the grip of his prick that matters had arrived at the point her own passions had most at heart, lifted him up, and said---
``Now I must put you in pickle, but as your great red buttocks are too large to be pickled, I shall pickle your prick instead. So come here, sir, and let me put this rampant fellow into my pickle tub, where, I promise, the salt brine will soon bring down his pride.''
I suppose this was the sort of childish yet lascivious talk which pleased them both, for uncle, who had risen, and who now presented a much finer weapon than I had given him credit for, pretended to fear this further punishment, and begged and entreated to be let off---he had been punished enough, \&c., \&c. Aunt, however, leading him by the prick to the bed, threw herself on the edge, and lying back, drew up her enormous thighs almost to her belly, and showed to my gloating gaze her tremendous salmon-coloured gash, all covered with spunk, for the operation had made her spend profusely. I never saw so large a cunt, nor such an extensive triangle as lay on the side of each lip between it and the commencement of the buttocks, beautifully covered with the fairest curls.
``There, sir, is your place of punishment, stoop and kiss it before I imprison your indecent cock within it.''
The doctor, nothing loath, stooped and gamahuched her so well that her mighty backside wriggled beneath his head, and made every thing in the room jingle; her hand pressed his head until I thought it would have been pushed in altogether. At last, she spent with a shout of delight. He hastily gobbled it all up, and rising, without more ado, thrust his stiff-standing weapon up to the hilt, I might almost say cods and all, in her longing and magnificent cunt. Here, he soaked for some minutes, and I could see by the convulsive movements of her backside how much aunt was enjoying it. They soon became bent on more active movements, for throwing her splendid legs over his back, she began an up and down movement, much more active than I could in any way have given her credit for. They went at it in real earnest for a longer time than I expected, but when the mighty crisis came, it was with an energy, and passionate struggles worthy of the strength and substance of the two love wrestlers. I could see her cunt all foam again around the roots of the increased size of uncle's very respectable prick, and then they lay in apparent apathy for full twenty minutes, but one could see by the convulsive throbs of their whole bodies what delicious transports of rapture they were enjoying. Uncle was the first to rise, but only to stoop and to greedily lick up all the foaming spunk which the wide-spread entrance to her glorious cunt exhibited. This being done, she, too, rose, and throwing her arms round the doctor's neck, drew his mouth to hers, and seemed to suck his slimy lips, and gain for herself as much as she could of the delicious spunk the doctor had been revelling in. This lasted some minutes. Then my aunt turned him down on the bed, and took a long suck at his prick, now hanging limp, but still of a goodly thickness. Then she thanked him for the great satisfaction he had given her, and declared it was almost as good as the first days of their union. Then after toying and cuddling on the bed for a time, she said they must now proceed to a little further castigation, on her bottom this time, as he had promised to give her a double dose.
``Yes, my love, but you know you promised I should take my choice of which temple I should make my sacrifice at.''
``My own John, you know, that after being once well fucked, the hinder hole is my preference, that is understood.''
They accordingly rose, and uncle, furnishing himself with the rod, desired aunt to kneel on the edge of the bed, and present her magnificent backside projecting out fair for his birching. This she immediately did, and being directly before my eyes, I had a full front view of her gloriously large wide-open cunt, and all the pinky brown \textit{aureola} around her charming bottom-hole, over which the little fair ringlets showed in great beauty. I need not say that my own John Thomas was in all the pride and panoply of prickdom, and ready to burst with excitement. My uncle took the rod in hand as soon as aunt was in position, and placing himself on one side, while his left hand passed under her belly to frig her clitoris, he had his right hand free to inflict any amount of whipping. And, I must say, neither one nor the other spared the rod; they laid it on right soundly, but drew forth no word or sign of complaint. My aunt soon began to wriggle her stupendous backside, in a way to show how very exciting the birching was to her. Her exquisitely creamy white skin began to see the scarlet of the blood rushing to the surface under the infliction received. The redder it became, so did the evident palpitating movement of her two resplendent orbs increase, until uncle, too, showed how the glorious sight was stimulating his less easily excited system, by the stiffening and uprising of his pego. Aunt's hand slipped down to it, and being well acquainted with its habits, pronounced it to be as equally ready as herself. Turning her body lengthways, but still on her knees, the doctor scrambled up behind her, and first stooping, licked up the foam on her cunt, for she had already spent once; and then, rolling his tongue about the beautiful indentation leading to her delicious bottom-hole, he thrust it in as far as he could there. Then rising on his knees, he first plunged his jolly good prick into her cunt for two or three shoves, and then drawing it out well lubricated, presented its point to her exquisite bottom-hole, and plunged it up to the hilt at a single thrust. Aunt gave a cry and shudder of delight as she felt it penetrate to her very entrails. The doctor, satisfied for the moment, lay soaking in the exquisite pressure that aunt's \textit{sphincter ani} was applying to his happy prick. He looked down upon her glorious buttocks, handling them with evident pleasure. I saw aunt's hand steal down to her cunt, and could observe that she was actively frigging her clitoris. She shortly cried out to uncle not to be so idle, but to commence the delicious movements she expected from him.
He did---they did; and such a scene of excitement it was to see so magnificent a woman with such a mighty backside in all the agonies of enjoyment that I could hold out no longer, but seizing my bursting prick in my hand, two or three rapid movements up and down, and tight graspings of the shaft, brought on the ecstatic rapture of so lascivious a spend that I actually fainted and fell heavily on the floor. It was fortunate that aunt and uncle were so hotly engaged that an earthquake might have shook the house without their being conscious of it. So as I only fell from my knees it never disturbed one moment of their pleasure. I must have been some minutes without consciousness, for when I came to my senses, and was able to resume my inspection, I found their crisis was past, but that uncle still lay soaking in the narrow cell he so delightfully occupied. He was gazing with evident pleasure on the still palpitating buttocks of the divine backside immediately below him. Neither was in any hurry, but they dwelt for a considerable space of time in this repose of lubricity. At last, his cock, reduced in bulk, slipped out of its close quarters. Then, rising, and helping aunt out of bed, they warmly embraced each other, kissed and tongued, and aunt thanked him for a most rapturous fuck. Aunt then sat down on her bidet, and uncle used the wash basin. After purifying themselves, and aunt showing all the extraordinary fine development of her glorious form, they put on their night-dresses, blew out the lights, and tumbled into bed. I immediately hastened to gain my sisters' room, with my cock standing stiffer than ever. I entered gently---they were all asleep. My two sisters lay reversed, with their heads between each pair of thighs; they had evidently fallen asleep after a mutual gamahuche in the very attitude in which they had spent. Miss Frankland had apparently waited for me, but feeling drowsy, had thrust her very fine hairy backside right out of bed, ready to attract my attention the moment I should come. So gently approaching, and bringing the light to bear on the beautiful sight, I spit upon and lubricated the end of my prick, and very gently introduced him into her ever delicious cunt. I managed to fully engulph it before applying my finger to her bottom-hole, and my other hand to her clitoris. She had already in her sleep involuntarily squeezed me with her usual force. Then, suddenly applying all my energies, I began an active movement, which instantly awoke her. She was as ready for the sport as I was, and in a very few minutes we ran a most rapturous course of intense delight, and spent with an energy which proved the strength of the excitement I had been under. As I was standing by the bedside, and she lying on it with her fine bottom projecting beyond the edge, it was not a position to remain long in; besides, I was still dressed. So, withdrawing, I undressed myself. My sisters had slept through all this, so first preparing everything for an excessive orgie, by getting out dildoes and birch rods, we awoke the two darlings, who, rising, stripped to the buff. The three dear creatures were all curiosity to know what had kept me so long---more than two hours and a half, and what had been done.
I recounted all the proceedings, except in so far as they had talked of initiating me, for neither Miss Frankland nor I wished my sisters to be acquainted with that matter. They laughed heartily, and little Lizzie said she must act aunt, first flog me and be fucked; then be flogged by me, and have my darling prick up her bottom-hole to follow. We laughed and humoured her, and that scene came off with considerable \textit{eclat}. Miss Frankland fucking Mary, for whom she had a great letch, in the cunt first, and in the bottom, after my example on Lizzie, in the second place. Lizzie and I then laved our parts and prepared for fresh encounters, and we then began a more regular course of the most lascivious lubricity, in which dildoes and rods played conspicuous parts, both becoming necessary under the excessive indulgencies of these last few nights. I stole to my room long after daylight, and slept soundly for an hour or two. You may be sure our lessons were of the lightest in these few days that were left us, and I was allowed to doze off during school hours.
Miss Frankland again walked with me alone in the garden, to give me, as she thought, last lessons in the way I should act with aunt, who she now felt more certain than ever would very soon attack and carry my person when she reached home and had the place and time all to herself. I listened with apparently great attention; as the reader knows, I was already an adept in the art she wished to indoctrinate---thanks to the admirable advice of my ever charming real first instructress, the lovely Mrs. Benson. But I could not help thinking how completely these two admirable women had the same wisdom and knowledge of the world with which they were so anxious that I, too, should become conversant.
The next night the doctor and aunt went quietly to bed, the doctor declaring that his previous night's doings would prevent any more that night. So I only had one more gaze at all aunt's magnificent beauties, which had a never failing effect on my excitable weapon, and which she sent away when her light was put out in a perfectly fit state for the work that awaited him in my sisters' room. I came upon them sooner than expected, and found the three rolled into one body, two gamahuching each other, and Miss Frank-land's clitoris in Mary's bum-hole. For a wonder they did not hear me as I gently opened the door, and I patiently waited till the lascivious crisis brought down a delicious spend from them all. When clapping my hands applaudingly, I cried---
``Bravo! bravo! encore!''
I was so far glad, for to confess the truth the pace was telling, and I began to require more and more of the rod. However, we had but this and the next night at our disposal, and the knowledge that we must soon cease our delicious orgies nerved us all to increased efforts.
Again our passions raged furiously, and broke out in spurts of foaming sperm. Every desire our lascivious lubricity could suggest was carried out to increase our pleasures or renew our exhausted resources, until time warned us again to separate.
The next day there was no school time---it was spent in packing and preparing for departure. My poor mother took it much to heart---she was a most affectionate creature, as innocent as a babe. I often wondered where we three got all the natural wantonness of our characters, for mamma had nothing of it. I suppose it must have come from our grandparents, as aunt had it in the fullest degree, and was almost the equal of the adorable Miss Frankland, who only excelled her in having Greek blood in her veins, which, doubtless, accounted for the extreme heat of her lubricity. Some day I will recount the chief events of her romantic story, which she herself, in after-time, fully related to me. The day was a sad one for us all, even sadder than the next, the actual day of departure. As often happens, the anticipation of evils is greater than the reality when they come.
That night my aunt and the doctor had another whipping bout, but this time she only succeeded in getting a single course out of the doctor. As before when all was over, I slipped away to pass the last delicious night with the dear creatures with whom I had now carried on the most rapturous orgies for more than two years past. My sisters were rapidly developing into remarkably handsome fine young women, especially Mary, who, having the advantage of a year and a half over Lizzie, was naturally more filled out and formed, although Lizzie promised in the end to be, and in fact became, the finest woman, and had also by far the hotter temperament of the two. We passed the night in orgies the most refined, interspersed with tears of regret at our parting, and soft endearments leading to perfect furies of lubricity, until I was nearly fainting with exhaustion. We tore ourselves asunder with difficulty, and the three angelic creatures held their door open, and with streaming eyes watched my receding form; twice, on looking back, I could not help returning again and again to throw myself into their arms for a last loving embrace; but like all things human it came to an end, and I reached my bed and sobbed myself to sleep.
It is needless to dwell on our parting next day. My mother accompanied us to the town where we were to take a coach. It drove up. My poor mother could hardly utter her blessing and farewell, and I saw the tears coursing down her venerable cheeks as she waved her handkerchief before the coach turned the corner that shut us from her view. Of course my heart was full, whose could be otherwise when quitting home for the first time. My aunt put her arm round my waist, and laid my head on her ample bosom, and comforted me as well as she could; but a full heart must vent itself. Fortunately, we had the inside all to ourselves. My aunt was very tender, and so was the doctor. I soon sobbed myself to sleep; even in the bitter grief of the moment I had some slight comfort in the idea of pressing those glorious orbs. My aunt frequently kissed me, and I returned it with full pouting lips, which I fancied rather pleased her. I slept until the coach stopped for supper, ate heartily, and, as may be supposed after my late week of hard work, soon again slept like a top.
I did not awake until it was broad daylight, and, like all heavy sleepers, was awake and sensible of what was going on before opening my eyes. I became conscious that a hand was gently pressing and apparently taking the size of my standing pego, which the pressure of water on my bladder had occasioned to be in an erection of the hardest. I lay quite still, continuing to breathe heavily, but unable to prevent sundry throbbings of my pego, occasioned by the soft hand of my aunt, who was gently following its form from the outside of my trousers. It appeared she had only just commenced her manipulations, not having previously observed the bulging out of its large dimensions under my trousers. She pressed her knee against that of the doctor opposite, who I presume, was dozing off, and in a whisper I heard her draw his attention to my extraordinary development.
``Feel it, my dear, but very gently, so as not to waken him, it is the largest prick I have ever felt, and altogether beats the late Captain of Grenadiers you used to be so jealous of.''
The doctor did feel, and I think aunt would have unbuttoned my trousers, had not the coach suddenly pulled up at the inn we were to breakfast at. So perforce they shook me up. I acted the suddenly awakened sleeper very well. As soon as we were out of the coach, I whispered to the doctor---
``If you please, uncle, I want to piddle very bad.''
``Come here, my dear boy.''
And taking me behind some wagons in the innyard, where we would not be seen, he said---
``Here, we can both piss down this grating.''
And, forsooth, to encourage me, pulled out his own standing pego. I saw what he wanted, and out with my own in all its length and strength.
``Good heavens, Charles, what an immense cock you have got---does it often stand like that?''
``Yes, uncle, every morning it hurts me so until I piddle---it gets worse and worse, and bigger and bigger---it was not half so big a year ago. I don't know what to do to cure myself of this hardness, which is very painful.''
``Ah, well, I must speak to your aunt, perhaps she can help you. Have you ever spoken to anybody else about it?''
``Oh, dear no! I should have been quite ashamed; but when I saw you also had the same hardness, I was very glad to ask your advice, dear uncle.''
``Quite right. Always consult me about that part of your body, whatever you may feel.''
We breakfasted, and I could see, on regaining the coach, that uncle and aunt had a satisfactory exchange of words on the subject. We got to the Rectory in Kent in time for dinner, at which I was the object of great and devoted attention of both, especially of my aunt.
Our previous long journey made an early retreat to bed a necessity for all of us. They both conducted me with much \textit{empressement} to my bedroom, a very comfortable one, having a communication at one end with a corridor, and, on the right-hand side entering, another door communicating with my uncle's dressing-and bath-room, and these opening into their bedroom, which had a similar dressing-room on the other side fitted up with wardrobes for female gear, and dedicated to my aunt's sole use. I was left to a quiet night's rest, which I most thoroughly enjoyed, and slept profoundly until late in the morning. I was awakened by my uncle drawing all the clothes off me. Of course, I was rampant, as usual. He gazed for a moment or two without speaking at my enormous cock at full stand. He then said it was nine o'clock, and breakfast was ready, that he had not liked to disturb me sooner, as I was in so sound a slumber, but now it was time for me to get up.
``I see,'' he added, ``that your doodle, as you call it, has got the hardness you spoke of yesterday.''
Then he laid hold of it, and gently squeezed it---it filled his grasp. He evidently enjoyed the pleasure of handling it, but contented himself with saying that my aunt must see to giving me some remedy the next day, when she should come and inspect it in the morning, so as to see how hard it was, and how it hurt me.
I replied that it would be very kind of aunt, but what would she think of my showing my doodle to her; mamma had told me, when I slept in her room, always to piddle in a corner, and never let anyone see it.
He laughed at my apparent simplicity, and said---
``Your mamma was quite right as to people in general, but it is quite a different thing with your aunt, whose close relationship authorises her doing what she can to relieve her dear nephew, in whom we both take such an interest; besides, I suppose your mamma never saw it in this size and hardness?''
He was gently handling it all the time of our conversation.
``Oh, no! mamma never saw it but at night, when it was quite shrunk up, and that is nearly a year ago, when I used to sleep in her room; it is since then it has grown so large and hurts so much, and throbs so violently as it is doing now in your hand. It makes me feel so queer, dear uncle, and I shall be so much obliged to dear auntie if she will but give me a remedy to relieve the pain I suffer.''
He laughed again, and said---
``I shall speak to your aunt, and we shall see---we shall see; but get up now, we shall find your aunt waiting for us. So make haste and dress; come down stairs, you will find us in the dining-room.''
He left me, and I could hear him laughing to himself, as he walked along the corridor, doubtless at my apparent innocent simplicity. I saw at once that I should be called upon to show myself a man next day; but I already felt the advantage of the advice both my admirable mistresses had given me, as to making all new conquests believe that they had my first fruits. I determined to adhere to the game I was playing, and I foresaw that the pleasure of supporting such a thing would greatly enhance the delight aunt would naturally take in being fucked by my really monstrous cock. I was soon down to breakfast, and was most warmly embraced by my gloriously beautiful aunt, who, in a graceful dishabille, looked more charming than ever. She hugged me for more than a minute in her arms, and devoured me with kisses. I have no doubt the doctor had recounted our interview, and by the sparkle of her eye, and the flush on her face, as she so closely embraced me, she showed that already her passions were excited, and she was longing for the hour in which she could indulge them. However, all that day, they were kept under restraint. The doctor had some parish business to attend to, and aunt leaving me for an hour after breakfast, while she attended to some necessary household affairs, afterwards took me all over the house and grounds, and then we had a walk through the village. The house was one of those snug rectory houses situated in their own grounds which abound in England, but few have so glorious a prospect as was seen from the front of the house. Leeds, in Kent, is situated on the ridge of hills running east and west, and commanding views over the rich and beautiful weald of Kent. The rectory faced the south, and the ground falling rapidly beyond the garden left a splendid landscape in full view. Although close to the village and the church, both were planted out by a thick belt of evergreen trees, which extended to north and east, sheltering the house and grounds from every adverse wind. The house itself was very commodious, but unassuming. The south front had a large projecting half-circle, with three windows in it and a window on each side of the half-circle; this formed the drawing-room below and my uncle's bedroom, and two dressing-rooms above. To the right, looking at the house, there was a wing with an open-arched passage leading to a greenhouse and vinery, while above ran a suite of three rooms, each with one good-sized window overlooking the garden. These were the three rooms kept for the same number of young gentlemen who might be taken in for preparation for the University---a number the doctor never exceeded. Of these rooms I was at present the only occupant. They were built so as to be shut off from all the rest of the house by a door on the landing, leading into the corridor, from which a door communicated with the doctor's dressing-room, and with each of the three rooms. At the end was a water-closet for general use. I have already mentioned the first of these rooms had a second door of communication with the doctor's dressing-room, and this was appropriated to me. Below these rooms, but looking north, and communicating with the village by a covered way and having a playground into which it looked, was the school-room, taking up about half the space of the rooms above. Beyond the covered way to the village was a quiet garden square, into which the doctor's study looked. This study was separated by a passage from the school-room, and had double baize doors both on the house and school-room sides. It was in fact the doctor's sanctum sanctorum, of which more will be told in the sequel. In this manner the school-room part of the house was quite shut off from the rest, and was nowhere overlooked. To return to the habitable part. The west front contained a small library, opening from the drawing-room, and beyond a comfortable dining-room, communicating with the kitchen and offices, which overlooked the courtyard of the entrance to the house, above these were the domestics' bedrooms, \&c. The entrance was from the north into a handsome entrance-hall, with a good broad staircase leading to the upper landing, which, turning westward, led to three extra bedrooms above the library and dining-room. It was thus a very convenient house and well-adapted for a clergyman adding scholastic duties to his other ministrations. I forgot to say that the first bedroom, in the west wing, had a door of communication with my aunt's dressing-room, which I afterwards found had often served for amorous propensities by making it the bedroom of some favoured lover. The grounds were charmingly laid out with a profusion of flowers. There was a perfectly shaded walk in the east shrubbery leading from the greenhouse down to a most charming summer house overlooking the very finest prospect, and perfectly secure from all observation. It was furnished very appropriately for amorous purposes, the couches being low, broad, and with patent spring-cushions. In the sequel it was the scene of many a bout of lubricity. My aunt took me through all that I have described. When we arrived at the summer house, I could see that it was with difficulty she restrained her great desire to possess me; I would most willingly have rushed into her longing arms, and fucked her to her heart's content, but prudence withheld. I had undertaken to act a part, and must go through with it. No doubt aunt was withheld by a similar motive. She and the doctor had resolved that nothing to alarm my modesty-heaven save the mark!---was to be attempted till the next morning. So with a deep sigh she led me away from the summer house into the village, where we met the doctor, and returned to luncheon. After luncheon the doctor took me for a walk again through the picturesque village along the ridge of hills, to enjoy the beautiful views of Leeds Castle, the doctor giving me very many interesting historical details connected with it. After a most pleasant and lengthened walk we returned in time to dress for dinner. I found that one of the rules of the house was that no matter, whether alone or with company the doctor invariably insisted on regular evening costume at dinner-time. This has many advantages. In the first place it gives at least half an hour's occupation, an object in itself worth something to persons living in the country, and then it gives a \textit{cachet} or rather \textit{chic} to your dinner party, however small it may be, and is in itself a certain amount of restraint on excessive exuberance of spirits, and thus may be considered as a disciplinary element of education tending to keep up that reserve and self-restraint characteristic of Englishmen.
Beyond a marked attention to me in every way, our dinner and evening passed without anything worthy of record. I was evidently high in their favour, probably for the reason that both began to have great hopes that I would serve their purpose in every way. We retired early to rest, and I thus obtained three nights of uninterrupted rest, recruiting me after all the excesses I had indulged in before quitting home. It was so far fortunate, that I was thus ready to satisfy the strong passions of my aunt, who was insatiable when once her lust was let loose. I awoke earlier than on the previous morning, and shortly afterwards, hearing a movement in the doctor's dressing-room, I feigned sleep. It was as I expected, the doctor coming to me in company with my aunt. They approached my bedside. I had laid myself on my back purposely to allow the thin summer-covering to be lifted up and bulged out by my stiff-standing pego. I heard the doctor whisper to aunt, to draw her attention to it. She gently slipped her hand under the clothes, and grasped it in her soft fat fingers, upon which it throbbed so violently that I thought it politic to waken at once. My aunt was not at all put out, but held it still in her hand with a gentle pressure. She said---
``My dear nephew, your uncle has brought me to see if I cannot relieve the extreme hardness and pain you feel in this immense thing of yours. Let me see it.''
She now threw off the coverlet, and brought to light my large prick in all the glory of the stiffest stand.
``My word! what a monster!'' she cried.
Her eyes sparkled, and her face flushed as the sight met her full gaze. The doctor approached, and also handled it with evident delight.
``My dear, will you be able to put it into your natural warm bath? It is so very large!''
``Oh! I have not the slightest doubt but that I shall be able to soothe and deliver it of all pain---poor fellow, how it throbs! Does it hurt much, dear Charles?''
``Oh, yes; your hand seems to make it even harder than before, but, at the same time makes me feel so very queer, as if I were going to faint. Do relieve me, dear auntie, the doctor says you can if you like.''
``I will do so, certainly, my dear boy; but the method is a great secret, known only to your uncle and myself; and you must assure me you will never mention it to any one, or tell how I cured you. It is only my strong affection for you that makes me anxious to do anything I can to relieve you. Do you promise to be discreet?''
``My dear aunt, you may be sure I shall be too much obliged to you ever to think of revealing your great kindness. Do, pray, do it at once; I feel so queer, and I am bursting with pain.''
``Well, then, make room for me beside you, and I shall lie down; the doctor will cover us up, and I shall soon reduce the stiffness.''
She got into bed, lay down on her back, pulled the sheet over us, laying bare her splendid belly, and, at the same time, opening her magnificent limbs and desiring me to get upon her, telling me she had a sheath in her body, which, when my hard doodle was put within, would soon relieve it of its stiffness. I got awkwardly upon her. She seized my standing prick, and placing its knob between the already very moist lips, told me to push it in as far as it would go. It glided into its delicious sheath up to the cod piece in a moment.
``Oh, heavens!'' I cried, ``how nice! Dear, dear, auntie, what shall I do now, I feel as if I were going to die.''
My apparent innocence seemed to add to her pleasure. She threw the sheet that covered us on one side, and with arms and legs clasped round my body, begged me to move my bottom up and down, so as to make my doodle go in and out. I followed her directions, and she seconded me with rare art, squeezing my instrument with wonderful pressures as I withdrew and she retired, to meet again the up and down shock with the most lascivious delight. I felt the hand of the doctor embracing my testicles and gently pressing them. I became aware that the crisis was approaching, and shoved home with a cry of rapture, but remembering my part, I exclaimed---
``Oh, I am dying, dear aunt; oh! oh! stop! stop! I---can't---can't---bear it.'' I sank away, but could hear aunt murmuring---
``Dear, darling, delicious boy, I never had such a glorious prick in me, or a better fuck before. I fear the dear child has fainted from the excess of pleasure, and the newness of the sensation, but his glorious prick still throbs deliciously within me---only feel its root, doctor, how stiff it is.''
I felt the doctor grasping it, making it throb violently as he did so.
``The dear boy is as stiff as ever. You will get another fuck out of him the moment he comes to himself. I am glad of that, for it is delightful to see you at it, especially with so splendid a prick operating upon you---it is the greatest treat you have ever given me in that way.''
``I don't wonder at that, my dear, for I never met with such a fine prick in my life before, and little thought my nephew could have had such a splendid one in his trousers when we first saw him. Oh, I am lewder than ever, and am spen---spen---spending. Oh!---oh!''
And she poured down another copious hot flow on my enraptured prick. I let her revel in the ecstasies of her second lascivious discharge until I found that her libidinous passions were again excited and longing for more active operations. I pretended not to know where I was, and began a faltering---
``Oh, where am I? What has happened? I have been in paradise!''
Lifting up my head, I apparently recognised aunt in surprise---
``Oh, dear; how came I here? Oh, remember, auntie, you promised to relieve my hardness, and it seemed so nice, but I feel it is harder than ever; you will try and relieve me again, won't you, dear auntie?''
``Certainly, my dear nephew, you must do as you did at first, move in and out, and I shall second you; and perhaps we shall succeed this time better than before.''
Of course, I was less gauche, and she more energetic. I felt the doctor insert a moistened finger up my fundament, and move it in unison with our thrusts. Aunt cried out to me to go on faster and faster, and we soon came to the grand crisis, dying away together in sobs and sighs of delighted enjoyment. I again sank on her noble panting bosom, really overcome with the rapture-giving delights of that most delicious cunt. On lifting my love-humid eyes to the face of my aunt, she seized my head in both hands, and drew my lips to her in a long, long kiss of satisfied lust, and thrust her tongue into my mouth, which I immediately sucked. She then begged me to give her mine. After tongueing together for a minute or two she asked if my doodle was in less pain, and if its hardness was reduced.
``A little, dear auntie, but I feel it is getting hard again---you must try once more, if you please---oh! it is so nice!''
And my prick throbbed up and stiffened to prove the truth of my words. But the doctor here interrupted us by saying that he must have his own stiffness reduced, at the same time presenting his really fine prick at full stand before our faces.
``You must get up, my dear boy, and your aunt will allay your new hardness in another way, in which she will be able to relieve both our hardnesses together.''
Reluctantly I rose, withdrawing my reeking prick at more than half stand. Looking down as I rose on the truly large and magnificent foaming gash from which I had just withdrawn, I cried---
``Oh, dear aunt, what a wonderful sight it is; I must kiss it for the efforts it has made to relieve me.''
I threw my head down upon it, kissed it, licked its wide open lips all foaming with fuck as they were, thrusting my tongue in as far as it would go. This evidently gave aunt great delight. But the doctor drew me off, told me to lie down on my back, and made aunt straddle over me. She took hold of my now completely standing prick, bent it back, and directing it aright, sank upon it until her ample bush of hair lay crushed on mine. She rose up and down two or three times in a slow delicious movement, and then bending forward, glued her lips to mine while I threw my arms round her glorious body.
I could feel the doctor getting up between my legs on his knees, and then felt his prick was rubbing against the lips of the cunt fully distended round my large pego, doubtless for the purpose of lubricating it before thrusting it into aunt's magnificent backside. I felt the rubbing of his prick against mine through the thin partition, as he glided slowly up into her entrails. We then began our joint movements, but aunt beat us both, and spent twice before joining in our final finish, which was ushered in by loud cries of delight from all three as the death-like ecstasy seized us, and we sank in that half unconscious state of supreme bliss. It was some time before any of us spoke a word. The doctor rose first, and without drawing his prick from the delicious orifice in which it had been engulphed, showed by the way it hung down its pendant head, that aunt had at all events allayed its stiffness. He desired aunt to rise also, but I felt by her throbbing cunt, and the pressure she put on my prick, as she rose from it, so that it came out with a loud flop, that she would fain once more have done me the service of allaying any stiffness that might re-arise. However, it was much limper than before, although still of a goodly thickness. When she got on her legs, she stooped forward, kissed it, took it in her mouth, and most lovingly sucked it, saying how delighted she would be to relieve me whenever it was troublesome. They begged me to get up and dress, and we should meet at breakfast. They then withdrew, to complete their own toilets. I lay for some minutes in the dreamy delight of thinking over the delicious event that had just taken place, and amused at the last remark of my aunt, which seemed to infer that she thought I was innocent of the real meaning of the performances that had just taken place. I determined to act as if it were so.
We met at breakfast, aunt kissed me most lovingly. I thanked her for her great kindness in relieving me from pain in so delicious a manner, and told her I could not help loving her more than I had ever loved any one before, and said I hoped she would kindly relieve me every morning, for I always suffered at that time from the painful hardness, though I should never be sorry for that, as long as she would so kindly allay it. I put my hands quite in a childish way on each cheek, and held up my mouth for a kiss, which was given to me in the lewdest way. She called me her dear boy, and told me that she would always help me as she had done that morning, as long as she found I was discreet, and never told how she did so. You may be sure that my promises were most earnestly reiterated. So we kissed again, and sat down to an excellent breakfast with sharpened appetites from our early exercise, and did full justice to the viands set before us. The doctor gave me a book of history, and desired me to read for a couple of hours, and said that at luncheon we would talk over the subject of my reading. I studied attentively for the time prescribed, and then aunt came to ask me to walk in the grounds with her. Insensibly or not, she led me to the summer house, and sat down on a low ottoman. I sat down beside her. She drew me to her, kissed me, and clasped me to her bosom, murmuring terms of endearment, and pressing me to her glorious bubbles. Of course, my unruly member fired up at once. To prevent her imagining it was lasciviousness that prompted me, I said---
``Oh, my dear aunt, I do so want to piddle, my doodle at once gets as hard as wood if I at all restrain the inclination to do so, just feel how stiff it has become; will you let me go and piddle?''
``My dear boy, I will go with you, and unbutton your trousers for you.''
We went among the trees. Her busy fingers undid my trousers, and helped to bring forward my lordly cock in its glory. Fortunately, I did want to piddle, and aunt held it up as I did so, her eyes sparkling with lust as she handled it, and her face flushed with her excited passions. She remarked what an astonishing size it was, gently rubbing it up and down. Of course, it became more rampant than ever. Throwing my arms round her stooping neck, I asked her if she could not again relieve the excessive hardness and pain it was in.
``To be sure, my dear boy. Come here again into the summer house, where we cannot be observed.''
We entered. She put a cushion on the floor for my knees, threw herself on her back, and lifted all her petticoats well over her belly, exposing her very hairy cunt, and its splendid pinky gash, already moist from her excitement. I threw myself on my knees, and stooping down, said---
``I must kiss the dear reliever of my pains.''
I kissed and tongued, until my aunt begged me to raise my body, and come upon her, that she might quickly put me out of pain. I rose, and slipped my stiff member up to the hilt in her longing cunt almost taking away her breath by the suddenness and completeness of the insertion. Her legs and arms were round me in a moment, and at it we went hammer and tongs, until we quickly spent with cries of delight, and sank in momentary oblivion, soon to recover our full sensations, and dash again on passion's furious course, this time aunt pouring down her hot boiling discharge before me, and again when she felt the torrent of my sperm shooting up to the top of her womb. Our final crisis was even more ecstatic than the first time, and we lay longer in the soft languor of the after-sensations. The excessively voluptuous nature of her inward pressures soon re-illuminated all my libidinous desires, and refired my prick with renewed force. We soaked for a short time, each indulging in the delicious inward throbbings, until our lust could stand no longer such mere preliminary work, and stimulated anew, we rushed with freshened passions into the fray. The fiery nature of my lustful aunt paid down two tributes to Priapus to my one. This time our sensations were so ecstatic in spending that we really lost all consciousness, and lay for long locked in the closest embrace. I could feel that we were both becoming re-excited, but my aunt begged me to rise, saying that was enough for the present, the stiffness was allayed, and my weight was too much for her to endure longer. I rose, but again buried my face in the wide gash of that glorious cunt, and before rising completely, I licked up the delicious foam, and even ventured to give, as it were, an accidental lick to her little knob of a clitoris, for she was not much distinguished in that way; she shivered with excitement, when I touched it, and even pressed my head down upon it, when she felt the pleasure pressure.
``My dear boy, what exquisite delight you give me! Continue for a little to keep moving your tongue on that hard projection.''
I did so. Her splendid backside wriggled below in the fullest enjoyment. She rapidly came to the ecstatic ending, nearly thrusting my whole face into her vast orbit, and spurting out a very torrent of sperm, all over my face and neck. She seized me by the shoulders to draw me up, that she might kiss me. My prick had regained its full vigour, and could not fail to slip in of itself into that most lascivious and gaping cunt when it reached the entrance. My aunt started at such an unexpected result, but was too much gratified to hesitate for an instant. Throwing legs and arms around me, her supple loins were in immediate action. I myself was equally in a state of wild lubricity, so that our course was even more rapid than at first, and we both spent and sank together in the delicious after-languor as soon as the ecstatic joy of the first rush of the exquisite discharge was over. My aunt, who could not but be most highly gratified, still kept up the appearance of relieving me, she desired me to rise, and said we must go, as luncheon time was at hand.
``But, my darling nephew, you must yourself endeavour to keep down your hardness, and not allow it to become stiff so often---you will injure me with your violence.''
``Oh, my darling aunt, you give me relief with such exquisite pleasure that my doodle seems to harden only for the purpose of your relieving it---see how it is again bulging out of my trousers,'' for she had buttoned it up. She put her hand upon it, and squeezed it, but said, with a deep sigh---
``Come along, come along, or I do not know what might happen.''
She drew me away, but by the manner in which she squeezed my arm, I could feel she was herself still greatly excited. Her prudence alone enabled her to resist further indulgence, as she seemed to think I was still unaware of the real nature of our proceedings. We found the doctor waiting for us at the luncheon table. He guessed by the flushed face of my aunt the nature of our late employment, and asked if I had been again troubled with my unnatural hardness.
``Yes, poor fellow,'' said my aunt, ``it appears that whenever he wants to piddle, and cannot do so at once, it troubles him in that way, and I have had some difficulty in allaying it. I succeeded at last, but I have told my dear nephew that he must endeavour himself to restrain it in the daytime, as it is not always in my power to relieve him.''
``Quite right, my love; my dear Charles, you must endeavour to follow the wishes of your of aunt.''
Of course I promised, and with such a look of innocence that I could see they exchanged smiles at it. We sat down to luncheon. Afterwards the doctor, seating himself by my side, began a conversation on the historical subject I had been studying. Our conversation became really very interesting. The doctor was a man of great erudition, and of varied knowledge, and had a manner, special to himself, of making almost any subject most interesting. Hours flew by, and it was only when aunt entered about five o'clock, to take a cup of tea, as was her wont, that we were aware how time had flown. The doctor praised my knowledge of history, and the pertinency of the questions I had put to him, in a manner highly flattering to me, and I could see that I had risen much in his estimation, quite apart from any erotic influences. He proposed a constitutional walk before dinner, and much interested me by his instructive conversation during it. Our dinner was most agreeable. In the drawing-room aunt, a most admirable performer on the piano, enchanted us with her skill and taste. The doctor challenged me to a game at chess. He was, of course, far superior to me, but he praised my style of play, saying I should become a great proficient with time and practice. We retired, as usual, about half-past ten, the doctor seeing me to my room, and promising to bring aunt in the morning to see if I was still troubled with that painful hardness. I thanked him warmly, but with much simplicity, as if quite unaware of the real nature of the application of the remedy. He left me to my repose. The quiet nights of sound sleep made my day efforts pass off without any exhaustion, and I felt my erotic powers increasing in force.
I slept soundly, and so long that I was only awakened by the caressing hand of my aunt on my stiff-standing pego. She had gently lifted off all the coverings, and I lay quite exposed to eye and touch.
``Oh, my darling aunt! how kind of you to come this early to relieve that troublesome thing.''
I held out my arms. She stooped down to kiss me. I clasped her to my bosom. Our lips met, and our tongues darted fiery lust into our bodies. She threw herself down by my side, I was onto her in a moment. The doctor took hold of my pego, and guided it into the delicious orbit of his wife. Dear aunt begged me to do as I did yesterday, if I wanted relief. Our action became fast and furious. Her legs and arms wound round me in loving pressures. Her active backside wriggled in delight. The doctor had introduced first one finger, and then two, into my fundament, and added greatly to the fury of my lust, so that I spent in an agony of pleasure, as quickly as the fiery lust of my aunt produced her hot and plentiful discharge. I sank on her charming bosom, panting with the force and fury of our coition, but like all very fast fucking, my virile member hardly flinched from his first vigour, and a very few of aunt's exquisitely delicious internal pressures sufficed to bring him up to the fullest stiffness. We were about to plunge again with renewed ardour into all love's wildest excitement, but the doctor insisted upon our first changing places, that he, too, might have his hardness allayed. Our change of position was instantly accomplished, and dear aunt, after impaling herself on my upright member, sank on my bosom and was clasped in my longing arms. The doctor scrambled up behind her, and lost no time in sheathing himself in her fine and beautiful bottom-hole, and then we ran a double course of delight, dear aunt taking the lead as usual, and deluging us with her hot and delicious discharges before we were ready to pour into her a double dose of delight, which again made her spend with fury and cries of rapturous enjoyment, in which we both joined, and then sank in love's exquisite inanimation. On recovering ourselves the doctor withdrew, but I was already as stiff as before. Aunt began a most effective and delicious movement above me, which soon brought on another grand finale, and we died away in mutual delight. I could feel that the doctor was gently handling my cods, both during and after our last combat. When, by our mutual throbbings, he saw that we were about to become fit to enter on another career, he begged his wife to rise from off me. But the idea of losing her and her extra pressures made my prick immediately resume an erect position, so that when she rose from off it, it was shown in a completely standing state.
``What! again, Charles?'' said the doctor. ``Your member is sadly unruly. My dear, you must again try to allay it, but put yourself this time on your knees, and we shall see if that position be better adapted for the purpose of relieving this immense object.''
He was gently and admiringly handling it all the time. His wife was quite aware of his object, and, indeed, so was I. Our last bout had helped to restiffen his prick, and although not yet quite rampant, it was evident that when my bottom was in full view, and so placed as to be got at with facility, it would be quite as stiff as necessary. When his wife had knelt down, and by lowering her head had exposed all the wondrous grandeur of the most superb backside that ever met my eyes, my prick bounded with joy. The doctor still grasping it, and feeling it throb so wildly, saw that his game was sure. He pointed out all the beauty of aunt's second orbit of love, and told me it was in that he had allayed his own hardness, and as the other orifice had not succeeded in quieting me, he recommended my entering within the narrow path of ecstasy. I professed no surprise, but seemed to take it quite as a matter of course in the simplest innocence of manner. Uncle continued to handle my tool as I mounted on my knees behind aunt. Guiding the almost bursting weapon into the delicious cunt in the first place, to be lubricated there, and then telling me to withdraw it, he directed it to the smaller orifice, and desired me to push gently and smoothly in. It glided in slowly up to the meeting of my belly against the enormous buttocks of that sublime backside. There I paused for a minute or two within the throbbing sheath. Aunt had pushed her bottom well out, and by the action of apparently voiding, had facilitated the entrance. She winced once or twice, but on the whole, as she told me afterwards, took in my enormous tool with less difficulty than she expected. After a few slow movements, during with I caressed and devoured with admiration the glorious orbs beneath my dearest gaze, uncle desired me to lean forward and embrace my aunt's splendid bosom. As soon as I did this, and began slowly to thrust in and out of the delicious sheath in which I was so rapturously engulphed, I felt uncle's hands wandering over my buttocks, followed by the introduction of two fingers into my anus. My throbbings on them showed how much he pleased me. He asked if it added to the pleasure I was enjoying.
``Oh, yes, dear uncle, immensely.''
``Then,'' said he, ``as I, too, am suffering from hardness, I shall try to allay it in your bottom, as you are doing in my wife's; don't be afraid, if I hurt you I shall stop.''
``Do just as you like, dear uncle, both you and aunt are so kind as to do all you can to relieve my pain, and I should be very ungrateful if I did not do all in my power to relieve you.''
``You are a darling boy, and I shall love you dearly.''
He knelt behind me, and spitting on his cock, presented it at my bum-hole, and pressing gently forward, soon sheathed it to the utmost depth. He did not hurt me at all, as I was too much used to be dildoed there to have felt any difficulty of approach, but I deemed it politic to beg him to be gentle from time to time, as if it were a virgin vale he was entering. He fancied as much, and that was just as good. When once he was fully within, after a few throbs, which were felt most deliciously on his delighted prick, we proceeded to more active work. Aunt, in the meantime, by more pressure on my prick, and by frigging her own clitoris, which I was quite aware she was doing, had spent profusely; and, as the case with all the mucous membranes of the body which sympathise with the cunt's discharge, her bottom-hole became quite moist and deliciously heated. The doctor and I then went at it with fiery force, and soon gave down nature's tribute, and mutually poured a flood of sperm up the entrails we were respectively belabouring.
We lay for some time after in all the luxury of soaking in the delicious apertures. I fell to nothing, and reluctantly withdrew. I had again become rampant, and keeping myself more erect, with a hand on either immense hip, I devoured with greedy eyes all the glories beneath my gaze. Fired by such a truly magnificent sight as these huge buttocks were, when in an entire state of wriggle, I again spent with cries of agonised delight, and in all the ecstasy of fully satiated lust, sank almost insensible on the broad and beautiful back of my aunt, who herself had spent several times, squealing like a rabbit, and eventually falling flat on her belly overcome with exhausted lust, drawing me with her still held a willing prisoner in her glorious and exquisite bottom-hole. We lay entranced for some time, until the doctor, who, during our last bout, had purified himself, told us we must now get up. With difficulty I tore myself from out of that delicious sheath, and rose with my cock at last pendant. The doctor congratulated me on the success of the last move. His wife lay still panting with all the delight of satisfied desire, and we had to help her up. She threw herself into my arms, and hugged me close to her heaving bosom, kissed me tenderly, and hoped she had relieved me of all pain. I was her own darling boy, and she would always be truly happy in relieving me of that inconvenience whenever it troubled me. I was internally amused at their continuing to keep up this idea, but I humoured them, and appeared the most innocent simpleton, notwithstanding all that had occurred. The day passed much as the previous one. After two hours' reading, aunt again proposed a walk, which, of course, ended at the summer house, where again a pressure of water brought on the painful hardness, which aunt succeeded in allaying after four most exquisite bouts of love, varied by a thoroughly good double gamahuche between the last two acts. Aunt must have spent at least ten times, and appeared thoroughly contented, but continued to attribute it to her gratification at having relieved me of my painful hardness. Again I passed hours in instructive conversation with my learned uncle and after a similar evening to the last, retired at our usual hour.
Next morning I was awakened by uncle alone, who told me that my aunt was somewhat poorly, and could not come.
``I am sorry it is so, for this little fellow is as hard as usual.''
``Oh, I am so sorry dear aunt is poorly, both on her account and my own. What shall I do, dear uncle? It is so hard and painful.''
``Well, my dear boy, I must try to allay it myself. I love you too dearly to leave you in this state. I am not so good at allaying this painful attack as your aunt, but as you know you were successfully relieved in her bottom, and I in yours, yesterday, we shall try to-day if I can accommodate this huge fellow, of which I have some doubts. Take off your night-shirt as I do mine, it will be more commodious.''
In an instant we were both stark naked. We threw ourselves into one another's arms and lovingly kissed each other. Our tongues met in a delicious sucking---our hands took each a prick, and we had a most exciting and loving embrace. The doctor then took my prick in his mouth, sucked it a little, and well lubricated it with his saliva, spitting on the lower part of the shaft and rubbing it round with his finger. He then knelt, and presenting a really beautifully rounded bottom of the fairest hue, he pushed it out, showing a light brown corrugated bum-hole, most tempting to look at. He desired me to wet it with my saliva. I stooped and applied my mouth and tongue to the appetising morsel, and thrust my tongue in as far as it would go---to his evident delight, leaving it well moistened. I then brought my prick to the entrance; he shoved his backside well out, and acted as if he desired to void himself. A firm but slow pressure quickly engulphed the knob. The doctor desired me to rest a moment, and drop some spittle on the shaft. Again it was firmly pushed forward, and gradually it won its way up, the belly against the buttocks, without much flinching on the doctor's part. After resting a while, he desired me to bend forward and feel his cock while I should move backwards and forwards in the sheath until I was relieved. I had a most delicious fuck. The doctor's bottom-hole was quite hot internally. His pressures with the sphincter were exquisitely delicious, and he had acquired the charming side wriggle so exquisite in quim fucking. Of course this was an old letch of his, which his position as schoolmaster had given him so many opportunities of indulging in, and the still greater pleasure of initiating others in it. At this very moment he was delighted with his delusion about me in that respect. Of course I never undeceived him, and he had all the extra delight of the idea. My younger and hotter passions had made me spend before he could; so after indulging me in a delicious soak after the ecstasy of the discharge, he drew my attention to the rigidity of his own member, which, he said, I must now allow him to allay in turn.
``Of course, my dear uncle, I am too sensible of your great kindness in relieving me to hesitate about giving you the same relief.''
I now withdrew. He rose for a mutual loving embrace, and then I stooped, and taking his fine milk-white prick with its lovely vermilion knob into my mouth, most deliciously sucked it, making my tongue tickle the entrance to the urethra, to his infinite delight. He murmured out soft terms of endearment; then getting exceedingly lewd, he begged me to kneel down as he had done. He then kissed and gamahuched my bottom-hole, making my prick stand and throb again with delight. Then spitting on his prick he quickly sheathed it in my glowing backside. After pausing to enjoy the exquisite pleasure of complete insertion, he stooped, and passing a hand round my belly laid hold of my stiff-standing prick with one hand, while he gently pressed the ballocks with the other. We then proceeded to active measures. He soon made me spend, which I did with loud cries of delight, giving him the most exquisite pleasure by the pressures the act of spending made me exercise on his pleased prick. He soon resumed his thrusts, and eventually we both spent together in the most ecstatic joy. I sank forward on the bed, dragging the doctor with me still imbedded in the rapture-giving aperture of my backside. We lay long in all the enchantment of delight. At last he withdrew completely reduced, but was surprised to see me still in a rampant state. When I got up he took my prick in his hand, praised its noble proportions, and again stooping, took it in his mouth, frigging the lower shaft with one hand; he then introduced two fingers into my bottom-hole, continued his suction and movement on my prick in unison with the working of his fingers up my bum-hole, and in this manner quickly produced a delicious discharge in his mouth. I had placed my hands mechanically on his head, and I nearly choked him as I thrust my prick halfway down his throat as I spent. He greedily swallowed every drop, and then rising, embraced me lovingly, telling me I had given him the greatest treat in the world, and he loved me dearly. After this he invited me into his dressing-room, and we both entered the bath together and mutually laved each other. Then dressing we joined aunt at breakfast. She had not the least air \textit{d'une malade}, but with a sly smile hoped the doctor had proved as efficient as herself.
``Oh, yes, my dear aunt, and I am so much obliged to both of you for your solicitude to relieve the pain I suffer in the morning, but it seems to me that it more frequently and more severely attacks me than ever. I only hope I shall not tire out your kindness by such frequent appeals to your aid.''
``Oh, my darling nephew, do not imagine anything of the sort. We are but too happy to be of any service to you.''
This was accompanied with a knowing smile cast at each other, caused by my apparent uncommon simplicity, but which they were evidently glad to see. We sat down and enjoyed a capital breakfast.
The day passed quite as the two preceding ones. Aunt asked me to walk with her, and as before ended by leading me to the summer house, where, after relieving my distress symptoms, as she called them, three times, and finding that the relief was still inefficacious, she proposed to try if by adopting my uncle's position she could not be more successful. So kneeling on the low ottoman, and throwing her clothes over her back, she exposed all the glories of that most splendid backside, and dazzled my sight with its huge magnificence and ivory-like surface, perfectly milk-white, the pureness of which was equally perceptible through the rich light curly hair that spread bush-like between her legs, and wandering beautifully upwards between the cheeks of the enormous orbs, stole round the charming corrugated aperture that I was about to penetrate, the rosy circle of which appeared all too small to admit my very large virile member. I threw myself on my knees, and first licking out the wide open lips of her wonderously fine cunt, and taking care to pay my respects to the small knob of her indurated clitoris, I transferred all my attention to the smaller and most charming orifice. After kissing it most lovingly, I thrust my tongue in as far as it would go, and rolled it about to her infinite delight, while with my left hand below I kept pressing and frigging at her excited clitoris. She wriggled her glorious backside in all the agonies of the delicious excitement until she spent most profusely, actually hurting my tongue with the tightness of the squeeze her sphincter muscle gave as she poured down her plentiful discharge over my chin and neck. In her grand excitement, and wild with the fury of her lust, she cried out---
``Oh! fuck me, my darling, and shove your glorious prick into my bottom-hole. Oh! fuck---fuck---fuck me directly!''
Inwardly delighted at this natural outbreak of her passions, naming matters by their more appropriate terms, I replied by acts, without any words at the moment. It may well be imagined I was myself in the most rampant fury of desire. So bringing my raging prick up to her magnificently large cunt, all foaming as it was with her recent discharge, I plunged with a furious bound up to the codpiece at once. She met my forward lunge with a backward push and a cry of delighted satisfaction. I moved a few times in and out, so that my prick was white with the foam of her delicious cunt. Then suddenly withdrawing, I presented it at the entrance of the more secret temple of Venus, and more gently pushed it home, she helping me with outthrust buttocks and outward straining of the entrance, so that I most charmingly glided slowly into the glowing furnace that was awaiting with such lascivious desire to engulph and devour my longed-for prick. For, as I have before observed, my dear aunt was gluttonous of a bottom-fuck, after being so fucked in cunt as I had already served her. It was so deliriously tight and hot that I lay in the exquisite rapture of complete insertion for some minutes. I had seen my aunt's arm move in a manner to convince me she was frigging her own clitoris, in fact, the movement of her hand frigging herself was felt by my codpiece. I let her continue, until finding by the involuntary wriggling of her bottom that she was about again to spend, I aided her with my prick, and had hardly made many moves before she poured down another tribute of lust, with a squeal of delight, and with such pressure on my prick as nearly drove him at once to a similar discharge. I did my best, and succeeded in not following suit. My aunt was insatiable, and I was glad to let her spend as often as possible, and I so managed matters that she spent again before joining me in the final crisis, which seized us together, and we died away in joyous cries of thoroughly, though but momentarily, satisfied desire. I sank on that magnificent back, as the languor that follows the ecstatic moment overtook me, but it was only for a short time. The exquisite internal pressures that my amorous and glorious aunt was exercising on my delighted prick were too exciting not to rapidly produce a reaction; nonetheless rapidly that it was in such a delicious retreat as the pleasure-giving aperture of that gloriously exciting backside. I was lying down on her broad back, so passing one hand round to her large but firm bubby, I took its nipple between my fingers. The other hand sought the knob of her still stiff clitoris. I excited both while making a very gentle move with my hardly fully standing prick. I felt at once how this gratified her, indeed, she often afterwards assured me that such frigging, with the movement of the softened prick gently working within her, was most exciting, and almost better than when it was in full force. I soon made her spend again. Another of her delights was to have a stiff prick shove away into her the instant after she had spent, when she herself was at the moment incapable of action on her part. She in after-days proved that her greatest pleasure was to have a fresh-standing prick near, to take the place of one that had made her spend, and had spent itself, and have it thrust into her with all the vigour and lust the sight of the previous fucking had inspired and fired it with. At this moment, as I had not spent, it was the exact counterpart her libidinous imagination could have desired. I fucked and frigged on until we both gave down in cries of joy our united tribute to Venus. We both sank this time down on the couch in utter forgetfulness of all but the ecstatic bliss with which we were overcome. We long lay soaking in all the delightful sensations my adorable aunt's convulsive clutchings of my prick with her delicious close pressures excited. At last she begged me to withdraw, although she could feel me now re-stiffening under the delights of that exquisite interior. I would fain have recommenced.
``You must not, my dear boy, it is more than nature can support, and I must consider your youth; you have delighted me even beyond previous delights---rise then my love, and let me embrace, thank, and love you as I shall always do.''
I rose, and we threw ourselves into each other's arms, lovingly kissing and tongueing each other. Aunt then buttoned me up, first kissing and taking a mouthful of my prick for a moment between her lips, and then putting him away, calling it ``my pretty doodle.'' I seized the expression, and said---
``Dear aunt, you called it my prick just now, and begged me to fuck you, and to shove it well into your cunt. Are these the real names for my doodle and your Fanny, and what does ``fuck'' mean, my darling aunt? Do tell me, dear auntie? and teach me the language I ought to use when you are so kindly relieving me of the pains of my now so frequent hardness. I don't know whether you have observed it, dear auntie, but I never enter this summer house with you, but it becomes painfully hard at once; to be sure you give me such exquisite pleasure in relieving me that I could wish to have constant hardnesses as long as you were near to calm them. Is this natural, dear aunt, or a disease? Pray tell me, and teach me all the endearing terms you so lavish upon me while I am reducing my hardnesses.''
My apparent simplicity evidently pleased her. She probably thought, too, that as I must sooner or later really thoroughly understand the nature of our intercourse, it would be much better she should, as it were, make a confidant of me, and attach me more securely to herself. She begged me to be seated and she fully explained everything to me. Of course I was even better acquainted than herself with all she communicated, but I confirmed the idea she evidently entertained of her being my first instructress by various naive remarks on all she was telling me. Of course I proved an apt scholar, and by my close-put questions brought out all her own knowledge, and left nothing for me to learn. At the end, I said---
``Do all women have such a delightful sheath---cunt I mean---between their legs as you have, dear aunt?''
``Yes, my darling; but you must never stray to others; you will find none so fond of you, or I may add, without vanity, so capable of satisfying this dear fellow; but come, I see it will be dangerous to allow him to stay here longer.''
She rose, but I quickly unbuttoned and produced my prick in an almost grander state than ever. I begged of her to let me have one more ``fuck'' now that I knew what it all really meant. I put it into her hand. Her own previous descriptive lesson had aroused her lasciviousness. She fondly grasped it, and stooping down, kissed it, saying she could not resist its eloquent look. Throwing herself back on the couch, with her clothes up, her feet on the edge, and her legs apart, her glorious cunt lay open in its moist magnificence. I threw myself on my knees and gamahuched her until she spent: and now, knowing her greatest letch, I instantly brought my bursting prick up to her foaming cunt, plunged in and began a furious movement, accompanying it with all the most endearing bawdy phrases she had just, as she thought, taught me.
``Oh, my most gloriously cunted aunt, do I fuck you? Wriggle your arse faster---that's it! Do you feel my prick up to the hilt in your delicious cunt? Oh! what pleasure you do give me!''
She replied as broadly. Passing her hand down she pressed my cods, and asked if thus squeezing my ballocks added to my pleasure.
``Oh, yes, my love, your cunt, your arse, your bubbles, are all delicious. Oh, I never before knew there could be such additional pleasure to our fucking as using these endearing words produce.''
We were both so excited by the bawdy terms we so profusely used that we went off in the utmost excess of ecstasy, and died away thoroughly satiated with our libidinous and most lasciviously delicious fuck. It was time to finish. So sliding off her, I again buried my face in her delicious gaping and foaming cunt, my mouth, lips, nose, and cheeks were covered with sperm; she drew me to her lips and licked it all off. Then repairing our disordered dress we returned to the house, and found the doctor impatiently awaiting us. Our flushed and excited faces at once showed that we had been indulging in the greatest excess. He joked aunt upon her skill in allaying such frequent attacks as I now appeared subject to.
Aunt informed him that she had inadvertently in her lust made use of expressions which had betrayed so much to me that she had found it necessary to leave me nothing more to learn, and I was now fully aware of the true nature of our connection; after luncheon he himself might further enlighten me, for she was certain that complete confidence would be the best policy to pursue; it must come about, sooner or later, and it was far better it should come from him than that I should learn it elsewhere. He said she was quite right, and that he would further instruct me after luncheon, so we set to work on the viands before us, to which I did ample justice.
I was thus, as they supposed, newly initiated in the mysteries of the coition of the sexes. I shall reserve further details of our more intimate and expansive experiences for the third volume of this true Romance of Lust, and still of Early Experiences.
\gutchapter{END OF VOLUME II.}
\section*{\raggedright VOLUME III.}
\gutchapter{CONTENTS}
Aunt Brownlow---Harry Dale---Mrs. Dale and Ellen---Mrs. D.---Ellen---Mrs. D., Dr. Brownlow, and Harry---
After the luncheon, which closed the last volume, a churchwarden occupied uncle for about an hour. When he had left off, uncle proposed a walk in the garden. I could see at once what this was meant to lead to, as he almost immediately turned in the direction of the summer house. When we got there he sat down on the couch, and begged me to sit beside him. He opened the subject at once by saying---
``My dear Charlie, I am very much pleased that your aunt has opened your eyes to the real nature of our actions with you, which your simple innocence had imagined to be a mere kindly relief to the overgorged vessels of your virile member. Accident might have made you acquainted with this through some less interested channel, and you might have innocently betrayed your future position. I believe you to possess a large fund of good sense and discretion, and the advice I shall give you as to the conduct to pursue in future will not only be received with confidence as meant for your future good, but listened to attentively and acted upon. The world, my dear boy, and by that I mean Society in general, condemns the practices we have lately been indulging in with you. Their narrow prejudices ignore the fact that nature alone prompts to these delightful acts, and that the great God of nature gifted us with the powers necessary for their performance. But, as the world has chosen to brand them with its censure, men of prudence, like myself, whilst apparently conforming outwardly to such stupid prejudices, know how in secret to fully enjoy them. I am blessed in your darling aunt with a wife who fully understands and humours my desires. She is rarely splendid in the glorious beauties of her body, and in temperament hot as the most erotic of our sex could desire. Even in your ignorance you must have felt the wonderful power of conferring carnal ecstatic pleasure she possesses, and have heard how, in the energy of her passion, she allowed her lust to betray her into the use of grossly bawdy terms, but which, as they have enlightened you when best prepared to receive such knowledge, is rather fortunate than otherwise. I speak thus frankly to you, my dear boy, because I have found you of a rare facility in giving and receiving erotic pleasures, and of a temperament worthy of the descent from the same stock as your aunt. You are worthy of each other, and formed to enjoy to the utmost each other's carnal delights, and I bless my happy star that has brought you both under my own roof. Henceforth there must be no secrets between us. It was at my earnest wish that your aunt relieved you; and, of course I had my own object in view. In the first place I require some extra excitement to be able myself to indulge in these delightful combats in love's domain. You and your aunt's copulations were to me more exciting than you can imagine. You will have observed, too, what is the real quarter to which, when excited, I pay my devoirs. Glorious as is the backside of your incomparable aunt, your young charms, virgin in that respect, excited me still more. I began by gentle touches, and then tried the insertion of my finger, when I saw you were far too busy operating within the orbit of your lustful and lusty aunt to observe or even feel what I was doing. I found a facility about your bottom as perfect for enjoyment as your truly magnificent prick or cock was fitted for operating in its way. It was then I suggested to your aunt to mount upon you, and afterwards made you aware that your aunt possessed another aperture which could equally well allay what you then looked upon as a source of pain. My object was to lead you to the same point. Your innocent docility lent itself with easy simplicity to all my desires. I saw that you entered readily into your aunt's glorious bum-hole, and allowed me to work with two fingers in your own. Finding that it rather gave you pleasure than otherwise, I proposed to abate my own stiffness in your bottom. Your affectionate docility enabled me to obtain unfailing ecstasy. Your after-fucking of me, while I was in my wife's bottom, conferred the utmost erotic bliss upon me, as you have experienced when operating and being operated upon. These---these are the moments of a felicity your stupid prejudiced worldlings know nothing of; and these are the pleasures which, now that we have initiated you into all their secret mysteries, we will enjoy to the utmost. To the true votaries of these love orgies grossness of language is a stimulant to passion. Fuck-frig---bugger---cunt---prick---ballocks---bubbles---arse-hole---are all sacred words only to be pronounced when in the exercise of love's mysteries. At all other times a guarded decency of word, act, and gesture is imperative, as enhancing the delight of an unbridled vocabulary in the voluptuary of raging lust. I shall from time to time inculcate sage precepts on this point---enough for the present. Let us now indulge in mutual embraces.''
So ending, he took me in his arms and glued his lips to mine. Our tongues met. Both our hands wandered, his on my prick, which immediately responded to the touch, my hand was placed on his prick, but which was only at half-cock. I rapidly unbuttoned, and brought it forth, then stooping I took it in my mouth, and sucked it and fingered the root with my hand. Then passing my other hand below, I sought to penetrate with my finger into the interior of his fundament. He rose to a standing position to enable me to enter his anus more easily. His prick quickly standing fiercely showed how much I excited him. I ceased not until he was in an agony of pleasure---forcing my head down on his prick until it entered almost completely into my mouth, and shooting his sperm right down my throat. I continued to suck and frig him until I produced somewhat of a restiffening of his prick. He begged me to rise, that he might take mine in his mouth, desiring me at the same time to take off my trousers and lie down on the couch. I did so. He knelt at my side, and first handling and examining it with loudly expressed admiration of my noble weapon, he took its head in his mouth, and then with his hand on its lower shaft, and finger up my fundament, brought on a similar crisis as that I had produced on himself. He just as greedily swallowed all. I had allowed my hand to fall down by the side of the couch, where it encountered his prick, which had resumed its pristine vigour.
``Come my dear uncle,'' said I, ``and let us put it into its favourite corner.''
I rose, and kneeling, turned my backside full in his face. He stooped, caressed, kissed, and tongued the rosy orifice. With the plentiful saliva with which the operation of sucking my prick had filled his mouth, he moistened my bum-hole and his own prick, and then easily glided up to the hilt within my delighted backside. Resting for a while in all the ecstasies of insertion, which I heightened by my internal pressures, he seized my prick which had stood again at once at the pleasing sensation occasioned by the introduction of his prick in my bottom. Thus frigging my prick and fucking my arse, with occasional pauses to lengthen out our pleasures, he at last brought matters to a most exquisite termination, and died away in cries of joy as we poured forth a mutual torrent of sperm. Uncle continued soaking in all the blissful after-sensations, which I did everything in my power to enhance by the delight-giving pressures of my sphincter muscle. When he withdrew and rose to his legs, he helped me up, and drew me to his bosom, and we had a long kiss of gratified desire, tongueing each other the while, and handling our ballocks with mutual gratification. My uncle was profuse in his praises of my docility and aptitude, declaring that his pederastic enjoyment of my person excelled all he had ever experienced in his long practice of the habit, and my delicious sideways wriggle was superior to the very fine bum-fucking his adorable wife had the art of giving him: then there was the further excitement of handling the very finest prick he had ever met with.
``It is no flattery to you, my dear Charles,'' he said, ``but mere justice to its superb dimensions and admirable power.''
Here he stopped, and sucked anew its reeking head, getting a few more drops out. We then purified ourselves---a basin with water was kept in a small cupboard purposely for such occasion, for I afterwards learned the place had been the scene of innumerable contests of the same kind with aunt and other boys. Having readjusted our disordered habiliments, we left the grounds, and took a long quiet walk in the fields; the good doctor inculcating admirable advice to me, whom he considered an innocent tyro in love's ways. Nevertheless, all he taught me only strengthened my high opinion of the wisdom of dear Mrs. Benson, and the adorable Frankland, whole opinion of what was likely to happen to me at the rectory had been so quickly realised. We returned in time to dress for dinner. The evening passed as the previous ones. I was conducted to my room, and left alone to recruit my forces by a quiet night's rest. I may here incidentally mention that it was a rule of uncle and aunt, very rarely departed from, to send their favourites to their lonely couches as a means of restoring their powers, and reinvigorating them for daylight encounters---both the dear creatures loving to have the fullest daylight on all the charms of their participants in pleasure, at the same time yielding an equally undisguised inspection of their own. This was their principle reason, but they also considered it advisable as a restorative, and a useful precaution not to overstrain the energies of the youths they both so much enjoyed. My late experiences at home had already taught me the advantage and utility of a quiet night's rest after frequent contests in the fields of Venus and Juno.
I slept on this occasion with a deep and continuous slumber, until I was awakened by my uncle, who came to summon me to the arms of his wife, who, in the splendour of her full-blown charms awaited me in her own bed, naked as the day she was born. Her arms outstretched, she invited me to the full enjoyment of her glorious person. The doctor drew my night-shirt over my head, and in a moment I was locked in the close embrace of that superb creature. We were both too hot to wait for further preliminaries, but went at it in furious haste, and rapidly paid our first tribute to the god of love. The doctor had acted postillion to both of us, with a finger up each anus. The exquisite pressures of my aunt's cunt reinvigorated me almost without a pause, and we proceeded at once to run a second course. Uncle got three fingers into her divine bottom-hole, as her legs were thrown over my waist, and her immense buttocks well thrown up enabled him to have full play between the cheeks of her backside. This double operation made the dear lascivious creature spend again in a very few movements, and giving her hardly time to finish her discharge, I fucked on with double force, and with prick as hard as wood, as fast I could work. This furious onset, which was the most exciting thing she knew of, rapidly caused a third discharge. To prevent my own prick from spending too quickly, I held somewhat back; then again we went at it fast and furious, and the dear lustful creature, with cries of joy, spent again with me, and fainted from excess of pleasure; but her glorious cunt continued to throb on my delighted prick, as if it would nip it off by the roots. I never met with so lusciously large a cunt, or one with a greater power of pressure. She could quite hold even an exhausted prick a complete prisoner in these most delicious and velvety folds. Great as was the power of Miss Frankland's cunt in that way, aunt beat her. I may here mention an occurrence that took place some time after this period. It was during a rare opportunity from an accidental absence of the doctor, when I was sleeping with my gloriously beautiful aunt. I had fucked her to her heart's content before we slept, and again on waking, in full daylight, after which we rose to relieve our natural wants. I laid myself down on the floor, that I might completely see my dear aunt piddle from her splendid cunt. It was a glorious sight, which instantly fired my passions and was at once followed by a fuck on the floor, my aunt's enormous backside being quite cushion enough, and we enjoyed the novelty of the thing amazingly. She was loud in her praises of my indefatigable prick, which, with its vigour and superb dimensions, was beyond all she had ever seen or felt, and just fitted her large and luscious cunt, which had never before been so well filled. This remark reminded me of a desire I had long had to have a thorough investigation of that immense and splendid object. I expressed a wish to that effect.
``My darling boy, anything you like, you could not have a better opportunity, my legs point to the window, so you have the fullest sight---look, feel, frig, fuck, or bugger, all is at your free disposition---only give me a pillow from the bed, as the floor is too hard for me to continue so long as you are likely to be.''
I jumped up and gave her two pillows. Then laying her limbs wide open, with knees bent, the magnificence of that luscious cunt lay in all its grandeur before me. I have before described what a large, but splendidly proportioned woman she was---small feet, and clear-run ankles, large, but admirably turned calves, very small knees, above which rose the very finest and fleshiest of thighs, worthy supporters of what I have already described as the largest and finest backside my eyes ever lighted on. Immense hips, and wonderfully and naturally small waist, above which were her superb, large, fine, and firm bubbles that stood out when naked, as hard and firm as those of the youngest of women; a charming neck, and well-posed head with most pleasing and beautiful features crowned the whole. Her arms were superb, and equal in proportion to her other grand and splendid limbs. The flesh was of the most delicious creamy white, without a spot or a blemish. The hair of the head, plentiful in the extreme, and so long and thick that when undone it fell all around her and below her superb buttocks, so that she could shake it out all round, and completely hide her nakedness. Often and often has she allowed me to pose her in every way, and shake it out all over her, and well she might, for no matter how often I might have fucked her previously, it was sure to produce at least three more encounters, one of which was always in her backside, a most favourite way with her and which she declared was by far the most pleasurable provided the other aperture had been previously well fucked. With such a taste, of course, her greatest pleasure was to have two pricks in her at once, the \textit{ne plus ultra} of erotic satisfaction. To return to the inspection I was about to describe, which was really the first at my full disposition, for although I had often gamahuched, felt and seen the beautiful object, it was when my passions were excited, and when the gratification of lust alone prompted me, a state of mind opposed to close observation of natural beauties. Now, repeated tributes to the god of lust had cooled my ardour for the moment, and left me to the perfect enjoyment of the sight before me, with the temper to inspect its full-blown beauties in the minutest way. I have said before that my aunt had one of the broadest, most prominent, and most beautiful mounts of Venus that I ever saw. It was thickly covered with beautiful silky fair curls, which did not hinder you from seeing her exquisite skin below. The sweep round, to pass between her thighs, was bold and graceful. In the middle was a well defined semi-circular depression, from whence the large, thick and beautifully pouting lips of her cunt commenced, which in her present position lay partially open. You could just see where the clitoris lay snug. I have already observed that this was not largely developed, nor were the inner labia of her cunt at all projecting, indeed, they were not visible, unless her legs, with bent knees, were stretched apart, as at present. On each side of these luscious pouting lips, and the long immense pinky gash, was a triangle of considerable space, such, in fact, as is only to be seen in a woman of the splendidly large proportions of my aunt; this was covered as much as her mount with fair silky curls, which ran down to her beautiful corrugated and rosy bottom-hole. Nothing could be finer or more beautiful than the sight, as she thus lay fully exposing every part in the broadest daylight. After handling and admiring all, I laid the lips well back and apart, and there they kept open. Nothing could be more charming than the interior of that most enchanting cunt, of an exquisite salmon-pink in colour, nothing was out of order. The clitoris, which bulged out in excitement from my touches of all the parts around, lay first in the upper partition of the pouting lips; then became below, slightly open, a charming entrance to the urethra, larger than usual, to allow the mighty rush of waters to pour from it when piddling; below this was the opening of the vagina, which I parted with my fingers, and could see even to the corrugated sides of that exquisite pleasure-giving sheath; then followed some sinuosity of pinky flesh, whose duty it was to stretch to allow the largest prick to penetrate. Half-an-inch beyond was the rosy orifice of her bottom. Such was the exquisite scene before my delighted eyes. I proceeded with my internal examination. Thrusting in three fingers of each hand, I forced open by literal pressure the lips, until I could see to a depth of four or five inches. It was a most beautiful sight. The sheath appeared to have ribs running round it about half-an-inch apart, and I could see they were the means of causing the exquisite pressures her cunt could so ecstatically exercise. Indeed, excited by my \textit{attouchements}, I could see them contracting and relaxing. It was, doubtless, these ribs that seemed to exercise a sort of peristaltic motion on the prick, when reposing at full stand in that glorious cunt. I was able so widely to open this splendid vagina that I thought I would try to get my hand altogether in. Projecting my fingers forward, with the first and fourth drawn under the middle ones and the thumb between, I pushed them forward, and as the whole cunt was reeking with my last discharge, and was well lubricated, I glided on; there was a little difficulty at the knuckles, but I exerted a slight, gentle pressure, and in all went. Aunt winced a little, and asked what I was doing. I told her.
``It is all in, my darling?''
``Yes, Auntie.''
She closed upon it, and squeezed it quite hard.
``Oh, how nice!'' she exclaimed, ``push it further in.''
I advanced, and could feel the end of her womb, which appeared like three points to fingers and the thumb drawn together, and looked at endways is something like what it felt---of course, without the nails. Aunt asked me if I could double my fist where it was. I had no difficulty, as the part yielded to the greater bulk. Aunt cried out---
``My darling boy, that is delicious; push it further in.''
I did so, and began working within her, backwards and forwards. She wriggled her splendid backside in ecstasy, and before I had made a dozen movements, poured down upon my hand and arm a torrent of almost boiling liquid, and went off with a cry of enjoyment. Her arms and legs relaxed, and she lay quite still in the utmost after-enjoyment, but with a pressure on my arm and fist quite wonderful. Knowing how she liked the movement to be continued at such a moment, I worked in and out slowly. She soon recovered, and again seconded my movements, and again went off in all the fury of lust, accompanied with shouts of excitement, urging faster movements, and again went off in all the fury of her most libidinous nature, and spent most profusely. All this had now brought me into as furious a state as herself. I wanted to withdraw and substitute my prick, not only from the state of excitement I was in, but also to experience the effects of such a well-stretched cunt upon my lesser-sized weapon. But so tight did my aunt hold my imprisoned hand that I could not withdraw. I begged her to let it go, as I wanted to fuck her instantly, but she prayed me to give her one more of such exquisite manoeuvres, it was a joy beyond anything she had ever before experienced, so she begged her darling boy to join. On I went as she desired, and a more exciting picture of furious lust never met my sight. I helped her final discharge by thrusting two fingers in her bum-hole. Never shall I forget the grip she gave my arm and fingers when she spent. It was positively painful, and showed the enormous force of passionate lust. She went off in such a fury of excitement that I thought she had fainted outright. But her pressures continued all the time. It was long before she recovered her senses, and my arm was aching, and my prick bursting. At last she exclaimed---
``Oh! where am I? I have been in paradise.''
``Dear aunt,'' I cried, ``do let me out. I am bursting to fuck you, and I can't get my arm out, if you don't relax your grip of my wrist.''
``I can't help it, my dear boy, it is involuntary, put your other hand on my mount, and pull steadily, but not with a jerk.''
I did so, and really it required considerable force to withdraw it, notwithstanding I had previously unclenched my hand. I jumped immediately upon her, and at one bound plunged into that vast cavity, up to the cods. It immediately closed upon me, and tight as she usually held me, she really appeared to do so this time tighter than ever, so wonderfully gifted was that longest, highest, and most luscious cunt I ever fucked. You may easily imagine the rapid ending of such raging lust. I spent with cries more like the braying of a donkey than any other sound, and then lay like one dead on that glorious belly, with head reposing between the firm and splendid bubbies, aunt clasping me to her bosom, panting with all it had just granted. We lay long in ecstatic trance of the delicious after-sensations. Our mutual internal throbbings gradually re-excited all our passions. With renewed ardour I quickly made my lascivious and libidinous aunt spend again on my delighted prick, which kept ramming at her during the swoon-like pause which spending produced; she had taught me this was exquisite delight to her. She soon resumed the full swing of her lust, but suddenly stopping, said---
``Charlie, my darling, withdraw, and shove it in behind.'' She quickly turned round, with great agility, prompted by the excess of her desires. I was behind her in an instant, and as my prick was reeking with the fuck she had just so plentifully bedewed it with, and the divine lower orifice had also received its tricklings, I had no difficulty in pushing firmly but not too forcibly right up to the meeting of her stupendous buttocks and my belly. She sighed deeply with delight, when she felt me fully imbedded, and began the delicious side wriggle, while I remained for some minutes quiet, that I might enjoy the superb beauty of those mighty orbs, in all their play of passion. Aunt grew furious with lust. Her hand was actively frigging both clitoris and cunt. She called out to me to shove on; two or three thrusts on my part, and the dear, lecherous creature again poured down her nature. I paused to restrain my own discharge, but made my prick throb within its most exquisite sheath, which never ceased responding most deliriously. It was but for a minute or two, when my own fierce passion drove me to very energetic action. My delighted aunt seconded my movements, fast and furious grew our sport, until, with cries of the wildest lust, we both spent deliciously together. I sank on her glorious bottom and back, and by embracing her superb bubbies with both hands, until her exquisite pressures again renewed my forces, and drove me on to another delightful career, in which again the hot lust of my aunt drew down from her several discharges to my one. At last we sank both together, in all the joys of fully satiated desire. Again I lay for some time on that broad and beauteous back, until aunt said I must withdraw, as she had great natural want. I instantly withdrew, out he came with a loud plop, followed immediately by a tremendous succession of farts. Aunt professed to be quite horrified, but I only burst into a loud fit of laughter and told the dear creature to fart, piss, or shit, whenever she felt inclined, I should only love her the better. She said she must at once do the latter, and was running off to the water closet as soon as she could hurry on some clothes. But I drew out the chamber, and begged her to sit down there at once. It would give me pleasure and excite me as well. She was too hard pressed to hesitate, so sitting down, she had a ``hell of a let-fly'' as a military friend of mine used to say. I stooped over her back, caressed her bubbies and when she turned up her delighted face, our lips were glued together in a loving kiss, while my nose sniffed the really delicious odour that came from her. When she had done, she begged me to hand her a towel to wipe herself.
``No, no, my darling aunt, nothing of the sort; stoop down forward on to your knees, and I will lick the delicious orifice clean with my tongue.''
She laughed, kissed me, and told me I was a darling boy, just after her own heart, but hardly expected I had already acquired the tastes of my uncle, the rector, whose letch lay in that practice. She let herself down on her knees as her sublime arse raised itself from the pot, and stooping her head low down, presented her immense buttocks before me, with the chink between well stretched open. I move the pot on one side, threw myself on hands and knees, and eagerly kissing the exquisite orifice, greedily licked it clean, and thrusting my tongue well within, rolled it about, to the great delight of dear aunt, whose passions were instantly aroused, and her divine backside began to wriggle. I shoved my thumb up her cunt, and frigged until she spent. Meanwhile my own unruly member had become distended to his full size, and was throbbing with desire. So raising my body erect, I brought him again to the rosy orifice I had just been tongueing, and to my aunt's infinite delight, again housed him as far as he could go, and again began active operations, which I continued until aunt's lasciviousness again made her spend. I paused a little after this, or otherwise I should have gone off myself. Stooping over the glorious bottom, I replaced her hand with my own, and began frigging her clitoris, till her passions, again excited, made her begin ecstatic movements, in which I joined until the grand crisis seized us both together, ushered in with cries of joy. We spent, and sank down sideways on the floor in quite a death swoon of ecstatic and satisfied lust. Here we lay quite exhausted for some time. At last aunt let me out, and begged me to rise.
``I must purify you, my darling boy, as you did me.'' And seizing my limp prick in her mouth she sucked it clean, until she began to feel symptoms of the resurrection of the flesh. She hastily rose, and said---
``No, Charlie, you have done far too much to-night. I must see you to your bed, that you may get at least a couple of hours sleep.''
She took up my night-shirt, threw it over me, led me to my bedroom, tenderly embraced me, and thanked me for such a night of pleasure as she had never in all her life enjoyed the equal. Then locking me in, she retired to her own bed. It may well be supposed that after such exertions, I slept the sleep of the just for many hours. My aunt had frequently come to look at me, but seeing me in so sound a slumber, would not have me disturbed---a politic proceeding, as it resulted in a fuller indulgence in the summer house that day than would have happened if my powers had not been restored by refreshing sleep.
This kind of life had been going on for nearly three weeks. The doctor became less easy to move. One morning I had fucked my aunt twice; the doctor's prick at the end of the second had stiffened to about half stand. I took it into my mouth, which, with handling his ballocks and postillioning his bottom-hole, brought him up to the-full standard. He proposed to bugger aunt while I did the same kind office to himself. A caprice seized me, and I proposed, on the contrary, that we should both fuck aunt's capacious cunt at once. Aunt, for form's sake, cried out against it, but the idea tickled the fancy of my uncle, who would not only enjoy all the beauties of my aunt's glorious backside in motion, but could postillion her as well. So I lay down on my back, aunt mounted me, and presented her splendid bum to the attack of her excited husband. He first thrust his prick up to the hilt in her luscious and well-bedewed cunt; when well lubricated, he withdrew, to allow me to make my place in full possession, then bringing his stiff-standing prick against the root of mine, pressing it well down, he gently shoved forward, and gradually sheathed himself within the well-stretched and capacious orbit of my aunt, who winced a little in pretended pain, but who, by the grip she immediately gave to the double fuck within her, showed how much gratified she was. After a pause of enjoyment, I gave the signal for exact joint movements, both pulling out gently, and sliding slowly in again. Two or three thrusts, aided by the doctor's finger in her bottom-hole, sufficed to make dear leacherous aunt spend profusely. We increased our speed, but still not fast, which quickly re-awakened all aunt's lust. Before we ourselves were ready, the dear lascivious creature again poured down her nature, boiling hot, on our delighted pricks. This produced such excitement upon us that we could no longer restrain our own desire to come to the ecstatic conclusion. Our movements became more rapid. We each felt the electric-like sensation of the approaching crisis. Aunt doubly felt the influence of our increasing speed and hardness, and was as ready as ourselves to pour down the tribute to the goddess of love or lust, holy Mother Venus. The novelty, the pressure, and the excess of pleasure declared itself in the loud cries of the last crisis, as we all died away in the enrapturing sensations produced by the intense satisfaction our desires had experienced. We lay long wrapped in the after-ecstasy; aunt's delicious internal movements began again. The doctor's prick had shrunk to a merry piece of inanimate dough, and he withdrew, begging us at the same time to change our position, and let him enjoy seeing me attack my aunt in rear. This inflamed me at once. Aunt rolled from off me. I took my place behind, and we ran a most delicious course, rendered much more excitable to me by the introduction of uncle's two fingers up my fundament, which kept time with my action in the delicious aperture of my aunt's most superb and glorious backside, the movements of which beneath my delighted gaze had not been the least stimulating part of the enjoyment. The crisis was most ecstatic, and I sank exhausted on her broad buttocks and beautiful back, to clasp her lovingly in my arms and sob out bawdy terms of the warmest endearment. The doctor, who had very much enjoyed the sight, but who pointed out the sadly downcast state of his prick, which had been in no wise excited by the scene, said to his wife---
``My darling, we must have recourse to the grand remedy, I will also initiate dear Charlie into a new mystery of love, of which he can have no idea.''
I guessed at once what he meant, but professing extreme ignorance, I begged him to tell me what it was. Aunt rose and said---
``My darling, your uncle requires his blood to be excited by flogging his buttocks with a birch rod.''
``How odd,'' said I, ``I never felt anything but the severest pain when I was flogged, and I took precious good care not to deserve it again. How then can it excite?''
``You shall see, my dear.''
She opened her wardrobe and produced a formidable rod of fine fresh-cut birch twigs. The doctor begged me to lie down on my back---he got over me, and we commenced sucking each other's pricks. Mine stood at once, as the doctor, in addition to sucking, thrust a couple of fingers up my bottom-hole, and frigged away as fast as he sucked. The doctor's buttocks were left at the mercy of aunt, who flogged away at them with no gentle hand. I spent before the doctor could quite get his prick to standing point, but the copious torrent I poured into his mouth, and his after-suction on my prick, in addition to the red raw state of his buttocks, at last brought him up to full stand. He wanted to put it into me when ready, but aunt said that as flogger she had herself become greatly excited, and must have it herself.
``While this dear prick,'' throwing herself on it, and sucking it, ``shall fuck me at the same time.''
I was quite ready, and she straddled across me, and guided my now longing prick into her luscious cunt. She soon stopped, and we tongued each other while the doctor was mounting to the assault on her delicious bottom-hole. As soon as he was housed, we began another charming course, in which aunt, as usual, spent frequently before our less lecherous natures were ready to join in one general and exquisite discharge. We went off in furies of delighted lust, and then sank exhausted in the delicious after-sensation. We long lay in the sweet inanition and luxury of satiated lust. At last we disconnected ourselves, rose, and laved each other with cold water, more as a restorative than as a purification. Aunt and I had two bouts after---one in front and one behind. The doctor would not allow a fresh application of the birch, as he said it would only produce so great an exhaustion as would require days to restore. I retired after this, but ever afterwards the doctor was regularly birched before he could even copulate once. Sometimes he required to flog my aunt's glorious bum to excite his fading powers, declaring that it was almost as exciting as being flogged. He even gave it me gently, although I hardly ever required it, but I professed my surprise at its efficiency.
The holidays were at an end, but I was as yet the only boarder. There were, however, some twenty or thirty youths from the neighbourhood, who were day scholars at the doctor's school. Among these the doctor had his pick in the flogging way, but he never allowed them to know anything of our other proceedings, or to imagine that the birching which took place was otherwise than as a punishment for faults or inattention. However, I was generally the chosen companion of these whippings, in which I acted as horse, or holder of the boy to be flogged. Of course I took good care to expose as much as possible their lovely cocks, as well as their plump bottoms, and as this excited me as well as the doctor, it often ended, after the culprit was dismissed, in my flogging the doctor, followed by a mutual rack off in each other's bottoms.
There was one fine, plump, girlish-looking youth, named Dale, who was here for the first half. He had not as yet been brought up for punishment, although the doctor had confided to me the letch he had taken to flog his fine fat bottom. One day, Master Dale brought a sealed note from his widowed mother, who lived about a mile from the village, in a charming cottage ornee. The doctor read the note. By chance I was looking at him, and saw a smile of joy light up his features.
``Come here, Master Dale,'' said he in a mild and gentle voice, ``your mother tells me that you have behaved in a most shameful manner to your pretty young cousin, who is residing with your mamma.''
Master Dale blushed scarlet, for he was not aware until now that any one had been a witness to the scene that had taken place between him and his pretty cousin.
This was what had passed. The cousin, a lovely girl of fifteen, was in a secluded spot in the garden, near an arbour, the preceding afternoon. She was bending down, tying up a flower close to the ground, which made her stoop to such a degree that she could only reach it with ease by having her legs wide apart. Her back was towards the walk by which young Dale was advancing. As he approached unheard by her, he could not fail to see peeping out between the stretched open expanse of snowy drawers the inner part of her well-rounded globes of dimpled ivory. Her shift had somehow worked upwards, and revealed all the charms of her delicate young bum and plump white thighs. The sight inflamed the youth beyond measure. He crept up noiselessly quite close to her, and, stooping down until his head was below the level of her raised petticoat, he feasted his eyes for some time with the lovely prospect before him, her little virgin rosebud slit, its pink and pouting lips, plump little mount already delicately shaded with a curly foliage that promised soon to be much more dense, together with the swell of her lovely young thighs and calves. All this was quite unsuspected by the object of his admiration, who was absorbed in her garden operations. At length, however, the excited youth could not resist the temptation of applying his soft warm hand to the parts he was admiring, which made Miss scream slightly---she thinking it was some insect up her petticoats-exclaiming---
``Oh, dear! oh, dear!''
But turning her head round, she discovered the delinquent.
``Forgive me, Ellen dear, but really you exposed so very pretty a sight while stooping that upon my word I could not help it.''
Now girls are curious as well as boys, perhaps more so; and if the truth must be told, Miss had for some time past longed for an opportunity to become better acquainted with things in general, and, therefore, thought here was a chance not to be thrown away. So, after some little show of resistance on her part, for decency's sake, it was agreed between them that he should have a good look at \textit{hers}, if he would afterwards show her \textit{his}. Miss Ellen had never seen a male ``diddle,'' as she and her young playfellows called it, not even that of a boy, and she was all excitement and expectation to feel with her own hand the ``funny thing,'' for so a communicative servant-maid had described it, who at the same time had fully explained the theory of its use, which made Miss long to obtain some practical knowledge also. So to the arbour they both adjourned. Miss Ellen first lay back upon the seat, while the young rogue unfastened her drawers, and pulling them down, feasted his eyes with a full view of her virgin charms as long as he pleased, for Miss Ellen was a lecherous little maid, who really felt a precocious pleasure in being thus exposed to the close observation and admiration of one of the opposite sex, although for form's sake she covered her blushing face with her delicate little hands. He touched \textit{it}, pressed \textit{it}, rubbed \textit{it} with his finger. Her thighs trembled and opened. Taught by nature, he imprinted a burning kiss on the lovely little quim before him. She sighed, and mechanically put her hand on his head and pressed it closer to her naked skin. Guided by his feelings, he indulged in movements of his lips and pressures which speedily excited the amorous little maid to such an extent as to make her give down with a deep and trembling sigh the first tribute of her virgin cunt. Feeling the warm liquid oozing from the pouting orifice on his closely pressed lips, he could not help tasting it with his tongue. This reawakened very quickly the sensibilities of the lecherous little thing, and awoke her to the desire to practise a like pleasure with his cock. So reminding him of his promise, she made him stand up before her, while she undid his trousers with her fairy fingers, all trembling with excitement, and drew out his stiff affair, which already gave promise of a very respectable future, now swollen to a size it had never before known. Delighted at the sight of so bewitching a plaything, she made him lie down as she had done, and kneeling beside him, with cheeks glowing with excitement, she closely examined every part of the rampant little member. Strange to say, no hand, hardly even its owner's, had as yet invaded its virgin precincts, and it had not yet had its ruby head fully uncovered, although he was upwards of fifteen. The delight caused by the touch of her warm hand pressing and encircling his stiffened cock was most exquisite. She was not long, however, before she became curious to see what could possibly be underneath the skin that covered its rounded head. In her toying she sought to draw the skin back over the head, a slight cry of pain from him caused her to stop. But when, be she young or be she old, is a woman's curiosity to be baulked. She had managed to draw it back a short way, and now it suddenly occurred to her that by the help of a little moisture her object might be accomplished without hurting the dear fellow. By an impulse of passion she stooped and took the rosy head into her delicious little mouth, closing her coral lips around it, and lubricating it with her tongue, to the intense gratification of the youth, who involuntarily wriggled his body about voluptuously, and could not help raising it up to her mouth. This movement, combined with the pressure of the lips, perfectly succeeded, without further pain, in completely unhooding the charming little cock she was so deliciously embracing in the soft folds of her lips. She lifted up her head to see the result. The tight foreskin had closed below the nut, and left the now fiery red head bursting with excitement, and visibly throbbing with intensity of passion. Her joy and delight at this full revelation of the ``funny thing,'' as she continued to call it, now knew no bounds. She drove him nearly frantic with her ardent caresses---she again drew the covering over the vermillion head, and still finding that it did not easily return again, she thrust her head down upon it, and with lips, mouth, and tongue began again her attempt to unhood it. Poor Dale was brought up to the wildest state of excitement, his hands involuntarily pressed down her head, his body rose to meet it, and at that ravishing instant the grand crisis seized him, and, with a cry of delight, he shot forth his first tribute to Venus within the delicious mouth in which he was enclosed. The ecstatic gush poured down the throat of the dear girl, and she gulped it all down by the mere effort to avoid choking. Poor Dale's hands fell down insensibly from her head, which she instantly withdrew, and gazed on the youth. To her great surprise she beheld the so lately rampant weapon drooping its head and withdrawing within its shell, while some few drops of a milky white creamy like liquid were slowly oozing from the small orifice off its head. While she gazed it reduced itself to a mere shadow of its former state, and the foreskin slowly covered again the so lately fiery and bursting head. She was lost in wonder, and was about to express her surprise at the strangeness of the whole affair, but they now became aware that footsteps were approaching. Fortunately for them, as they thought, the noise of the gravel underfoot was distinguishable at such a distance that they had time to arrange their clothes, and when Dale's mother appeared at the arbour, she found them, on entering, quietly seated and talking together; and thanks to the youth's discharge in his cousin's sweet mouth, without any tell-tale flushings of his face. Little did they suspect she had already seen all.
This then was Master Dale's misconduct, and this it was that had been fully and minutely detailed in the note sent by this mother to the doctor, with a request that he should punish him well in whatever way he should think proper. The note further begged to know the terms for his becoming a boarder with the doctor, as she could no longer have him residing in her house with her orphan niece, whose guardian she was. You may easily imagine the double delight of the doctor. Another boarder, a point of some consequence to him after a previous scandal which, although hushed up, had deprived him of house pupils; and now with two, he foresaw a quick return to his full number; and then his delight at having to flog young Dale, and the erotic pleasure of drawing from him the exciting description of their young loves and voluptuous actions.
``Now Master Dale,'' said the doctor, ``you and I have an account to settle, follow me.''
And without a word further he led him into his private room, where, as was generally understood in the school, he birched the worst offenders. Arrived in the room alone with the young culprit, he locked the door, and taking a large cutting birch rod from a closet, sat down on a sofa. He called the youth to him and told him to unbutton and let down his trousers, and tuck up his shirt well under his waistcoat. This being done, the doctor said---
``Now, Master Dale, we shall see if this birch will cool your itching for feeling your pretty cousin's private parts.''
Poor Dale had never yet been punished more severely than by the hand of his mamma, and certainly trembled at the sight of the formidable birch rod threatening his now bare bottom; yet, notwithstanding his fears, the allusion to the pretty private parts of his cousin so fired his imagination that his cock instantly stiffened and stood out, to the infinite delight of the doctor, who augured therefrom a future further felicity. Keeping him standing close beside him, and enjoying the sight of his youthful charms so deliciously exposed immediately before his eyes, he proceeded---
``So, Master Dale, you have, it seems, been gratifying yourself with looking and feeling between the legs and thighs of a pretty young girl of fifteen, your cousin, is she not?''
``Yes, sir,'' sobbed out the youth.
The doctor's gaze was fixed upon the stiff and rampant member of the youth, watching the throbbings produced by every allusion to the luscious scene of the day before.
``Now, come, tell me all about it,'' said he, putting his arm round the sobbing boy's waist, and making him stand still closer beside him. ``Was she such a very pretty girl?''
Another throb of the rampant member.
``Yes, sir.''
``And you saw all her legs, thighs, plump little bottom, rosy pouting little \textit{slit}''---(\textit{throb}---\textit{throb}---\textit{throb})---``moist with the dew of excitement and amorous play eh?''
The little machine seemed ready to burst at the thought of it.
``And did it have the same effect as I see it now has on \textit{this?} Dear me, how naughty it is.''
And here the lecherous doctor took the rampant little cock in his hand and pressed it.
``And what did she do to this, did she touch it so?'' squeezing it gently.
``Yes, sir,'' stammered out the youth, who was getting greatly excited.
``And so?'' said the doctor, passing his hand gently and caressingly up and down the stiffened and throbbing little cock.
``Yes, sir.''
``And so, too, I'll be bound,'' pushing back the foreskin off the head, and as quickly replacing it, several times.
``Ye---es. Oh! sir; oh! oh!''
The boy's feelings, as the doctor swiftly frigged his cock, began to be exquisite. The doctor could not resist the temptation of bringing things to a crisis. Clasping the youth tightly with one arm, he continued his toying with the plaything in a quick exciting manner, apparently without reflecting upon what he was about, uttering, at intervals---
``Dear me,'' and ``how naughty it was of you; but how pretty she must have been to tempt you to do so, was it not?''
The lovely youth was now in paradise. In the exquisite sensations of pleasure he felt he was losing all consciousness when the doctor suddenly stopped short and said---
``This is indeed, a pretty piece of business, seducing your young cousin; you must be cured of such doings in future by means of a good flogging with an excellent birch rod, and on this your saucy bottom.''
Here he let fall the arm that had clasped the boy's waist, and let his hand wander over the plump, hard, and lovely orbs. The doctor now took up the rod which he had previously dropped to occupy his hand with the charming young prick he had just been so deliciously frigging. Shaking the rod angrily at the now trembling youth, he exclaimed, in a fierce voice---
``Now, you young rascal, down, down on your knees, and beg to be flogged.''
The poor boy was forced, trembling, to obey. This over, the doctor commanded the delinquent to lie down across the sofa. Reluctantly he complied, but at last he straddled across it with his snow-white plump backside fully bared to view, and a fair mark to the threatening rod. He looked like a young Adonis displaying his beauties to a satyr. The doctor was greatly excited at the lovely prospect, and gloated his eyes on the beautiful display, and, then and there, there sprang up a determination to fully enjoy the ravishing of these virgin charms before many weeks were over. Lifting the rod on high, he cried out---
``Now, you young villain, I'll teach you to look up young ladies' petticoats again, I warrant.''
Clasping him tight round the waist, the rod descended smartly on the lovely hillocks of the boy's charming backside.
``There! there!'' cried the doctor at each blow.
``Oh! sir, oh! Do pray forgive me!'' screamed the beautiful youth, as he felt the stinging strokes descend on his hitherto virgin posteriors.
``Oh sir. Oh! sir. I'll never do so any more. Oh! sir. I won't indeed. Oh! pray, sir, have mercy.''
The doctor, whose erotic passions were fully aroused, was deaf to all his entreaties, and kept on flogging harder and harder, and faster and faster while the poor boy's bottom bounded and tossed upon the sofa; but he was kept in a firm position by the strong arm of the doctor, who, to hold him the faster, had laid hold of the stiff cock still at full stand.
``No, no,'' said he, ``you shall not escape my cuts, I assure you,'' cutting at the lovely delicate buttocks with all his might.
``Oh! oh! sir; mercy, mercy; I can't bear it.''
``You must bear it, you young rascal you shall have no mercy until I have made your bottom bleed for your crime.''
The poor youth, from the pain he was suffering, bounded up and down on the sofa as he felt the stinging blows descend upon his bottom. This action caused his standing cock to rub up and down in the doctor's warm hand, who seemed to have accidently laid hold of it. The effect was such that the poor youth hardly knew whether he felt most pleasure or pain, for while his posteriors were of a burning heat, the warmth flew to the opposite part, which was so deliciously clasped in the doctor's caressing hand. He ground his teeth with pleasure and pain, he ceased to cry out, but sobbed and moaned with the excess of indefinable feelings. The doctor never ceased lecturing him, dwelling continually on the beauties of his lovely young cousin and the scene in the arbour. He thought but of her, of her lovely pink little slit so sweetly shaded with soft downy short curls; how she had so charmingly caressed his cock, until he felt a heave and a shudder, another, a sensation as if he was going to expire, a short cry, a catching of his breath. Wildly and vigorously he thrust his member down on the doctor's warm hand, he shut his eyes, he felt not the rod, although the doctor redoubled the strokes with all the force of his arm, and drew blood at every cut. A bound, a convulsive start, and he felt as if his lifeblood were coming from him---out it spurted in large drops on the sofa and on the doctor's hand. The youth had with difficulty shed another tribute to Venus. For a moment or two he felt as if in paradise, but a sharp cut from the rod quickly aroused him. He was soon fully alive again to its tortures.
``Why, you young rascal, what is this you have been doing on my best sofa, eh, sir?'' said the doctor.
Another sharp stroke demanded a reply.
``I, sir. Oh! sir; indeed I---that is---indeed, I don't know.''
``No lies or evasions here, sir, for they will not avail you. Your bottom shall pay for this nastiness. Why, what is it? What can it be? I never saw the like of this in my life, I declare,'' and he examined it with his eyeglass, saying more to the same effect.
Poor Master Dale was, we know, quite in the dark as to what it could be, or how it came there.
``Has this ever occurred to you before,'' asked the doctor.
``Yes, sir, yesterday, when my cousin was caressing it in her mouth,'' replied the frightened youth, ``but I really don't know how it happened, and did not mean any harm.''
``Oh, indeed!'' said the doctor, ``your mother did not mention that, did she see you?''
``No, sir, it happened just as she was coming through the shrubbery, and was all over before she reached the arbour.''
``And so your cousin took it into her mouth, why did she do that?''
``She was curious to see what was under the skin of its head, and finding it would not go back without hurting me, she put it into her mouth to wet it and make it go back easier, which it did for the first time; she pulled it forward again, and again put it into her mouth to push the skin back with her lips, when I felt funny all over, and something came out of me into her mouth.''
``Indeed! Well, you must tell me all about that another time. This flogging will be enough for the present, but I shall punish you for your nastiness some other time. Put up your trousers, in a day or two I shall want you in this room to pay for your dirty conduct.''
The poor boy retired, sobbing hysterically.
The second day after this the doctor sent for Master Dale, who, in the meantime, had occupied the bedroom next to mine. The doctor was in his private room in his dressing-gown, long and flowing, so that for the moment it concealed the fact that he had nothing but his shirt on below it. He received Master Dale somewhat sternly, saying---
``Now, sir, for your punishment for your last nasty misconduct.''
``Oh, sir,'' said the frightened and trembling youth, ``I really could not help it,'' and he began to cry. ``Oh! pray, sir, don't flog me so hard again.''
``The more trouble that you give me, the harder will be the flogging. Now take off your jacket and waistcoat.''
The youth did so.
``Now come close to me.''
The doctor then lowered down Dale's trousers, and raising his shirt, contemplated with great pleasure the pretty belly of the lovely boy, and then turning him round, under pretence of seeing if the weals of the last flogging were still visible, he gazed on his white firm backside and swelling thighs, examining the marks still left from the previous punishment. He then turned him round, and inspected the pretty little cock, which, under the mortal fear he was in, hung down its head in a limp and pitiable state.
``And so this is the little offender,'' said he, applying his hand to it, and squeezing and pressing it gently. ``What a naughty little thing it is!''
The youth could not avoid showing the pleasure these lascivious caresses gave him, and smiled.
``Oh, don't smile, sir, this is no laughing matter. Look at the marks of the mess you made on my sofa,'' pointing to it. ``I can't have my furniture spoiled in this manner, so if your little cock is to be naughty again, I must flog you upon my knees, but first come here; take off these trousers, which hanging about the legs are only in the way---there. Now sit down on my knee, and tell me all about this naughty little thing.''
He drew aside his robe, so that the boy's bare bottom came in naked contact with his brawny naked thighs, and the youth could feel the doctor's prick swelling up, although that part was still covered with his shirt. The doctor, taking hold of the youth's now standing prick, asked if it had ever behaved so badly before the scene with his pretty young cousin.
``No sir, never. I never thought of it until I got sight of her bare bottom and other parts by accident.''
The doctor continued his toyings, caressing the young balls, and feeling all over the plump and firm backside.
``Why, he is going to be naughty again!'' said the doctor, as the youth's prick throbbed under his exciting touches. ``I must flog your bottom for all this, for it is very naughty and improper. Why, you seem to take a pleasure in it.''
``Oh, sir, I never felt anything so delightful,'' said Master Dale.
``The more reason I should punish you, but remember, you bad boy, if you are to do that dirty thing again, you must do it on my knee, and not on the sofa.''
The doctor then took the birch in hand, and with his arm round the boy's waist, drew him to him, but before laying him across his knee, he slipped the boy's shirt over his head, leaving him stark naked in all the glorious fairness of skin and beauty of form. The doctor's eyes gloated over the charming sight, but becoming too excited to pause longer, he drew up his own shirt, displaying his fine pego at full stretch. He bent the boy's warm body over his brawny thighs, and with his arm pressed his glowing form against his own rampant pego---Dale's young stiffened cock rubbing against the naked thigh he lay on. The doctor now raised the rod, and said---
``Now, sir, for your punishment, I must flog this round, hard, little bottom, till it reddens again.''
Whack, whack, went the birch rod, but with much less force than on the previous occasion, but still sufficiently stinging to cause the youth to move up and down, rubbing his cock against the doctor's thighs, and causing him such ecstasies as hardly to allow him to feel the blows. His warm soft flesh, too, rubbing against the doctor's large, stiff tool, soon put them both in a delirium of delight. The doctor then changed his position, and drew the boy more over his belly, so that his great prick could get between the boy's thighs, rubbing under his balls in the trough between the buttocks, while the boy's cock rubbed against the doctor's belly.
``Now,'' said the doctor, ``I have got you fast, and must teach you not to play such naughty tricks in future.''
Whack---whack, again went the rod, causing the most delicious movements of the boy's backside upon the doctor's excited prick, and not less upon his own, which was rubbing against the doctor's belly, giving fresh pleasure at each repetition of the blow. But neither of them had spent yet. The boy's bottom was now red with a glowing heat, and his cock was in a state of intense excitement, and the doctor's tool was as stiff and randy as possible. The doctor now ceased his flogging, and squeezing the boy tight against his person, said---
``Well, you have not done that naughty thing to-day---the flogging has done you good.''
The lovely boy looked up and smiled. He had felt the doctor's large prick working away between his thighs, and pressing against the cleft of his buttocks. As the doctor relaxed his hold, the boy turned half round, thus releasing it from its confinement. Looking down, he beheld the large stiff monster imbedded in a forest of dark curly hair, presenting a startling contrast to his own small member, which was as yet hardly fledged with a silky down around it.
``Ah!'' said the doctor, who observed the flush of excitement the sight of his superior prick gave the boy, ``what a shame it is of you to compel me to flog you in this manner, without my trousers. I must give you a lecture---so sit on my knee, thus,'' placing him so that his lovely bottom should press against the huge prick. Taking the boy's cock in his hand, he said---
``How stiff it is.''
``Yes, sir, I can't help it.''
``Well, you must not play such naughty tricks. I can't allow it. You are too young yet.''
The doctor worked the skin of the sweet boy's cock up and down.
``Was that the way your pretty cousin played with it?''
``Yes, sir, and then she took it in her mouth.''
``And did you like to have it done to you, you bad boy?''
``Oh! yes, sir, it is such pleasure.''
``Does it really give you so much pleasure?''
``Yes indeed, it was most delicious.''
``Dear me, I must try if it would do so to me, take hold of my cock and rub it up and down, as she did, that I may know how it feels.''
``The dear boy had already longed to do so, but had been afraid to say so. He now seized with avidity the noble prick, so stiffly standing beside him. He could hardly grasp it in his hand, and worked the skin up and down in the most delicious manner. The doctor was in ecstasies.
``Oh, you naughty boy, to teach your master such bad things.''
``Is it not very nice, sir?'' said the charming youth, as the doctor's buttocks responded to every stroke of his hand.
``Well, it is, indeed, very nice, I could not have believed it; but if ever I catch you at it again, you may be sure I shall flog you.''
And the doctor responded to every rub upon his prick with another rub upon the boy's cock, until almost at the same moment a most delicious mutual spend was the result of their lascivious toyings.
``Now,'' said the doctor, ``put on your clothes, and remember you must avoid such naughty tricks in future, or your bottom will pay for it.''
The doctor made me aware of this, and arranged for a meeting of us three, under the pretence of inattention which I was to simulate and draw young Dale into some fault that would require punishment. It was also arranged that I was to initiate him still more into the secret pleasures of mutual satisfaction, so as to prepare him for still greater gratification to the lecherous doctor, who liked nothing better than ``teaching the young idea how to shoot.''
Accordingly, after passing a delicious night with my lovely and glorious aunt and the doctor, in which we practised every delightful method of enjoyment, and in which the doctor stimulated himself by recalling and describing the exciting interview with the innocent youth, I left them, and entered young Dale's room. He had insensibly kicked off all the clothes, and lay on his back exposed, with his promising young cock at full stand, throbbing from time to time; and by the involuntary movements of his body, and the smile on his face, he was evidently realising, in his sleep, the scene he had enacted with his pretty young cousin. He was quite charming to look at, his young and throbbing prick was deliciously fair, and you could see the blue veins coursing through it, the top was only partially uncovered, the point of the head showing its vermillion tip in fine contrast to the creamy white and crossed blue veins of the stiffened shaft. His balls were as yet not fully developed, but such as they were, they formed a closely drawn up little bag, crimped and wrinkled, and felt as hard as stones. I gently handled them, which made him heave his bottom in evident ecstasy. It was all so beautiful and enticing, and I could not resist stooping down, and taking the delicious morsel in my mouth. Pressing the glorious head with my lips, I thrust, to his infinite delight, the foreskin back, his buttocks instinctively rose to meet my voluptuous and lascivious proceeding. He awoke on the instant, but in that dreamy state that made him think he was only realising the previous dream. His hands embraced my head, and pressed it down closer on the delicious prick which already touched the back of my mouth. He cried out in an ecstasy of delight.
``Oh, my darling Ellen, what a joy you are giving me. Oh! oh! it is greater than I can bear.''
I felt by the electric stiffening of his young cock that the crisis was close at hand. I tickled his tightened balls with my hand, and pressed a finger hard against his bottom-hole, but without entering more than the depth of the nail, at the very instant that he poured his young tribute into my longing mouth. I immediately swallowed the greater portion, lubricating the still throbbing shaft with a part. For some minutes he lay on his back, with closed eyes, in all the after-enjoyment, heightened by the continued suction of his still throbbing prick, which I kept up for a short time. At last he opened his eyes. It was broad daylight, and when I lifted my head, his eyes seemed almost to start out of their sockets in a sort of incredulous surprise, at finding it was not his dear young Ellen, but me, his school companion. For a minute or two he was speechless with consternation, until taking hold of his fast receding little cock, I asked if I had not given him quite as much pleasure as his darling Ellen had previously done.
``Is it you? and Ellen! how do you know anything about my cousin?''
``Your cousin, is she? I did not know that, but when I came in, you were dreaming of her, and muttering in your sleep the delight she gave you by sucking your prick; so I thought I would give you the true pleasure of the thing, and thus realise your dream; besides, I, myself, not only love to suck a prick, but also to have my own sucked, and I could neither resist the opportunity, nor fail to be delighted that you should already have practical knowledge of its enjoyment---did I not give you the greatest pleasure?''
``Oh, yes; it was most delicious, and then I thought it was my pretty cousin, even after I awoke, which made it doubly delightful, for I had no idea it would be so nice with another boy.''
``Why not? see this charming little fellow is already raising himself up again at the mere thought; look how its head is showing its ruby face, and how it throbs. Ah! I must suck it again---it is so delicious.''
I threw myself upon it, and devoured it at once, rapidly moving my head up and down, and titillating the orifice of the urethra with my tongue. I quickly drove him half mad with excitement. My mouth was full of saliva. I slobbered some out on my fingers, and lubricated all about the aperture of his charming backside, and then, as he became still more furious in the upward lunges of his bottom, and downward pressures of his hands on my head, I thrust my middle finger up his fundament, and worked away, frigging it in unison with the movements of my mouth. I drove him half frantic with pleasure, the ecstasy again seized him, and with a cry of agonised delight, and a convulsive shudder, he poured a still more copious draught of love's essence into my mouth, which, as before, I greedily swallowed. He lay panting in ecstatic joy for a much longer period than before, with convulsive upward thrusts of his still half-stiffened prick within my mouth, which:' still continued its pressures and suctions to his infinite delight. At last I rose. He held out his arms. I precipitated myself into them; our lips met in sweet embrace. I thrust my tongue into his mouth, and solicited him to do the same, and we had some delicious tongueing, nature having at once achieved his love education. We were closely entwined in a loving embrace. I had become terribly excited notwithstanding the hard work I had undergone during the night, and my prick stood stiff as iron pressing against his belly. Suddenly the thought occurred to him that he ought to gratify me in like manner as I had done him. He proposed it, and begged me to turn from off him, and lie on my back. I immediately complied, and pulling up my shirt, displayed my immense splitter in all its glory.
``Good heavens!'' he cried, ``what an enormous cock! Why, it is bigger than the doctor's.''
``Oh! you have seen the doctor's, have you?'' He blushed, and acknowledged it. I drew from him an account of their proceedings, which I already knew, but I was at the same time delighted to have surprised the acknowledgment from himself, in his wonder at seeing my large proportions. I made him show me all he had done to the doctor, and the doctor to him; as all this was paving the way for future proceedings with the doctor---in fact, the innocent youth was already playing into our hands. His admiration of and handling my prick was meanwhile exciting me up to the greatest pitch. As I had continued questioning him regarding his proceedings with the doctor, he could only play with my prick in his hands. Now that matters were getting too warm for further discussion, he stooped down, but could only get the head and a small portion of the upper part of the shaft into his mouth. His lips closed beneath the gland in the most exquisite manner. I begged him to grasp the lower part of the shaft with one hand, and to thrust a finger of the other up my bottom-hole, which I had already lubricated by spitting on my fingers and conveying the saliva in the desired direction. He obeyed with the docility of an apt apprentice---and thus working in unison, quickly brought on the ecstatic crisis. I seized his head in my hands, and, at the final discharge thrust it down on my delighted prick, as I poured out a perfect torrent of sperm, nearly choking the poor youth with the length of prick I thrust into his mouth. He was obliged to withdraw for an instant to take breath, but I was pleased to see that he instantly resumed his delicious sucking of my prick, which he continued to do until it gradually shrunk up to very diminished proportions. I then drew him upon me, and we had another sweet embrace of lips and tongues, and then, side by side, we held a long converse on erotic matters. He told me all the tale of his affair with his cousin, and, although I was already well acquainted with it, I was glad to draw all the particulars from himself. I had seen the note his mother wrote to the doctor. The minuteness and undisguised description she had therein given struck me as very strange, and I augured that she herself must be a lewd and lecherous person, to have done more than merely hint at the affair, instead of dwelling, doubtless in erotic delight, on such details. So I pumped him as to what sort of woman his mamma was. His description showed that she was a fine, full grown woman, old, in his opinion, but in reality in the prime of life, between thirty-five and forty. He had not scanned her proportions with any erotic thought and did not seem to attach the idea of the woman to her---only that of the mother. But I drew out of him that she was broad in the shoulders, full in the bosom, with a small waist, small feet, and small hands, a very fine head of hair, and fine eyes---evidently a desirable woman. Already I had set my imagination in play, and began to hope I might some day work my way into her favour. It will be found how well I succeeded, as these true memories will describe when I arrive at the period of my success with her. For the present I had advanced the erotic education of the dear youth considerably, and thus prepared him for further initiation at the hands of the doctor and his glorious and magnificent \textit{cara sposa}, who had already determined to enjoy his first fruits in cunny land. We had a similar enjoyment, carried somewhat further the next morning, in which we practised more fully the frigging of the bottom, and discussed the pleasure it produced. I was gradually leading him on. That morning I purposely arranged that we should be late in entering the school-room. The doctor sternly reprimanded us, and told us we must attend him in his private room after twelve o'clock. Poor Dale turned pale as he heard this, dreading the punishment to come of which his experience was so recent and so severe.
At twelve we, apparently ruefully, entered the doctor's \textit{sanctum sanctorum}. He had preceded us by some few minutes, and had already donned his long dressing-gown, by which I was certain that he had at the same time doffed his pantaloons.
``Now then, boys, you must prepare for your punishment, I cannot allow this evidently wilful inattention. Off with your clothes except your shirts and stockings.''
We hesitatingly stripped; poor Harry Dale weeping at the thought of the dreaded punishment. I deemed it prudent also to draw a long face. The doctor spread a towel over his sofa, saying we had such naughty cocks that we were constantly dirtying his sofa. He then desired us to kneel on it with our heads down, and our tails well up. He then rolled up our shirts, and tucked them in above the small of our backs, in doing which he indulged in various lascivious touchings, which excited us as well as himself, and all our three cocks were at full stand. Harry Dale turned his head to gaze at mine, and could not resist putting his hand upon it, and gently pressing its large stiff shaft. Young Dale's smaller, but very beautiful member, which was daily developing itself in a striking manner, also excited me, and I reciprocated his caresses.
``This will never do,'' said the doctor, ``I must flog this evil spirit out of you.''
He threw off his dressing-gown to be more at his ease, he said, and taking rod in hand, applied it gently in turns to each of our projecting bottoms. It was not for punishment but for excitement that he operated upon us. He quickly threw our bottoms all in a glow, and our excitement became intense, and we wriggled our bums in evident delight. This was the point the doctor wished to attain, that he might arrive at his desired object, which was the possession of young Dale's bottom-hole.
``Stop, stop, my dear boys, I see you are at your naughty tricks again, but there must be no spending yet; get up. We must all strip to the buff, and I shall show you how they used to flog me when I was at school. Stand up, Charles.''
I did so, and the doctor for an instant handled, in evident delight, my huge stiff-standing pego, drawing young Dale's attention to its much larger dimensions than his own.
``Now, lean half forward on the sofa. Dale, put your arms round his waist, and stow away this charming rampant little fellow between the cheeks of Charles' buttocks. Charles, do you spit on your hand, and moisten between the cheeks, and then press with your hand his throbbing young prick against the cleft.''
I did as directed. Young Dale felt so deliciously sheathed that he thrust his cock well forward.
``Now,'' said the doctor, ``you are properly horsed, as we used to say, and now, for a little more flogging of these fine hard, rosy mounts,'' and he lasciviously caressed them before applying the rod.
Whack---whack---whack---fell the strokes, sufficiently sharply to make Master Dale wince and wriggle his bottom to and fro. Quickly the exciting pleasure overcame all pain, and his lust rising, he thrust furiously in the artificial channel he was operating in. I now shortened the grasp of my hand on his shaft, and pressing it somewhat upwards, raising my buttocks at the same time, I directed it so fairly upon the aperture that at the next push it entered fully two inches within; then again favouring his return stroke, he completely sheathed himself up to the meeting of his belly against my buttocks. I gave him a pressure which had an instant effect, and he began to thrust fast and furiously, evidently enjoying it to the utmost. I let him feel the full enjoyment of his new quarters, only telling him to lay hold of my cock and frig me; and then I cried out to the doctor---
``Flog him well, sir, he has thrust his cock into my bottom-hole.''
This was the very thing the doctor most wished for. So he continued his flogging only to such an extent as to still more inflame the lust of the now lecherously excited boy, who shortly brought on the final crisis and died away in delight as he shot his first tribute within the divine temple of Priapus. At the moment of the crisis coming on, the doctor had ceased his flogging and wetting two fingers gradually introduced them into the bottom-hole of young Dale, and frigged him in unison with his movements into me, so that the ecstasy was almost more than the poor boy could bear. He lay almost inanimate on my back, but his still throbbing half-standing cock responded to the inward pressures I was exercising upon it. The doctor had ceased his flogging to admit and caress the well-formed posterior of the charming boy. Becoming greatly excited, he drew him off me, and closely embraced him, but professing at the same time to be greatly shocked; his prick, meanwhile, gloriously stiff, pressed hard against young Dale's belly. The doctor then relaxing his hold, young Dale gazed, with pleased enjoyment, on the size and stiffness of the doctor's cock, and, by an impulse of passion, took it in hand, knelt before him, and put it into his mouth, and sucked it lasciviously. The doctor placed his hands on Dale's head, and pressed it down for a minute or two, and then begged the dear youth to rise, as he did not yet wish to spend---thanking him for the exquisite pleasure he had given him.
``Now,'' said he, ``it is your turn to flog, so Charlie, you must be horsed upon me, and Harry Dale shall take a first lesson in the art of flogging upon your posteriors.''
The pose was arranged as before. My formidable weapon was placed between the cheeks of the doctor's fine fat backside. His hand pressed my cock as I had done to Dale's. Dale took the rod in hand, and at the very first cut made me wince, for the young rogue laid on with a will. The doctor had applied a good quantity of saliva to the end of my prick, and thrusting out his buttocks, he quickly guided it into the longing orifice, in which I vigorously sheathed myself to the utmost extent. I seized his cock and squeezed it gently, but he begged me not to make him spend, but to enjoy myself to the utmost, crying out, at the same time to young Dale---
``Flog him well, Harry, for he has thrust his great tool up my bottom-hole; it is wonderful how it ever could get in.''
Indeed so little could young Dale believe in the possibility of such a thing that he stopped flogging to assure himself of the fact, by both touch and sight. I drew my prick out and in that he might be perfectly satisfied of the truth, and the doctor wriggled his backside to and fro to show what pleasure it gave him. Of course, all this was preliminary to the grand attack he meant afterwards to make on the virgin aperture in young Dale's bottom. After Harry had assured himself of the fact, he pitched into my poor bottom with redoubled vigour, which, though it greatly excited me at the moment, made my poor bum smart for days afterwards. I quickly sent a torrent of sperm far into the entrails of the doctor, to his great delight, but he tenaciously avoided spending lest his powers should fail to overcome the natural obstacle of a virgin bottom-hole, especially in one so young as Dale. Consequently, after retaining me for some few moments in the delightful pressures of the internal folds, he allowed me to withdraw, all reeking with my own sperm. It was now the doctor's turn to be flogged by me, while he was horsed on Harry's loins. As Harry had already found out what pleasure a bottom-hole gave to the plugger of it, and had also seen how the doctor seemed to enjoy, and so easily engulphed, the much larger weapon I possessed, he had no idea there could be any pain accompanying it, and consequently he lent himself entirely to every direction that was given him. He placed himself in the easiest position, stuck his bottom well out, stretching wide the channel between the orbs, and exposing a charming little rosy aperture most tempting to the sight; indeed, the doctor instantly knelt to pay his devotions to it, devouring it with kisses, and thrusting his lecherous tongue within its tight little folds, taking the opportunity to thoroughly lubricate it with his spittle. This preliminary, followed by a little frigging with his middle finger, which produced nothing but pleasurable sensations in the dear youth, completely captivated him. The doctor wisely informed him that the first attack was sure to be somewhat painful, but that if he felt it so, he was not to draw away his body, but simply to complain, and the doctor would instantly remain quiet without withdrawing, and he would then find that the strange sensation would rapidly pass off, and allow a further progress, which would be again arrested if the pain was renewed. In this way he would eventually find that the pleasure would become indescribably delicious, as he had seen how both Charlie and himself had enjoyed it. Poor Dale assured the doctor he might proceed at once, and he would be perfectly docile. So the doctor first asking me to suck his cock a little to moisten it well, put the charming youth in the best position, telling him to strain as if he wished to void himself, then applying his well-lubricated pego to the rosy orifice, by gentle pressure, he succeeded, with hardly a twinge of pain to the dear boy, in housing the head and about two inches of the shaft within the delicious receptacle. Here the pain became so great that young Dale would have withdrawn himself away from the doctor had the latter not taken the precaution to seize him by the two hips, and hold him as if in a vice, but without attempting a further insertion then.
``Keep still, my dear boy, and I will not move, and you will find in a minute or two that the strange sensation will pass away.''
Turning his head to me, he said---
``Charlie, gently frig the dear boy.''
I immediately did so, which rapidly had the effect of exciting him up to a pitch that made him forget all pain, and he even thrust his bottom further back, and as I had taken the opportunity of the pause to drop some more spittle on the lower shaft, a further gentle pressure forced it in almost up to the hilt. Here, again, young Dale cried out to stop, it was so painful.
The doctor paused again. I continued caressing his now inflamed and stiffened prick. His convulsive twitches, caused by my lascivious caresses, were followed by involuntary wrigglings, which of themselves completed the entire insertion of the doctor's excited prick. He still continued quiet, allowing the passions of the youth to become still more excited. Then gradually and gently withdrawing, and as gently again thrusting within, he went on until the youth's movements betrayed the raging lust that possessed him---then the doctor increased his pace. I frigged on fast and furious, and in a few minutes they both died away in wild excess of the most ecstatic joy. As to Dale, his gaspings and wild cries of delight proved that the final joys were almost too great for him to bear. The doctor had drooped his head upon his chest, and closed his eyes, in all the gratification of having ravished the first fruits of this charming youth's beauteous bottom, and I could see by his momentary convulsive thrusts, and the pressures of his hands on Dale's hips to draw the bottom more completely against his belly, as well as by the broken sighs that heaved his bosom, how exquisitely he was enjoying his triumph. Gradually his cock reduced its dimensions, but even when quite down and soft, it left the tight sheath it was in with a ``plop'' showing how well and close those delicious folds had embraced it. The doctor would not allow young Dale to rise until he had embraced and kissed the lovely bottom that had just yielded him such intense satisfaction. Then, drawing the youth to his bosom, he embraced him most tenderly, and thanked him for the heroic manner in which he had borne the attack, and told him he would never suffer so much in after-attacks as he had done in this first taking of the virginity of his bottom-hole.
It was thus this dear youth was initiated into our mysteries, and henceforward he became an apt disciple, and by being introduced into our interior circle, added much to the variety and enjoyment of our orgies. For, as may well be supposed, my glorious and most lecherous aunt thoroughly enjoyed the taking of his first tribute in the legitimate temple of holy Mother Venus. I was present on the occasion, which was supposed to be unknown to the doctor. The first coup was on her belly, the sight of which and her truly magnificent cunt wildly excited Dale, and his cock stood stiffer and really bigger than ever. It was quite surprising how rapidly it developed when once he got thoroughly into hardness. He fucked aunt twice, spending as rapidly as she herself, lecherous as she was at all times. I acted postillion to them both. I stopped further combats until I too could enter the field. So aunt mounted upon him, and falling forward lent her divine backside to all my fantasies. Twice we ran a course without changing. Then aunt herself claimed my big prick for the contentment of her randy cunt. We quickly changed positions. I, on my back, received dear aunt's delicious cunt on my stiff-and-hard-as-wood-standing pego. She straddled over me, and sank her luscious orbit down upon me until our two hairs were crushed between us. Here, by rising and falling, she had another delicious discharge before bending down to be embraced by my loving arms. She then presented her most glorious bottom to the wonder and admiration of dear Harry, who had been caressing and kissing it, and at the critical moment had thrust a frigging finger in, and turning his head in front had greatly increased the pleasure of my loved and lecherous aunt by sucking the large nipple of her wondrously fine bubby. When once she was fairly down on my belly, Harry scrambled up behind, and quickly inserted his already fine but still comparatively small prick, which, of course, found ready entrance where my splitter had previously opened and greased the way; but he gave a cry, almost of pain, or at least of surprise, on finding the sudden grip which my aunt, with her wonderful power of pressure, instantly gave him. At it we went, fast and furious, until again the grand crisis overtook my lascivious aunt, who spent deliriously. We boys both paused a second or two to allow her to enjoy her discharge to the utmost; then recommencing with increased vigour and speed, we soon both discharged at one and the same time our freights into the delighted vessels that were conferring such exquisite enjoyment upon us. Aunt, too, did not fail to join us at the ecstatic moment. We lay for many minutes panting in all the after-sensations of the most exquisite joys humanity can revel in. We kept it up for several hours, aunt sucking young Dale's toothsome prick while I gamahuched and postillioned her to her infinite satisfaction. In this way, and with repeated changes from one receptacle to the other but always both occupied at once, we at last gained a reprieve, and retired to well-deserved repose. The doctor, who had kept out of the way on this our first bout with my glorious aunt, afterwards apparently surprised us together, and, after giving us and receiving a pretty sharp flogging, he joined in all the ecstasies of our orgies. He especially delighted in being into my bottom while I fucked his wife, and he himself had the double pleasure of having young Dale's fast growing pego into his bottom at the same time. It was some time longer before I succeeded in completely sheathing my huge prick in the delicious bottom-hole of the dear youth, but at last I succeeded to the utmost extent of my wishes, and although I continued to hurt him for some weeks after the first attack, he could at last entertain me with perfect ease, and we were thus enabled to play successively into each other's bottoms, and every one of us enjoyed the exquisite delight of fucking and being fucked at the same time.
As we grew more lasciviously intimate, I often turned the conversation on his mother and cousin. At last I told him, I thought from his description that his mother would be a good fuck, and that if ever I had the opportunity I might cover his attack on his cousin by fucking his mother; only we must lead her to believe that she took my virginity. The idea pleased him. He began to think his mother must be a desirable woman for me, as I was so largely hung; and then the opportunity that I would give him to enjoy his longing for his cousin was an inducement to second my views to the utmost. Towards the close of the half-year his birthday occurred, and his mother could not do less than have him home for the day. She felt that her niece would be in greater security when Harry begged she would allow him to bring with him the doctor's nephew---myself, to wit---telling her that we had become very close friends as well as schoolfellows. I had previously told him I should play the complete innocent, but should take care some time or the other during the day to put myself in such a position that his mother should get a glimpse of my prick, so that if not immediately successful, I might pave the way for future success. His birthday fell on a Saturday. We were only asked to spend the day, with the intention of returning in the evening. Accordingly, on the happy day we made our appearance after breakfast. I have before said that his mother lived in a very pretty cottage ornee, about a mile and a half from the parsonage. We were most kindly received by her. She first lovingly embraced her son, wishing him many happy returns of the day, declaring that he was much improved, \&c. She then turned to me, and gracefully and kindly bade me welcome. The niece was a charming girl, just budding into womanhood. She blushed greatly in welcoming her cousin, and bashfully did the same to me. We spent the earlier hours in conversation; the mother having much to ask and to hear from her son, from whom she had never before been separated. I had thus time to scan her well. She was a fine, broad built, well standing up woman, with broad shoulders, and hips that gave promise of good form beneath. Without being beautiful, her face was a well formed oval, with really fine eyes, to which her son's description had hardly done justice. It appeared to me that a good deal of suppressed passion lurked in their expression, and I already began to think she would be a real \textit{bonne bouche} if once we could come to close quarters. After luncheon we strolled in the garden. The leaves had already fallen, but the afternoon was bright and warm for the end of November. I told young Dale to keep close to his mother, and not show any wish to stray away with his cousin---feeling certain that if she became anxious about their movements I should have no chance to play off my little game. All went as I could wish, we threw his mother off her guard, and she then began to show closer attention to me. I acted the ingenuous and innocent youth to perfection, but at the same time, in thinking of her charms, I let my prick get up to half-stand, so as to show its large proportions under my trousers. I very soon perceived that it had struck her notice, and her attention became concentrated upon me. She questioned me a good deal, and especially sought to find out if \textit{peculiar} intimacy existed between her son and me. I played the innocent, and professed that the utmost intimacy existed; but when she tried to find out if it had gone to what she really meant, I gave such an innocent character to our intimacy that she was quite convinced of my thorough ignorance of all erotic tendencies, and she became more endearing in her manner of addressing me.
Harry and I had previously agreed that after I addressed to him some particular frivolous remark, he should seize the first occasion near a shrubbery to go on more ahead, and alarm his mamma by turning round a corner. Our stratagem succeeded. She immediately hastened to follow them. As soon as she had turned the corner I drew out my tool, now at full stand, and placed myself so that when she returned she should see it fully developed, while I would take care not apparently to see her, but be intent upon piddling. To the utmost of my wish it fell out. She had told her son to stop and returned to join me. My eyes being turned downwards did not let her become aware that I was watching for her, but I could see the bottom of her petticoats as she turned the corner, and also that she came to a sudden stop, which must have been at the moment she caught sight of the noble proportions before her. I took care to pass my hand once or twice backwards or forwards while pissing, and then shook my prick deliberately, and exposed the whole length and breadth of it for a minute or two before buttoning it up, during which I could see she stood perfectly still, rooted to where she had first stopped. After I had buttoned up, I stooped down, apparently to tie my shoe, but in fact to give time for it to be supposed I had not seen her previous approach. So when I rose up she was already at my side. There was a flush on her cheek and a fire in her eye that showed the bait was swallowed. My role was to play the perfect innocent, and appear quite unconscious of her having seen me.
She took my arm, and I could feel that her hand trembled. She led me along, hastily at first, until we joined her son and niece. After that she became uncommonly endearing in her manner to me, making such remarks as she thought would show her that I was not so innocent as I looked, if my replies had jumped with her expectations. But I was in reality too experienced not to pay her off in kind, and ended in making her believe that she had a perfect virgin to deal with. We walked on, she was evidently much preoccupied, becoming at times quite silent for a minute or two, and then, gently pressing my arm, she would make some endearingly flattering remark, at which I would look lovingly but innocently up to her face to thank her for her kind opinion. On these occasions her eyes sparkled in a peculiar manner, and her colour went and came. After a while, her hand left my arm and rested on the opposite shoulder, in a half embrace, which became warmer and warmer, her conversation became more affectionate. She was profuse in her congratulations that her son had found so charming a schoolfellow; and here she halted, and turning half in front of me, said that she felt that she could love me as if I were indeed her own dear son; and, stooping slightly, she sought a kiss of maternal affection. I threw my arms round her neck, and our lips met in a long and loving kiss---very warm on her side, but a simple though affectionate kiss on mine.
``Oh!'' I said, ``how happy I shall be to call you my mamma, and I will love you as if you were it, it is so good of you to allow me to do so. This half-year has been the first time in my life that I ever was separated from my mother---and, although my dear aunt is as kind as possible to me, still I can't call her mamma. My guardian won't allow me to go home for the Christmas holidays, but now I shall have a dear, kind new mamma to make me happy.'' Here I again raised my lips for an embrace, which was given with even more than the previous warmth. Her arm had fallen to my waist, and she pressed me with energy to her bosom, which I could feel was unexpectedly firm, and even hard. I had great difficulty in keeping my unruly member down, that she might think I took her warm embraces as nothing more than affectionate friendship. I succeeded, however, and this, of course, more than ever convinced her of my entire ignorance of carnal desires. As I closely embraced her, and glued my lips to hers, she became greatly agitated, trembled visibly, sighed convulsively, and then pushed me from her, and seemed suddenly to recover herself, seized my arm, and hurried on after her son. For, as may well be supposed she had purposely loitered behind to allow them to get out of sight, before she indulged her uncontrollable desire to embrace me. She spoke not a word until we came in sight of them, apparently sauntering along, innocently enough. But Harry afterwards told me that having seen how his mother had halted to gaze at my prick, which he knew beforehand I meant she should see, he had watched us through the shrubbery, and afterwards had noticed her warmth of manner to me, and the loitering of her walk. He had turned a corner some distance ahead of us, and was out of sight when his mother stopped to embrace me, as described above. He guessed she would be in no hurry to follow him. So rapidly advancing with his cousin, he got some way before us, and choosing a place where he could see us through the bushes when we did follow, he sat down on a garden seat, and drew his cousin on his lap, asking her if she did not regret their hasty separation after their last delicious interview, and telling her his mother had seen them, which was the cause of his being sent as a boarder to the doctor. She was much surprised to hear this, as her aunt had never breathed a word of it to her; and she had been greatly distressed at his being sent away from home. Of course his hands were not idle; but first unbuttoning his trousers, he put his cock, now much increased in size, into her hand. She at once observed how much larger it had become, and began to caress it. He meanwhile was busy frigging her little clitoris. He found that she was already quite moist, and he had hardly frigged her a minute, when a sigh and an ``Oh! how much more pleasure you give me than my aunt does.'' She spent profusely, grasping his prick with painful tenacity. Her breath was taken away for some minutes. When she recovered a little, and was gazing lovingly with half-closed eyes upon him, he at once recurred to her unexpected confession.
``When does my mother do this to you?''
``Ever since you were sent away; your mother took me to sleep with her, as she said, she felt so lonely after you left. For some time she used to embrace me very lovingly, and hold me close pressed to her bosom. As I always went to bed before her, I was generally sound asleep when she joined me. I used at first to wonder how when I awoke in the early morning my chemise was drawn up close to my neck, and your mother's was in the same state, and our two naked bodies closely united by the embracing arms of your mother. I even one morning found that my hand was held by hers against that part which you are now feeling so nicely. She had fallen asleep in this position, but I could feel that she was as moist there as you have just made me. I could not help feeling it was very nice, and gently removing her hand, I began to feel all over her in that part and, do you know Harry, she is all covered with such thick and curly hair there. In groping about, I felt the lips pouting and thick, and on trying I found I could get my fingers in. I pushed on, I got up to the knuckles, when I felt it give such a convulsive pressure upon them, and her body was projected towards me with a heave of her bottom, then drawn back, and pushed forward again, while her arms pressed me closer to her, and she commenced some loving expressions in her sleep. I felt something grow hard against my thumb, it was just what you have been feeling.---'Oh! go on,' she cried.''
``I renewed my tickling operations again, and I made her spend,'' Harry continued. ``As she came to her senses, I gamahuched her; I thrust my tongue up her sweet little cunt, and licked up all the delicious spendings. As I rose, with prick erect and standing stiff out of my trousers, she seized it in her mouth, and, with very little sucking, made me spend to excess, and the dear girl swallowed it with all the luxury of the utmost voluptuousness. We had no time for more at that moment, as I caught sight of mamma's dress through the trees. I buttoned up hastily, and we strolled along, as if nothing had happened. It was in our after-walk, when we had allayed mamma's suspicion, that my dear Ellen continued her confessions.''
The stiff thing pressing against her thumb was mamma's clitoris, which, by her account, is wonderfully developed. She, knowing from her former experience with me that it was the point of most exquisite enjoyment, turned her finger upon it, and began awkwardly playing with it. It was at this moment that the greater excitement awoke mamma, who finding to her surprise what Ellen was doing, seized her hand, and pressing and rubbing it with more art against her clitoris, continued its action with exclamations of delight, declaring that Ellen was her dear precious loved girl, and then with a positive cry of delight, spent profusely over Ellen's hand. After panting for some time in perfect bliss, she turned and took Ellen in her arms, kissing her most warmly, and thrusting her tongue into Ellen's mouth, and then demanding hers in return. After much embracing, mamma asked her how she came to do what she found her doing when she awoke. Ellen described how she found her hand held against it, and then two naked bodies pressed against each other---that she was surprised at this, and wondered how it came so; that on moving her hand she felt mamma give a throb down there, and a push of her body forward, which made her finger slip easily in, this still more surprised her, as she had tried often if her fingers could get into her own, but it hurt her so much that she had given it up as impossible; and now she had found one where all her fingers, up to the knuckles, slipped in quite easily; the inside movements, and the heavings of her aunt's body, showed that it gave her pleasure. In continuing her movements she had felt a hard body at the upper part pressing against the side of her hand; she withdrew her fingers to feel this strange thing, and in doing so aunt awoke.
``And you know the rest, dear auntie, I was so glad that I had given you so much pleasure.''
``Dear, dear girl!'' her aunt replied, ``I shall love you more dearly than ever; yes, and you, too, shall have the utmost pleasure. I have long wished to initiate you into the secrets of womanhood, but thought you too young to be able to keep secret such intimacy as we may indulge in. Often in your sleep with your lovely naked charms exposed to me, and pressed against my own lascivious person, have I enjoyed you, and even made use of your own hand all unconscious in sleep, to excite me to a still greater pitch; last night I had enjoyed you to the utmost, kissing your lovely budding and hidden charms, and must have unconsciously dropped off to sleep with my hand still pressing yours against my secret charm. But now I must initiate you into the same joys, even in a more exquisite way.''
``Upon this she begged me to throw off my chemise, while she did the same. We stood up to do this, and your mother took the opportunity to pose me in every way, admiring and kissing me all over. I did the same to her, and I can assure you, dear Harry, your mother is far better made that I am, both in the bosom and the bottom, and with such firm thighs and legs, and her affair is so well developed and pouting, and with such silky curls all around it. I can feel you passing your fingers through the curls of mine; but though it has more than it had when last you felt and caressed it, it is nothing to dear auntie's. When she had much excited me, and was evidently herself greatly so, she desired me to lie across the bed on my back, and to draw my knees up so as to let my feet rest on the edge. She then placed a footstool in front, and kneeling upon it, after first feeling and caressing me down there, she glued her lips to it, and after sucking a while began to play with her tongue upon what you have been so deliciously rubbing. She licked me most exquisitely, and soon made me die away in ecstasy of delight. She sucked it for some time after, while I lay in a languid state of joy. When at last she rose, she threw herself on the bed, and our two naked bodies became closely united in the most loving embrace. Her lips were wet with the moisture that had escaped from me, its peculiar aromatic odour \textit{m'enivrait} and I could not help licking the creamy juice from off her lips.
```Oh, my beloved aunt,' I cried, 'you have given me the joy of paradise, I must try and do as much for you.'
``'My darling Ellen, you will make me positively adore you. I now only regret that I had not sooner taken you into my confidence, as I at once perceive I might have done so in perfect safety. Yes, my darling, you shall indeed try, and I shall instruct you as we advance how to obtain the greatest amount of pleasure from our libidinous and lascivious enjoyments, delights that are without risk, and from which we shall have no anxieties as to fatal results, which are the consequence of connection with the opposite sex, who only make use of us for their own sensual enjoyment, and abandon us at the very moment they ought to console and cherish us the most.'
``Dear aunt, again embracing me tenderly, threw herself in the same position I had previously lay in. I knelt on the cushion as she had done. But before proceeding to do as she had done to me, I could not help pausing to gaze with delight on her natural charms. Oh! dear Harry, you cannot imagine the beauty of that part of your mamma. Her stomach is of the purest white, smooth and firm, round and beautiful. Below a crease commences a large plumped out swelling seen through the fair and thick silky curls that so much adorn it, then grandly rounded sinks down between her thighs, and the beautifully pouting lips rise richly tempting through the thickest of hair, that goes far beyond between the large rounded orbs that project behind. At the upper part of the lips, where they form a deep indented half-circle, I could distinguish a stiff projecting object, as long and thicker than my thumb. I now know that this is the centre of exquisite joy. Your mother had since taught me to call it her clitoris, and says that although seldom so strongly developed as in her case, it exists in every woman and becomes stiff and excited as the final crisis of joy approaches. I glued my lips around this charming object, and sucked it, and played with my tongue around its point. Your mother, in an ecstasy of delight, wriggled her bottom below me, and with both hands pressing my head down on the excited point, gave utterance to the most loving and sensual expressions. She begged me to pass the flat of my hand under my chin, and introduce my thumb within the lips below, where I was sucking, and move it backwards and forwards as much as I could. I did so, and immediately found that it added greatly to your mother's delight. Faster and faster grew her movements, until, with a cry of delight, a firm pressure of my hand against her affair, and still firmer pressure on my thumb, she suddenly ceased all movement, her hands relaxed their hold of my head, the stiffness left her clitoris, and beyond convulsive graspings of the interior of her affair upon my thumb, she lay for some time inanimate. At last she recovered her senses, she seized me under the arms, and drew me upon her belly, her hands pressed my bottom down close upon her person, until I found that my affair was nestled in the rich profusion of curls that so finely adorned hers. She thrust her tongue into my mouth, and sucked off all the rich creamy substance that had flowed from her in such abundance. She blessed the happy chance that had led her to give me her confidence; told me that for long she had only enjoyed the unsatisfactory delight of lonely self-gratification, and said that now we should revel in mutual delight of every sensual indulgence that woman can have with woman. We lay for some time enjoying such delicious communings, until compelled to rise by the lateness of the hour. We have since practised every method of enjoyment given to two of the same sex. Your mother has often introduced her stiff excited clitoris within the lips of my affair as far as it would go, but I have always longed, my dear Harry, for you to penetrate still further with that larger and longer thing you have got, although what I have seen to-day of its increased size has made me greatly fear it can never get in.''
Thus ended her ingenuous description. Harry, of course, promised that he would never hurt her, that those parts were made to yield, that, doubtless, his mother's large clitoris had hurt her at first, but had given her great pleasure afterwards.
Yes, that was so, and it was that that gave her courage, and if they could only get the opportunity she would allow him to do anything he pleased.
It may well be supposed this account of Ellen's intercourse with her aunt fired my imagination and made me resolve to have her. Indeed, I began to conceive that there would be no occasion for me to make any effort, that all would be done by dear mamma herself. We had returned to the house after this agitated walk. Mamma was evidently greatly preoccupied, but at length she appeared to have come to a final determination, for she told Ellen to go up to her room, and begged us two boys, as she called us, to go out and amuse ourselves for an hour. It was during this interval that Harry narrated his interesting conversation with his cousin. Her lively description had set his imagination on fire, and he now declared his regret that it was not to be he who would enjoy his lasciviously sensual mother. Neither of us had any doubt but that she would now find an opportunity of enjoying me. If we had, our doubts were solved on re-entering the house. Mamma first, for form's sake, kissing her son, and then far more warmly kissing me, informed us that she had written to the doctor that we had been such good boys that she would feel greatly obliged if he would allow her son to remain with her until Monday, and also leave his nephew to keep him company and prevent any of his former misbehaviour which, she was happy to say, he appeared to have forgotten, but still it would be better he should have the safeguard of so intelligent and discreet a friend as she was glad to see he had found in the doctor's nephew. My uncle, without knowing exactly what to make of this note, had consented. Hence her joy in being able to communicate the pleasing intelligence---doubly so to me, as I immediately augured the downfall of my assumed virginity. Dear mamma was all radiant with joy, and conveyed me at once to where she intended I should sleep. I marked that it was in an out of the way room, easy of access, but not likely to be interfered with by passers-by.
``And here, my dear son, for you know in future you are always to call me mamma, I hope you will find yourself comfortable, and that you will not be alarmed because you are in an out of the way part of the house, but in case you should, before I go to bed, I shall come to see that you are comfortably asleep.''
Here she kissed and embraced me warmly. I repaid her most affectionately, but apparently in all innocence. She sighed, as I thought with regret, that she could not at that moment go further, and then led me away.
The afternoon, the dinner, and the evening passed away without anything worthy of remark, except that mamma was frequently absent and preoccupied. She sat by me on the sofa while Ellen played to us; her hand sought mine, and frequently squeezed it affectionately. Harry sat by Ellen, which enabled me often to raise my head and pout my lips for a kiss in a boyish way. It was never refused. She dwelt on my mouth sensuously with half-opened lips, but apparently afraid to tip me the velvet of her tongue. She frequently gave a shudder and trembled, and was evidently greatly excited. In the course of the afternoon, Harry and I had had an opportunity of exchanging ideas. I told him I was certain his mother would come to me that night, and he might be sure if she did that she would remain till daylight. I advised him to watch her, and when he saw her leave her bedroom to come to me, then he could slip into his cousin's room, and effect his purpose, but to be sure to retire at the first dawn. I said that if that time his mother wanted to leave me, I would keep her another quarter of an hour to enable him to put matters to rights with his cousin, and regain his own room. I advised him also to put a towel under his cousin's bottom, as he was sure to make her bleed, and he must take it away in the morning to prevent any traces of what he had done being perceived by his mother, and to tell Ellen to feign deep sleep on his mother's return, and to appear quite unconscious in the morning of her aunt having been absent. A little before ten o'clock mamma thought it time for her children, as she called us, to go to bed. Her son and niece both kissed her, and I, too, claimed a kiss of my new mamma. It was taken and returned in quite a passionate way, her lips seemed loath to leave mine, and her arms encircled me in a very loving embrace. ``Dear mamma,'' I said, ``I shall love you ever dearly.''
``My darling boy, I already love you as if you were indeed my son.''
She sent the others to their bedrooms, but escorted me herself to mine. I could see that she trembled greatly, and was evidently glad to put down the candlestick. She turned down the bedclothes for me, hoped I would sleep well, and, with considerable agitation, again embraced me most passionately. I could feel that her tongue would fain have thrust itself between my lips. I had great difficulty in restraining myself, but somehow I managed to do so. She at last left me, saying she would give a look in to see that I was comfortable before she herself went to bed. I told her it was very kind of her, but that there was no necessity for her doing so, as I always went to sleep like a top the moment I lay down.
``I am glad of that, my dear child, but nevertheless I will look in, lest the strange bed should prevent your sleeping.''
And again she hugged me passionately against her firm and well-formed bosom, kissing me with a long, long kiss. Quitting me with a deep sigh, at last she said good night, and shut the door, apparently going away. But I fancied that she stopped short, and that I could hear her gently stealing back, probably in the hope of seeing me undress, and of catching a view of my huge pego. So I determined she should have her curiosity indulged. I hurried off my clothes, and before putting on one of Harry's night-shirts, which had been laid on the bed for me, I took up the chamber pot, and turned fronting the key hole, stark naked, and cock in hand. It was at half-cock, but when I had piddled I made it throb and raise its head, and gave it a rub or two, and a shake very deliberately, so that she might be still more bent on possessing it. I took up the night-shirt, and turning to the light, was very awkward in getting it on, so as to give time for a good sight of my prick at full stand against my belly. I then blew out the light, and tumbled into bed very quickly. I listened attentively, and could hear a deep half-suppressed sigh, and then footsteps stealing quietly away. I lay awake cogitating as to how I should receive her, whether to feign profound sleep, and so let her take all the initiative, or whether to pretend that the novelty of the bed, and thinking over her affectionate kindness to me had kept me awake. I decided upon pretending to be sound asleep, chiefly that I might see how she would carry out her designs, and also as allowing me to play the surprised one.
In little more than half an hour after all had retired to rest I saw the glimmer of light through the key hole. I had studied a pose that would facilitate matters. I lay on my back, the clothes partially thrown off my breast, and the hand next to the side on which she must approach, placed above my head. Of course my cock was at full stand and as I had thrown off the heavy counterpane, it easily lifted up, and bulged out the sheet and light blanket. I closed my eyes, and breathed heavily. The door was gently opened, and she entered. She turned to close it, and I gave a peep through a half-opened eye, and saw that she had only on a loose \textit{robe de chambre}, which was thrown open in turning, so that I could see there was nothing but her shift below. I even caught sight of her beautiful bosom, which at once caused my prick to throb almost to bursting, so that when she came to my side, it stood up most manfully. She paused, evidently intent on the sight. She then held the light towards me, and spoke in an undertone, asking me if I was awake. Of course I only breathed the heavier, and lay with my mouth half open, as if in the very deepest first slumber. She then turned her attention to the bulging-out substance, and ventured to touch it gently; then, growing bolder, she still more gently grasped it from above the clothes, and then turned the light on my face, but I gave not a sign. She then put the candle down, and, taking a chair, sat down close to the bed. Here she again spoke to me in a subdued tone. Finding no cessation of the deep breathing, she gently insinuated her hand below the already favourably turned-down bedclothes, and with great care slipped it down to my prick, which she grasped softly. I could now feel her whole body tremble, her breath came fast and short. She passed her hand gently up from the root to the head, its size evidently greatly exciting her. When she grasped the head, it gave a powerful throb. She eased her hand, and, I felt certain, turned to see if it had disturbed me. But I slept on profoundly. She seemed to gain more confidence, for both hands were now applied, and it was evident she had assumed a kneeling posture, the better to favour her designs. I could feel her pass one hand over the other, until she found the head was still partially above the third grasp. I heard her give an involuntary exclamation of surprise at its size. Her curiosity growing by what it fed on, she now commenced with the utmost caution gently to remove the bed-clothes, that she might see, as well as feel. When this was accomplished, she rose and brought the light, again passed it before my eyes, and then moved it down towards my prick. Being sure she was now far too deeply engaged to turn her eyes towards mine, I half opened them, and beheld her bending close over the great object of attraction. I heard her exclaim half aloud---
``How wonderful! I never could have imagined such a thing, and in such an innocent boy, too. Oh! I must possess-yes---I must possess it.''
Here she grasped it more forcibly than before. Then, rising, she put the candle on the pot stand, which she removed to the foot of the bed. Then taking my prick in both hands, she gently rubbed it up and down, and even stooped and fondly kissed the nut. It throbbed more violently than ever at this, and I thought it time to start, and appear to awake. She instantly quitted her hold of it, and stood up, but was too agitated to think of covering me. I opened my eyes in apparent great surprise, but recognising mamma, I said---
``Oh! is that you, dear mamma? I was dreaming such a nice dream about you. Oh, do kiss me,'' purposely not seeming to know that my person was all naked.
She stooped and kissed me tenderly, saying---
``My dear, darling boy. I came to see if you were comfortable, and found you lying uncovered, and with this extraordinary thing sticking up.''
She had seized it with her left hand, as she stooped to kiss me. On the instant, I determined to play off the same game that had succeeded so well with my aunt.
``My dear mamma, I should not have dared to speak to you about \textit{that}, but it does give me much pain, by becoming so hard that it throbs, as you may feel, at the least touch. I don't know what to do; and it makes me feel so queer too, especially at the gentle pressures you have just given it; dear mamma, can you tell me how I can cure it, and I will love you so dearly.''
Here she stooped and kissed me very luxuriously, actually thrusting her tongue into my mouth. I sucked it, and told her how sweet it was. But my prick becoming perfectly outrageous, I implored her to tell me what I could do to relieve it. She looked at me long and intently, blushing and turning pale by turns.
``Yes, my dear boy, I could relieve you, but it is a secret that I hardly dare confide in one so young.''
``Oh! you may trust me, my dear mamma, you know I am becoming a young man, and men must know how to keep secrets, or they would be despised, besides, so dear and loving a mamma as you are to me would doubly make me keep secret anything you confide to me on those terms.''
``I will trust you, my darling boy, but you will at once see by what I shall do, how completely I sacrifice myself to do you good.''
Upon this, she threw off her robe, and sprang into bed by my side.
``Oh! how nice of you, dear mamma,'' said I, as I took her in my arms, and kissed her lovingly. ``Feel, mamma, how much harder it is, so tell me at once how I am to relieve it.''
``Well, my dear child, we women are made to relieve such stiffnesses as this; we possess a sheath to put it in, and then it gradually softens.''
``Oh! where---where---dearest mamma, do tell me?''
She took my hand and put it down on her cunt, already quite wet with the excitement she had been in.
``There, feel that, do you not find an opening?''
``Oh, yes, but how am I to get in there---won't it hurt you?''
``I will show you.''
She turned on her back, opened her legs, and desired me to mount on her belly, with my legs between hers, then guiding my rampant pego, and rubbing its great head up and down the lips to moisten it, she told me to push gently downwards, for it was so large that I would otherwise hurt her. Playing the novice to perfection, I awkwardly but gently soon thrust it in, up to the codpiece. She uttered an ``Oh! oh!'' when it was fairly hilted; then throwing her legs over my loins, and her arms round my waist, she begged me to move my bottom backwards and forwards, always thrusting it in as far as I could. Three or four pushes finished me off, in the great excitement I was under. She, too, died away with a great convulsive sigh. I took care to cry out---
``Oh! my dear mamma---oh! stop. I am dying---I---I---am dy---dy---ing.''
Her convulsive internal pressures were delicious, and quickly roused my prick up again. She also had come to, and had glued her lips to mine---giving her own, and then asking in return for my tongue to suck.
``Oh! what heavenly joys, my dear mamma, you did, indeed, reduce its hardness, but just feel---it has got hard again, you must reduce it once more.''
``My beloved boy, I shall always be ready to do so, but it must be the most sacred secret between us, or I should never be able to do it again.''
You may well suppose my protestations were of the strongest. At it we went again, and again, and again. Mamma declared that I was a most apt scholar. Four times did I pour into her foaming and fiery cunt torrents of sperm. At last she insisted upon my withdrawing, saying it would injure my health to indulge any more. So I withdrew, and we embraced each other most lovingly. I now expressed a wish to see the wonderful place that had given me the ecstasies of paradise. She lent herself with admirable grace and ease to my boyish curiosity, and even threw off her shift, making me do the same, that she too might admire the undisguised beauties of my form. There was no pretence in the great admiration I expressed for her really superb form, but I expressed it in a naive and innocent way, that made her laugh heartily, and confirmed her idea that she was not only the first naked woman I had seen, but that she was the first I ever knew, or who had taught me what sensual pleasure meant, and great was her delight in thinking she had taken my virginity, and been the first to initiate me in love's delightful mysteries. Of course, I did everything I could in order to carry on the deception she was so much pleased with, and I may add this was the last time I ever did so, for daily becoming more of a man, I took \textit{things} by the forelock at once, and rarely failed to succeed. We got up, and she turned herself round in every way for me to see the rare beauties of her person---herself explaining to me where she was well made-bosom, buttocks, belly so white and smooth, without a wrinkle, although she had had a son. She was, indeed, one of those rare cases where nothing remains to tell of such an event. Her bosom, without being so large as aunt's, was gloriously white and firm, with such pink nipples, larger than in a maid, but sticking out hard and inviting a suck. Then her cunt---for she laid herself on her back, opened her legs, and allowed me the closest inspection. I have already alluded to her clitoris, as described by Ellen to Harry; it was charmingly developed, about half the length of Miss Frankland's, and not so thick. As I felt her cunt and introduced my fingers to hold it open, she got excited, and Master Clitoris raised his head, and came out of his corner in full stand. I professed great surprise to find she had a little doodle of her own. I purposely used the boyish expression. I began to play with it.
``Oh!'' I said, ``I must kiss it.''
I did so, and began to suck it. She got dreadfully lewd, and seizing upon my now-again-standing prick, drew me upon her, and introduced once more my master weapon. With greater slowness until the final crisis drew near, we had another delicious fuck. She was a woman of very warm passions, and the long pent-up seclusion she had kept herself in with regard to our sex being once broken, now that the flood-gates were opened, there was no resisting the torrent of her lascivious passions. Twice again did we fuck without withdrawing. Then, after hugging and thanking me for the ecstasies I threw her into, she rose for a natural purpose, and advised me to do the same, and we would then both lave ourselves with cold water to restore our nerves. She laved me and I her. She then insisted on my lying down on my back, while she admired what she called the masterpiece of Nature. From seeing and feeling, she soon came to sucking. Up he got in a moment. Playing the ignoramus, I asked if it was not possible that we could both enjoy that pleasure at once.
``Oh, yes, my dear boy. I am so delighted to find that this pleases you! Lie on your back, I shall get over you in the reverse way, and while I suck this enormous jewel, whose head I can hardly get into my mouth, you shall do as you like with my notch.''
``Is that what you call it, dear mamma?''
``That is one name, and it has many others, but you men generally call it cunt, as we call yours prick, it is just as well you should know their ordinary names, as children only call them Fanny and Doodle.''
``Prick and cunt---oh! I shan't forget, so let me have that beautiful cunt to suck.''
We had a mutual gamahuche, and both greedily swallowed the double result, and continued our caresses of both parts, until they were again in full vigour, and inspired with a desire for more solid enjoyments.
``My darling boy, you are so apt and excellent a scholar that I must show you there are several ways of allaying the stiffness of this dear fellow, who seems as desirous as ever to have his hardness taken out of him. I shall show you how my husband liked best to enjoy me.''
She scrambled up on her knees, and presented her very fine bottom, told me to kneel behind and give her my prick in her hand, which she thrust out backwards between her thighs. I did so. She told me it would appear to get further in this way, and, in fact, it did. After it was all in until thighs and buttocks met, she told me to admire, praise, and handle the splendid cheeks of her bottom, and said that such praise greatly excited her. Of course I did so, admiring not only their size and fairness, but also the beautiful curly silk meshes that ran between the cheeks, covered her beautifully pink bottom-hole, charmingly puckered as it was, and ran up to the flat of her back. After I had so excited her, she begged me to lean forward, and to handle one bubby while I should play with her clitoris with the other hand. I did all this tolerably well, but with somewhat of awkwardness. She said I would soon be perfect. We ran again two courses before she fell forward dragging me down without withdrawing, and then turning on our sides, still intertwined we fell off into a deep slumber, and did not awaken till daylight. Mamma jumped out of bed, unseating me by the act. She was alarmed lest the hour should be late enough for the household to be up. I tried hard to persuade her to reduce once more the hardness which had again seized me as she might see and feel for herself.
``No, my dear boy, we must not be imprudent, my niece may have awakened and grown anxious at my absence, and she may rise to seek me; so good-bye, my darling, go to sleep again.''
She embraced me tenderly, but I could not prevail upon her to go further, although she promised to seek an opportunity during the day, and to give me as much as I liked the next night. She left me, and I pondered over the lucky chance that had put so desirable and fine a woman into my arms, and also congratulated myself on the stratagem by which I had fully convinced her that she was my first instructress in the art of love, a circumstance ever dear to the ardent imagination of the darling sex. I easily fell asleep again, wondering how Harry in the meantime had got on with his cousin. My dear mamma would not allow me to be disturbed. She had entered my room once or twice, and found me sleeping soundly.
At last she again entered, just as I had satisfied a natural want for which I had risen. To rush to her, to embrace her tenderly, to fasten the door, and compel her, not much against her will, to come towards the bed, to beg her to lie on her belly on the bedside, to cant her petticoats up, to kneel and gamahuche her cunt from behind until she begged me to rise and fuck her, was but the work of a minute or two. And then my stiff-standing pego, aided by the mouthful of thick saliva occasioned by the gamahuche, was directed at her cunt, and driven home as far as the buttocks or her fine backside would allow. My prick being fairly sheathed, I paused for a moment to handle and praise the beauty of her posterior orbs. Then, stooping, I nibbled at her bubbles with one hand, and frigged her clitoris with the other. Sharp set, with my long rest and refreshing sleep, I rapidly ran a first course, but not quicker than the lascivious nature of dear mamma, who joined me in a copious discharge with the most ecstatic joy, and the most delicious inward pressures. For she was a perfect and most accomplished actor in the combats of love, and in her own way was worthy of my glorious aunt and my loved Miss Frankland, and as thoroughly accomplished as they in all the abandon of lust and lubricity, although at the moment I had only proved things in an ordinary way. Her exquisite internal suctions almost prevented the slightest relaxation in my delighted pego, and after a minute or two of indulgence in the after joy, I began again almost before dear mamma had recovered her senses, when she tried to tear herself away. But before she knew where she was I had succeeded in again firing her ardent and lascivious nature, and she became as eager for a second course as myself. This was naturally longer than the first fiery one. I raised myself upright on my knees, contemplated with the utmost delight the uncommon active play of her loins, and the exquisite side wriggling of her very fine backside. I loudly praised her delicious manoeuvres, and seconded them to the utmost, until getting more and more excited, fast and furious grew our movements. I bent down to second her by frigging her clitoris, and the final crisis seized us both with its agony of joy, and I sank almost insensible on her back. We lay for a short time lost to every thing, until mamma, remembering the risk we ran of discovery, begged me to withdraw, and let her go away. She rose and threw herself into my arms, glueing her lips to mine with a most loving kiss. Then stooping, she gave my now pendant prick a most delicious suck, making her tongue play into and around the mouth of the urethra. This was so delicious that the delighted member instantly showed its appreciation of the pleasure by starting up in full swing. Mamma gave it a pat, and said he was a most charming and delicious boy, who did not know how to behave himself. Again she kissed me, and tore herself away, but I could easily see the regret was as great on her side as mine. She told me her son had been as lazy as I was, and said that breakfast was waiting for us both. I quickly finished my toilet and found them all at the breakfast table.
Ellen blushed deeply when she saw me. A glance from Harry assured me he had succeeded, and that Ellen not only knew what I had been about, but also that I knew what she had been doing. Hence her high colour when she saw me. I smiled, and nodded to her knowingly, and as she had observed the intelligent glance that passed between Harry and me, it did not tend to put her at her ease.
Mamma, of course, knew nothing of what had passed in her bed while she was with me, and was all affectionate attention to the whole party, but with a marked tendency to pay me more particular attention. Our breakfast was late, so we had to hurry ourselves for church. Mamma drove Ellen in a small pony phaeton, while Harry and I took a short cut across the fields.
Harry told me how he had watched his mother and had quietly approached my door, and as the bed was exactly opposite the key hole, had seen and enjoyed her proceedings, especially as he knew that I only pretended sleep.
``By Jove,'' he said, ``what a fine woman mother is! I could not tear myself away, and remained until you both went at it again, stark naked. My mother's beautiful hairy cunt, fine bubbies and backside, nearly drove me mad with desire. I could have violated her if she had been alone. And, then, her energy in fucking was superb. I could hold out no longer, but rushed to dear Ellen's side. She was asleep. I took her in my arms, and awoke her by feeling her delicious young cunt. She opened her eyes, and thinking it was mamma she turned round to repay the compliment, and started on having hold of my pego.''
```Why! Harry dear, how came you here? We shall be caught by mamma.'
``'Oh, no, my love, mamma is better engaged, and has slipped away to Charlie's room to get done to her what I am going to do to you.'
``She was too much alarmed to believe me, and I was obliged to bring her to your door. I first peeped, and saw you were still at it. Mamma's legs and arms thrown around you allowed me to see your great big thing rushing in and out, and driving home with immense vigour. I whispered to Ellen to peep. While she did so, stooping, I sat down on the floor and gamahuched her. She spent almost immediately, and was so excited that it quite filled my mouth. I rose on my legs, and bringing my prick against her cunt, made an entrance as far as over the nut, but was myself so excited with all I was doing, and all I had previously seen, that I went off in an agony of delight and with a suppressed cry, which must have been heard by you and mamma if you had not been so busily engaged. Ellen had been so excited and so intent on the to her new scene enacting before her eyes, that she had never ceased gazing on it, and left me to do whatever I pleased, but my cry alarmed her, especially as in my last forward push I had sent her head with some noise against the door. She rose, and so unseated me from the slight hold I had got of her cunt. She turned round to embrace me most excitedly, and whispered that we must go elsewhere. I took her round the waist, and we quickly regained mamma's bed. The light enabled me to find a towel. I told Ellen it was to prevent any moisture betraying our acts. She was far too excited and wishful for the article to make the slightest resistance, or even pretence of refusal. I begged her to throw off her shift, as she had seen both Charlie and mamma were quite naked. She at once complied, being now as eager for the fray as myself. I, too, threw off my night-shirt. For a moment we embraced each other's naked bodies. My cock was as stiff as iron. She lewdly laid hold of it, while I handled her charming young cunt. I helped her onto the bed, she at once lay down on her back, and threw open her legs as she had seen mamma do. I stopped and gave her cunt, all oozing with her own and my spunk, a warm kiss, and with a lick or two on her budding clitoris, I fired her even to greater excess than she had yet been in.
``'Oh, come to my arms, my dear Harry, and let us do as they were so delightedly doing.'
``She had noted with what rapture mamma was enjoying you, and she had noted, too, what a much larger cock yours was than mine; so she had naturally reasoned that if one so big gave her aunt so much delight, my smaller one could not possibly hurt her, hence her eagerness to have me at once. I did not baulk her, but throwing myself between her widespread thighs, I soon brought the point of my prick to the longing lips of her little virgin cunt. I rubbed it up and down in between the pouting and self-opened lips, partly to moisten it, and partly to still more excite her lust. I then gently pressed it forward, and introduced just its head, and drawing it in and out, made her beg me to go further. I did so, slowly, until I found there was an impediment. I knew that I must burst through this and that it would hurt her, so I continued withdrawing and re-entering without going further until she became so voraciously lewd as to throw her legs around my loins and heave her bottom up to meet my thrusts. I seized the fortunate moment and with one downright violent thrust burst through every barrier and buried my prick in her up to the very hilt. The attack and its result was so unexpected by Ellen that when she felt the knife-like thrust of agony she gave a shriek of pain, and made an immediate effort to throw me off. I was too firmly seated for any other result of her struggles than the still more complete rupture of her maidenhead, which my forward thrust had partially effected. I lay for some time quite tranquil, and when her immediate pain wore off I commenced a gentle in and out movement, which, without exactly exciting her, produced a pleasing sensation. I then went on faster and faster until the crisis came upon me, and I shot into her a torrent of boiling sperm that by its balmy nature mollified the previous smarting; so that when I had recovered from the delicious ecstasies of my first success, and my prick gradually resumed its former vigour, I found by the somewhat increased pressure upon it that her passions were re-awakening. Three times did I fuck her before I withdrew, the last one appeared to give her more pleasure, but still she complained of a smarting pain as I passed over and over the shattered hymen. I advised her to rise and lave herself as a relief, and to wash away the stains of blood from her thighs. The towel was a fortunate thought on your part, but, in fact, I had followed in all my movements the sage counsels you had given me from the experience you had had in taking the virginities of Mrs. Vincent and your two sisters, or I should otherwise probably have bungled the matter, although my experience with your magnificent aunt had naturally put me up to all the art of fucking. I had some difficulty in persuading Ellen to let me put it in again, as she declared she had endured perfect agony when I broke through her maidenhead. However, I gamahuched her well, got her passions up, moistened the shaft well, and was very gentle in entering and in my first movements. I spent without making her do so. But the well-greased sheath now allowing more easy movements, she gave down her nature with considerable pleasure as I spent the second time. Still there was fear and restraint---fear lest mamma might come back---so I thought it advisable to retire to my own room, being quite certain that now the road is open her lascivious nature will not be long in enabling her to enjoy the sport to the utmost. By the way, she could not help wondering how mamma could take in your immense pego; why, she said, it was as thick as her wrist and much longer than her hand, and yet it seemed to slip into mamma with ease and pleasure, 'while yours, dear Harry, which is not thicker than my two forefingers, and hardly much longer, has given me such pain.' I assured her it was only for the first night, and that if she would bathe it with warm water two or three times during the day, and put up a little glycerine as far as where it hurt, which her finger could easily reach, she would find that to-morrow night there would no longer be any pain felt, and she would enjoy it as much as she had seen mamma do. With this advice I left her to her repose, and gained my own room unobserved.''
After this we concerted together as to what we should say to the doctor, who was sure to question us. Mrs. Dale's cottage was not in our parish, but she had driven over to our church, partly to throw off all suspicion from the doctor's mind, and also to thank him for allowing us to stay with her.
We, therefore, knew that we should have to go to the rectory and stay for luncheon. We agreed that we should not on this occasion take the doctor into our confidence, but that we should tell him we had purposely been very quiet and discreet, so as to throw Harry's mother off her guard. That Ellen slept with her, so that it became doubly necessary to gain her confidence. This being arranged before we reached church, we entered. After service we all adjourned to the rectory. The doctor escorted Mrs. Dale, Harry, Ellen, and I my aunt. Aunt, pressing my arm, asked me if I had had Mrs. D., as she seemed a fine woman worth having.
``Oh, dear no. I have had no opportunity, even if she would have consented. I have been playing the ingenuous youth to help Harry with his cousin. I thought we had somewhat thrown her off her guard, but she was still jealous and watched him closely. Ellen slept with her, which rendered things more difficult for Harry. She has closely examined me as to the sort of intimacy existing between us. I threw such an air of candour and innocence over my replies that she was quite delighted Harry had met with such a companion. I fully expect she will break out in praises of my \textit{modest} and discreet conduct.''
Indeed, so it turned out, and Mrs. Dale did it with such an air of candour that aunt was quite convinced nothing as yet had occurred between us. While the ladies discussed the dresses and bonnets of all who had appeared in church, uncle took Harry and me for a walk in the garden until luncheon was ready. Here he began, as aunt had done, to question us as to our proceedings, and the reason for Mrs. D. asking permission for us to stay. The same replies that had satisfied aunt satisfied him that nothing as yet had taken place beyond my gaining the confidence of Mrs. D.
``My dear Charlie,'' said uncle, ``you have only now to manage somehow or other to let her see your great big cock without apparently your being aware of it, and I will warrant, from my knowledge of woman's nature, that she will find a way to have you, only mind you play the innocent, and be very awkward, and let her appear to teach you, which will give double pleasure and prevent any questioning as to how you have gained your instruction, if she thought you instructed.''
I smiled inwardly at these sage directions, and thought how completely all persons knowing in the ways of the world gave the same advice. But little did uncle then think that I had acted up to the very letter what he was advising for my future conduct. We re-entered the house on luncheon bell ringing. Mrs. Dale complimented the doctor on the advance her son had made both in manners and instruction, and quite naturally congratulated herself on his finding so very modest and gentlemanly a companion in the doctor's nephew---myself to wit.
Returning home, Ellen begged she might be allowed to walk, doubtless calculating on having Harry for a companion. But mamma, while agreeing to her request, was still sufficiently on her guard to take Harry in the phaeton, and leave me to escort Ellen. Here was a chance! Ellen blushed, but took my arm as we left the rectory. Uncle gave me a knowing look, and a glance at Ellen, as we parted, as much as to say, I guess what will happen. We walked away steadily enough until the first hedge hid us. I stopped, and embraced Ellen tenderly, saying how glad I was to be able to congratulate her on the happy chance her aunt had given her, by coming to me for the night. She was a good deal confused at thinking that I should know how she passed the night. I rallied her upon this, told her that no secret existed between Harry and me, and that, in fact, if I had not lent myself to the game, she would not have had the opportunity for the great pleasure she must have had in Harry's arms. I knew she had not had much, but I wished to draw her on, and to make her open out as to her feelings, being determined to make the most of any confidence on her part. She replied that, indeed, she had done nothing but suffer, and would not have allowed Harry to do what he did if she had known the pain it would give her, she had been deceived by seeing how much aunt had seemed to enjoy what was so greatly superior in size to what Harry had. I smiled at her allusion to the size of my pego, and knowing that her curiosity must be creating in her a desire to see it, I told her it was well for her, in the first instance, to have had the smaller weapon to penetrate her, and that now she would never again suffer, even by the introduction of so large a one as mine.
``Oh, but when I think of the immense size of yours, I could never dare to allow you to try, although aunt did seem to enjoy it, when you pushed it in with such force.''
``My dear Ellen, it was the size alone that tempted mamma, and if I had not been larger than Harry, I doubt if ever she would have come to me at night.''
``But how could she dare to do so?''
``Curiosity to enjoy an unusually large cock, my dear.''
``Did you know she was coming?''
``Yes, and no. I saw that her passions were excited, when I had once \textit{accidentally} allowed her to see my large proportions.''
``Yes, Harry told me what you were about, but I hardly expected aunt would have dared to come to you---how did it happen?''
``Well, if you will promise \textit{never} to let your aunt know that I have told, I shall tell you. She came and found me \textit{apparently} asleep, first felt me, and seeing I did not awake, carefully uncovered me, looked at, handled and kissed it, upon which, as my cock was nearly bursting, and I could stand it no longer, I awoke, and innocently complained of the stiffness I suffered from in that part, and begged her to tell me if there was any means of relieving it. She told me there was, but it was a great secret she hardly dared trust me with---and even if she could do so, she was afraid of a great long thing like \textit{that}, three of her hands long below the head! but that if I promised secrecy, she would try. Then she lay down and taught me how to put it in, and I know you afterwards enjoyed the sight of our being in full action, quite naked---did you not enjoy it, dear Ellen?''
``Well, dear Charlie, it was very exciting, and made me feel queer all over; but is it really three hands and a head long?''
I was delighted at the question, as it showed me she was ready for what I intended should be done. Curiosity once excited was sure to go to the utmost length, if it had the opportunity. I had purposely been hurrying on to gain a dense copse through which our path lay, and I knew there was a snug glade, where we would be in perfect security. It was the dinner hour of the peasantry, and no one else was likely to come that way. Just as we entered the copse, she had put her last question. I told her I would show her, if she would step a few yards beyond the foot path. She objected, for form's sake saying---
``What would Harry say?''
``There is no occasion for him to know anything about it, but even if he did, has he not himself shown you mamma and me in full enjoyment of her sweet charms; but, unless you tell him you may be sure I never shall, it will not take a minute, and as we have already walked very fast, we have plenty of time, and our absence will not be observed.''
With professed reluctance she allowed me to lead her where I wished. Having arrived at the favourable spot, I sat down on a gentle slope, and begged her to sit down beside me. As you may well suppose, my prick was rampant, and almost bursting open my trousers, so that as soon as I unbuttoned, out it flew in all its splendour. She gave a half scream of surprise as she gazed upon its large proportions, and declared it looked larger than when she had seen it with aunt. Her face flushed, and her eyes sparkled as she gazed, but she seemed half frightened to touch it. I took her hand and placed it on it. She immediately grasped it convulsively, but sighed deeply. I had lain back on the grass that it might stand out boldly before her and I told her to try if it was not three hands and a head long. She immediately passed one hand over the other from the root, and said it was really monstrous, and she wondered how aunt could have got it into her.
``Oh, my darling, I hope some day you will find that you can take it all in with the utmost delight, but I should not think of trying until you have had some more practice with Harry.''
Meanwhile she was handling it with great excitement, and while saying she was sure I could never succeed with her, she was evidently longing to be able to take it in. I saw I must work her up more---so I said---
``Dear Ellen, you know what pleasure it gave you and Harry to play with each other with your mouths, it is now your turn to let me see your dear little thing---and then you must lie over me reversed, so that we may enjoy ourselves with tongues and mouths.''
She let me at once pull up her petticoats, but said she feared that even for that she was still too sore from Harry's work last night. I asked if she had bathed it in warm water and put glycerine up. ``Oh, yes.''
At first it smarted, but before going to church, she had done it three times, and no longer felt any pain, but still was afraid of my finger going up. I was introducing it at the moment. It passed in its full length without hurting her.
``Now, pull up your petticoats well, and lie down on me, while I do the same with this charming little cunt; my tongue can only give it the utmost pleasure.''
She herself was now so much excited that she was ready enough to comply with my desires. She got upon me, her petticoats well canted over her back. She glued her lips to my prick, and sucked and frigged it with an energy that proved how highly her passions were fired. Her cunt was already in a foam of spendings, which I first licked up. Then sucking her tiny clitoris, stiffly projecting slightly out, I thrust my middle finger up her cunt, and by the wriggling of her backside, saw how much she enjoyed it. Introducing a second finger to moisten it, I withdrew both, and, turning my hand sideways, made each finger enter a separate aperture. She was already nearly in the grand crisis; it came upon her before I was ready. She poured a greater discharge into my mouth than I thought the young thing could have spent. It took her breath away, and she released her suction of my prick for a minute. But on my begging her to continue sucking, she did so with increased energy, and I poured out a torrent of sperm that shot down her throat and nearly choked her, but the dear girl never let go for all that, and sucked away until not only was there not a drop left, but by her delicious titillations she had brought my prick up to its utmost vigour again. I, too, had reawakened her passions. She wanted to renew the sport in this way again, but I begged her to allow me to rub the head of my prick up and down between the well-moistened lips of her cunt, and then to spend with the point, or at most the head, within it. She asked if she could trust me to stop if it hurt her.
``Of course, my darling,'' I said, ``nothing shall be done, or rather everything shall cease the moment you tell me to stop.''
Half afraid, yet wishing to try, she changed her position to a kneeling one. I canted her petticoats well over her back, and first kissing and handling her hard and plump buttocks, which promised a future perfection, I stooped and again licked her charming pouting cunt with all its budding fair young curls. Then applying my surcharged mouthful of saliva to my already well-moistened prick, I lubricated it completely from point to root, and then applied it to the half-opened lips. Rubbing it up and down here, and over the clitoris, I excited her to the greatest pitch.
``Oh! Charlie dear,'' she cried, ``try if its head will go in now, and I will try to bear it.''
I was only too glad of the permission, and very rapidly got it in over the nut, but it was very tight. I drew it half out again, and then, on repeating this five or six times, found I was imperceptibly gaining ground.
``Oh! dear Charles, it is delicious! Try on, gently.''
I did so, and had got rather more than half way in when she went off in an agony of delight, deliciously pouring her warm liquid over my enchanted prick, giving, at the same time, such a push backwards, which, meeting a firm, though gentle forward movement on my part, joined with the natural relaxation following her discharge, drove me up to the hilt in the very tightest little cunt it has ever been my good fortune to sheath myself in. I seemed to fill every cranny, and to have stretched every part to its utmost distention. My aunt with her great cunt had a power of pressure that seemed almost to nip off your prick, Miss Frankland, too, was great in that way. But this was more like a very well made first-rate kid glove, two sizes too small for your fingers, yet giving way without bursting, and fitting every irregularity of the nail or finger; just so her little cunt fitted my prick exactly like a glove, and it was truly most ecstatic. A gentle withdrawing, and then as gentle resheathing, so excited me that I shot a torrent of sperm up into her very womb. She gave quite a cry of ecstasy, and I could feel the tight sheath exercising a running movement along the whole length of my prick, and still more tightly closing all round it---if that were possible. It was so exquisitely delicious that both of us were almost instantly in readiness for another course.
She asked if I was all in.
``Oh, yes, my dear, do you think you could have taken any more?''
``Oh, no, it appears to fill me to bursting, and to be up to my very heart. I could not have supported more, but could hardly believe I had it all, as I did not think it possible, and was afraid there was more to come.''
``Did it give you any pleasure?''
``Oh, yes; and does so still---push on, dear Charlie, and don't spare me, it is heavenly.''
She wriggled and heaved her backside. I seized her by each hip and favoured her side movements by, as it were, drawing her off and on; faster and faster we moved, until at last the crisis seized us both together. Her head sank with a deep sigh, or rather cry of ecstasy. She would have fallen forward on her belly, but that my grip of her hips held her bottom close up to my belly, with my prick thrust into the innermost end of her cunt, until I felt the three points of the opening of her womb, like the nailless ends of three fingers grasping, as it were, the very point of my prick, and opening themselves to receive the whole discharge of my sperm within its innermost recesses. Nothing could be more delicious, and as I held her fast, I was myself in a state of perfect ecstasy. At last addressing some endearing expressions, and getting no reply, I found that the dear girl had quite fainted away, and was insensible in every respect except in the continued convulsive throbs of her delicious tight cunt. However, finding that she did not recover her senses, I gently withdrew my still stiff prick. Very little sperm followed the withdrawal. I wiped her cunt dry with my handkerchief, and was glad to see there was no blood stains. I laid her gently down on her back, ran to a little stream, and taking two handfuls of water, came back, threw some on the still throbbing cunt, and sprinkled her face with the drops that still adhered to my palms. This had the desired effect; she opened her eyes, raised herself on her bottom, and threw her arms round my neck as I knelt by her side. Telling me I had made her taste of the joys of heaven, she kissed me, and then burst into a hysterical flood of tears. I comforted her as best I could and asked why she wept.
``I don't know, dear Charles, but the last time made me feel both sick and faint just after you had given me such ecstasy as I never dreamt was possible. I believe I then fainted, and even now, I don't know why but I feel quite hysterical.''
I kissed her tenderly, begged her to rise and come to the spring, where she could drink and said if she sat down on her feet I would bathe and cool her dear little cunt, which would probably put all to rights. She did so, and was quickly quite restored to herself again. She said she supposed it was my enormous size.
``But it did not hurt me, dear Charles, it only gave me too much pleasure; but you will do it to me another time whenever we have any opportunity, will you not, dear Charlie?''
I assured her I should always be too glad to do so, but that we must neither let her aunt nor Harry know of our proceedings. This being arranged, and she having quite recovered from the pallor her fainting fit had caused, we resumed our course homeward, and so hurried on that Harry, who had come to meet us found us getting over the stile of the last field, and was even disappointed that we had got so far, for we were now in sight of the cottage. He had hoped to find us much further back, and that I might have favoured his having a go at his cousin before reaching home. Ellen squeezed my arm. I said it was just as well as it was, for any imprudence might have awakened his mother's suspicions, and prevented a night of pleasure, which would be far better than any uncomfortable field affair.
When we arrived at home, mamma thought that Ellen looked fatigued, and advised her to go and lie down on her bed, and take an hour's siesta. She told us boys we had better do the same, as she had some private matters to attend to. Harry and I saw immediately what was meant, and we betook ourselves to our respective rooms, I to expect mamma, who did not fail to come, and Harry to watch her, and then made the most of the opportunity with his cousin. I quickly undressed, and when mamma came I found she had divested herself of stays and undergarments; so when she undid her gown, and let fall her shift, she stood in all the naked glory of her beautiful form. I flew to embrace her most lovingly. Both our hands wandered and being both in full heat, we were at it in a moment fast and furious. I drove on, admirably seconded by dear mamma, and we quickly both gave down at the same instant a most delicious libation on the altar of Venus, and then died away in all the after-enjoyment. We lay for nearly a quarter of an hour soaking in the delicious bliss of satisfied desire. Mamma, on coming to her senses, kissed me most tenderly, and declared she had never believed it possible that she could have had such exquisite delight.
``But then, my dear Charles, I never dreamt that any man, let alone a boy like you, could be so magnificently hung. Oh, it is also such joy to me to think I have first taught you the real joys of coition, and tasted the first sweets of that most glorious weapon. My dear Charlie, I must contemplate its beauties in this full light; withdraw the dear fellow and turn on your back.''
I did so. She rose, and turning in the reverse position, brought her lovely foaming cunt right down on my mouth. I sucked up all the delicious foam oozing from the aperture. Then drawing into my mouth her half-stiffened clitoris, which was then pendant like a little boy's cock, I soon sucked it into its utmost rigidity, frigging her rich pouting cunt with two fingers, the while. She, on her part, was not idle, first playing with my prick, covering and uncovering its head, which soon made it stand up in all of its glory. She was profuse and loud in its praises. Then getting too excited for mere admiration, she took it in her mouth and sucked it, and manipulated it with one hand, fingering my codpiece with the other. I then found her fingers were feeling and tickling my bottom-hole. She took her mouth from off my prick, and paused a moment; then again applied her finger to my fundament, and made it gently penetrate as far as it would go. The previous pause had evidently been for the purpose of moistening her finger with her saliva that it might slip in easily. I was delighted to find that she had come to this, but pretending ignorance, I stopped my proceedings to ask her what she was doing to my bottom, which could give me such exquisite delight.
``It is my finger, my dear Charles, my late husband was always delighted with my doing this, and used also to add greatly to my pleasure by doing the same to me.''
``Shall I do so to you, dear mamma?''
``Oh, yes, my darling boy; moisten your finger first and then do it in my bottom-hole, as you have been doing it in my cunt.''
``But I think I can do both at the same time, they are so close together.''
``You are a delightful darling; do so, and it will be double pleasure to me.''
So I immediately commenced to postillion her to her and my extreme gratification. We soon spent with the utmost delight, and both swallowed all we could get, continuing our suctions until the passions of both were again excited. I now declared I must fuck her again in the kneeling position, in which she had before given me such exquisite pleasure. As she drew a little higher up, I flung my arms round her fine backside, and glued my lips to her bottom-hole, and thrust my tongue in and out.
``Oh, Charles, dear, what are you doing? Oh! how delightful.''
And she wriggled her backside over my mouth in a most voluptuous and lascivious way.
``Oh, rise my darling, and fuck me; you have made me so very lewd.''
I drew myself up on my knees behind her, and was into her with wild ferocity that made her cry out with joy as she felt the mighty instrument rush within her. I stooped and frigged her clitoris at her desire, but wishing to contemplate the glorious movement of her backside I begged her to frig herself that I might be able to do so. Seizing hold of her lips, I drew her splendid bottom off and on my stiff and glowing prick with such immense delight to her that she went off and spent profusely, the hot stream bathing my delighted prick. But having already fucked Ellen so shortly before, and having spent twice at the present time, I remained for a while quiet, with mamma's exquisite cunt deliriously throbbing round it to the infinite enjoyment of my cock. I stooped and nibbled with my fingers at one of her nipples. I played with and frigged her very fine clitoris, which was soon in stiff-standing excitement again. Being cool myself, I soon worked her up into the wildest state of excitement by my frigging and the throbbings of my prick, aided with occasional long slowly drawn-out movements, and then as slowly regaining ground until within the last three inches, when it was thrust vigourously forward, and kept there for her convulsive pressures on it. I kept this up until she was almost wild with lust, and cried out for more vigorous movements. I did not immediately comply, but continued my exciting proceedings until she bit the pillow in the madness of her lust. Then I drove on fast and furious, amid cries of delight and ecstasy on her part, until the grand crisis overtook us both at the same instant in a perfect fury and agony of delight. I had previously left the frigging to herself, and had seized her lips and enjoyed the glorious sight of the furious contortions of her bottom under the excessive lubricity of her wildly excited lasciviousness. She died away in such excess of ecstasy that she would have fallen on her belly but for the grip I had upon her hips, and the pressure with which I drew back her glorious bottom against my belly. I threw back my head in the agony of delight, and brayed like a donkey as I had done once before when fucking the luscious Frankland, and felt the three pointed entrance to her womb close upon and nibble at the point of my prick so delightfully, just as dear Ellen's had done in the wood. As I came to my senses I spoke to dear mamma, and found that she too had fainted away, and was quite insensible to everything but the convulsive inner movements of her delicious cunt. I withdrew and laid her gently down on her side, bringing a tumbler of water, a sponge, and towel, I opened her splendid thighs, sponged and bathed her cunt, which showed but little of the torrent of sperm I had just poured into it. I then sprinkled her face, and she came to with a deep sigh. Her first utterance was to bless me for the joy I had given her, which was in fact too much, and then she burst into tears and became quite hysterical. I thought it odd that I should have produced the same effect upon her more accustomed and more developed organs as I had done on dear Ellen. I comforted her in my boyish way, and asked how it was that the effect should have been different from anything she had previously experienced with me.
``Ah! my dear boy,'' she said, with a deep sigh, ``you have caused me such extreme sensations that I fear you must have got me with child, you seemed to penetrate my very womb, and to excite me far beyond anything I ever previously remember.''
``My loved mamma, can I possibly get a child?''
``Get a child, indeed!'' she replied. ``Yes, a dozen, with such a great monster of a cock, that so excites us poor women.''
I embraced her most tenderly, and said I was so happy to think I should be the father of a child of hers.
``Alas! my dear boy, it may be joy to you, but what a sorrow it will be to me if such should be the case; think how I should lose position in the world if it should be known, and even if by going abroad I could hide my shame from the public, still what shifts and contrivances I should be put to to ensure secrecy; but never mind, my darling, I would run twice such risk to enjoy your person, and secure your affection; you must ever cherish and love me, my Charlie, for I risk good name and fame for you; but now I must be gone, or we shall be sought for; try and sleep a little, my dear boy, for I am sure you need it after your exertions, and remember you must gain strength to renew them this night.''
She kissed me lovingly, rose, put on her things and left me to repose. But I could not help thinking of what she had said about fearing that that peculiar fuck in which she had fainted portended fructification. If so, I thought dear Ellen will probably be in the same predicament, for the result was precisely the same with her. I may here observe that mamma's fears became certainties, both in her case and Ellen's. Eventually they both left the country together, when staying would have brought on discovery. And, curiously enough, they were both delivered of daughters on the same day. Of both I was the happy father, although Harry had the credit of Ellen's child, but she herself always asserted to me that it was the delicious fuck in the wood that did the mischief. And from the peculiar effect produced on both mothers on that day, I never had any doubt of the real paternity, besides, the child grew up my very image. Mamma's daughter was superbly developed when she became a young woman. She had even a larger clitoris than Miss Frankland, with which she absolutely deflowered her sister's cousin at the age of fourteen. I may also incidentally observe that at the age of fifteen I had both their maidenheads, as far as the male sex was concerned. And Harry and I often fucked them together in every way; and my darling daughter with her long and large clitoris has often fucked my bottom, while I was doing the same to her sister, with Harry below fucking her whom he believed to be his daughter. But this belongs to my latter experiences, and has nothing to do with the present period of my life, though, perhaps, I may be tempted hereafter to enter into all the details of my middle age and later experiences.
Dear reader, pardon me this digression. To resume, I slept soundly for an hour, then rose, and strolled in the garden with Harry, who related to me how he had taken advantage of mamma's occupation to steal into Ellen's room. She had been much afraid, the sly pussey, to allow him to enter again, but when once he got within, and she found it did not hurt her, but the contrary, she got extremely lewd, and they had two splendid fucks. Then stealing along to my door to peep as to how we were getting on, we so excited them again that he had another from behind, while she stooped and peeped all the time, for it was when I was fucking mamma from behind, on my knees, and they concluded it would be our last for the present. When they had brought matters to a finish they separated, and mamma had found Ellen fast asleep.
``But, by Jove, Charlie,'' said Harry, ``how splendid mother fucks, I quite envied you, and I shall never rest until I get into her myself; how gloriously she wriggles her backside, and how lusciously she enjoys fucking; to be sure such a mighty prick as yours is enough to stir up every passion; it astonished, and I think made Ellen more lewd, although she is sure she could never take in such a monster.''
I smiled at thinking how easily the very youngest of the fair sex deceives us, but I took care not to let Harry know my opinion.
We re-entered for dinner, and spent a pleasant evening, which was the forerunner to the delights of the night. Mamma came as soon as she thought Ellen fast asleep, which Ellen took very good care should soon be the case. In a moment, she was quite naked, and clasped to my equally naked body. I had been expecting her, and thinking over the delights of our last fuck, so that I was rampant before her arrival. She was equally eager for the fray, and at it we went hammer and tongs, and soon brought the first bout to a close, in mutual ``ah's!'' and ``oh's!'' of delight. We soaked for some time in the delicious enjoyment. Then mamma scolded both herself and me for our precipitation, saying that we threw away all the luxury and abandon of fucking when we went at it in such haste; it was in that way mere animal instinct, and wanted all the lascivious delight of lubricity and skill in fucking. She said, now that the edge was taken off our appetites, we must begin again with a mutual gamahuche. She rose first to piddle, and allowed me to see the rush of water from her delicious cunt. Then lighting two more candles, she placed two at the foot of the bed, and two at the head, by which we should both have the advantage of seeing all we were caressing. Then I lay down on my back, and she mounted on me, in reverse, thus bringing her bottom down over my face. I thrust my tongue up her cunt, and licked up the delicious spunk oozing down from the inside. Her piddle had washed all away from the pouting lips. Then taking her charming clitoris in my mouth, I sucked it up to its greatest stiffness. I had thrust three fingers into her cunt, and when I found she had thrust hers into my bottom, I transferred them all into her beautiful pink bum-hole. They were very greasy from my sperm coming down upon them when in her cunt, and as she favoured their entrance by pushing out her bottom, all three slipped in, without, apparently, her thinking it was more than one. I was delighted to see how easily it stretched out, for this gave me great ground to hope that I should be able to manage to get my large pego within, which I was fully resolved upon doing, but it required a little artfulness to do so without raising her suspicions that it was no new road to me. She brought matters to a conclusion much as before, and when re-excited, mamma proposed to teach me a new way, which was by her mounting on me, and staking herself on my standing pego. Like others before her, she did not stoop down upon me until she had made herself spend where she was, while I saved myself for further fucking.
When she died off, she sank on my bosom. I clasped her waist with one arm, sucked the bubby nearest my mouth, and reaching round my other arm, I brought my hand over her bottom to the delightful orifice, first moistening my finger with her spending which was oozing out between the lips of her cunt and my standing prick. I thrust my finger into her bottom-hole, and worked it in and out, to her infinite satisfaction. She cried out in the excess of her lewdness---
``Oh! my dear boy, that is just as my dear husband used to do, and it gave me great pleasure, but not near so much as you do, for your dear prick is twice as large as his was, and fills me with an excess of pleasure which was never approached with him.''
All this led up to a superb and lascivious fuck, in which we both died away in mutual ecstasy, with cries of voluptuousness, and then lay soaking in delight until her weight forced me to beg her to turn on her side. We then had a long sweet chat of love. Turning the talk on her suspicions of having got with child at the morning prayers, I remarked that she had had only one child by her husband, and as he had lived many years after Harry's birth, and from what she said, she had continued to be enjoyed by him, it was, therefore, not probable she would now be got in the family-way.
``That appears probable, my dear boy, but then he took precaution not to get any more children.''
``But what precautions could he take, and how did he do so?''
``You are a curious boy, but I shall tell you. He used to continue long at it, making me spend two or three times before he did, and then when he felt it coming he used to withdraw, and his prick being all moist, he would slip it into my bottom, and spend there as soon as ever he got the head of it inside.''
``And did that give you any pleasure, mamma?''
``He had excited me, and made me spend several times before he did so; and beyond slight irritation I did not feel much pleasure, as he was generally so near the crisis that he could scarce do more than get its head in when off he went.
``Did he ever get it in altogether---and then did it give you pleasure?''
``Sometimes he did when he had drawn it out of my cunt too soon; in such cases he used to pause until by rubbing my clitoris he got me into a renewed state of lewdness, and then the pleasure was peculiar and great.''
``Oh, my dear, mamma, you must let me too fuck you in that way, and then you know we shall get no children.''
``My darling Charlie, it is impossible that this great big thing could ever get into that orifice, my late husband's was not half your size, and he had great difficulty unless I had already spent three or four times and relaxed all those parts. I should not dare to let you attempt it.''
``Oh, yes my darling mamma, you will let me just get its point in and spend there. I should so like to try. We will fuck two or three times first, and then after the third time I shall frig you till you spend first, and so I shall be ready just to put in the point for you to try how it feels.''
``But, my dear boy, the least throb on my part will push it out, unless it is in over the nut, and only look what a size it is. I can hardly grasp it, and although it is so velvety it is quite hard. Oh, the dear fellow, let me kiss it, and then do you fuck me again, my darling.''
She bent her body, gave me a delicious suck, then throwing herself on her back, and opening her beautiful thighs, invited me to mount her. Before doing so I also bent and sucked her charming and well-developed clitoris, until she squealed again with pleasure, and begged me to put it to her. I threw myself on her belly, and with one vigorous shove drove my rampant prick up to the hilt, making her all shake again. She was so hotly wound up that she spent with the single shove, and poured a flood of hot liquid over my delighted prick. I, too, would have gone off in two more thrusts had she not thrown her arms and legs around me, and slipping her hands over my buttocks, held me tight pressed against the pouting and greedy lips of her salacious cunt as if she would shove in ballocks, buttocks and all, if it were possible. So keeping it tightly thrust in up to the lowest hair, which lay all crushed between us, I let her indulge in all the delight of perfect conjunction, responding to her delicious throbbing cunt with powerful throbs of my own highly excited prick. For more than a quarter of an hour did she lie panting and convulsively sobbing in the perfect ecstasy of enjoyment. At last she drew my mouth down to hers, and thrust her sweet tongue into my mouth; I sucked it, and her hands relaxing the pressure of my buttocks, against her cunt, I began a slow in and out movement that soon renewed her utmost lubricity. Most actively and divinely did she second me with an art quite her own. Fast and furious grew our movements, until, like all things human, they came to an end in a death-like agony of delight, in which my very soul seemed to take flight, and we lay all unconscious, for I don't know how long, enjoying all those exquisite after-delights which a prick soaking in the cunt of a beautiful and lewd woman so enchantingly confers. When we recovered, we rolled over sideways, and still intertwined and conjoined in the sweet priapic bonds, we lay billing and cooing with all those soft loving murmurings and bitings so befitting such moments. At last both were again ready, and longing for the fight. I proposed the delicious kneeling position. She saw at once my object, and said I was a little traitor, who wanted to surprise her bottom-hole.
``But, my darling boy, it is really impossible.''
I embraced, flattered, cajoled, and implored her until at last she promised that if I would engage on honour not to go further, she would try and support the entrance of my prick as far as over the nut, but that I must really withdraw it if it was too painful for her. So these preliminaries being arranged, she got into position. First stooping to lick out her delicious cunt, and give a suck or two at her charming clitoris, I brought my eager prick to the pouting and longing lips of her delicious cunt, and after two or three rubs, thrust it in with a rush that made my belly smack against her glorious backside. We then lay quiet, throbbing mutually in the luxury of voluptuousness. I passed a hand under her belly, and frigging her clitoris quickly, made her come in an ecstasy of delight. I only gave her time for one or two throbs of my prick, and knowing that nothing so much delights a lecherous woman as quick movements almost immediately after spending, I commenced rapid series of thrusts, shoving my prick well up to the hilt every time, and talking grossly all the while, such as---
``Does not that shove make you quiver? There you have it to the ballocks in your lascivious and delicious cunt,'' \&c.
She grew madly lewd, called me her own dear delightful fucker.
``Yes, yes; I feel it is up to the root. I have it well in, my dear boy. Your dear, great big prick, it kills me---kills---kills me-with-joy. Oh! oh! oh!''
She squealed again with all the lewdness of the most delicious spend. She had hardly gone off, and was yet in all the throes of delight when I, too, feeling I could hold out no longer, suddenly withdrew the reeking shaft, and bringing it to bear against the corrugated and beautiful orifice of her bottom, attempted to introduce it. Notwithstanding the fury of my excitement, I was sufficiently gentle to push in without force, and sheathed it over the nut without difficulty or drawing a murmur from dear mamma, who fulfilled her promise, and did her utmost to help me by pushing out her big bum, and offering no resistance with her sphincter muscles. I was so highly wound up that even if I had promised to be content with the insertion of the head, I could not have gone on further, as the access seized me with such killing sweetness that I melted away, shooting a torrent of sperm far up into her entrails, and then losing all power of even the slightest further thrusts. I suppose it was the long holding back to let mamma spend two or three times that had wrought me up to such a high pitch of nervous excitement that when I spent I seemed to lose all power of further advance. This was the first time I ever felt this momentary impuissance, but it was by no means the last; it generally follows the holding back your spending powers in the fuck that leads to it. The delicious throbbings of dear mamma's luscious cunt, which were repeated in her arms, soon reawakened my momentarily dormant powers. My prick had gone down more than usual, so that it was only a soft half stiffness that ensued, but enough to enable me to give it a forward movement, and it slipped almost imperceptibly in quite as far as he could go before dear mamma had recovered from the ecstasy of her last discharge. As she came to, I continued convulsively catching my breath, as if I were still in that exquisite sensation of half consciousness. I felt her pass her hand between her thighs, and heard her murmur---
``Why, I declare he is up to the hilt!''
Her gentle touch on my cods, which she took in her hand and fondly caressed, made my prick stiffen sensibly. She felt this, and caressed them more until she made it stand as stiff as ever, still imbedded to the utmost in that delicious bottom, which by its increased throbbings, seemed rather to welcome the stranger than repulse him. I pretended now to recover my full consciousness, and cried out---
``Oh, where am I? I have never known such heavenly joy.''
She raised her face up from the pillow---
``Why, you naughty boy, you have actually gone in up to the hilt; ah, you have broken your promise; but I forgive you, only don't move yet.''
I assured her I did not know how it got there, as I had spent and lost consciousness as soon as ever his head was within.
Here I throbbed, and was met by as delicious a pressure. I passed my hand round her belly, and found her clitoris stiff and excited. I rubbed with the fingers of the other hand at one of her hard projecting nipples. She soon grew madly lewd, and began a side wriggle on my rampant prick. I lay still, determined to let her passions demand movement of my part. I had not to wait long. She begged me to try a gentle movement, I obeyed, and slowly withdrew but a short way, and as slowly returned. Soon her lubricity got beyond all bounds. She begged me to draw out further and somewhat quicker---then quicker and quicker, until we both were in an excess of furious lust, which knew no bounds. We rushed on to the final crisis with mutual cries of agonised delight; indeed, mamma squealed so loud that I afterwards thought she must have been heard. Her pleasure was of the wildest, and when I poured a flood of sperm up her entrails at the very moment she herself was spending, we both fell forward and fainted away. I was too much lost in ecstatic joy myself to observe this, but lay long a tightly held prisoner engulphed in that most exquisite joygiving aperture. At last I became aware that mamma had really fainted. So drawing my prick out with somewhat of a good pull, for he was most tightly held, and came out with a flop, I rose and brought some water to mamma. I sprinkled her face, and she opened her eyes, which beamed the intensest love upon me. Her lips murmured something, I put the tumbler to her mouth, she drank with avidity. Then looking at me again with the most loving expression, she said---
``My darling boy, you will kill me with delight. Never---oh, never---have I known such joy. It was too much for me, and I fear I am also injuring you. We must be more moderate in future. Help me up, for I must rise. Your last coup requires me to absent myself for a few minutes.''
She rose, threw her robe over her shoulders, and left the room to go to the water closet. I hoped that she would not go into her own room and discover how matters were going on there. Fortunately she was afraid of awakening Ellen, and so prevent our continuing bedfellows for the rest of the night. She returned. I had purified myself in the meantime, and now acted as her \textit{femme de chambre}, and laved all the parts.
``My dear boy, we must not do this again for some time, do you know I have passed blood, and was very sore when relieving myself.''
We got again into bed. She would not allow of any further fuckings, but tenderly embracing me, and putting my head on her bosom, we soon fell asleep. She awoke me at dawn with kissing me and feeling my stiff-standing pego. She laid herself on her back, and we had two most delicious fucks without withdrawing. I knew that if I did withdraw she would take herself off. Nevertheless, she took most kindly to the second, as it would be our last until we had another opportunity of meeting. She exerted all her wonderful skill and her movements were of astounding agility. She twined herself round me almost serpent-like. Our mouths and tongues were equally engaged, and the final crisis was beyond description exquisite. I tried hard for a third course, but we had already prolonged our sports to so dangerous an hour, for we could hear them opening the lower window shutters, that she gave me a sweet kiss of thanks and tore herself away. I lay thinking over the joys of that ecstatic night, and then rose and dressed quickly, as we were to breakfast and then walk home, where we were expected at nine o'clock. However, after breakfast, mamma drew me into her sanctum, a house store-room, to give me some directions. Of course, no sooner was she there, than pushing her towards the table, I canted up her petticoats over back, and gave her a good fuck, getting in from behind. She yielded with a good grace, notwithstanding her protestations that it was not for that she had come, as if it had been for anything else! Oh! woman, woman! how thou seekest to deceive, even when gaining the very object thou hast in view.
Harry told me they had peeped in and seen what we were at but he was not so ready as me, and had not been able to go and do likewise.
We loitered all too long, and did not get back to school until after ten o'clock. The doctor sternly ordered us to attend him in his sanctum at twelve o'clock. We knew what that meant---a good flogging, and then the doctor enjoying the account of our successes. At twelve o'clock we entered the doctor's room, who followed us immediately after. He scolded us sternly for being late, and said he meant to flog us both well for our idleness and, he had no doubt, debauchery. We knew immediately that he meant to lay on. From time to time he was fond of really seriously flogging some one and we now saw that such was his present intention, although we also knew it would end in an orgie, after we had excited him sufficiently by recounting the details of the fucking which he no doubt felt certain had taken place. He made us all strip, and choosing to take Harry first he made me the horse to flog him on. When all was ready, he began by some real sharp cuts on Harry's backside, and then commenced his remarks.
``So, young gentleman, you have been seducing your cousin, have you?''---whack---whack---whack---``and then making that the excuse for neglecting your school.'' Whack, whack, whack. ``I thought I had formerly whipped out all idea about fucking your cousin.'' Whack, whack, whack.
Poor Harry writhed in real pain.
``Oh, sir, I'll never do it again without your leave.''
``My leave indeed!'' Whack, whack, whack.
The doctor now laid on for some time most unmercifully until the revolution of pain turned to lubricity, and Harry's cock began to stand, rapping fiercely at my bottom as he writhed under the sharp infliction of the rod. Upon seeing the expected effect, the doctor relaxed his severity, and changing the rod to his other hand, afterwards only tickled the bottom to keep up the excitement. Taking hold of the standing prick, he said---
``So this is the article that has been doing all the mischief.''
He frigged it a little, stooped and gave it a suck.
``Ah, yes, I find it still tastes of cunt, and smells the true odour of it; so you have been at it this morning again. Let me hear how it happened.''
Here Harry was let go. The doctor seated himself, Harry stood before him, while the doctor in delight, handled his stiff-standing pego.
``Now, let me hear.''
``Well, sir, when Charlie occupied mamma---''
``Oh, that is it, is it?'' cried the doctor, ``we shall have all that out of him, by and by, go on.''
``I slipped into Ellen. She made some difficulty for fear mamma should catch us; but I took her and showed her through the key hole, how she was having Charlie into her. Ellen was astonished at Charlie's immense size, and seeing how easily and delightedly mamma accommodated him, she thought that my smaller size could not hurt her, and she let me do it. But I made her scream and bleed when I got in far enough to reach her maidenhead. She tried to shake me off, but I was too firmly seated for that, and I fucked her then, and again before I withdrew. I laved her cunt and applied some glycerine, and this morning did it again without hurting her any more. And she liked it so much that afterwards she would kiss and suck it, and made me spend in her mouth, and then got me up again for a final go.''
``Upon my honour, a very pretty affair,'' cried the doctor. ``Now suck my prick, as she sucked yours.''
This Harry did, till the doctor was rampant. He then made him cease, but ordered me to mount on Harry's back. I knew I should catch it sharp, as the doctor was just excited enough to wish to be more so. And preciously he gave it me---interpolating questions as to how I had accomplished my wicked ends. I told him it was his own advice to me to let her see my prick, which I did, and the bait took. Whack---whack---whack.
``And did you act this innocent sin?''
``Oh, yes! do spare me, sir, and don't lay on so hard.''
Whack---whack---whack.
``Spare you, indeed! and how did she fuck?''
``Oh, most splendidly, sir.''
Whack---whack---whack.
``How often did you do it?''
``I hardly know, sir; we were at it all night, and again this morning.''
``Did she suck your prick?''
``Oh, yes, sir.''
Whack---whack---whack.
``What did she think of it?''
``She said it was the finest she had ever seen, and that I must keep it for her only.''
``Well, that will do, now suck my prick, as she did yours.''
He was soon excited up to the top of his bent. He made Harry take the rod, and belabour his backside, and I had to stoop over the table, while he fucked and frigged me, repeating all the time the account we had given him of our fucking. After he spent, he dismissed us, having gained his object. Shortly before our Christmas holidays commenced, dear Mrs. Dale informed me, while sleeping with her one Saturday night, that she found from the stoppage of certain things, she was in the family-way by this sad rogue of a fellow, taking my large though at the moment soft and inert instrument into her caressing hand.
``Oh, my darling mamma, is it so indeed?''
My prick rose to bursting point at the very idea, and in an instant I was on her, and we ran a most delicious course, in which both died away in rapturous insensibility. Being thus cooled, mamma began to discuss the probabilities, and what ought to be done, if it should turn out as she feared. She explained to me that as yet she could not speak with certainty, but remembering the fainting on the first night, and the cessation of her monthlies, the nature of which she explained to me, little dreaming that I was perfectly \textit{au fait} of the whole matter, she had every reason to dread that her fears where too well founded. This would make it necessary for her to go abroad, when she would be so far advanced as to be likely to draw observation. But she said it would not do to distress ourselves about that until we were more certain of the event. However, the very idea nerved me to renewed efforts, and again, and again, we rushed into all the ecstasies of passion in every form and way, especially did I gamahuche and suck up her precious balm, and in like manner she, too, sucked me until exhausted nature laid us both in the lap of Morpheus. We renewed our delightful pastimes when morning light awoke us after our refreshing slumbers. Several times during the Sunday we adjourned to mamma's bedroom for the same purpose, and again had a glorious night of it before separating on the Monday morning. The following Sunday, after another Saturday night of bliss, we all went over to church, which heavy rain had prevented on the previous week, and after service went to the rectory for luncheon. Here, in course of conversation, Mrs. Dale mentioned that business would require her presence in London for some days, and that she proposed starting on the following Thursday, which was the day after our breaking up for the holidays. She said also that she would take her son with her to London. The doctor here observed that he, too, must go to London, to see a gentleman who had some idea of sending his son to the rectory, and if Mrs. Dale could defer the departure until Saturday, it would be very agreeable to him to be her companion on the journey. This was readily acceded to, and my dear aunt, who guessed to what this tended, and who had herself taken a great fancy to Ellen, and longed to embrace her young charms and gamahuche her, chimed in with a proposal that as the dear girl would thus be left quite alone, she would be most happy if she would accept her invitation to occupy the bedroom that opened out of her own room during Mrs. Dale's absence. The latter, who little dreamt of my connection with her dear niece, and thought that the protection of my aunt would be a safeguard to her, jumped at the invitation, and expressed her gratification and thanks for so kind a consideration on my aunt's part. I have not alluded to Harry all this time, but of course, whenever his mother and I were occupied in amorous alliance, he was equally engaged in the same delicious pastime with Ellen. And, I may add, that once or twice I had seized a favourable opportunity of gratifying the little lecherous creature with what she called a feast of my noble prick. She, of course, was delighted at my aunt's proposition as she at once foresaw how she would have me all to herself for more than a week. A single glance from her explained all this; and when, on leaving, she found an opportunity of taking my hand, her pressure of it was most eloquent. So all parties were delighted, for Harry, when we got together alone said---``By Jove, Charlie, I am so jolly glad; I'll bet you anything I'll fuck my mother before I come back. You know how I long to be in the delicious cunt that bore me; the moment I heard she meant to take me with her, my cock stood ready to burst.''
My uncle, too, who also longed to fuck Mrs. Dale, had his intentions in that direction favoured by the arrangement concluded. The following night, when I was in bed with aunt and him, in the interval of a charming little orgie, and after fucking me while I was in aunt's bottom, and for the moment he could do no more, the conversation turned on the coming journey. He expressed the pleasure he felt at the opportunity it gave him of indulging in a long desired object. The lecherous old fellow also alluded to a future opportunity it would give him of enjoying the younger charms of the niece.
``Of course, you and my dear wife between you will break her in to allow of any action on my part; and, by the way, my dear, I would suggest that you should surprise Charlie in the act, and tear them asunder in pretended rage---that Charlie should seize you, and say he would make you by force a participator in the act, on the pretence of shutting you up for finding fault: you must break from his arms, and fly to your own bed, he must catch you as you try to enter it, and push his great big cock into you, on which you must cry for help, and call upon Ellen to come to your succour; she will come, but I do not judge her right, if she will not rather assist Charlie, by holding you, than otherwise. You must afterwards appear much offended; but it may be safely left to the influence of Charlie's great prick to reconcile you to the incest, then relaxing, as if gained over by it, you can join in their sports.''
Thus this admirable man, with his great knowledge of the world and sex, gave us excellent advice, which, as I shall state in the sequel, we followed pretty exactly. Meanwhile aunt, excited by expectation, had taken my prick in her mouth, and sucked it into firmness, then mounting upon me, she began such an exciting action, wriggling her magnificent backside, that it fired my uncle anew. Finding his prick stood sufficiently stiff, he knelt between my legs, and greatly to the satisfaction of my darling aunt, gave her the double pleasure of two pricks fucking her at the same time, one before and the other behind.
My guardian had desired that I should continue with my uncle during the holidays, and I was to leave him altogether at the end of the next half. I did not know his object at the time, but I found that he himself went down to my mother, and stayed for a fortnight, paying great attention to Miss Frankland. He announced his wish that my sisters should go to a first-rate finishing school in London in the summer, and seeing Miss Frankland look somewhat disappointed, he sought an interview with her, and laid himself and his fortune at her feet; expressing a wish that if she accepted him, their marriage should take place on her separation from her pupils. This was too good an offer to be refused, and after the usual grimace of being perfectly unprepared for such a proposal, and desiring to have a day or two to consider it, she accepted the offer. I at once anticipated immense gratification from this connection. I should naturally, when in London, have every opportunity of enjoying that adorable creature, and it will be seen in the fourth volume of these memoirs, to what delicious orgies this connection led. You may be sure that my loved mistress, the adorable Benson, and the no less lascivious Egerton, would welcome so glorious a creature as Miss Frank-land, at that time become Mrs. Nixon, and how the Count's eyes glistened, when he beheld her in all the majesty of her superb and hairy form; how the two women gamahuched her splendid clitoris, and how the Count and I strove which should most fully satisfy her lascivious and lustful passions. But all this will be seen in its proper place in the sequel.
Meanwhile the day arrived for the departure of uncle and Mrs. Dale, with Harry. As the coach passed through our village, Mrs. Dale drove over bringing Ellen with her, to leave her at the rectory, as arranged. All the proprieties were duly observed. They departed, Harry going outside, with only the doctor and Mrs. Dale in the interior, I squeezed my uncle's hand, and gave him a knowing look, which he returned, with a meaning wink---and off they went. When we returned to the house, and aunt took Ellen up to the room adjoining her own, with which there was a door of communication, and which, I have before observed, had been made use of by uncle on more than one occasion. When they came downstairs, with kind consideration, for she could see by the protrusion in my trousers the state I was in, aunt said---
``My dear, I have some household duties to arrange, so you must excuse me; meanwhile Charlie will show you our grounds, and amuse you for an hour or two. When luncheon is ready I shall order the large bell to be rung for you.''
Ellen had not yet removed her bonnet, and taking up her shawl, we sallied out. You may be sure we lost no time in reaching the summer house, already known to you as arranged for and dedicated to the service of Venus. A fire was always kept laid, which I immediately lighted, but as it was a bright sunny day, and the place looked south, it was not at all cold. While I was occupied at the fire, Ellen threw off her bonnet and shawl, and undid her belt---she wore no stays. I seized her in my arms, and gently laid her on the couch---her petticoats were freely canted up, showing her beautiful belly and now more fully fledged cunt. I stooped and gamahuched her at once. She was so excited that in two minutes she sighed deeply, pressed my head down to the lips of her cunt, and gave down her sweet and balmy sperm. I myself was already so rampant that not waiting to lick it up, I brought my huge pego to the charming orifice, and plunged in one effort up to the hilt, quite taking away her breath. But she recovered herself in an instant, and with all the energy of her younger lubricity, quickly brought us both to the grand final ecstasy in which soul and body seem to die away in a joy too great for poor humanity to bear. We remained locked in each other's embrace, and lost to all around for some time. On coming to our senses I rose, and said we must go to work more lasciviously the next time. The fire having burnt up, and the room being small, it was already of a pleasant temperature. So begging Ellen to strip, I threw off my own clothes, and we quickly stood in all the beauty of nature, admiring each other. Some delicious preliminaries preceded our next encounter, which we procrastinated till passion could no longer be restrained, and again we died away in all the raptures of satisfied lust, and sank once more into the soft languor of the after-enjoyment. Next we had a mutual gamahuche, and then a final fuck for the present, as it was time to dress and be ready when called to luncheon. As soon as our toilets were finished, I took her on my knee, and told her how I should steal along to her bedroom at night, so that she must not lock her door. I told her also that we must be as quiet as possible, as aunt slept in the next room. She was delighted with the prospect of having me all to herself for the whole night, naively telling me that I gave her so much more pleasure than Harry did, that I seemed to fill her whole body with a joy almost too intense, and now that she was to have me every night, she hoped her aunt would stay away for a month. Here the dear creature threw her arms round my neck, and kissing me, thrust her sweet little tongue into my mouth. You may be sure I reciprocated, and putting a hand up her petticoats, and a finger up her charming little cunt, was just about to turn her on the sofa, when my aunt opened the door, and stopped further proceedings. She pretended not to see Ellen's confusion, hoped I had amused her, and told us to return to the house, as luncheon was ready. We, of course, obeyed. With sharpened appetites, produced by our late warm exercise, we indulged in a plenteous meal, aunt taking care to ply me with Champagne, for which, as may well be imagined, she had her object. She afterwards ordered me to my room, to do the daily task the doctor had set for me and which, as she said, she was to see to the doing of---giving me a sly wink.
``Ellen, my dear,'' she added, ``you must keep up your practice at the piano daily, for an hour and a half at least.''
She thus separated us. I went to my room, lay down, and fell fast asleep, but in about half an hour, was awakened by the warm embrace of my glorious and wantonly lustful aunt. She stooped down, and taking my limp prick in her mouth, rapidly sucked it into its accustomed firmness. As soon as that was accomplished, she begged me to rise and undress. She herself had come only in a loose morning dressing-gown, which she instantly threw off, and jumped on my bed, where she lay stark naked, in all the splendid development of her superb form. I was naked in a jiffey, but knowing she would want some extensive fucking, I threw myself upon her cunt, and gamahuched her until she spent twice before I mounted upon her, and introduced my large tool into her longing cunt. Here, also, I played with her, and did not spend myself until she had twice given down her own contribution. This encounter was on her belly, with her magnificent legs twisted above my loins for a fulcrum to her splendid action, for few women could equal her in the delicious wriggle of her glorious backside. After we had soaked for some time in all the ecstasies of the after-languour. I withdrew, to place her on her hands and knees for the next bout, but took advantage of her position to gamahuche her again into spending twice before I withdrew my insidious tongue. Then turning round, and gazing in rapture on that most noble and massive bottom, which, as I have before remarked, I never saw equalled by any woman, I stooped, and closely embraced and kissed its divine orifice, tickling her into wild excitement by thrusting my tongue therein, so much so that she begged me to fuck her at once. I mounted behind, her hand passed under her belly and guided me into her throbbing hot and longing cunt. I gave one violent lunge, and sent my prick at the first thrust up to the hilt. This so excited the dear creature that in one or two delicious wriggles on my stationary prick, and with a pressure that seemed as if it would nip it off, she spent profusely, squealing all the time like a rabbit. I was very glad to give her so many discharges, without myself being forced to spend, for I wished to be able to do my duty by Ellen at night. Aunt lay for several minutes panting and throbbing on my prick most deliciously, until I could no longer bear to be inactive, although the pleasure of looking down on the glorious and palpitating orbs below me had given me the greatest satisfaction. But now stooping down upon her, I passed one hand under to excite her clitoris, and with the other took hold of one of her beautiful large and hard bubbies, and began manipulating its nipples---a proceeding most powerfully exciting to dear auntie. It awoke all her lust and the dear lascivious creature again spent before I was ready to follow suit. The pause that followed allowed my excitement to subside a little, and enabled me to hold out until her lust recovered its wonted energy. She again, with her pressures and movements, soon compelled me to more rapid action, but this time I determined to enjoy the exquisite delights of her delicious bottom-hole. So when she became very hot, I suddenly withdrew, and, happily, hitting at once on the delicious orifice, plunged at the first thrust up to the cods, taking dear aunt's breath away, but she instantly recovered, and loving sodomy to her heart's core, I could not have done anything better suited to her libidinous passions. It was glorious to see the energy with which she met and responded to my thrusts, her superb buttocks working with surprising energy, and giving me, at each stroke, when I buried my prick to the hilt, the most exciting pressures. Both being so lustfully excited, matters were not long in coming to the final ecstasy. I felt as if my whole soul was poured into her, when with loud cries of the liveliest enjoyment, I spent with fury, in the very heart of her entrails. She was perfectly overcome with delight, and sank senseless on her belly, dragging me down with her, for her grip by the sphincter was too strong to let anything out that was within. We both became insensible to everything but the delicious death-like languor of the after-enjoyment. We lay long in this trance of joy, and when dear auntie came to her senses, she begged me to rise, as she must go downstairs. I did so, and when she rose from the bed, she took me in her loving arms, and kissing me tenderly, thanked me for the enormous pleasure I had given her, and said no one in the world was my equal, and that I ought to thank her much, that she allowed any one else to participate in my exquisite power of fuck. She gathered up her gown, and left me to dress. I soon was downstairs, and found Ellen, who looked as if she expected me to find an opportunity to fuck her at once. But after the encounters I had already had, both with-her and with aunt, though I had kept myself from excess with the latter, I felt no inclination to press matters again to a conclusion, especially seeing that I intended passing the night with her. So assuring her we should be likely to be caught if imprudent and so lose all chance of night work, she was satisfied to be quiet and reasonable. Aunt coming in, we spent the afternoon in pleasant conversation, and a walk together in the garden. After dinner I fell sound asleep on the sofa. The two women, each with the same object, left me to my deep repose, and only awoke me when it was time for all to retire. Thus refreshed, I was all ready for the night's work before me. I allowed half an hour to elapse, that all the house might to be in their bedrooms, and then, with merely a loose dressing-gown on, I stole along to dear Ellen's room, opened the door and entered. She was already in bed, impatient for my arrival; she had left both lights burning, as well as a cheerful blaze from a good fire. I dropped my robe, and was in an instant stark naked, and in her longing arms. Under our mutual impatience, our first was a rapid course. Then followed a long enjoyment of the after-languor, and then a more prolonged and rapturous embrace. After soaking in bliss for some time, we rose, and I posed her before the fire, gazing delightedly on all her young charms. The hair on her cunt had become much more developed than before, her bosom too was filled out, even her hips and bottom seemed enlarged, doubtless owing to the fucking she had had since I first knew her, which naturally hastened her ripening into womanhood. I grew very excited by this inspection of her increasing charms, and determined to have a fuck on the rug before the fire. In order to enjoy it the more, I drew forward a cheval glass, projected it forward, and lying down, directed her to move it until I was satisfied I could see all the play of her bottom in the position I meant to fuck her. So lying down on my back, I made her stride across my head and settle down on her knees, and bringing forward her delicious little cunt over my mouth, I gamahuched her until she had twice given down her balmy essence. Then she shifted her position lower down, until just above my prick, which by this time was rampant with desire. I guided its point to the rosy-lipped orbit, and bringing her own weight to bear upon it, she sank delightfully impaled upon the upright stake. I made her rise and fall a few times, that I might enjoy the sight of its entrance and exit. Then gently drawing her down upon me, I folded one arm round her slender waist, and turning my head, found that the cheval glass, inclined forward, reflected as it were from above her beautiful bottom and back, and of course her cunt stretched to the utmost with my huge prick, and above it the sweet little corrugated pink aperture of her bottom. With my free arm I embraced one hip, and bringing my hand round, moistened it with the plenteous spunk of her cunt, and insinuated a finger into the smaller abode of bliss. Her excitement grew furious, and knew no bounds. The action of her backside was glorious to see reflected in its active risings and fallings. I let her do all the work, which enabled me to hold back my own, until she approached a second discharge, when the heat of her cunt seemed to fire me with additional powers, and the action of both our backsides became fast and furious, and soon brought down the ecstatic discharge, which instantly laid us low, panting with all the wild passions we had just allayed. We lay long locked in each other's arms in the ecstasy of blissful enjoyment. Then rising, we embraced tenderly, and retook us to bed. I would have excited her and myself to another effort, but she begged off, saying that she felt quite exhausted and overcome with the day and night's work we had already enjoyed. Indeed, I did not wonder at it, for I had made her spend seven or eight times more than myself. Nor did I regret her resolution, as I knew the morning would bring my aunt into the field, and then the two would try my powers to the utmost.
We slept profoundly, and morning was already advanced before we awoke. From a displaced chair I saw that aunt had been in to look at us, so I knew she was on the watch. I threw the clothes off dear Ellen that I might gaze on all her young charms. The want of covering awoke her. She lovingly looked up at me, and throwing her arms round my neck as I bent over her, drew my head down to hers, and impressed a loving kiss on my lips. Our tongues interlaced---a hand slipped down and encircled my rampant and throbbing prick. I turned, and placing my knees between her legs, was about to penetrate love's bower when the door leading to my aunt's room flew open. My aunt entered, gave a scream of surprise---well acted---and cried out---
``Good gracious! What do I see? Who would have thought it-''
And, apparently to save Ellen, she rushed forward, seized me by the arm, and with a certain degree of willingness on my part, drew me out of bed, saying---
``I am horrified beyond measure. How dare you commit such a sin and crime as to seduce a young girl under my care? Cover yourself up, sir, directly, and go to your own room.''
I boldly declared I would do no such thing; on the contrary, as she had spoiled my sport with Ellen, I was determined she should pay for it herself.
``How dare you talk to me, you dreadful boy?''
``Not dreadful at all, dear aunt, look at this poor dumb thing, and see how he longs to be into you.''
Upon this I seized her in my arms as if to throw her on the bed. She made a pretended struggle, during which she gave a tender squeeze to my rampant prick. Then, breaking from me, she fled to her own room, pretending to endeavour to shut the door in my face but taking care to give way and hasten towards her bed. I caught hold of her as she bent forward as if to get into it, and canting up her chemise, the only article of dress she wore, I was into her longing and luscious cunt from behind up to the hilt in one thrust. She gave a subdued scream, and called to Ellen to come and prevent me from violating her. Ellen came, but wisely would only look on while I worked away manfully.
``Ellen, why don't you pull him away---he is ravishing me---and oh, horror!---committing incest.''
She pretended to struggle greatly, but cleverly did so to her own profit, by wriggling her backside so as to send me further up into her cunt.
``Oh, Ellen, Ellen, do help me.''
``Ah, no,'' said Ellen, ``I shall let him do it, and then you cannot tell upon me.''
My aunt seemed greatly distressed at this, and actually managed to shed tears, then buried her face in the bed as if in despair, but all the time most actively seconding me. As the crisis drew near, she raised her head, and said---
``Heaven pardon me, this mere simple schoolboy is exciting me to such pleasure as I never before felt.''
She then gave way to all her lubricity, and we brought matters to a crisis in the utmost ecstasy of enjoyment. Aunt's head sank on the bed, while the rapturous inward pressures of her cunt soon began to raise my prick to its pristine vigour. She felt its throbs and responded to them, but no doubt thinking that an immediate repetition would betray our previous intimacy, she turned her face and body suddenly round, and completely unseated me, my prick coming out with a plop. She began again to weep, women can do so at pleasure, and to scold me for the dreadful crime I had committed; to do so to her was incest---here followed sob upon sob. I threw my arms round her neck, and kissing her tears away, laid all the blame on that rampant fellow---taking her hand and placing it on my still stiff prick. She drew her hand away quickly, but not before she had given it a gentle squeeze. She told me I was a dreadful boy, and that I must go away and leave her and Ellen to think over what could be done in such an awful dilemma.
Here Ellen came forward, and tenderly kissing her begged her not to send me away.
``I do so love him, dear madam, and I do so long to have him now---it was so exciting to see him having you, that I shall die if you don't let me have him now.''
``Dreadful! dreadful!'' said aunt. ``Why, I thought I was just in time to save you.''
``Oh, no, he had slept with me all night, and has often had me before, but he was not the first who had me, so that there was no violation nor seduction.''
``Then you must have seduced him, you wicked minx, for a more innocent boy never was known.''
Poor Ellen, confounded at the accusation, repelled it as untrue, and said she knew well enough who seduced me.
Aunt for the moment felt this as a home thrust, for be it remembered, she fancied she had had my maidenhead.
``What do you mean by that? I insist upon you speaking out.''
Ellen gave way and said it was Mrs. Dale who first had me.
``She had accidentally seen how powerfully Charlie was armed, and then could not resist teaching him how to use his weapon. I saw them doing it, and hence I longed for it myself. Look, dear madam, what a noble one it is. I am sure, if you had known of it, you could not yourself have resisted having it, try it, try it once more, and I am sure you will forgive us, and share our joys.''
I seconded this good advice. Aunt seemed to be afraid of me, and jumped into bed. While she was on her hands and knees I also jumped up, and catching her round the waist, held her fast until I could also kneel behind her and bring my prick into play. With all her apparent attempt at resistance everything was done in such a way as to facilitate rather than prevent matters going forward. Of course I was in her in a moment, and then remained quiet for a few minutes to let her enjoy her inward pressures for which she was so famous. She had buried her head in the pillow, crying out---
``It is dreadful!-it is dreadful!''
Ellen came and leant over the bed embracing her, and telling her not to resist, but to take it in freely, and then she was sure it would give her the utmost pleasure.
``It is that which horrifies me, my dear, I never felt anything so exquisite in my life before, but then think of the sin---with my own nephew! it is quite an incestuous connection.''
``What does that matter, dear aunt? for I shall call you aunt too, you are so loveable and so beautiful. Oh, it was such a pleasure to see him doing it to you and you are so gloriously fine a woman, I longed to be a man to have you.''
She had embraced aunt's splendid bubbies, than which nothing could more please her, and now she begged to be allowed to suck one. Aunt gave way, and was delighted. She slipped the hand next to Ellen down to her charming cunt-Ellen opened her legs---Aunt's fingers began frigging her.
``Ah, my dear, how I loved to embrace my own sex at your age, our tongues acted instead of men, and I could still delight in a fine fresh one like this, it would almost reconcile me to what this bad wicked boy is doing.''
``Oh, that would be charming!---do let us do it at once. Charlie can withdraw for a moment while I get under you, and while you lick me I can excite you and see the glorious work above me.''
``You tempt me much, my dear girl, but what would your aunt say if she knew?''
``But she never will know,'' said Ellen, who was all the time arranging herself on the bed.
Aunt moved aside to allow Ellen to get under her, who then begged aunt to throw off her chemise that both their bodies might be in close contact. Aunt was longing to do so, yet made some grimaces about it. She at length complied, and striding across Ellen, threw herself with avidity on the delicious young cunt below, and began to gamahuche her a mart. I instantly resumed my position. Ellen guided my prick into aunt's burning cunt, then frigged aunt's clitoris, and worked a finger in my fundament, while aunt was so delightfully gamahuching her. We all rapidly came to the grand finale, with an excess of lubricity rarely equalled. We were all somewhat exhausted by this bout, and, as it was getting late, we rose. Aunt pretended to forgive my violating her for the pleasure I afterwards afforded her. She embraced Ellen tenderly, and said she had so enjoyed her person she hoped to renew such a delight. Then taking hold of my prick she kissed it and sucked it until it stood upright, and said---
``I don't wonder, my dear, at your having it when once you had seen it, and I envy Mrs. Dale the pleasure of having first enjoyed such a monstrous thing. If I had known he was so wondrously provided, I doubt if I could have resisted the temptation to teach him how to make use of it myself---my only wonder is how such a little thing as you have got could ever take it in.''
Ellen laughed, and said that her cousin Harry had opened the way, or she doubted if ever she could have admitted it, but I was so gentle while getting in, and when once in, it filled up every crevice so deliciously, that she should grieve much if she were refused access to it in future.
``So, dear aunt, I hope you will let him do it to us both. I can do to you what you have just done to me, because before we had him and Harry, aunt and I used to amuse ourselves in that way. Aunt is immense in that particular, she could put it a little way into me, and gave me great pleasure, and she said that I sucked it better than either her late husband or any of half-a-dozen schoolfellows who used to amuse each other; so, dear aunt, you must let me do it to you while Charlie is in me, and then you will do it to me while he is in you. Only fancy how nice it will be.''
``Oh, you dear little coaxer, you are enough to seduce an angel.''
So all was arranged that Ellen should come from her room and I from mine, and meet in aunt's bed at night. We did so meet, and a most glorious eight days we spent. I showed aunt that I could get into Ellen's bottom-hole, and thereby gave her immense pleasure, and with more reason the same result would occur with her. She gave an apparently reluctant consent, and, that done, there was no bridle to the utmost lubricity that the most wanton lust could devise. Aunt took immensely to Ellen, and gamahuched her \textit{\`{a} mort}, while the other repaid her in kind. I did not regret this for it relieved me from too excessive work. Thus we passed a most delightful eight days before the absent ones joined us. Both uncle and Harry had succeeded in their desires. From each I had the fullest details, but as their stories would in some particulars repeat themselves, I shall relate the events in a connected narrative.
Uncle and Mrs. Dale had the inside of the coach to themselves, Harry riding outside. Uncle began by praising Harry; and then reverting to the time he was first sent to the rectory, and the note Mrs. Dale sent with him; he asked, not without a knowing smile, if the intimacy she had formerly feared had been at all renewed, because he had observed that Harry appeared worn and pale on his return on the Mondays, and was dull and stupid that day. Mrs. Dale seemed somewhat alarmed at hearing of this, probably she began to think that something might have occurred between the cousins while she was busied with me, uncle observed her uneasiness, and, guessing the cause, said---
``My dear Mrs. Dale, if anything has taken place, and anything comes of it, I am a man of the world, and you may rely upon my assistance and discretion to take such steps as may tend to keep it from the knowledge of the world.''
She thanked him, and said she would be glad to accept his aid if any unfortunate event should have happened---but she hoped not.
Uncle saw that her fears were excited, so he held onto the subject, so at last she avowed the she feared there might have occurred some passages between the two cousins, for she had foolishly trusted that all thought of that had gone out of their heads, and she might not have taken such precautions as she ought to have done.
``Well, my dear madame, my services are at your disposal in case of any necessity, I am not in reality strait-laced, although, in my position, I am obliged to appear so. I feel certain that my experience would be able to suggest the best way of hushing up the scandal if such should be likely to occur.''
Mrs. Dale was profuse in thanks, and the doctor became warmer in his discourse, saying that for such a woman as herself, whom he had long admired and coveted, he would do anything.
``For, my dear madam, though I am in the church, something of the old Adam still adheres to me, and the sight and touch of one who has so charmed me as you have done makes a young man of me again.''
Here his arm glided round her charming little waist. He drew her to him, and with some coyness and words of refusal, she yielded her lips to his embrace. His other hand, lifting up her petticoats, sought to feel her beauteous cunt. Again resistance of hand and tongue, but a yielding for all that, and the doctor soon got possession of her lovely cunt. Finding her large and fine clitoris in a state of stiffness, he knew that her passions were excited. So opening her legs, he got between them down on his knees, and as he previously unbuttoned his trousers in readiness and the fresh cunt stimulating his powers, he pulled out his prick fully erected, and quickly established himself up to the hilt within, the lady up to the last declaring she could not allow him, but wriggling her bottom to perfection as soon as she felt the doctor's very fine prick working within her. She then hugged and seconded him, kissing and tongueing to his heart's desire. They soon brought things to the ecstatic conclusion, to the great satisfaction of both parties.
Of course, after this there was no difficulty in arranging for a comfortable meeting in London. Indeed, it was resolved that they should lodge in the same house and have contiguous apartments. On their arrival in town they put up at one of those large lodging houses in Norfolk Street, Strand, and were fortunate in finding the first-floor bedrooms vacant. The house was a double one, or rather two houses opening into each other. The doctor's bedroom was in the front, and a former door of communication with the back room was locked on one side and bolted on the other. Mrs. Dale took the back room, from whence opened a small room with a bed in it, where Harry was lodged. The doctor had thus easy access when the lady chose to withdraw the bolt on her side. After consultation it was thought more advisable that she should go into the doctor's room, so that Harry might not by any possibility, hear any love exclamations that might happen to escape them in the excess of their amorous amusements. Of course, the doctor, who knew all about Harry's great desire to fuck his mother, and that he meant to do so by one way or another in London, communicated his intention of having Mrs. Dale into sleep with him that night, and, therefore, begged Harry to defer his attempt until after the first night, and then the doctor would aid him in his efforts.
The wily doctor fully intended, after Harry had perfectly succeeded, to become the future companion of their incestuous intercourse. Harry's bedroom door had one of those old-fashioned brass locks that were screwed onto the inside of the door, with a brass covering for the bolt at the side---not morticed as is now usual. Mrs. Dale locked her son in after he retired to bed. Harry noticed the circumstance and smiled to think how easily he could foil her but as he had promised the doctor to make no attempt on his mother that night, he went to bed and slept soundly. Next day he provided himself with a turn-screw and a small phial of sweet oil. When mamma was busy at cards, he slipped upstairs and easily unscrewed the brass receiver of the bolt, he oiled the screws and worked them in and out until they went freely and then screwed the covering on again, and felt secure of entering mamma's room whenever he pleased. It had been combined between the doctor and him that by means of gamahuching and frigging, mamma should be put into a state of great excitement without allowing her to be satisfied, so that her passions might be in favour of being fucked, no matter by what prick. For this purpose the doctor was to keep her with him till dawn. At night Harry watched through the key hole, and when he saw his mother pass into the doctor's room, he at once unscrewed the covering, shot back the bolt, and screwed the cover on again. He was thus all ready for any event, and if his mother was astonished at his entrance, he could say he found the door open, and she must have forgotten to lock it. Thus prepared he went to bed and slept soundly. He was awake before seven o'clock, and gently opening the door a little, he could see by the opposite open door, and the light in the doctor's room, that mamma had not yet left him. He drew on his woollen socks, and sitting where the light flashed through the key hole, awaited his mamma's return, which occurred very shortly after. The shutting off the light by closing the door of communication told him that she had returned to her own room. He heard her sit down on the pot, and the force of the flow of water proved how healthy she was. He heard her rustle into bed. Then throwing off his dressing-gown and socks he opened the door and approached his mother's bed. Being awake, she instantly saw him in the half-daylight that came from the unshuttered window.
``Harry! What on earth brings you here, and how did you open the door?''
``I heard you moving, dear mamma, I could not sleep for the cold. I got up and tried the door, it was not locked, you must have omitted to turn the bolt, but I should have rapped and called to you, if it had not been open. I want you to let me get warm in your nice warm bed, and you will cuddle your poor Harry---will you not, dear mamma?''
``If you will be quiet, and speak lower, for the doctor may hear you, you may come in, and if you turn your back, I will warm you.''
Harry lost no time in lying down by her side, and being really very cold, and even shivering, he was glad enough to do as she bid him, and turn his back, and cuddle his bottom into his mother's belly. She said---
``Poor boy, he is indeed cold, now go to sleep in mamma's arms.''
Of course, he had no such intentions. Speedily getting warm, he turned his face to mamma, and whispered, in the same tone she used---
``Oh, how I love my beautiful mamma.''
Pressing his belly against hers, and letting her feel his prick standing against her mons Veneris.
``Harry! What do you mean by embracing me in that way---don't you know I am your mother, sir?''
He had seized with one hand her beautiful firm bubbies, and was evidently in full amorous excitement, as she could feel by the stiff pego pushing against her mount of Venus.
``My darling mamma, if you knew how much I love you and how I have longed to embrace your beautiful body.''
``Go along, you impudent boy, do you not know it would be sinful to indulge in such sentiments with your mother-leave me directly.''
``Oh, no, mamma, I can't, indeed, my own mamma. I mean to possess you, what harm can there be in returning to whence I came.''
Here he transferred his hand from her bubby to her splendid mons Veneris, and showed what his words meant. She pretended to be very angry, and endeavoured to push him away, but he held her round the waist with his other arm too well.
``Desist this instant, or I shall cry out.''
She really appeared very angry but, nevertheless, did not excite a whisper during all the colloquy before or after. Harry now thought of his best argument.
``Why do you attempt to repulse me in this way, dear mamma? Why should you not let me enjoy your person as much as you like Charlie to do it?''
She gave a start at this home thrust.
``What do you say, you naughty boy? and where did you hear such a falsehood as that? is that one of your friend Charlie's inventions, after all the kindness I have shown him?''
``My darling mamma, Charlie never opened his lips to me on the subject. I speak from what I saw with my own eyes.''
``What do you mean? Tell me directly.''
``Well, my loved mamma, do you remember the first Saturday night that Charlie and I slept at home: after retiring to my room, I was obliged to go downstairs to the water closet, where I went in my stockings, and without a light, not to disturb you. I was coming up again, when a sudden flash of light shone out in the upper passage. Mounting the stairs, and when my head was on a level with the upper floor, I saw you going towards Charlie's room. I went into my own, but left the door open to see when you would return; finding you did not come back, I crept softly along the passage, until I came to the turning that led to Charlie's room. The light shone through the key hole. I quietly approached. You know the bed exactly faces the door---and there, my darling mamma, I saw you initiate Charlie into what was to him a previously unknown pleasure. Oh! my beloved mother, the sight of your naked charms, of the delicious way in which you were giving him his first lesson in love, maddened me with desire. I was almost tempted to come in upon you and violate you, if you would not consent. It was in that state I remembered that Ellen was asleep in your bed. I ran there, and throwing off the little I had on, I lay down beside her, and began feeling her private parts. She awoke and said---
```Dear aunt, do you wish me to do the same to you?'
``Her hand passed down to my erect member, she gave a cry of astonishment. I whispered it was only me.
``'Oh! you must leave me directly. Aunt can only have gone to the water closet, and will be back directly.'
``She was not to be pacified until I convinced her that there was no chance of your speedy return, so I was obliged to bring her along to Charles's door; we saw you quite naked, rising and falling on the enormous weapon that Charles has. I never before saw it erect and could scarcely believe my eyes; nor was it less wonderful the way in which you so charmingly took it in. It greatly excited Ellen, as well as me. We returned to your room---the fire still burned. I laid her down on the rug before it, and took her maidenhead. She had seen how Charlie's monstrous affair went easily into you, and felt how much less mine was, so she never dreamt of it hurting her, and she let me get fairly within the lips; then, while making her spend, I suddenly thrust it through all impediments, and the affair was done; she gave a scream, as it hurt her, but I had shut the door and none of you heard it. I let her sleep after this, and did not do it again till morning. The next night we again watched your delicious proceeding. Ellen was less sore, and we repeated your example several times. She continues to this day to wonder at the enormous size of Charlie's tool, and is surprised at your taking it in so easily. But, oh, my mother, how my passions have been excited by your glorious charms. What is Ellen compared with you? She did very well to relieve my agony of desire to possess you, when I knew you were better occupied, and that I could not do so---but that is all. It is you, and you alone, my beloved mother, whom I adore, and I wildly long to' possess this dear and magnificent cunt beneath my hand.'' Mrs. Dale was perfectly flabbergasted at this recital. ``You abominable boy, how dared you to follow me, and be a spy upon your mother, and to make it known to Ellen, too; doubtless you have been boasting of it, and telling others.''
``No, indeed, mamma, Ellen and I were on oath that we would never reveal to any mortal the delicious sight we had seen---so you see, darling mamma, that you can fully trust your own boy. Oh, do let me do it; feel how my poor thing throbs.''
Here I must give you Harry's own account of what took place.
``I took her hand with very little resistance, and I could feel her fingers gently clasped my prick, before she withdrew her hand.
``'But no---it cannot be---it would be incest.''
``She twisted her body round, so that her magnificent bottom came against my belly. As she turned, I slipped my hand down, and laid hold of her shift, so that in turning, it left her bottom bare, and sticking out against me. I lost not an instant, and before she had quite settled down, I brought my stiff-standing pego against her delicious cunt from behind, and as it was reeking from her previous spendings produced by the rector's gamahuching, I plunged it at one shove as far as her buttocks against my belly would allow, at the same time dropping my hand from her waist to her cunt, so that when she sprang forward, as if to turn me out, I met her clitoris, it was quite stiff, showing her to be really in a state of amorous excitement. This attack on the clitoris made her as quickly move back, which double movement thoroughly engulphed me. I lost no time in proceeding to the most active movements in and out. This was too much for her, she could not resist entering into the encounter with all the force of her passions, and we ran a very rapid course, ending in the most ecstatic delight, and with sighs of joy we lay clasped together in all the delicious after-languor. I could feel by her exquisite internal pressures that her lust was not yet alleviated, and this nerved me to fresh efforts. After a feigned resistance, dear mamma passed her hand behind her, and putting it on my buttocks, assisted in sending me further in at each home thrust. We were longer this bout, and enjoyed it more. After the usual indulgence in the after-joy, she turned, and embracing me tenderly said---
``'Oh, my dear child, this is very wrong, but very delicious. You must be very discreet, my dear Harry, for if it were known it would disgrace us both for ever.'
``'My sweet mamma, do not fear; have you ever seen anything like indiscretion in the last six weeks, although I was madly longing for you? Oh, kiss me, my beloved mother.'
``The sweetest of kisses followed, our tongues met, her hand wandered; already she found my pego standing.
``'My darling, I must kiss it, it is so much more developed than I could have expected, and as hard as iron.'
```Not as large as Charlie's, mamma.'
``'That is true, my dear; but it is the stiffness, and not the size, that gives the real pleasure. Of course, when both are combined, as with Charlie, they are irresistible.'
``Meanwhile I was feeling her cunt: her clitoris, which you know is largely developed, stood stiff.
``'Mamma, darling, what a size this is. Ellen told me you could put it into her.'
```Oh! the bad girl, to tell tales out of school.'
```Never mind, mamma, I must suck it while you play with mine.'
``I turned on my back with my heels up---mamma lay down upon my belly reversed. I sucked her clitty while frigging her cunt, and she sucked my prick until we both spent, and each licked or sucked all the balmy sperm that issued from the other. We continued our caresses until my prick showed its readiness for another encounter. Mamma took me on her belly this time, and as soon as I was engulphed, threw her legs over my loins, and, by the most lascivious actions, contributed to our enjoyment. Her glorious bottom heaved in unison with mine, our tongues were interlaced, and at last with sweet murmurs of delight, we died away in each other's arms in the most luxurious ecstasy of thoroughly gratified desire. We lay long insensible of all around, throbbing in pressures of lascivious delight, which would have soon led to another love bout, but that mamma whispered it would be imprudent to continue, for the sun was up, and breakfast time had arrived. I withdrew from the sweet cunt with great regret, and in slipping out of bed brought my mouth down to it, and gave it a loving kiss and suck, played with the magnificent covering of bushy ringlets, and then tore myself away with difficulty. Thus ended my first possession of my adored and glorious mother, which was followed by night upon night of the most lascivious enjoyment. I returned to my room, and was dressed and downstairs before her. The doctor took an opportunity to inform me that she had excused herself from joining him the next night on the pretence of not feeling well, but in reality it was to have me all to herself for the whole night; and a most delicious night it was. She displayed and exercised her libidinous passions to the utmost. Never before had I such a treat. It was, perhaps, the closeness of the relationship that added to the excitement, but it appeared to me that she beat even the doctor's splendid wife. Oh, she was so loving, too. The way she fondled me in her arms and caressed me was irresistible. I can't tell how often we did it---we were at it all night. The next night, under pretence of fearing to exhaust me, she forced me to retire to my room after two fucks, and locked me in. I had previously been informed by the doctor that he had bespoken her for that night, and he begged me to fuck her first, that the pleasure of gamahuching her might be enhanced. I, therefore, did not do more than make a feigned resistance to her when she told me I must go to my own bed. She said she would let me have one embrace before she rose in the morning but that one was converted into two exquisite spends. The next night the doctor wished to repose, as he purposed surprising me in the morning. I laid myself out for this, and when mamma was asleep I rose as if to piddle. I unbolted the door and shook up the doctor, and then returned to bed. I had agreed with him to make more noise than usual in the final ecstasy; he was to wait long enough to allow of the after-enjoyment, as if he was taking time to clothe himself a little, and was then to come in with a light. My mother still slept. It was about four o'clock in the morning. I began feeling her glorious buttocks, and, sliding under the clothes, turned her legs apart---she insensibly slipped upon her back, I took her charming clitoris between my lips, and soon sucked it into stiffness. The excitement awoke her---she had dreamed I was fucking her---and so was hot and randy. She drew me upon her bosom, threw the clothes off, and her glorious limbs clasped my loins---her two hands pressed on my buttocks, as if to drive me further home, and we ran a most delicious course, I feigned to be even still more excited than I really was, and almost brayed at the ecstatic moment of ejection. Mamma herself was too far gone in delight to notice the loudness of my braying. She lay panting and throbbing on my prick, almost in a state of insensibility to aught else beside. Her eyes were closed, so that she did not observe the entrance of the light carried by the doctor. It was not until he was standing by the bedside, and made an exclamation of surprise, that she was aware of his presence.
``She gave a scream---though not very loud---and covered her eyes with her hand. I scrambled off her. The doctor, with great politeness, begged her pardon for his intrusion, but hearing what appeared to him an unearthly noise, he had feared she was taken ill.
``Here the usual resource of woman---tears---fell plentifully from mamma. The doctor most affectionately begged her to calm herself.
```My dear madam,' said he. 'I do not in any way blame you for this. I am a man of the world, and I know that incest is practised to a far greater extent than is at all imagined, and to prove that it in no way offends me, I may at once tell you that it was my own mother who initiated me into these delightful mysteries. I see that this dear boy looks terribly frightened at my being a witness to the delight he must have had; but to put him at his ease, we may as well inform him that we, too, have indulged in that delicious game. I may add that this is not the first time I have joined in orgies with more than one man or woman, and nothing gives me more pleasure than to embrace one reeking from the arms of another, especially if I have been a witness to the previous encounter. See, my dear madam, how this dear instrument stands stiff in proof of what I say, and to insure my silence dear Harry must not object to my enjoying you after and before him.'
``So saying he dropt off his trousers and jumped into bed. He was met with feeble remonstrances from my mother at doing it before her son: but I assured her that I rather preferred to see her at work, as she knew, than otherwise, especially as she evidently enjoyed it so much. So the doctor forthwith mounted her. There could be no doubt that she enjoyed it equally with him. My cock stood at the sight. I put it into her hand, and she squeezed it lovingly---then stooping I sucked one nipple, and you know how this excites her, and slipped a hand behind the doctor, and after gently tickling his ballocks, acted postillion to his bottom-hole. They ran a most exciting course and died away in mutual raptures. No sooner did he turn off than I jumped up into his place, and in one moment was up to the cods in that overflowing cunt. Mamma feebly expostulated, but the doctor begged her to let him have the pleasure of witnessing the vigour of the youth. I knew that in heart mamma was delighted, for all women especially enjoy having a fresh prick into them immediately after a previous one has been withdrawn.''
This is quite true---witness my own dear Benson in our early days; her greatest delight was to have me the instant B. retired, and she avowed that nothing could give her greater pleasure. I knew a lady in after-life whom I and three others used to have together, and no sooner was one off than another was on and sometimes two at once. She used to tell us how she deceived her husband. When at Florence she had eight lovers, and she had had them all on the same night without any of them knowing of the others. She managed it in this way. She made them come---two at ten o'clock, two at half-past ten, two at eleven, and two a half-past eleven. They were put in four different rooms with convenient sofas. She ran to No. 1 in a merely loose robe, which was instantly thrown off. She was a magnificently made creature, the sight of whose charms would inflame any one. She rapidly got two goes from the first without withdrawing. Then saying that her husband would be seeking her if she did not leave him, she rang for her German valet, who used to fuck her himself, and who afterwards confirmed her story to me, who showed my gentleman out of the room. Off she ran to No. 2, told him she had only got away by letting her husband have a go, and that he thought she had only gone to the water closet so he must do one good and leave her. Of course the cunt full of fuck only excited him the more, and he very soon racked off to her great satisfaction, and was dismissed, leaving the rooms vacant for the two at eleven. As there was not five minutes to spare she ran to No. 3, where another lover was waiting. The same pretence was made as to the last, but as he was largely hung, she got two coups from him and then packed him off, and in the same way ran to the others, always with the same story, getting two coups out of three running, who were the best fuckers, and waiting with the last until he could do no more.
The same lady told me that once while living at Dieppe her husband ran over to England for a few days. During his absence she had four young men to supper every night, and made them all fuck her on the sofa squabs laid on the floor, accommodating one in her bottom at the same time. During the day her landlord, a married man, used to come in and rack her off besides. At one time she was left alone at Mannheim, where she made acquaintance with an officer, who introduced a second, and a third, until she knew eight in all. She had the whole lot once to supper, and they all fucked her three times each. She was a wonderfully fine woman, and could take no end of fucking. Her father had initiated her at twelve years of age. She was of Greek origin, and actually was hairy and menstruated at that early age. But all women are rakes in their hearts, and numbers never encumber them.
During Harry's encounter with his mother, the doctor stood beside them, and handled Harry's ballocks and acted postillion to him. Mamma took to it most kindly. The sight again inflamed the doctor, the incestuous idea enhancing the excitement. As Harry withdrew, he begged Mrs. Dale to get on her hands and knees, to let him put it in from behind. He would rather have gone in behind but did not think she was as yet quite prepared to allow that. He only said that the movement below his eyes of such a fine bottom as hers added to the excitement. He further proposed that she should kneel over Harry's body reversed, so that she might gamahuche him, and he frig her beautiful clitoris.
``You mean to kill me between you,'' she said, but all the same complied. She sucked Harry's prick and he spent in her mouth, which she swallowed with great gusto, spending herself at the same moment in advance of the doctor. Harry kept frigging her clitoris with one hand, while the other was frigging the doctor's bottom-hole. It was a long bout, she made Harry spend twice in her mouth, while she spent thrice to the doctor's once, all dying away together in the final fuck. They lay long lost to everything, and when they recovered, they separated and retired to their own rooms.
The ice being thus broken, the remaining days were passed in the most refined lasciviousness. The doctor had his way with her bottom, and asked her leave to have Harry's after Harry had had his mother's bottom-hole, while the doctor was fucking her, and had fucked the doctor upon another occasion, the doctor crying out---``Hi, hi, hi!'' as if it hurt him, and he was losing his maidenhead. He professed immense satisfaction, when she let him have Harry, declaring that he could not tell whether having her both ways, having Harry, or being had himself, was the greatest pleasure. Mamma declared that to have both apertures filled at the same moment was the most delicious. It was then the doctor said he would try. So fucking mamma in a kneeling position he presented his great backside to Harry and was well fucked. It was after this complete initiation that they returned home, and after such proceedings, the transition to a general entry into our orgies was easily arranged. As they were to arrive to a late dinner, it was resolved that Mrs. Dale should stay the night, and we would see what that would bring forth. They arrived accordingly. Mrs. Dale went to Ellen's room, taking Ellen with her to help her at her toilet. Here ensued an explanation between them. Mrs. Dale felt that there must be an explicit avowal on both sides. She admitted to Ellen that Harry had come to her bed, and only succeeded in his horrible purpose by telling her how he and Ellen had seen her operations with Charlie, and had followed her example.
``And now, my dear Ellen, as there must be no secrets between us, tell me if you and Charlie have got together.''
``Well, yes, we have. You know I had seen how immense he was, and yet with what pleasure you took him in. So curiosity made me give way one day that we were in the summer house, and he slept with me afterwards.''
``Does the doctor's wife suspect?''
``Oh, yes, she knows all about it. I forgot to bolt the door one night; in the morning Charlie made too much noise. She came in, merely in her chemise, ran up and pulled him off me, without imagining she ran any risk herself. Charlie seized her in his arms, and swore he would do as much to her, to prevent her telling. She was horrified, and fled to her own room, but had not time to shut him out; he forced the door open, she ran to her bed, intending to ring for the servant, he caught her as she had one knee up on the bed, and was into her from behind before she could accomplish her purpose. She cried out to me, to come and pull him away. I went, but told her Charlie was right, as it would prevent her splitting upon us. I rather think that Charlie's large proportions gave her much pleasure, for she soon ceased to struggle, indeed she had her back to him, and his strong arms round her waist prevented her using her hands. She cried much afterwards, and talked about the greatness of the crime. She had then got into bed. Charlie followed, to coax and console her, and, of course, got into her again. I thought she enjoyed the second, for her bottom heaved to meet him. She afterwards accused him of the crime of seducing a young lady, her guest, but I stopped that, by avowing that my cousin had had me previously. Then she accused me of seducing Charlie, and here, I must implore your pardon, for I let out inadvertently that you had initiated him, for I had seen you having him.''
``Oh, you bad girl, how could you be so cruel and imprudent?''
``Well, dear aunt, there is no great harm done. Charlie's aunt was soon quite appeased and regularly joined us after this. She is as fond, if not fonder, of gamahuching me as you used to be; she has grown greedy for Charlie's immense cock, envies your having had the first of him, and says that if she had known of his wonderful proportions, she could not have resisted initiating him herself. She hopes that, through my means, she will become more intimate with you. I have told her of your beautiful clitoris. She dotes on the gamahuche, and vows that she will never be happy till she has done it to you.''
This explanation was a great relief to the widow, who knew she was all right with the doctor, and now foresaw that it would be all right with his wife also and they would have complete freedom to indulge in the wildest lubricity. So having dressed, they descended to dinner. The doctor had explained all their London doings to his wife, so that after dinner the three ladies exchanged confidence. Aunt was so eager to see and suck Mrs. Dale's large clitoris that they adjourned to aunt's bedroom, where the doctor discovered them in the midst of their operations. Mrs. Dale was stretched on her back, with extended thighs---aunt, with her head pressed by Mrs. Dale's hands down on her cunt, was sucking at the splendid clitoris, and working some fingers in and out her cunt. They were too intent on their pleasure to notice his entrance. Aunt's petticoats were above her hips, as she knelt. The old boy's cock stood, he advanced, knelt down, got between her legs, and fucked her as she was, begging her to continue her lascivious operations on Mrs. Dale. When he had finished, he congratulated both ladies on the intimate friendship established between them, said it was the best wish of his heart. He assured Mrs. Dale that his wife was the best woman in the world, and never grudged him a little variety.
``So I have acknowledged my infidelity with you, and it appears my nephew has been taking my place, in my absence. She tells me you instructed Charlie, and that he is monstrous when in erection, as big again as me, or as a certain Grenadier Captain, once a favourite of my wife's. I am curious to see it. She tells me also that he has been sleeping with your charming niece Ellen, who, I must confess, has raised in me a great desire to possess her. Now, my dear madam, if you will consent to invite Charlie to sleep with you and Ellen, I could come in, after you have each had a turn or two out of Charlie, and take Ellen, while you would have Charlie all to yourself. My wife won't object, and I hope you will give your consent.''
``Well, my dear doctor, after what has occurred between us, I can refuse you nothing, but I think dear Harry should have some comfort. I suppose, my dear madam, that the doctor has told you of my son surprising and violating me. Your husband reconciled me to his caresses, and I can assure you that without the enormous size of Charlie, he has a charming way that may please any woman. From what the doctor says, you are free from prejudices, why should you be left out in the cold, while we are all enjoying ourselves, why should you not go to his bedroom, and see what he is made of. I, his mother, can strongly recommend him to your favour.''
So it was arranged.
During the evening, Mrs. Dale whispered to me to come to them after the domestics had gone to bed. I went and fucked them both three times, twice in front and once behind, the one who was being fucked always gamahuching the other. When I began to tail off, Mrs. Dale arose, unbolted the door of communication with uncle's room, and invited him to Ellen's arms, who was very glad to have a little further experience of another man's prick. Uncle gallantly gamahuched her before fucking her, then begged to see my wonderful prick, pretended to be perfectly astonished at its monstrous dimensions, and wondered how Ellen's little cunt could ever have taken it in. It was a tight fit certainly, but the dear creature liked it none the worse for that. Before fucking Ellen, he begged Mrs. Dale to let him guide my large prick into her. After enjoying our first movements, and being excited into sufficient consistency, he proceeded to fuck dear Ellen; even he had some difficulty in entering, notwithstanding the libations I had previously poured into her, but being once fairly hilted, he declared it was one of the tightest little cunts it had ever been his good fortune to fuck. After this, we proceeded each our way, and with a longer interval, brought matters to the exquisite conclusion, panting and throbbing for some time afterwards.
The doctor now retired, and we arranged ourselves for repose. We were awakened in the morning by the entrance of aunt and Harry. He flew to his mamma's arms, who lay down on Ellen to gamahuche her while Harry was fucking her. Aunt and I coupled in the old-fashioned way. Uncle entered while we were in full operation, and seeing the tempting backside of Harry, scrambled up behind and fucked his bottom. After we had done, aunt pretended to be shocked at his attack on a boy's bottom---a woman's was a different thing.
``Well, then, my dear, get upon Charlie the next time, and I shall fuck your truly magnificent bottom.''
And he drew Mrs. Dale's attention to the glorious proportions, not only of aunt's bottom, but of her body, and all her limbs.
``Oh, it is indeed glorious,'' said she. ``I must, my dear madam, gamahuche you. I have not forgotten the exquisite pleasure you gave me in that way.''
``Willingly,'' cried my aunt, ``provided you give me your clitoris to occupy me.''
``Certainly, that will suit me admirably; but you must lie upon me, that I may have the pleasure of gazing on that magnificent bottom, and caressing the immense rotundities of your buttocks.''
Oh, it was a glorious sight to see these two wanton lascivious women in the full enjoyment of each other. It set us all on fire, and the moment they had done I slaked the fire within me in the capacious but tight cunt of my aunt, while uncle fucked her bottom-hole. Mrs. Dale lay under Ellen, while Harry fucked Ellen from behind, and Ellen gamahuched her aunt, who herself guided her son's prick into Ellen's cunt, and tickled her clitoris, at the same time acting postillion to her son's bottom-hole.
Oh! it was a splendid bout---we were all so excited and it was also the first meeting of us all in one orgie. We all died away in a perfect heaven of ecstasy, and lay long in the after-enjoyment. Our previous night's work made this the last for the time, and we all separated to seek some welcome repose before breakfast time.
Mrs. Dale remained our visitor for three days, during which we met in the doctor's bedroom every night, and renewed our delicious orgies. Mrs. Dale carried off her son and niece, and I promised to come over to her cottage on the following Saturday, when Harry and I took turn and turn about with the two dear creatures, sometimes fucking one between us two at once. When the school resumed its work, Mrs. Dale and Ellen always dined at the rectory on Sunday and slept there, when we made a general orgie in the old style.
This continued until our midsummer holidays, when I was to leave the rectory for King's College. Mrs. Dale's and Ellen's pregnancies, daily becoming nearer to the period of parturition, were getting more difficult to conceal. We had long discussions with uncle as to what was best to be done. It was at last arranged that they should leave the cottage as if for a tour on the continent, but in reality should only go to Paris, and take apartments in the house of a good \textit{accoucheuse} in the environs, and remain quiet there till the period of delivery. It was not necessary for them to go before we broke up, and the doctor and Harry and I could accompany them, and after I had seen my guardian on my return to London, I had no doubt of getting his leave, and the necessary means to visit the continent up to the middle of October, when the classes would begin. It all fell out as arranged. Nothing of the pregnancy was visible, thanks to the full robes worn.
We accomplished our journey, found a capital \textit{accoucheuse} in a beautiful neighbourhood, with a large garden. Harry, uncle, and aunt remained with them, while I returned to London. I saw my guardian, who, after putting me through an examination, expressed himself much pleased with my progress, said the visit to the continent would expand my mind, and that he would furnish me with the means. He recommended that I should visit my mother first for a fortnight, and announced that at about the end of that time the girls would come up to London to enter a first-rate finishing school. He further told me he had proposed to and had been accepted by Miss Frankland, and they were to be married at the same time; my sisters were to be bridesmaids, and I could be present at the marriage before going abroad. All this being arranged, I ran down home. My mother was delighted to see me, and thought me grown and much improved. It is needless to say how glad my sisters and Miss Frankland were to see me. They had had no fucking except by tongue or dildo, so you may imagine the fury with which they set upon me the first two or three nights. We resumed all our lascivious operations of former days. My sisters had developed into splendid women, the youngest still the most libidinous. Dear Miss Frankland, on my congratulating her on her intended marriage, lovingly told me that it was the prospect of being near me that had reconciled her to it. We spent a most delicious fortnight, which passed like a day.
I found an opportunity of fucking my old governess, Mrs. Vincent that was. My son was a fine little fellow, toddling about and talking already. His mother loved me as much as ever, and was become a finer developed woman, more amorous and lecherous than she used to be. She said no one could be kinder or more loving than her husband, and she had never been unfaithful to him but with me, whom, as her own formation, she must always love, and would never refuse me anything I asked when it could be safely done. At the sole opportunity I had I fucked her three times without drawing, and finished with a bottom-fuck. I may here mention that a little girl followed for nine months from that period, which she always assured me was mine.
My mother, the girls, and Miss Frankland all came up with me to London. The marriage went off with \textit{eclat}. My guardian made very handsome presents to my sisters, and gave me a gold watch, chain, and seals, together with a handsome cheque for my travelling expenses. He and his bride, whom I fucked just before she went to church, departed for Scotland, to return by the English lakes, for their honeymoon trip. A few days afterwards, having had two or three nights excellent fucking with my sisters, mamma and I conducted them to their school, and left them with tearful adieux. My mother was to remain in town for a week until uncle and aunt's return, when she intended to accompany her sister to the rectory and remain there until I returned from the continent. I was quickly again in Paris. We took rooms near the two darlings, where uncle and aunt remained for the week they had yet to stay. We took mamma and Ellen several times to the play, and they slept with us every night. Uncle and aunt left at the end of the week, but we kept on the apartments for the dear women to come to us, fucking them as much as we could. It seemed as if their pregnancy stimulated their lubricity, for we could hardly satisfy them. We had at least always to take them on hands and knees, although neither of them ever showed much in front---their babes lying just between---but, by Jove, their hips expanded splendidly. Dear mamma measured a yard across, and her backside projected almost as much as my aunt's. She loved to be fucked in her bottom-hole to the last. We actually had them both up to the night before the day they were each confined. Nothing could be more favourable than their time. As I formerly stated, each had a little daughter.
On the ninth day afterwards they were both able to rise, but as it would have been very prejudicial to renew our intercourse before another three weeks had elapsed, Harry and I went off for a walking excursion in Switzerland, which we traversed in all directions, with continual delight at the glorious scenery. We did not touch a single woman. When very sharp set we fucked each other, but very little even of that, so that we renovated our constitutions and returned in robust health, ready to do justice to the charms of the two darlings, who had impatiently awaited our arrival.
It is needless to repeat the description of the delicious fucking with which they welcomed us. They appeared more lovely than ever, especially Ellen, who had developed into womanhood. We made arrangements to leave the two darling children in the hands of a healthy wet nurse, and set out on an expedition down the Loire to Tours, Bordeaux, and the Pyrenees, returned at the end of September by Montpellier, Nismes, Avignon, and Lyons.
The two babes were in excellent health. Arrangements were made for their remaining with their foster mother for a year, and we all returned to London together.
We had three nights' delicious fucking before they returned to the country, and promises were made that they would come to town from time to time to renew our orgies. My mother and aunt came up to see me settled in my lodgings, which were taken in Norfolk Street, and I was entered at King's College.
I passed a delicious night with aunt before she left; and ran down with my mother to see her safe home. On my return I found my guardian had returned. I called to pay my respects to his wife. I found her alone, and we managed her first piece of adultery, which, as you may suppose, was not the last. But as this third volume is already a long 'one, I shall here close it.
The fourth will introduce us to London, and renew the delicious intercourse with Mrs. Benson, as well as with my guardian's wife, and our dear friend MacCallum, as well as many other friends.
END OF VOLUME III.
\gutchapter{VOLUME IV.}
\section*{\raggedright CONTENTS}
Jane---Ann---Mrs. Nichols---The Benson, the Egerton, and the Count---Ann,
the Nichols, and MacCallum---Aunt, Uncle, Harry, the Frankland and the
De Grandvits---Carl---The Count---The Frankland
I concluded my last volume by saying that I had taken lodgings in
Norfolk Street, Strand, for the convenience of being near King's
College. It was at the house of a Mrs. Nichols, tall, powerfully built,
masculine, but a kind and motherly looking widow of fifty-two---an
attentive and bustling landlady, looking herself to the better cooking,
and having a plain cook, who was also a general servant, to help her
downstairs, and two nieces to do the waiting and attendance on her
lodgers upstairs. The younger was there alone when I entered the
lodgings; her elder sister had had what they called a ``misfortune,'' and
was then in the country until she could be unburthened of it. She was
expected back in about six weeks. Meanwhile, as the winter was not the
season, I was the only lodger, and the younger had only me to attend
to; her name was Jane; she was but a little thing, but very well made,
good bubbies and bottom, which I soon discovered were firm and hard,
projecting fully on both sides. She was fairly good looking, but with a
singular innocent manner of freedom about her that made me imagine she
had as yet had no chance of a ``misfortune.'' In a week we became
intimate, and after often praising her pretty face and figure, I
snatched a kiss now and then, which at first she resented with an
attractive yet innocent sort of sauciness. It was in her struggles on
these occasions that I became aware of the firm and hard bosom and
bottom.
Up to this time my flirtations were without ulterior object, but the
reality of the attractions of these hidden charms raised my lustful
passions. I gradually increased my flatteries and caresses, squeezed
her bubbies, when I sometimes drew her on my knee and was kissing her,
and as at first she resisted my drawing her to my knee, I took occasion
to lay hold of her buttocks, which I found more developed than I could
have supposed. Gradually her resistance to these little liberties
ceased and she would quietly sit on my knee and return the kiss I gave.
Her dress was a little open in front, so from feeling her bubbies
outside, I gradually got to feeling their naked beauties inside. I now
thought I could attempt greater familiarities, so one day when seated
on my knee with one arm round her waist, I pressed her to my lips, and
while so engaged, whipt my free arm up her petticoats, and before she
had become aware of the movement, had got my hand upon her mount, a
very nicely haired one. She started up to a standing position, but as I
held her close clasped round the waist she could not get away, and her
new position enabled me the easier to get my hand between her thighs
and thus to feel her charming pouting little cunt. I began attempting
to frig her clitoris, but stooping she drew her cunt away, and looking
at me with a droll innocent expression of alarm, and with a perfect
unconsciousness of the import of her words, cried,---``Oh! take care
what you are at. You don't know how a lodger this last summer suffered
for seizing me in that way and hurting me very much. I screamed out,
aunt came up, and, do you know, he had \pounds50 to pay for his impudence.''
I could not but smile at the extraordinary innocence of the girl.
``But I do not hurt you, dear Jane,'' said I, ``and don't mean to do so.''
``That was what he said, but he went on in a most horrible way, and not
only hurt me very much, but made me bleed.''
``It would not be with his hand, you see I only gently press this soft
hairy little thing. I am sure that don't hurt you.''
``Oh, no! if that was all I should not mind it, it was when he pushed me
on the sofa, and pressed upon me, that he hurt me terribly, and you
must take care what you are about, or you, too, will have to pay \pounds50.''
There was a curious air of innocence in all this; it was evident to me
the fellow had got into her, and broken her hymen with violence, and
then her screams had prevented his finishing his work. Her manner
convinced me that she was really not aware of the consequences, or
rather had not as yet really had her sexual passions aroused.
``Well, my dear Jane, I neither intend to hurt you or make myself liable
to pay \pounds50, but you will not refuse me the pleasure of feeling this
nice little hairy nest, you see how gentle I am.''
``Well, if you will do me no more hurt than that I shan't refuse you,
because you are a nice kind young gentleman, and very different from
the other rough fellow, who never chattered with me and made me laugh
as you do---but you must not push your fingers up there, it was
something he pushed up there that hurt me so.''
I withdrew my finger, and as, at my request, she had opened her thighs
a little, I felt and caressed her very nice little cunt, and with a
finger pressed externally above her clitoris, I could see that she
flushed and shivered on feeling me there. However, I did no more than
gently press and feel all her hairy mount and fat pouting cunt; she
said I must let her go, or her aunt would be coming up.
The first step was now gained. Gradually I progressed further and
further; felt her charming bare arse as she stood before me, got her to
let me see the beautiful curls she had got on her cunt, then came to
kissing it, until at last she opened her thighs and let me tongue it,
to her most exquisite delight. I made her spend for the first time in
her life, and soon she came to me for it. I had gradually introduced a
finger up her cunt while licking her clitoris and exciting her so much
that she was unconscious of my doing it; then two fingers, and after
she had spent deliriously, I made them perform an imitation of a throb,
which made her jump and ask what I was doing. I asked if she did not
feel that my fingers were inside of her sweet Fanny.
``You don't say so. It was there I was so hurt.''
``But I do not hurt you, dear Jane?''
``Oh, dear no, it makes me feel queer, but it is very nice.''
``Well, now you know that I have two fingers inside, I will use my
tongue again against your charming little clitoris, and work the
fingers in and out.''
I did so, and she soon spent in an agony of delight, pressing my head
down hard on her cunt, and crying---``Oh! oh! it is too great a
pleasure!'' and then died off, half insensible. Another time I repeated
this she told me not to forget to use my fingers. Having made her spend
twice I took her on my knee, and told her that I possessed an
instrument that would give her far more pleasure than tongue or finger.
``Indeed?'' said she, ``where is it? I should so like to see it.''
``You won't tell.''
``Oh, no!''
So pulling out my stiff-standing prick, she stared in amazement. She
had really never seen a prick, although it was evidently a prick that
had deflowered her, for with my fingers I had explored her cunt, and
found no hymen there. I put her hand upon it, she involuntarily grasped
it firmly.
``This enormous thing could never get into my body, look, it is thicker
than all your fingers put together, and only two fingers feel so tight.''
``Yes, darling, but this dear little thing stretches, and was made to
receive this big thing.''
I was exciting her clitoris with my finger, she grew evidently
lasciviously inclined, so saying, ``Just let me try, and if it hurts you
I will stop; you know I am always gentle with you.''
``So you are, my dear fellow, but take care not to hurt me.'' She lay
down on the bed, as I desired, with feet up and knees laid open. I spat
on my prick, and wetted the knob and upper shaft well, then bringing it
to her cunt, well moistened by my saliva in gamahuching her, I held
open the lips with the fingers of my left hand, and half buried its
knob before getting to the real entrance.
``Don't flinch, dearest, I shall not hurt.'' And I got it well over the
knob, and buried it one inch further.
``Stop!'' she cried, ``it seems as if it would burst me open, it so
stretches me.''
``But it does not hurt you, dearest?'' I had immediately stopped before
asking the question.
``No not exactly, but I feel as if something was in my throat.''
``Rest a little, and that will go off.'' I slipped a finger down on her
clitoris, and as I frigged it she grew more and more excited, giving
delicious cunt pressures on my prick, it gradually made its way by the
gentle pushing I continued to make without other movements. It was more
than half in when she spent, this not only lubricated the interior, but
the inner muscles relaxing, a gentle shove forward housed it to the
hilt, and then I lay quiet until she recovered from the half fainting
state her last discharge had produced; soon the increased pressures of
the inner folds showed that her passions were awakening afresh. She
opened her eyes and, looking lovingly, said I had given her great
pleasure, but she felt as if something enormous was stretching her
inside to the utmost. Had I got it all in?
``Yes, dearest, and now it will be able to give you greater pleasure
than before.'' I began a slow withdrawal and return, frigging her
clitoris at the same time, for I was standing between her legs. She
soon grew wild with excitement, nature prompting her, her arse rose and
fell almost as well as if she was mistress of the art. The novel
combination of prick and finger quickly brought on the ecstatic crisis.
I, too, was wild with lust, and we spent together, ending in an
annihilation of all our senses by the extreme ecstasy of the final
overpowering crisis. We lay panting for some time in all the
after-joys. Dear Jane begged me to give her some water, as she felt
quite faint. I withdrew, still almost in a standing state, got her some
water, helped her up, seated her on the sofa and kissed her lovingly as
I thanked her for the exquisite joy she had given me. She threw her
arms round my neck, and with tears in her eyes told me I had taught her
the joys of heaven, and she should always love me, and I must always
love her, for now she could not live without me. I kissed and dried her
eyes, and told her we should in future enjoy it even more when she got
accustomed to it.
``Let me see the dear thing that gave me such pleasure.''
I pulled it out, but it was no longer at the stand; and this surprised
her. I explained the necessity of its being so, but said she would
quickly see it rise and swell to the former size if she continued to
handle it so nicely. It rose almost before I could say as much. She
fondled it, and even stooped and kissed its ruby head. We should
quickly have got to another bout of fucking if the ringing of the call
bell had not brought us to a sense of its imprudence; so after
arranging her hair and dress, she hastily descended with some of the
breakfast things.
Of course, so good a beginning led to constant renewals and Jane
quickly became extremely amorous, and under my instruction a first-rate
fucker.
As all my dear friends were not in London, I was fortunate in having
such a \textit{bonne bouche} to comfort me. My sisters passed every Sunday
with me, and both got some good fucking out of me in every way, without
raising any suspicions in the house.
A month after I had taken up my residence at Mrs. Nichols's, Jane's
sister arrived. She was a much finer woman than Jane, broad shouldered,
wide-spread bosom, which, in after-days, I found had not suffered by
her ``misfortune,'' but then she had not suckled it. Her hips were widely
projected, and she was grand and magnificent in her arse. Naturally of
a very hot temperament, when once she had tasted the magnificent weapon
I was possessed of, she grew most lasciviously lustful, and was one of
the best fuckers I ever met with. Her power of nip almost equalled by
beloved aunt's. Jane was fair, Ann was dark, with black locks and black
hairy cunt---a very long cunt, with a small tight hole in it, and above
it a wide-spread projecting mount, splendidly furnished with hair. Her
clitoris was hard and thick, but with little projection. She also
became madly fond of arse-fucking, and particularly liked me to spend
therein. This was partly to prevent any consequences leading to a
second ``misfortune.''
On her first arrival Jane was much afraid she would discover our
connection and we took every precaution, although I, in my heart,
wished this might occur, for as she occasionally waited on me, I grew
lecherous upon one whose charms, even covered, excited me greatly. I
always flattered and praised her magnificence of figure whenever she
came alone to me, but as Jane generally was running in and out, I did
not attempt further action. One morning I overheard Mrs. Nichols tell
Jane to put on her bonnet and go to Oxford Street on some errand; I
knew thus that Ann would attend on me, and there would be no chance of
interruption from Jane, so I determined to come at once to the point.
We had become on friendly, chatty terms, and when she had laid
breakfast I asked her to help me on with my coat, which done, I thanked
her and with one arm round her waist drew her to me and kissed her.
``Hallo!'' said she, ``that is something new,'' but did not attempt to
withdraw, so giving her another kiss, I told her what a glorious woman
she was, and how she excited me---just see. I held one of her hands, and
before she was aware, placed it on my huge prick, that bulged out of my
trousers as if it would burst its way through.
She could not help squeezing it, while she cried---``Goodness, gracious!
what an enormous thing you have got!''
Her face flushed, her eyes sparkled with the fire of lust that stirred
her whole soul. She tried to grasp it.
``Stop,'' said I, ``and I will put it in its natural state into your hand.''
So pulling it out, she seized it at once, and most lasciviously gazed
upon it, pressing it gently. She evidently was growing lewder and
lewder, so I at once proposed to fuck her, and thinking it best to be
frank, and put her at her ease, I told her that I knew she had had a
``misfortune,'' but if she would let me fuck her I should be on honour to
withdraw before spending, and thus avoid all chance of putting her
belly up.
She had become so randy that she felt, as she afterwards told me, she
could not refuse so splendid a prick of a size she had often dreamt of,
and longed for.
``Can I trust you?'' said she.
``Safely, my dear.''
``Then you may have me---let me embrace that dear object.''
Stooping, she kissed it most voluptuously, shivering at the same time
in the ecstasy of a spend produced by the mere sight and touch. She
gave one or two ``oh's,'' and drawing me to the bed by my prick, threw
herself back, pulling her petticoats up at the same time. Then I beheld
her splendid cunt in all its magnificence of size and hairiness. I sank
on my knees and glued my lips to the oozing entrance, for she was one
who spent most profusely, her cunt had the true delicious odour, and
her spunk was thick and glutinous for a woman.
I tongued her clitoris, driving her voluptuously wild. So she cried---
``Oh! do put that glorious prick into me, but remember your promise.''
I brought it up to that wide-spread, large-lipped, and immense cunt. I
fully expected that big as I was I should slip in over head and
shoulders with the greatest ease. So you may imagine my surprise to
find the tightest and smallest of entrances to the inner vagina I
almost ever met with, it was really with greater difficulty I effected
an entrance than I had with her little sister, whose cunt presented no
such voluptuous grandeur. It was as tight a fit as Ellen's was to me on
our first coition. Tight as it was, it gave her nothing but the most
exquisite pleasure, she was thoroughly up to her work, and was really
one of the most voluptuous and lascivious fuckers I have ever met with,
excellent as my experience has been. I made her, with fucking and
frigging, spend six times before I suddenly withdrew my prick, and
pressing its shaft against her wet lips, and my own belly, spent
deliciously outside. Shortly after it rose again, and this time after
making her spend as often as before, for she was most voluptuously
lustful, when I withdrew, she suddenly got from under me, and seizing
its shaft with one hand, stooped and took its knob between her lips,
and quickly made me pour a flood of sperm into her mouth, which she
eagerly swallowed and sucked on to my great delight.
We should have had a third bout but for the necessity of her going down
to her aunt.
I breakfasted, then rang to take away. Again we had a delicious fuck,
and a third when she came to make the bed and empty the slops. This
third time I begged her to kneel on the sofa, and let me see her
gloriously grand arse, and when I had to retire I would show her a way
that would continue both our pleasure. So after fucking her from
behind, and making her spend far oftener than me, I withdrew, and
pushing it up between the lips over the clitoris, with my hand round
her waist, I pressed it tightly against her cunt and clitoris, and
continued to wriggle my arse, made her spend again as I poured a flood
all up over her belly. She declared it was almost as good as if inside.
After this very shortly I proposed to push its nose into her
bottom-hole, and just spend within.
With reluctance at first, it ended in her not only liking the point
there, but deliciously enjoying my whole prick within, and eventually
it was always the receptacle of a first discharge induced by fucking,
and a second fuck completely carried on in that more secret altar of
lust. She became a first-rate \textit{enculeuse}.
It soon happened that both sisters knew of the other enjoying me, and
it ended in their slipping down from their attic, where both slept in
the same bed, to my room, and we had most delicious fucking and double
gamahuching.
Ann was by far the finest and the most lascivious fuck, but little Jane
had a certain charm of youth and also of freshness, which got her a
fair share of my favours.
We carried this on for several weeks until use made us careless and
noisy.
The aunt, when no lodgers occupied the room, slept overhead, and,
probably being sleepless one morning, when it was early daylight, heard
our voices, came down and surprised me in the very act of fucking Ann
and gamahuching Jane, who stood above her and presented her cunt to my
lecherous tongue. A loud exclamation from their aunt roused us up at
once.
``Get to bed, you dreadful hussies.''
They fled without a moment's hesitation.
Mrs. Nichols then began to remonstrate with me on the infamy of my
conduct. I approached the door apparently to get my shirt, for I was
stark naked, but in fact to shut and lock my door, and then to turn on
Mrs. Nichols, who apparently had quite forgotten she had only her short
shift on, which not only allowed the full display of very fine, firm,
and ample bubbles, but not falling below the middle of her thighs,
showed remarkably well made legs and small knees, with the swelling of
immense thighs just indicated.
My stiff-standing prick in full vigour, and if anything, still more
stimulated by the unexpected beauties shown by Mrs. Nichols, I turned
upon her and seizing her round the waist from behind, pushed her
forward, and before she could recover herself I had hauled up her
``cutty sark,'' seen a most magnificent arse, and into her cunt---not
without somewhat painful violence, before she could recover from the
surprise of the attack.
She screamed out murder, but there was no one who could hear but the
girls, and they knew better than to interrupt me. I kept fucking away
in spite of cries, and passing an arm round her body, with my finger I
got to her clitoris, which sprang out into considerable proportions. My
big prick and the frigging of her clitoris produced their natural
result. In spite of herself she grew full of lust. I felt her cunt
pressures, and knew how her passions were rising. Speedily, in place of
resisting, she began to cry, ``Oh, oh,'' and breathe hard, and then most
gloriously wriggled her splendid arse, and as I spent she too was taken
in the delicious ecstasy of the final crisis. She lay throbbing on my
delighted prick until it stood as stiff as before. I began a slow
movement, she made no resistance, except crying out, ``Oh! dear, oh!
dear,'' as if in spite of regrets, she could not help enjoying it;
indeed, at last she said---
``Oh! what a man you are, Mr. Roberts; it is very wrong of you to do
this, but I cannot resist enjoying it myself. It is years since I did
such a thing, but as you have done it, it makes me wish you should do
it again. Let us change position.''
``Very well, but you must throw off this tiresome chemise, or I won't
withdraw.''
As her lust was so excited, she made no objection, so withdrawing we
stood up; she drew her shift over her head, and displayed a far more
splendid form, with an exquisitely fair and dimpled skin, than I could
have thought possible.
``My dear Mrs. Nichols, what a fine perfect form you have got, let me
embrace you in my arms.''
She was nothing loath, flattered by my praise. She laid hold of my cock
with one hand, and closely clasped me with the other arm, while I threw
an arm and hand round on her truly magnificent arse, and with my other
hand pressed on a wonderful pair of bubbles as hard and firm as any
maid of eighteen. Our mouths met in a loving kiss, our tongues
exchanged endearments. She said---
``You have made me very wicked, let me have this enormous and dear
fellow again.''
I said I must first gaze on all her beauties, especially on her
gorgeous and enormous bottom. She turned herself round in every way,
delighted to find that I so ardently admired her.
She then lay down on her back, and spread wide her legs, and called to
me to mount and put it in.
``First I must kiss this beautiful cunt, and suck this superb clitoris.''
Her mount was covered with closely curled brown silky locks; her cunt
was large with grand thick lips and well-haired sides. Her clitoris
stood out quite three inches, red and stiff. I took it in my mouth,
sucked it, and frigged her cunt with two fingers, which went in with
the greatest ease, but were nipped tightly the moment the entrance was
gained, and I frigged and sucked until she spent madly with absolute
screams of delight. I continued to suck and excite her, which quickly
made her cry out---
``Oh, darling boy, come and shove your glorious prick into my longing
cunt.''
I sprang up and buried him until our two hairs were crushed between us.
She held me tight for a minute without moving, then went off like a
wild \textit{Bacchante}, and uttered voluptuous bawdy expressions.
``Shove your delicious prick further and harder. Oh, you are killing me
with delight.''
She was a perfect mistress of the art, gave me exquisite pleasure, and,
I may add, proved afterwards a woman of infinite variety, and became
one of my most devoted admirers. Our intrigue continued for years,
while her age, as is the case with good wine, only appeared to improve
her. Her husband was not a bad fucker, but having only a small prick,
had never stimulated her lust as my big splitter had done.
We had on this first occasion three other good fucks, which she seemed
to enjoy more and more.
As I had previously fucked the girls pretty well, my prick at last
refused to rise and perform. We had to stop fucking, but I gamahuched
her once more after again posing her, and admiring her really
wonderfully well made and well-preserved body. She had a good suck at
my cock, without bringing him up again.
At last we separated, but not before she made a promise that she would
sleep with me that night, and a glorious night we had. I had the more
difficult task of reconciling her to my having her nieces. I used to
have them one night, and sleep with her the next.
Ann, as I have said, was one of the lewdest and most lascivious women I
had ever known. I had told them of the beauty of their aunt's whole
person, and of her wonderful clitoris, and how she liked me to
gamahuche it. This awakened the tribadic passions of Ann to gamahuche
her aunt.
I, at last, persuaded her to let Ann join us, and both were afterwards
extremely glad I had done so, for both were thorough tribades, and
lasciviously enjoyed each other, while being fucked by me in turns.
Mrs. Nichols too, once she got used to arse-fucking, delighted in it,
and we had the wildest orgies together.
Meanwhile, my very dear friend MacCallum had returned to town. He lived
in the outskirts, but had taken a small set of chambers at Lyon's Inn,
a sitting-room and bedroom, where he had a complete library of bawdy
books and pictures to excite to new efforts passions palled with
excess. It was here I took my sisters, and every Sunday we four,
stripped to the buff, indulged in every excess the wildest lust could
prompt.
At Christmas, uncle, aunt, the Dales, and Ellen all came to town, and
taking the same rooms with others that uncle and Mrs. Dale and her son
had formerly down in Norfolk Street, we had the most glorious orgies.
I confessed that I had debauched my sisters during the weary months I
had been left alone with them, and advised their initiation into our
society. Uncle greedily snatched at the idea, so did aunt and Harry
Dale, but his mother and Ellen rather discouraged it. However, the
majority had it, and aunt went to the school, and took them away for
the holidays. I had instructed them to keep up the idea of a late
initiation by me, and how much they liked it when done, carefully
avoiding the least reference to former freedoms.
They afforded a very effective aid to the wild variety of our orgies.
Uncle especially affected them, and was never tired of fucking, sucking
or gamahuching their splendid charms. Aunt, whose lech was for fresh
young women, was unbounded in her admiration and tribadic use of their
bodies.
I made a confident of Harry Dale about our re-unions at MacCallum's,
and, with the latter's leave, introduced him to our orgies in the Inn.
MacCallum took greatly to the fine tight arse of young Dale. He also
wished to have Ellen introduced. I took occasion to break the matter to
her, and in the end she made a delicious addition to those private
orgies. In March Mrs. Benson, Mrs. Egerton, and husbands came up to
town.
I had written to the Benson, and got a note from her the moment she
arrived. I called immediately, and finding her alone, her husband
having gone to the city, was received with delight. After flying into
each other's arms, nature was too fierce for any amorous preliminaries.
A sofa received our ardent bodies, and before one could think, legs
were opened, cunt invaded, and a most rapid fuck, too rapid for luxury,
was run off. Then while recovering from our first delirium of pleasure,
we had time for a few words of mutual praise and admiration of
improvements in both; but it was not until I had fucked her four times,
and made her spend at least twice as often, that we found time to enter
into close converse upon past events.
I had known by letter of the intrigue with the Count, Mrs. Egerton, and
herself, and now heard, from her own mouth, more exciting details. She
told me how Mrs. Egerton was eager to possess my unusually great prick,
adding---
``By the way, she must be alone at this hour. Come along, we may have
some fun to-day.''
I had not seen Mrs. Egerton for many years, in fact, for long before I
had fucked Mrs. Benson. We went. Her reception was all I could wish.
Mrs. Benson told us to lose no time, but to run off at least an
introductory embrace when the field was so clear. Mrs. Egerton made no
objection; the Benson acted mistress of the ceremonies, pulled out my
prick and lifted the Egerton's petticoats, turning both sides to view,
and making the Egerton handle and admire the nobleness of my prick,
then telling her to kneel and present her fat arse to my lustful gaze,
guided my longing prick into her really delicious cunt; and a most
excellent fuck we had, which, as Mrs. Benson said, would put us at our
ease in an interview she had planned for next day, in which the Count
was to join us, and telling me I should have to show my mettle to rival
the Count.
We met next day at a quiet house in Percy Street, Tottenham Court Road.
The ladies had gone to the Soho Bazaar, leaving their carriage in Soho
Square, going out by another entrance in a back street, and driving up
in a cab to us in Percy Street.
At an evening call I had made, to be introduced to Mr. Egerton, I had
met and been introduced to the Count. We had walked home as far as his
apartments, in Berners Street, and arranged to meet in Percy Street,
before the arrival of our beautiful and dear friends. Thus we were
impatiently awaiting their coming when they arrived.
It is needless to say no sooner had they entered, and the mere embrace
and kiss of welcome been given, than they retired to another room,
opening into the one where we 'were, to take off all encumbrances to
the wildest lust, while we, too, disencumbered ourselves of all our
clothes. We were quicker than they were, and the Count was in the act
of handling and admiring the grandeur of my prick when the two
beautiful creatures entered in nature's only robe, and well might we
exclaim---
``Woman, when unadorned is adorned the most,'' for two more beautiful
women or more perfectly lovely in shape could hardly be seen. Women,
too, as voluptuous and lascivious in their passions as any of their sex
could be, and it was now our delight to enjoy and satisfy their ardent
lust by fucking them in every way, as well as for the first time giving
them the joy of having two real pricks in them at once. The charming
Benson, as my original initiator in love's mysteries, claimed my first
embrace, the Count fucking Mrs. Egerton. We were so placed that each
could see the other, and thus enjoy the excitement of the scene. The
dear creatures spent thrice to our once.
Then the Egerton claimed me while the Count refilled the cunt I had
just quitted.
Again we made them spend thrice to our once. They preferred these
preliminary encounters to the more lascivious excesses we were about to
enter upon as exciting and preparing their passions for more voluptuous
embraces.
Both the dear creatures loved a prick \textit{in culo} from time to time, but
as yet they had not had the opportunity of having a prick in each
aperture at once.
The Egerton, to whom my prick was as yet a novelty, said she must have
it in her cunt while the Count planted his lesser but very fine prick
in her arse.
The Count's prick was quite as long, or nearly so, as mine, and even
thicker close to the roots, but tapered up to a small pointed knob, so
that for the \textit{enculage} he had greater facility, than my huge-knobbed
affair, whose head was as thick as any part of it. This difference of
formation made the dear creatures both prefer my prick in front while
the Count attacked them in the rear. They generally each got two, with
me below and the Count above. But, although it was at first somewhat
painful when my huge prick took the rearward side with the Count in
front, they soon got accustomed to it, although invariably beginning,
after our preliminary fucking, with the Count first \textit{in culo}.
The Egerton, as I said before, made her first trial of two pricks
fucking her at once, by having me below her. I had laid down on my
back, she straddled over me, the Benson claimed the place of
conductress to the instruments of pleasure, and, first giving a suck to
my prick, she guided it into the delicious cunt of her friend, who sank
down upon my stiff-standing prick, deliciously impaling herself
thereon, and went off in a voluptuous discharge on feeling its huge
head engulphed to the utmost; she rose and fell upon it in an upright
position, until she had spent a second time, and had brought up her
passion to the wildest rage of lust, then falling into my longing arms,
she called out to the Count to shove his prick at once into her arse.
The Benson had, meanwhile, sucked and moistened the Count's fine prick,
making him as eager as the Egerton to be into her beautiful arsehole.
The Benson conducted it to the divine entrance of that rapture-giving
receptacle, which he entered at first with little difficulty, but as
the thickening of his prick by its further entrance began to stretch
the tender folds between our two pricks, the Egerton cried out for a
momentary pause, as it was producing the strange sensation that one
prick alone produces in the earlier stage of sodomitic embraces.
The Benson came to her aid by desiring the Count to withdraw about half
the distance he had gained and having whipt up some warm soapsuds she
well wetted his lower shaft and then he more easily recovered lost
ground, and gained a complete lodgement within the tremendously
stretched affair, for as I have said, the lower part of his shaft was
thicker than I could grasp.
The Egerton felt as if the two apertures were about to be torn into
one, and cried out for a few minutes' cessation.
We both lay still, beyond the involuntary throbbing of our pricks,
pressed as they were against each other, for the at-all-times-thin
membrane dividing cunt from arsehole was now stretched to the fineness
of gold leaf, and to our sensations did not appear to exist at all.
These double throbbings soon stirred up all the wild lubricity of the
Egerton's nature, first showing itself in the responsive inward
pressures of the delicate widely stretched folds of both receptacles,
then increasing in fiery lust, she cried out for us to begin gently our
first movements. We drew in and out in unison together, at first
slowly, but the Egerton finding that we were producing the most
excessive delight to her double-gorged receptacles, cried out---
``Oh! oh! It is heavenly; fuck faster, you angelic fuckers. I---Oh!
faster, faster. Oh! oh! it is too much.''
She spent in such an agony of ecstasy as to faint clean away.
We were not aware of this, not having ourselves spent as we had only
paused to let her enjoy her most heavenly discharge to the utmost.
Then, first with throbs, and then with in and outward movements, we
soon recovered her from her trance of excessive joy. Her passions were
more violently stirred than before. She wriggled her arse convulsively
sideways, she raved in the grossest bawdy terms, and so excited us that
we all three came to the final crisis in wild cries of the grossest
lust, and died away in an agony of bliss, so overpowering that we lay,
almost insensible, soaking in the sacred vases in which were compressed
our well-satisfied pricks. Meanwhile the Benson, wildly excited by the
scene enacted below her eyes, sought relief by kneeling beyond my head,
for we always fucked on the floor with mattresses spread widely around;
she then backed her splendid arse over my head, and brought her cunt to
my mouth, and I had gamahuched her continuously until my own delicious
spending annihilated all power of movement for the time being.
The Egerton, in the agony of her pleasure at the moment of the last
spend, had fastened her teeth on the glorious arse of the Benson before
her, and bit so hard as actually to draw blood and make the Benson
spring forward with a sudden start and cry. But we were all too lost in
the ecstatic joys to even hear the cry of pain she uttered.
At last the Egerton gave signs of returning life. The Benson had risen
and was eager for her turn, but Mrs. Egerton implored that she might
have once again a taste of these more than heavenly joys while both
pricks were still engulfed within her and thus avoid the pain of
entrance.
This was so reasonable that the Benson yielded with a good grace.
The Count, to indemnify her, begged her to stride over our two bodies,
so as to bring her delicious cunt to his mouth, which, as he was
kneeling, was just at the proper level; so he gamahuched, and embracing
her splendid arse, postillioned her at the same time; thus we were a
chain of delight.
This bout was drawn out to great length.
The Egerton must have spent half-a-dozen times, and when we both at
last jammed our pricks up in the ferocity of lust, making all three
give down life's essence in an almost killing ecstasy, she really
fainted quite away, and so alarmed us that we withdrew to use such
remedies as were at hand to bring her to; even then she was quite
hysterical. We laid her in the bed; she was relieved by a copious flood
of tears, which she assured us were those of joy at the exquisite and
overpowering delights we had conferred upon her. She begged us now to
gratify the Benson with the same ecstatic joys we had bestowed upon
her, and she would be a quiet, and delighted spectator of our doings.
It was now my own loved Mrs. Benson's turn to experience the
inexpressible delights of the double junction. From her love of my
splendid splitter, of which she had taken the first sweets, and which
had been initiated in her deliciously adulterous cunt into the divine
mysteries of love, and the still more sacred and secret joys of the
second altar dedicated to the worship of Priapean unutterably sensual
raptures; from this circumstance and the constant use of the rear
receptacle practised by her husband, whose prick was a very fine one,
the initiation into the \textit{double jouissance} was less nervously effected
than with the less used arsehole of the more delicate Egerton, but at
the same time two such pricks operating at once made her wince a little
before we were fairly engulphed to the cods, the banging together of
which in their close proximity added greatly to the stimulating of our
lust.
The sweet Benson lent herself most readily to the work, and seconded us
by her art in wriggling her arse and the delicious cunt and sphincter
pressures; enjoying herself at once and more rapidly than the Egerton,
she got four delicious discharges before our somewhat more sluggish
senses would allow us to come to the grand final crisis, which seemed
to stimulate the divine Benson to a point of raving lust, which showed
itself in cries of the grossest bawdy; shouting to us to shove our
pricks in further and faster, calling us all the loudest blackguard
names she could put her tongue to---absolutely roaring as the final
discharge seized her in the very same instant that we poured floods of
sperm into both interiors, she then sank, annihilated by the excess of
the voluptuous delights conferred upon her, but lay throbbing and
pulsating in all the after-joys of the utmost venereal satisfaction. We
lay long in this delicious inanition of such voluptuous excesses.
The darling Benson exercised her delicious ``nippers'' in both orifices,
which soon had their expected effect, and shortly the flesh gave
symptoms of its ``resurrection'' to mundane joys, after having passed
through the heavenly delights of Paradise, truly rising from the most
delicious graves in which they were lying so exquisitely buried. Like
her lovely predecessor she was eager for more, and if it were possible
our second course was superior to the first, at all events it was
longer drawn out, for the previous draughts on our slackening
appendages made the further delivery an effort requiring longer
pumping, and thus swelled the amount of pleasure by lengthening the
process before arriving at the grand final crisis.
The Benson, much more ungovernable in her passion than we were, must
have spent six or seven times in our last effort, and died away in, if
anything, greater abandon than in our first course, and eventually sank
completely overcome by the entire satisfaction of her raging lust for
the time being. We, too, both wanted a respite, so we all rose.
The two dear creatures when once on their legs found instant necessity
to evacuate their rear receptacles of the double cargo taken in, and
disappeared for a few minutes.
We all purified ourselves and well laved everything with ice cold water
to reanimate them the sooner. We then sat down to a stimulating
refreshment, in which we at least all drank a bottle of Champagne
apiece, in the midst of delightful and exciting bawdy wit and obscene
stories, in which our darling \textit{fouteuses} showed a witty proficiency.
In half an hour we began to take up our positions. It was my turn to
take them in the rear, but both begged off for that day. The pause for
refreshment had given time to make them feel sore after the great
stretching they had undergone for the first time, so my turn was
delayed for three days, that being the usual delay in their orgies to
prevent suspicion by too frequent absences, but not excluding any
opportunity that might occur for a rack-off in the mean time. So we
only each fucked them once and closed our exquisite orgy for that day;
parting with every expression of fully satisfied desires, and the
warmest kisses and embraces.
The Count and I walked to his apartments to refresh ourselves there
with hot tumblers of toddy; whiskey being a great favourite of his,
and, in his opinion, the best restorative after our exhausting efforts
with the two insatiable creatures.
He congratulated himself on my accession to these orgies, as being a
great relief to the burden he had had in satisfying both in both ways
when all alone with them.
However, the Count was an indefatigable and an unwearied fucker, but
two such insatiable cunts often had tried his powers to the utmost, and
was more than he liked to do at the interview, so he had found whiskey
toddy a remedy at once efficacious and agreeable. I myself with my
private excesses at home was glad to know so pleasant a restorative.
The Count and I became the most intimate and attached friends; through
him I perfected myself in Italian, and not many years after this,
passed some happy months with him in Italy after he had been amnestied,
returned to his country and recovered part of his once large property,
but of that hereafter.
I called next day on my adored Benson, who had developed into a
glorious woman, more lovely and lustful than ever.
We had but a moment to ourselves, and could not use it for amatory
purposes but as we both had much to relate we agreed to meet at our
house in Percy Street the next day.
This house was taken furnished for love purposes only, and merely an
old woman was kept to take care of and arrange matters when we were
gone; it was held in the Count's name but paid for by the two fair
users of it. They had latch keys each, and the place was kept ready for
everyday use.
The dear libidinous Benson avowed that she used it for other lovers
unknown to the Count or the Egerton; paying the old woman liberally,
she had all her own way.
We met there the next day, rushing into each other's arms, and then
assisting in the undressing we had three exquisite fucks, during which
the delighted Benson spent seven times, and then we could have a long
and uninterrupted talk over old times, and my after-doings. I told her
all, and how the Vincent, my elder sister, Miss Frankland, my aunt, and
Mrs. Dale had all thought me an innocent, receiving his first lesson in
their delicious cunts, and how true and wise had been her sage
counsels. She listened in wonder and delight, drew from me descriptive
pictures of our conjunctions and thrice interrupted my narrative to
have a delicious fuck to calm the excitement raised by the lascivious
descriptions of my acts with all those most glorious women. I told her
also of my intrigue at my lodgings with the two sisters and the Count.
My description of the latter set her off in raging lust, and produced
another most excellent fuck. But afterwards she told me I must find
other quarters in some place where Mrs. Egerton and she, or either of
them, could call and receive comfort without observation.
I told her I was inscribed for chambers in the Inner Temple, which I
had reason to believe I should get in a week or two. This much pleased
her, and it will be seen that I succeeded in getting just such a set as
exactly suited the great object in view, approachable without being
under the observation of others; commodious and agreeable, where all
that the dear Benson wished to be added to our set were brought
together, and the wildest orgies of the most insatiable lust were
carried on.
My description of my aunt, of Mrs. Dale, and especially of Miss
Frankland, now Mrs. Nixon, excited all the tribadic passions for which
the dear Benson was so famous.
Her clitoris, which was formerly prominent, was more so now, and she
dearly loved to gamahuche her own sex. In that way she took a great
fancy to my sisters, especially Eliza, who had all the same instincts
very decidedly pronounced. So we had the prospect of the most
consummate orgies in 'near view, and most gloriously in the end we
realised our wildest expectations.
In men we were more restricted; the Count would only consent to have
Harry Dale and my uncle in any orgy of which he formed part. He was
nervously timid about his sodomistic tendencies being known to many,
and only yielded on account of the relationship and the closer ties of
Harry Dale and myself, who eventually shared my chambers, and we lived
together, so that perforce he was obliged to put up with his presence.
He soon came to delight in having Harry's prick in his bottom when
fucking others at our orgies. It will thus be seen that the Count's
timid exclusiveness shut out from these family orgies my dear and
esteemed friend and master MacCallum More. However, in a certain sense,
it was an advantage, as we had at least the pick of the young ones, in
my two sisters and Ellen, who wanted very little persuasion to join our
Lyon's Inn orgies. And our excellent friend had some of his own set,
both male and female, to meet us either with one, two or all, for we
could not always manage to have the whole of the dear creatures
together. En revanche, dear MacCallum had several youthful ganymedes,
whose tight young bottom-holes were a great solace when cunts were
altogether absent.
We thus had two distinct and separate sets of orgies, which had all the
natural effect of novelty, and by exciting comparison, making us turn
from one to the other with renewed passions and power of enjoyment.
As my sisters could only come on Sundays, that was our exclusive day,
and we made an entire day of it, but I, in the end, persuaded Ann to
join our orgies with MacCallum, and she proved a first-rate addition in
every way.
I have already stated that she was of a most libidinous temperament,
and developed into one of the most lascivious and lustful of women one
could possibly find, and as she had rare beauties and splendour of
form, she was made to raise the most raging lust in man or woman, for
she, too, was as fond of tribadic indulgencies as my aunt or the
delicious Frankland. Her position as a servant prevented our
introducing her to Ellen or my sisters. As a matter of worldly prudency
it was best not to trust her with the knowledge of their complying with
all our lustful demands on their charming persons.
The Count, myself, and our two charming lovers met on the appointed day
to renew our delicious orgies. After both of us had fucked each dear
creature came the \textit{double jouissance}.
We took the adorable Benson first, that the scene of our erotic sports
might stir the lust of the darling Egerton to a greater heat. It was my
turn to lay my offering on the secret altar of Priapus, while the Count
filled her cunt with delight.
As I have before said, the arsehole of the Benson was much more used
that that of the Egerton, whose husband never dreamt of such a horror,
as she would call it. Mr. Benson, on the contrary, delighted in it, and
seldom passed a night without paying his devoirs to that delicious
aperture. So, although it was but the second time she had indulged in
the \textit{double jouissance}, yet her lust enabled her to take in with
greater ease my big prick in her arsehole, with the Count's fine prick
in front, than when our parts were reversed. She revelled in the wild
fury of raging lust, created by the glorious ecstasy of having a prick
in each aperture---screamed with wild cries of heavenly joy, spent
furiously, and eventually died away in an overpowering and
indescribable felicity. She soon recovered her senses, and begged for
another bout before withdrawing. Of course there was immediate
compliance and another more soul-killing encounter was run off with the
usual death-like termination.
I had continuously gamahuched the Egerton who straddled over the two
bodies below her, and brought her delicious cunt to my mouth, while my
arms encircled her beautifully formed and cream-like coloured buttocks,
at the same time acting postillion with two fingers to increase her
lustful gratification.
We purified ourselves after this, and drank some Champagne, then
standing stiff at the prospect of now possessing the lovely body of the
Egerton, we took up the same position as before, the Count under, in
cunt, the Egerton above, with her deliciously fair arse exposed to my
embraces first, and my big prick afterwards.
The adored Benson gave it a suck first, and well wetting the knob,
guided it to the narrow entrance of love's secret bower. Its head was
soon housed, and although still creating strange feelings, the previous
day's attack had made the entrance more facile.
With little halting we drove on to the first delicious discharge. The
second bout was all divine pleasure, and ever after the delicious
Egerton enjoyed it completely.
These delicious orgies with these two lovely women were indulged in on
every third day.
I became a favourite with both their husbands, thanks to a kind of
sheepish innocence that I had the power of putting on.
At the same time as my education had been well attended to and as I
myself was fond of study, attentive to my college instruction, and
anxious for a knowledge of foreign languages, I had become fairly
proficient in German and Spanish, and well read in French and Italian.
The latter was perfected by the Count's friendship, as we were much
together and spoke nothing else. Perhaps it was this which led to a
greater friendship for me on the part of Mr. Egerton, who was an
excellent Italian scholar. His wife's intrigue with the Count had also
perfected her, so that when we all four dined together Italian was the
only language spoken among us.
The dear Benson, too, was a perfect mistress of the Count's tongue, as
well she might, having it so often in her mouth; and as it is a soft
language that lends itself to love and lust, it became ours in all our
orgies.
The delicious Frankland, now Mrs. Nixon, returned to town with the
spring. By that time I was established in my chambers in the Inner
Temple, and had them simply furnished, but with every accessory for
love's combats in couples, or in the wildest orgies. The adorable
Benson inaugurated and dedicated them to the service of holy mother
Venus and her son Cupid, as well as the more lustful Eros.
The Egerton and the Count afterwards came to consecrate them to the
worship of Priapus, and we had a most delicious orgy on that sacred
celebration.
It was on this occasion that those two wild lustful creatures insisted
on seeing the Count and me in conjunction together. The Benson guided
me into the Count's bottom, while he was in the bottom of the Egerton,
and the Egerton conducted the Count into my bottom while I was
luxuriating in the delicious arsehole of my adored Benson. It satisfied
a longing desire on their parts to see man with man, and did not
displease either the Count or myself, who, in our secret hearts, had
each wished to possess the other.
The Count was a powerful and very hairy man, and had an especially very
full hairy arsehole, which to me was wildly exciting.
In that I differed from my dear friend MacCallum, who loved bare-arsed
youths with no hair there, telling me that coarse hairy arsed men
rather disgusted him, and although in his wide sodomitic experience he
had had such, it was with a certain repugnance that went against the
grain.
In that I differed from him entirely, the hairier and the coarser a
man's arsehole was the more it excited me. In that respect the Count
was exactly to my taste. He was very hairy all up the chink of his
arse, and had a very coarse skin and an almost black arsehole, so deep
a brown it was, the very sight of which always drove me mad with lust.
He as much loved me from another cause. His great letch was to frig a
fine prick while buggering the possessor, hence, as he had never met
with so fine a one as mine, he was insatiably fond of being into me and
frigging me at the same time.
We thus had two points of private attraction, that made us become the
closest of friends, but we did not let any of our dear female
participants know of the mutual joys of which they were not
participants.
The superb Frankland, now my guardian's wife, also came alone to my
chambers, and we had a renewal of all our wildest experiences. She told
me it was such a comfort to her, for although her husband, Mr. Nixon,
was very loving, and did all he could, still it was nothing but
exciting her to long for others, especially for my own huge prick, of
which she never knew but that she had been the first initiator of it
into love's delicious recess in either sacred grove.
So fresh and eager as she was for the fray you may easily imagine the
wild excess we indulged in, sucking, gamahuching, fucking, and
buggering. I cannot tell how often in every way her exciting and
glorious body carried me away to an excess beyond anything I could have
thought myself capable of.
When fairly exhausted, and we could uninterruptedly talk over all that
had occurred since I had left my mother's house, she heard in full
detail, for the first time, all my adventures.
I had given her, at the time of her marriage, a hint of how matters had
gone, yet without any details, which now she was voracious to hear. I
told her of my aunt's and uncle's apparent seduction of me, nor did I
hide our goings-on with young Dale, and my after-possession of Ellen
and his mother, who was the last to believe herself my seducer, for as
I told the delicious Frankland (I can never bear to call her Nixon), I
had followed her sage advice, and up to the Dale had played off the
innocent game with perfect success; but now that I was a man I threw
all that overboard.
``Indeed,'' said she, ``and who have you been throwing it overboard with.''
I laughed at her ready taking of me up, and then went on to a full
confession of all my intrigues.
She did not like my having taken up with the two servants, the nieces
of my late landlady, thinking it derogatory in one endowed with a prick
that any lady would be too glad to possess, but she was very much
struck with my description of the superb body and wonderful lubricity
of the Nichols.
It excited her much, especially when I told her that she had given me
the idea of her near approach in body and wantonness to herself.
It will be seen hereafter to what a closer alliance with the Nichols
this led. Pressing her enquiries, I acknowledged my intrigue with the
Benson, Egerton, and Count. This evidently excited her lust, as I could
see by the wild sparkle of her eye. It led to an immediate and
delicious fuck, and when recovered from its ecstatic finish, to closer
and more searching enquiry as to how I got into such intimacy, but I
had expected this somewhat jealous scrutiny, and was quite prepared for
it. I led her to believe they had been here nearly all the winter. I
told her my mother had desired me to call and see the Ben-sons as
friends of hers. I had done so. The Bensons quickly observed how
largely I was furnished, very soon gave me encouragement, of which I
did not want much after the late intercourse I had had with herself,
aunt, and Mrs. Dale.
Thus matters came quickly to their natural conclusion. She was
perfectly astonished at my powerful weapon, and as she and her dearest
friend already shared lovers, I was quickly introduced to her friend
Mrs. Egerton, and they had me together, and let me into the secret of
their intrigue with the Count, which was followed by my initiation into
their orgies.
My praises of these two ladies, and my saying how glorious it would be
for her to make a fifth, and my description of the exquisite body and
the tribadic tendencies of Mrs. Benson, fired her wild imagination, and
woke up all her tribadic lusts, and it ended in her begging me to give
a luncheon at my chambers to the Benson and the Egerton, that she might
be introduced to them, more especially as they really moved in a
society somewhat higher than Mr. Nixon's connections, although, in
point of wealth, the Nixons were far superior.
The little luncheon came off most agreeably. The ladies all took to
each other most warmly; seeing which, I boldly broke the ice, and
telling the Benson and Egerton that dear Mrs. Nixon was my first
initiator in love's mysteries, and as had both of them, the wisest
thing we could do would be to throw away all restraint and have a
jollification all round. To set them at their ease---for there was a
momentary hesitation---I pulled out my prick at full stand, and said---
``There's a prick worthy of all your exquisite cunts, and one, too, that
has enjoyed them all, and been enjoyed by every one of you. So throw
away all hesitation and let him enjoy you all again. Who is to have it
first?''
They laughed, and all approached and handled it, interchanging their
opinions upon its being the very finest one that any of them had ever
seen.
``Ah, now,'' said I, ``that is just the thing, you are at once put at
ease, then let us do it with ease; strip is the word, and let us have
it luxuriously.''
They laughed, kissed each other, and said the dear fellow must have his
way, and all at once proceeded to undress. The glorious and wonderfully
hairy body of the Frankland perfectly astonished them, and raised their
tribadic passions to fever heat, especially the Benson, who threw
herself on that glorious form in an ecstasy of delight, more especially
as the Frankland's passions being excited, her long red clitoris stood
out from the dense black mass of hair which covered not only her belly
and mount, but all down and around her cunt. Nothing would satisfy the
Benson but an immediate mutual gamahuche, for, with true tribadic
instinct, these two beautiful and libidinous women divined their mutual
letch for that particular lascivious inclination, and at once
proceeded, one on the top of the other, to wildly gamahuche each other.
The Egerton and myself seized the opportunity of having a delicious
fuck together, which we brought to a conclusion before the others had
satisfied their immediate desires.
The Frankland, who at first was under, was now above, and as she knelt
and pushed out her stupendous arse to bring her cunt over the Benson's
mouth, the sight of its hairy arsehole roused my desire to fuck it, and
my cock responded instantly, so kneeling behind her, I introduced it to
the well-known receptacle, and to her infinite additional delight,
sodomised her to perfection. This was another means of putting them all
at ease, and I fucked and buggered them all until neither handling nor
suction could get my prick to raise his head again.
You may easily imagine after this how delighted they were to make the
glorious Frankland a participator in our orgies with the Count. Nor
shall I forget the wild gaze of surprise and lust when the Count first
beheld the splendid and hairy form of the glorious Frankland when she
entered the room in all the dazzling splendour of her perfect
nakedness. These two natures were made for each other, both salacious
to a degree, both vigorous in body and untiring in the most libidinous
excesses of the wildest lust. Both hairy to a degree, showing the
meaning of that vast display all over both their bodies. They were
instantly attracted to each other, flew into the closest of embraces,
and sinking on the floor where they met, two strokes were racked off
before they came to a state of more moderation, amenable to our general
operations. It had been all the same an exciting scene to us.
The Benson was madly stimulated by the sight of the Frankland's superb
body; her long red clitoris, not satisfied with the double rack-off
with the Count, appeared only to be more excited, and stirred the whole
soul of the adorable Benson. She threw herself in reverse upon the
Frankland before she had time to raise herself, seized with her mouth
the wonderful clitoris, called upon me to fuck her from behind, and
then with fingers up arsehole and cunt worked furiously. The dear
Frankland responded on the fine clitoris of the Benson, and
postillioned me at the same time. We ran off two bouts in this
delicious position, and then with more regulated passions rose to form
more general combinations.
The Count had fucked the Egerton while we were engaged above the divine
Frankland. Our first pose was suggested by the Egerton, who had been as
yet less fucked than any. She had been also greatly taken with the
glories of the Frankland's superb body, and especially struck with her
extraordinary clitoris, and had taken the curious letch of wishing to
have it in her bottom-hole while riding St. George on my big prick. We
all laughed at her odd choice, but agreed at once, especially the
Frankland, whose greatest letch was to fuck very fair young women with
her long and capable clitoris. A fairer creature than the lovely
Egerton could not be found. The Frankland admitted that in her inmost
heart she had longed thus to have the Egerton from the moment she had
first seen her, and her delight and surprise at finding the dear
Egerton had equally desired to possess her, fired her fierce lust with
increased desire. I lay down, the Egerton straddled over, and feeling
the delight of my huge prick when completely imbedded, she spent
profusely with only two rebounds. Then sinking on my belly she
presented her lovely arse to the lascivious embraces of the salacious
Frankland, whose first act was to stoop, embrace, kiss, and tongue the
beautiful little pinky aperture, wetting it with her saliva, she
brought her fine long clitoris, stiff as a prick, and plunged within.
The letch that both had taken for the same indulgence lent enchantment
to the act, and their wild imaginations created an excess of joy that
the smaller size of the Frank-land's clitoris, in comparison with the
dimensions of our longer pricks, might not have led one to suppose
possible.
Twice we indulged in this excess, the women going off half a dozen
times to my once.
I had aided the Frankland by using a double dildo, which at once filled
both apertures. This excellent instrument was an invention of the
Frankland, which she had suggested to a Parisian dildo maker, and had
had it made in two or three sizes. It became very useful in our orgies,
as from disparity of numbers an odd couple were left out, when the
\textit{double jouissance} was in operation, and then the two outsiders, with
tongues and dildoes, could gamahuche with great satisfaction.
During our tribadic junction, with the Egerton fucked by the Frankland
in the arse, the Count had first fucked and then sodomised the Benson
to their mutual satisfaction. We all rose, purified, and refreshed with
wine and biscuits, while discussing what our next move should be. The
Count had not yet had the Frankland \textit{in culo}, and suggested, as it was
her introductory meeting, that the greater honours should be conferred
on her on this happy occasion, so I was to fuck her while he enjoyed
her in the rear quarters. The Egerton and the Benson should use double
dildoes to each other, or in any other way amuse themselves.
This was a most exquisite encounter, and with such unutterable
enjoyment that we hardly paused between the first and second, and it
was not until we had deluged thrice both interiors that we withdrew.
The delighted Frankland had never ceased spending, but so vigorous a
nature could easily have taken twice as much; but the other dear
creatures had now to be conciliated.
The Count next took the Benson in cunt while I blocked the rear
aperture, and the Frankland once more enculed the Egerton, who dildoed
herself in cunt at the same time; all of us running two courses. We
then rose, purified, and refreshed. When our pricks were ready it was
the Egerton who took me in front and the Count behind, and the Benson,
who had grown lewd on the Frankland's clitoris, was sodomised by her
and dildoed by herself. The Egerton still suffered a little in the
double stretching, so that we ran but one exquisite bout, enabling us,
whose powers began to fail to be re-excited, to finish with the \textit{double
jouissance} in the glorious body of the Frankland.
We carried this on until the midsummer holidays, when at their desire I
introduced the Benson, Egerton, Frankland, and the Count to my uncle,
aunt, Mrs. Dale, Ellen, and Harry, and we had some glorious orgies in
my chambers.
The splendour of my aunt's arse captivated the Frankland and the Count.
The latter soon got into young Dale's arse, which he did one day when
arriving for the very purpose half an hour before the appointed time of
all meeting. I was present, and was so excited at the sight that I
seized upon the Count's arse and delightedly astonished him by giving
the double enjoyment.
It was after this, as Harry remained to live with me, that he was
introduced to our general orgies, and thus we occupied all the dear
creatures at once, and most voluptuous and lascivious meetings we all
enjoyed, the Count occasionally giving us a private visit.
Meanwhile Ellen had been put to the same finishing school where my
sisters already were, with permission to go out with them on the
Sundays, when we always had a delicious orgy at our dear friend
MacCallum's. He, like the Count, had taken a peculiar fancy for the
tight young arsehole of Harry Dale, without altogether deserting the
women, especially my sister Eliza, whose delight in rear sports was
supreme, and she never would be fucked but when she had a prick in each
aperture, preferring mine in her cunt with either Dale or MacCallum
operating in the rear.
Knowing the hours when I could not be interrupted by any of my lady
friends, I did not neglect the superb Nichols, but had her and Ann to
come together for an hour and a half, from half-past nine to eleven
a.m., and most delicious fucking I had with both. I had equally
initiated them into the mysteries of rear delights, and both took it
with great gusto. Upon finding this I gradually descanted on the
exquisite delights of the \textit{double jouissance} with two male pricks,
filling with ecstasies indescribable the two apertures at once.
When once I had excited their desires on this point, I mentioned my
dear friend MacCallum More, as one in whom we could all confide, and
with some little hesitation obtained their consent to introduce him. I
had already mentioned the matter to him; told him he might think the
Nichols too old, but she was gloriously superb in body, and so
extraordinarily well preserved that her body was twenty years younger
than her face and her lust and fucking powers were far superior to a
woman of twenty-five. Besides, I hinted that he might persuade Ann, and
perhaps her sister Jane, to join our Lyon's Inn revels.
We met by appointment on a given morning. I advised MacCallum to come
sooner, and when the women came, under the pretence of his not being
able to join us that morning, I would get them stript, and when all was
ready he should appear in buff, and so break any \textit{mauvaise honte} they
might have at first undressing before him.
He was wonderfully struck with the superb body of the Nichols, and, as
the stranger, we gave him his choice. He clasped her in his nervous
arms, devoured her with kisses, and incontinently laying her down on
the mattressed floor, proceeded to fuck her in the good old English
fashion, with legs and arms around her body. Ann and I gazed for a
little on the splendid action of her aunt's arse, and the evident way
in which she milked the teat as it withdrew each time he heaved his
arse to re-enter with exciting vigour. We could hold no longer and each
ran a course of ecstatic delight ending in all the frenzy of lust to
die inanimate the next instant.
Our charming partners had spent repeatedly during our encounter. They
wanted an immediate renewal, but MacCallum suggested a change of
partners and of position, that is to say, fucking them on their knees
with their splendidly developed buttocks turned up but taking them in
the cunt.
This change was rapidly effected. We placed ourselves in such a
position that each could see all the action of the other. It was a
splendid fuck, and as our edge was taken off we drew it out a
considerable length, giving the dear recipients the opportunity of
spending four or five times to our once.
After recovering from the soaking after-joys of this delicious
encounter, we had some champagne and some smutty talk, as well as
outspoken praise of their splendid power of fuck; feeling their cunts
and they our pricks, till renovated and renewed, we arranged for
further action. As it was their introductory lesson in the double
enjoyment, the splendid Nichols had, of course, first choice. She chose
me for cunt, and to his intense delight, our dear friend for the rear
attack. Ann was to straddle over her aunt and me, and be gamahuched
both in cunt and arsehole by our friend. We had no difficulty in
hilting ourselves to the cods in both apertures, but so excited was the
Nichols that with the mere throbbings of our pricks on completely
housing ourselves, she spent, squealing like a rabbit. We gave her time
to fully enjoy it, and then commenced a slow, regulated movement, which
quickly drove the Nichols into a state of furiously raving lust, and
again she spent in an almost killing agony of delight, screaming with
excess of ecstasy. Again we paused to allow of the utmost enjoyment,
but renewed when her delicious cunt and arse pressures announced a
return of craving appetite. These pauses enabled us to bring on seven
overpouring discharges on her part, until she was quite exhausted,
especially when we both came together in an excess of joy that ended in
perfect inanition, on recovering from which we relieved the Nichols of
the double cargo within her.
She had already almost strangled me with her embraces in the
unutterable joys I had procured her. Rolling off on her side she drew
MacCallum also to her, to embrace him for the intense gratification he
had afforded her. We again refreshed the inward man after a
purification and laving with cold water, as a restorative. Then Ann
took up her position in her turn, for she, too, wished to try the novel
experiment with the smaller prick in her arsehole.
The Nichols felt exhausted for the moment, so lay on the sofa and
enjoyed the sight of our three persons in all the delirium of raging
lust and sodomy. The experiment enchanted Ann as it had overpoweringly
enchanted her aunt. She, too, spent seven or eight times before joining
us in our soul-killing discharge. The Nichols had laid still for about
two thirds of the time this bout continued, she then rose to straddle
across Ann and me, and was about to present her magnificently large
cunt to be gamahuched by MacCallum, but he begged her to turn her
bottom to him and heave it well up, while resting her hands on Ann's
shoulders. He then could first contemplate and handle her huge superb
buttocks, then transferring his hands to her clitoris and cunt, he
licked and tongued the grand aperture of her arse---rough, brown, and
corrugated, just my taste.
We had a most glorious bout, ending in all the ecstatic joys of
spending and after-delights. Ann was as greatly gratified with the
\textit{double jouissance} as her aunt had been before. We again laved and
refreshed, and closed this most delicious orgy with MacCallum first in
the Nichols' cunt, with my big and doted-on prick in her arse, which,
now she was used to it, pleased her more than ever.
In the same order we double-fucked Ann, although she expressed her
greater gratification of MacCallum in her arse and my splitter in her
cunt. Again we gamahuched them both, as time would not allow of our
resurrection, then they left us.
My guardian, at his marriage, had bought a house in Portland Place, but
the lease of its then tenant only expired on the 20th March this
spring, and before being occupied it had to be entirely new painted and
decorated, so that July was nearly at an end before they could
comfortably take up their residence in it. Meanwhile they had
apartments at a hotel near Hyde Park corner.
When once they were completely housed, which was not the case until the
middle of August, my guardian desired his wife to send the carriage for
the girls every Sunday morning. Hearing that Ellen was their intimate
friend, she became included in the invitation. This put an end to our
Sunday orgies in our friend MacCallum's chambers, much to our mutual
regret.
As far as Harry and I were concerned the ever thoughtful and delicious
Frankland came to our aid. Pretending that the girls must need walking
exercise, she always after luncheon proposed they should walk down to
their brother's chambers in the Temple, take him and Harry as their
further companions up to Kensington Gardens or the ``Zoo,'' and bring all
back to dinner.
As my guardian always took a siesta on Sundays after luncheon, for
being too old to fuck his wife every night, Saturday night, or rather
Sunday morning, when he had nothing in the way of business to trouble
him, was dedicated to two or three hours of extra dalliance with his
adored wife. She told me he was very amorous upon her, could not do
much fucking, indeed, she thought his efforts that way were even more
than he ought to do at his age, but he was never tired of gamahuching
her and posing her in every attitude when stark naked; of course she
lent herself to every wish of the old man, and had, even after great
persuasion, which only her love and attachment to him could have even
made her consent, allowed him the honours of her beautiful arse-hole.
This requiring, as he said, an extra firmness of prick, she further did
him the extra favour of toying and sucking his prick up to the utmost
stiffness. So she had made him absolutely adore her, and she could turn
him round her little finger. Her word and will was law, so she could do
as she liked.
She told me on several occasions that she thought he was exerting his
erotic powers to too great an extent, and that she did all she could to
moderate his excitement, but all to no purpose; he was infatuated with
the glorious charms of her body, or what is called cunt-struck, perhaps
the strongest passion that can seize on man and dangerous for a man of
advanced years. Well, his Sunday afternoon's siesta was long, and left
the Frankland at liberty to come to my rooms with my sisters, where
strip was the word, and fucking in every variety followed.
I soon found we must have other help; the pace I was going at was
beginning to tell, so with the consent of the darling Frankland I made
a confidant of the Count, and asked him to join our Sunday's orgy. You
may imagine with what joy he accepted, for apart from his delight in
seeing me in incestuous connection with my sisters, their young charms,
especially Eliza's, had great attraction for him, and then the
Frankland, so similar in lust and temperament. We had thus most
delicious orgies every Sunday afternoon, until the end of October of
the following year, when my sisters had finished their schooling, and
I, too, had left college, entered at the Middle Temple, and had been
for three months in a conveyancer's office, reading up previous to
being called to the bar.
It was then that Mr. Nixon's health gave symptoms of serious
disturbance, and his doctor recommended him to pass the winter in a
warmer climate. His wife suggested the advantage travelling would be
both to the girls and myself; she had only to express the wish to have
us all together, and we were warmly invited to join them.
We passed through Switzerland, Milan, and Florence to Rome, where we
took up our residence for four months.
The Egertons and Bensons happily spent the same winter at Rome.
My rooms were in an adjoining palace to where Mr. and Mrs. Nixon and my
sisters resided, there not being accommodation for me. I thus had a
charming entresol of five rooms all to myself; one of which looked on
and over the Tiber, and was in no way overlooked. To this room we
constantly resorted for orgies.
The Egertons had passed some winters in Rome, and she had two or three
clerical lovers, and these had introduced two others to the Benson on
her former visits, and all had been accustomed to general orgies. You
may imagine the delight of these priestly debauchees when they found
themselves introduced to our circle of three fresh cunts, and such
splendid ones, and all without any mock-modest prejudices but up to
every excess of lubricity. So to five women we thus had six men, and
eventually a very handsome young priest, debauched by the others,
joined our party, and we carried on the wildest and most extravagant
orgies of every excess the most raging lust could devise. We made
chains of pricks in arseholes, the women between with dildoes strapped
round their waist, and shoved into the arsehole of the man before them,
while his prick was into the arsehole of the woman in his front.
These holy fathers had immense resources in the way of infinite
variety, stimulating to excesses of debauchery that very soon brought
the rod into requisition.
We all from time to time enjoyed the double coition, the women
invariably so at every meeting.
These holy fathers had all very fine pricks, but none so large as mine,
and many of them loved to have my prick in their arses when opportunity
offered. In such delights the winter passed rapidly away.
In the spring Mr. Nixon's health seemed very precarious, and we moved
to Naples, where from necessity our extreme indulgence in venereal
excesses was much curtailed.
In May we returned to England, but poor Mr. Nixon was evidently fucked
out. The Frankland told me that the more his health failed the more
lewd he seemed to grow. His passion for gamahuching her cunt had
increased, and even his prick seemed to gather new life as life ebbed
away, for hardly a night passed without his fucking her, at night in
the cunt, and at morning, in full daylight, kneeling and feeling her
splendid arse, he took her in the rear aperture. He and she too felt it
was killing him, but his infatuation was overpowering, and he declared
if it did kill him he could not die a happier death. In fact a month
after we returned he had an apoplectic fit actually when his prick was
spending in her arsehole. He lived but a month afterwards. He left all
his property to his wife absolutely, with legacies of \pounds2,500 to each of
my sisters, and \pounds1,000 to me.
This sad event cast a gloom for some time over all our pleasures.
The Frankland took my sisters to reside with her, but all went down to
spend the first three months of mourning quietly with my mother. She,
too, took ill when we were with her, and died before the three months
were up. This drew me down to home, now mine, and the dear Frankland
continued to stay with us for two months longer, and then left for
London. We three orphans remained for all that winter in our old home,
settling a variety of things.
My sisters now with their succession to some \pounds600 apiece, the \pounds1,000
left them by our uncle, and the \pounds2,500 by Mr. Nixon, and the \pounds400 which
I promised them as a marriage present, and with their great beauty of
form and face, for both had grown into remarkably fine young women,
became very eligible matches.
Many country families sought us out after the first three months of our
mourning, and several offers were made to the girls. They were both
somewhat fastidious after the life they had led, but eventually both
were married. Mary to a very nice fellow, who proved, as she told me, a
first-rate fucker. He got her with child, and they had a son, a fine
boy, in the tenth month of their marriage. She was very happy, now and
then coming to see me, and getting a jolly good fuck from my renovated
prick, for now that he was lying fallow, my somewhat exhausted system
was getting quite recruited.
Alas! poor Mary lost her husband by cholera in the second year of their
marriage. He had a handsome estate, and left her well off, and sole
guardian to his son, who grew up a very fine fellow, and when at
puberty became the solace of his widowed mother, who had initiated him
into all love's mysteries.
Eliza was not quite so fortunate as her sister in her husband; he was a
good sort of man who, one would have thought, would just have suited
the hot temperament of Eliza, well and powerfully built, and with an
air of being a man of erotic passions; but he turned out to be of a
languid unimpassioned nature, who could not imagine any other manner
than simply mounting on a woman's belly and fucking her once a night,
and with no conception of using either preliminaries or aids to her
passions. So that he left poor Eliza only in a state of excitement
instead of giving any satisfaction to her lascivious nature. She did,
eventually, work him up to good night and good morning, but for her
full satisfaction she used too seek elsewhere, and even to content
herself with the embraces of a man servant, who, if not good looking,
proved to have a splendid and powerful prick, and nearly daily gave her
comfort. She also occasionally came to me, when she had both apertures
well exercised, and left me much comforted.
She never had any children, and so managed her intrigues as never to be
found out.
I returned to London in the spring, and was called to the bar.
I went the western circuit for odd assizes, and then abandoned the bar
as a profession.
Harry Dale, with more perseverance, as well as greater necessity for
exertion, continued in the profession, was duly called to the bar, and
eventually became a rising and successful barrister, and at this period
of our old age is now a distinguished judge.
But to return to our earlier days.
Harry and I carried on our intrigue with the Nichols and Ann, aided by
our dear friend MacCallum. Also from time to time with the Benson,
Egerton, and Count, to which generally the darling Frankland brought
her exquisite charms to intoxicate us with pleasure.
This delightful reunion was sadly affected by the loss of the Count,
who received an amnesty---I think I before have said he was a political
exile---returned to his own country, and we never again had his
delightful aid in our sadly shortened orgies.
The Count and I met in a future year at his old castle on the hills of
Pied, of which I shall have much more to say on a later occasion.
It was a sad loss, especially for the Egerton, who dearly loved the
Count. He had been her first lover, indeed, her initiator in the real
mysteries of Venus. It will be remembered that her husband was one of
those old insensible natures that think it is only necessary to hastily
``piss their tallow,'' as Falstaff says, as quickly as they can, and
leave a poor woman just sufficiently excited to be madly anxious for a
thorough good fucking. It is these insensate cold-blooded husbands who
raise, without satisfying, their wife's erotic passions, and drive them
perforce to seek salacious comfort in other arms.
Oh! how many women if only fucked with some regard to their own
naturally lewd feeling, would have never committed adultery or made a
scandal. Many are the women who have told me, with tears in their eyes,
of the cold insensible conduct of their husbands, who, never fucking
them but when their sluggish natures felt the want, then turning upon
them without the slightest preparatory handling or embracing, mount,
shove it in, give a few in-and-out movements, spend, and then withdraw,
just as they have done enough to excite their poor wives' passions
without satisfying them, and thus leaving them a prey to inordinate
longing that forces them to seek the relief to their passions the
selfish brutes of husbands had only raised without allaying.
I remember an intrigue I had with an Italian Countess. Her husband, a
tall and very capable man, was an extreme bigot, who thought it deadly
sin to indulge in any caresses or carnal excitement, or even for his
wife to expose any naked flesh to raise concupiscent ideas, so she had
to have her nightgown closed up to her throat, with long sleeves and
skirts, in the centre a slit through which he performed his duty when
in want of relief to himself. He never kissed or embraced her body at
any time, but lay like a log by her side, with his back turned to her.
When his own passions prompted him to fuck, which was very seldom, he
was naturally quite ready and rapidly finished his coup. He used to
turn to her, waken her up with a shake, cry out, ``Marietta, porgemi il
vaso generative'' (Marietta, reach me the generative vase), upon which
she stretched herself on her back, he got on her without lifting her
petticoats or feeling her cunt, but opening the slit, pointed his prick
to her cunt, thrust it up to the hilt, and being himself in want of
spermatic relief, in a very few strokes spent, just staying in long
enough to ``piss all his tallow,'' and then withdraw, turning his back
again to sleep, leaving his wife just sufficiently excited to have
enjoyed it, and thus left her madly longing for the satisfaction he did
not afford. She said he was quite capable, too, of giving satisfaction
if his bigotry had allowed him. We used to fuck at a tremendous rate,
and I always commenced with a ``Marietta, Marietta, porgemi il vaso
generative,'' and then proceeded to fuck and laugh like mad.
Of course, irritated as her hot passions were by her booby of a
husband, she resorted, not only to me, but to whomsoever she could get
to satisfy the cravings of her irritated cunt.
The Bensons and Egertons again left in the autumn for Rome.
The Frankland, not yet out of her year's widowhood, did not go much
into society, and we saw much more of her than before. She came at
least three times a week to my chambers, when Harry and I gave her the
comfort she so much required; first each fucking her singly twice over,
and then three double-pleasure fucks, with change about in the
apertures: finishing off with a mouth fuck from one or the other, and a
double gamahuche.
About once a week the amorous and delicious-fucking Nichols with Ann
would come of a morning, when we managed to send both away satisfied
for the day.
When winter drove our friend MacCallum home from his fishing, we
renewed some excellent orgies at his chambers, where Ann, and
afterwards Jane, occasionally came. By the way, Jane's arse had
developed in an extraordinary manner, and became one of the most
exciting delights of our orgies at MacCallum's. He also now joined in
our morning encounters with the Nichols and her niece.
At Christmas time the Frankland, Harry, and I all went down by
invitation to the Rectory, where uncle welcomed with great delight the
glorious and exciting Frankland. Mrs. Dale and Ellen joined our party.
Dear aunt positively devoured me with her caresses, and before I was
shown up to my room, had drawn me into her little room downstairs, had
a suck at my prick, leant her body on the table, stuck out her immense
arse, and had me into her cunt for a rapid rack-off; but this only
excited me to an immediate renewal, for the touch and sight of her
splendid buttocks instantly produced a stiffness, she herself in the
middle of my movements in front, drawing my prick out of her cunt and
guiding it into the inviting entrance to the secret altar of Juno and
of Venus Callipyge. Both courses were run off at a gallop, and were a
momentary allaying of the insatiable salacity of my most lewd and
lascivious aunt. She then next conducted the Frankland, I can never
call her Nixon, into her bedroom, under pretence of showing her to it.
She no sooner had her there than up went her petticoats, and aunt glued
her lips to the wonderful clitoris of the divine Frankland, and using
fingers up both apertures, made the Frankland quickly give down her
first offering to the obscene god.
As soon as aunt's tribadic rage to possess the Frankland was thus
abated for the moment, she allowed Mrs. Nixon to remove bonnet and
shawl, but then as quickly demanded and obtained a double gamahuche.
The Frankland the more readily consenting as she knew aunt had taken
the keen edge off my lecherous appetite, and she would revel in the
thick raging sperm I had shot into both orifices. These preliminaries
settled, we were able to be much more tranquil all the afternoon.
The Dale and Ellen came to dinner; I slipt into their room when all
were dressing for dinner, and had a delicious rack-off in both their
lecherous and longing cunts. Uncle had equally enjoyed the tight
favourite arsehole of Harry Dale, he having conducted him to the
well-known summer house for that purpose as soon as we arrived.
We could all thus peaceably enjoy the good things set before us, and
during our wine after dinner exchange accounts of all events that had
passed since last we met, and they were varied, for Mr. Nixon's death
and legacies to my sisters and myself were subjects of congratulation,
while the death of my mother was, on the contrary, one of condolence
and sympathy.
By ten o'clock we all broke up, but with the whispered request to all
to repair to aunt's bedroom half an hour after the household had
retired. We were all too interested in the delicious orgy there to take
place to fail. Blazing fires in both that and the adjoining room had
been kept up all the afternoon; plenty of lights were burning so as to
illuminate all sides at once. We all met in mere night wrappers, and as
soon as we were assembled and the word given ``off,'' they were thrown
aside, and we all stood in nature's lovely nakedness. Aunt, in her
eager and lascivious inspirations, flung herself on my naked body, drew
me to the bed, and had me into her longing and delicious cunt at once,
and with legs and arms thrown round me, was instantly pressing
furiously forward, notwithstanding the remonstrance of my uncle, who
wished to arrange a general plan of operations so as to include all at
once. Aunt's voluptuous eagerness produced a rapid discharge on her
part. Seeing this, while she was in the momentary ecstasy of spending,
he was enabled to drag me from her arms, fortunately before I had
weakened my powers by spending for a fourth time that day. Aunt, too,
was now in a condition to listen to reason, and bring her ideas of our
after-combinations into play.
As we had brought the Count with us for a week's stay, we were just
four cocks to four hens; so we could couple in the first instance on an
exact equality, it being necessary by previous good fucking to bring
the women's passions up to a boiling heat of lust to make them enter
into our greater excesses with all the wild energy of the most
salacious lubricity. Aunt had taken a great fancy to the Count when up
at midsummer.
Uncle was most lecherous on the glorious Frankland. I took most readily
to the luscious and lascivious Dale, who was equally eager to repossess
the prick which she firmly believed she had initiated into all the joys
of cunt, and a most delicious fuck we had, she spending furiously and
frequently to my once.
Harry was equally pleased to pair off with his loved cousin, whose
maidenhead he had undoubtedly taken.
The women would gladly have had each fucker run a second course without
drawing. But both aunt and uncle opposed this, as both more exhausting
and less variety. So aunt chose me, uncle took the exciting young cunt
of Ellen, Harry turned on to his mother's cunt, from whence he had
originally come into the world, and the Count got the glorious
Frankland, of whom he was never tired. This course was more prolonged
by the men than the first, with the object of somewhat allaying the
insatiable lust of the women by making them spend infinitely oftener
than their fuckers.
We so managed matters that we all came together or nearly so, and the
women followed suit at the last final crisis, which was ushered in with
wild cries of lust, and then a sudden overpowering silence fell on all
as they lay panting in all the after-joys that follow the ecstatic
discharge of life's essence.
We rose for a general embrace of our naked bodies, then a romp, and a
mutual slapping of arses and seizing of pricks and cunts, a very
exciting game, which soon brought evidences of renewed vigour in all
except poor uncle, who required a longer pause and an extra excitement
before he could indulge in a third encounter.
The Count took the delicious arsehole of the Frankland, who begged for
me as her fucker. Aunt got Mrs. Dale under for a double gamahuche,
while Harry crammed his prick up aunt's arse. Uncle enjoyed a delicious
gamahuche with Ellen, who sucked his limp prick all the time without
any success.
This was a delicious bout for us all, and ended in heavenly raptures.
Our second double couplings were: myself in my aunt's cunt, which
incest stimulated uncle to a stand, and he took to his wife's arse
while her nephew incestuously fucked her cunt. The Count took to the
delicious and most exciting tight cunt of the Dale, while her son
shoved his prick into his mother's arse, to her unspeakable
satisfaction. Ellen and the Frankland amused themselves with tribadic
extravagances.
This bout was long drawn out, and afforded inexpressible ecstasies to
all concerned. And after the wild cries and most bawdy oaths that
instantly preceded the final ecstasy, the dead silence and long
after-enjoyments were drawn out to a greater length than before. After
which we all rose and purified, and then took refreshment of wine and
cake, while discussing our next arrangement of couples.
Uncle had, fortunately for him, managed not to spend in the last bout;
he, therefore, was still capable of entering an arsehole, and he chose
the delicious arse of the Frankland to receive this final offering, for
after that he was done for that night. I was below engulphed in the
exquisite cunt of the Frankland. The Count fucked Ellen while Harry was
into her behind. Aunt and Mrs. Dale mutually gamahuched and dildoed
each other. This, too, was a long-drawn-out affair and ended in
perfectly convulsive ecstasies and cries of the wildest sensuality that
our most salacious passions could prompt.
I then took my aunt's arse while the lecherous Dale was underneath
gamahuching and dildoing her, and by putting the Dale close to the edge
of the bed, the Count stood between her legs, which were thrown over
his shoulders, and thus he fucked her, having taken a letch to fuck her
cunt, which was an exquisite one for fucking: her power of nip being
nearly equal to the Frankland, and only beaten by aunt's extraordinary
power in that way. We thus formed a group of four enchained in love's
wildest sports together.
The Frankland was gamahuched by uncle while having Harry's prick in her
arse, Ellen acting postillion to Harry's arse while frigging herself
with a dildo.
The closing bout of the night was the Count into aunt's arse, my prick
into the Frankland's arse, Harry enjoying an old-fashioned fuck with
his mother, and Ellen under aunt to dildo and be gamahuched and dildoed
by aunt. We drew this bout out to an interminable length, and lay for
nearly half an hour in the annihilation of the delicious after-joys. At
last we rose, purified, and then restoring our exhausted frames with
Champagne, embraced and sought well-earned sleep in our separate
chambers.
I slept the sleep of the just, and awoke late to find aunt sucking my
stiff-standing prick at the very instant it was filling her mouth with
a deluge of creamy spunk. She sucked up to the get all out, and in
doing so brought him up to the scratch again, so jumping out of my low
bed I made her kneel on it, stick out her enormous arse, and licked her
reeking cunt until I could stand it no longer. Then bringing my huge
prick I plunged in a single vigorous thrust up to the very top of her
cunt, and made her squeal and spend with that alone. Pausing to let her
enjoy it, I recommenced and ran a delicious course in that most
exquisite cunt, and would have done so a second time, after a pause of
ecstasy, if Harry Dale had not rushed into the room to say that all
were impatiently awaiting me to sit down to breakfast. Aunt just stayed
to give a final suck to my prick, and then vanished.
I hastened to wash and dress, having sent Harry off to beg they would
not wait for me.
On joining them the sly jokes they cut at my apparent laziness proved
that they knew of the cause of detention. I looked at dear aunt, and at
once saw by the air of gratification on her dear plump face, that she
herself had been boasting of her exploit, for it was all her own doing.
Being Sunday, we all went decorously to church. The doctor gave us a
very unctuous sermon on the goodness of virtue and chastity. It was a
really fine sermon, and delivered with an unction that forbade the
possibility of supposing that the preacher could be in reality the very
reverse of his doctrine. It much pleased some of the country families,
and one or two with their wives waited for the doctor leaving the
church, to compliment him on his eloquence and admirable teaching. The
flattered doctor ended by inviting two rather distant residents to
luncheon at the Rectory, so that we formed a numerous party, all on our
best behaviour. It was quite edifying to hear the pious and virtuous
remarks of the admirable Frankland, and the no less virtuous and
correct Dale. It gained them the entree into the exclusive set of both
these high country families, and eventually led to an excellent
marriage for the dear little Ellen. So much for the success of
dissimulation. Vice playing the part of virtue, and succeeding to
perfection. So goes the world. One thing is certain, that on this
occasion it enforced chastity, in one sense at least, that we had no
opportunity of practising vice that afternoon. The charming
Frankland-Nixon made a great impression on the wives as well as
husbands, to be sure it was well known that she was a very wealthy
widow, and they may have had some design of securing her for a son,
nephew, or at least having the chance at it. She thanked them with that
grace and charming ease of manner which so distinguished her and made
her so captivating, excusing herself from visiting, during the first
year of her widowhood, anywhere but among family friends, and as her
late husband was Charles Roberts's and his sisters' guardian, she
considered his family as almost her own. They hoped to have the
pleasure of seeing her some future day.
The whole visit passed off very pleasantly, and left us only an hour
for a stroll in the garden and time to dress for dinner. It will be
recollected that the doctor was a great exacter of full evening dress
at dinner, as tending to keep up proper appearances.
We met at the accustomed hour at night in aunt's room, in the full
dress of Adam and Eve before they munched the apple.
This night was dedicated chiefly to sacrifices to Venus Apostrophia,
for the doctor commenced by having the Count while he was fucking his
wife, and when able to get his fine old cock in for another go, would
only again have it in my arse, while I was doing the same to aunt's
glorious immensity with the Count below fucking her.
That was the end of poor uncle's powers for that night, but he
gamahuched all the women at the finish of their encounters with us
three men. We gave them all the \textit{double jouissance}, while those
unoccupied carried on their own little game with tongue or dildo.
It was again a night of most exquisite enjoyment.
The following and remaining nights of our visit brought into
requisition the rod before uncle could get his dear old prick to stand,
and myself tailed off on the next Sunday night, the last of our visit,
so that uncle seeing what he called the laziness of my prick, seized
the rod, and gave me as sound a flogging as ever he had done in my
schoolboy days. The fact was that he had been longing to renew on my
arse his letch for giving a really severe whipping. He had already by
dint of the same punishment fucked the arses of the Count and the
divine Frankland, and was now so excited anew that his prick stood as
stiff as ever it did; and my red excoriated arse excited and renewed
his very fine prick; but first I insisted upon moving to aunt's arse,
who at the moment was having a last fuck from the Count, and this
incestuous group closed our orgies on this occasion, for we left for
town the next day.
After breakfast in the morning I slipt into the Dale's room, and had a
parting fuck both with her and Ellen. Harry came in while we were at
work, Ellen under the Dale gamahuching her, and I above administering a
rear adieu. Harry stopped us for a moment until he could withdraw Ellen
and take her place, that he might have a parting fuck with his loved
mother, who thus had the two pricks she most loved in the world into
her together. We drew our pleasure out to the utmost length our lust
would allow of and spent in the most ecstatic joy that poor human
nature could support.
Aunt had gone to the Count's bedroom at the very time we were meeting
in Mrs. Dale's. Notwithstanding which, her insatiable cunt made her
draw me into her sanctum downstairs for a final fuck at the last moment
of our parting.
Harry Dale staying behind to pass a week at home with his mother, the
adorable Frankland, the Count and I returned to town together. On the
journey up we agreed to dine at Very's in Regent Street, and have a
comparatively quiet night all together at my chambers, which we did,
luxuriating in having the glorious naked body of the delicious
Frankland between us. After we had each bedewed both her front and rear
orifices with our life's balmy essence, we slept soundly till morning,
when we renewed our double offerings on those glorious and delicious
altars, then breakfasted.
This was the last occasion but one of our having the Count, whose time
for departure to his own country was drawing near.
He left that day on a visit to a family in Scotland, whose son and heir
was really the fruit of his loins.
On his return some fortnight later we again passed a night with our
exquisite friend the Frankland, and being both fresh from the country,
we administered so many delicious coups to both apertures as quite
contented her salacious love of prick. We parted next day with our
loved friend the Count, but not for the last time, as I shall relate in
its proper place---a delightful visit we paid to him in his old
ancestral castle, and an after-rencontre with him and his sister in
Turin.
I saw my loved Frankland to her home and left town myself the same
afternoon for my home in the country, to arrange for various repairs
and alterations required on the property.
I took my dear friend MacCallum with me. We spent a pleasant ten days,
varied with a visit first from one of my sisters and then from the
other, for two nights each, and jolly nights we spent fucking in every
way.
Mary's belly was up, but she declared it only made fucking more
delicious than ever to her, still more with the \textit{double jouissance}, in
which she preferred the smaller prick of MacCallum to mine in her arse.
When Eliza came she stayed a third night, and taxed our powers to the
utmost; she was such a glutton for fuck on this occasion, declaring
that her husband's want of power, as well as tact, left her more lewd
after his fucking her than she was before, so that she had been forced
by the excess of her unsatisfied lust produced by her husband to have
recourse to the fine prick of her footman, a powerful young fellow,
otherwise very plain, and not likely to inspire jealousy to any
husband, but with whom she rarely could do more than get a rack-off in
a hurry, which was far from satisfaction sufficient for her hot
passions. It was this that made her revel with such insatiable desires
in the possession of our almost untiring pricks. Differing from Mary in
her love of rear-fucking, it was my big prick she loved best to have in
her arse, while MacCallum's lesser shaft satisfied her less exacting
cunt. She was certainly one of the lewdest creatures ever made, wildly
lascivious and full of variety. She had the most engaging ways with
her, and could raise a prick from the dead. She was a worthy pupil of
the Frankland, and had all the love and longing for prick, and cunt
too, that our deliciously insatiable aunt was so famous for. She grew
older, and becoming one of the most desirable women, I never tired of
fucking her in both orifices whenever the opportunity presented itself.
I returned to town just in time to have a parting orgy with the Count
and the Frankland in my chambers, which I before said was a night of
the wildest orgies.
MacCallum was called to the country by the illness of some of his
family, and was absent for six months, so I was left with Harry to have
occasional orgies together with the Frankland three times a week, and
with the Nichols and Ann or Jane once a week by way of variety, but as
they only came for a morning visit, these were not exhausting
encounters, so that we lay comparatively in fallow, till the return of
the Benson and Egerton, when they and the lovely Frankland taxed us to
the utmost twice or thrice a week.
Thus time progressed. The Frankland had been a widow for nearly two
years when she proposed to travel for two or three years without
returning in the interval to England. She wished me to accompany her,
and made a most surprising and unexpected proposition to me.
She said, ``Charlie, my own darling, I love you more dearly than ever.
It is true I am considerably older than you, but you are now
twenty-five years of age, and, therefore, a full-grown man. I wish to
endow you with all my great wealth, and I offer you my hand in
marriage. Do not suppose I want to monopolize this dear prick.'' (We
were in bed naked, and had just concluded a most exquisite fuck.) ``No,
with our love of variety we will still seek it out, but as husband and
wife we can do so with perfect ease and safety; whereas if not married
and travelling together we should be compromised at every city we stop
at. What say you, my darling Charlie?'' Here she threw herself on my
bosom with loving eyes upraised to mine.
``Say, beloved of my soul! Why, look how the very idea has raised my
prick to instant life. If anything in the world could delight me more
than another it is your generous noble offer. To dedicate my life to
the woman I love more than any other is a joy greater than I can
express. I thank you from my soul, adorable creature as you are. Oh!
come to my arms as my future wife and let us revel in the glorious
idea.''
Such was the way in which this happiness was conferred on me, which
endured for long years, although, alas, my widowed heart now all
hopelessly ever regrets that most lovable of women and best of wives.
Oh, what happiness it was as long as I possessed her.
We were married in a few days after this by special license.
The Benson and the Egerton were present and Harry Dale was my best man.
We adjourned to her house, now ours, to breakfast. They also stayed to
dinner and slept at our house, that we might celebrate our marriage
with a parting orgy, for we announced to our friends that in marrying,
so far from renouncing our orgies, we meant our union to promote ever
varying ones, and that on our return we would renew the exquisite ones
we had so often enjoyed with them.
Harry and I did all we could on that happy occasion to satisfy three of
the finest women in the world, whose delicious power of fucking was
never surpassed and rarely equalled.
Oh! we had such a delicious night. As to the women, their amorous
gamahuching of each other was ever renewed, and was most exciting to
see.
After breakfast that morning they stopped to see us off, and threw old
slippers after us for luck.
We posted down to aunt's for a day and a night on our way to the
continent.
They were, of course, delighted with my marriage as bringing great
wealth into the family, indeed, my darling presented aunt with a cheque
for \pounds1000.
Mrs. Dale and Ellen came over, and we had another delicious night's
orgy, in which all exerted themselves to the utmost.
We parted from dear aunt and uncle, Mrs. Dale and Ellen, after
luncheon, and posted down to Dover; slept at Birmingham's Hotel, where
we had our real first night's fucking all to ourselves, enjoyed it in
moderation but in every endearment that two lovers could devise.
We crossed to Calais next day.
The sea was smooth at first, but we found it after passing the Foreland
very rough. My dear wife suffered severely; fortunately I myself never
felt better, and was thus able to devote every attention to the dear
sufferer. It left her even after we landed with nausea and a severe
headache, so that night at Devaux's Hotel we slept each in one of the
separate beds in the same room, as is usual in French hotels, and
indeed in continental hotels in general.
My darling wife was far from feeling well next morning, but fancied
that posting on to Abbeville would rather tend to recovery than
otherwise. We accomplished this easily between breakfast and dinner,
found a very comfortable hotel with very fair cooking and excellent
wines. My wife enjoyed her dinner, and felt something like herself
after it. We slept together by bringing the two beds side by side, but
only took a single fuck before sleeping, and next morning a double one.
We spent the day at Abbeville, wandering through its quaint streets and
seeing its fine unfinished cathedral. The following day we posted to
Amiens, visiting its very beautiful cathedral, posted the following day
to Beauvais, again slept, passed next day there, and on the following
day posted up to Paris, and drove to Meurice's Hotel in the rue de
Rivoli.
We had previously written for a set of rooms \textit{au premier}, overlooking
the Tuileries Gardens, with orders to have dinner ready at a given
hour. We arrived just in time to change our travelling costume and to
sit down to a luxurious dinner. Here, as we had ordered, our bedroom
contained a proper large bed for both to sleep in. This hotel being
much used by the English was furnished with French taste but English
comfort.
The dilatory manner of our journey, the agreeable breaks we had made at
different interesting towns had quite restored my beloved wife to all
her accustomed health, energy, and lubricity. The comfort of the bed,
the stimulating cheer, and the excellent wine also nerved me to meet
her utmost lasciviousness, and we had a night such as we used to have
when I first had her in my mother's spare bedroom.
We recalled those happy days, and revelled in every lascivious act of
the hottest lust. My adored wife excelled herself, and I myself was
fully up to the mark; we fucked ourselves to sleep, with prick left
soaking in her deliciously tightest of cunts, so that on awaking in
full daylight I found my cock stiff standing in her cunt, which was
giving it most delicious pressures, quite involuntarily, for the
darling was not yet awake. I roused her by gentle movements, and the
frigging of her long delicious clitoris, so that she awoke to joys of
which we never tired. On this occasion natural wants compelled a
temporary withdrawal to relieve our distended bladders. We found that
it was already past ten o'clock, so she smacked my bare bottom and sent
me off to my dressing-room, that both might get ready for breakfast,
for which our appetites were already craving. I slipt on a
dressing-gown, went into our sitting-room, rang for a waiter, and
ordered breakfast to be got ready immediately, so that by the time we
were dressed it was on the table all smoking hot, and we sat down and
did full justice to it.
We spent several days in visiting the wonders of Paris.
I had heard of a famous bawd residing at No. 60, rue Richelieu, and
another, Madame Leriche, in the rue de Marc, where they had rooms, from
which, through cleverly arranged peep-holes, any operation in the next
room could be distinctly seen.
Madame Leriche's girls were instructed to get the finest men they could
see in the street, to bring them in, and there to pretend to be so
struck with their beauty that they would not be content without having
them quite naked, stripping themselves also. When quite naked they
caressed their pricks, waltzed round the room, taking care to stop
exactly opposite each hidden opening, and there caress, handle, and
show the standing prick to any looker-on, eventually fucking in such a
position as all peepers could fully see and enjoy.
The fun of the thing was the perfect unconsciousness of the men as to
the purpose of all these gyrations. They took it proudly as a homage to
their virility, and the power of their charms over their new conquest,
and were doubly lustful in consequence, little imagining it was all a
well-acted scene, got up for an exhibition to please others, and show
all their virile gifts. Sometimes both man and girl were very
attractive, and I used to fuck my loved Florence while in the act of
peeping.
The place where we sat to see was a small narrow room, with just space
for a couch on one side and two chairs at the end, next to each
peep-hole. Three other similar narrow rooms looked into the same
operating room.
One day we had an exciting fuck from the exhibition of a very fine man
fucking his girl with a splendid prick. We were kneeling on the couch
with my prick soaking in the quietude of the after-joy. We heard a
scuffling with suppressed bawdy exclamations on the other side of the
thin partition next to us. We, too, had made use of bawdiness. I had
whispered to Florence how deliciously tight her hairy cunt was, and how
splendidly her enormous arse moved below my eyes as I fucked her.
We now discovered that the couple next to us had overheard us, for we
could just hear her ask if her arse's movement and size pleased him as
much as their neighbour's seemed to have done.
``Oh, yes, my angel, you wriggle your immense arse to perfection, and
your cunt is almost too tight.''
``Then fuck on with your splendid prick as hard as our neighbours were
at it.'' A happy thought seized me. I put my finger to my lip to give
the hint to Florence, slipped out into the passage and peeped through
the keyhole, which commanded the whole of the narrow room. I beheld a
handsome man fucking a superbly stout woman, kneeling with her head
down low, but towards the door. Her arse uncovered and held aloft was a
remarkably fine one, wriggling indeed to perfection.
I slipped back, described it to my dear wife, and suggested our
speaking to them through the partition as soon as they were done, to
avow that we had heard all their goings-on, as they had ours, and to
propose that we should form a \textit{partie carr\'{e}e}.
Florence jumped at the idea, just as their sighs and shaking of their
couch against the partition announced the grand final crisis.
We allowed them some minutes for the after-satisfaction; we then heard
the lady beg him to do it again as she felt his cock was stiffening
within her cunt.
``No wonder,'' said he, ``when your delicious tight cunt is giving me such
exquisite pressures.''
We thought this a happy moment, as they were both in a state of
lasciviousness; so tapping at the partition, and raising my voice just
sufficient to be clearly heard, I said---
``You have been following our example, and seem as lustful as we are,
suppose we join parties and exchange partners. I am sure you must be
two desirable persons, and you will find us worth knowing. It will be a
novelty exciting to all, and will lead or not, as it may be, to a
further acquaintance or just a momentary caprice. What say you?''
A pause and a whisper was followed by---
``Eh! bien, nous acceptons.''
``Come to us, for I am half undressed,'' cried the gentleman.
We rose and went in unto them, even in a biblical sense. My slight peep
had given me an idea of two handsome persons, but a full view proved
them to be eminently so. He was still up to the hilt from behind. She
lifted her head to look at us on entering, but left her splendid arse
exposed, and did not for the moment alter her position. We handled and
pressed it. The gentleman feeling my wife's arse cried out to his
dearie---
``Here's an arse equal to yours.''
Meanwhile, as I stood by her side feeling hers, she slipped her hand
into my flap, and in answer to his exclamation, said---
``There's a prick bigger than yours. Oh, I see we shall all be
delighted.''
She rose and pulled out my standing prick to show it to her husband,
for like us they turned out to be a most salacious couple of married
people.
My wife laid hold of the husband's prick, and declared it to be a very
fine one, and a delicious variety which was always charming.
I proposed, as the room and couch could only accommodate one couple,
that I should take his wife into our room, and leave mine with him, and
as the two couches were close to the partition between, we could excite
each other by our mutual sighs and bawdy exclamations. This was at once
agreed to.
We all of us stripped to the buff; my new companion was magnificently
made---very much of my aunt's figure, with a splendid arse, although not
so enormously developed as dear aunt's. Her cunt was delicious, a grand
mons Veneris, sweetly haired with silky curls; her pouting cunt had the
true odour, and was very tight, and her pressures and action left
nothing to desire.
I gamahuched her first---her clitoris was well defined and stiff. Her
bubbies were superb, and stood firmly apart, face charming with lovely
and lovable blue eyes, full of the sparkle of lust; lips red and moist,
inviting a tongue.
We indulged in delicious preliminaries; she had a good look at my
prick, declared she had thought her husband's could not be beaten, but
admitted mine was longer and larger. She sucked its head. Then lying
back on the couch she begged me to mount on her belly, as she liked to
commence in that pose. I mounted upon her, got my prick gradually up to
the crushing of the two hairs, and then alternately tongue-ing her
sweet mouth or sucking a nipple of her lovely bosom, ran a most
delicious course, making her spend thrice to my once.
Our other equally occupied couple had evidently got a course ahead of
us, and were changing into the position in which we had first fucked
our wives.
We, too, followed in the same attitude, and really the fine arse of my
\textit{fouteuse}, her naturally small waist, seen to perfection in this
position, and her noble shoulders beyond could hardly be excelled, and
were most inviting and inciting. I plunged with one fierce thrust up
her reeking cunt, and by the very violence of my attack made her spend
on finding it up to the cods, giving me at the same time a cunt
pressure almost equal to my loved wife's.
She was so delightful a fucker that I fucked her thrice more before
drawing out of that exquisite receptacle.
On comparing notes afterwards I learnt that my wife's fucker had just
done as much, and though not so cunt-satisfying a prick as mine, the
variety and novelty gave it an extra charm that more than made up for
any diminution of size.
We were thus all mutually delighted with our change of partners. An
acquaintance begun so delightfully led to a warm friendship and a
constant interchange of these most agreeable refinements, including
every variety of the gamahuche and \textit{la double jouissance} to all
parties.
We all went together to witness some rear-operations between two men,
for which the old bawd's house, No. 60, rue de Rivoli, was quietly
known to be the rendezvous. I made a first visit alone to see if it
would be worth our while; had an interview with the old bawd, a bold
masculine woman of a certain age, who must have been very desirable in
her younger years, for even now many who frequented her house finished
off in her fully developed charms. Her habit being, as I was told, to
come in to the man after one of the girls had left him to purify
herself, and herself to lave his prick from mere love and excitement of
handling a prick, and from long practice she had an art of doing it in
a way to raise another perpendicular, which led to its being allayed in
the full-blown charms of the bawd herself.
I was shown into her sanctum, and there I told her that I knew she
could arrange an exhibition of sodomy. I said that I only wanted to see
the operation, as it appeared to me impossible, and I should like the
two fellows to be well hung and good looking, if such she could procure.
``I have the very thing for you under my hand if you cart wait a quarter
of an hour.''
As that exactly suited my purpose I said I would.
She rose, rang the bell, and when a tap came to the door, went out and
gave some orders. When she came back she said to me, ``I have some very
fine girls, all entirely without prejudices, would you like to have one
up? I have them of all ages, from twelve to twenty-five; and also one
or two handsome boys to have in company with them, to excite the slower
powers of elderly men or those who like such additions.''
I thanked her, but told her my only object at present was to see an
actual scene of sodomy. So to occupy me she opened a small cupboard,
and took out some bawdy books, admirably illustrated. The examination
of these was exciting; her experienced eye detected the effect in the
distention of my trousers, the extent of which seemed so to astonish
her that she laid her hand upon it, gave an exclamation of surprise at
its size, and said she must see so noble a prick, unbuttoned my
trousers, and pulled it out. She handled it charmingly and looked so
lewd that I don't know what might have happened, for I had already
slipped a hand up to an enormous big and hard arse, when a tap came to
the door, and a voice announced simply that all was ready. This at once
recalled me to myself, although the bawd would willingly have made me
before adjourning to the other room.
She said, ``What a pity not to let me have this magnificent prick into
me. I wish the fellows had not come so soon, I am certain I could have
got it if we had not been interrupted, and I can tell you you would
have found me as good a fuck yet as the finest young woman you could
meet with.''
I laughed, and to quiet her, said, ``We may have that another time, for
you are a very fine and desirable woman.'' With this placebo she rose
and accompanied me to the room where the two men awaited us. They were
two tall, good-looking young men, evidently \textit{gar\c{c}ons de caf\'{e}s}, a class
much addicted to this letch, and acting as paid minions to those
wanting them.
They naturally concluded that such was my object. They were already
stript, and both their very fair pricks were nearly at full stand. They
each turned themselves round, and asked which arse I wished to operate
on, and which prick was to operate upon me.
The old bawd, whose interest it was to induce me to have them, handled
their pricks with great gusto, and pointed out the firmness and
attractiveness of their arses, bid me feel how hard they were, as well
as the stiffness of their pricks and the rough crispness of their
ballocks.
I felt them, and would gladly have had them both, but I knew they had
an infamous habit of \textit{chantage}, that is of denouncing to their gang
well-to-do men who were got within their meshes, and go where he would
in Europe he was sure to be waited on and money screwed out of him by
threatening to denounce his practices; so shaking my head and refusing
to let the old bawd pull out my prick, which might then have become too
unruly, I firmly told her she knew I only came there to see what the
operation was like, and had no idea of having my own person handled by
them.
A mutual glance of disappointment was exchanged between the bawd and
them, but they put themselves at my disposal, and asked which was to be
the recipient and which the operator. I pointed out the largest prick
as the operator. They drew a sofa into the best light, and one knelt on
it, presenting a very tempting arsehole to his fellow minion; after
moistening it and spitting on it, the old bawd, with apparent relish,
guided the prick of the other to the aperture, and it glided with all
ease into the well-accustomed receptacle.
I was seated by their side with my eyes on a level and close to the
point of junction. A very exciting scene, for he went up to his cods,
and fucked right earnestly while the recipient wriggled his arse to
perfection, and seemed really to enjoy it. They spent with cries of joy
in great delight, it excited me very much, and the observant old bawd
could see my prick bounding within the confinement of my trousers.
Hoping to overcome my reluctance to take part in the programme, she
stimulated them to change places, and the recipient became the
operator, and the other the recipient. I was awfully lewd, but resisted
even that; after they had done I gave them a Napoleon apiece in excess
of the price paid to the bawd, and left them to dress, and retired with
the bawd to make other arrangements.
On shutting the door and entering the corridor I perceived at once some
doors opening upon small rooms adjoining the operating room, I guessed
their destination; on attempting to open one the bawd seized my arm in
great alarm, and said---
``You must not go there.''
I smiled and said, ``Oh, I understand, come along.''
When once more in her sanctum I said, ``I see you have had peepers
watching the operations, so it is well I resisted any complicity in the
action, but the discovery that you have the peep-holes already
simplifies my object. I have come here to report upon the effect of
this scene of sodomy. A friend who dares not do as much requires such a
stimulant to enable him to fuck a woman he much desires to have, and
who is my mistress. Now it so happens I want very much to fuck his
mistress, and we have made a compact that if this scene is likely to
excite him, we are to come to your peep-holes, and while he is thus
enabled to fuck my woman I shall fuck his. I am thus explicit that you
may know our real object. I suppose that now the witnesses to our
operations to-day have left, so let me see the rooms that I may judge
how far they will suit and which will most favour our object.''
The old bawd complied directly, but still longing to have my big prick
into her, pulled her petticoats up to her navel, showing an enormous
mons Veneris, thickly haired, and turning round a still finer arse,
said, would I not like to assuage my excited prick in one or other of
her really splendid attractions.
I said not at present, thank you. And tightening my trousers over it,
showed her that it had quite drooped its head, and was no longer in the
humour.
She undertook to raise it very quickly, but I politely declined, on the
play of want of time, to thoroughly enjoy so splendid a woman.
With a sigh of disappointment, for the size of my prick had evidently
raised her lewdness to fiery zest, she led the way. Two or three of the
peeping rooms were too small for four, but one was arranged for a
\textit{partie carr\'{e}e}. I made an arrangement for the second day from then,
and requested, if possible, to have four buggers together, to do it in
various positions, and once at least in a chain of three pricks into
the arses before them at the same time; I paid in advance half of the
high price we were to pay, and fixed the hour of one o'clock in the
afternoon, in order to have plenty of daylight to see and thoroughly
enjoy all the excitement.
I left but allowed the old bawd just on going away to take out my prick
and give it a suck by the way of allaying a little the great desire she
had for it. She doubtless expected to raise such a heat as would compel
my passions to satisfy her, but I had now sufficient command of it to
keep it down.
Our grand scene of sodomistic encounters took place as arranged, the De
Grandvits, as our new friends were called, and ourselves, with a basket
containing two bottles of champagne, biscuits, and glasses, betook
ourselves to No. 60, and were installed in the chosen chamber some five
minutes before the arrival of the sodomites. We saw them undress, slap
each other's arses, and feel each other's pricks to get them in
fighting order.
The old bawd was there and lent an effective helping hand where wanted.
They all declared it would much assist their operations if she would
strip and let them see her flitting round and aiding in their efforts.
She knew herself to be much more attractive in body than face, and
complied directly, and really added much to the excitement of the scene.
They began by coupling in threes, so that one after the other held the
delicious position of middle man---be fucked and fucking. The spare
fourth fucking the old bawd, much to her gratification and ours.
The first outsider was now placed in the middle, and the previous
recipient became his attacker in the rear, while the previous
rear-fucker became the recipient of the outsider. The late middle man,
instead of fucking the bawd, buggered her to her apparent satisfaction.
This was just what I wanted, for we had not as yet in our \textit{parties
carr\'{e}es} with the De Grandvits indulged even in bottom-fucking the
women, but this, as we afterwards found, was equally indulged in
private by them as well as ourselves. Now we enjoyed the sight of the
old bawd wriggling in delight with loudly uttered exclamations of
pleasure at having her bottom well fucked, for it was the largest prick
of the four and a very fine one that was into her arse.
We had already racked off in the cunts of each other's wives at the
first display.
``Let us try that,'' whispered my wife to De Grandvit, ``it appears to
give the old woman vast delight.''
It was what De Grandvit had been longing for in his inward soul. At the
proposition of my adored wife his cock sprung into its utmost
stiffness. She knelt on a chair before a peep-hole. De Grandvit brought
his fine prick, which he had moistened with his spittle, up to her
delicious hairy arsehole and with very little effort housed it to the
hilt.
At the whispered wish of my darling Florence he did not press to a
rapid conclusion, but drew the fuck out to a most exciting length, and
to an ecstatic ending in which they had great difficulty in suppressing
exclamations of the delight afforded.
I had followed the lead given, the delicious big arsed De Grandvit had,
like her husband, a long previous wish to be so fucked, and from
practice of that divine coition had no difficulty in taking in with
vast pleasure my bigger prick.
We both ran two exquisite courses in their delicious arse-holes, and
then separated that all might see the grand finale of the four
sodomites each in the arse before him, and the fourth front man into
the immense and magnificent arse of the old bawd. This ended their
exhibition.
I should add that all in turn had either fucked or buggered the old
bawd to her infinite gratification both in person and nurse for she
claimed afterwards and received a good additional douceur for the extra
sight of her own fine body, naked and in double action.
When they were dressed she got out the liqueur bottle and gave them all
a dram and a biscuit. We, too, partook of our champagne and biscuits
while discussing the charms of the scenes just witnessed.
My darling wife chimed in with the remark of how much more the middle
man had seemed to enjoy it than the two outsiders.
I added a remark that I had heard that such a position was the \textit{ne plus
ultra} of delight.
``Then why should we not try it?'' said the glorious De Grandvit.
``I quite agree with you,'' said the husband.
``Who shall begin?'' I thought, as the idea of trying it was first
suggested by his wife, she ought to have her own idea first realised in
her magnificent person. I would fuck her while he went into her arse.
This was immediately adopted. I lay down on my back, the delicious De
Grandvit mounted on me, rose and fell, and spent before she stooped on
my belly and presented her splendid arse to her impatient husband, who
for some minutes knelt prick in hand behind her. With gentle care and
well-moistened prick he got housed at last in his wife's delicious
arsehole, and then slowly at first, but more energetically afterwards,
we ran a most exquisite course.
As no one was now in the adjoining chamber, no restraint was put upon
our lascivious exclamations. The De Grandvit was in such a delirium of
ecstasy that she screamed again and died away in absolute annihilation
of all sense but that of the utmost satisfied lust.
My darling wife had straddled across us and been gamahuched deliciously
by De Grandvit while \textit{enculant} his wife. We next changed the venue; he
fucked my wife while I plunged into her glorious arse; the De Grandvit
straddled across the others, was gamahuched by me as her husband had
gamahuched my wife. This course, too, was run in an ecstasy of
enjoyment to all concerned, and ended our orgy on that occasion. We
arranged our dresses, finished our champagne, called up the old bawd,
satisfied her demands, and thanked her for the exciting scene she had
procured us. On asking her, she admitted that the other peeping rooms
had been occupied by couples, and that one elderly gentleman had had
two of her page boys to operate and be operated upon while the scene
before him excited him to the necessary extent to take a part in it
himself. He had just left, having stayed to listen to our proceedings
and had told her the two gentlemen had in consequence of the scene
witnessed initiated the women into the \textit{double jouissance}, and the
excitement of listening and hearing had enabled him once more to get
into the handsomest boy, and have the other in him.
She hinted that we ought to come again, and have the boys in, for she
said the gentlemen, that is ourselves, would find an immense additional
pleasure in letting the boys penetrate their bottoms while they were
into their ladies in both apertures.
We laughed, and said we should consider her offer, but for the present
we were fucked out.
We did not forget the bawd's proposal of having a boy to fuck us while
\textit{enculant} the dear women. A hint to my darling wife brought this out
at our next meeting. After the dear creatures had both enjoyed the
\textit{double jouissance}, my wife said to Madame De Grandvit---
``We are really quite selfish, here are our two loved husbands giving us
the unutterable joys of the double junction, and yet not enjoying it
themselves. You remember how the old woman at No. 60 spoke of the
raptures the addition of her boys would be to the bottoms of our
husbands while administering the double coition to us. Why should they
not try the same on themselves, and give us the delight of seeing them
in all the ecstasies their double embrace confers upon us? We know how
they delight in being postillioned, which shows how much they would
like the real thing if they dared avow it. It is for us to break down
the barriers of prejudice and false shame. Here, Charlie, let me
dedicate your bottom to the lust of our dear friend De Grandvit.''
My beloved wife was at the moment handling the prick of De Grandvit,
and whose full-standing stiffness showed he was ready to face any
difficulty.
I pretended a fear of its size being too great to allow of entrance
into that narrow path of bliss without great pain to the recipient.
``You can never know that till you try,'' cried my darling wife.
In all this she was only acting a part prompted by myself, for I was
most anxious not only to have De Grandvit into my own arse, but was
longing to be into his great, coarse, hairy, corrugated deep-brown
arsehole. In this I differed greatly from our dear friend MacCallum,
who loved the delicate unfledged arseholes of youths, while to me it
was necessary to be the very reverse of the fair sex, whose arseholes
in general are of a delicate pink with puckered-up charming little
orifices, which, of course, have their charm; but when with men to me
it was twice as exciting to find them like my dear friend the Count's,
quite contrary to those of the fair sex. Dark-brown, roughly
corrugated, and coarse hairs all round them were the arseholes that
raised all my lust, and made sodomy a delicious contrast to merely
fucking the arse-holes of women; such an arsehole as I most loved to
fuck was M. De Grandvit's.
I had suggested to my wife to tempt him with mine for the sole object
of getting into his. He bit at the bait, so I shoved my prick into his
wife's arse, my wife conducted his prick into my delighted bottom. I
made some affected grimaces, but of course took him in with the
greatest ease. My darling wife had acted postillion to him, and had
frigged his wife with her other hand, so we ran a delicious course of
the wildest lust.
As we had already served out our wives too many fucks one trial was
sufficient for the moment. De Grandvit was in ecstasies at the delight
I had afforded him, especially as he appeared to be revenging the
affront I gave to him by being into his wife \textit{in culo}.
My adored wife, with her happy art of handling and exciting a prick,
nonetheless willingly that she was getting it up to go into her own
hungry and delicious arsehole, soon brought De Grandvit to the
necessary stiffness.
I wanted no other stimulant than the expected satisfaction of a letch I
had long had to be into his fine, rough, hairy arsehole. As soon as he
was fairly hilted in my adored wife's splendid bottom, his better half
took my prick in hand, put it into her mouth to suck and moisten it,
and then guided it into that narrow abode of bliss I so longed to
possess. It really was the first time De Grandvit's arsehole had ever
been penetrated by a prick, although he had long wished for such an
experience; there was therefore some real grimacing, for mine was not a
prick of the ordinary dimensions, that might penetrate any arsehole,
but a prick of the biggest, so I was obliged to be very gentle and make
frequent halts.
My darling wife was obliged to exert all her delicious means of keeping
his fine prick in her arse at full stand by cunt pressures and her
delicate handling of his ballocks; at last I was fully engulphed, and
pausing until all strange feelings had subsided, a gentle movement and
my darling wife's admirable seconding enabled us to end the course in
the wildest ecstasies of the most delicious delight, and to sink on the
broad back of my splendid wife, completely annihilated by the most
exquisite joys of satiated lust.
Once this delicious practice had been indulged in, you may be sure it
did not end with a single experience, but as thereafter the \textit{bonne
bouche} or finish of all our after-orgies.
My beloved wife, whose eye for a capable man was infallible, had
observed a genteel, tall, good-looking young German waiter in the
hotel, who looked superior to his place. He turned out to be the son of
a wealthy hotel proprietor at Frankfort, who had sent his son to
Meurice's in a sort of apprenticeship, to learn how a large Parisian
hotel was managed. In such a situation they receive no wages and have
even in general to pay a premium for the privilege---this practice,
which is general with German innkeepers, accounts for the number of
genteel-looking waiters that are met with in the large hotels of great
capitals, and who are found to be of superior education and information
when spoken to in a friendly and familiar manner.
This was eminently the case with our friend Carl. My wife had taken
rather a fancy to him, not at first erotic, but observing that after
she had talked to him familiarly that he began to be very deferential
to her and with a certain manner that she, with the instinct of a
woman, saw at once arose from amorous admiration. Casting her eye
downwards she detected the effect produced in his trousers whenever she
was kindly civil to him. She increased her familiar conversation, which
evidently allayed any fear he might have had, and she could soon see by
the increased bulging out of his trousers, not only that he was growing
more lewd upon her, but that he was evidently very well furnished.
Learning that he was the son of a wealthy father, well educated, only
now placed in the position of a servant, in order to know, by obeying,
how to command, and also to gain the experience which large and
well-frequented hotels alone could teach how best to conduct his own
hotel hereafter.
She told me all about it, and thought he might be moulded to our
purposes. Even if not she had taken a caprice to him so that in any
case it would be a gratification to her to possess him.
So I lent myself to aid her by purposely absenting myself either at
breakfast or luncheon, under pretext of going to take one or the other
with bachelor friends.
As Carl was told off to especially attend upon us, and no other servant
ever came near unless rung for, my wife had easy opportunity, and with
her practised skill in seduction, had him into her on the second day.
He proved an admirable stallion; grew passionately lewd on the splendid
person of my wife, and became in fact cunt-struck upon her, probably
the strongest bond that can entangle a man. It becomes an infatuation
that makes him the slave of the cunt that has attracted him. There are
few men of hot temperament who have not experienced this overmastering
infatuation, and they know that even supposing the object becomes
perfectly unworthy, unfaithful, abusive, and with every vice indulged
openly before them, they may wince, they may thoroughly despise her,
but the chain holds them fast in adamantine bonds, which neither the
persuasion of friends nor their own knowledge of the perfect
unworthiness of the object can tear asunder.
Such became the fate of Carl, and my wife moulded him, with all her
wily skill, to our lascivious purposes. When once under her
enchantment, I made a run over to England on some urgent
matters---purposely leaving the field open---my wife completed her
conquest, had had him in every way, had postillioned him, and wormed
out of him that at college he had indulged in sodomitical practices
with young students like himself; but knowing how prejudicial it would
be to him in his profession, he had weaned himself from the habit with
men, but dearly loved the \textit{enculage} with women, and doubly adored my
wife when he found her extraordinary and exquisite talent in that way.
She also, after much apparent hesitation, in answer to his eager and
continual questioning, omitted that her husband was much addicted to
worshipping her bottom, and had taught her its divine use. She even
cautioned him against any imprudence on my return, for she said she had
her suspicions that I had a letch for men, and if I discovered their
liaison, would be apt to avenge myself that way.
``Oh, if he would still allow me to possess your enchanting person he
might make what use he pleased of me.''
This was the point aimed at from the beginning. My wife wrote to me,
and we arranged that I should announce my return for a certain morning,
and that she should have Carl to sleep with her the previous night.
I arrived in the middle of the night, walked into the room, found him
in bed, played the angry husband, swore I must have revenge, and that
as he had cuckolded me I must avenge the affront in being into his
person.
He objected, for form's sake, but said he would yield to anything if I
would not drive him away from the adorable Madame.
``That will depend upon the manner in which you satisfy my desires.''
``Oh, do what you like, dear sir, if only you will allow me to love
Madame.''
``We shall see, we shall see; let me look at your prick. Oh, a good
size, even when down. Let me see it at full stand.''
My wife here interfered, and said Carl was so good that she was sure he
would prove a satisfaction to me. She took his prick in hand, and with
her art of handling a prick, had it at full stand in a minute and asked
me if she could possibly have refused so handsome a prick as that. And,
indeed, it was a very fine one.
Carl was a very fair young man, with a most beautiful and satiny skin.
His prick was exquisitely white, and the blue veins showed themselves
coursing through in a most tempting way---it was seven and a half inches
long, by quite six in circumference, was thick up to the vermilion nut,
although gently diminishing from the roots, the glans was smaller than
the shaft close up to it; a hollow, like what you sometimes see in the
neck of a bottle, ran all round the edge of the nut, and thus made it a
head to the shaft. My wife declared that its shape gave her great
pleasure in both orifices. It certainly was a very attractive prick,
and now that it was at full stand I made him lie on his back on the
bed, took it in my mouth, sucked and frigged it until he spent in an
agony of delight.
I then made him turn over on his belly, that I might admire his
ivory-like buttocks, which I caressed and kissed in every way. My wife
slipping her hand under his belly soon recovered the stiffness of his
prick. I now desired him to kneel that I might be into his bottom.
His exquisitely white buttocks, marble-like in polish, hardness, and
coldness to the touch, were most attractive to women as well as to me.
While thus kneeling with head low, and the chink between the buttocks
well spread open, his exquisite small, pink, corrugated arsehole with
almost invisible fair, short ringlets around it was truly lovely and
exciting.
As a rule, I like to fuck a rough, hairy-arsed man, but I can all the
same appreciate the delight in such an exquisite arse-hole as Carl
possessed. To me also it had the attraction of its first possession.
When thus first fully displayed to my delighted eye, I flung myself on
my knees, kissed and tongued the exquisite and delicious orifice, and
speedily got furiously lewd upon it; and rarely have I fucked an arse
more deliciously incentive to sodomy.
``Oh, poor fellow,'' cried my wife, ``you must let this fine object (his
prick) be housed in me first, and then he will less feel the
introduction of your large instrument.''
I immediately consented, on which he cried out in delight---
``Oh do what you like with me, as long as your adorable lady will permit
me to possess her.''
``Well,'' said I, ``see, her cunt is reeking with your spunk, so I will
first bathe my prick therein, to make it go easier into your arse.''
We took up kneeling positions. He filled the delighted cunt of my wife,
and presented his really beautiful arse to my raging lust. I humoured
the entrance a little, but once within over the nut, I plunged
recklessly forward, somewhat too roughly, for it made him wince, and he
would have escaped from me if he had not been doubly imprisoned. The
pause I gave him after being fully engulphed calmed the strange
sensation, and we gradually increased our movements until both died
away in excessive delight, especially to him, for it was his first
experience of \textit{la double jouissance}, and it gave him such exquisite
enjoyment that he begged me not to withdraw, but to run a second
course. My darling wife thinking it would increase his lewdness if she
changed his prick from her cunt to the more divine orifice, withdrew
it, and placed it in the grove sacred to the secret rites of Priapus.
He enjoyed the ecstasies of paradise on this last occasion, and we all
fell on the bed completely overcome by the soul-killing joys of the
discharge, and lay soaking in all the after-pleasure for some time,
until my darling wife begged us to relieve her of our overpowering
weight. We rose and purified ourselves, and then I posed him standing
up, admiring the really fine proportions and beauty of his handsome
fair form. I sucked his prick until it stood, and then told him he must
give me the pleasures of the middle, which he was so highly praising as
the utmost exquisite enjoyment he had ever experienced.
My darling wife was delighted. She got on her knees. I entered her
delicious cunt in the first bout, and I quickly housed Carl's prick in
my arse.
We ran an exquisite course, and then a second with only a change of my
prick to Florence's arse instead of cunt. Carl was after this obliged
to leave us, as the morning was getting on.
I sent him away the happiest of men by telling him as long as he placed
his arse at my disposal, he should have my wife always at the same time.
Thus we had secured another fine prick to our general orgies. We told
the Grandvits of our fortunate \textit{trouvaille}.
Monsieur made some difficulty about his being a servant, and the fear
of discovery of our orgies through his indiscretion; but hearing that
he was much superior to a servant he consented to his introduction.
After they had seen and admired him, they expressed their extreme
satisfaction at the result of his joining us---for both Madame and
Grandvit loved to have him into all their orifices. We could now fuck
both women at once, and the double pleasure could be given to either
sex without there being any outsider.
Every third night they slept in our hotel, and that night we never
ceased conjunctions in every variety, with pauses for refreshment,
purification, pleasant bawdy talk, fun, and frolic. For a month longer
this delicious existence lasted, and then it was time for us to proceed
southward. We parted from the Grandvits with much regret, but promised
to return in the spring and visit them at their country house. I may
here add that we did so, and enjoyed our visit to the utmost; and, in
the second year of our absence, they accompanied us into Germany, where
at last we left dear Carl. He had begged us to let him go as my valet
with us to Italy.
His intended stay in Paris was within a month of its termination; he
wrote to his father that the opportunity of travelling through Italy
under the offer we had made was too advantageous to be lost. His father
consented, and thus for eighteen months he was our constant companion
and participator in all our lascivious conjunctions.
Carl accompanied us to London on our first return home, and resided
with us for three months. I told the Benson and the Egerton of our good
fortune in discovering him, and the exquisite addition to our party of
us and the De Grandvits he had been.
They were instantly alive to the delight of possessing him.
I had continued the occupation of my chambers in the Temple, in which
Harry Dale still resided; it was there we erected our altar to the
Apostrophian Venus, and held our orgies.
Carl delighted our old friends, who were never tired of having him one
way or another, while Harry or I administered to the \textit{double
jouissance}.
A new prick to a woman is like a fresh cunt to a man, and for the time
gives additional zest to the lust which rages in us. So it is with the
darling Benson and the lovely Egerton. They revelled in the possession
of Carl. They knew they could only have him but a short time, and they
made the most of him.
My beloved wife, with that kindly consideration for every one which
distinguished her, quite abandoned Carl to these two dear insatiable
cunts, and contented herself with presiding over our orgies, dictating
new and exciting poses to our two friends, leaving Carl and me to their
embraces, and consoling herself with a fuck now and then from Harry
Dale, when we two were simply fucking each his dame. She told them, ``I
can have Carl and Charlie whenever I like at home, so must leave them
to you for the three months that Carl can only give us.''
We met thrice a week. My wife used to drive to the dear creatures and
take them up, the husbands being much gratified at the affection shown
by my wife to them, and never having the slightest suspicion of the
object my wife had in taking them out. As to our own servants they knew
the chambers belonged to their master, and they knew we lunched there,
but they never imagined their mistress would take ladies to share in
their master's embraces. So that we carried on our intrigue in perfect
safety and impunity.
It was a sad day when we left with Carl, who never again returned to
England. Our darling companions had become much attached to him, and
parted with close embraces, and with bitter tears bade him adieu.
We parted from him at Frankfort, where his father, retiring to a
country life, left him proprietor of a capital hotel, to which in
after-years we often resorted when going to and from the German spas,
and always stayed some days to renew the orgies we all so loved. His
love for my adored wife's cunt endured for ten or twelve years, when an
advantageous marriage softened it, perhaps more through the jealousy of
his wife who, suspecting, caused us to desist from using his hotel. He
had also got a family of a boy and two girls growing up, which
completely ended our acquaintance.
To return to the time of our conducting him to Frankfort with the
Grandvits, they afterwards accompanied us in a tour in Switzerland, but
left us at Sion, when we turned our steps across the Simplon to Italy.
We were invited by our friend the Count to visit him for a month at his
old castle in the hills of San Giovanni, overlooking all the ground of
Bonaparte's earlier battles in his first Italian campaign.
We followed the right bank of Lake Maggiore to Arona and Allessandria,
and thence by Acqui gained the castle of the Count on the hill above.
It was situated in the midst of glorious scenery. From the summit of a
hill near the glorious line of the \textit{Alps} could be seen Monte Rosa,
Mont Blanc, Mont Cenis, Monte Giovi, and thence round the Apennines,
while the Gap leading to Savona gave a view of the sea, the southern
suburb of Genoa, and the line of coast leading to Spezia.
It was a glorious view, and we often directed our steps to the summit
from whence it was seen during our month's stay with our loved and
delightful host.
His old castle was only partially ruinous, but quite habitable.
However, his father had built a comfortable house in the garden, at the
base of the rock.
The castle crowned a perfect perpendicular detached mass of rock, half
round which rushed a mountain torrent, the approach being a very steep
zigzag with now ruinous defences, a very steep and difficult ascent. It
is true from a low entranced cave at the foot a secret stair led up
from the garden, of which I shall have more to say in relating some
incidents of the Count's earlier history, as confessed to us in our
close and intimate intercourse.
We were warmly welcomed by our dear friend, who, leading us to our
rooms, had a rack-off of his waste steam in the ever delicious cunt of
my loved wife, who, it will be recollected, had a great penchant for
the Count, when she used to prefer him at our Percy Street orgies. When
the Count retired, I plunged my excited prick into the balmy bath he
had prepared for me in my wife's cunt, fucking her fast and furiously
the instant he retired, a change she loved above all things; this
calmed us for the moment, and enabled our waiting for night.
We had expected to find a young sister of the Count with him but at our
orgy at night he told us that since his return home he had had this
sister, and that in fact at that moment she was staying with an
accoucheuse at Turin, and he expected to hear of her delivery by every
post. We congratulated him on finding so delicious a bit of incest to
his hand on his return to his country.
``Ah!'' said he, ``it is much more delicious than you think.''
``Indeed, how is that?''
``She is my own daughter as well as sister.''
``What a delicious idea!'' cried I, ``what a cockstand, and what a fuck it
must have been to you! But you must have had your own mother to bring
about such a delicious result. Do let us hear all about it, my dear
Count, it will excite us all to renewed efforts, as incest always does.''
This conversation occurred during a long pause we had made in our first
night's orgy when quietly seated after purification, restoring our
powers with Champagne and some slight refreshments prepared by our host
for the occasion. We had already had three hours of the most delicious
fucking in every possible combination, being all, especially the Count,
fresh and in excellent order for a thorough excess. So we all were glad
of a respite, and listened to the exciting story of the Count's
delicious double incest. As we did not hear all at that sitting, I will
finish an account of our doings, and then give a connected narrative or
sketch of that strange intrigue, and some other of his earlier
escapades, merely adding that his account of his affair with his mother
set us all off in such an excitement of lust, followed by such an
excess of fucking in bouts of \textit{double jouissance}, in which not only my
adored and most lascivious wife came in for her full share, but both
the Count and myself enjoyed the double bliss in our turn. We carried
on to such an excess that we were quite knocked up, and were so
overpowered with sleep the next evening that by common consent we
quietly went to bed, and deferred till morning any fresh deeds in the
fields of love and lust.
We found this so refreshing to our powers of fucking that we regularly
adopted the system of lying fallow the earlier portion of every other
night.
We passed a most agreeable time with walks and rides through the lovely
scenery, and explorations of the old castles.
The Count himself had two, but the one immediately above his house was
by far the most interesting and was the original seat of his ancestors,
wild robber barons of their day; and a black deed was reported in the
traditions of the peasantry around.
The castle, although in a valley between the hills, stood on a high
perpendicular isolated rock some hundred-and fifty feet above its base;
it was crowned with a very high building to make up for want of space
at the foundation, and had besides a very lofty and bold round tower,
rising high enough above the sides of the valley to serve as a lookout
beyond them. The habitable part was reached from the main gate by a
steep stair, at one of the landings was a trap door opening upon a
profoundly deep shaft; tradition said that this was a trap for personal
enemies, who, on pretence of reconciliation were invited to the castle;
on passing over the trap it opened, and they were precipitated to the
bottom. It was the common tradition of the peasantry that wheels with
scythes attached chopped them to pieces at the bottom.
It is a curious fact, and one showing how tradition may preserve a
truth where least expected. Our friend the Count for six months lay
hidden in the secret recesses of this old castle at the time a price
was set on his head for treason. This had led him to all sorts of
explorations, in which he had discovered many hiding places.
Knowing of this tradition about the cutting up of bodies at the bottom
of this deep shaft he got his two younger brothers to let him down by a
long cord, and really found the remains of machinery and wheels with
rusty blades attached.
After he had finally escaped, a more regular search was made, and it
was discovered that a communication with the torrent on a former higher
level had let the water pass underneath the castle, and turn a water
wheel which cut up the bodies and made them float away by the outlet.
Human skulls and bones were found, singularly verifying the truth of
tradition.
At the time the Count was a fugitive hiding therein, the old apartments
were used as a granary to store the rent in kind of his father's
tenantry. As there were suspicions of his having taken refuge here, the
place had been two or three times ransacked by the police without their
discovering him---thanks to the ingenious hiding places he had
discovered.., But for this very reason every precaution had to be
taken, and no beds, bedding, or plates, knives, chairs, or tables were
there; he slept on the corn, spread three feet thick on the floor, or
sat on it when tired. His mother, with provisions under her petticoats,
would saunter in the garden, and, when unobserved, slip into the low
cavern and ascend by the secret stairs, and seated on the corn by his
side, would wait until he had done, to take everything away, and leave
not a trace of any one being provisioned up there. These details are
explanatory of what follows. The Count had been one of the Royal Guard
for two years at Turin, and being a handsome young fellow, had as much
fucking at command as he could wish for. When shut up for months in his
asylum the passions that had been kept under by constant gratification
began to torment him; from the loopholes of the castle he could see the
peasant women working on the mountainside, and, in stooping, showing
their legs even up to the bare skin, and this used to drive him mad
with desire. He did not frig himself, but at night stole down to the
garden, secured a large pumpkin or two, took them up to his retreat,
cut small holes in their sides, and then thrust his stiff-standing
prick into them, forcing the hole to the size of his prick, and then
working the pumpkin with both hands till he spent deliciously; he used
to get six or seven fucks in these artificial cunts, then throw away
the finished one on the torrent side of the castle. This was so far a
relief, but his lust grew fiercer every day, and on one occasion became
uncontrollable.
His mother, who had married at fifteen, was now a fine ripe woman in
her thirty-sixth year. One day, after setting down the things she had
brought up, she lifted her outer gown that she might not show she had
been sitting on corn; the Count was already seated much below her body
on the low corn. His mother accidentally on this occasion drew up all
her clothes, showing the whole of her fine arse, and in stooping
backwards to seat herself all her fine hairy and gaping cunt was
visible to his lower sight. This was too much for the Count, in a
moment his prick sprang to the fiercest stand, he instantly unbuttoned
his trousers; his mother finding she had brought her bare arse onto the
corn, leant over on the side opposite to her son to tuck her petticoats
under her arse, but the Count seized her round the waist with one arm,
with his body pressed on her already bent body, forced her quite down
on her side and was into her cunt up to the hilt, he thrust it up so
fiercely as not only to make her shriek with surprise, but also with
pain. She struggled to be free, but was held down with all the energy
of his ferocious lust. Very few thrusts in and out were required to
bring down the first rush of his sperm; this lubricated her cunt, his
prick never, yielded, but stood as stiff as ever, and with hardly an
instant's pause he recommenced a more delicious action than the
previous one. His mother, however, was much distressed in mind at the
first horror of the incest, but being a ripe woman of hot lubricity,
could not feel a fine prick deliciously belabouring her cunt without
having her lust excited in spite of herself. As all pain of the
unprepared forcing of her cunt had passed away, and the plentiful rush
of her son's spunk lubricated all the passage, she soon could not
control her passions, and seconded him with an art which left nothing
to desire. His long deprivation fired him to unusual efforts, and he
fucked her five times before he withdrew.
When she sat up she said, ``Oh! Ferdinand, what have you done! How could
you do so? Violate your own mother. It is dreadful.''
The poor Count, seeing her much distressed, burst into tears, threw his
arms round her neck, and weeping told her he could not help it.
She patted his head, and said. ``Poor fellow, poor fellow.''
On this he lifted his head to kiss her. She, too, wept, and they
mingled tears and caresses together; this almost instantly restored his
prick to its pristine stiffness. He bent his mother back on the corn,
and although she resisted a little, and said it was too dreadful his
wanting to commit such a sin again, she opened her legs when he got
over her, and did not prevent his pulling up her petticoats.
He was into her this-time-well-moistened and really longing cunt, for
her passions were now become lascivious.
Thrice more did he fuck her, each time more delicious than the others,
and in all seconded by the most splendid action of his mother's arse,
and the most exciting pressures of the inner folds of her really
delicious cunt.
At last she left him, but after so delightful a commencement every day
saw a renewal of these delicious encounters.
His mother proved an adept in every resource of lust. Being a
splendidly made woman, and salacious in the extreme, when once she had
given way to her lubricity, she indulged in every whim of lust. She
always, after a few days' fucking, came very lightly dressed, with no
stays or other encumbrances, so that they used to strip and fuck at
ease in every way. The Count assured us that much as he had since
enjoyed some of the finest women, never had one given greater pleasure
than his delicious, lewd, and salacious mother, doubtless the fact of
it being incest added to the usual gratification given by a ripe,
well-made, luscious-cunted woman.
After the first week of their delicious encounters, his mother said to
him, ``My dear Ferdinand, we are very imprudent, you may get me with
child if we do not adopt precautions. Your father does not wish to have
any more children, and takes care not to get them.''
``How does he prevent it, my dear mamma?''
``Well dearest, he goes slowly to work, and while he has it in me rubs
his finger on the point where you are now feeling (he was gently
rubbing up her clitoris, a well-developed one) until he has made me
enjoy it several times, and when he finds he is about to discharge he
suddenly withdraws it, and pushes the head of it into my bottom and
spends there. You must do the same, but you must not put all this long
thick fellow in. Oh! come to my arms, my son, you have excited me until
I must have it immediately.''
Upon which the Count mounted and fucked so deliciously that with arms
and legs round his body and loins, devil a bit would she allow him to
withdraw, but spent with him most ecstatically, and quickly called for
more, so that it was not until the third time of his being about to
spend, that throwing her fine legs high in the air, and bringing her
arse with a heave well up, and taking his prick out with her hand, she
guided it to the delicious smaller orifice, and as all was reeking with
the previous discharge, slipped it in, not the head only, but the whole
shaft. She cried out, ``Not so far, not so far,'' but as he began shoving
in and out she quickly got excited, and wriggled her arse with all her
accustomed skill, and spent deliciously again as he shot his spunk
right up into her incestuous entrails.
He passed a hand between their bodies to press a finger on her
clitoris, this made her cunt throb, which was felt by his prick, and
quickly sent him up upon another delicious enjoyment of the tight
recess of obscene lust, and a second most exquisite and luscious course
was run, equally to his mother's as to his satisfaction. Then he
withdrew to relieve her body of the weight she had so long sustained,
they mutually embraced their naked bodies, and sweetly conversed on the
exquisite joys they had just participated in. His mother declared his
father gave her nothing like the lascivious joys she received from his
dear son. They toyed and kissed until, handling his prick with skill,
she got two more delicious fucks, one in each receptacle, and parted
for the day.
By the second month she discovered that what she dreaded had happened.
Her son had got her with child; she wept when she communicated this
unfortunate result, but the Count, like me, always stood fiercely at a
woman's tears. Several splendid fucks followed, all in the cunt---the
mischief was done, and precautions were no longer necessary.
His mother abandoned herself to him with a greater excess of lust than
she had ever yet done, and fucked with an excellence, vigour, and
energy that drew from him eight discharges in a wonderfully short time.
The fact of his having put a baby into her appeared to stimulate both
their passions. She declared she never in her life had enjoyed fucking
more. They used the grossest bawdy terms in their intercourse, as if it
was one barrier more broken down between them, and made their
incestuous love more exciting and a greater destruction of all natural
ties between them.
Before parting they consulted about how best to fix the parentage on
her husband.
He was a man of fifty-five, and, therefore, past the ardour of
passion---taking even his fucking coolly---and, therefore, more difficult
to hoodwink.
She knew that he awoke with a cockstand, although that did not always
lead to a fuck. Upon this they founded their hopes, and at last
arranged she should drug his coffee, and when still asleep in the
morning she should handle his prick, get him up, turn her bum, put it
into her cunt, work him gently, make him spend which would awake him,
hold him in, pretend she herself was in the acme of delight, but on
coming to her senses, upbraid him with having spent inside.
This all happened as planned, he did awake on spending, but his wife
exerted such unusually delicious pressures upon his delighted prick,
that he got so excited as to fuck her in and she took care he should
spend inside a second time-'she pretended to be carried away by passion
as much as he was. But remonstrating afterwards upon the imprudence of
what he had done, especially in having so excited her that she could
not help spending at the instant he did, which made it more dangerous.
She did not know how it was but she had never before seemed to receive
such pleasure from him as he had given her that morning.
``Well, my darling, it is a curious coincidence, but you never seemed to
me more delicious or more lasciviously excellent in your fucking than
you did just now. As it is but once let us take more care in future,
and hope nothing will come of this little and delightful imprudence.''
But of course there did, as the Count related to us, and seven months
after this morning fucking my mother gave birth to a daughter. ``I had
already been in exile for five months when this event came off, I had
letters from my mother after she got about and for some years
afterwards, telling me that my sister was a beautiful child, and
growing up \textit{the image of her father}, underlining those words for me,
to put the true construction on them. Poor darling mamma, she died four
years ago, and my father followed her two years later. I never saw
either of them again.
``Before I escaped from Italy I had passed five months in the constant
possession of my beloved mother. As her pregnancy advanced her
salacious avidity for my embraces seemed to increase. She was
insatiable, but with such variety of charm and art that I never failed
to answer to her call. Every refinement and excess of the wildest and
grossest lust was practised by us.
``My father possessed a small collection of the grossest bawdy books; my
adored and salacious mother purloined from time to time the lewdest, we
read and excited ourselves in the realisation of the wildest and
grossest scenes therein depicted.
``My mother was an instance of a woman getting once out of bounds and
then stopping short of no excess, and became boundlessly corrupt. There
was no horror we two could possibly commit that we did not indulge in.
``My father, when once the pregnancy was undoubted, was less reticent of
his fucks. My mother at my request used to stimulate him to fuck her
just before coming up to me, so that I used to shove my prick into the
paternal sperm, sometimes in her cunt, and sometimes in her arse, and
eventually used to lick it up before fucking her either way. The incest
of her son upon the immediate fuck of her husband was, she said, the
most stimulating to her excessive lust of anything I could possibly do.
``My father was obliged to go to Turin for ten days; it was the time of
new moon, when nights were dark. My mother used to put on a dark cloak
and come up to me; we lay down on her cloak, and, stark naked, gave
ourselves up to the wildest lust until dawn, when mother slipt away to
the house and left me well inclined to sleep until she returned with my
food.
``Oh! it was a happy time, its combinations of solitude and incest,
combined with my lusty youth, for I was only nineteen years old at that
time, made me be constantly at her call, and she never went away before
her excessive lust had been satisfied for the moment. Had circumstances
permitted her to stay with me longer than she usually did, she would
have got more frequent fucks out of me; at night, when she could come,
she got ten and sometimes eleven discharges from me, and probably
herself spent twice as often. I was indefatigable.
``In all her after-letters to me she constantly avowed grief that she
had lost her most loved son; that she was inconsolable, punning on the
con in the word, which is French for `cunt.'
``Various allusions of that sort were in all her loving letters. Often
and often when I have been slack in fucking a woman, and my prick not
answering when called on, I had only to conjure up some of these scenes
with my mother when my cock would spring to the stand instantly, to the
immense satisfaction of my momentary \textit{fouteuse}, and it is so yet, a
thought of her reanimates it at once.''
Here my adored wife slipped her hand under his dressing gown, and found
his prick standing fiercely, she seized it, and pretending to be his
mother, cried out---
``Come, oh, come! my beloved Ferdinand, into your own loving mother's
arms.''
She fell back on the couch, he got between her legs, kneeling on the
floor, having thrown off his robe, exhibiting his fine hairy arse---one
of those I so dearly loved. The sight fired my salacious prick, so
kneeling behind, I guided it into his arsehole, and while he fucked my
adored wife, I sodomised his superb arse. We ran two delicious courses,
then my wife took me in her cunt, while the Count buggered his supposed
mother, for that stimulating idea was kept up. A second fuck followed
in the same pose, with both her apertures filled to satiety.
This concluded that delicious orgy; we had a half night's rest the
following night, as usual, to recruit, that we might better enjoy a
perfect excess on the subsequent night.
It was in this way we kept up our powers, and only near the end of our
visit had we any occasion to apply the birch, and that to no great
excess.
It was in the middle of the second night that the Count continued his
recital of the result of the intrigue with his mother. His
sister-child, for she was both, was born in his first year's exile.
Beyond his mother's description of her, that she was growing up a
beautiful girl, the image of her father, meaning her son, the Count, he
had no other intelligence of her. She had just turned eleven when her
mother died; for two years after that sad event she kept house for her
father.
He then dying, the second brother took possession of the property. As
the state had deprived him of all civil rights, the property was given
up to the brother. On his return, after being amnestied, the Count had
to go to law with his brother to get back his property. His
sister-daughter, who had been unhappy with her brother's wife, gladly
left them to keep house with the Count. She was then in her seventeenth
year, splendidly developed in bosom and bottom, lovely and lustful
deep-brown eyes, the very image of her father, although she only knew
him as her brother. The recollection of the fierce joys he had had with
his own and her mother, drove him wild with lust to possess the
incestuous fruit of his intrigue with his own mother. He used of an
evening after dinner to have her sit on his knee while he related his
adventures abroad, intermingled with kissing and toying. He praised her
splendid bubbles and felt them; he said he could not believe that her
immense prominence behind was real unless he felt the bare skin. With
little resistance this was permitted once, then indulged in, until from
less to more he got to feeling and frigging her cunt, while he put his
own standing prick into her caressing hand. There could be but one end
of this. He took her maidenhead, and then she crept into his bed every
night. He initiated her into every excess of venery, and ended by
getting her with child. It was concealed as long as possible, and then,
on pretence of a visit to a friend at Turin, to see some fetes, he
conducted her to an \textit{accoucheuse}, and left her there until her
parturition was over.
I may here mention that just five weeks after that event came off we
met them at Turin, on our way home from Venice. She was a beautiful
girl. The Count introduced us as old friends, with whom every thing
could be done in common.
We stopped a fortnight, and initiated her into all the mysteries and
extravagancies of the wildest lust, and she proved apt a scholar that
she almost equalled in action and enjoyment the greater experience of
my beloved wife.
The Count had taken apartments at Turin for the winter, and finding his
sister-daughter so facile a pupil he intended getting up a \textit{partie
carr\'{e}e} to continue these delightful orgies. His child was a lovely
fruit of double incest, and gave promise of being a lovely woman. Her
mount was charmingly plump, and the pouting lips of her delicious
little cunt were already lust-exciting. The Count hoped he would be
able to fuck her when old enough and promised me a participation when
the time came.
I may here add he had her always to bed with him, and his sister-mother
every morning, and in the bath with him.
She grew up admirably developed. From between seven and eight years old
he gamahuched her delighted cunt; at eight began rubbing his prick on
her clitoris, and by nine had gradually stretched it that he could
enter nearly his whole length, and spend there.
We long knew each other, and he always said he was practising the
lesson my adored wife Florence had instructed him in, when relating to
us the incidents of her earlier days, and of her gradual violation by
her own father.
I shall defer this story that I may at once describe the after-fate of
this beautiful child, whom I and my wife have since often enjoyed
between us, when she was entrusted to us by her father.
After a visit to us in England he left her to perfect her English for
six months with us. We certainly perfected her erotic education while
she perfected herself in English by her own ready talent for language,
for although only in her sixteenth year, she spoke five languages
perfectly, besides all the local dialects of Italy, which differ
greatly from each other. Her stay with us was much prolonged, for at
the time she was about to leave us she proved to be with child by me.
In due course of time she was safely delivered of a daughter.
Her father, who came over to take her home after the advent, ceded the
dear little object of my connection with her mother to my wife's
prayers.
We had no children of our own, and she would adopt her. The Count, who
in his heart was delighted at the proposition, left her with us. He
afterwards had a son by this beautiful and charming daughter and
granddaughter of his at one and the same time.
It is now long years ago, and that son legally adopted is now Count in
succession after his father's death.
We paid many visits during these years to each other, during which the
Count related to us some of the episodes in his life, which I give in
his own words---
``You ask me to relate my first experiences. My earliest initiation into
the secret mysteries of love's recess was rather a curious one, and one
which ended very disagreeably for the fair nun who sought to teach me
the gentle art of love.
``You must know that after Bonaparte's first conquest of Northern Italy,
when he had turned the Alps by the Savona depression, and by the
battles of Montenotte and others in that neighbourhood, gained the
interior plains and carried all before him, Piedmont was annexed, and
after the then French fashion, all church property was seized. Monks
and nuns were turned loose in the world, with a promise of small
pensions which never were paid. A nun of a convent in our neighbourhood
was one thus thrown on the world. To sustain life she opened a little
school for boys and girls of tender age. The neighbouring gentry,
willing to assist a worthy creature reduced to poverty by no fault of
her own, sent their children to her for primary instruction; my mother
had taken a great fancy to Sister Bridget, as she was called, and I was
sent to her school. I had just entered into my twelfth year, but was a
fine grown boy of my age, and I can remember that my prick when
standing in the morning had already shown proofs of fair development,
which gave promise of its future prominence I think I was the biggest
boy in the school, all the others being two or three years my juniors.
I was in perfect ignorance as to the relation between the different
sexes. The nun seemed to have taken a fancy to me, she used to embrace
me with her arms, and kiss me with very pouting lips, and I could feel
that she seemed to suck in my breath. She made me stand very close to
her in repeating my lessons, her arms or elbows, apparently by
accident, were always pressed against the spot where my, at first
insensible, prick lay hid. Without knowing how it came about, these
sort of accidental pressures at last excited it to stand, which she, no
doubt on the watch, was delighted to perceive. Seeing how she could now
excite it to the point she wished to arrive at, she said aloud one
day---'Fernandino, you must stay to repeat that lesson after the school
rises. You want a little extra instruction which I cannot give you
while occupied with all the class.' I thought this a kindness on her
part, but her object was very different. When all had gone and we were
left alone, she desired me to come nearer, the elbow played its usual
game, my cock stood, she pressed harder against it, then cried out,
`Dear me! what is that hard thing in your trousers? let me see.' She
unbuttoned them, put in her soft hand, and drew out my prick. 'How
curious that is. Is it always so?' `No, not always.' 'When how comes it
so now?' 'I don't know, but sometimes in moving to show me my lesson
your elbow touches it, and it gets into that state.' All this time she
was handling my prick in the gentlest and most exciting manner, indeed
she very quickly produced the spasmodic joys of heaven thus brought
down to mortal man, of course with only the nervous result. This was
all that was attempted the first time, when she told me to button up,
saying that it was a very bad thing to encourage that habit, and I must
be prudent and not let others know of its being improperly hard and
stiff.
``This sort of thing continued for a day or two. Finding I had said
nothing about it to any one, she proceeded to effect her grand object.
I was kept in as before. She excited me as usual, and soon had it out
stiff-standing. `Now,' said she, 'I will initiate you into love's
mysteries. I see you are discreet and can be trusted; lie down on your
back on this school form.' I did so. She lifted my shirt, my trousers
were already down on my legs, she felt the shaft and appendages, then
kneeling by my side she sucked it deliciously until it felt as if it
would burst. She then rose and straddled over the low form and my body,
pulled her petticoats up to her navel, and to my great surprise showed
an immense thick mass of hair, covering the whole of her lower belly.
Guiding my prick to the entrance of her cunt, she gradually engulphed
the little object by letting her body descend upon it. I felt a certain
smarting of pain in her first movements, and my prick partially
softened, but quickly regained all its stiffness by the pleasure she
gave me by her up and down movements on it. I went off as before in a
paroxysm of choking delight; she, too, spent, for I was conscious of a
stream of warm liquid flowing on my cock. She tightly held me where I
was, and by cunt pressures quickly brought it up to full stiffness
again, and a second delightful paroxysm followed.
``After this I fairly shrank to nothing, and dropped out. On rising I
saw that there was a wetness streaked with blood all over my cock and
cods; boylike, the sight of blood frightened me, and I began to cry,
she wiped it all off, and skinned back my prick to wipe under it but
here the raw surface made it painful, and even drew a show of blood;
previously my foreskin had been attached to the projecting edge of the
nut, her action of sinking on it had torn it off and forced it down on
the shaft, doubtless this is the maidenhead of a boy, and hence the
first smarting pain and the slight loss of blood that followed. She
tried to detain me that she might get some warm water, which she told
me would put it all to rights. I was too frightened, and ran off home
crying all the way, and like a stupid lubberly boy, sought my mother
and told her all what Sister Bridget had done and showed how sore she
had made my cock. My mother, enraged, ran at once to the school, where
in a back room Sister Bridget resided-berated her well, and in her
anger let it all out, so that the poor woman, lost all her scholars,
and was reduced to perfect poverty. However, a young Count in the
neighbourhood, who had been long trying to have her, now persuaded her
to accept his protection; she had the wisdom to make him settle
indefeasibly a pension upon her, so as to be safe from future
abandonment. I, of course, soon regretted the stupidity of my conduct.
As soon as cured of the slight soreness of my cock, my imagination
recurred to the pleasure her handling and sucking had given me, and the
delicious paroxysms she had produced, but, alas! all too late. However,
now I was awakened to the true use of a prick, and our women servants
and the peasant girls in the neighbourhood, who knew of my affair with
the nun, gave me encouragement, and I fucked them right and left, in
the fields, under the bushes, in stables or lofts, and carried on this
for a year; but at last I was discovered by my father, and sent off to
college at Savona. Colleges in Italy have schools attached for younger
students like your King's College, in London.
``Here I found a youngster but six months older than me, the son of a
friend of my family. I told him the story of my affair with the nun. We
used to get leave to go to the water closet from different masters, so
as not to be supposed to go together by design. From feeling our cocks
and frigging ourselves until we spent, which we both now could do, my
friend suggested that I should put it into his bottom, which a young
usher in his first school had taught him to do. He was a plump,
good-looking lad, with wonderfully large buttocks, and with an arsehole
which from the usher's practice, whose cock was full grown, was so
widened and sunk in that it really looked more like a vulva than an
arsehole. By this time my cock was nearly as large as it is now,
notwithstanding it entered up to the hilt without difficulty, and I
used to fuck him most deliciously. It is a curious fact that he liked
to be the recipient, and to be frigged by me at the same time. Although
he got into my arsehole a few times it was merely from curiosity; his
letch was to be fucked and frigged. While at college together this
quite satisfied us, and we never sought the dangerous intercourse of
the strumpets of the town, and so avoided the horrible diseases that so
many of our fellow students suffered from, many for all their lives
after. For years this agreeable intercourse lasted, and was only cut
short by my exile.
``Meanwhile, on my return home for the vacation, I had not forgotten
Sister Bridget, and longed intensely to renew my acquaintance with her.
I easily discovered her abode; meeting her one day she scowled at me,
and turned off in another direction. But I found out she had a
favourite walk in a lonely direction. I hid myself until she approached
too near to get away, seized her hand, implored her to forgive the
folly of a mere boy, who had ever regretted his ignorant stupidity, but
who was now a man, and longed to prove his devotion to her. Here I had
unbuttoned my trousers with the other hand, and pulled out a very fair
prick, at full stand.
```There!' cried I, 'see how the recollection of the paradise I lost
grieves him to the heart, let the poor dumb creature plead for me.'
``I placed the hand I held upon it, she grasped it tightly---
``'O! Fernandino, I always loved you, and but for your indiscretion
should have had you all myself for months.' I threw my arms round her
neck, our mouths met in a loving kiss, her tongue darted fire into my
soul. I drew her, a willing participator, into some side bushes. She
sank on the ground, her legs fell apart; I lifted her petticoats, her
rich fleece and palpitating cunt were irresistible, I flung myself upon
her, I gamahuched her until she spent twice, and then fucked her three
times before I withdrew. I would willingly have continued the delicious
junction, but that she implored me for prudence' sake to rise. We
parted, but not before arranging for other meetings, which took place
in woods and barns, wherever most convenient. Her protector going for a
week to Turin during one of my vacations, I was admitted to her room at
night by climbing the roof of an outhouse, and then stark naked we
indulged in every excess. She was hot and lewd to the utmost, a
splendidly made woman, with an insatiable cunt when once our sports
began. She was, as I before mentioned, most hairy, had a well-developed
clitoris, and fucked with as much pleasure in the rear attack as in her
tight delicious throbbing cunt. She loved above all things to gamahuche
a prick, sucked it most charmingly, but with greater art licked around
the hollow below the nut, and down the under side of the prick, with an
occasional lick of the ballocks, all in so exciting a manner that no
matter how often I had fucked her, she was sure to get another and
another. This charming intrigue continued until I went to Turin.
``During my connection with Sister Bridget I learnt the whole history of
her convent life. She was forced to take the veil by her family, much
against her will, for she even then felt the prickly sensation of
desire, making her cunt throb at the idea of coition with the male sex.
She quickly found a friend with similar desires, but more experience,
who first taught her all the art of tribadism, and then confessed to
having connection with the youngest father confessor. This priest came
once a week to confess the nuns, to confess their liaison, and to let
him inflict what penalty he liked. He told her he would flog her, and
then punish her where she sinned, which, in fact, meant putting his
prick into her cunt when in a kneeling position. This sort of thing was
done to see if she took it in with gusto, and when it was found that
was the case, their hour of confession was a scene of every excess,
stark naked, for neither wore aught but the frock of monk or nun. This
delicious indulgence lasted until the dissolution of the convent, and
you know the rest.''
Another recital of the Count much amused us. The Count was admitted at
seventeen into the Royal Guard, where each private was born a
gentleman, and held the rank of sub-lieutenant in the army. Here he had
many intrigues, and took the maidenhead of a charming and beautifully
made girl, who was being brought up for the stage as an opera dancer,
for which she showed early capacity. She proved a great success when
brought forward. She dearly loved our friend, and was supposed to be
faithful to him, although she had developed excessive wantonness and
lubricity under his able tuition. His flight and exile separated them.
Years afterwards he met a lovely, magnificent, fully developed woman,
splendidly attired, walking in the Regent's Park, He did not recognise
her, but was looking at her with longing eyes, when suddenly she seized
him by the arm, and exclaimed in the patois of Piedmont, ``Ces tu si!
Buzaron.'' (Is that thou thyself, Buzaron). This latter word is a
familiar expression of carnal affection, but, literally, is ``big
bugger.''
Their intercourse became of the warmest, she was now a first-rate
\textit{danseuse}, very highly paid.
The Count had first had her, she really loved him, and in London stuck
faithfully to him, for love alone, for she never would accept even the
smallest present. She, of course, had plenty of splendid offers from
noblemen, but as long as the Count would have her she was faithful to
him. When, which a knowing woman's tact senses, she saw a falling off,
she released him, and, although never refusing her person to him, took
to others as well. She was a very lovely bird, and used to relate the
erotic experiences of her previous years. Many of these were most
amusing, but one in especial showed the ardent nature of her
temperament. She had accepted, when dancing at Genoa, an eligible offer
from the Lisbon Opera proprietors, and had to take passage on an
Italian brig; she was the only passenger, and her berth was in the same
open cabin as that of the captain and mate. On the second day out the
captain showed signs of wishing to have her. She was already longing
for a fuck, to which she had been daily habituated on shore, so she
lent herself most willingly to his desires; from him to the mate, and
eventually to all the ship's company, without any jealousy of captain
or mate; for the system in those days made captain and crew all equally
interested in the success of the voyage from the terms of their
agreement.
The captain, mate and carpenter were owners of the vessel. The crew of
a boatswain and four picked men received food, mostly dried fish, but
no wages. They were entitled to a certain share of the profits of the
voyage, and thus were interested in its success, and on very different
terms of intimacy with the captain to what ordinary sailors could be.
The voyage lasted six weeks, and during all that time she had every man
in the ship into her every day, and from fair front-fucking had
eventually satisfied them in both apertures, and often had had one in
each orifice, and sucked a third to spending point, which she
deliciously swallowed; she had even taken the \textit{premisses} of the little
twelve-year-old cabin boy, and she declared that she never enjoyed so
complete a satisfaction of her excessive lascivious lust as in that
happy voyage of six weeks' duration.
The Count, who had split his sides with laughter as she recounted this
extraordinary indulgence in every enjoyment of lust, related in the
amusing patois of Piedmont, told us that notwithstanding such excessive
indulgence in both orifices, and by pricks, many of which were of
immense dimensions, not the slightest appearance of such ample
stretching could be detected on the closest examination, and that in
either orifice she could almost nip your prick off. One of those
exceptional constitutions and splendid forms that no excess injures,
and who are ready for any number of pricks and reducing them all to
inanition, while she remained as ready as ever to recommence the utmost
excess of lust as soon as any one set had exhausted themselves.
As a sequel to the Count's confessions, I shall here give my adored
wife's account of her early life in the form of a narrative, for when
it was told to me it was interrupted by various lustful encounters
produced by the lascivious and exciting nature of her revelations.
She was the daughter of a Greek mother, married to a high clergyman of
the Church of England, a man of great erudition, who had taken the
highest honours at Oxford. When Fellow of his college he was tutor to a
great nobleman's son, had travelled for years with him, and hence his
wide acquaintance with the languages of modern Europe. In Greece he had
fallen over head and ears in love with her mother, had tried to seduce
her, and, failing that, married her. He was a man of most lustful
propensities, her mother was of a beauty most attractive and exciting
to such a man, having lustrous and most lustful eyes, extraordinary
wealth of hair, which when undone reached to her heels; thick and
meeting eyebrows, and a well-defined moustache, all enough to drive a
sensualist like her father mad. So failing all other means to have her,
he married her, and, as far as she could afterwards learn from him, was
in all voluptuously lewd, carnal acquirements, every thing the wildest
imagination of lust could desire. It was from her mother she inherited
all that deliciously haired body, and from both parents her intensely
lascivious passions. She lost her mother just as she had attained her
eighth year. During her mother's life she had generally crept into
their bed in the mornings to have a cuddle, and had often been a
witness to the fucking of her mother by her father, and had, at other
times, played with his prick until it stood, and even made him spend
with her toyings. She owned to a sense of sensual gratification in
this, but at that early age without any idea of the possibility of its
being put into her. She always accompanied papa to his bath, and he
invariably dried her and finished by kissing her mount and her cunt,
and without tongueing it.
After her mother's death he always had her to sleep the whole night
with him, and when in her ninth year he had commenced by gamahuching
her clitoris, which even at that early age he declared gave promise of
exceeding in projection the fine one with which her mother had been
provided.
In this manner he soon awakened all the latent lubricity of her nature.
Afraid to force an entrance at that early age, after exciting both her
and himself, he used to rub his great prick between the lips of her
cunt, and against her clitoris, until worked up to spending point, when
he transferred his prick to her mouth, and spent therein, he having
taught her to practise that voluptuous and delicious method.
It was naturally impossible to stop short \textit{dans un tel beau chemin},
and it ended by his first getting the knob of his prick into her small
tight slit and spending there, gradually forcing his way further and
further in, until she, driven mad by such excitement, felt the utmost
desire to have it into the deepest recess of her longing cunt, and
begged him to shove it in harder and further.
With such a spur to his passions, unable to control himself, he burst
through all obstacles, and completely deflowered her, giving her
greater agony than she expected, which was subsequently completely
alleviated and converted into the most exquisite sensations. Once he
had fairly fucked her, he continued to do so constantly until the age
of puberty, which declared itself by the coming-on of her monthly
courses even before she was twelve years of age. Already an extensive
moss-bed of sable silky short curls adorned her mount and body.
At this period her father told her he must take precautions against
getting her with child; at first he drew out and spent in her mouth,
which she dearly loved, but becoming lewd on her bottom-hole, which he
constantly fingered, he declared it was too much derangement of
position to get it into her mouth, and suggested merely driving the
knob into the arse-hole, and spending therein, which he could do by her
merely heaving up her arsehole as high as her cunt had been, and so
entering without any change of position on his or her part. Of course
it soon came from the knob only to the utmost length of his prick in
her arse, and gradually she came so to like it that often the entire
encounter of three or four coups was delivered in her arsehole to her
infinite satisfaction; and thus her father enjoyed the first fruits of
every aperture in her body.
He it was that instructed her so deeply in classic literature as well
as modern languages, but always choosing such lewd works to carry out
her education, such as Meursius and Suetonius in Latin, Athenaeus with
his supper conversations in Greek, especially drawing her attention to
his chapter on boy love, Boccaccio and Casti in Italian, the
uncastrated editions, the adventures of Casanova, and the hundreds of
other French bawdy books, with the most exciting illustrations of all
these works and many others besides. The lecture on them always led to
good fucking in one aperture or the other, practising the particular
description that excited their lewdness.
He thus depraved her mind that she soon longed for other experiences
than all he could give, and she cast about for an \textit{aide-du-con}. This
she first found in their young and handsome footman, who proved not
only discreet, but completely up to his work, and uncommonly well
furnished.
They occupied in every voluptuous excess the hours papa had to attend
to the extensive and rich parish of which he was the rector. I must
tell the rest in her own words; she said---
``Continued immunity in our excesses led to excessive in-cautiousness,
and caused the discovery of our intrigue by my father, who appeared
shocked and distressed at the discovery, but he was quickly reconciled,
as it ended in his having the youth himself, and his introduction into
our incestuous orgies, in which he both fucked and was fucked by my
father when not giving me the exquisite delight of having both
together. And for five or six years I had but these two charming
satisfiers of my lust.
``At this period a beautiful youth of fourteen, the son of a younger
brother of my father, and, consequently, my first cousin, came to live
with us. He was an orphan, left by his mother under the guardianship of
my father. I was some three years his senior and he took to me as an
elder sister, was very loving in that character only, and used to
embrace and kiss me most affectionately. I, for my own part, soon began
to have other feelings.
``On his first arrival, in grief at the loss of his only surviving
parent, he feared to go to bed alone, so I used to accompany him, and
help to undress him. He was all innocence, his mother, up to her recent
death, had done the same, so he had no \textit{mauvaise honte}, and I helped
off his shirt and helped on his night-gown, and even witnessed his
diddling before he got into bed, which I tucked him in and kissed him
before leaving.
``Of course with my then complete knowledge and practice of every art of
lust, I could not but look for and discover all his secret charms, then
always in a state of repose, but promising a future development. I grew
lewd upon him one morning, after an orgy with papa and the footman, who
had not altogether satisfied me. I was tempted to go along to my cousin
Henry, to waken and cuddle him, knowing that he would probably awaken
with a cockstand, as usual with youths, and even men.
``I slipt along, and at once saw, as I expected, for only partially
covered with the sheet, the prominence of his prick was unmistakable. I
gently removed the sheet, and was delighted to see that his instrument,
insignificant enough when down, was of a very respectable volume when
erect, and quite capable of giving any woman perfect satisfaction from
its excessive hardness. I gently took it in my hand to feel it, it
throbbed at the touch, and felt like a piece of wood in hardness, with
a velvet covering.
``I got into his bed by his side without awakening him, taking care to
pull up my chemise so as to let him feel the contact of my bare skin. I
pulled the sheet over us, took him in my arms, and woke him with a kiss.
``He was surprised and delighted at finding me by his side, but as yet
had no idea but that of cuddling and caressing me. In throwing my arms
around him I had taken care to pull his night-gown up to his loins, so
that his naked body pressed against mine as we embraced.
``In apparent surprise I cried out what is that pressing so hard against
my body; at the same time moving my hand and laying hold of it. It
throbbed violently to the touch. I threw the sheet off to see what it
could be.
```Dear me,' said I, 'how is this? What a change! it was not like that
when I put you to bed last night. How has it become in this strange
state?'
``'It is so, dear cousin, when I want to pee in the morning, and goes
down afterwards.'
```Then jump up and pee, and I want to do the same.'
``He took the pot and piddled. I took another and piddled, standing with
legs wide apart, and holding the chamber pot partly between and partly
under my thighs, so that he could perfectly see the whole of my cunt,
and the flow of water from it.
``He stared with astonishment; it was really the first time he had any
knowledge that women were differently formed down there than he was.
```How funny,' cried he, 'you piddle from a chink, and have no doodle. I
should like to see it nearer.'
``I told him I should lie down on my back on the bed, and he could look
as much as he pleased, but he must never tell any body what he would
see, because it was a great secret.
``He promised, of course. I lay down on my back, having first thrown off
my chemise, stretched wide my legs, told him he would see better if he
knelt between my legs, some slight distance from the object to be seen.
``He got up and began a close examination, admiring the immense quantity
of hair I had already got, opened the lips, caressed what he called the
little doodle, my clitoris, which was rampant with lewdness. I told him
to feel inside with his middle finger; he pushed it up---I nipped it, to
his astonishment, so that he could hardly withdraw it. Nature, unknown
to him, acted her part; his cock, which had gone down after piddling,
stood stiffer than ever. I laid hold of it, and said---
```How comes this, Henry? You can't want to piddle again.'
``'No, no, but I feel queer all over, I don't know why, and it seems to
have raised my doodle as you see.'
```If you will keep it secret I will show how it comes about.'
``He promised that he would never, never, tell any thing I should teach
him. So I said---
``'Come to my arms, lie down on my belly, and I will teach you. There
that is it.'
``His cock beat fiercely against my cunt. I passed my hand down, guided
it into my longing cunt, then placing my hands on his buttocks pressed
down and forced his charming shaft up to the hairs of my longing cunt,
foaming with my father's and the footman's sperm, so that he slipped in
with the greatest ease; but no sooner was he hilted than one of my
exquisite cunt-pressures made him cry out with unexpected pleasure,
while I spent with the delicious conviction that I was enjoying the
first fruits of a beautiful youth. I told him how to move in and out,
nature did the rest the moment he knew what to do. A very few thrusts
brought down his first tribute on the altar in the exquisite recess of
Venus, the voluptuous goddess of love. I joined in the delicious
discharge.
``Once experiencing the joys of coition the dear boy fucked me five
times before I could get him to withdraw, and it was only the fear of
discovery that induced him at last to get off me. We had a delicious
cuddle, and I promised to come every morning I could do so with safety.
Impressing upon him the absolute necessity of secrecy and caution, if
he wished to have any repetition of the delightful lesson I had given
him, I returned to my room gratified beyond measure in having taken a
maiden tribute. Women who have the luck of such good fortune alone know
the exquisite delight of initiating a virgin prick into love's
mysteries and our longing cunts.
``We carried on this delicious intercourse for months before it was
discovered, but use begets want of caution, and my father at last
discovered it. Poor Henry thought himself happily excused by allowing
my wanton parent to take possession of his bottom while fucking me. My
warm embraces enabling him to support the great and curious pain and
pleasure attending a first penetration of that delicious narrow
aperture, dedicated to the obscene god. It ended in his complete
initiation into our orgies with the footman. His addition to the orgy
enabling more complex and lustful combinations than two men and a woman
alone could indulge in.
``My father, who lived quite up to his income, died and left me with a
very small capital at his death, which happened after the coming of age
of my cousin Henry, to whom I had become violently attached. Indeed, it
was my first love, and had all the devotion and ardour of that passion.
He had a small independence, and we lived together for two years after
my father's death, secretly sleeping together.
``The interference of relatives who, without suspecting our real sensual
intercourse, preached upon what the world would say, \&c., induced me to
undertake a governesship, for which the great instruction I had
received from my papa more than fully qualified me. I saw the
reasonableness of this, and also thought it was more likely to
strengthen Henry's love than otherwise. But the parting was a great
trial. He had grown a fine man, with a superb prick, although far
inferior to this monster,'' laying hold of mine at the moment standing
stiff and wanting but her touch to make me bend her back and fuck her
off hand, so exciting had been her recital.
She resumed after this episode, by saying her system of teaching was
eminently successful. From time to time she was comforted by interviews
with her loved Henry, besides satisfying the lust of both the father
and sons of the families she lived with, teaching and taking the
maidenheads of several youths, but in none receiving the gratification
her loved Henry had given her, until, as she flatteringly said, she had
the good fortune to enter our family and find such a jewel as I
possessed.
She had occasionally found girls of such a warm temperament that she
was induced to initiate them into the art of gamahuchery. It was in
this character of instructress that she had first used the rod to the
bottoms of her pupils, and it was seeing the erotic effect produced on
them as recipients that first gave her the letch of being herself
birched. After this she had had a vast variety of youths, fathers of
families, and old worn-out patients, whom she birched into action.
From one situation to another she had arrived at ours; since which time
I knew all her doings.
The Count's son and my daughter meanwhile grew up to puberty. We
watched their progress with great interest. They were both initiated in
all love's delicious mysteries by their respective parents.
My lovely little Florentia, for we christened her in my adored wife's
name Italianized, which became familiarly Entee, was a great comfort to
us. From childhood she always came to cuddle us in bed before we rose.
She was so beautifully made that we used to strip her naked and kiss
her whole body, which always gained my dear wife an extra fuck,
especially after she reached her tenth year, when her form was rapidly
developing into puberty. Being from infancy familiar and accustomed to
be always stript by us, she had no shyness; indeed she became so
exciting that often I grew rampant and fucked my dear wife while she
was present. She grew to like to see us do it, and used to play with my
big cock, and bring him up to the scratch. It ended as it was sure to
end, in my gradually toying with her from one excitement to another,
until she was completely fucked in her thirteenth year.
Ten years after that epoch I lost my beloved wife, and would have been
quite inconsolable but for the sympathizing endearments of this darling
child, who became so necessary to my existence that twelve months after
my adored wife's decease I married her. She was a perfect Italian
beauty, and no one supposed she was other than an orphan adopted by my
late wife.
Now, in my old age, she is the comfort of my life and the mother of my
beautiful son, whom we have named Charley Nixon, in memory of both my
first adored wife and my guardian, through whom he will inherit great
wealth. The dear little fellow is now eighteen years of age, handsome,
well grown, and very well furnished, although not so monstrous in that
way as his father. His dear mother has initiated him in every delight,
and he has all the fire of lust that his old father had before him. He
often comes to us at night, indeed, it is the only thing that enables
me from time to time to get a cockstand and a fuck at his mother. To
see them in all the agonies of lust, fucking furiously before my
delighted eyes, so excites me now and then, for, alas, it has come to
be a gratification few and far between. But occasionally to suck up his
young sperm after the excitement of their love combat produces a
stiffness for my beautiful wife to mount upon me and then have our
charming son to put his prick into her bottom for this, too, is
necessary to my failing vigour, and the contact of his vigorous young
prick against the thin filmy substance separating us feels as nothing.
I am long in spending, and his delighted mother gets two and sometime
three delicious discharges in her arse before my lazy prick deluges her
cunt with my incestuous sperm.
We are thus a happy family, bound by the strong ties of double
incestuous lust. It is necessary to have these loved objects to fall
back upon, for alas! all the earlier partakers of my prick are now dead
and gone. Aunt and uncle, the Dales, the Nichols, my beloved Benson,
and her friends the Egertons.
I have already mentioned the Count's death, and both my sisters have
left me alone, and I should have been a dreary and solitary old man but
for my beloved wife and son, who solace me and replace the void in my
heart I should otherwise have so sadly felt.
I shall here end this long tale of my erotic life.
A curious event has happened lately, the divorce of a Mr. Cavendish
from his wife for adultery with the young Count de la Rouchefoucalt.
The details brought before the court were of the most scandalous
nature, especially the letters exchanged between them when the Count
had to go to Rome, where he was attach\'{e} to the French Embassy. When the
husband's counsel handed up the letters with the sworn notary's
translation, he remarked that he thought they were too horribly
scandalous to be read in court. The judge scanned a few of them, and,
addressing the counsel said---
``I am perfectly of your opinion, my learned brother, I shall take them
home and make a point of them in my address to the jury.''
It will be seen that they were of such a nature that doubtless the old
judge, who was no other than my dear old chum Harry Dale, gave his wife
two or three extra fucks on the strength of the lust produced by those
exciting and extraordinary lascivious letters from a young man of only
twenty-one years of age, showing quite as early an initiation into all
the luxury of the utmost depravity as any of my own details of my early
experiences with my darling old aunt.
Some of the letters are a string of imaginary events as to how far they
could carry their imaginations. The Count constantly alludes to the
inferiority of his descriptions to those given in her replies. Alas! as
he possesses those exciting replies of the lady, they cannot be got at,
but from his descriptions, and the remarks on certain gross
familiarities, it's evident she was gifted with as lascivious and
lustful a temperament as either my aunt or the divine Frankland.
A chance threw these interesting letters into my possession, and I can
assure the reader they are the veritable sworn translations of the
letters found in Mrs. Cavendish's davenport when it was broken open by
her husband, and produced on the trial. The Count had evidently dreaded
such an event, and it will be seen he constantly implores her to
destroy his letters as soon as read. But, with the infatuation of her
sex, she kept them to furnish the sole evidence by which she lost her
place in society and became a lost woman. It is added that she was a
woman of forty-five, and the mother of several children, but it is
these randy voluptuous matrons who have the most attractions to a young
man who feels flattered and is proud of, as he thinks, conquering a
woman in a good position in society. It is evident enough that she was
no tyro in every depravity of lust, and probably had passed through
many hands before he gained her. He appears to have been really
cunt-struck, which, as I have before observed, is one of the strongest
infatuations that a man can have.
\gutchapter{END OF VOLUME IV.}
\section*{\raggedright ADDENDA}
\gutchapter{LETTERS}
\section*{\raggedright PRODUCED IN THE DIVORCE-CASE}
\gutchapter{CAVENDISH}
\textit{v}.
CAVENDISH AND ROCHEFOUCAULT.
Rome, \textit{Saturday}
\textit{August 6th, 1859, 10 o'clock}.
I tried last night, my angel, to write you a half sheet, but it was as much as I could do to read your letter a second time, and it was only by making a great effort that I was able to write a few lines. However, this morning I will try and continue, in order to reward you, not for that one which you have deprived me of from pique, but for those bewitching ones which I have lately received.
I have just received your letter 17, begun August 3rd, 11 o'clock at night, and bless you for the idea of addressing it to Pal. B., it is infinitely preferable, and there is no fear of any risk ("indiscretion'' in original) either now or later.
I am delighted when I think of the pleasure you derived from what I sent you the other day. I only decided upon it in fear and trembling. I do not understand what you mean by letter direct to Albert. If you do not send it per Embassy bag I should not have it here till Monday; you would have done much better to have put it in the parcel. All last night I slept very badly, no doubt in consequence of a presentiment I had that I should not receive a half sheet, and that you were annoyed at my going to Albano, and I thought of a mass of things as disagreeable as they are painful. Of your birthday, for instance, the 1st of October, which will be an opportunity for II * [* II stands for her husband] to make you a present in return for the set of studs which you gave him on his birthday, when you, no doubt, will give him something.
As to your brown cloak which II gave you, \&c. \&c., I request that on your birthday when he makes you his usual present, whatever it may be, you accept it and say, ``I thank you,'' and, without even looking at it, put it upon the table, immediately speak of something else, and when he has left the room, put it away out of sight without ever speaking of it again, or appearing to know what has become of it.
I have just been interrupted for an hour and a half by M. de Fiennes---very agreeable is it not? You must forgive me if I am unable to write to you at length; what I have said to you above is for the future, but the past is over since he has those studs. I forbid you to give him something in future, unless you cannot possibly do otherwise; and, in that case, you must give cigarettes or anything which does not last. I will see what is to be done about your shawl, was it not II who gave it to you? Thanks, my treasure, to walk so far from II; it is so good of you to give up to me that walk, which I hate when you take it with him.
Ah! new projects again, but let us hope these will be the last, how I pity you. You were so well you told me two days ago, and now you are already obligated to take some powders---it is II system. Nice health you seem to have; you have good reason for believing that the regimen you have hitherto followed is a good one, it succeeds so well! Poor darling I can comprehend how uncomfortable these frequent agitations must make you. I suffer from them so often myself.
I will make you some drawings later. I have not the time to-day. Those uncertainties of your mother are terrible. Oh, yes, I am in despair at that departure, particularly before my lot is decided, and knowing, as I do, that you are unhappy. But, my child, do not fear to let it be known in every direction that you cannot endure II, and that you have taken a disgust to him. Do not hesitate to give the true reasons when you refuse to do anything, simply, ``Yes, or No, the hand, but with ........... it is not necessary. I can dispense with it, nothing of that sort is necessary.'' And then, when that has produced the desired effect, add, ``We can only live under the same roof upon those conditions, for sooner would I go away altogether than that it should be otherwise.'' Speak in this manner; it won't answer very well at first perhaps; but he will soon get accustomed to it, ``How do you do?'' in the morning, and ``Good night,'' at night. Then gradually get into the way of saying ``Mr. C.'' when talking of, or speaking to him. You may be told it is not the custom. Answer you don't care, it is not the custom to be such an idiot as he is. Ah, you are too sad, poor child, all that is charming, and all our superstitions. Moreover, one must think of what has been, not of what will be, and compare it with what is. The progress is very delightful and consoling.
Do not be unhappy about my horse, he did not go very well, and then I do not care about driving in a carriage when you are on foot.
I have made two drawings, one prettier than the other, and I have had a copious emission.
Mrs. S. has made no tentative overtures towards me. She is often that way inclined, and with everybody. Be calm then; but, after all, you are perfectly so, only you pretend to be otherwise. God bless you for speaking so often of your pretty rose-coloured silk stockings. I like them so much, and adore you for wearing \textit{them}, although it is not the custom, above all in the day time. Doubtless it is very coquettish, pretty, and wondrously exciting. Even only to think of them gives me an erection. And that rice powder! how divine you must look. It is to be hoped that the powder in your hair will not give ideas to II and embolden him---take care. Thanks for thinking so often of me, my idolized angel. Adieu, my good, my best treasure, I love and embrace you tenderly. I will have my revenge, for I, too, had prepared a half sheet, but will not send it till to-morrow.
Rome, \textit{Saturday, for Sunday's Post}
\textit{August 6th, 1859, 2 o'clock}.
I wish to give you a little surprise, my own dear little darling, in sending you this letter, which you will receive with a half sheet upon which you had not reckoned on Tuesday morning, so as to supply the place of Sunday's post. It was to give you this little surprise, and in no way of retaliation, that I did not send a half sheet in my letter of this morning. It was very unkind of you not to send yours upon the pretext that I was at Albano, but you will have been ashamed of it since. Besides, even supposing that I had been there, I should not have committed any indiscretion with your envelopes, which are so excellent, and, if one had felt inclined to do so, your letter was sufficient to make me indifferent to it. I suspect you of not having prepared what is necessary, I shall be sure to see if it be so; to-morrow's letter ought to contain two. I continue your letter 17, and I perceive with rapture that you have had a thick cream-like emission of enjoyment. How delicious it would be in my tea. How I should like to send you some like it also. It is a good thing that my letter to the little girl was successful. Will you tell Madame de Delmar that I am sorry to hear that she is suffering, particularly as her ordinarily detestable disposition only becomes more thick and more execrable. Suppress this latter part if you think it better.
Ah! you think that Madame Salvi has played her cards well and in what way, I ask? You are too bad, too implacable. I do not like that in you. I have told you that your suspicions wounded me, and I think you can believe me when I tell you that I have completely changed my conduct in that respect. Besides, what can I possibly do. I am very uncomfortable here. The Abdol don't want me; besides, the Duke has given me to understand that I ought occasionally to go and see his wife, and the Borgh bother me with all their children.
Thanks, my good angel, for the letter Des Pierre. If it be decided that you leave, I shall go for a few days to Civita---sad and mournful consolation. Why do you tell me that you will go barefooted when I go to see you. I am quite of your opinion that your feet are only too delicious. The costume rather disgusted me than otherwise, without, however, producing any effect upon me. To-morrow I shall pay the Duchess de Grano a visit, and since it seems to put you out, shall not return again to Albano.
Heaven knows that the pleasure is not great, and that I care very little for it. The other day I did not even find it any cooler there. The Duchess of St. Alban's leaves on the 20th for Schwalback and England on account of the apprehensions about war---another subject of uneasiness for me---such is life. I can go and live with the Duchess de Grano and Salvi. No one would say anything about the one, and not much about the other, whatever you yourself might say, but that annoys me exceedingly, and disgusts me, and I dare not do so with you. You might, however, have been my ambassadress, see what it is to be so seductive, so graceful, so pretty, so kind and gentle. Just fancy, dearest, that I have not answered Madame Rudiger. I must really do so to-day. She is a person one must be careful with.
I have always this phrase before my eyes. ``I prepared a divine half sheet yesterday evening, but dare not send it!'' Very agreeable, and very kind of you! well, I do not complain.
They have been so heavenly during the last three days, and mine are so shameful. How nice it is for me to think that I should have sufficient influence over you to get you to sit perfectly naked at my table. Long ago might you have had that influence over me, and even have enforced requirements more depraved, and more degrading than that if you had wished it, and with what rapture! Adieu, my angel, what a happiness to give you this trifling pleasure.
When I shall have undressed my adorable little mistress it will be nine o'clock, she will be mad with desire, delirious from passion and rapturous exactions (exigencies), her maddening look exciting me in the highest degree will arouse all the strength I possess, and enable me to exhaust her so completely that she herself will attain the height of happiness; the greater the refinement and delicacy of my caresses the greater will be your happiness, the more languishing will your eyes become, the more will your pretty mouth unclose itself, the more will your tongue become agitated, the more will your bosoms, firm and soft as velvet, become distended, and their nipples grow large, red, and appetizing; then will your arms grow weaker and then will your angelic legs open themselves in a voluptuous manner, and then seeing ourselves reflected on all sides in the mirrors, shall I take you in my arms in order to excite you (\textit{branler}, frig) with my hand, whilst your little rosy fingers will similarly excite me with vigour, and I shall suck your divine nipples with passion. When the agitation of your little legs, of your lovely little bottom (\textit{derri\`{e}re}), of your head, and those murmurs of pleasure (\textit{rugissements}) prove to me that you are at the point of emission, I shall stop and carry you to a piece of furniture made to sustain your head, your back, your bottom, and your legs, and having near your cunt (\textit{con}) an opening sufficiently wide to allow my body to pass erect between your legs; then shall I fuck (\textit{enfiler}) you with frenzy with my enormous and long member, which will penetrate to the mouth of your womb; being squeezed by your pretty legs, which will bring me closer to you, I shall wriggle (\textit{remuerai}) my strong pretty member, which you love, with more vigour than ever; my private parts (\textit{organes m\^{a}les}, testicles) will touch your little bottom, and this contact will provoke such an abundant flow of the essence of love in your little cunt that I shall be as if I were in a bath.
How I fear to leave off there! But we shall see. Do not write to me by the night post, it is useless! It is true that when I am near you in a carriage I have difficulty in remaining quiet. Oh, no, do not alarm me by your insatiability, mine is much greater than yours, there is not the slightest comparison to be drawn between us in a physical point of view, but as far as our moral nature and heart is concerned we can rival each other, and I am very happy on that account.
1:40. I was most annoyingly interrupted by the luncheon bell, and afterwards I played a game of Fourreau (a game all the fashion at Verteuil), and here I am again. I have just refused to accompany my father and mother in a drive in the neighbourhood, so that I shall be able to write to you more at length, unless, indeed, I write to Fallenay.
You tell me that you like the little costume, but that is all you say, and you give me no details as to the colours, the length and shape. I will believe my treasure, my jewel, that your bosoms will be white, swollen and soft as velvet, and it is very nice of you to tell me that my hands will have difficulty in holding them and putting their ruby lips to my mouth.
You are quite right in saying that you will develop my virility, it is you who have made my member what it is now. I repeat, on my word of honour, perhaps you will not like to hear these details, but, nevertheless, I shall say it, you are the first woman in the world who has stimulated that essence which flows from my prick (\textit{queue}), which your kisses have rendered so pretty, and it is you who have plucked the flower of my virginity. Never have I had (\textit{bais\'{e}}) any other woman, and whatever may be the misfortunes to which I may be destined, it will always be an immense and ineffable happiness to me to think that I have given and lost it through the luscious draughts you offer (\textit{par tes d\'{e}lices}). It is, and it will be, perhaps, the greatest blessing, and the only consolation of my life. But before God it is a great one, and my enjoyment has not been such as one can expect to find in this world. I do not believe that he who had the madness to rob you of yours was as pure as myself, and as for voluptuous pleasures, if there be any greater than that which I know, I promise you never to learn or seek it, although I don't require this at your hands. I do not wish to have any other woman spoken of, they all disgust me, even to look at them. You know it, and you know that there is nothing, absolutely nothing, in you to disgust me, but all that belongs to you maddens me, and I love and adore all; it has become a madness, and you know it; for when you are kind you give at least the idea by letter of that which you would not do if you had the slightest doubt.
You know that I have sucked you between the legs at those delicious moments when you made water, or when you had your monthly courses, and that my happiness will be complete when you will allow me, and when circumstances will allow you, to let me lick (\textit{passer la langue}) at that ineffable moment when your little love of a jewel of a bottom has just relieved itself. In you every thing appears different and pure, the purity which reigns in your every feature, the excess of refinement which exists in your whole body, your hands, your feet, your legs, your cunt, your bottom, the hairs of your private parts, all is appetizing, and I know that the same purity exists in all my own desires for you. As much as the odour of women is repugnant to me in general, the more do I like it in you. I beg of you to preserve that intoxicating perfume... but you are too clean, you wash yourself too much. I have often told you so in vain. When you will be quite my own, I shall forbid you to do so too often, at most once a day. My tongue and my saliva shall do the rest.
If it is necessary let the doctor cauterize you (\textit{toucher}), that is to say with his instrument, and mind he does not fall in love with you; I bet he has never before seen anything so seducing, so pretty, or so perfect. It is to be hoped that the irritation does not proceed from the size of my member.
You did quite right to go to the play, and I regret sincerely to have spoilt the pleasure you had in going, it shall not happen again.
As to the place George had, that is perfectly indifferent to me.
Ah! you think that the portrait was done afterwards. You are not sure of it, but it is a matter of no moment, my much loved one. I shall not be the less happy to have the photograph if you are good enough to give it to me, not too much in miniature. I shall be very grateful for it.
If I said that Galitzin was clever, I was wrong; he has a kind heart, and is very fond of me. Now that he has lost his mother, I shall be more kind to him. He is a person one can depend upon; his letters are silly productions. Those Russians have always the imagination easily excited.
Yes, my father has always the same answer. Thanks for your obliging offer of gloves, my mother must settle about it.
I shall still have lavished the following caresses upon you, angel of my delight, were I a little calmer. I had a dream, such as it was, about it last night, and only remember it just now by way of explanation of my mad excitement of this morning. I saw you as I was asleep, you were by my side frigging me with your fingers of love, and you heard me say to you, ``I see you there.'' You are as lovely as Venus, your lusciousness and lasciviousness are at their very height, your body is completely perfumed with your urine, in which I forced you to bath yourself for my enjoyment, so that I might lick you. You have painted the most seductive parts of your person. Your shoulders are white, your rosy bosoms reveal themselves through a rose-coloured gauze, trimmed with bows of the same hue. Your thighs, as well as your navel and your heavenly bottom, are revealed through a heavenly gauze, your legs are clad in rose-coloured stockings. The sperm flows; but how much I needed it! This is true, for my testicles were swollen in an alarming manner.
Oh, my child, my pretty little mistress, if you only knew how much I suffer from the excessive heat, and the privation in which I live! Without exaggeration, my testicles are enormous. My member is as large, straight, and stiff as my arm. I am mad from desire for you. I had the unhappy idea of going to bed again. My mind was full of a dream I had had, and of which you were, of course, the subject. Then I thought of the caresses which you would have been obliged to submit to, and at last, in consequence of your yesterday's half sheet, so pretty at the beginning and at the end, but yet quite beside the question, and found myself engaged in the act of rubbing myself with frenzy, and of stroking myself and of frigging my prick (\textit{la pine}) until I was exhausted, before I could discharge the merest drop; that was too much for me, and now I desire you like a mad man. If a delicious half sheet does not arrive by the Embassy bag, I know not what will become of me. I have had an emission. I am saved. I shall feel myself so relieved. You have forbidden my going with other women. You are determined that I shall not have a discharge with any one but yourself, and that I have fucked (\textit{bais\'{e}}) no one but you. Oh! how I must love you.
It is two o'clock in the morning, I have violated and well worked you, kissed, frigged, licked, and sucked you obliged you to yield to my desires, the most debauched, the most shamelessly degrading during the whole of the afternoon. All the afternoon, too, I have got you to suck my member and my testicles. I have made you pass your tongue between my toes and under my arms. I have compelled you to paint your body, to drink my urine. I was almost on the point of getting you sucked and licked by a pretty Lorette, perfectly naked, between your legs, and to make you piss into her cunt in order to make the depravation more debased than ever. I have had discharges from jealously. I have discharged at least forty times; and when, after having left you to go to my club, I returned home, and finding you fast asleep from exhaustion, I awakened you and insisted upon your frigging me with your rosy fingers, all the while licking my several parts. You implore me. You are wearied, but I am intractable. You must do it in order to excite you as much as I am myself excited. I suck your breast with frenzy. The sucking that I have given your bosoms, and the fear you have lest I should fetch a young girl to violate you with her breasts in your cunt, filling your womb with her milk, excite your senses, and then you hear a voice whose sound alone so pleasingly tickles your womb, saying to you, ``My pretty mistress, I implore you to abandon your (?) to me. I will love you so fondly. I will be too kind and gentle, I am so handsome, I will do all you can possibly wish. I know so well how to have and suck a woman, my member is enormous, it is beautiful, rose-coloured, large, long, hard and vigorous. Yield yourself to me.''
Tell me if you like this one.
When you are ready you will call me so that I may come and say my daily ``How do you do?'' You will begin by taking my------out of my trousers, then half opening your gown, you will lift up your pretty chemise with one hand, and will pass your other arm, soft as satin, round my neck. I shall embrace you tenderly, then I shall lick your snow-white shoulders, your bosoms, which seem to be bursting from the imprisonment of your rose-coloured stays embroidered with lace. I shall lick between your legs, over your divine little bottom, your nymph-like thighs being at that moment on my knees; then you will place your angelic little feet, with your stockings on, one after the other in my mouth. After this you will send me into the dining-room, in order to get rid of the servants, and, by this time, filled with an amorous and impassioned languor, each of your movements breathing forth the frenzy and voluptuousness of passion, you will come and join me. There will be only one chair, and the table will be laid for only one person. We shall each of us have only one hand free, I the right, and you the left; then you will sit upon my left leg, which you have found the means to make naked; you will have unfastened your gown in such a way that it will hang down behind, and your right hand will caress and stroke my enormous prick, which you will have taken between your legs without putting it into your angelic cunt, whilst my left arm will wind itself round your lovely waist in order to bring you still nearer to me.
After breakfast, which will have lasted till half-past twelve, and which will have given you strength, we will go into the little rose-coloured boudoir. I shall place myself in a low narrow chair, and as I shall be very much excited by your enchanting looks, my enormous member will come out of its own accord from its prison, and you will sit astraddle upon me, introducing, with the greatest difficulty, my pretty and vigorous prick into your pretty girl-like cunt, when wriggling about from sheer enjoyment you will stop its movements every time I tell you I am on the point of discharging, so as to increase my desires and my transports of happiness. Then in half an hour's time you will get up and place yourself upon the sofa, whilst I, at your desire, shall dip off all my clothes; then you will get up from the sofa and take off your dressing-gown only keeping on what you have underneath. In my turn I will stretch myself on the sofa, getting every moment more delirious with passion, for your dress, betraying the delicious outlines of your figure, without revealing them entirely, will render me almost beside myself, and will make my prick so long and so stiff that you will hardly be able to sit on its point without being fucked, in spite of its size, which will force from you sighs and murmurs of rapture. At last, when once seated, fucked by my manly and powerful prick, you will throw yourself backwards. I should lean my enraptured legs against your bosoms, in order that you might lick my feet, while you would pass your amorous and divine legs, softer, whiter, and more rose-tinted every day, over the whole breadth of my chest, placing your tiny goddess-like feet in my mouth. As our desires would augment at every moment, you would allow me, would even ask me to take off your garters, your pretty stockings, and your slippers, in order to procure me the luxury of licking every part of your body there, and of realising in the most perfect manner the intense enjoyment arising from the contact of the most delicate, the most woman-like, the most voluptuous member of your body. My hands would frig your little love of a member, my manly prick would kiss your celestial womb, and my thighs would caress your delicious bottom. When I have worked you in this way for hours, ceasing every moment you were on the point of emission, I should, as I withdrew my member, let you at last discharge, and then an immense stream of love would flow into my mouth, which suddenly and as if by enchantment would find itself in the place of my member while your bosoms would be covered with that white essence of which you are the only source in my eyes (I had never known it before Homburg), and which would escape from my amorous member.
Every day after dinner, reclining voluptuously on a couch, you would snatch a few moments of repose while I was taking off all my clothes. When I had finished, and when I, filled with love, had shown myself to your contemplation, you would give up to me your place upon the sofa, and assuming the most seductive, the most coquettish, and the most graceful attitudes, would come and play with my member, whose vigour would arise solely from the sight of your pretty costume, which, I am convinced, would render you more delicious than the most graceful fairy. You would love me so deeply that I should cease to have any power of will, you would have exhausted me, sucking me completely dry, nothing would remain in my prick, which would be more full of desire, more enormous, and stiffer at every moment. My languishing eyes, gentle as love itself, surrounded by large dark blue circles caused by your look, your tongue, your bosom, your cunt, your member, your heavenly little bottom, your legs, your fingers, and your angelic little feet would tell you how complete was my happiness, my intoxication, my ecstasy, and my faint, exhausted but happy voice would give you the same assurance, would murmur with rapture in your ears---``Oh how I love you, my lady love, my divine little virgin, caress me yet once more, again, still again, it is a dream. Thank you, oh, thank you and yet again. Oh I am in heaven, do not pause, I implore you, suck me harder than ever; lick me well; oh! what rapture; ask me what you will, it shall be yours. You are my mistress, no other but you in the whole world can transport me in this way. Frig me with your knees. Oh! oh! oh! I am going to discharge,'' and my half-opened mouth would prove to you my enjoyment, and the thirst I had for the bliss you could confer.
Then, more full of passion than ever woman lover had ever been, and enraptured as you listened to my voice, so completely beneath your sway, listening only to your own love, you would raise your little coquettish petticoat, and pressing dear little loves of calves more closely together, for you could be on your knees, resting upon my little blue veins, you would frig me in this manner, with greater vigour than ever sitting down every now and then upon your fine little heels, in order the better to release my beautiful prick, perfectly straight and rudely swollen and inflamed with passionate desires, from between your divine thighs, as soft as satin, and as white as snow, to better introduce the wet tips of your lovely and velvet like bosoms into the seductive little hole of my member, whilst my knees raised slightly behind would gently caress your bottom, so as to give you some little satisfaction in your turn; and at last, unable any longer to retard the moment of emission, you would bend forward, resting upon both your hands, to increase my desire, and keeping yourself back a little distance from me, while your petticoats would now cover my head, and act almost like an electrical conductor upon me, you would intoxicate me with the perfume exhaled from your legs, from your member, from your cunt, from your bottom, and lastly, you would slack my thirst and complete the celestial transport by pissing, with eager rapture, between my burning lips some of that woman's nectar which you would alone possess, and which, emanating from you alone in the world, is worthy of the gods. It would be half-past eight.
You cannot form any idea of my excitement at this moment. I hope you will like this, and will answer me prettily. Am I sufficiently in love? And do you believe there will be another woman in the whole world beside yourself for whom I shall have any desire? Oh, how wild is the longing that I have for you at this moment; and this nectar I have spoken of, from whom else could I care for it, could I endure it even, whilst from you what mad delight! Tell me, do you believe this? You know it perfectly well, I am sure; these are not mere words. Tell me that you will piss into my mouth again when I ask you. I am now going to try to sleep, but what chance of doing so with this love that consumes me. I must await your pretty letter of to-morrow morning, for it is that alone which will excite the flow and stream.
At half-past eight you would like to conform to the usages of this room of mirrors, and as your desires have become greatly inflamed by my own state, and by the soft and sensual temperament of our bodies, you would ask me to undress you, in order that, being completely naked, I might the more easily overwhelm you with my most passionate caresses. I should then strip you of every thing, except that in order that your feet might not come into immediate contact with the looking glasses upon which we should be walking, I would slip on your feet a pair of tiny little slippers, with little silk soles, at a distance they would hardly be visible.
Some one is coming. Adieu till to-morrow.
And larger and stouter than that of my little darling, and so indifferently shod with shoes. (Their boots are pretty.)
Adieu, my angel, I finish this so as to be enabled to add a few lines to the picture---it is late. I love you with all my soul, with love, respect, and adoration. Nothing yet has been heard about de L. R. It is very bad weather, and my father is still no better.
I would take you for a drive either in a pretty barouche or in a phaeton, your toilette would be beautiful but simple. I would only insist upon your wearing a veil, for my love and happiness would render me somewhat egoistical with regard to others. We should not be serious all the time of our drive, for at every instant I should steal a kiss, and your feet would be resting on mine.
We should return home about half-past five to dress for dinner. You would change every thing, and without paying any attention to what our servants might think, I should put on a loose pair of trousers, prettier than what I had worn this morning but, like them, opening in the front. As for you, my own love, I should insist upon your dressing yourself as a ravishingly pretty little \textit{danseuse}, with some little difference, however, in my favour. Your hair would be in curls, falling all round your head, upon your beautiful naked shoulders. You would crown them with a pretty garland of flowers, such as I like for Aim\'{e}e. You should wear a light-coloured muslin dress, very low and very short, up to the knees, your arms bare, and the skirts exceedingly full (the body of which would be transparent, and refine and reveal the divine shape of your angelic bosoms), your legs, perfectly naked, would be visible amongst a mass of folds of muslin, and would be covered by little open-work stockings of rose-coloured silk, fastened at the instep by bows, like the dress, and on your tiny virgin feet you would have little satin shoes, without soles. To pass into the dining room, so as to avoid catching cold, and also prevent the servants revelling in the sight of my treasure, you would envelope yourself from head to foot in a long veil. During dinner I would try to remain tolerably quiet so that you might eat and strengthen yourself for the evening, which would be a fatiguing one. Our servants would have directions not to enter until we rang; during each course you would open your veil, and turning towards me (for you would be on my right hand), you would place your pretty legs across mine; immediately my manly prick, which your love would render daily more and more delicious, would display its vivacity, and you would caress it with your lovely satin-like calves, your chair enabling you to do this, being tolerably large, with only one arm on the right, while mine would be much lower, that would not fatigue you much, and this is what you would say to me, ``Am I not bewitching and delicious? Do you not think me voluptuous? and regard me as your mistress, holding you under my entire subjection? I am very happy to please you this way.'' And I should answer, ``Yes, I am your slave; you give me the greatest enjoyment that can be had; there is not a woman in the world who possesses the attractions you have; you make me do anything, you are the queen of voluptuousness, of enjoyment. No one knows how to make love as you do.'' At last at the dessert you would glide gently upon my lap, allowing your petticoats to flow behind. I should suck your bosoms, for as the servants would be getting their own dinners, I should have thrown your veil quite off, and you would then appear enveloped in all your many charms. Then I should give you your dessert, which would consist of a biscuit moistened with that white essence which you alone in the whole world have known and know how to produce in me, and for my reward you would allow me to make my wine for dessert. I would then place my wine-glass between your legs, opened voluptuously wide, and you would let that delicious urine flow into it. The intoxication that this fragrant liquor would produce would be the signal for my most passionate caresses.
You would begin by placing yourself astride me, and I should thrust with the greatest difficulty my virile member between your legs. In this position we should leave the dining-room, I carrying you along by the stiffness of my member, while every step I took would make you wild with excess of enjoyment. We should go into a pretty boudoir, the floor of which would be completely covered with looking glasses, and filled with furniture intended by their shape and softness to augment the voluptuousness of our embraces. No costume whatever would be put on in this room. Nudity alone would have a right to remain there. There would be pieces of furniture to excite the senses and whereon to recline, others enabling us to suck each of our members, to lick, to frig, to kiss, to enjoy, to complete our performance, to discharge, to fuck, in one word, to supplement and promote the extremest refinements of the most celestial and most perfect of all enjoyments.
The continuation on some future occasion My fear of exciting you will depend somewhat upon my letter of this evening or to-morrow, and particularly upon the frank and sincere reply for which I ask you for the day after to-morrow.
Send me back the beginning.
You cannot have the faintest idea of my dread when one of these sheets is on its way.
Why do you trouble yourself to pay so much attention to style and writing-that takes time. I never read mine over, and that is so much time gained.
\gutchapter{ENGLISH TRANSLATION OF THE LETTER WRITTEN BY THE COUNT ALMOST ENTIRELY IN CYPHER}
Here is the response of my heart, my beloved adored one.
Thou shall have it as soon as I shall dare to send it to you.
Thou shalt belong to me entirely one day, perhaps in eighteen months, and then here is the existence which you shalt have the grief to be compelled to lead.
In the apartment which I depicted to you the other day, and with the toilette that I require of my beloved lady, my lady mistress is to render herself every day between eleven o'clock and noon.
She will find there thy loving husband, all fresh and in every respect desirable (\textit{gentil}), clothed in a dressing-gown of very light texture.
From noon until three o'clock this is the programme.
At noon thou wilt stretch thyself on thy easy chair, thou wilt loosen a little thy girdle and open thy pretty dressing-gown. I on my bent knees at your side shall lick you with my tongue, while my arm shall encircle thy divine waist and thy two naked arms shall encircle my neck; afterwards softly widening thy virgin legs thou will cast aside all that which hides from the eyes, and you will place me between those divine legs.
Successively I shall lick with voluptuousness thy neck, thy shoulders, under thy arms, thy breasts. I shall suck with force those chaste little bosoms, which by their swelling would desire to escape from the pretty little rose-coloured stays; then passing to thy intoxicating cunt, I should suck it with such an amount of frenzy that thou wouldst discharge for the first time in my mouth.
This done it will have so much excited me that, taking thy place, it will become your turn to mount between my legs, and licking all my chest thou wilt finish by frigging with passion my prick, which will become longer and straighter than ever.
As soon as thou shalt feel the enjoyment coming thou wilt cease, in order to lick the parts adjoining.
At one o'clock thou wilt want to make water, then my mouth adhering between thy legs, thou wilt allow me to swallow all, then lying down again on thy little belly, I shall lick with fury thy bottom so voluptuous, and thy delicious legs.
Afterwards it will be thy turn to continue thy caresses upon me.
At two o'clock both of us elevated in a supreme degree, lifting up thy little chemise in front we shall do the business, that is to say, that surrounding me with vigour with thy legs, thou wilt make efforts in order to fuck thyself (\textit{enfiler}), but my member will be to such a degree enormous that we shall have all the trouble in the world (the delights corresponding to the efforts). At last, once entered thou wilt procure, by my movements and my pauses, such enjoyments that I shall hear you uttering the softest murmurs of thy voice, and so that thou wilt wriggle thyself on my ravished prick which will still further augment thy transports.
Thou wilt enjoy thyself thus three times. At the third time I shall suck thy breasts with such passion that thy eyes depicting a heavenly languor and a divine abandonment, thou wilt empty out upon me thy delirium-causing seminal fluid.
That will last until half-past two o'clock, then we shall sleep together thus until three o'clock, and at three o'clock thou wilt go to dress thyself in order to go out or to receive visits.
Behold, the following part shall come to you if the commencement pleases you.
Mem. The commission herein is returnable in Paris, 24th June, 1866.
\gutchapter{End of the Project Gutenberg EBook of The Romance of Lust, by Anonymous}
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\def\ttablerule{\ \vrule\ }
\def\dbhfill{\penalty-100\discretionary{}{}{}\hfill}
\def\qnAA{(a) State the triangle inequality for real numbers.
(b) Use it to show that, for all real numbers $x$ and $y$
(i) $|x-y|\le|x|+|y|$ (ii) $|x|-|y|\le|x-y|$ (iii) $|x|-|x-y|\le|y|$
(iv) $||x|-|y||\le|x-y|$.
\dbhfill[AA]}
\def\ansAA{(a) $|x+y|\le|x|+|y|$ for all real numbers $x$ and $y$.
\medskip
(b)(i) $|x-y|=|x+(-y)|\le|x|+|-y|=|x|+|y|$.
\quad(ii) $|x|=|(x-y)+y|\le|x-y|+|y|$; subtracting $|y|$ from both
sides, $|x|-|y|\le|x-y|$.
\quad(iii) By (ii), $|x|-|x-y|\le|x-(x-y)|=|y|$.
\quad(iv) By (ii), $|x|-|y|\le|x-y|$; similarly,
$|y|-|x|\le|y-x|=|x-y|$; so
\Centerline{$||x|-|y||=\max(|x|-|y|,|y|-|x|)\le|x-y|$.}
}%end of ansAA
\def\qnAB{(a) For a set $A\subseteq\Bbb R$, what is meant by (i) an
upper bound of $A$ (ii) $\sup A$?
(b) Give examples
of sets $A$ for which (i) $\sup A$ exists and belongs to $A$ (ii)
$\sup A$ exists and does not belong to $A$ (iii) $\sup A$ does not
exist.
(c) For sets $A$, $B\subseteq\Bbb R$ write
$A+B=\{a+b:a\in A,\,b\in B\}$.
\quad(i) Find $[0,1]+[-1,2]$, $[1,2]+\coint{0,1}$,
$\ooint{0,1}+\ooint{0,1}$. (Hint: they are all intervals; but you
have to decide whether they are open or closed.)
\quad(ii) Suppose that $A$, $B\subseteq\Bbb R$ have upper bounds $a$,
$b$ respectively. Show that $a+b$ is an upper bound of $A+B$.
\quad(iii) Suppose that $A$, $B\subseteq\Bbb R$ and that $c$ is an upper
bound of $A+B$. Show that $x\le c-y$ whenever $x\in A$ and $y\in B$.
\quad(iv) Suppose that $A$, $B\subseteq\Bbb R$ have suprema $a$, $b$
respectively, and that $c$ is an upper bound of $A+B$. Show that
$a\le c-y$ for every $y\in B$, and therefore that $c-a$ is an upper
bound of $B$.
\quad(v) Show that if $A$, $B\subseteq\Bbb R$ have suprema $a$, $b$
respectively, then $a+b$ is the least upper bound of $A+B$.
\dbhfill[AB]}
\def\ansAB{(a)(i) An {\it upper bound} of $A$ is an $a\in\Bbb R$ such
that $x\le a$ for every $x\in A$.
\quad(ii) $\sup A$ (when it exists) is the least upper bound of $A$.
\medskip
(b)(i) $A=[0,1]$ (ii) $A=\ooint{0,1}$ (iii) $A=\emptyset$ {\it or}
$A=\Bbb R$.
\hfill(There are many alternatives.)
\medskip
(c)(i) $[-1,3]$, $\coint{1,3}$, $\ooint{0,2}$.
\quad(ii) If $z\in A+B$, then $z=x+y$ for some $x\in A$, $y\in B$. Now
$x\le a$ and $y\le b$, so
\Centerline{$z=x+y\le a+y\le a+b$.}
\noindent As $z$ is arbitrary, $a+b$ is an upper bound of $A$.
\quad(iii) Since $x+y\in A+B$, we know that $x+y\le c$; subtracting $y$
from both sides, $x\le c-y$.
\quad(iv) Take any $y\in B$. By (iii), $x\le c-y$ for every $x\in A$,
so $c-y$ is an upper bound of $A$ and $a\le c-y$. So $a+y\le c$ and
$y\le c-a$. This is true for every $y\in B$, so $c-a$ is an upper
bound of $B$.
\quad(v) By (ii), $a+b$ is an upper bound of $A+B$. On the other hand,
if $c$ is any upper bound of $A+B$, then $c-a$ is an upper bound of $B$,
by (iv); but this means that $b\le c-a$, so $a+b\le c$. So $a+b$ is
the least upper bound of $A+B$.
}%end of ansAB
\def\qnAC{Which of the following are equivalent to the sentence
`$a=\sup A$'? (You may take it for granted that
$a\in\Bbb R$ and $A\subseteq\Bbb R$.)
\inset{(i) For every $y\in\Bbb R$, $y\ge a$ whenever $y$ is an upper
bound of $A$.
(ii) For every $y\in\Bbb R$, $y\in\coint{a,\infty}$ iff $y$ is an upper
bound of $A$.
(iii) For every $y\in\Bbb R$, $y$ is an upper bound of $A$ whenever
$y\ge a$.
(iv) $A$ is not empty and for all $x\in A$, $y\in\Bbb R$, $x\le y$ iff $a\le y$.
(v) $\forall\,y\in\Bbb R,\,a\le y\iff\forall x\in A,\,x\le y$.}
\noindent For those which are {\it not} equivalent, give examples to
show that they are not.
\dbhfill[AC]}
\def\ansAC{(i) False; try $a=0$, $A=[0,1]$.
(ii) True.
(iii) False; try $a=2$, $A=[0,1]$.
(iv) False; try $a=1$, $A=[0,1]$.
(v) True.
}%end of ansAC
\def\qnAD{Find negations of the following sentences. (In each case,
move any quantifiers `$\forall$' and `$\exists$' as far to the left as
possible, and any negative assertions $\notin$, $\not\le$ as far to the
right as you can; do not use $\not\forall$ and $\not\exists$. You may
suppose throughout that $A$ and $B$ are subsets of $\Bbb R$.)
\halign{\qquad##\hfil\quad&##\hfil\cr
(i) $\forall\,x\in A,\,x\le 2$
&(ii) $\exists\,x\in A,\,x>0$\cr
(iii) $\forall\,x\in A\Forall y\in B,\,x\le y$
&(iv) $\exists\,x\in A\Exists y\in B,\,x\le y$\cr
(v) $\forall\,x\in A\Exists y\in B,\,x+y>0$
&(vi) $\exists\,x\in A\Forall y\in B,\,x+y>0$\cr}
\dbhfill[AD]}
\def\ansAD{(i) $\exists\,x\in A,\,x>2$.
(ii) $\forall\,x\in A,\,x\le 0$.
(iii) $\exists\,x\in A\Exists y\in B,\,x>y$.
(iv) $\forall\,x\in A\Forall y\in B,\,x>y$.
(v) $\exists\,x\in A\Forall y\in B,\,x+y\le 0$.
(vi) $\forall\,x\in A\Exists y\in B,\,x+y\le 0$.
}%end of ansAD
\def\qnAE{One of the following is true, and the other is false. Which
is which? Justify your answer.
\quad(i) if, for every $x\in A$, there is a $y\in B$ such that $x\le y$,
then there is a $y\in B$ such that $x\le y$ for every $x\in A$.
\quad(ii) if there is a $y\in B$ such that $x\le y$ for every $x\in A$,
then, for every $x\in A$, there is a $y\in B$ such that $x\le y$.
\dbhfill[AE]}
\def\ansAE{(i) is false. For instance, if $A=B=\Bbb R$, then for every
$x\in A$ there is a $y\in B$ such that $x\le y$ (just take $y=x$), but
there is no $y\in B$ such that $x\le y$ for every $x\in A$.
\medskip
(ii) is true. For if $y_0\in B$ is such that $x\le y_0$ for every
$x\in A$, then, for every $x\in A$, there is certainly a $y\in B$ such
that $x\le y$: just take $y=y_0$.
}%end of ansAE
\def\qnAF{Which of the following are true, and which are false?
\halign{\qquad\qquad##\hfil&\quad##\hfil&\quad##\hfil\cr
(a) $\pi=3.142$
&(e) $-\pi>-3.142$\cr
(b) $\pi\ge 3.142$
&(f) if $a\le b$ and $c\le d$ then $a+c\le b+d$\cr
(c) $\pi\le 3.142$
&(g) if $a\le b$ and $c\le d$ then $a-d\le b-c$\cr
(d) $-\pi<-3.142$
&(h) if $a<b$ and $c<d$ then $a+c\le b+d$.\cr
}
\dbhfill[AF]}
\def\ansAF{(a) False.
(b) False.
(c) True.
(d) False.
(e) True.
(f) True.
(g) True.
(h) True.
}%end of ansAF
\def\qnAG{In each of these possible positions in noughts-and-crosses,
the {\tt X} player moved first, and now must move again. In each case
say what the result of the game will be if both players play
correctly, and copy the position out with a further correct move
by {\tt X}.
\sbAG
\dbhfill[AG]}
\def\ansAG{Possible correct moves are marked * .
\sbAG
}%end of ansAG
\def\sbAG{\def\hs{\hskip 1.2em\relax}
\def\hss{\ifquestions\hs\else\enskip*\enskip\fi}
\def\O{{\tt O}}
\def\X{{\tt X}}
\def\w{\ifquestions\else\hs {\tt X} wins\fi}
\def\d{\ifquestions\else\hs draw\fi}
%\showthe\pagetotal
$$\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil
&##\hfil\cr
\hss&\hss&\O&\d\cr
\noalign{\hrule}
\hss&\X&\hss\cr
\noalign{\hrule}
\hss&\hss&\hss\cr
}}
\qquad\qquad\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil
&##\hfil\cr
\hss&\O&\hss&\w\cr
\noalign{\hrule}
\hss&\X&\hss\cr
\noalign{\hrule}
\hss&\hs&\hss\cr
}}
\qquad\qquad\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil
&##\hfil\cr
\O&\hs&\X&\w\cr
\noalign{\hrule}
\hs&\hs&\hss\cr
\noalign{\hrule}
\hss&\hs&\hss\cr
}}
\ifquestions\else$$\par$$\fi
\qquad\qquad\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil
&##\hfil\cr
\hss&\hs&\X&\w\cr
\noalign{\hrule}
\hs&\hs&\hs\cr
\noalign{\hrule}
\O&\hs&\hss\cr
}}
\quad\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil
&##\hfil\cr
\hss&\X&\hs&\w\cr
\noalign{\hrule}
\hs&\hs&\hs\cr
\noalign{\hrule}
\O&\hs&\hs\cr
}}$$
}%end of sbAG
\def\qnBA{(a) In the Analysis Game, the first player says `$\lim_{n\to\infty}\Bover{(-1)^n}n=0$'. Who plays next? If both play correctly, will he win? If so, give a possible winning move.
(b) In the Analysis Game, the first player says `$\lim_{n\to\infty}\Bover{(-1)^n}n=0{\cdot}01$'. Who plays next? If both play correctly, will he win? If so, give a possible winning move.
(c) In the Analysis Game, the first player says
`$\lim_{x\to\infty}\sin x=0$'. Who plays next? If both play correctly, will he win? If so, give a possible winning move.
(You may take it for granted that $\sin$ behaves as you have been taught that it does.)
(d) In the Analysis Game, the first player says
`$\langle\Bover{n+1}{n}\rangle_{n\ge 1}$ converges'. Who plays next? If both play correctly, will he win? If so, give a possible winning move.
\dbhfill[BA]}
\def\ansBA{(a) Second player plays next, and loses.
(b) Second player plays next, and should win; $\epsilon=0{\cdot}005$ wins. (Any $\epsilon\in\ooint{0,0{\cdot}01}$ will do.)
(c) Second player plays next, and should win; $\epsilon=\bover12$ wins. (Any $\epsilon\in\ooint{0,1}$ will do.)
(d) {\it First} player plays next, and should win; he must say
`$\lim_{n\to\infty}\bover{n}{n+1}=1$'.
}%end of ansBA
\def\qnBB{For a real function $f$ and real numbers $a$ and $b$, what is meant by saying that $\lim_{x\downarrow a}f(x)=b$?
Show from your definition that if $\lim_{x\downarrow a}f(x)=b$
and $\lim_{x\downarrow a}g(x)=c$ then
$\lim_{x\downarrow a}f(x)+g(x)=b+c$.
\dbhfill[BB]}
\def\ansBB{$\lim_{x\downarrow a}f(x)=b$ means:
\Centerline{$\Forall\epsilon>0\Exists\delta>0,\,
f(x)$ is defined and $|f(x)-b|\le\epsilon$ whenever $0<x-a\le\delta$.}
If $\lim_{x\downarrow a}f(x)=b$
and $\lim_{x\downarrow a}g(x)=c$, let $\epsilon>0$.
Then $\bover12\epsilon>0$, so there are $\delta_1$, $\delta_2>0$ such that
\Centerline{$f(x)$ is defined and $|f(x)-b|\le\bover12\epsilon$ whenever $0<x-a\le\delta_1$,}
\Centerline{$g(x)$ is defined and $|g(x)-c|\le\bover12\epsilon$ whenever $0<x-a\le\delta_2$.}
\noindent Now set $\delta=\min(\delta_1,\delta_2)>0$. Then whenever $0<x-a\le\delta$, $f(x)$ and $g(x)$ are both defined and
$$\eqalign{|(f(x)+g(x))-(b+c)|
&=|(f(x)-b)+(g(x)-c)|
\le|f(x)-b|+|g(x)-c|\cr
&\le\Bover12\epsilon+\Bover12\epsilon
=\epsilon.\cr}$$
\noindent As $\epsilon$ is arbitrary, $\lim_{x\downarrow a}f(x)+g(x)=b+c$.
}%end of ansBB
\def\qnBC{For a real function $f$ and real numbers $a$ and $b$, what is meant by saying that $\lim_{x\to a}f(x)=b$?
Show from your definition that if $\lim_{x\to a}f(x)=b$
and $\lim_{x\to a}g(x)=c$ then
\ifdim\pagewidth=390pt\break\fi
$\lim_{x\to a}f(x)g(x)=bc$.
\dbhfill[BC]}
\def\qnBCA{For a real function $f$ and real numbers $a$ and $b$, what is meant by saying that $\lim_{x\downarrow a}f(x)=b$?
Show from your definition that if $\lim_{x\downarrow a}f(x)=b$
and $\lim_{x\downarrow a}g(x)=c$ then
\ifdim\pagewidth=390pt\break\fi
$\lim_{x\downarrow a}f(x)g(x)=bc$.
\dbhfill[BCA]}
\def\ansBC{$\lim_{x\to a}f(x)=b$ means:
\Centerline{$\Forall\epsilon>0\Exists\delta>0,\,
f(x)$ is defined and $|f(x)-b|\le\epsilon$ whenever $0<|x-a|\le\delta$.}
If $\lim_{x\downarrow a}f(x)=b$
and $\lim_{x\downarrow a}g(x)=c$, let $\epsilon>0$. Set $\eta=\min(1,\Bover{\epsilon}{1+|b|+|c|})>0$. Then there are $\delta_1$, $\delta_2>0$ such that
\Centerline{$f(x)$ is defined and $|f(x)-b|\le\eta$ whenever
$0<|x-a|\le\delta_1$,}
\Centerline{$g(x)$ is defined and $|g(x)-c|\le\eta$ whenever
$0<|x-a|\le\delta_2$.}
\noindent Set $\delta=\min(\delta_1,\delta_2)>0$. Then for any $x$ such that $0<|x-a|\le\delta$, $f(x)$ and $g(x)$ are both defined and
$$\eqalign{|f(x)g(x)-bc|
&\le|f(x)-b||g(x)-c|+|f(x)-b||c|+|b||g(x)-c|\cr
&\le\eta^2+\eta|c|+|b|\eta
\le\eta(1+|b|+|c|)
\le\epsilon.\cr}$$
\noindent As $\epsilon$ is arbitrary, $\lim_{x\to a}f(x)g(x)=bc$.
}%end of ansBC
\def\ansBCA{$\lim_{x\downarrow a}f(x)=b$ means:
\Centerline{$\Forall\epsilon>0\Exists\delta>0,\,
f(x)$ is defined and $|f(x)-b|\le\epsilon$ whenever $a<x\le a+\delta$.}
If $\lim_{x\downarrow a}f(x)=b$
and $\lim_{x\downarrow a}g(x)=c$, let $\epsilon>0$. Set $\eta=\min(1,\Bover{\epsilon}{1+|b|+|c|})>0$. Then there are $\delta_1$, $\delta_2>0$ such that
\Centerline{$f(x)$ is defined and $|f(x)-b|\le\eta$ whenever
$a<x\le a+\delta_1$,}
\Centerline{$g(x)$ is defined and $|g(x)-c|\le\eta$ whenever
$a<x\le a+\delta_2$.}
\noindent Set $\delta=\min(\delta_1,\delta_2)>0$. Then for any $x$ such that $a<x\le a+\delta$, $f(x)$ and $g(x)$ are both defined and
$$\eqalign{|f(x)g(x)-bc|
&\le|f(x)-b||g(x)-c|+|f(x)-b||c|+|b||g(x)-c|\cr
&\le\eta^2+\eta|c|+|b|\eta
\le\eta(1+|b|+|c|)
\le\epsilon.\cr}$$
\noindent As $\epsilon$ is arbitrary, $\lim_{x\downarrow a}f(x)g(x)=bc$.
}%end of ansBCA
\def\qnBD{Consider the following strategy for playing noughts-and-crosses.
\inset{\noindent If you have an open pair (that is, two of your symbols in a line with the third place empty), complete the pair to win
\inset{\noindent else if your opponent has one or more open pairs, block one (chosen at random)
\inset{\noindent else if the centre square is empty, fill it
\inset{\noindent else if one or more corners is empty, play in one (chosen at random)
\inset{\noindent else play anywhere}}}}
\noindent fi fi fi fi}
(a) If you are first player and follow this strategy, will you always play correctly?
(b) If you are first player and you know your opponent will follow this strategy, can you be sure of beating him?
\hfill [BD]}
\def\ansBD{(a) No. Suppose the game goes like this:
\def\hs{\hskip 1.2em\relax}
\def\O{{\tt O}}
\def\X{{\tt X}}
$$\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil\cr
\hs&\hs&\hs\cr
\noalign{\hrule}
\hs&\X&\hs\cr
\noalign{\hrule}
\hs&\hs&\hs\cr
}}
\quad\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil\cr
\hs&\O&\hs\cr
\noalign{\hrule}
\hs&\X&\hs\cr
\noalign{\hrule}
\hs&\hs&\hs\cr
}}
\quad\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil\cr
\X&\O&\hs\cr
\noalign{\hrule}
\hs&\X&\hs\cr
\noalign{\hrule}
\hs&\hs&\hs\cr
}}
\quad\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil\cr
\X&\O&\hs\cr
\noalign{\hrule}
\hs&\X&\hs\cr
\noalign{\hrule}
\hs&\hs&\O\cr
}}$$
\noindent as it very well could, since you played in the centre for your first move and in a corner for your next. Then you ought to be able to win (play in bottom left corner). But the rule allows you to pick a corner at random, and if you pick top right then it's a draw (unless your opponent is a total idiot).
\medskip
(b) Yes. Play as follows:
$$\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil\cr
\X&\hs&\hs\cr
\noalign{\hrule}
\hs&\hs&\hs\cr
\noalign{\hrule}
\hs&\hs&\hs\cr
}}
\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil\cr
\X&\hs&\hs\cr
\noalign{\hrule}
\hs&\O&\hs\cr
\noalign{\hrule}
\hs&\hs&\hs\cr
}}
\vbox{\offinterlineskip
\halign{\hfil##\hfil\strut
&\ttablerule\hfil##\hfil
&\ttablerule\hfil##\hfil\cr
\X&\hs&\hs\cr
\noalign{\hrule}
\hs&\O&\hs\cr
\noalign{\hrule}
\hs&\hs&\X\cr
}}$$
\noindent Your opponent's next move will be in a corner and you can hardly help
winning.
}%end of ansBD
\def\qnBDA{Can you find a simple strategy which will give a correct move in noughts-and-crosses from any position?
\dbhfill[BDA]}
\def\ansBDA{I have no convincing candidate. I did once write a
noughts-and-crosses program, which always played correctly (I think), but in effect I gave it a list of possible positions \& told it what to do in each.
}
\def\qnBE{Put the following nine sentences in such an order that they form a correct proof of the theorem that if $\sequencen{x_n}\to b$ and $\sequencen{x_n}\to c$, then $b=c$.
\medskip
\inset{
If $\epsilon=0$ then $|b-c|=0$ and $b=c$, so we can stop.
Also there is an $n_2\in\Bbb N$ such that $|x_n-c|\le\epsilon$ for every
$n\ge n_2$.
Set $\epsilon=\bover13|b-c|$, so that $\epsilon\ge 0$.
Because $n_0\ge n_1$, $|x_{n_0}-b|\le\epsilon$.
So $|b-c|\le|x_{n_0}-b|+|x_{n_0}-c|\le 2\epsilon=\Bover23|b-c|$.
Because $n_0\ge n_2$, $|x_{n_0}-c|\le\epsilon$.
If $\epsilon>0$, then there is an $n_1\in\Bbb N$ such that $|x_n-b|\le\epsilon$ for every $n\ge n_1$.
Therefore $\Bover13|b-c|\le 0$ and $b=c$ in this case also.
Set $n_0=\max(n_1,n_2)$.
}
\medskip
\noindent (Collaboration on this question is strongly recommended.)
\dbhfill[BE]}
\def\ansBE{
Set $\epsilon=\bover13|b-c|$, so that $\epsilon\ge 0$.
If $\epsilon=0$ then $|b-c|=0$ and $b=c$, so we can stop.
If $\epsilon>0$, then there is an $n_1\in\Bbb N$ such that $|x_n-b|\le\epsilon$ for every $n\ge n_1$.
Also there is an $n_2\in\Bbb N$ such that $|x_n-c|\le\epsilon$ for every
$n\ge n_2$.
Set $n_0=\max(n_1,n_2)$.
Because $n_0\ge n_1$, $|x_{n_0}-b|\le\epsilon$.
Because $n_0\ge n_2$, $|x_{n_0}-c|\le\epsilon$.
So $|b-c|\le|x_{n_0}-b|+|x_{n_0}-c|\le 2\epsilon=\Bover23|b-c|$.
Therefore $\Bover13|b-c|\le 0$ and $b=c$ in this case also.
\noindent[Of course the point of this proof is that it is {\it impossible} for
$\epsilon$ to be non-zero, so that all the deductions after the
`if $\epsilon>0$' are counter-factual. But we don't know this until we come to the end.]
}%end of ansBE
\def\qnBF{Show that if $f$ and $g$ are real functions and $\lim_{x\to a}g(x)=b$ and $\lim_{y\to b}f(y)=f(b)=c$ (so that $f$ is defined near $b$ and continuous at $b$), then $\lim_{x\to a}f(g(x))=c$.
\dbhfill[BF]}
\def\ansBF{Let $\epsilon>0$. Then there is an $\eta>0$ such that
$|f(y)-c|\le\epsilon$ whenever $0<|y-b|\le\eta$; but since we know also that $f(b)=c$, we see that $|f(y)-c|\le\epsilon$ whenever $|y-b|\le\eta$. Now there is a $\delta>0$ such that $|g(x)-b|\le\eta$ whenever $0<|x-a|\le\delta$. If we now take any $x$ such that $0<|x-a|\le\delta$, then $|g(x)-b|\le\eta$ so (because of the way we chose $\eta$) $|f(g(x))-c|\le\epsilon$. As $\epsilon$ is arbitrary, $\lim_{x\to a}f(g(x))=c$, as claimed.
}%end of ansBF
\def\qnCA{Show that if $\sequencen{x_n}$ is a sequence converging to $b$, and
$f$ is a real function such that $\lim_{y\to b}f(y)=f(b)=c$, then $\lim_{n\to\infty}f(x_n)=c$.
\dbhfill[CA]}
\def\ansCA{Let $\epsilon>0$. Then there is an $\eta>0$ such that
$|f(y)-c|\le\epsilon$ whenever $0<|y-b|\le\eta$; but since we know also that $f(b)=c$, we see that $|f(y)-c|\le\epsilon$ whenever $|y-b|\le\eta$. Now there is an $n_0\in\Bbb N$ such that $|x_n-b|\le\eta$ whenever $n\ge n_0$. If we now take any $n$ such that $n\ge n_0$, then $|x_n-b|\le\eta$ so (because of the way we chose $\eta$) $|f(x_n))-c|\le\epsilon$. As $\epsilon$ is arbitrary, $\lim_{n\to\infty}f(x_n)=c$, as claimed.
}%end of ansCA
\def\qnCB{A sequence $\sequencen{x_n}$ is {\it non-increasing} if
$x_{n+1}\le x_n$ for every $n\in\Bbb N$. {\it Without} quoting any theorem concerning non-decreasing sequences, show that a bounded non-increasing sequence is convergent.
\dbhfill[CB]}
\def\ansCB{Let $\sequencen{x_n}$ be a bounded non-increasing sequence. Then
$A=\{x_n:n\in\Bbb N\}$ is a non-empty bounded set, so has a greatest lower bound; set $b=\inf A$. For any $\epsilon>0$, $b+\epsilon$ is not a lower bound of $A$, so there is an $n_0\in\Bbb N$ such that $x_{n_0}<b+\epsilon$. If now $n\ge n_0$, $x_n\le x_{n_0}\le b+\epsilon$, because the sequence is
non-increasing; but also $b\le x_n$, because $b$ is a lower bound for $A$. Thus $|x_n-b|\le\epsilon$. As $\epsilon$ is arbitrary, $\lim_{n\to\infty}x_n=b$ and $\sequencen{x_n}$ is convergent.
}%end of ansCB
\def\qnCC{(a) Show by induction on $n$ that $(1+\eta)^n\ge 1+n\eta$ for every $n\in\Bbb N$ and $\eta\ge-1$.
(b) Show that $\lim_{n\to\infty}x^n=0$ whenever $|x|<1$. ({\it Hint\/}: given $\epsilon>0$, take $n_0\ge\Bover1{\epsilon(1-|x|)}$.)
(c) Show that $\sum_{k=0}^nx^k=\Bover{1-x^{n+1}}{1-x}$ for every $n\in\Bbb N$ and every $x\in\Bbb R\setminus\{1\}$.
(d) Show that $\sum_{k=0}^{\infty}x^k=\Bover1{1-x}$ whenever $|x|<1$.
\dbhfill[CC]}
\def\ansCCa{Base step: if $n=0$ then LHS is $(1+\eta)^0=1$ and RHS is $1+0\cdot\eta=1$, so the induction starts.
Inductive step: if $(1+\eta)^n\ge 1+n\eta$, then
$$\eqalignno{(1+\eta)^{n+1}
&=(1+\eta)(1+\eta)^n\ge(1+\eta)(1+n\eta)\cr
\displaycause{because $(1+\eta)^n\ge 1+n\eta$, by the inductive hypothesis, and $1+\eta\ge 0$}
&=1+(n+1)\eta+n\eta^2
\ge 1+(n+1)\eta,\cr}$$
\noindent so the induction proceeds.}
\def\ansCCb{If $x=0$ then $x^n=0$ for every $n$ so $\lim_{n\to\infty}x^n=0$.
If $0<|x|<1$, then, given $\epsilon>0$, take $n_0\in\Bbb N$ such that
$n_0\ge\Bover1{\epsilon(1-|x|)}$. If $n\ge n_0$, then
$$\eqalignno{\Bover1{|x^n|}
&=\bigl(1+(\Bover1{|x|}-1)\bigr)^n
\ge 1+n(\Bover1{|x|}-1)\cr
\displaycause{by (a)}
&\ge n_0(\Bover1{|x|}-1)
\ge\bover{\bover1{|x|}-1}{\epsilon(1-|x|)}\cr
&=\Bover{1-|x|}{\epsilon|x|(1-|x|)}
=\Bover1{\epsilon|x|}
\ge\Bover1{\epsilon},\cr}$$
\noindent so $|x^n-0|=|x^n|\le\epsilon$. As $\epsilon$ is arbitrary, $\lim_{n\to\infty}x^n=0$.}
\def\ansCCc{$$\eqalign{(1-x)\sum_{k=0}^nx^k
&=\sum_{k=0}^nx^k-x^{k+1}
=\sum_{k=0}^nx^k-\sum_{k=0}^nx^{k+1}\cr
&=\sum_{k=0}^nx^k-\sum_{k=1}^{n+1}x^k
=x^0-x^{n+1}=1-x^{n+1};\cr}$$
\noindent dividing both sides by $1-x$, we have the result.
\medskip
{\it Alternatively\/}: Induce on $n$. If $n=0$ the LHS is 1 and the RHS is $\Bover{1-x}{1-x}=1$, so the induction starts. For the inductive step, given that $\sum_{k=0}^nx^k=\Bover{1-x^{n+1}}{1-x}$, then
$$\eqalignno{\sum_{k=0}^{n+1}x^k
&=\sum_{k=0}^nx^k+x^{n+1}
=\Bover{1-x^{n+1}}{1-x}+x^{n+1}\cr
\displaycause{by the inductive hypothesis}
&=\Bover{1-x^{n+1}+x^{n+1}-x^{n+2}}{1-x}
=\Bover{1-x^{(n+1)+1}}{1-x}\cr}$$
\noindent and the induction proceeds.}
\def\ansCCd{$$\eqalignno{\sum_{k=0}^{\infty}x^k
&=\lim_{n\to\infty}\sum_{k=0}^nx^k\cr
\displaycause{by the definition of $\sum_0^{\infty}$}
&=\lim_{n\to\infty}\Bover{1-x^{n+1}}{1-x}\cr
\displaycause{by (c)}
&=\lim_{n\to\infty}\Bover1{1-x}-\Bover{x}{1-x}x^n
=\Bover1{1-x}-\Bover{x}{1-x}\lim_{n\to\infty}x^n
=\Bover1{1-x}.\cr}$$
}%end of ansCC
\def\qnCD{(a) Show that if $\sequencen{x_n}$ and $\sequencen{y_n}$ are bounded sequences, then $\sequencen{x_n+y_n}$ and $\sequencen{x_ny_n}$ are bounded.
(b) Give an example of a bounded sequence $\sequencen{x_n}$ such that no $x_n$ is zero and $\sequencen{\Bover1{x_n}}$ is not bounded.
(c) Show that if $\sequencen{x_n}$ and $\sequencen{y_n}$ are non-decreasing sequences then $\sequencen{x_n+y_n}$ is non-decreasing.
(d) Show that if $\sequencen{x_n}$ and $\sequencen{y_n}$ are non-decreasing sequences and $x_n\ge 0$, $y_n\ge 0$ for every $n$, then $\sequencen{x_ny_n}$ is non-decreasing.
\dbhfill[CD]}
\def\ansCD{(a) Let $M$, $M'\in\Bbb R$ be such that $|x_n|\le M$ and
$|y_n|\le M'$ for every $n\in\Bbb N$. Then $|x_n+y_n|\le|x_n|+|y_n|\le M+M'$ and $|x_ny_n|=|x_n||y_n|\le MM'$ for every $n$, so $\sequencen{x_n+y_n}$ and $\sequencen{x_ny_n}$ are bounded.
\medskip
(b) Try $x_n=\Bover1{n+1}$ for every $n$.
\medskip
(c) For each $n\in\Bbb N$,
$$\eqalignno{x_n+y_n&\le x_{n+1}+y_n\cr
\displaycause{because $x_n\le x_{n+1}$}
&\le x_{n+1}+y_{n+1}\cr}$$
\noindent because $y_n\le y_{n+1}$.
\medskip
(d) For each $n\in\Bbb N$,
$$\eqalignno{x_ny_n&\le x_{n+1}y_n\cr
\displaycause{because $x_n\le x_{n+1}$ and $y_n\ge 0$}
&\le x_{n+1}y_{n+1}\cr}$$
\noindent because $y_n\le y_{n+1}$ and $x_{n+1}\ge 0$.
}%end of ansCD
\def\qnCE{A sequence $\sequencen{x_n}$ is {\it monotonic} if it is either
non-increasing or non-decreasing.
(a) Give an example of a sequence which is both non-increasing and
non-decreasing.
(b) If $\sequencen{x_n}$ and $\sequencen{y_n}$ are both monotonic, does $\sequencen{x_n+y_n}$ have to be monotonic? Justify your answer with a proof or a counter-example.
\dbhfill[CE]}
\def\ansCE{(a) $x_n=0$ for every $n$.
\medskip
(b) No. Try $x_0=0$, $x_n=1$ for $n\ge 1$ (so that $\sequencen{x_n}$ is
non-decreasing) and $y_0=y_1=1$, $y_n=0$ for $n\ge 2$ (so that $\sequencen{y_n}$ is non-increasing) while $\sequencen{x_n+y_n}=(1,2,1,\ldots)$ is not monotonic.
}%end of ansCE
\def\qnCF{(a) Define {\it Cauchy sequence}, and state the General Principle of Convergence.
(b) Let $\sequencen{x_n}$ and $\sequencen{y_n}$ be two series such that $0\le x_n\le y_n$ for every $n\in\Bbb N$. Set $s_n=\sum_{k=0}^nx_k$, $t_n=\sum_{k=0}^ny_k$ for every $n$.
\quad(i) Show that $|s_m-s_n|\le|t_m-t_n|$ for all $m$, $n\in\Bbb N$. Hence show that if $\sequencen{t_n}$ is Cauchy, so is $\sequencen{s_n}$.
\quad(ii) Show that if $\sequencen{y_n}$ is summable, so is $\sequencen{x_n}$.
\dbhfill[CF]}
\def\ansCF{(a) A sequence $\sequencen{x_n}$ of real numbers is a Cauchy sequence if for every $\epsilon>0$ there is an $n_0\in\Bbb N$ such that
$|x_m-x_n|\le\epsilon$ for all $m$, $n\ge n_0$.
The General Principle of Convergence states: a real sequence is convergent iff it is Cauchy.
\medskip
(b)(i) If $n<m$ then
$$\eqalignno{|s_m-s_n|
&=|\sum_{k=0}^mx_k-\sum_{k=0}^nx_k|
=|\sum_{k=n+1}^mx_k|
=\sum_{k=n+1}^mx_k\cr
\displaycause{because $x_k\ge 0$ for every $k$}
&\le\sum_{k=n+1}^my_k\cr
\displaycause{because $x_k\le y_k$ for every $k$}
&=|\sum_{k=n+1}^my_k|
=|t_m-t_n|.\cr}$$
If $m=n$ then $|s_m-s_n|=0=|t_m-t_n|$.
If $m<n$ then
\Centerline{$|s_m-s_n|=|s_n-s_m|
=\sum_{k=m+1}^nx_k\le\sum_{k=m+1}^ny_k=|t_m-t_n|$.}
Now suppose that $\sequencen{t_n}$ is Cauchy. In this case, given $\epsilon>0$, there is an $n_0\in\Bbb N$ such that $|t_m-t_n|\le\epsilon$ for all $m$, $n\ge n_0$. But this means that $|s_m-s_n|\le|t_m-t_n|\le\epsilon$ for all $m$, $n\ge n_0$; as $m$ and $n$ are arbitrary, $\sequencen{s_n}$ is Cauchy.
\medskip
\quad(ii) If $\sequencen{y_n}$ is summable, that is, $\sequencen{t_n}$ is convergent, then $\sequencen{t_n}$ is Cauchy, so $\sequencen{s_n}$ is Cauchy, therefore convergent, and $\sequencen{x_n}$ is summable.
}%end of ansCF
\def\qnCG{Give examples of
\inset{(i) a continuous function $f:\ooint{0,1}\to\Bbb R$ which is not bounded;
(ii) a bounded continuous function $f:\ooint{0,1}\to\Bbb R$ which does not attain its supremum;
(iii) a continuous function $f:\coint{0,\infty}\to\Bbb R$ which is not bounded;
(iv) a bounded continuous function $f:\coint{0,\infty}\to\Bbb R$ which does not
attain its infimum.}
\noindent In each case explain (with reference to theorems proved in lectures) why your function is continuous. In parts (ii) and (iv), say (without proof) what the supremum and infimum of your function are.
\dbhfill[CG]}
\def\ansCG{(i) $f(x)=\bover1x$ for $x\in\ooint{0,1}$. ($x\mapsto x$ is continuous, and the reciprocal of a continuous function is continuous.)
\medskip
(ii) $f(x)=\Bover{x}{1+x}$ for $x\in\ooint{0,1}$. ($x\mapsto x$ and
$x\mapsto 1$ are continuous, and the sum of two continuous functions is continuous, so $x\mapsto 1+x$ is continuous; the reciprocal of a continuous function is continuous, so $x\mapsto\Bover1{1+x}$ is continuous; the product of two continuous functions is continuous, so $x\mapsto x\times\Bover1{1+x}$ is continuous.)
The supremum is $\bover12$ and the infimum is $0$.
\medskip
(iii) $f(x) = x$ for $x\ge 0$.
\medskip
(iv) $f(x)=\Bover1{1+x}$ for $x\ge 0$; supremum is 1, infimum is 0.
(Of course there are many alternative answers.)
}%end of ansCG
\def\qnCH{A real function $f$ is {\it uniformly continuous} if
\inset{for every $\epsilon>0$ there is a $\delta>0$ such that
$|f(x)-f(y)|\le\epsilon$ whenever $x$, $y\in\dom(f)$ and $|x-y|\le\delta$.}
\noindent Show that if $f$ and $g$ are uniformly continuous real functions then $f+g$ is uniformly continuous. ({\it Hint\/}: use ideas from the proof that the sum of two continuous functions is continuous.)
\dbhfill[CH]}
\def\ansCH{Let $\epsilon>0$.
Because $f$ is uniformly continuous, there is a $\delta_1>0$ such that
$|f(x)-f(y)|\le\Bover{\epsilon}2$ whenever $x$, $y\in\dom(f)$ and
$|x-y|\le\delta_1$.
Because $g$ is uniformly continuous, there is a $\delta_2>0$ such that
$|g(x)-g(y)|\le\Bover{\epsilon}2$ whenever $x$, $y\in\dom(g)$ and
$|x-y|\le\delta_2$.
Set $\delta=\min(\delta_1,\delta_2)>0$. If $x$, $y\in\dom(f+g)$ and
$|x-y|\le\delta$, then
\Centerline{$|(f+g)(x)-(f+g)(y)|
\le|f(x)-f(y)|+|g(x)-g(y)|
\le\Bover{\epsilon}2+\Bover{\epsilon}2
=\epsilon$.}
\noindent As $\epsilon$ is arbitrary, $f+g$ is uniformly continuous.
}%end of ansCH
\def\qnDA{Recall that a real function $f$ is uniformly continuous if
\inset{for every $\epsilon>0$ there is a $\delta>0$ such that
$|f(x)-f(y)|\le\epsilon$ whenever $x$, $y\in\dom(f)$ and $|x-y|\le\delta$.}
\noindent Show that if $f$ and $g$ are uniformly continuous real functions then $f\frsmallcirc g$ is uniformly continuous.
\dbhfill[DA]}
\def\ansDA{Let $\epsilon>0$. Let $\eta>0$ be such that
$|f(x)-f(y)|\le\epsilon$ whenever $x$, $y\in\dom f$ and $|x-y|\le\eta$. Let $\delta>0$ be such that $|g(x)-g(y)|\le\eta$ whenever $x$, $y\in\dom g$ and
$|x-y|\le\delta$. Now if $x$, $y\in\dom(f\frsmallcirc g)$ and
$|x-y|\le\delta$, $x$ and $y$ belong to $\dom g$ so $|g(x)-g(y)|\le\eta$, while also $g(x)$ and $g(y)$ belong to $\dom f$ so
\Centerline{$|(f\frsmallcirc g)(x)-(f\frsmallcirc g)(y)|
=|f(g(x))-f(g(y))|\le\epsilon$.}
\noindent As $\epsilon$ is arbitrary, $f\frsmallcirc g$ is uniformly continuous.
}%end of ansDA
\def\qnDB{Show that, for any real function $f$ and any $a$, $b\in\Bbb R$,
$\lim_{x\to a}f(x)=b$ iff $\lim_{x\downarrow a}f(x)=\lim_{x\uparrow a}f(x)=b$.
({\it Hint\/}: you will probably need to set $\delta=\min(\delta_1,\delta_2)$.)
\dbhfill[DB]}
\def\ansDB{(a) Suppose first that $\lim_{x\to a}f(x)=b$.
\quad(i) For any $\epsilon>0$, there is a $\delta>0$ such that
$|f(x)-b|\le\epsilon$ whenever $0<|x-a|\le\delta$. In particular,
$|f(x)-b|\le\epsilon$ whenever $a<x\le a+\delta$. As $\epsilon$ is arbitrary, $\lim_{x\downarrow a}f(x)=b$.
\quad(ii) For any $\epsilon>0$, there is a $\delta>0$ such that
$|f(x)-b|\le\epsilon$ whenever $0<|x-a|\le\delta$. In particular,
$|f(x)-b|\le\epsilon$ whenever $a-\delta\le x<a$. As $\epsilon$ is arbitrary, $\lim_{x\uparrow a}f(x)=b$.
\medskip
(b) Now suppose that $\lim_{x\downarrow a}f(x)=\lim_{x\uparrow a}f(x)=b$. Let $\epsilon>0$. Then there are $\delta_1$, $\delta_2>0$ such that
\quad $|f(x)-b|\le\epsilon$ whenever $a<x\le a+\delta_1$
\quad $|f(x)-b|\le\epsilon$ whenever $a-\delta_2\le x<a$.
\noindent Set $\delta=\min(\delta_1,\delta_2)>0$. If $0<|x-a|\le\delta$, then
{\it either} $x>a$ so $a<x\le a+\delta\le a+\delta_1$ and $|f(x)-b|\le\epsilon$
{\it or} $x<a$ so $a-\delta_2\le a-\delta\le x<a$ and $|f(x)-b|\le\epsilon$.
\noindent Thus $|f(x)-b|\le\epsilon$ whenever $0<|x-a|\le\delta$; as $\epsilon$ is arbitrary, $\lim_{x\to a}f(x)=b$.
}%end of ansDB
\def\qnDC{(a) Suppose that $\lim_{n\to\infty}x_n=b$, $\lim_{n\to\infty}y_n=c$ and that $x_n\le y_n$ for every $n\in\Bbb N$. Show that $b\le c$.
(b) If we know that $x_n<y_n$ for every $n$, can we conclude that $b<c$?
\dbhfill[DC]}
\def\ansDC{(a) \Quer\ If $b>c$, set $\epsilon=\bover13(b-c)>0$. Then there are an $n_1\in\Bbb N$ such that $|x_n-b|\le\epsilon$ whenever $n\ge n_1$, and an $n_2\in\Bbb N$ such that $|y_n-c|\le\epsilon$ whenever $n\ge n_2$. Set $n=\max(n_1,n_2)$; then $x_n\ge b-|x_n-b|\ge b-\epsilon$ and
$y_n\le c+|y_n-c|\le c+\epsilon$, so
\Centerline{$x_n-y_n\ge (b-\epsilon)-(c+\epsilon)=b-c-2\epsilon=\epsilon$.}
\noindent But $x_n-y_n\le 0$, so this is impossible.\ \Bang
We must therefore have $b\le c$, as claimed.
\medskip
(b) No; consider $x_n=0$, $y_n=\Bover1{n+1}$ for every $n$, $b=c=0$.
}%end of ansDC
\def\qnDD{(a) For $\lambda\in\Bbb R$, consider the polynomial
$f(x)=x^5-2x^4+\lambda x^3-4$. Show that for any $c\in\Bbb R$ there are $a$, $b\in\Bbb R$ such that $a\le b$ and $f(a)\le c\le f(b)$.
({\it Hint\/}: show that if $|x|\ge 1$ then $|f(x)-x^5|\le (6+|\lambda|)x^4$, so that if $x\ge |c|+6+|\lambda|$ then $f(x)\ge |c|$.)
(b) Now suppose that $f(x)=x^5+\lambda_4x^4+\lambda_3x^3+\lambda_2x^2+\lambda_1x+\lambda_0$. What changes must you make to the argument above to show that for any $c\in\Bbb R$ there are $a$, $b\in\Bbb R$ such that $a\le b$ and $f(a)\le c\le f(b)$?
(c) Show that for any polynomial $g$ of {\it odd} order there is an $x\in\Bbb R$ such that $g(x)=0$.
\dbhfill[DD]}
\def\ansDD{(a) If $|x|\ge 1$, $|x|^4\ge|x|^3\ge 1$ so
\Centerline{$|f(x)-x^5|\le 2|x|^4+|\lambda||x|^3+4
\le 2|x|^4+|\lambda||x|^4+4|x|^4=(6+|\lambda|)x^4$.}
\noindent Consequently
\Centerline{$f(x)\ge x^5-(6+|\lambda|)x^4=(x-6-|\lambda|)x^4$}
\noindent whenever $x\ge 6+|\lambda|$. Setting $b=|c|+6+|\lambda|$ we have $f(b)\ge|c|\ge c$. Similarly,
\Centerline{$f(x)\le x^5+(6+|\lambda|)x^4=(x+6+|\lambda|)x^4\le x+6+|\lambda|$}
\noindent whenever $x\le -6-|\lambda|$. Setting $a=-|c|-6-|\lambda|$ we have $f(a)\le -|c|\le c$.
\medskip
(b) Take $\max(1,\sum_{k=0}^4|\lambda_k|)$ instead of $6+|\lambda|$.
\medskip
(c) Express $g(x)$ as $\sum_{k=0}^n\lambda_kx^k$ where $n$ is odd and $\lambda_n\ne 0$. Set $f(x)=\Bover1{\lambda_n}g(x)$. Then by the arguments of (a)-(b),
$|f(x)-x^n|\le\sum_{k=0}^{n-1}|\Bover{\lambda_k}{\lambda_n}||x|^{n-1}$ whenever $|x|\ge 1$. So if $b=\max(1,\sum_{k=0}^{n-1}|\Bover{\lambda_k}{\lambda_n}|)$ and $a=-b$, $f(a)\le 0\le f(b)$. By the Intermediate Value Theorem, there is an $x\in[a,b]$ such that $f(x)=0$ and $g(x)=0$.
}%end of ansDD
\def\qnDE{Let $f$ and $g$ be real functions, and suppose that $x_0\in\dom g$ is such that $f$ is differentiable at $g(x_0)$ and $\lim_{x\to x_0}g(x)=g(x_0)$.
(a) Define a real function $h:\Bbb R\to\Bbb R$ by setting
$$\eqalign{h(x)&=\Bover{f(g(x))-f(g(x_0))}{g(x)-g(x_0)}
\text{ if }x\in\dom(f\frsmallcirc g)\text{ and }g(x)\ne g(x_0)\cr
&=f'(g(x_0))\text{ for other }x\in\Bbb R.\cr}$$
\noindent Show that $\lim_{x\to x_0}h(x)=f'(x_0)$.
(b) Hence show that if $g$ is differentiable at $x_0$ then $f\frsmallcirc g$ is differentiable at $x_0$.
\dbhfill[DE]}
\def\ansDE{(a) Let $\epsilon>0$. Let $\eta>0$ be such that
$|\Bover{f(y)-f(g(x_0))}{y-g(x_0)}-f'(g(x_0))|\le\epsilon$ whenever
$0<|y-g(x_0)|\le\eta$.
Let $\delta>0$ be such that $|g(x)-g(x_0)|\le\eta$ whenever
$0<|x-x_0|\le\delta$. Now if $0<|x-x_0|\le\delta$, $g(x)$ is defined and $|g(x)-g(x_0)|\le\eta$. If $g(x)=g(x_0)$ then $h(x)=f'(g(x_0))$, so
$|h(x)-f'(g(x_0))|=0\le\epsilon$. Otherwise, $f(g(x))$ is defined and
\Centerline{$|h(x)-f'(g(x_0))|
=\Bover{f(g(x))-f(g(x_0))}{g(x)-g(x_0)}-f'(g(x_0))|\le\epsilon$.}
\noindent Thus in either case $|h(x)-f'(g(x_0))|\le\epsilon$. As $\epsilon$ is arbitrary, $\lim_{x\to x_0}h(x)=f'(g(x_0))$.
\medskip
(b) Now suppose that $g$ is differentiable at $x_0$, so that
$\lim_{x\to x_0}\Bover{g(x)-g(x_0)}{x-x_0}=g'(x_0)$. Then
\Centerline{$h(x)\cdot\Bover{g(x)-g(x_0)}{x-x_0}
=\Bover{f(g(x))-f(g(x_0))}{x-x_0}$}
\noindent whenever $x\in\dom(f\frsmallcirc g)$, because if $g(x)=g(x_0)$ then both sides are zero, and otherwise the left-hand-side is
\Centerline{$\Bover{f(g(x))-f(g(x_0))}{g(x)-g(x_0)}
\cdot\Bover{g(x)-g(x_0)}{x-x_0}
=\Bover{f(g(x))-f(g(x_0))}{x-x_0}$.}
\noindent Since $\lim_{x\to a}h_1(x)h_2(x)
=\lim_{x\to a}h_1(x)\cdot\lim_{x\to a}h_2(x)$ whenever both limits on the right are defined, we have
$$\eqalign{\lim_{x\to x_0}
\Bover{(f\frsmallcirc g)(x)-(f\frsmallcirc g)(x_0)}{x-x_0}
&=\lim_{x\to x_0} h(x)\cdot\Bover{g(x)-g(x_0)}{x-x_0}\cr
&=\lim_{x\to x_0} h(x)\cdot\lim_{x\to x_0}\Bover{g(x)-g(x_0)}{x-x_0}\cr
&=f'(g(x_0))g'(x_0),\cr}$$
\noindent and $(f\frsmallcirc g)'(x_0)$ is defined (and equal to $f'(g(x_0))g'(x_0)$).
}%end of ansDE
\def\qnDF{Let $f:[a,b]\to\Bbb R$ be a continuous function, where $a\le b$ in $\Bbb R$.
(a) Show that its set of values is a closed interval $[c,d]$.
({\it Hint\/}: you will need to use {\it two} of the theorems from lectures in Week 16.)
(b) Now suppose that $f$ is injective. Show that {\it either} $f(x)\le f(y)$ whenever $a\le x\le y\le b$ {\it or} $f(x)\ge f(y)$ whenever $a\le x\le y\le b$.
({\it Hint\/}: consider the cases $f(a)\le f(b)$, $f(a)\ge f(b)$ separately.)
\dbhfill[DF]}%end of qnDF
\def\qnDG{Suppose that $f:[a,b]\to\Bbb R$ is injective and that $f(x)\le f(y)$ whenever $a\le x\le y\le b$. Show that $f^{-1}$ is continuous.
\dbhfill[DG]}
\def\ansDF{(a) Because continuous functions on closed bounded intervals are bounded and attain their bounds, there are $z_1$, $z_2\in[a,b]$ such that $f(z_1)\le f(x)\le f(z_2)$ whenever $x\in[a,b]$. Setting $c=f(z_1)$ and $d=f(z_2)$, we see that $[c,d]$ includes the set of values of $f$. On the other hand, if $y$ is any point of $[c,d]$, it lies between $f(z_1)$ and $f(z_2)$, so is of the form $f(x)$ for some $x$ lying between $z_1$ and $z_2$, by the Intermediate Value Theorem; and the set of values of $f$ is the whole of $[c,d]$.
\medskip
(b) Suppose $f(a)\le f(b)$. \Quer\ If there are $x$, $y\in[a,b]$ such that $x\le y$ and $f(x)>f(y)$, then we must have $x<y$. Now
either $f(a)<f(x)$ or $f(y)<f(b)$. If $f(a)<f(x)$ set $w=\max(f(a),f(y))$, so that $f(a)\le w<f(x)$ and $f(x)>w\ge f(y)$. By the IVT, applied to the intervals $[a,x]$ and $[x,y]$, there are $z\in[a,x]$ and $z'\in[x,y]$ such that $f(z)=f(z')=w$; but as neither $z$ nor $z'$ can be equal to $x$, $z<z'$ and $f$ is not injective.
Similarly, of $f(y)<f(b)$, set $w=\min(f(x),f(b))$; then $f(y)<w\le f(x)$ and $f(y)<w\le f(b)$, so there are $z\in[x,y]$ and $z'\in[y,b]$ such that $f(z)=f(z')=w$, and again $f$ is not injective.\ \Bang
So $f(x)\le f(y)$ whenever $a\le x\le y\le b$.
}%end of ansDF
\def\ansDG{We know that $\dom f^{-1}$ is an interval $[c,d]$ where $c\le d$, and that $f(x)\le f(y)$ whenever $a\le x\le y\le b$, so that $c=f(a)$ and $d=f(b)$ and in fact $f(x)<f(y)$ whenever $a\le x<y\le b$ (since $f$ is injective). Let $v\in[c,d]$ and $\epsilon>0$. Set $x=f^{-1}(v)\in[a,b]$. Define $\delta_1$, $\delta_2>0$ by setting
$$\eqalign{\delta_1
&=v-f(x-\epsilon)\text{ if }x-\epsilon\ge a,\cr
&=1+d-c\text{ otherwise},\cr
\delta_2
&=f(x+\epsilon)-v\text{ if }x+\epsilon\le b,\cr
&=1+d-c\text{ otherwise}.\cr}$$
\noindent Because $\delta_1$ and $\delta_2$ must both be greater than $0$.
Set $\delta=\min(\delta_1,\delta_2)>0$. If $w\in[c,d]$ and $|v-w|\le\delta$,
then
$$\eqalignno{f(\max(a,x-\epsilon))
&=\max(c,v-\delta_1)\cr
\displaycause{because if $a\le x-\epsilon$ then $f(x-\epsilon)=v-\delta_1$, while if $x-\epsilon<a$ then $v-\delta_1<c$}
&\le w\le\min(d,v+\delta_2)
=f(\min(b,x+\epsilon))\cr}$$
\noindent because if $x+\epsilon\le b$ then $f(x+\epsilon)=v+\delta_1$, while if $b<x+\epsilon<a$ then $d<v+\delta_1$. But this means that $f^{-1}(w)$ must lie in the interval $[\max(a,x-\epsilon),\min(b,x+\epsilon)]$ and
$|f^{-1}(w)-x|\le\epsilon$. As $\epsilon$ is arbitrary, $f^{-1}$ is continuous at $v$; as $v$ is arbitrary, $f^{-1}$ is continuous.
}%end of ansDG
\def\qnEA{ Recall that a real function $f$ is uniformly continuous if
\inset{for every $\epsilon>0$ there is a $\delta>0$ such that
$|f(x)-f(y)|\le\epsilon$ whenever $x$, $y\in\dom(f)$ and $|x-y|\le\delta$.}
(a) Show that the function $x\mapsto x:\Bbb R\to\Bbb R$ is uniformly continuous.
(b) Show that the function $x\mapsto x^2:\Bbb R\to\Bbb R$ is not uniformly continuous. ({\it Hint\/}: take $\epsilon=1$ and $x=\Bover1{\delta}$.)
(c) Show that $|\sin^2 b-\sin^2 a|\le 2|b-a|$ for all $a$, $b\in\Bbb R$. (You may assume that $\sin$ is differentiable and that $|\sin x|\le 1$, $|\sin'(x)|\le 1$ for every $x$.)
Show that $\sin^2$ is uniformly continuous.
(d) Show that if $f$ and $g$ are bounded uniformly continuous real functions then $f\times g$ is uniformly continuous.
\dbhfill[EA]}
\def\ansEA{(a) Let $\epsilon>0$. Then (writing $f(x)=x$ for $x\in\Bbb R$) $|f(x)-f(y)|\le\epsilon$ whenever $|x-y|\le\epsilon$, so $f$ is uniformly continuous.
\medskip
(b) Write $f(x)=x^2$ for $x\in\Bbb R$. Set $\epsilon=1$. If $\delta>0$ and $x=\Bover1{\delta}$, $y=x+\delta$ then $f(y)-f(x)=\delta^2+2>\epsilon$. So there is no $\delta>0$ such that $|f(x)-f(y)|\le\epsilon$ whenever
$|x-y|\le\delta$, and $f$ is not uniformly continuous.
\medskip
(c) Set $f(x)=(\sin x)^2$. Then
$f'(x)=\sin x\sin'x+\sin'x\sin x=2\sin x\sin'x$, so $|f'(x)|\le 2$ for every $x$. If $a<b$ there is an $x\in\ooint{a,b}$ such that
$f'(x)=\Bover{f(b)-f(a)}{b-a}$, so that $\sin^2b-\sin^2a=(b-a)f'(x)$ and $|\sin^2b-\sin^2a|\le 2|b-a|$. Similarly, if $a>b$,
$|\sin^2b-\sin^2a|=|\sin^2a-\sin^2b|\le 2|a-b|=2|b-a|$, while if $a=b$ we have
$|\sin^2b-\sin^2a|=0=2|b-a|$.
Now let $\epsilon>0$. Set $\delta=\bover12\epsilon>0$. Then if $x$, $y\in\Bbb R$ and $|x-y|\le\delta$,
\Centerline{$|\sin^2x-\sin^2y|
\le 2|x-y|\le 2\delta=\epsilon$.}
\noindent As $\epsilon$ is arbitrary, $\sin^2$ is uniformly continuous.
\medskip
(d) Let $\epsilon>0$. Let $M_1$ be such that $|f(x)|\le M_1$ for every $x\in\dom f$. Let $M_2$ be such that $|g(x)|\le M_2$ for every $x\in\dom g$. Set $\eta=\min(1,\Bover{\epsilon}{1+M_1+M_2})$. Let $\delta_1>0$ be such that $|f(x)-f(y)|\le\eta$ whenever $x$, $y\in\dom f$ and $|x-y|\le\delta_1$. Let $\delta_2>0$ be such that $|g(x)-g(y)|\le\eta$ whenever $x$, $y\in\dom g$ and
$|x-y|\le\delta_2$. Set $\delta=\min(\delta_1,\delta_2)>0$. Now if $x$, $y\in\dom(f\times g)$ and $|x-y|\le\delta$, then
$$\eqalign{|f(x)g(x)-f(y)g(y)|
&\le|f(x)-g(x)||f(y)-g(y)|\cr
&\qquad\qquad+|f(x)||g(x)-g(y)|+|f(x)-f(y)||g(x)|\cr
&\le\eta^2+M_1\eta+\eta M_2
\le\eta(1+M_1+M_2)
\le\epsilon.\cr}$$
\noindent As $\epsilon$ is arbitrary, $f\times g$ is uniformly continuous.
}%end of ansEA
\def\qnEB{(a) Use the Intermediate Value Theorem to show that for every $x\in\coint{0,\infty}$ there is a $y\in\coint{0,\infty}$ such that $y^2=x$.
Show that this $y$ is unique.
(b) For $x\in\coint{0,\infty}$ let $\sqrt{x}$ be the unique number in $\coint{0,\infty}$ such that $(\sqrt x)^2=x$. Show that
\quad(i) if $0\le x\le y\le x+\delta$ then
$|\sqrt{y}-\sqrt{x}|\le\sqrt{\delta}$;
\quad(ii) $\sqrt{\phantom{x}}:\coint{0,\infty}\to\coint{0,\infty}$ is uniformly continuous.
\dbhfill[EB]}
\def\ansEB{(a) Setting $f(y)=y^2$ for $y\in\Bbb R$, we know that $f$ is continuous and that $f(0)\le x$, while $f(\max(1,x))=\max(1,x^2)\ge x$. By the Intermediate Value Theorem, there is a $y\in[0,\max(1,x)]$ such that $y^2=x$.
If now $y_1^2=y_2^2=x$ we must have $(y_1+y_2)(y_1-y_2)=0$, that is, at least one of $y_1+y_2$, $y_1-y_2$ is zero. If $y_1$ and $y_2$ are both greater than or equal to $0$, this can happen only when $y_1=y_2$.
\medskip
(b) We have $(\sqrt x+\sqrt\delta)^2=x+\delta+2\sqrt x\sqrt\delta\ge x+\delta$,
so $(\sqrt x)^2\le y\le(\sqrt x+\sqrt\delta)^2$ and there is a
$z\in[\sqrt x,\sqrt x+\sqrt\delta]$ such that $z^2=y$; since positive square roots are unique, $z=\sqrt y$ and $\sqrt x\le\sqrt y\le\sqrt x+\sqrt\delta$, so that $|\sqrt y-\sqrt x|\le\sqrt\delta$.
Now let $\epsilon>0$. Set $\delta=\epsilon^2$. If $x$, $y\ge 0$ and
$|x-y|\le\delta$, either $x\le y\le x+\delta$ or $y\le x\le y+\delta$. In either case, $|\sqrt y-\sqrt x|=|\sqrt x-\sqrt y|\le\sqrt\delta=\epsilon$. As $\epsilon$ is arbitrary, $\sqrt{\phantom{x}}$ is uniformly continuous.
}%end of ansEB
\def\qnEC{Rewrite the following, completed and corrected to form a proof that if $\lim_{x\downarrow x_0}f(x)=\lim_{x\downarrow x_0}g(x) =0$ and $\lim_{x\downarrow x_0}\Bover{f'(x)}{g'(x)}=b$ then $\lim_{x\downarrow x_0}\Bover{f(x)}{g(x)}=b$.
Let $\epsilon>0$. Let $\delta_0\ge 0$ be such that $\Bover{f'(x)}{g'(x)}$ is defined and $|\Bover{f'(x)}{g'(x)}-b|\le\ldots$ whenever
$x\le x_0+\delta_0$. Note that this implies that $f'(x)$ and $g'(x)$ are defined, and that $g'(x)\ne 0$, for $x_0<x$; so that $f(x)$ and $g(x)$ are defined for $x_0<x\le x_0+\delta_0$, and $g$ is continuous at every point of $\ocint{x_0,x_0+\delta_0}$. By Rolle's theorem, $g(x)$ cannot be zero for more than one value of $x$ in $\ocint{x_0,x_0+\delta_0}$. So there is a $\delta\in\ocint{0,\delta}$ such that $g(x)\ne 0$ for
$x_0<x\le x_0+\delta$.
Take $x$ such that $x_0<x\le x_0+\delta$.
Then $\lim_{y\downarrow x_0}\Bover{f(x)-f(y)}{g(x)-g(y)}
=\ldots$. Let $\delta_1>0$ be such that
$\Bover{f(x)-f(y)}{g(x)-g(y)}$ is defined and
$|\Bover{f(x)-f(y)}{g(x)-g(y)}-\ldots|\le\ldots$ whenever
$x_0<y\le x_0+\delta_1$. Since we cannot have $g(y)=g(x)$ for
$x_0<y\le x_0+\delta_1$, $x_0+\delta_1>x$. Now examine $y=x_0+\delta_1$. Because $y<x$ and $f$ and $g$ are differentiable (therefore continuous) at every point of $[y,x]$, $\ldots$ tells us that there is a $z\in\ooint{x_0,y}$ such that $\Bover{f(x)-f(y)}{g(x)-g(y)}=\Bover{g'(z)}{f'(z)}$. So
\Centerline{$|\Bover{f(x)}{g(x)}-b|
\le|\Bover{f(x)}{g(x)}-\Bover{f(x)-f(y)}{g(x)-g(y)}|
+|\Bover{f'(z)}{g'(z)}-b|
\le \ldots\le\epsilon$.}
\noindent As $\epsilon$ is arbitrary,
$\lim_{x\downarrow 0}\Bover{f(x)}{g(x)}=b$.
\dbhfill[EC]}
\def\ansEC{Let $\epsilon>0$. Let $\delta_0\pmb{>}0$ be such that $\Bover{f'(x)}{g'(x)}$ is defined and
$|\Bover{f'(x)}{g'(x)}-b|\le\pmb{\bover12\epsilon}$ whenever
$\pmb{x_0<}x\le x_0+\delta_0$. Note that this implies that $f'(x)$ and $g'(x)$ are defined, and that $g'(x)\ne 0$, for $x_0<x\pmb{\le x_0+\delta_0}$; so that $f(x)$ and $g(x)$ are defined for $x_0<x\le x_0+\delta_0$, and $g$ is continuous at every point of $\ocint{x_0,x_0+\delta_0}$. By Rolle's theorem, $g(x)$ cannot be zero for more than one value of $x$ in $\ocint{x_0,x_0+\delta_0}$. So there is a $\delta\in\ocint{0,\pmb{\delta_0}}$ such that $g(x)\ne 0$ for
$x_0<x\le x_0+\delta$.
Take $x$ such that $x_0<x\le x_0+\delta$.
Then $\lim_{y\downarrow x_0}\Bover{f(x)-f(y)}{g(x)-g(y)}
=\Bover{f(x)-0}{g(x)-0}=\pmb{\Bover{f(x)}{g(x)}}$. Let $\delta_1>0$ be such that $\Bover{f(x)-f(y)}{g(x)-g(y)}$ is defined and
$|\Bover{f(x)-f(y)}{g(x)-g(y)}-\pmb{\Bover{f(x)}{g(x)}}|
\le\pmb{\bover12\epsilon}$ whenever
$x_0<y\le x_0+\delta_1$. Since we cannot have $g(y)=g(x)$ for
$x_0<y\le x_0+\delta_1$, $x_0+\delta_1\pmb{<}x$. Now examine $y=x_0+\delta_1$. Because $y<x$ and $f$ and $g$ are differentiable (therefore continuous) at every point of $[y,x]$, \pmb{Cauchy's Mean Value Theorem} tells us that there is a $z\in\ooint{\pmb{y,x}}$ such that
$\Bover{f(x)-f(y)}{g(x)-g(y)}=\pmb{\Bover{f'(z)}{g'(z)}}$. So
\Centerline{$|\Bover{f(x)}{g(x)}-b|
\le|\Bover{f(x)}{g(x)}-\Bover{f(x)-f(y)}{g(x)-g(y)}|
+|\Bover{f'(z)}{g'(z)}-b|
\le\pmb{\bover12\epsilon+\bover12\epsilon}=\epsilon$.}
\noindent
As $\epsilon$ is arbitrary, $\lim_{x\downarrow 0}\Bover{f(x)}{g(x)}=b$.
}%end of ansEC
\def\qnED{Suppose that $\sequencen{x_n}$, $\sequencen{y_n}$ are two series, of which $\sequencen{y_n}$ is absolutely summable; set $s_n=\sum_{k=0}^nx_k$ for each $n$.
(a) Show that if $|x_n|\le|y_n|$ for every $n\in\Bbb N$, then $\sequencen{x_n}$ is summable. ({\it Hint\/}: show that $\sequencen{s_n}$ is a Cauchy sequence.)
(b) Show that if, for some $n_1\in\Bbb N$ and $M\ge 0$, $|x_n|\le M|y_n|$ for every $n\ge n_1$, then $\sequencen{x_n}$ is summable.
(c) Show that if $\lim_{n\to\infty}\Bover{x_n}{y_n}$ is defined in $\Bbb R$ then $\sequencen{x_n}$ is summable.
\dbhfill[ED]}
\def\ansED{(a) Set $t_n=\sum_{k=0}^n|y_k|$ for $n\in\Bbb N$. Then $\sequencen{t_n}$ is convergent, therefore Cauchy. Now $|s_m-s_n|\le|t_m-t_n|$ for all $m$, $n\in\Bbb N$. \Prf\ If $n<m$, then
\Centerline{$|s_m-s_n|=|\sum_{k=n+1}^{m}x_k|\le\sum_{k=n+1}^{m}|x_k|
\le\sum_{k=n+1}^{m}|y_k|=t_m-t_n=|t_m-t_n|$.}
\noindent If $m<n$, then
\Centerline{$|s_m-s_n|=|s_n-s_m|\le|t_n-t_m|=|t_m-t_n|$.}
\noindent If $m=n$ then $|s_m-s_n|=0=|t_m-t_n|$.\ \Qed
Let $\epsilon>0$. Then there is an $n_0\in\Bbb N$ such that
$|t_m-t_n|\le\epsilon$ whenever $m$, $n\ge n_0$. So $|s_m-s_n|\le\epsilon$ whenever $m$, $n\ge n_0$. As $\epsilon$ is arbitrary, $\sequencen{s_n}$ is Cauchy. By Cauchy's General Principle of Convergence, $\sequencen{s_n}$ is convergent, that is, $\sequencen{x_n}$ is summable.
\medskip
(b) Set $t_n=\sum_{k=0}^n|y_k|$ for $n\in\Bbb N$. Then $\sequencen{t_n}$ is convergent, therefore Cauchy. Now $|s_m-s_n|\le M|t_m-t_n|$ for all $m$,
$n\ge n_1$. \Prf\ If $n<m$, then
\Centerline{$|s_m-s_n|=|\sum_{k=n+1}^{m}x_k|\le\sum_{k=n+1}^{m}|x_k|
\le\sum_{k=n+1}^{m}M|y_k|=M(t_m-t_n)=M|t_m-t_n|$.}
\noindent If $m<n$, then
\Centerline{$|s_m-s_n|=|s_n-s_m|\le M|t_n-t_m|=M|t_m-t_n|$.}
\noindent If $m=n$ then $|s_m-s_n|=0=M|t_m-t_n|$.\ \Qed
Let $\epsilon>0$. Then there is an $n_0\in\Bbb N$ such that
$|t_m-t_n|\le\Bover{\epsilon}{M+1}$ whenever $m$, $n\ge n_0$. So
$|s_m-s_n|\le\epsilon$ whenever $m$, $n\ge\max(n_0,n_1)$. As $\epsilon$ is arbitrary, $\sequencen{s_n}$ is Cauchy. By Cauchy's General Principle of Convergence, $\sequencen{s_n}$ is convergent, that is, $\sequencen{x_n}$ is summable.
\medskip
(c) Set $b=\lim_{n\to\infty}\Bover{x_n}{y_n}$. There is an $n_1\in\Bbb N$ such that $|\Bover{x_n}{y_n}-b|\le 1$ for every $n\ge n_1$, so that $|\Bover{x_n}{y_n}|\le |b|+1$ and $|x_n|\le(|b|+1)|y_n|$ for every $n\ge n_1$. By (b), $\sequencen{x_n}$ is summable.
}%end of ansED
\def\qnEE{Let $\sequencen{x_n}$ be a sequence such that $\lim_{n\to\infty}|\Bover{x_{n+1}}{x_n}|=b<1$.
(a) Set $c=\bover12(1+b)$. Show that there are
$n_0\in\Bbb N$ and $M\ge 0$ such that $|x_n|\le Mc^n$ for every $n\ge n_0$. ({\it Hint\/}: show that there is an $n_0$ such that $|x_{n+1}|\le c|x_n|$ for every $n\ge n_0$.)
(b) Show that $\sequencen{x_n}$ is absolutely summable. ({\it Hint\/}: use question ED above and question CC from Problem Sheet 3.)
\dbhfill[EE]}
\def\ansEE{(a) Because $\bover12(1-b)>0$, there is an $n_0\in\Bbb N$ such that $|\Bover{x_{n+1}}{x_n}-b|\le\bover12(1-b)$ for every $n\ge n_0$, so that $|\Bover{x_{n+1}}{x_n}|\le b+\bover12(1-b)=c$ for every $n\ge n_0$. (Note that $b$, being the limit of a sequence of non-negative terms, is
non-negative.) This means that $|x_{n+1}|\le c|x_n|$ for
$n\ge n_0$. Set $M=|x_{n_0}|c^{-n_0}$. Then $|x_n|\le Mc^n$ for every
$n\ge n_0$. \Prf\ Induce on $n$. For $n=n_0$ we have
$Mc^n=|x_{n_0}|c^{-n_0}c^{n_0}=|x_{n_0}|$, so the induction starts. For the inductive step, given that $n\ge n_0$ and $|x_n|\le Mc^n$, then
$$\eqalignno{|x_{n+1}|&\le c|x_n|\le cMc^n\cr
\displaycause{by the inductive hypothesis}
&=Mc^{n+1},\cr}$$
\noindent so the induction continues.\ \Qed
\medskip
(c) We know that $\sum_{n=0}^{\infty}c^n=\Bover1{1-c}$, so $\sequencen{c_n}$ is absolutely summable; by the comparison test, so is $\sequencen{x_n}$.
}%end of ansEE
\def\qnEF{(a) Let $f$ be a real function such that $f'$ is defined and continuous on $[a,a+h]$, where $a\in\Bbb R$ and $h>0$, and $f''$ is defined on $\ooint{a,a+h}$.
\quad(i) Show that there is an $h_1\in\ooint{0,h}$ such that
$\Bover{f(a+h)-f(a)-hf'(a)}{h^2}=\Bover{f'(a+h_1)-f'(a)}{2h_1}$. ({\it Hint\/}: apply Cauchy's Mean Value Theorem to $\Bover{g(h)-g(0)}{h^2-0^2}$ where $g(x)=f(a+x)-f(a)-xf'(a)$.)
\quad(ii) Show that there is an $h_2\in\ooint{0,h_1}$ such that
$\Bover{f(a+h)-f(a)-hf'(a)}{h^2}=2f''(a+h_2)$.
(b) Now suppose that $f''$ is defined and continuous on $[a,a+h]$ and $f'''$ is defined on $\ooint{a,a+h}$. Show that there is an $h_3\in\ooint{0,h}$ such that $f(a+h)=f(a)+hf'(a)+\bover12h^2f''(a)+\bover16h^3f'''(a+h_3)$.
({\it Hint\/}: use $g(x)=f(a+x)-f(a)-xf'(a)-\bover12x^2f''(a)$.)
\dbhfill[EF]}
\def\ansEF{(a)(i) We have $g'(x)=f'(a+x)-f'(a)$ defined and continuous on $[0,h]$ while $g''(x)=f''(a+x)$ is defined on $\ooint{0,h}$. By Cauchy's Mean Value Theorem, there is an $h_1\in\ooint{0,h}$ such that
$\Bover{g(h)-g(0)}{h^2-0^2}=\Bover{g'(h_1)}{2h_1}$, that is,
$\Bover{f(a+h)-f(a)-xf'(a)}{h^2}=\Bover{f'(a+h_1)-f'(a)}{2h_1}$.
\medskip
\quad(ii) Now $g'$ is continuous on $[0,h_1]$ and differentiable on $\ooint{0,h_1}$, so by Cauchy's Mean Value Theorem again there is an $h_2\in\ooint{0,h_1}$ such that
\Centerline{$\Bover{g''(h_2)}{2}=\Bover{g'(h_1)-g'(0)}{2h_1-2\cdot 0}
=\Bover{g'(h_1)}{2h_1}$,}
\noindent that is,
\Centerline{$\Bover12f''(a+h_2)=\Bover{f(a+h)-f(a)-hf'(a)}{h^2}$.}
\medskip
(b) Set $g(x)=f(a+x)-f(a)-xf'(a)-\Bover12x^2f''(a)$. Then $g$ is defined and continuous on $[0,h]$ and differentiable on $\ooint{0,h}$. So there is an $h_1\in\ooint{0,h}$ such that
$$\bover{f(a+h)-f(a)-hf'(a)-\bover12h^2f''(a)}{h^3}
=\bover{g(h)-g(0)}{h^3-0^3}
=\bover{g'(h_1)}{3h_1^2}.$$
\noindent Now apply (a) to the function $g'$; there is an $h_3\in\ooint{0,h_1}$ such that
\Centerline{$g'(h_1)=g'(0)+h_1g''(0)+\Bover12h_1^2g'''(h_3)
=\Bover12h_1^2g'''(h_3)$,}
\noindent so that
$$\bover{f(a+h)-f(a)-hf'(a)-\bover12h^2f''(a)}{h^3}
=\bover16g'''(h_3)=\bover16f'''(a+h_3).$$
}%end of ansEF
\def\wideblock{\vrule height7pt depth0pt width30pt}
\def\block{\vrule height7pt depth0pt width10pt}
\def\qnGA{(a) Define the expressions
\Centerline{$\lim_{x\to a}f(x)=b$,
\quad$\lim_{x\downarrow a}f(x)=b$,
\quad$\lim_{x\uparrow a}f(x)=b$}
\noindent where $f$ is a real function.
\medskip
(b) Show from your definitions that
\quad (i) $\lim_{x\to a}f(x)=b$ iff $\lim_{x\downarrow a}f(x)=b$ and $\lim_{x\uparrow a}f(x)=b$;
\quad (ii) if $\lim_{\downarrow a}f(x)=b$ and
$\lim_{x\downarrow a}g(x)=c$ and $f(x)\ge g(x)$ for every
$x\in\dom f\cap\dom g$, then $b\ge c$.
\medskip
(c) Show that if $\lim_{x\to 0}f(x)=0$ and $g:[-1,1]\to[-1,1]$ is any function then $\lim_{x\to 0}f(x)g(x)=0$.
\dbhfill[GA]}
\def\ansGA{ (no solution written out)}
\def\qnGB{(a) What is meant by saying that a real function is {\it continuous}?
\medskip
(b) Show from your definition that
\quad(i) if $f:\Bbb R\to\Bbb R$ is continuous and $\sequencen{x_n}$ is a convergent sequence then $\sequencen{f(x_n)}$ is convergent;
\quad(ii) the function $x\mapsto|x|:\Bbb R\to\Bbb R$ is continuous.
\medskip
(c) What is the {\it infimum} of a subset of $\Bbb R$? Which subsets of $\Bbb R$ have infima? Give examples of (i) a set
$A_1\subseteq\Bbb R$ with no infimum (ii) a set $A_2\subseteq\Bbb R$ with an infimum belonging to $A_2$ (iii) a set $A_3\subseteq\Bbb R$ with an infimum not belonging to $A_3$.
\medskip
(d) Complete and correct the following to render it a proof that any continuous function $f:[a,b]\to\Bbb R$, where $a\le b$ in $\Bbb R$, is bounded.
\inset{Set $A=\ldots$. Then
$a\in A$ and $A$ is bounded above by $b$. Accordingly $c=\inf A$
is defined and $c\in[a,b]$. There is therefore a $\delta>0$ such that
$|f(y)-f(c)|\le\ldots$ whenever $y\in\dom f$ and $|y-c|\le\delta$. Set
$z=\min(b,c+\delta)$. Then $z\in A$. To see this, note that there is
an $x\in A$ such that $\ldots$. Now $[a,z]=[a,x]\cup[x,z]$.
Because $x\in A$, there is an $M$ such that $|f(y)|\ge M$ for every
$y\in\ldots$; because $[x,z]\subseteq\dom f\cap[c-\delta,c+\delta]$,
$|f(y)|\le|f(c)|+1$ for every $y\in[x,z]$; so
$|f(y)|\le\ldots$ for every $y\in[a,z]$, and $f$ is bounded on
$[a,z]$. This means that $z\le c$. So $z=b=c$ and $b\in A$; which is what we set out to prove.}
\dbhfill[GB]}
\def\ansGB{ (no solution written out)}
\def\qnGC{(a) Show that any bounded non-decreasing sequence is convergent.
\medskip
(b) Define {\it Cauchy sequence}. Show that
\quad(i) every convergent sequence is Cauchy;
\quad(ii) every Cauchy sequence is bounded.
\medskip
(c) Complete the following sentences and arrange them in such an order that they form a correct proof that any Cauchy sequence $\sequencen{x_n}$ is convergent.
\inset{
1. For each $n\in\Bbb N$, $A_n=\ldots$ is a non-empty subset of
$[-M,M]$, so has an infimum $y_n$ say, with $-M\le y_n\le x_n\le M$.
2. There is an $n_0\in\Bbb N$ such that
$|x_i-x_n|\le\epsilon$ whenever $i$, $n\ge n_0$.
3. Let $\epsilon>0$.
4. Because $\sequencen{x_n}$ is Cauchy, it is bounded; let $M$ be such that $-M\le x_n\le M$ for every $n\in\Bbb N$.
5. As $\epsilon$ is arbitrary, $\lim_{n\to\infty}x_n-y_n=0$.
6. Thus $\sequencen{y_n}$ is a bounded non-decreasing sequence and has a limit $y$ say.
7. For each $n\in\Bbb N$, $A_{n+1}\ldots A_n$ so $y_n$ is a lower bound of $A_{n+1}$ and $y_n\ldots y_{n+1}$.
8. If $n\ge n_0$, $A_n\subseteq[x_n-\epsilon,x_n+\epsilon]$ so
$y_n\ge x_n-\epsilon$ and $|x_n-y_n|=x_n-y_n\le\ldots$.
9. So $\lim_{n\to\infty}x_n=0+y=y$ and $\sequencen{x_n}$ is convergent.
}
\dbhfill[GC]}
\def\ansGC{ (no solution written out)}
\def\qnGD{(a) What is meant by saying that a real function is {\it differentiable}?
\medskip
(b) Show from your definition that the product of two differentiable functions is differentiable.
\medskip
(c) State Rolle's theorem and the Mean Value Theorem, and prove the latter.
\medskip
(d) Show that if $f:\Bbb R\to\Bbb R$ is a differentiable function and
$f':\Bbb R\to\Bbb R$ is bounded, then there is an $M\ge 0$ such that $|f(x)|\le M(1+|x|)$ for every $x\in\Bbb R$.
\medskip
\inset{(In this question, standard theorems concerning limits may be used without proof, but should be clearly stated.)}
\dbhfill[GD]}
\def\ansGD{ (no solution written out)}
\def\qnGE{Recall that a real function $f$ is {\it uniformly continuous} if for every $\epsilon>0$ there is a $\delta>0$ such that $|f(x)-f(y)|\le\epsilon$ whenever $x$, $y\in\dom f$ and $|x-y|\le\delta$.
Rearrange the following sentences to form a proof that if $a\le b$ in $\Bbb R$ and $f:[a,b]\to\Bbb R$ is continuous, it is uniformly continuous.
\inset{1. Let $\epsilon>0$.
2. Set $A=\{x:a\le x\le b,\,\exists\delta>0,\,|f(y)-f(y')|\le\epsilon$ whenever $y$, $y'\in[a,x]$ and $|y-y'|\le\delta\}$.
3. So $z=\sup A$ is defined and $a\le z\le b$.
4. $a\in A$ because if $y$, $y'\in[a,a]$ then $|f(y)-f(y')=0$.
5. Because $z=\sup A$, there is an $x_1\in A\cap[z-\bover12\delta_0,z]$.
6. Because $z\in[a,b]$, $f$ is continuous at $z$; let $\delta_0>0$ be such that $|f(y)-f(z)|\le\epsilon$ whenever
$y\in[a,b]\cap[z-\delta_0,z+\delta_0]$.
7. Set $\delta=\min(\bover12\delta_0,\delta_1)>0$.
8. Let $\delta_1>0$ be such that $|f(y)-f(y')|\le\bover12\epsilon$ whenever $y$, $y'\in[a,x_1]$ and $|y-y_1|\le\delta_1$.
9. $A$ is bounded above by $b$.
10. If $\min(y,y')\ge z-\delta_0$, then $|f(y)-f(y')|\le|f(y)-f(z)|+|f(y')-f(z)|
\le\bover12\epsilon+\bover12\epsilon=\epsilon$.
11. If $\min(y,y')\le z-\delta_0$ then $\min(y,y')\le x-\bover12\delta_0$; since $|y-y'|\le\bover12\delta_0$, both $y$ and $y'$ belong to $[a,x]$; since
$|y-y'|\le\delta_1$, $|f(y)-f(y')|\le\epsilon$.
12. Suppose that $y$, $y'\in[a,\min(b,z+\delta_0)]$ and $|y-y'|\le\delta$.
13. Since $z$ is an upper bound for $A$, $\min(b,z+\delta_0)\le z$, so in fact $z=b$ and $b\in A$.
14. Since $y$ and $y'$ are arbitrary, $\min(b,z+\delta_0)\in A$.
15. This means that there is a $\delta>0$ such that $|f(y)-f(y')|\le\epsilon$ whenever $y$, $y'\in[a,b]$ and $|y-y'|\le\delta$. As $\epsilon$ is arbitrary, $f$ is uniformly continuous.}
\dbhfill[GE]}
\def\ansGE{Let $\epsilon>0$.
Set $A=\{x:a\le x\le b,\Exists\delta>0,\,|f(y)-f(y')|\le\epsilon$ whenever $y$, $y'\in[a,x]$ and $|y-y'|\le\delta\}$.
$a\in A$ because if $y$, $y'\in[a,a]$ then $|f(y)-f(y')=0$.
$A$ is bounded above by $b$.
So $z=\sup A$ is defined and $a\le z\le b$.
Because $z\in[a,b]$, $f$ is continuous at $z$; let $\delta_0>0$ be such that $|f(y)-f(z)|\le\epsilon$ whenever
$y\in[a,b]\cap[z-\delta_0,z+\delta_0]$.
Because $z=\sup A$, there is an $x_1\in A\cap[z-\bover12\delta_0,z]$.
Let $\delta_1>0$ be such that $|f(y)-f(y')|\le\bover12\epsilon$ whenever $y$, $y'\in[a,x_1]$ and $|y-y_1|\le\delta_1$.
Set $\delta=\min(\bover12\delta_0,\delta_1)>0$.
Suppose that $y$, $y'\in[a,\min(b,z+\delta_0)]$ and $|y-y'|\le\delta$.
If $\min(y,y')\ge z-\delta_0$, then $|f(y)-f(y')|\le|f(y)-f(z)|+|f(y')-f(z)|
\le\bover12\epsilon+\bover12\epsilon=\epsilon$.
If $\min(y,y')\le z-\delta_0$ then $\min(y,y')\le x-\bover12\delta_0$; since $|y-y'|\le\bover12\delta_0$, both $y$ and $y'$ belong to $[a,x]$; since
$|y-y'|\le\delta_1$, $|f(y)-f(y')|\le\epsilon$.
Since $y$ and $y'$ are arbitrary, $\min(b,z+\delta_0)\in A$.
Since $z$ is an upper bound for $A$, $\min(b,z+\delta_0)\le z$, so in fact $z=b$ and $b\in A$.
This means that there is a $\delta>0$ such that $|f(y)-f(y')|\le\epsilon$ whenever $y$, $y'\in[a,b]$ and $|y-y'|\le\delta$. As $\epsilon$ is arbitrary, $f$ is uniformly continuous.
}%end of ansGE
\def\qnGF{(a) Define the expressions $\lim_{x\to\infty}f(x)=b$,
$\lim_{x\to-\infty}f(x) = b$.
\medskip
(b) What is meant by saying that a real function is {\it continuous}?
Show from your definition that the composition of continuous functions is continuous.
\medskip
(c) For which of the following sets, if any, is there an unbounded continuous function $f:I\to\Bbb R$? Give an example in each case in which one exists.
\quad(i) $I=\coint{1,\infty}$.
\quad(ii) $I=\ocint{-1,1}$.
\quad(iii) $I=[-2,2]$.
\medskip
(d) Show that if $f:\Bbb R\to\Bbb R$ is continuous and $\lim_{x\to\infty}f(x)$ and $\lim_{x\to-\infty}f(x)$ are both defined in $\Bbb R$, then $f$ is bounded.
\dbhfill[GF]}
\def\qnGG{(a) What is meant by saying that a real function is {\it differentiable}?
\medskip
(b) Show that a differentiable function is continuous. Give an example of a continuous function $f:\Bbb R\to\Bbb R$ which is not differentiable.
\medskip
(c) State Rolle's theorem and the Mean Value Theorem, and prove the latter from the former.
\medskip
(d) Let $f:\ooint{-1,1}\to\Bbb R$ be a continuous function. Suppose that
$\lim_{x\to 0}f'(x)=b$. Show that $f'(0)$ is defined and equal to $b$.
\dbhfill[GG]}
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nist.gov
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| 104,563,766 | 656 |
\[(-1)^{n-k}\mathop{s\/}\nolimits\!\left(n,k\right)=\sum_{1\leq b_{1}<\dots<b_{n%
-k}\leq n-1}b_{1}b_{2}\cdots b_{n-k},\]
|
http://ida.darksky.org/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%2C%20area%20FROM%20refs%20WHERE%20serial%20%3D%201506%20ORDER%20BY%20first_author%2C%20author_count%2C%20author%2C%20year%2C%20title&client=&formType=sqlSearch&submit=Cite&viewType=&showQuery=0&showLinks=1&showRows=5&rowOffset=&wrapResults=1&citeOrder=&citeStyle=APA&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg=
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darksky.org
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CC-MAIN-2020-45
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crawl-data/CC-MAIN-2020-45/segments/1603107898577.79/warc/CC-MAIN-20201028132718-20201028162718-00332.warc.gz
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%&LaTeX
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\begin{document}
\begin{thebibliography}{1}
\bibitem{Morag+Ohlsson2016} Morag, I., \& Ohlsson, A. (2016). Cycled light in the intensive care unit for preterm and low birth weight infants. \textit{Cochrane Database Syst Rev}, \textit{8}, Cd006982.
\end{thebibliography}
\end{document}
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https://dlmf.nist.gov/2.4.E21.tex
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nist.gov
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| 285,977,915 | 717 |
\[w=z^{-1/3}v-a,\]
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https://git.bettercrypto.org/ach-master.git/blobdiff_plain/73abc462de0f52307c4e568ff1c38f3bc1c81ac3..4e7b3fa42eabef05f4e1a7647e1fc1d1698a4d64:/src/practical_settings.tex?ds=sidebyside
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bettercrypto.org
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| 310,115,841 | 1,378 |
X-Git-Url: https://git.bettercrypto.org/ach-master.git/blobdiff_plain/73abc462de0f52307c4e568ff1c38f3bc1c81ac3..4e7b3fa42eabef05f4e1a7647e1fc1d1698a4d64:/src/practical_settings.tex?ds=sidebyside
diff --git a/src/practical_settings.tex b/src/practical_settings.tex
index 7e9988b..74588bc 100644
--- a/src/practical_settings.tex
+++ b/src/practical_settings.tex
@@ -680,35 +680,14 @@ Adi?? }
\subsubsection{XMPP / Jabber}
\subsubsection{IRC}
+\input{DBs}
+
\subsection{Intercepting proxy solutions}
\todo{write this!! }
\input{proxy_solutions}
-%\subsection{Database Systems}
-%\subsubsection{MySQL}
-%
-%\paragraph*{my.cnf}\mbox{}\\
-%
-%%Tested with Debian 7.0 and MySQL 5.5
-%
-%\begin{lstlisting}[breaklines]
-%[mysqld]
-%ssl
-%ssl-ca=/etc/mysql/ssl/ca-cert.pem
-%ssl-cert=/etc/mysql/ssl/client-cert.pem
-%ssl-key=/etc/mysql/ssl/client-key.pem
-%ssl-cipher=EECDH+aRSA+AESGCM:EECDH+aRSA+SHA384:EECDH+aRSA+SHA256:EDH+CAMELLIA256:EECDH:EDH+aRSA:+SSLv3:!aNULL:!eNULL:!LOW:!3DES:!MD5:!EXP:!PSK:!SRP:!DSS:!RC4:!SEED:!AES128:!CAMELLIA128:!ECDSA:AES256-SHA
-%\end{lstlisting}
-%
-%After restarting the server run the following query to see if the ssl settings are correct:
-%\begin{lstlisting}[breaklines]
-%show variables like '%ssl%';
-%\end{lstlisting}
-
-
-
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "applied-crypto-hardening"
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https://ctan.mines-albi.fr/macros/latex2e/contrib/univie-ling/tex/univie-ling-poster.cls
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mines-albi.fr
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% File univie-ling-handout
%%
%% Scientific Posters in (Applied) Linguistics at the University of Vienna
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https://people.maths.bris.ac.uk/~matyd/GroupNames/401/C3%5E4sC5_char.tex
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bris.ac.uk
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CC-MAIN-2021-49
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crawl-data/CC-MAIN-2021-49/segments/1637964358774.44/warc/CC-MAIN-20211129134323-20211129164323-00421.warc.gz
| 519,058,860 | 1,273 |
\documentclass[11pt]{amsart}
\begin{document}
\begin{table}
$$
\begin{array}{c|rrrrrrrrrrrrrrrrrrrrr}
\rm class&\rm1&\rm3A&\rm3B&\rm3C&\rm3D&\rm3E&\rm3F&\rm3G&\rm3H&\rm3I&\rm3J&\rm3K&\rm3L&\rm3M&\rm3N&\rm3O&\rm3P&\rm5A&\rm5B&\rm5C&\rm5D\cr
\rm size&1&5&5&5&5&5&5&5&5&5&5&5&5&5&5&5&5&81&81&81&81\cr
\hline
\rho_{1}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1\cr
\rho_{2}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&\zeta_5&\zeta_5^2&\zeta_5^3&\zeta_5^4\cr
\rho_{3}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&\zeta_5^3&\zeta_5&\zeta_5^4&\zeta_5^2\cr
\rho_{4}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&\zeta_5^2&\zeta_5^4&\zeta_5&\zeta_5^3\cr
\rho_{5}&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&1&\zeta_5^4&\zeta_5^3&\zeta_5^2&\zeta_5\cr
\rho_{6}&5&\frac{-5-3\sqrt{-3}}{2}&\frac{1-3\sqrt{-3}}{2}&2&-1&\frac{1-3\sqrt{-3}}{2}&2&\frac{1+3\sqrt{-3}}{2}&-1&\frac{-5+3\sqrt{-3}}{2}&\frac{1+3\sqrt{-3}}{2}&-1&2&-1&2&-1&-1&0&0&0&0\cr
\rho_{7}&5&\frac{1-3\sqrt{-3}}{2}&\frac{-5+3\sqrt{-3}}{2}&-1&2&-1&-1&-1&2&\frac{1+3\sqrt{-3}}{2}&\frac{-5-3\sqrt{-3}}{2}&\frac{1-3\sqrt{-3}}{2}&2&\frac{1+3\sqrt{-3}}{2}&2&-1&-1&0&0&0&0\cr
\rho_{8}&5&\frac{1-3\sqrt{-3}}{2}&-1&2&-1&\frac{-5+3\sqrt{-3}}{2}&2&\frac{-5-3\sqrt{-3}}{2}&-1&\frac{1+3\sqrt{-3}}{2}&-1&\frac{1+3\sqrt{-3}}{2}&-1&\frac{1-3\sqrt{-3}}{2}&-1&2&2&0&0&0&0\cr
\rho_{9}&5&\frac{1+3\sqrt{-3}}{2}&\frac{-5-3\sqrt{-3}}{2}&-1&2&-1&-1&-1&2&\frac{1-3\sqrt{-3}}{2}&\frac{-5+3\sqrt{-3}}{2}&\frac{1+3\sqrt{-3}}{2}&2&\frac{1-3\sqrt{-3}}{2}&2&-1&-1&0&0&0&0\cr
\rho_{10}&5&-1&2&\frac{-5-3\sqrt{-3}}{2}&-1&-1&\frac{-5+3\sqrt{-3}}{2}&-1&-1&-1&2&2&\frac{1+3\sqrt{-3}}{2}&2&\frac{1-3\sqrt{-3}}{2}&\frac{1+3\sqrt{-3}}{2}&\frac{1-3\sqrt{-3}}{2}&0&0&0&0\cr
\rho_{11}&5&-1&\frac{1+3\sqrt{-3}}{2}&-1&2&\frac{1-3\sqrt{-3}}{2}&-1&\frac{1+3\sqrt{-3}}{2}&2&-1&\frac{1-3\sqrt{-3}}{2}&\frac{-5+3\sqrt{-3}}{2}&-1&\frac{-5-3\sqrt{-3}}{2}&-1&2&2&0&0&0&0\cr
\rho_{12}&5&2&2&\frac{1-3\sqrt{-3}}{2}&\frac{1-3\sqrt{-3}}{2}&-1&\frac{1+3\sqrt{-3}}{2}&-1&\frac{1+3\sqrt{-3}}{2}&2&2&-1&-1&-1&-1&\frac{-5-3\sqrt{-3}}{2}&\frac{-5+3\sqrt{-3}}{2}&0&0&0&0\cr
\rho_{13}&5&-1&-1&\frac{1-3\sqrt{-3}}{2}&\frac{1+3\sqrt{-3}}{2}&2&\frac{1+3\sqrt{-3}}{2}&2&\frac{1-3\sqrt{-3}}{2}&-1&-1&2&\frac{-5-3\sqrt{-3}}{2}&2&\frac{-5+3\sqrt{-3}}{2}&-1&-1&0&0&0&0\cr
\rho_{14}&5&\frac{-5+3\sqrt{-3}}{2}&\frac{1+3\sqrt{-3}}{2}&2&-1&\frac{1+3\sqrt{-3}}{2}&2&\frac{1-3\sqrt{-3}}{2}&-1&\frac{-5-3\sqrt{-3}}{2}&\frac{1-3\sqrt{-3}}{2}&-1&2&-1&2&-1&-1&0&0&0&0\cr
\rho_{15}&5&\frac{1+3\sqrt{-3}}{2}&-1&2&-1&\frac{-5-3\sqrt{-3}}{2}&2&\frac{-5+3\sqrt{-3}}{2}&-1&\frac{1-3\sqrt{-3}}{2}&-1&\frac{1-3\sqrt{-3}}{2}&-1&\frac{1+3\sqrt{-3}}{2}&-1&2&2&0&0&0&0\cr
\rho_{16}&5&-1&2&\frac{-5+3\sqrt{-3}}{2}&-1&-1&\frac{-5-3\sqrt{-3}}{2}&-1&-1&-1&2&2&\frac{1-3\sqrt{-3}}{2}&2&\frac{1+3\sqrt{-3}}{2}&\frac{1-3\sqrt{-3}}{2}&\frac{1+3\sqrt{-3}}{2}&0&0&0&0\cr
\rho_{17}&5&2&-1&-1&\frac{-5+3\sqrt{-3}}{2}&2&-1&2&\frac{-5-3\sqrt{-3}}{2}&2&-1&-1&\frac{1+3\sqrt{-3}}{2}&-1&\frac{1-3\sqrt{-3}}{2}&\frac{1-3\sqrt{-3}}{2}&\frac{1+3\sqrt{-3}}{2}&0&0&0&0\cr
\rho_{18}&5&2&-1&-1&\frac{-5-3\sqrt{-3}}{2}&2&-1&2&\frac{-5+3\sqrt{-3}}{2}&2&-1&-1&\frac{1-3\sqrt{-3}}{2}&-1&\frac{1+3\sqrt{-3}}{2}&\frac{1+3\sqrt{-3}}{2}&\frac{1-3\sqrt{-3}}{2}&0&0&0&0\cr
\rho_{19}&5&-1&\frac{1-3\sqrt{-3}}{2}&-1&2&\frac{1+3\sqrt{-3}}{2}&-1&\frac{1-3\sqrt{-3}}{2}&2&-1&\frac{1+3\sqrt{-3}}{2}&\frac{-5-3\sqrt{-3}}{2}&-1&\frac{-5+3\sqrt{-3}}{2}&-1&2&2&0&0&0&0\cr
\rho_{20}&5&-1&-1&\frac{1+3\sqrt{-3}}{2}&\frac{1-3\sqrt{-3}}{2}&2&\frac{1-3\sqrt{-3}}{2}&2&\frac{1+3\sqrt{-3}}{2}&-1&-1&2&\frac{-5+3\sqrt{-3}}{2}&2&\frac{-5-3\sqrt{-3}}{2}&-1&-1&0&0&0&0\cr
\rho_{21}&5&2&2&\frac{1+3\sqrt{-3}}{2}&\frac{1+3\sqrt{-3}}{2}&-1&\frac{1-3\sqrt{-3}}{2}&-1&\frac{1-3\sqrt{-3}}{2}&2&2&-1&-1&-1&-1&\frac{-5+3\sqrt{-3}}{2}&\frac{-5-3\sqrt{-3}}{2}&0&0&0&0\cr
\end{array}
$$
\caption{Character table of $C_3^4{\rtimes}C_5$}
\end{table}
\end{document}
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http://www.robotikk.com/student/thesis.tex
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robotikk.com
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CC-MAIN-2021-39
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crawl-data/CC-MAIN-2021-39/segments/1631780057787.63/warc/CC-MAIN-20210925232725-20210926022725-00638.warc.gz
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\documentclass[english]{report}
\usepackage{babel}
\usepackage{epsfig}
\usepackage{ifthen}
\setlength{\textwidth}{14cm}
\setlength{\textheight}{21cm}
\setlength{\oddsidemargin}{1.2cm}
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%\includeonly{appendix}
\begin{document}
\begin{center}
.
\end{center}
\vspace{3cm}
\begin{figure}[htb]
\begin{center} \epsfig{file=fig/logo.eps,width=3cm} \end{center}
\title{{\vspace{-1.5cm} \huge First-order \\frequency \ds modulation} \\ {\large Dr. Scient thesis}}
\author{\em Mats E. H\o vin}
\date{January\\2000}
\maketitle
\end{figure}
\newpage
.\thispagestyle{empty}
\newpage
\include{preface}
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.\thispagestyle{empty}
\newpage
\addtocounter{page}{-1}
\tableofcontents
\include{introduction}
%\chapter{Traditional \ds analog-to-digital conversion}
\include{ds}
%\chapter{Basic FDSM techniques}
\include{basic}
%\chapter{Extended low-pass techniques}
\include{undersampling}
\include{sc}
\include{triangular}
\include{pointer}
\include{paralell}
%\chapter{Exess noise}
\include{phase}
\include{pattern}
\include{dither}
%\chapter{Summary}
\include{summary}
%\chapter{Publications}
\include{publications}
%\chapter{Appendix}
\include{appendix}
\include{references}
\end{document}
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https://www.math.bgu.ac.il/he/teaching/fall2023/courses/introduction-to-differential-equations-c.tex
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bgu.ac.il
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CC-MAIN-2023-06
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\renewcommand{\href}[2]{\IfBeginWith*{#1}{http}{\hhref{#1}{#2}}{\hhref{\rooturl#1}{#2}}\extrahref{#1}}
\usepackage{polyglossia}
\usepackage{longtable}
%% even in English, we sometimes have Hebrew (as in course hours), and we
%% can't add it in :preamble, since it comes after hyperref
%%\usepackage{bidi}
\setdefaultlanguage{hebrew}
\setotherlanguage{english}
\setmainfont[Script=Hebrew,Ligatures=TeX]{Libertinus Serif}
\newfontfamily{\hebrewfonttt}{Libertinus Serif}
\SepMark{.}
\robustify\hebrewnumeral
\robustify\Hebrewnumeral
\robustify\Hebrewnumeralfinal
% vim: ft=eruby.tex:
\begin{document}
\pagestyle{empty}
\pagenumbering{gobble}
\pagestyle{fancy}
\begin{center}
\huge{המחלקה למתמטיקה}\\[0.1\baselineskip]
\Large{סמסטר 23--2022--א}\\[0.2\baselineskip]
\end{center}
\begin{description}
\item[שם הקורס]
מבוא למשואות דיפרנציאליות ג
\item[מספר קורס]
\LRE{201.1.9271}
\item[עמוד הקורס ברשת]\mbox{}\\
\url{https://www.math.bgu.ac.il//he/teaching/fall2023/courses/introduction-to-differential-equations-c}
\item[מרצה אחראי]
ד``ר מתן זיו-אב,
\nolinkurl{<[email protected]>},
חדר מינוס 123
\item[שעות קבלה]
\url{https://www.math.bgu.ac.il/he/teaching/hours}
\end{description}
\section*{תקציר}
\section*{דרישות והרכב ציון הקורס\footnote{דרישות הקורס יכולות להשתנות במהלך השבועיים הראשונים של הסמסטר, ויש לשים לב להודעות באתר הקורס}}
\section*{נושאי לימוד}
\begin{enumerate}
\item{} משוואות דיפרנציאליות רגילות: פתרונות מפורשים למשוואות דיפרנציאליות מסדר ראשון. משוואות דיפרנציאליות מסדר שני. משוואות דיפרנציאליות מסדר גבוה, מערכות של משוואות דיפרנציאליות רגילות.
\item{} טורי פורייה: חזרה על טורי פונקציות. פיתוחי פורייה ותכונות של טורי פורייה, התכנסות של טורי פורייה, תופעת גיבס. שימושים למשוואות דיפרנציאליות רגילות מחזורית.
\item{} טרנספורם לפלס, שימושים למשוואות דיפרנציאליות רגילות.
\end{enumerate}
% vim: ft=eruby.tex:
\end{document}
% vim: ft=eruby.tex:
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%%
%% Der LaTeX-Begleiter, zweite Auflage (September 2005)
%%
%% Beispiel 2-2-1 von Seite 28.
%%
%% Copyright (C) 2005 Frank Mittelbach, Michel Goossens,
%% Johannes Braams, David Carlisle, and Chris Rowley
%%
%% Uebersetzung: Copyright (C) 2005 Claudia Krysztofiak,
%% Rebecca Stiels und Frank Mittelbach
%%
%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
%%
\documentclass{lb2exa}
\pagestyle{empty}
\setcounter{page}{6}
\setlength\textwidth{173.4372pt}
\StartShownPreambleCommands
\renewcommand\thesection{\Alph{section}}
\StopShownPreambleCommands
\begin{document}
\section{Anders als sonst}
\subsection{Auch hier}
Aufgrund der Standardeinstellungen verändert
sich nicht nur die Abschnittsnummerierung;
auch Gliederungsbefehle niedrigerer Ebenen
übernehmen dieses Format der Abschnittsnummer.
\end{document}
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% \iffalse meta-comment
%
% Copyright 1997, 1998, 2008 2015 LaTeX Project and Peter Breitenlohner.
%
% This file (etex.sty) may be distributed and/or modified under the
% conditions of the LaTeX Project Public License, either version 1.3 of
% this license or (at your option) any later version. The latest
% version of this license is in
% http://www.latex-project.org/lppl.txt
% and version 1.3 or later is part of all distributions of LaTeX
% version 2003/12/01 or later.
%
% This work has the LPPL maintenance status "maintained".
%
% The Current Maintainer of this work is Peter Breitenlohner.
% \fi
\NeedsTeXFormat{LaTeX2e}
\ProvidesPackage{etex}
% [1997/08/12 v0.1 eTeX basic definition package (DPC)]
% [1998/03/26 v2.0 eTeX basic definition package (PEB)]
% [2015/03/02 v2.1 eTeX basic definition package (PEB,DPC)]
% [2015/07/06 v2.2 eTeX basic definition package (PEB,DPC)]
[2015/07/08 v2.3 eTeX basic definition package (PEB,DPC)]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% A basic interface to some etex primitives, closely modeled on
%% etex.src and etexdefs.lib provided by the core etex team.
%% The etex.src `module' system is not copied here, the standard
%% LaTeX package option mechanism is used instead,
%% however the package options match the module names.
%% (Currently grouptypes, interactionmodes, nodetypes, iftypes.)
%% The individual type names are different too: We use, e.g.,
%%
%% `\bottomleveltype' and `\simplegrouptype' instead of
%% `\grouptypes{bottomlevel}' and `\grouptypes{simple}'.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Other Comments...
%% The names of the `interactionmodes' are not too good.
%% In particular \scroll and \batch are likely to clash with existing
%% uses. These names have been changed into \batchinteractionmode,
%% \scrollinteractionmode etc.
%% Similarly, the names of the `groupetypes' have been changed, in
%% particular \mathgroup would conflict with the LaTeX kernel.
%% \etex logo could have the same trick as \LaTeXe to pick up a bold
%% epsilon when needed. (Not done here, I hate wasting tokens on logos.)
%% This version does have a \m@th not in the original.
%% The \globcountvector, \loccountvector, etc. allocation macros are
%% not (yet) implemented.
%% Currently if run on a standard TeX, the package generates an error.
%% Perhaps it should instead load some code to try to fake
%% the new etex primitives in that case???
%% Likewise, the package generates an error when used with e-TeX V 1
%% The etex.src language mechanism is not copied here. That facility
%% does not use any of the etex features. LaTeX should be customised
%% using the same hyphen.cfg mechanism as for a format built with a
%% standard TeX.
%% David Carlisle
%% Upgraded for e-TeX V 2.0
%% Peter Breitenlohner
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\ifx\eTeXversion\@undefined
\PackageError{etex}
{This package may only be run using an\MessageBreak
etex in extended mode}
{Perhaps you forgot the `*' when making the format with (e)initex.%
}
\fi
\ifnum\eTeXversion<2
\PackageError{etex}
{This package requires e-TeX V 2}
{You are probably using the obsolete e-TeX V 1.%
}
\fi
% 2.2
% Check if the new latex 2015/01/01 allocation is already using
% extended reisters. If so it is too late to change allocation scheme.
% Older versions of LaTeX would have given an error when the classic
% TeX registers were all allocated, but newer formats allocate from
% the extended range, so usually this package is not needed.
\@tempswafalse
\ifnum\count10>\@cclv\@tempswatrue\else
\ifnum\count11>\@cclv\@tempswatrue\else
\ifnum\count12>\@cclv\@tempswatrue\else
\ifnum\count13>\@cclv\@tempswatrue\else
\ifnum\count14>\@cclv\@tempswatrue\else
\ifnum\count15>\@cclv\@tempswatrue
\fi\fi\fi\fi\fi\fi
\if@tempswa
\PackageWarningNoLine{etex}{%
Extended allocation already in use.\MessageBreak
etex.sty code will not be used.\MessageBreak
To force etex package to load, add\MessageBreak
\string\RequirePackage{etex}\MessageBreak
at the start of the document}
\expandafter\endinput\fi
% End of 2.2 addition.
% 2.3 move option handling after the above error checks.
\DeclareOption{grouptypes}{\catcode`\G=9}
\DeclareOption{interactionmodes}{\catcode`\I=9}
\DeclareOption{nodetypes}{\catcode`\N=9}
\DeclareOption{iftypes}{\catcode`\C=9}
\DeclareOption{localalloclog}{\let\et@xwlog\wlog} % the default
\DeclareOption{localallocnolog}{\let\et@xwlog\@gobble} % be quiet
\DeclareOption{localallocshow}{\let\et@xwlog\typeout} % debugging
% End of 2.3 addition.
\def\eTeX{%
$\m@th\varepsilon$-\TeX}
\def\tracingall{%
\tracingcommands\thr@@ % etex
\tracingstats\tw@
\tracingpages\@ne
\tracinglostchars\tw@ % etex
\tracingmacros\tw@
\tracingparagraphs\@ne
\tracingrestores\@ne
\tracinggroups\@ne % etex
\tracingifs\@ne % etex
\tracingscantokens\@ne % etex
\tracingnesting\@ne % etex
\tracingassigns\@ne % etex
\errorcontextlines\maxdimen
\showoutput}
\def\loggingall{%
\tracingall
\tracingonline\z@}
\def\tracingnone{%
\tracingonline\z@
\showboxdepth\m@ne
\showboxbreadth\m@ne
\tracingoutput\z@
\errorcontextlines\m@ne
\tracingassigns\z@
\tracingnesting\z@
\tracingscantokens\z@
\tracingifs\z@
\tracinggroups\z@
\tracingrestores\z@
\tracingparagraphs\z@
\tracingmacros\z@
\tracinglostchars\@ne
\tracingpages\z@
\tracingstats\z@
\tracingcommands\z@}
%% Register allocation
%% We have to adjust the Plain TeX / LaTeX register allocation counts
%% for our slightly modified book-keeping, but first we allocate our
%% insertion counter \et@xins, because \insc@ount of Plain TeX / LaTeX
%% will be used differently.
\newcount\et@xins
\advance\count10 by 1 % \count10=23 % allocates \count registers 23, 24, ...
\advance\count11 by 1 % \count11=10 % allocates \dimen registers 10, 11, ...
\advance\count12 by 1 % \count12=10 % allocates \skip registers 10, 11, ...
\advance\count13 by 1 % \count13=10 % allocates \muskip registers 10, 11, ...
\advance\count14 by 1 % \count14=10 % allocates \box registers 10, 11, ...
\advance\count15 by 1 % \count15=10 % allocates \toks registers 10, 11, ...
\advance\count16 by 1 % \count16=0 % allocates input streams 0, 1, ...
\advance\count17 by 1 % \count17=0 % allocates output streams 0, 1, ...
\advance\count18 by 1 % \count18=4 % allocates math families 4, 5, ...
\advance\count19 by 1 % \count19=0 % allocates \language codes 0, 1, ...
\et@xins=\insc@unt % \et@xins=255 % allocates insertions 254, 253, ...
%% To ensure working in LaTeX 2015 release do define \newcount etc
%% with their pre 2015 LaTeX definitions
\def\newcount{\alloc@0\count\countdef\insc@unt}
\def\newdimen{\alloc@1\dimen\dimendef\insc@unt}
\def\newskip{\alloc@2\skip\skipdef\insc@unt}
\def\newmuskip{\alloc@3\muskip\muskipdef\@cclvi}
\def\newbox{\alloc@4\box\chardef\insc@unt}
\def\newtoks{\alloc@5\toks\toksdef\@cclvi}
\def\newread{\alloc@6\read\chardef\sixt@@n}
\def\newwrite{\alloc@7\write\chardef\sixt@@n}
\def\new@mathgroup{\alloc@8\mathgroup\chardef\sixt@@n}
\let\newfam\new@mathgroup
\def\newlanguage{\alloc@9\language\chardef\@cclvi}
%% When the normal register pool for \count, \dimen, \skip, \muskip,
%% \box, or \toks registers is exhausted, we switch to the extended pool.
\def\alloc@#1#2#3#4#5%
{\ifnum\count1#1<#4% make sure there's still room
\allocationnumber\count1#1
\global\advance\count1#1\@ne
\global#3#5\allocationnumber
\wlog{\string#5=\string#2\the\allocationnumber}%
\else\ifnum#1<6
\begingroup \escapechar\m@ne
\expandafter\alloc@@\expandafter{\string#2}#5%
\else\errmessage{No room for a new #2}\fi\fi
}
%% The \expandafter construction used here allows the generation of
%% \newcount and \globcount from #1=count.
\def\alloc@@#1#2%
{\endgroup % restore \escapechar
\wlog{Normal \csname#1\endcsname register pool exhausted,
switching to extended pool.}%
\global\expandafter\let
\csname new#1\expandafter\endcsname
\csname glob#1\endcsname
\csname new#1\endcsname#2%
}
%% We do change the LaTeX definition of \newinsert
\def\newinsert#1{% make sure there's still room for ...
\ch@ck0\et@xins\count{% ... a \count, ...
\ch@ck1\et@xins\dimen{% ... \dimen, ...
\ch@ck2\et@xins\skip{% ... \skip, ...
\ch@ck4\et@xins\box{% ... and \box register
\global\advance\et@xins\m@ne
\unless\ifnum\insc@unt<\et@xins \global\insc@unt\et@xins \fi
\allocationnumber\et@xins
\global\chardef#1\allocationnumber
\wlog{\string#1=\string\insert\the\allocationnumber}}}}}}
\def\ch@ck#1#2#3#4%
{\ifnum\count1#1<#2#4\else\errmessage{No room for a new #3}\fi}
%% And we define \reserveinserts, so that you can say \reserveinserts{17}
%% in order to reserve room for up to 17 additional insertion classes, that
%% will not be taken away by \newcount, \newdimen, \newskip, or \newbox.
\outer\def \reserveinserts#1%
{\global\insc@unt\numexpr \et@xins \ifnum#1>\z@ -#1\fi \relax}
% Now, we define \globcount, \globbox, etc., so that you can say
% \globcount\foo and \foo will be defined (with \countdef) to be the
% next count register from the vastly larger but somewhat less efficient
% extended register pool. We also define \loccount, etc., but these
% register definitions are local to the current group.
\count260=277 % globally allocates \count registers 277, 278, ...
\count261=256 % globally allocates \dimen registers 256, 257, ...
\count262=256 % globally allocates \skip registers 256, 257, ...
\count263=256 % globally allocates \muskip registers 256, 257, ...
\count264=256 % globally allocates \box registers 256, 257, ...
\count265=256 % globally allocates \toks registers 256, 257, ...
\count266=1 % globally allocates \marks classes 1, 2, ...
\count270=32768 % locally allocates \count registers 32767, 32766, ...
\count271=32768 % ditto for \dimen registers
\count272=32768 % ditto for \skip registers
\count273=32768 % ditto for \muskip registers
\count274=32768 % ditto for \box registers
\count275=32768 % ditto for \toks registers
\count276=32768 % ditto for \marks classes
% \count registers 256-259 and 267-269 are not (yet) used
% \def \et@xglob #1#2#3#4% <offset>, <type>, <method>, <register>
% \def \et@xloc #1#2#3#4% <offset>, <type>, <method>, <register>
\def \globcount {\et@xglob 0\count \countdef}
\def \loccount {\et@xloc 0\count \countdef}
\def \globdimen {\et@xglob 1\dimen \dimendef}
\def \locdimen {\et@xloc 1\dimen \dimendef}
\def \globskip {\et@xglob 2\skip \skipdef}
\def \locskip {\et@xloc 2\skip \skipdef}
\def \globmuskip {\et@xglob 3\muskip \muskipdef}
\def \locmuskip {\et@xloc 3\muskip \muskipdef}
\def \globbox {\et@xglob 4\box \mathchardef}
\def \locbox {\et@xloc 4\box \mathchardef}
\def \globtoks {\et@xglob 5\toks \toksdef}
\def \loctoks {\et@xloc 5\toks \toksdef}
\def \globmarks {\et@xglob 6\marks \mathchardef}
\def \locmarks {\et@xloc 6\marks \mathchardef}
\let\newmarks=\globmarks %% this used to be \newmark for e-TeX V 1.1
\def\et@xglob#1#2#3#4%
{\et@xchk#1#2{% make sure there's still room
\allocationnumber=\count26#1%
\global\advance\count26#1\@ne
\global#3#4\allocationnumber
\wlog{\string#4=\string#2\the\allocationnumber}}%
}
\def\et@xloc#1#2#3#4%
{\et@xchk#1#2{% make sure there's still room
\advance\count27#1by\m@ne
\allocationnumber=\count27#1%
#3#4=\allocationnumber
\et@xwlog{\string#4=\string#2\the\allocationnumber\space(local)}}%
}
%% The allocation messages for local allocations use \et@xwlog, such that
%% these messages can easily be switched on/off
\let\et@xwlog=\wlog
\def\et@xchk#1#2#3%
{\ifnum\count26#1<\count27#1 #3\else\errmessage{No room for a new #2}\fi}
% Next we define \globcountblk, \loccountblk, etc., so that you can
% say \globcountblk\foo{17} and \foo will be defined (with \mathchardef)
% as the first (the zeroth?) of a block of 17 consecutive registers.
% Thus the user is intended to reference elements <\foo+0> to <\foo+n-1>,
% where n is the length of the block allocated.
% \def \et@xgblk #1#2#3#4% <offset>, <type>, <register>, <size>
% \def \et@xlblk #1#2#3#4% <offset>, <type>, <register>, <size>
\def\globcountblk {\et@xgblk 0\count }
\def\loccountblk {\et@xlblk 0\count }
\def\globdimenblk {\et@xgblk 1\dimen }
\def\locdimenblk {\et@xlblk 1\dimen }
\def\globskipblk {\et@xgblk 2\skip }
\def\locskipblk {\et@xlblk 2\skip }
\def\globmuskipblk {\et@xgblk 3\muskip }
\def\locmuskipblk {\et@xlblk 3\muskip }
\def\globboxblk {\et@xgblk 4\box }
\def\locboxblk {\et@xlblk 4\box }
\def\globtoksblk {\et@xgblk 5\toks }
\def\loctoksblk {\et@xlblk 5\toks }
\def\globmarksblk {\et@xgblk 6\marks }
\def\locmarksblk {\et@xlblk 6\marks }
% \def\et@xchkblk#1#1#3#4% <offset>, <type>, <size>, <action>
\def\et@xgblk#1#2#3#4%
{\et@xchkblk#1#2{#4}% make sure there's still room
{\allocationnumber\count26#1%
\global\advance\count26#1by#4%
\global\mathchardef#3\allocationnumber
\wlog{\string#3=\string#2blk{\number#4} at
\the\allocationnumber}%
}%
}
\def\et@xlblk#1#2#3#4%
{\et@xchkblk#1#2{#4}% make sure there's still room
{\advance\count27#1-#4%
\allocationnumber\count27#1%
\mathchardef#3\allocationnumber
\et@xwlog{\string#3=\string#2blk{\number#4} at
\the\allocationnumber\space(local)}%
}%
}
\def\et@xchkblk#1#2#3#4%
{\ifnum#3<\z@
\errmessage{Negative register block size \number#3}%
\else\ifnum\numexpr\count26#1+#3>\count27#1%
\errmessage{No room for new #2block of size \number#3}%
\else #4\fi \fi
}
\catcode`\G=14
\catcode`\I=14
\catcode`\N=14
\catcode`\C=14
\ProcessOptions
%% Declare names for `grouptypes'
G \chardef \bottomleveltype = 0 % for the outside world
G \chardef \simplegrouptype = 1 % for local structure only
G \chardef \hboxgrouptype = 2 % for `\hbox{}'
G \chardef \adjustedhboxgrouptype = 3 % for `\hbox{}' in vertical mode
G \chardef \vboxgrouptype = 4 % for `\vbox{}'
G \chardef \vtopgrouptype = 5 % for `\vtop{}'
G \chardef \aligngrouptype = 6 % for `\halign{}', `\valign{}'
G \chardef \noaligngrouptype = 7 % for `\noalign{}'
G \chardef \outputgrouptype = 8 % for output routine
G \chardef \mathgrouptype = 9 % for, e.g, `^{}'
G \chardef \discgrouptype = 10 % for `\discretionary{}{}{}'
G \chardef \insertgrouptype = 11 % for `\insert{}', `\vadjust{}'
G \chardef \vcentergrouptype = 12 % for `\vcenter{}'
G \chardef \mathchoicegrouptype = 13 % for `\mathchoice{}{}{}{}'
G \chardef \semisimplegrouptype = 14 % for `\begingroup...\endgroup'
G \chardef \mathshiftgrouptype = 15 % for `$...$'
G \chardef \mathleftgrouptype = 16 % for `\left...\right'
%% Declare names for `interactionmodes'
I \chardef \batchinteractionmode = 0 % omits all stops and omits terminal output
I \chardef \nonstopinteractionmode = 1 % omits all stops
I \chardef \scrollinteractionmode = 2 % omits error stops
I \chardef \errorstopinteractionmode = 3 % stops at every opportunity to interact
%% Declare names for `nodetypes'
N \chardef \charnode = 0 % character nodes
N \chardef \hlistnode = 1 % hlist nodes
N \chardef \vlistnode = 2 % vlist nodes
N \chardef \rulenode = 3 % rule nodes
N \chardef \insnode = 4 % insertion nodes
N \chardef \marknode = 5 % a mark node
N \chardef \adjustnode = 6 % an adjust node
N \chardef \ligaturenode = 7 % a ligature node
N \chardef \discnode = 8 % a discretionary node
N \chardef \whatsitnode = 9 % special extension nodes
N \chardef \mathnode = 10 % a math node
N \chardef \gluenode = 11 % node that points to a glue specification
N \chardef \kernnode = 12 % a kern node
N \chardef \penaltynode = 13 % a penalty node
N \chardef \unsetnode = 14 % an unset node
N \chardef \mathsnodes = 15 % nodes that occur only in maths mode
%% Declare names for `iftypes'
C \chardef \charif = 1 % \if
C \chardef \catif = 2 % \ifcat
C \chardef \numif = 3 % \ifnum
C \chardef \dimif = 4 % \ifdim
C \chardef \oddif = 5 % \ifodd
C \chardef \vmodeif = 6 % \ifvmode
C \chardef \hmodeif = 7 % \ifhmode
C \chardef \mmodeif = 8 % \ifmmode
C \chardef \innerif = 9 % \ifinner
C \chardef \voidif = 10 % \ifvoid
C \chardef \hboxif = 11 % \ifhbox
C \chardef \vboxif = 12 % \ifvbox
C \chardef \xif = 13 % \ifx
C \chardef \eofif = 14 % \ifeof
C \chardef \trueif = 15 % \iftrue
C \chardef \falseif = 16 % \iffalse
C \chardef \caseif = 17 % \ifcase
C \chardef \definedif = 18 % \ifdefined
C \chardef \csnameif = 19 % \ifcsname
C \chardef \fontcharif = 20 % \iffontchar
\catcode`\G=11
\catcode`\I=11
\catcode`\N=11
\catcode`\C=11
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\title{After Nice attack — ISIS and Pegida want a clash of civilisations — lets resist their project}
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\author{Andrew Flood}
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At least 84 dead in Nice as a truck is deliberately driven through the huge crowds marking the anniversary of the French Revolution. The driver who was shot be police after plowing through the crowd for 1.2km is reported to be a French citizen of Tunisian descent. The method of attack which is very similar to one advocated in the ISIS magazine strongly suggests an ISIS \Slash{}Daesh attack.
The massive ISIS bombings in Baghdad that killed nearly 300 this month on 3\textsuperscript{rd} July followed the ISIS defeat in Fallujah and this attack comes on the eve of their defeat in Manbij. There the Kurdish YPG as part of the SDF are engaged in street fighting in the city center and the next stage of their offensive should cut ISiS off from its resupply routes into Turkey. This is probably why I’m the Turkish state has been making noises about normalizing relations with the Assad regime, their next best hope of crushing the revolution in Rojava.
ISIS also lost their top commander, Omar al-Shishani, again their worst terror attacks often follow military setbacks. There are two reasons for this, the first is simply to have something to distract their international followers from their defeats. But the second is because they badly need new recruits to travel to the region and such attacks in Europe are designed with the hope of triggering a racist backlash.
So far after every attack the European far right have tried to give ISIS what it wants but have largely failed to generate the racist programs they also desire. But we can be sure in the days ahead they will try again through Pegida and other front organizations. Trump will play the same game the US. They must not succeed.
It’s probable that the attack on the celebrations of the French Revolution were not just about the date but also the opportunity of crowds on the street. Nevertheless as ISIS sufferers defeat after defeat in Syria at the hands of today’s revolutionaries there it’s useful to understand the symbolism of that date as part of centuries old struggle between those desiring true human freedom for all, and the regressive far right ideologies from ISIS to Pegida.
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The Anarchist Library (Mirror)
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Anti-Copyright
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Andrew Flood
After Nice attack — ISIS and Pegida want a clash of civilisations — lets resist their project
July 15, 2016
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Retrieved on 13\textsuperscript{th} August 2021 from \href{http://www.anarkismo.net/article/29439}{www.anarkismo.net}
\bigskip
\textbf{usa.anarchistlibraries.net}
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\begin{document}
\title{Integral Cayley graphs generated by distance sets in vector spaces over finite fields}
\author{
Nguyen Ngoc Dai\\
\small School of Applied Mathematics and Informatics \\[-0.8ex]
\small Hanoi University of Science and Technology\\[-0.8ex]
\small\tt [email protected]\\
\and
Nguyen Minh Hai\\
\small Faculty of Mathematics, Mechanics and Informatics\\[-0.8ex]
\small Hanoi University of Science\\[-0.8ex]
\small Vietnam National University, Hanoi\\[-0.8ex]
\small\tt [email protected]\\
\and
Do Duy Hieu\\
\small Faculty of Mathematics, Mechanics and Informatics\\[-0.8ex]
\small Hanoi University of Science\\[-0.8ex]
\small Vietnam National University, Hanoi\\[-0.8ex]
\small\tt [email protected]\\
\and
Le Anh Vinh\footnote{This research was supported by Vietnam National Foundation for Science and Technology Development grant 102.01-2012.29.}\\
\small University of Education\\[-0.8ex]
\small Vietnam National University, Hanoi\\[-0.8ex]
\small\tt [email protected]
}
\date{\empty}
\maketitle
\begin{abstract}
Si Li and the fourth listed author (2008) considered unitary graphs attached to the vector spaces over finite rings using an analogue of the Euclidean distance. These graphs are shown to be integral when the cardinality of the ring is odd or the dimension is even. In this paper, we show that the statement also holds for the remaining case: the cardinality of the ring is even and the dimension is odd, by showing a sufficient condition for Cayley graphs generated by distance sets in vector spaces over finite fields to be integral.
\end{abstract}
\section{Introduction}
Let $\Gamma$ be an additive group. For $S \subseteq \Gamma$, $0 \not \in S$, and
$S^{- 1} =\{- s : s \in S\}= S$, the Cayley graph $G = C (\Gamma, S)$ is the
undirected graph where the vertex set $V (G) = \Gamma$ and the edge set $E (G) =\{(a,
b) : a - b \in S\}$. The Cayley graph $G = C (\Gamma, S)$ is regular of degree
$|S|$. For any positive integer $n > 1$, let $\mathbb{Z}_n =\mathbb{Z}/n\mathbb{Z}$ be the finite cyclic ring of $n$ elements. Notice that one can identify $\mathbb{Z}_n$
with $\{0, 1, ..., n-1\}$. The unitary Cayley graph $X_n = C (\mathbbm{Z}_n,
\mathbbm{Z}_n^{*})$ is defined by the additive group of the ring $\mathbbm{Z}_n$ of integers modulo
$n$ and the multiplicative group $\mathbbm{Z}_n^{*}$ of its units. So $X_n$ has vertex set $V
(X_n) = \mathbbm{Z}_n =\{0, 1, \ldots, n - 1\}$ and edge set \[E (X_n) =\{(a, b) : a, b
\in \mathbbm{Z}_n, \gcd (a - b, n) = 1\}.\] The graph $X_n$ is regular of degree $|\mathbbm{Z}_n^{*} |
= \phi (n)$, where $\phi (n)$ denotes the Euler function. Unitary Cayley
graphs are highly symmetric and have some remarkable properties connecting
graph theory and number theory. More information about the unitary Cayley
graphs can be found in Berrizbeitia and Giudici \cite{berrizbeitia}, Dejter and Giudici \cite{dejter},
Fuchs \cite{fuchs 1}, and Klotz and Sander \cite{klotz}.
In \cite{si-vinh}, Si Li and the fourth listed author studied higher dimensional unitary Cayley graphs over
$\mathbbm{Z}_n^d$ using an analogue of the Euclidean distance. Precisely, they
defined for positive integers $n$ and $d$ the unitary Euclidean graph $T_n^{(d)}$ with vertex
set $V (T_n^{(d)}) = \mathbbm{Z}_n^d$ and edge set
\begin{equation}
E (T_n^{(d)}) = \left\{ (a, b) : d (a, b) = \sum_{i = 1}^d (a_i - b_i)^2 \in
\mathbbm{Z}_n^{*} \right\} .
\end{equation}
Note that the Euclidean graph $E_R^{(d)}(r)$ over a finite ring $R$, $r \in R$, is the Cayley graph with vertex set $V(E_R^{(d)}(r)) = R^d$ and the edge set
\begin{equation*}
E (E_R^{(d)}(r)) = \left\{ (a, b) : d (a, b) = \sum_{i = 1}^d (a_i - b_i)^2 = r\right\} .
\end{equation*}
In \cite{medrano 1}, Medrano, Myers, Stark and Terras studied the spectrum of the Euclidean graphs over finite fields and showed that these graphs are asymptotically Ramanujan graphs. In \cite{medrano 2}, these authors studied the same problem for the Euclidean graphs over rings $\mathbbm{Z}_q$ for an odd prime power $q$. They showed that over rings, except for the smallest case, the graphs (with unit distance parameter) are not (asymptotically) Ramanujan.
In \cite{bannai}, Bannai, Shimabukuro and Tanaka showed that the Euclidean graphs over finite fields are always asymptotically Ramanujan for a more general setting (i.e. they replace the Euclidean distance by nondegenerated quadratic forms). The fourth listed author recently applied these results to several interesting combinatorial problems, for example to the Erd\"os distance problem \cite{vinh 1}, Szemeredi-Trotter type theorem and sum-product estimate \cite{vinh 2}, singular matrices with restricted entries over finite fields \cite{vinh-dm} and explicit constructions of existentially closed graphs \cite{vinh-dam}.
Si Li and the fourth listed author \cite{si-vinh} showed that the spectrum of unitary finite-Euclidean graphs consists entirely of integers when $n$ is odd or the dimension $d$ is even. This property seems to be amazingly widespread among Cayley graphs on abelian groups. One of the first papers in this direction is due to L. Lov\'asz \cite{lovasz 1}, who proved that all Cayley graphs, (cube-like) graphs, on $\mathbbm{Z}_2^d$ are integral where $\mathbbm{Z}_n$ is the ring of integers modulo $n$. In this paper, we extend this result by showing a sufficient condition for Cayley graphs generated by distance sets in vector spaces over finite fields are integral. We would like to remark that our result is closely related to the result of W. So in \cite{so}.
\section{Cayley graphs generated by distance sets}
For any $\mathcal{U}\subseteq \mathbb{Z}_n$, the distance graph generated by $\mathcal{U}$ over $\mathbb{Z}_n^l$, $G(\mathbb{Z}_n^l,\mathcal{U})$, is the undirected graph that has the vertex set $V(G)=\mathbb{Z}_n^l$ and
the edge set
\[E(G)=\{(\tmmathbf{a},\tmmathbf{b}): d(\tmmathbf{a},\tmmathbf{b})= \sum_{i=1}^{l}(a_i-b_i)^2 \in \mathcal{U}\}.\]
Let
\[S_{l,n}(\mathcal{U})=\{\tmmathbf{a}\in \mathbb{Z}_n^l: d(\tmmathbf{a},\tmmathbf{0}) = a_1^2 + \ldots + a_l^2 \in \mathcal{U} \}, \]
then the graph $G(\mathbb{Z}_n^l,\mathcal{U})$ is the Cayley graph $C(\mathbb{Z}_n^l,S_{l,n}(\mathcal{U}))$.
Recall that the eigenvalues of Cayley graphs of abelian groups can be computed easily in terms of the characters of the group. This is an old result and easy to prove: the characters of a finite abelian group are eigenfunctions for the convolution operators on the group (see also \cite{medrano 1}). This implies that the eigenvalues of the graph $G(\mathbb{Z}_n^l,\mathcal{U})$ are all the numbers
\begin{equation}
\lambda_{\tmmathbf{b}} = \sum_{\tmmathbf{x}\in S_{l,n}(\mathcal{U})} e_n (^t\tmmathbf{b} \cdot \tmmathbf{x}),
\end{equation}
where $\tmmathbf{b} \in \mathbbm{Z}_n^{l}$ and the exponential $e_n (z) = \exp \{2 \pi i z / n\}$.
In particular, the unitary Euclidean graph $T_n^{(d)}$ is the Cayley graph generated by the distance set $\mathcal{U} = \mathbb{Z}_n^{\ast}$. Using properties of Gauss and Ramanujan's sums over cyclic rings, Si Li and the fourth listed author \cite{si-vinh} showed that $T_n^{(d)}$ is an integral graph if $n$ is odd or $d$ is even.
\begin{theorem} \label{old-thm} (\cite[Theorem 3.6]{si-vinh})
If $n$ is an odd integer or $d$ is an even integer then all eigenvalues of the unitary Euclidean graph $T_n^{(d)}$ are integers.
\end{theorem}
They conjectured that the same result also holds in general.
\begin{conjecture} \label{old-conj} (\cite[Conjecture 3.7]{si-vinh})
For any positive integers $n$ and $d$ all eigenvalues of the unitary Euclidean graph $T_n^{(d)}$ are integers.
\end{conjecture}
From Theorem \ref{old-thm}, the remaining open case of Conjecture \ref{old-conj} is: $n$ even and $d$ odd. In this paper, we will give a simple proof of this conjecture. More precisely, we have the following sufficient condition for Cayley graphs generated by distance sets in vector spaces over finite fields are integral. We conjecture that the given condition is also the necessary condition.
\begin{theorem} \label{main}
Suppose that $d_1,d_2,...,d_r$ are divisors of $n$. Let
\begin{equation} \label{e2} \mathcal{U}=\cup_{j=1}^{r}G_n(d_j),\end{equation}
where \[G_n(d_j)=\{ y \in \mathbb{Z}_n : gcd(n,y)=d_j \}.\] Then we have
$\lambda_{\mathcal{U}}(\tmmathbf{b})\in\mathbb{Z}$
for all $\tmmathbf{b} \in \mathbbm{Z}_n^l$.
\end{theorem}
\begin{proof}
Let $d$ be a divisor of $n$. Note that $\tmmathbf{x}\in S_{l,n}(G_n(d))$ then $k\cdot \tmmathbf{x}\in S_{l,n}(G_n(d))$ for any $k\in \mathbb{Z}_n^*$, and if $\tmmathbf{x}\not= \tmmathbf{x'}$ then $k\cdot\tmmathbf{x}\not=k\cdot \tmmathbf{x'}$ for every $k\in \mathbb{Z}_n^*$. Since $S_{d,n}(G_n(d))$ is finite, it follows that
\begin{equation*}
kS_{l,n}(G_n(d)) \equiv S_{l,n}(G_n(d)).
\end{equation*}
Hence, $k S_{l,n}(\mathcal{U}) \equiv S_{l,n}(\mathcal{U})$ when $\mathcal{U}$ is of the form (\ref{e2}).
We write $n = p_1^{r_1}...p_t^{r_t}$ for the prime decomposition of $n$. For any nonempty subset $I \subseteq \{1,...,t\}$, set \[ p_I=\prod_{i\in I} p_i \,\,\, \mbox{and}\,\,\, n_I=n/p_I.\]
We have
\begin {eqnarray*}
\lambda_{\mathcal{U}}(\tmmathbf{b})&=&\sum_{\tmmathbf{x}\in S_{l,n}(\mathcal{U})}e_n(^t\tmmathbf{b}\cdot \tmmathbf{x})\\
&=&\frac{1}{|\mathbb{Z}_n^*|}\sum_{k\in \mathbb{Z}_n^*}\sum_{\tmmathbf{x}\in kS_{l,n}(\mathcal{U})}e_n(^t\tmmathbf{b}\cdot\tmmathbf{x})\\
&=&\frac{1}{|\mathbb{Z}_n^*|}\sum_{\tmmathbf{x}\in S_{l,n}(\mathcal{U})}\sum_{k\in \mathbb{Z}_n^*}e_n(k^t\tmmathbf{b}\cdot\tmmathbf{x})\\
&=&\frac{1}{|\mathbb{Z}_n^*|}\sum_{\tmmathbf{x}\in S_{l,n}(\mathcal{U}),^t\tmmathbf{b}\cdot\tmmathbf{x}=0}|\mathbb{Z}_n^*|+\frac{1}{|\mathbb{Z}_n^*|}\sum_{\tmmathbf{x}\in S_{l,n}(\mathcal{U}),^t\tmmathbf{b}\cdot\tmmathbf{x}\not=0}\sum_{k\in \mathbb{Z}_n^*}e_n(k^t\tmmathbf{b}\cdot\tmmathbf{x})\\
&=& \#\{ \tmmathbf{x}\in S_{l,n}(\mathcal{U}),^t\tmmathbf{b}\cdot\tmmathbf{x}=0\} \\ && +\frac{1}{|\mathbb{Z}_n^*|}\sum_{\tmmathbf{x}\in S_{l,n}(\mathcal{U}),^t\tmmathbf{b}\cdot\tmmathbf{x}\not=0} \sum_{k\in\mathbb{Z}_n}e_n(k^t\tmmathbf{b}\cdot\tmmathbf{x})\\
&& + \frac{1}{|\mathbb{Z}_n^*|}\sum_{\tmmathbf{x}\in S_{l,n}(\mathcal{U}),^t\tmmathbf{b}\cdot\tmmathbf{x}\not=0}\sum_{I\subseteq\{1,...,t\}}(-1)^{|I|}\sum_{s\in\mathbb{Z}_{n_I}}e_n(p_Is^t\tmmathbf{b}\cdot\tmmathbf{x}).
\end{eqnarray*}
Since $^t\tmmathbf{b}\cdot\tmmathbf{x}\not=0$, from the orthogonality of exponential sums, we have
\[\sum_{s\in\mathbb{Z}_{n_I}}e_n(p_Is^t\tmmathbf{b}\cdot\tmmathbf{x})=1\]
if $n=p_1...p_t$ and $I=\{1,2,...,t\}$; and
\[\sum_{s\in\mathbb{Z}_{n_I}}e_n(p_Is^t\tmmathbf{b}\cdot\tmmathbf{x})=0\]
otherwise.
This implies that $\lambda_{\mathcal{U}}(\tmmathbf{b})\in \mathbb{Q}$ for any $\tmmathbf{b}\in\mathbb{Z}_n^l$.
Furthermore, the characteristic polynomial of the adjacency matrix of $G(\mathbb{Z}_n^l,\mathcal{U})$ is monic with integer coefficients, so $\lambda_{\mathcal{U}}(\tmmathbf{b})\in\mathbb{Z}$ for any $\tmmathbf{b}\in\mathbb{Z}_n^l$. This concludes the proof of the theorem.
\end{proof}
Note that Theorem \ref{main} implies Conjecture \ref{old-conj} by setting $r = 1$ and $d_1 = 1$.
\begin{remark}
For any distance function $f : \mathbb{Z}_n^l \times \mathbb{Z}_n^l \rightarrow \mathbb{Z}_n$ and the distance set $\mathcal{U} \subset \mathbb{Z}_n$, define the generating set
\[S_{l,n}(f,\mathcal{U})=\{\tmmathbf{a}\in \mathbb{Z}_n^l: f(\tmmathbf{a},\tmmathbf{0}) \in \mathcal{U} \}.\]
The above proof of Theorem \ref{main} works transparently for any distance function $f$ that satisfies the condition $k S_{l,n}(\mathcal{U}) \equiv S_{l,n}(\mathcal{U})$ when $\mathcal{U}$ is of the form (\ref{e2}). For example, Theorem \ref{main} still holds if we use the distance function $d(\tmmathbf{a},\tmmathbf{b})= \sum_{i=1}^{l}(a_i-b_i)^h$ for any $h \ge 1$. Let $l = h = 1$, we obtain alternative proofs of \cite[Theorem 16]{klotz} and \cite[Corollary 4.5]{so}.
\end{remark}
\begin{thebibliography}{99}
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\bibitem{mccarthy} P. J. McCarthy, \textit{Introduction to arithmetical functions}, Universitext. Springer-Verlag, New York, 1994.
\bibitem{medrano 1} A. Medrano, P. Myers, H. M. Stark and A. Terras, Finite analogues of Euclidean space, \textit{Journal of Computational and Applied Mathematics}, \textbf{68} (1996), 221--238.
\bibitem{medrano 2} A. Medrano, P. Myers, H. M. Stark and A. Terras, Finite Euclidean graphs over rings,
\textit{Proceedings of the American Mathematics Society}, \textbf{126} (1998), 701--710.
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W. So, Integral circulant graphs, \textit{Discrete Mathematics}, \textbf{306} (2006), 153-158.
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L. A. Vinh, Explicit Ramsey graphs and Erd\"os distance problem over finite Euclidean and non-Euclidean spaces, \textit{The Electronic Journal of Combinatorics}, \textbf{15} (1), 2008, R5.
\bibitem{vinh 2}
L. A. Vinh, Szemeredi-Trotter type theorem and sum-product estimate in finite fields, \textit{The European Journal of Combinatorics}, \textbf{32}(8), 1177-1181 (2011)
\bibitem{vinh-dm} L. A. Vinh, Singular matrices with restricted rows in vector spaces over finite fields, \textit{Discrete Mathematics} \textbf{312}(2), 413--418 (2012).
\bibitem{vinh-dam} L. A. Vinh, An explicit construction of $(3,k)$-existentially closed graphs, \textit{Discrete Applied Mathematics} (to appear).
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\section{Author’s note}
\emph{ This is not a definitive statement, merely a personal account, and seeks in general terms to explain what is meant by anarcho-primitivism. It does not wish to limit or exclude, but provide a general introduction to the topic. Apologies for inaccuracies, misinterpretations, or (inevitable) overgeneralizations.}
\section{What is anarcho-primitivism?}
Anarcho-primitivism (a.k.a. radical primitivism, anti-authoritarian primitivism, the anti-civilization movement, or just, primitivism) is a shorthand term for a radical current that critiques the totality of civilization from an anarchist perspective, and seeks to initiate a comprehensive transformation of human life. Strictly speaking, there is no such thing as anarcho-primitivism or anarcho-primitivists. Fredy Perlman, a major voice in this current, once said, “The only -ist name I respond to is ‘cellist’.” Individuals associated with this current do not wish to be adherents of an ideology, merely people who seek to become free individuals in free communities in harmony with one another and with the biosphere, and may therefore refuse to be limited by the term ‘anarcho-primitivist’ or any other ideological tagging. At best, then, anarcho-primitivism is a convenient label used to characterise diverse individuals with a common project: the abolition of all power relations — e.g., structures of control, coercion, domination, and exploitation — and the creation of a form of community that excludes all such relations. So why is the term anarcho-primitivist used to characterise this current? In 1986, the circle around the Detroit paper Fifth Estate indicated that they were engaged in developing a ‘critical analysis of the technological structure of western civilization[,] combined with a reappraisal of the indigenous world and the character of primitive and original communities. In this sense we are primitivists\dots{}’ The Fifth Estate group sought to complement a critique of civilization as a project of control with a reappraisal of the primitive, which they regarded as a source of renewal and anti-authoritarian inspiration. This reappraisal of the primitive takes place from an anarchist perspective, a perspective concerned with eliminating power relations. Pointing to ‘an emerging synthesis of post-modern anarchy and the primitive (in the sense of original), Earth-based ecstatic vision,’ the Fifth Estate circle indicated: We are not anarchists per se, but pro-anarchy, which is for us a living, integral experience, incommensurate with Power and refusing all ideology\dots{} Our work on the FE as a project explores possibilities for our own participation in this movement, but also works to rediscover the primitive roots of anarchy as well as to document its present expression. Simultaneously, we examine the evolution of Power in our midst in order to suggest new terrains for contestations and critique in order to undermine the present tyranny of the modern totalitarian discourse — that hyper-reality that destroys human meaning, and hence solidarity, by simulating it with technology. Underlying all struggles for freedom is this central necessity: to regain a truly human discourse grounded in autonomous, intersubjective mutuality and closely associated with the natural world. The aim is to develop a synthesis of primal and contemporary anarchy, a synthesis of the ecologically-focussed, non-statist, anti-authoritarian aspects of primitive lifeways with the most advanced forms of anarchist analysis of power relations. The aim is not to replicate or return to the primitive, merely to see the primitive as a source of inspiration, as exemplifying forms of anarchy. For anarcho-primitivists, civilization is the overarching context within which the multiplicity of power relations develop. Some basic power relations are present in primitive societies — and this is one reason why anarcho-primitivists do not seek to replicate these societies — but it is in civilization that power relations become pervasive and entrenched in practically all aspects of human life and human relations with the biosphere. Civilization — also referred to as the megamachine or Leviathan — becomes a huge machine which gains its own momentum and becomes beyond the control of even its supposed rulers. Powered by the routines of daily life which are defined and managed by internalized patterns of obedience, people become slaves to the machine, the system of civilization itself. Only widespread refusal of this system and its various forms of control, revolt against power itself, can abolish civilization, and pose a radical alternative. Ideologies such as Marxism, classical anarchism and feminism oppose aspects of civilization; only anarcho-primitivism opposes civilization, the context within which the various forms of oppression proliferate and become pervasive — and, indeed, possible. Anarcho-primitivism incorporates elements from various oppositional currents — ecological consciousness, anarchist anti-authoritarianism, feminist critiques, Situationist ideas, zero-work theories, technological criticism — but goes beyond opposition to single forms of power to refuse them all and pose a radical alternative.
\section{How does anarcho-primitivism differ from anarchism, or other radical ideologies?}
From the perspective of anarcho-primitivism, all other forms of radicalism appear as reformist, whether or not they regard themselves as revolutionary. Marxism and classical anarchism, for example, want to take over civilization, rework its structures to some degree, and remove its worst abuses and oppressions. However, 99\% of life in civilization remains unchanged in their future scenarios, precisely because the aspects of civilization they question are minimal. Although both want to abolish capitalism, and classical anarchism would abolish the State too, overall life patterns wouldn’t change too much. Although there might be some changes in socioeconomic relations, such as worker control of industry and neighbourhood councils in place of the State, and even an ecological focus, basic patterns would remain unchanged. The Western model of progress would merely be amended and would still act as an ideal. Mass society would essentially continue, with most people working, living in artificial, technologised environments, and subject to forms of coercion and control. Radical ideologies on the Left seek to capture power, not abolish it. Hence, they develop various kinds of exclusive groups — cadres, political parties, consciousness-raising groups — in order to win converts and plan strategies for gaining control. Organizations, for anarcho-primitivists, are just rackets, gangs for putting a particular ideology in power. Politics, ‘the art and science of government,’ is not part of the primitivist project; only a politics of desire, pleasure, mutuality and radical freedom.
\section{Where, according to anarcho-primitivism, does power originate?}
Again, a source of some debate among anarcho-primitivists. Perlman sees the creation of impersonal institutions or abstract power relations as the defining moment at which primitive anarchy begins to be dismantled by civilized social relations. In contrast, John Zerzan locates the development of symbolic mediation — in its various forms of number, language, time, art and later, agriculture — as the means of transition from human freedom to a state of domestication. The focus on origin is important in anarcho-primitivism because primitivism seeks, in exponential fashion, to expose, challenge and abolish all the multiple forms of power that structure the individual, social relations, and interrelations with the natural world. Locating origins is a way of identifying what can be safely salvaged from the wreck of civilization, and what it is essential to eradicate if power relations are not to recommence after civilization’s collapse. What kind of future is envisaged by anarcho-primitivists? Anarcho-primitivist journal “Anarchy; A Journal of Desire Armed” envisions a future that is ‘radically cooperative \& communitarian, ecological and feminist, spontaneous and wild,’ and this might be the closest you’ll get to a description! There’s no blueprint, no proscriptive pattern, although it’s important to stress that the envisioned future is not ‘primitive’ in any stereotypical sense. As the Fifth Estate said in 1979: ‘Let us anticipate the critics who would accuse us of wanting to go “back to the caves” or of mere posturing on our part — i.e., enjoying the comforts of civilization all the while being its hardiest critics. We are not posing the Stone Age as a model for our Utopia[,] nor are we suggesting a return to gathering and hunting as a means for our livelihood.’ As a corrective to this common misconception, it’s important to stress that that the future envisioned by anarcho-primitivism is sui generis — it is without precedent. Although primitive cultures provide intimations of the future, and that future may well incorporate elements derived from those cultures, an anarcho-primitivist world would likely be quite different from previous forms of anarchy.
\section{How does anarcho-primitivism view technology?}
John Zerzan defines technology as ‘the ensemble of division of labor\Slash{} production\Slash{} industrialism and its impact on us and on nature. Technology is the sum of mediations between us and the natural world and the sum of those separations mediating us from each other. It is all the drudgery and toxicity required to produce and reproduce the stage of hyper-alienation we languish in. It is the texture and the form of domination at any given stage of hierarchy and domination.’ Opposition to technology thus plays an important role in anarcho-primitivist practice. However, Fredy Perlman says that ‘technology is nothing but the Leviathan’s armory,’ its ‘claws and fangs.’ Anarcho-primitivists are thus opposed to technology, but there is some debate over how central technology is to domination in civilization. A distinction should be drawn between tools (or implements) and technology. Perlman shows that primitive peoples develop all kinds of tools and implements, but not technologies: ‘The material objects, the canes and canoes, the digging sticks and walls, were things a single individual could make, or they were things, like a wall, that required the cooperation of many on a single occasion \dots{} Most of the implements are ancient, and the [material] surpluses [these implements supposedly made possible] have been ripe since the first dawn, but they did not give rise to impersonal institutions. People, living beings, give rise to both.’ Tools are creations on a localised, small-scale, the products of either individuals or small groups on specific occasions. As such, they do not give rise to systems of control and coercion. Technology, on the other hand, is the product of large-scale interlocking systems of extraction, production, distribution and consumption, and such systems gain their own momentum and dynamic. As such, they demand structures of control and obedience on a mass scale — what Perlman calls impersonal institutions. As the Fifth Estate pointed out in 1981: ‘Technology is not a simple tool which can be used in any way we like. It is a form of social organization, a set of social relations. It has its own laws. If we are to engage in its use, we must accept its authority. The enormous size, complex interconnections and stratification of tasks which make up modern technological systems make authoritarian command necessary and independent, individual decision-making impossible.’ Anarcho-primitivism is an anti-systemic current: it opposes all systems, institutions, abstractions, the artificial, the synthetic, and the machine, because they embody power relations. Anarcho-primitivists thus oppose technology or the technological system, but not the use of tools and implements in the senses indicated here. As to whether any technological forms will be appropriate in an anarcho-primitivist world, there is debate over this issue. The Fifth Estate remarked in 1979 that: ‘Reduced to its most basic elements, discussions about the future sensibly should be predicated on what we desire socially and from that determine what technology is possible. All of us desire central heating, flush toilets, and electric lighting, but not at the expense of our humanity. Maybe they are all possible together, but maybe not.’ What about medicine? Ultimately, anarcho-primitivism is all about healing — healing the rifts that have opened up within individuals, between people, and between people and nature, the rifts that have opened up through civilization, through power, including the State, Capital, and technology. The German philosopher Nietzsche said that pain, and the way it is dealt with, should be at the heart of any free society, and in this respect, he is right. Individuals, communities and the Earth itself have been maimed to one degree or another by the power relations characteristic of civilization. People have been psychologically maimed but also physically assaulted by illness and disease. This isn’t to suggest that anarcho-primitivism can abolish pain, illness and disease! However, research has revealed that many diseases are the results of civilized living conditions, and if these conditions were abolished, then certain types of pain, illness and disease could disappear. As for the remainder, a world which places pain at its centre would be vigorous in its pursuit of assuaging it by finding ways of curing illness and disease. In this sense, anarcho-primitivism is very concerned with medicine. However, the alienating high-tech, pharmaceutical-centred form of medicine practised in the West is not the only form of medicine possible. The question of what medicine might consist of in an anarcho-primitivist future depends, as in the Fifth Estate comment on technology above, on what is possible and what people desire, without compromising the lifeways of free individuals in ecologically-centred free communities. As on all other questions, there is no dogmatic answer to this issue.
\section{What about population?}
A controversial issue, largely because there isn’t a consensus among anarcho-primitivists on this topic. Some people argue that population reduction wouldn’t be necessary; others argue that it would on ecological grounds and\Slash{}or to sustain the kind of lifeways envisaged by anarcho-primitivists. George Bradford, in \emph{How Deep is Deep Ecology?}, argues that women’s control over reproduction would lead to a fall in population rate. The personal view of the present writer is that population would need to be reduced, but this would occur through natural wastage — i.e., when people died, not all of them would be replaced, and thus the overall population rate would fall and eventually stabilise. Anarchists have long argued that in a free world, social, economic and psychological pressures toward excessive reproduction would be removed. There would just be too many other interesting things going on to engage people’s time! Feminists have argued that women, freed of gender constraints and the family structure, would not be defined by their reproductive capacities as in patriarchal societies, and this would result in lower population levels too. So population would be likely to fall, willy-nilly. After all, as Perlman makes plain, population growth is purely a product of civilization: ‘a steady increase in human numbers [is] as persistent as the Leviathan itself. This phenomenon seems to exist only among Leviathanized human beings. Animals as well as human communities in the state of nature do not proliferate their own kind to the point of pushing all others off the field.’ So there’s really no reason to suppose that human population shouldn’t stabilise once Leviathanic social relations are abolished and communitarian harmony is restored. Ignore the weird fantasies spread by some commentators hostile to anarcho-primitivism who suggest that the population levels envisaged by anarcho-primitivists would have to be achieved by mass die-offs or nazi-style death camps. These are just smear tactics. The commitment of anarcho-primitivists to the abolition of all power relations, including the State with all its administrative and military apparatus, and any kind of party or organization, means that such orchestrated slaughter remains an impossibility as well as just plain horrendous.
\section{How might an anarcho-primitivist future be brought about?}
The sixty-four thousand dollar question! (to use a thoroughly suspect metaphor!) There are no hard-and-fast rules here, no blueprint. The glib answer — seen by some as a cop-out — is that forms of struggle emerge in the course of insurgency. This is true, but not necessarily very helpful! The fact is that anarcho-primitivism is not a power-seeking ideology. It doesn’t seek to capture the State, take over factories, win converts, create political organizations, or order people about. Instead, it wants people to become free individuals living in free communities which are interdependent with one another and with the biosphere they inhabit. It wants, then, a total transformation, a transformation of identity, ways of life, ways of being, and ways of communicating. This means that the tried and tested means of power-seeking ideologies just aren’t relevant to the anarcho-primitivist project, which seeks to abolish all forms of power. So new forms of action and being, forms appropriate to and commensurate with the anarcho-primitivist project, need to be developed. This is an ongoing process and so there’s no easy answer to the question: What is to be done? At present, many agree that communities of resistance are an important element in the anarcho-primitivist project. The word ‘community’ is bandied about these days in all kinds of absurd ways (e.g., the business community), precisely because most genuine communities have been destroyed by Capital and the State. Some think that if traditional communities, frequently sources of resistance to power, have been destroyed, then the creation of communities of resistance — communities formed by individuals with resistance as their common focus — are a way to recreate bases for action. An old anarchist idea is that the new world must be created within the shell of the old. This means that when civilization collapses — through its own volition, through our efforts, or a combination of the two — there will be an alternative waiting to take its place. This is really necessary as, in the absence of positive alternatives, the social disruption caused by collapse could easily create the psychological insecurity and social vacuum in which fascism and other totalitarian dictatorships could flourish. For the present writer, this means that anarcho-primitivists need to develop communities of resistance — microcosms (as much as they can be) of the future to come — both in cities and outside. These need to act as bases for action (particularly direct action), but also as sites for the creation of new ways of thinking, behaving, communicating, being, and so on, as well as new sets of ethics — in short, a whole new liberatory culture. They need to become places where people can discover their true desires and pleasures, and through the good old anarchist idea of the exemplary deed, show others by example that alternative ways of life are possible. However, there are many other possibilities that need exploring. The kind of world envisaged by anarcho-primitivism is one unprecedented in human experience in terms of the degree and types of freedom anticipated \dots{} so there can’t be any limits on the forms of resistance and insurgency that might develop. The kind of vast transformations envisaged will need all kinds of innovative thought and activity.
\section{How can I find out more about anarcho-primitivism?}
The Primitivist Network (PO Box 252, Ampthill, Beds MK45 2QZ) can provide you with a reading list. Check out copies of the British paper Green Anarchist and the US zines Anarchy: A Journal of Desire Armed and Fifth Estate. Read Fredy Perlman’s \emph{Against His-story, Against Leviathan!} (Detroit: Black \& Red, 1983), the most important anarcho-primitivist text, and John Zerzan’s \emph{Elements of Refusal} (Seattle: Left Bank, 1988) and \emph{Future Primitive} (New York: Autonomedia, 1994). How do I get involved in anarcho-primitivism? One way is to contact the Primitivist Network. If you send two 1\textsuperscript{st} class postage stamps, you will receive a copy of the PN contact list and be entered on it yourself. This will put you in contact with other anarcho-primitivists. Some people involved in \emph{Earth First!} also see themselves as anarcho-primitivists, and they are worth seeking out too.
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This past May, the Italian state began another round of repression against anarchists. The more sporadic state harassment anarchists had experienced over the past few years gave way to what is clearly a more focused attack. This reflects the growing repressive reality around the globe, the needs of the Italian state in the present circumstances and its attempts to find ways to strike the anarchist movement and all anti-institutional movements of resistance and revolt in their totality. Thus, these events demand more than news and calls for solidarity; they require an attempt to analyze the situation on several levels in order to develop meaningful and substantial forms of active solidarity.
\section{The May arrests}
Perhaps it would be most accurate to see the arrest of Francesco Gioia in Barcelona Spain on May 11 as the beginning of this wave of repression. Francesco is an anarchist from the Italian group II Silvestre. He was placed under house arrest last year in relation to actions in the region of Pisa claimed by the COR (Revolutionary Offensive Cells). He escaped from house arrest last summer and went on the lam to Spain. Although “subversive association with the aim of terrorism” had not been one of the original charges at the time the arrests were first made, it was added to the charges, a fact that made the recapture of Francesco more imperative to the judiciary officials of Italy. So a joint operation between Italian \emph{cambinieri}\footnote{The \emph{carbinieri} are the Italian national police force, a part of the military. They operate abroad as well as in Italy. For example, they have carried out military operations in Somalia and the Balkans, and the current Italian troops in Iraq are \emph{carabinieri}.} and Spanish police under the aegis of Europol captured Francesco in Barcelona on May 11.
On May 12, in the region of Lecce, a force of 150 cops from several police forces (including special anti-terrorism and bomb units) carried out “Operation Nighttime.” They raided and searched dozens of anarchists’ houses and spaces, making five arrests and issuing ten more notices of investigation. Cristian Pal-ladini, Salvatore Signore, and Saverio Pellegrino were taken to prison, while Annalisa Capone and Angela Marina Ferrari were placed under house arrest. The most significant charge is “subversive association with the aim of terrorism and subversion of the democratic state.” In addition, there were charges for attacks against the Cathedral and the home of the priest, Father Cesare Lodeserto (who had run the local Center for Temporary Residence (CPT)\footnote{CPTs are concentration camps for undocumented immigrants, who are held there awaiting deportation.} until he was arrested for “private violence” and kidnapping of inmates at the center); attacks against the Banca Intesa where the Regina Pacis Foundation (through which the CPT is run) keeps its funds; instigation to revolt due to a march in front of the CPT that ended in conflicts with the authorities; “harassment” and “defamation” of collaborators in the operation of the CPT and “dirtying” the entrance of one of these collaborators’ houses; organizing an unauthorized march against the Benetton corporation that has taken over a large portion of Patagonia where the Mapuche people have been living for centuries; vandalism against a Benetton store and against an Esso station;\footnote{The European branch of Exxon, suppliers of fuel for the Coalition forces in the war in Iraq and Afghanistan.} occupying the abandoned space station Capolinea; and writing on walls. Along with the searches and arrests in Lecce, there were searches throughout the rest of Italy as well.
A week later on May 19, another police operation occurred in Sardinia. Police searched over fifty anarchist houses and spaces and placed seven anarchists under house arrest. (These seven have not made their names public.) Charges include the infamous “subversive association with the aim of terrorism and subversion of the democratic state” and “propaganda favoring and celebrating subversive practices.” All of those arrested are under investigation in relation to parcel bombs that were sent to a \emph{carabinieri} barracks in 2003 and an arson attack against the electoral office of the \emph{Forza Italia}, party\footnote{Right-wing party to which Italian prime minister Berlusconi belongs.} that happened in June 2004. They are charged as “members or sympathizers” of the FRARIA Center for Anarchist Studies in Caliari. Twenty-six other anarchists have been placed under investigation for “subversive association” and “propaganda and celebration of subversive practices.”
On May 26, operations were carried out in the vicinities of both Bologna and Rome, involving 110 searches, 10 arrests and 21 others being put under investigation. All of those arrested were charged with “subversive association” with the usual aims. The seven arrested by the Bologna justice system (Lucia Rippa, Mat-tia Bertoni, Elsa Caroli, Teo Tavernese, Marco Foresto, Danilo Cremonese, and Valentina Speziale) were accused of being members of an organization that claimed certain bomb attacks, the Informal Anarchist Federation (FAInformal) and of spreading subversive propaganda through the Anarchist Black Cross bulletin. The DIGOs’\footnote{Italian political police} closed down the local ABC website. Meanwhile, the Roman public prosecutor issued arrest warrants for five anarchists, two of whom — Danilo and Valentina — also had arrest warrants in •Bologna. The other three are Stefano Del Moro, Massimo Leonard!, and Claudia Cospito. Along with the usual “subversive association” charge, they have charges relating to bomb attacks against the minister oTjustice of Viterbo and to the sacking of a McDonald’s that happened toward the end of a demonstration in 1999.
The arrests have come up before the so-called Courts of Liberty of the various regions where the judges determine the legitimacy of the arrest warrants. These courts upheld the arrests everywhere except in Bologna where the judge in charge concluded that the evidence was simply too weak. So five of the comrades arrested in Bologna were freed. Danilo and Valentina, who also had arrest warrants in Rome, remain in prison.
Various solidarity activities have been going on steadily since these arrests, ranging from benefit concerts and dinners to marches and demonstrations to occupations of buildings to attacks against police property and the property of Italian multinationals like Benetton and Fiat. Alongside the ongoing activity in Italy, anarchists in Greece and Spain have been particularly active in support of those arrested. It is not surprising, in this light, that the repression spread beyond the borders of one country.
On June 25, in Barcelona, anarchists held a demonstration in solidarity with the people arrested in Italy. Police attacked the demonstrations, causing many injuries. Seven anarchists were arrested and beaten. Two were released that night, and over the next couple of weeks, all but one were released. Alberto Bettini, an Italian anarchist, remains in the men’s prison in Barcelona.
\section{The context of the repression}
This round of repression against anarchists is not just a state whim of the moment. There are reasons why it is happening now and being carried out in the way it is. First of all, over the past few decades the capitalist social order has undergone changes creating precariousness and a lack of place within the existing social world for more and more people and threatening all except those at the top. Newly dispossessed indigenous people, undocumented immigrants, the homeless, temporary workers, unemployed youth, refugees from war, poverty and disaster, inhabitants of shantytowns that represent the place of greatest growth in the burgeoning metropolises of this world — here is a growing underclass that does not fit within the framework of citizenship or consumerism and thus has no place in this world. Inevitably, this underclass is restless.
It is impossible to claim that the social consensus necessary to the democratic state is there, unless these undesirables are excluded. This process of exclusion includes both the blatant physical creation of spaces of exception, such as the CPTs in Italy and their equivalents throughout the world, and the more subtle use of propaganda to drive wedges between those at the bottom and those who still manage to maintain a precarious hold on a place within this society. Undocumented aliens are particularly singled out as threats to the jobs of the “native-born” exploited and as potential terrorists. In this way, the state and its media seek to keep the exploited from seeing their common condition and finding the ways to intertwine their struggles against this condition.
Nonetheless, the restlessness that precariousness imposes has led to rebellion. In Italy, immigrants both within and outside of the CPTs have been rebelling. And anarchists have been there acting in solidarity. This is not at all surprising. If undocumented immigrants have had the condition of being strangers everywhere in the world thrust upon them, it is a condition that anarchists, who desire a life so utterly different than that which is imposed on us, know quite well. And the recognition of this potential (and frequently actualized) complicity between undocumented immigrants and anarchists certainly plays a role in the state’s attempt to isolate anarchists from the general populace as well, portraying them as “terrorists.”
In Italy, the democratic state has its allies in the various neo- and “post”-fascist groups. These groups help to raise hysteria about the “invasion” of foreigners, often participate in police raids in immigrant neighborhoods and, over the past few months, have made several attacks against anarchist spaces and individuals. It is important to recognize here that these fascists are serving the interests of the \emph{democratic} state to keep the exploited in their atomized roles as citizens, producers and consumers, and to demonize those of the exploited who cannot or will not fit into these roles.
In short, the precariousness that defines every aspect of life in the present world also puts the state and capital at risk. They can only operate on a perpetual defensive, going from one emergency measure to the next. If the restlessness that dominates most of existence is not channeled into ethnic, racial, or religious conflicts, representing the fear of the unknown, and portraying the state as the purveyor of stability, it might break out as rebellion against the horror of the known, a rebellion that recognizes the unknown as the realm of new possibilities for creating a different way of living. And this is what the state fears.
In this light, we can understand the focus of the Italian state on anarchists and others who choose to oppose it in a consistently self-determined, anti-institutional manner. It is certainly not the strength of anarchists that it fears, but its own weakness, and the capacity of anarchists to see their own condition in that of all of the excluded and to bring to that condition an embrace of the unknown that can move rebellion beyond sheer desperation to the exploration of possibilities for action, complicity and the creation of life. Since the huge and growing number of those who do not fit into the social order has made it impossible for the democratic state to maintain social consensus, an increasingly heated social war is inevitable. Thus, the democratic state is forced more and more to operate within a “state of exception” — so much so that the “exception” is proving to be the rule. And this could easily lead to the situation where the lie of rights and the “rule of law” are exposed as simply one specific form of the arbitrary rule of the state. Only by the creation of the “other” — the supposed threat of the foreigner, the criminal, the terrorist, etc. that can justify this situation in the eyes of those who continue to see themselves as “included” — can the state put off this exposure of its nature. And those who are quick to point out its real nature, like anarchists, must be among those who are made “other.” The rise of the “war on terrorism,” the new laws passed in Italy and in Europe increasing police and judiciary powers in the face of the “terrorist threat,” the complete complicity of mass media with the state and the various associative laws already on the books, provide the state with the framework for pursuing this endeavor.
The methodology of the repression
There is a common thread that runs through all of these arrests as well as through the notices of investigation issued to so many anarchists who were not arrested. That is the charge of “subversive association with the aim of terrorism and subversion of the democratic state.” In fact, the evidence supporting any of the other charges is quite flimsy, but this charge is broad enough to require very little evidence. So it’s worth our while to look at it more closely.
The law against subversive association is article 270 of the Italian legal code. It was put into effect by Rocco, attorney general under Mussolini. The law was not rescinded when Italy reestablished a democratic state. Instead it has been amended a couple of times to give it more bite. One such amendment, add ed in the 1970s to counter the increasing use of insurgent violence against the Italian state, is paragraph 270bis dealing with “subversive association” with the aim of terrorism and ”the subversion of the democratic order.” Nothing in the wording of this law requires those charged with it to ever actually carry out an act of violence against the state in order to be convicted. They only have to propose such actions. Thus, there is a certain flexibility to the law that is useful to the state.
The law has not yet been used successfully against anarchists, but starting with the Marini trial, its use has been increasing rapidly over the past several years, and not just against anarchists. In fact, more than 8,000 people have been charged with subversive association for various-struggles that have happened in the past two years, and as social conflict heats up in Italy, this number could go into the tens of thousands.
The Italian state has had a problem trying to strike the anarchist movement in the country as a whole. The piecemeal approach involving convictions for specific acts was not enough. When Marini decided to go after the anarchists, he decided to make associative crimes the center of his investigation and prosecution. This was the first use (at least in recent times) of paragraph 270bis against anarchists. His hope (and that of the Italian state) was that this would prove to be a tool for striking the entire anarchist movement at once. Although the Marini trial certainly drained a lot of energy in anarchist circle in the years and years that it dragged on, ultimately no one was convicted of 270bis or any other associative charge. The law had been written in a different time, when highly structured organizations were the most common form for insurgent groups to take. So with the disintegration of those forms of organization, the law became more difficult to apply.
In current struggle in Italy, whether anarchists are involved or not, the activity is carried out mostly by individuals or small, barely structured, usually temporary groups. As they had been previously applied, article 270 and paragraph 270bis were not able to deal with such a phenomenon. But those in power are experimenting in order to refine this repressive law. Interior minister Pisanu wants to add a new paragraph to article 270 that would specifically criminalize anarchists and other revolutionaries who cannot be repressed with the three existing paragraphs.
Certain steps have already been taken to refine this law. Since no large association exists, it is essential to find small associations. Thus, in 2001 in Taranto, some revolutionaries were prosecuted for a local 270bis. It was claimed that a local organization existed to commit crimes in the local area. This went almost unnoticed at the time within anarchist circles, and that is too bad. The situation was a testing ground for trying out the capacity for resisting a qualitative leap in the field of repression. The non-response among anarchists and other rebels opened the door to further experiments by the state.
That same year, a further step was taken in the investigation called “Rebel South.” Rather than speaking of structures, the state spoke of a “coordination” among various organizations that shared a few objectives, the chief one supposedly being to bring the real conflicts of southern Italy and the world in general to Genoa. Though some resistance developed against this repressive operation, it faded quickly, and soon local investigations for 270bis began to proliferate.
Public prosecutor Vitello decided to take the experiment in a rather different direction. He opened the Cervantes investigation based on his hypothesis that there is already a national association for carrying out attacks, but that it is not organized according to the old norms. Rather, he claims, it functions according to the model of affinity groups. To build up the evidence for this “national association” Vitello encouraged public prosecutors to open local investigations in their regions. Vitello opened the first such investigation on February 17, 2004, against the Citizens’ Committee Against Prison and Social Repression in Viterbo. Besides the usual 270bis, he claimed that the comrades were involved in the crimes cited in the Cervantes investigation. Then he included some of the anarchists of the Committee in the Cervantes investigation. And so public prosecutors all over Italy began to open local investigations for 270bis with the aim of combining them into one great investigation that could open the door to a specific law against “anarcho-insur-rectionalists,” as the prosecutors, Pisanu and the media love to label these rebels.
The media plays its role by portraying the “anarcho-insurrectionalists” as terrorists second only to Al-Qaeda in terms of the danger they threaten to let loose. The authorities, in fact, play on the “war on terrorism” throughout, both because it permits the use of draconian measures and because it creates the image of the anarchist as a dangerous “other” against which the state defends its good citizens.
Even if the attempts to prosecute anarchists for 270bis should fail again, those arrested on this charge can be held in prison for up to two years without a trial, thus effectively taking them away from their lives and struggles, and this at a time when the deterioration of social conditions is stoking the fires of social conflict in Italy and elsewhere.
There is much more about of the methods of the Italian state and why they are doing this now that could be examined, but I am hoping that Italian comrades will write in to correct and supplement these comments.
\section{Showing solidarity}
Solidarity is not an obligation, but a choice based in mutuality. If I choose to express solidarity with any struggles, comrades or prisoners, it is because I see my struggle to take back my life and live it on my terms within them. This is why the most essential aspect of solidarity is to continue the struggles and revolts we share with our comrades where we are.
Understood in this way, solidarity is never with the suffering of others — that would merely be pity, not true solidarity. Rather we act in solidarity with the ferocity that they have shown in refusing to accept their suffering. This is why questions of guilt or innocence are of no importance in relation to solidarity with arrested and imprisoned comrades. What matters is that we know that they are fighting the state and its servants and that currently the state has chosen to strike them fiercely for attacking it.
‘The comrades arrested and placed under investigation in Italy, along with those whose homes were invaded by cops, all recognized what their daily rebellion shared in common with other exploited people. All of these comrades acted in their own way to express their complicity and solidarity with the rebellions of those in the CPTs and throughout Italy, in Patagonia, in Iraq, and throughout the world.
In the same way, solidarity with the anarchists arrested in Italy would need to start from a recognition of complicity and mutuality, recognizing our own rebellion in theirs. Thus, the greatest acts of solidarity would be to find the places where our struggles interweave with those they were involved in. Thus, also to find where they interweave with the revolts of wildcatting transit workers, fare strikers, undocumented immigrants, all those fighting dispossession, the portions of Iraqi resistance that remain free of sectarian and nationalist rackets\dots{} And so the threads weave farther and farther.
And the forces of domination, exploitation, and repression are the same here as in Italy and the rest of the world even if the specific methods of their functioning vary due to specific circumstances. We can find the links in the chain of exploitation that connect us with the comrades in Italy and with all the exploited and dispossessed in revolt and aim our attacks at these points. And this is true solidarity, which gives substance to any support we may choose to give the arrested comrades, showing its basis in complicity rather than charity or duty.
\begin{quote}
\emph{An anarchist stranger in an alien world}
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The Anarchist Library
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Anti-Copyright
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Anonymous
State Repression Against Anarchists in Italy
2005–2006
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Originally published in “Anarchy: A Journal of Desire Armed” \#60, Fall\Slash{}Winter, 2005–06, Vol. 23, No. 2.
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\textbf{theanarchistlibrary.org}
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\section{Efficiency of the Parabix}
\label{section:baseline}
In this section we analyze the energy and performance characteristics
of the Parabix-based XML parser against the software XML parsers,
Xerces and Expat. For our baseline evaluation, we compare all the XML
parsers on a fixed platform, the \CITHREE{}.
%some of the numbers are roughly calculated, needs to be recalculated for final version
\subsection{Cache behavior}
The approximate miss penalty in \CITHREE\ for L1, L2 and L3 caches is
4, 11, and 36 cycles respectively. The L1 (32KB) and L2 cache (256KB)
are private per core, while the 4MB L3 is shared by all the
cores. Figure \ref{cache_misses} shows the cache misses per kilobyte
of input data. Analytically, the cache misses for the Expat and Xerces
parsers represent a 0.5 cycle per XML byte processed. This overhead
does not necessarily reflect in the overall performance of these
parsers as they experience other overheads related to branch
mispredictions. Parabix's data reorganization significantly improves
the overall cache miss rate. We experience 7$\times$ less misses than
Expat and 25$\times$ less misses than Xerces at the L1 and 104$\times$ less misses than
Expat and 15$\times$ less misses than Xerces at the L2 level. The improved cache
utilization keeps the SIMD units busy and prevent memory related
stalls. Note that cache misses also cause increased application energy
consumption due to increased energy required to access higher levels
in the cache hierarchy. We estimated with microbenchmarks that the L1,
L2, and L3 cache misses consume approximately 8.3nJ, 19nJ, and 40nJ
respectively. For a 1GB XML file Expat and Xerces would consume over
0.6J and 0.9J respectively due to cache misses alone.
%With a 1GB input file, Expat would consume more than 0.6J and Xercesn
%would consume 0.9J on cache misses alone.
\begin{figure}
\subfigure[L1 Misses]{
\includegraphics[width=0.32\textwidth]{plots/corei3_L1DM.pdf}
\label{corei3_L1DM}
}
\subfigure[L2 Misses]{
\includegraphics[width=0.32\textwidth]{plots/corei3_L2DM.pdf}
\label{corei3_L2DM}
}
\subfigure[L3 Misses]{
\includegraphics[width=0.32\textwidth]{plots/corei3_L3CM.pdf}
\label{corei3_L3DM}
}
\caption{Cache Misses per kB of input data.}
\label{cache_misses}
\end{figure}
\subsection{Branch Mispredictions}
\label{section:XML-branches}
In general, reducing the branch misprediction rate is difficult in
text-based XML parsing applications. This is due to (1) variable
length nature of the syntactic elements contained within XML
documents, (2) a data dependent characteristic, and (3) the extensive
set of syntax constraints imposed by the XML. Traditional
byte-at-a-time XML parser's performance is limited by the number of
branch mispredictions. As shown in Figure \ref{corei3_BR}, Xerces
averages up to 13 branches per XML byte processed on high density
markup. On modern commodity processors the cost of a single branch
misprediction is incur over 10s of CPU cycles to restart the processor
pipeline. The high miss prediction rate in conventional parsers add
significant overhead. In Parabix the transformation to SIMD operation
eliminates many branches. Further optimizations take advantage of
Parabix's data organization and replace condition branches with {\em
bit scan} operations that can process up to 128 characters worth of
branches with one operation. In many cases, we also replace the
branches with logical predicate operations. Our predicates are cheaper
to compute since they involve only bit parallel SIMD operations.
As shown in Figure \ref{corei3_BR},
Parabix processing is almost branch free. Parabix exhibits minimal
dependence on source XML markup density; it experiences between 19.5 and
30.7 branch mispredictions per thousand of XML byte. The cost of
branch mispredictions for the Expat parser can be over 7 cycles per
XML byte (see Figure \ref{corei3_BM}) ---this cost alone is higher
than the average latency of a byte processed by Parabix.
\begin{figure}
\subfigure[Branch Instructions]{
\includegraphics[width=0.5\textwidth]{plots/corei3_BR.pdf}
\label{corei3_BR}
}
\hfill
\subfigure[Branch Misses]{
\includegraphics[width=0.5\textwidth]{plots/corei3_BM.pdf}
\label{corei3_BM}
}
\caption{Branch characteristics on the \CITHREE\ per kB of input data.}
\end{figure}
\subsection{SIMD Instructions vs. Total Instructions}
In Parabix, bit streams are both computed and
predominately operated upon using the SIMD instructions of commodity
processors. The ratio of retired SIMD instructions to total
instructions provides insight into the relative degree to which
Parabix achieves parallelism over the byte-at-a-time approach.
Using the Intel Pin tool, we gather the dynamic instruction mix for
each XML workload, and classify instructions as either vector (SIMD)
or non-vector instructions. Figure~\ref{corei3_INS_p2} shows the
percentage of SIMD instructions for the Parabix XML parser. The ratio of executed
SIMD instructions over total instructions indicates the amount of
parallel processing we were able to extract.
%(Expat and Xerce do not use any SIMD instructions)
The Parabix instruction mix is made up of 60\% to 80\% SIMD
instructions. The markup density of the files influence the number of
scalar instructions needed to handle the tag processing which affects
the overall parallelism that can be extracted by Parabix. We find
that degradation rate is low and thus the performance
penalty incurred by increasing the markup density is minimal.
%Expat and Xerce do not use any SIMD instructions and were not
%included in this portion of the study.
% Parabix gains its performance by using parallel bitstreams, which
% are mostly generated and calculated by SIMD instructions. We use Intel
% pin, a dynamic binary instrumentation tool, to gather instruction
% mix. Then we adds up all the vector instructions that have been
% executed. Figure \ref{corei3_INS_p1} and Figure \ref{corei3_INS_p2}
% show the percentage of SIMD instructions of Parabix1 and Parabix
% (Expat and Xerce do not use any SIMD instructions). For Parabix1,
% 18\% to 40\% of the executed instructions consists of SIMD
% instructions. By using bistream addition for parallel scanning,
% Parabix2 uses 60\% to 80\% SIMD instructions. Although the ratio
% decrease as the markup density increase for both Parabix1 and
% Parabix2, the decreasing rate of Parabix2 is much lower and thus the
% performance degradation caused by increasing markup density is
% smaller.
\subsection{CPU Cycles}
Figure \ref{corei3_TOT} shows overall parser performance evaluated in
terms of CPU cycles per kilobyte. The Parabix parser is 2.5$\times$
to 4$\times$ faster on document-oriented input and 4.5 to 7 times faster
on data-oriented input. Traditional parsers can be dramatically
slowed by dense markup, while Parabix is affected much less. The
results presented are not entirely fair to the Xerces parser since it
first transcodes input from UTF-8 to UTF-16 before processing. In
Xerces, this transcoding requires several cycles per byte. However,
transcoding using parallel bit streams is significantly faster and
requires less than a single cycle per byte.
\begin{figure}[htbp]
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[width=\textwidth]{plots/corei3_INS_p2.pdf}
\caption{SIMD Instruction Percentage}
\label{corei3_INS_p2}
\end{minipage}%
\hfill
\begin{minipage}{0.5\linewidth}
\centering
\includegraphics[width=\textwidth]{plots/corei3_TOT.pdf}
\caption{Performance (CPU Cycles per kB)}
\label{corei3_TOT}
\end{minipage}
\end{figure}
\subsection{Power and Energy}
In this section, we study the power and energy consumption of Parabix
in comparison with Expat and Xerces on \CITHREE{}. The average power
of \CITHREE\ is about 21 watts. Figure \ref{corei3_power} shows the
average power consumed by each parser. Parabix, dominated by SIMD
instructions which uses approximately 5\% additional power. While the
SIMD functional units are significantly wider than the scalar
counterparts; register width and functional unit power account only
for a small fraction of the overall power consumption in a processor
pipeline. More importantly by using data parallel operations Parabix
amortizes the fetch and data access overheads. This results in minimal
power increase compared to the conventional parsers. Perhaps the
energy trends shown in Figure \ref{corei3_energy} reveal an
interesting trend. Parabix consumes substantially less energy than the
other parsers. Parabix consumes 50 to 75 nJ per byte while Expat and
Xerces consume 80nJ to 320nJ and 140nJ to 370nJ per byte respectively.
Although Parabix requires slightly more power (per instruction), the
processing time of Parabix is significantly lower.
\begin{figure}
\subfigure[Avg. Power (Watts)]{
\includegraphics[width=0.5\textwidth]{plots/corei3_power.pdf}
\label{corei3_power}
}
\hfill
\subfigure[Energy Consumption ($\mu$J per kB)]{
\includegraphics[width=0.5\textwidth]{plots/corei3_energy.pdf}
\label{corei3_energy}
}
\caption{Power profile of Parabix on \CITHREE{}}
\end{figure}
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\documentstyle[dutch,a4wide,12pt]{article}
\pagenumbering{niks}
\begin{document}
\subsubsection*{11-11-1992 (voor 11-11) Michiel}
Nu, we hebben dus afgesproken elke dag een 'idee' te schrijven. Nou dan doe ik
het ook maar. Het is trouwens best moeilijk om iets te bedenken. Opeens
herinnerde ik me echter een uitspraak van Herman (de Aanstichter) waar ik nog
wel even over kan door 'idee\"en'. Het is wiskunde dus is beweeg mij op
gevaarlijk terrein en ben dus gaarne bereid eventuele onjuiste uitspraken die
gaan volgen terug te nemen.
De uitspraak was dat je {\em niets} over een lege verzameling kunt zeggen. Dit
wil ik tot mijn spijt tegenspreken. En dit doe ik door het te doen, er iets
over zeggen bedoel ik:
\begin{enumerate}
\item Een lege verzameling heeft nul elementen; was dit immers niet zo en had
hij/zij (wil ik vanaf zijn, gebruik gewoon hij) er meer dan nul bijvoorbeeld n
($ n>0, n \in N$) dan was hij immers niet meer leeg, maar een
verzameling met n elementen.\\
\item Ik ben eigenlijk InfiC aan het leren en durf nu te beweren $sup(\emptyset)=
- \infty $\\
\item Als $S$ een willekeurige verzameling is dan:\\
$S \cup \emptyset = S$\\
$S \cap \emptyset = \emptyset$ \\
$\emptyset \subseteq S $?\\
\end{enumerate}
Zo vind ik het wel weer genoeg. Ik hoop voor mijn idee van morgen op meer
inspiratie want dit was niet echt daverend orgineel natuurlijk, maar toch wel
acceptabel zou ik zeggen.\\
\pagebreak
\subsubsection*{12-11-1992 (voor 12-11) Michiel}
Het tweede stukje in deze cyclus. Tijdens het practicum van deze middag kwam ik
hier op. Deze keer gaat het over taal. Dit is een erg leuk onderwerp vind ik
zelf en ik er in het verleden dus ook al van alles over bedacht, hiermee kan ik
dus vooruit als ik niets weet. Nu weet ik echter wel wat en ik denk dat het
hoewel waarschijnlijk onzinnig vast wel orgineel is.
Het idee was als volgt. Het is zo in het nederlands dat je vele meervouden uit
de enkelvouden vormt door er op \'e\'en of andere manier '--en' achter te
zetten. Ik vroeg mij af of dit niet eigenlijk verkeerd was (vast niet) omdat je
veel makkelijker zegt aan het eind '--e'. Zo zeggen veel mensen het ook in de
dagelijkse praktijk. Mensen die wel '--en' zeggen zou je ervan kunnen verdenken
dat ze dat zeggen omdat je het ook zo schrijft, wat natuurlijk de omgekeerde
gang van zaken is, schrijven met een alfabet is bedoeld als 'opteken zoals je
het zegt'.
In het nederlands moeten ook veel meervouden, zoals ook in bijvoorbeeld in het
frans en engels, gevormd worden door er een 's' achter te plakken. Nu is het zo
dat hoewel de 'e' en de 's' niet zoveel op elkaar lijken --De een is
bijvoorbeeld klinker de ander medeklinker, een tamelijk groot verschil lijkt
me-- ze wel een bepaalde relatie moeten hebben. Dit is te zien in het feit dat
veel franse woorden op de nederlands lijken als je de 'e' waarmee het begint
vervangt door een 's' (bijvoorbeeld:'\'ecole' wordt dan 'scole' wat lijkt op
'school' en 'ecrire'?(is dit frans?) wordt 'scrire' lijkt op 'schrijven'. De
vervanging zal wat volediger misschien eigenlijk moeten zijn 'ec' wordt
'sch'. Mijn frans is echter niet van een zodanig nivo dat ik dit met een
tegenvoorbeeld (hetzij dat e $\rightarrow$ s ook zo bestaat hetzij dat c
$\rightarrow$ ch ook apart een verwisseling kan zijn) dit kan aantonen). Dit
verschijnsel komt waarschijnlijk door dat in de geschiedenis van de evolutie
van de taal er een periode is geweest dat de fransen of nederlanders (in dat
geval waarschijnlijk vroeg en niet door nederlanders maar door germanen in het
algemeen want bijv. duits en engels kennen ook de 's' i.p.v. 'e') zoiets in hun
uitspraak hebben 'omgezet'.(Dit is dan net zoiets als de 'p' $\rightarrow$
'v' of 'f' wat ook is voorgekomen (latijn $\rightarrow$ germaans)
bijvoorbeeld:'pater' naar 'vader'/'father'. Dit zal trouwens ook wel te maken met
de 'ph' die je als 'f' uitspreekt.)
Nu tenslotte is mijn opmerking dus dat die 'meervouds-e' (hoewel in schrift dus
'meervouds-en') in het nederlands misschien toch wel afstamt van de 'meervouds-s' in talen als engels, frans en
ook gedeeltelijk dus nederlands (duits ook geloof ik). Dit ook in analogie met
die s-e overgang die dus zeker al bestaat. (Die is een stuk zekerder omdat je
dan goed kunt zien dat '\'ecole' en 'school' eigenlijk de zelfde woorden zijn. Dit
geval is lijkt mij echter veel onzekerder omdat die meervoudsuitgangen niet
echt bij het woord horen en je dus niets kunt herkennen, meervouden worden meer
volgens vaste regels gevormd denk ik. )
\end{document}
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\bibitem{Huettel_etal2013} Huettel, M., Berg, P., Kostka, J.E., 2013. Benthic Exchange and Biogeochemical Cycling in Permeable Sediments. Annual Review of Marine Science 6 (1), 130829112540002.
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\title{Nice Shit for Everybody}
\date{March 24th, 2016}
\author{L.A. ONDA, Hostis}
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We hereby reject any form of self-imposed austerity. We posit that we want nice shit for everybody and that is not only feasible but desirable. We will not put forth graphs announcing how much work (or not) will require such a project but will state that such a project is part of our desire for communism. We hereby reject all forms of feigned punk slobbiness, neo-hippie shabby chic, or pajamas in the outdoors. We see the stores of the bourgeois parts of town (\& the newly-gentrified ones too) and say that we want that shit and even more. Capitalism is that which stands in the way of us having the shit we want with its hoarding of commodities only to sell them to highest bidder.
We’ve been told to live with less and less by not only Green Capital, but by the Church, by our liberal “friends,” and even by fellow comrades. Fuck that shit. Nah; if we’re going to be putting our shit out on the line it’s definitely not going to be so that I can live \emph{simply.}
Is this commodity-fetishism? Yes, of the \emph{worst kind}. Mainly, it is the kind that does not want to maintain capitalist social relations but one that seeks to destroy them. We’ve been living without and we want to remedy this situation. Do we also want to live with the deepest, most sensual set of social relations: YES. But why must we choose between the two? The destruction of capitalism\emph{, for communism}, will leave us with so much time to cultivate ourselves, our tastes, our desires. Pre-capitalist peoples did not dress themselves in tunics of ash gray or shave their heads \emph{en masse}. It is capitalism which has made our self-fashioning so impoverished; though glimmers of \emph{indulgent} self-fashioning sometimes does grace the streets; sadly only to be homogenized, recuperated and sold back to an indiscriminate consumer.
It is capitalism which has accustomed us to bland food \& drink, or tricked us into paying top dollar at the co-op. It is capitalism which has us moving our IKEA furniture from apartment to apartment. We imagine all the home furnishings to be plundered. Capitalism in its poverty of ideas, by way of colonialism, plunges itself into our indigenous cultures and sells us back what it took from us. We still remember that we used to build structures that still stand while cheap buildings kill so many now in disasters. We still remember that European colonialism spread its tentacles across the world because it was without and we lived in such wealth (after it had plundered its own).
“I want to shed myself of my \emph{first world privilege} and not live confined by how capitalism wants me to.” If only it were so simple. We’ve actually read this sentence (though its intent we’ve seen many, many times). This is \emph{pure reactionary thought}. To run and do the opposite just because capitalism displays certain social features does not make one an anticapitalist. It makes you a \emph{petit-bourgeois bohemian}. We all want to not pay rent, or pay for food, or have to work so many hours of our lives but \emph{there is no outside of capitalism}. Asceticism is not revolutionary. Even those nodes of autonomy scattered around the globe, like among the \emph{Zapatistas}, or Marinaleda, Spain still have to contend with the fact that Capital has them surrounded.
But we will not squat our way to a revolution. Squatting, dumpster-diving, train-hopping, stealing from work, work slowdowns are not acts of revolt but of \emph{resistance}. Thus we understand that the nice shit will not come until capitalism is done with, because little acts of appropriation will not really \emph{get the goods} as we see fit\emph{.}
This is no mere provocation – it is part of our intent. Communism, for us, is not as we were taught in schools: the general immiseration of everyone. Rather, as Marx so eloquently put forth in 1845, “the real movement that abolishes the present state of things.” The \emph{present state of things} is poverty, hunger, work, racialized social death, gendered violence, the unmitigated murder of transgender people, the free movement of goods but \emph{not} people and the general \emph{immiseration of everyday life}.
Further, a critique of consumerism (\& likewise Capital) that only asks us to \emph{consume less} misses the trees for the forest. Capital would have us consume less only to appease our consumer guilt. Let us not be fooled, Capital necessitates eternal growth and this growth is done on terms that will destroy us regardless of how much (or little) we buy. Capital has made a sin of our desires because they inevitably know that it cannot satisfy. To each according to their need, and to each according to their desire. We contend with capitalist logic and aim for the unreasonable because capitalist logic would have us cut ourselves from our ludic, indulgent dreams.
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L.A. ONDA, Hostis
Nice Shit for Everybody
March 24th, 2016
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\textbf{lib.anarhija.net}
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%%
%% Der LaTeX-Begleiter, zweite Auflage (September 2005)
%%
%% Beispiel 6-5-13 von Seite 331.
%%
%% Copyright (C) 2005 Frank Mittelbach, Michel Goossens,
%% Johannes Braams, David Carlisle, and Chris Rowley
%%
%% Uebersetzung: Copyright (C) 2005 Claudia Krysztofiak,
%% Rebecca Stiels und Frank Mittelbach
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%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
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\subfloat[Zwei]{\fbox{Abbildung II}}
\caption{Drei Abbildungen}
\end{figure}
\pagebreak % <-- zur Darstellung
\begin{figure}[!ht]
\centering \ContinuedFloat
\subfloat[Drei]{\fbox{Abbildung III}}
\caption[]{Drei Abbildungen (Forts.)}
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%%%
\documentclass[a4paper,amsmath]{oblivoir}
\setkormainfont{HCR Batang LVT}
\setkorsansfont{HCR Dotum LVT}
\usepackage{ifthen}
\newcommand{\numone}{①}
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\altsel{#3}\> \altsel{#4} \\
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\= \kill
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{\setlength{\altaltseldimen}{\linewidth}}%
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\else\ifdim\altaltseldimen>.18\linewidth
\def\ALTSELtype{2}%
\else\def\ALTSELtype{3}%
\fi \fi
}
\newcommand\munje[6]{%
\par
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\noindent\llap{\fbox{\themunje.~}} #1
\par
\setlength{\altaltseldimen}{0pt}%
\getaltseldimen{#2}{#3}{#4}{#5}{#6}%
% {\sffamily\tiny 제일 긴 줄 : \the\altaltseldimen} %%% debug
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\fi
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\begin{document}
\munje{100+200은 얼마인가?}%
{200}{300}{400}{500}{600}
\munje{다음 중 그 계산의 결과가 나머지 하나와 다른 것은 무엇인가?}%
{$10+20+30+40$}{$120-100+80+3+2-5$}%
{$25\times4+25-5-20$}{$500-100-100-100$}
{$(2\pi-\pi)\times\dfrac{100}{\pi}$}
\munje{평행사변형의 넓이를 구하는 방법은 무엇인가?}%
{평행사변형의 밑변과 높이를 구하여 더한다.}%
{평행사변형의 밑변과 높이를 구하여 곱한다.}%
{평행사변형의 밑변과 높이를 구하여 더한 뒤 2로 나눈다.}%
{평행사변형의 밑변과 높이를 구하여 곱한 뒤 2로 나눈다.}
{삼각형 두 개로 쪼개어 각각의 넓이를 구하여 합한다.}
\munje{평행사변형의 넓이를 구하는 방법은 무엇인가?}%
{평행사변형의 밑변과 높이를 구하여 더한다.}%
{평행사변형의 밑변과 높이를 구하여 곱한다.}%
{평행사변형의 밑변과 높이를 구하여 더한 뒤 2로 나눈다.}%
{없음}
{같은 크기의 색종이를 오려서 사각형으로 만들어 넓이를 구한다.}
\munje{200+300은 얼마인가?}%
{200}{300}{400}{500}{600}
\end{document}
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\[R_{n}(x)=\mathop{R_{{n}}\/}\nolimits\!\left(x;\alpha,\beta,\gamma,\delta\,|\,q%
\right)=\sum_{{\ell=0}}^{n}\frac{q^{\ell}\left(q^{{-n}},\alpha\beta q^{{n+1}};%
q\right)_{{\ell}}}{\left(\alpha q,\beta\delta q,\gamma q,q;q\right)_{{\ell}}}%
\*\prod_{{j=0}}^{{\ell-1}}(1-q^{j}x+\gamma\delta q^{{2j+1}})=\mathop{{{}_{{4}}%
\phi_{{3}}}\/}\nolimits\!\left({q^{{-n}},\alpha\beta q^{{n+1}},q^{{-y}},\gamma%
\delta q^{{y+1}}\atop\alpha q,\beta\delta q,\gamma q};q,q\right),\]
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%% TEMPLATE for articles submitted to the full-tex econf proceedings
%%
%%
%% Please do not remove lines commented out with %+
%% these are for the editors' use.
%%
%% Questions? Send email to : [email protected]
%%
\documentclass[12pt]{article}
\usepackage{epsfig}
\usepackage{lineno}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% Do not change these:
\textwidth=6.0in \textheight=8.25in
%% Adjust these for your printer:
\leftmargin=-0.3in \topmargin=-0.20in
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% personal abbreviations and macros
% the following package contains macros used in this document:
\input econfmacros.tex
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
% To include an item in the INDEX of the conference volume,
% flag it with \index{<item name>}
% The use of this macro is illustrated in the text.
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\def\Title#1{\begin{center} {\Large {\bf #1} } \end{center}}
\begin{document}
%\linenumbers
\Title{Fourth generation searches at ATLAS}
\bigskip\bigskip
%+\addtocontents{toc}{{\it D. Reggiano}}
%+\label{ReggianoStart}
\begin{raggedright}
{\it Dennis Wendland (Humboldt-Universit\"at zu Berlin, Germany)\index{Wendland, D.}\\
On behalf of the ATLAS Collaboration}
\bigskip\bigskip
\end{raggedright}
%\section{Introduction}
%The Standard Model of particle physics describes the elementary particles and their
%interactions. The fermions are grouped into families where three generations have been experimentally
%verified. However, a fourth family can not be excluded
%at this point, neither based on theoretical arguments nor by experimental constraints.
%The Standard Model of particle physics can not exclude a fourth family of fermions,
%neither based on theoretical arguments nor by experimental constraints.
In the following, four analyses with searches for heavy quarks of a 4th generation\index{fourth generation} are presented.
The analysed data ($1$~fb$^{-1}$) %$\mathcal L = 1$~fb$^{-1}$)
were taken with the ATLAS detector~\cite{atlas} at the LHC %Large Hadron Collider
at $\sqrt{s} = 7$~TeV. %~\cite{lhc}.
All analyses assume QCD pair-production of these Dirac fermions as a simple extension of the three
generations Standard Model (SM), where the up-type quark is noted as $t'$ and the down-type quark as $b'$.
%The analyses assume $100\%$ branching fraction for the heavy quark decays.
%If these particles exist they
4th generation quarks could play an interesting role in electroweak symmetry breaking~\cite{Holdom:2006mr}.
%Compared to SM they would also allow
%for a %significantly
%higher Higgs-boson mass~\cite{Kribs:2007nz}.
%A SM with four generations (SM4) has also the interesting property that gauge
%coupling unification can be generated at a scale of order $10^{15}-10^{16}$~GeV
%in the simplest non-supersymmetric Grand Unification model $SU(5)$.%~\cite{Hung:1997zj}.
A SM with four generations %(SM4\index{SM4})
has also the property to generate gauge coupling
unification at a scale of order $10^{15}-10^{16}$~GeV~\cite{Hung:1997zj}.
There are also discussions in the literature if a %SM4
fourth generation of quarks could play a central role
in baryogenesis~\cite{Hou:2008xd}.
%Direct Higgs searches have the power to exclude a sequential fourth generation:
%if a Higgs boson exists with a mass around 126 GeV a fourth generation extension
%of the Standard Model can be excluded if the Higgs signal is standard-model like
%in all decay channels (see e.g.~\cite{Djouadi:2012ae,Eberhardt:2012sb,Eberhardt:2012ck}).
%Moreover, lattice gauge theory studies suggest that very heavy fermions of order 600
%to 700 GeV constrain the Higgs mass, due to the large Yukawa coupling, to lie within
%a mass range of 500 to 800 GeV~\cite{Gerhold:2010wv}). If the Higgs boson is found at
%126 GeV a fourth generation might still be in accordance with experimental constraints
%though when extending the Higgs sector.
%\section{Physical object selection}
%\section{Results}
In the presented analyses, events are selected passing single electron or muon triggers.
Electrons are selected with a transverse energy of $E_T>25$~GeV, %$|\eta^{Cluster}|<2.47$, tight
tight quality selection criteria, calorimeter isolation,
and must be within $|\eta|<2.47$ ($\eta=-\ln\tan\theta/2$), %$|\eta| \notin [1.37,1.52]$), %
excluding $1.37<|\eta|<1.52$. Muons are selected with
a transverse momentum $P_T>20$~GeV, $|\eta|<2.5$, isolation and
a veto on cosmic muons. %cosmic rejection. % (in case of di-muon events).
%In addition, m
Muons overlapping with jets are removed. Jets are reconstructed with an
anti-$k_T$ algorithm and are required to have $P_T>25$~GeV (unless otherwise stated) and $|\eta|<2.5$. Jets
overlapping with electrons are removed. $H_T$ is defined as the scalar sum of jets
and leptons $P_T$. The %analyses with dilepton final-states
dilepton analyses apply cuts on the invariant mass $m_{inv}$
of %two same-flavor
the leptons ($ee/\mu\mu$) of $m_{inv}>15$~GeV and $m_{inv} \notin [81,101]$~GeV.
$m_T = \sqrt{2E_T^{Miss}P_T^\ell \left( 1-\cos \left( \Delta\Phi \left( E_T^{Miss},P_T^\ell \right) \right) \right)}$
describes the transverse mass
of lepton $\ell$ and neutrino in the single lepton analyses.
%\section{Results}
%
In \cite{singlep_bprime_note} $b'$ quarks are searched for in single lepton final states
($b'\rightarrow tW$, branching ratio BR=$100\%$), where the events are required to
have at least six jets, missing transverse energy $E_T^{Miss}>35$~GeV and
$m_T>25$~GeV ($e$ events), $E_T^{Miss}>20$~GeV and $E_T^{Miss}+m_T>60$~GeV
($\mu$ events) and %assuming high $P_T$ W decays $W\rightarrow jj$.
high-$P_T$~$W$ decays into two jets are identified.
The signal is extracted by a binned maximum-likelihood fit
of nine bins in ($N_W$,$N_{Jets}$), where $N_W=0,1,\geq2$ describes the number of reconstructed $W$ bosons and
$N_{Jets}=6,7,\geq 8$ the number of jets. No excess is found and a limit of $m_{b'}>480$~GeV (Fig. \ref{fig:limit_singlep})
is set.
In \cite{ss_note} $b'$ quarks are searched for in same-sign dilepton final states
($b'\rightarrow tW$, BR=$100\%$). The
events are required to have at least two jets ($P_T>20$~GeV), $H_T>350$~GeV and $E_T^{Miss}>40$~GeV. The signal is
extracted by a single-bin counting experiment. No excess is observed and a limit of
$m_{b'}>450$~GeV %(Fig. \ref{fig:limit_dilep})
is set.
In \cite{singlep_tprime_note} $t'$ quarks are searched for in single lepton final states,
($t'\rightarrow bW$, BR=$100\%$) where the events are required to
have at least three jets (one with $P_T>60$~GeV), at least one $b$-jet, $E_T^{Miss}+m_T>60$~GeV,
$E_T^{Miss}>35$~GeV ($e$ events) and
$E_T^{Miss}>20$~GeV ($\mu$ events). The reconstructed $t'$ mass $m_{reco}$ is used as discriminant in the signal extraction.
No excess is found and a limit of $m_{t'}>404$~GeV %(Fig. \ref{fig:limit_singlep})
is set.
In \cite{os_note} heavy 4th generation quarks $Q$ (benchmark model: $t'$) are searched for
in opposite-sign dilepton final states
($Q\rightarrow qW$), where the
events are required to have at least two jets, $H_T>130$~GeV ($e\mu$ events) and
$E_T^{Miss}>60$~GeV ($ee/\mu\mu$ events).
The signal is extracted by a fit of $m_{coll}$, which is the reconstructed heavy quark mass assuming the neutrinos
from %$Q\rightarrow q+W, W\rightarrow \ell+\nu$
$Q\rightarrow qW\rightarrow q\ell\nu$ are approximately collinear to the charged leptons.
No significant excess over expected background is observed and a limit of
$m_{Q}>350$~GeV %(Fig. \ref{fig:limit_dilep})
is set.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% use this format to include an .eps figure into your paper
%%
%\begin{figure}[htb]
%\begin{center}
%\begin{minipage}{0.3\textwidth}
%\epsfig{file=osdileptons_limit.eps,width=1.0\textwidth}
%\end{minipage}
%\begin{minipage}{0.32\textwidth}
%\epsfig{file=sstop_limit.eps,width=1.0\textwidth}
%\end{minipage}
%\caption{\textit{Left:} Limit on $Q$ mass in opposite-sign final states~\cite{os_note}.
% \textit{Right:} Limit on $b'$ mass in same-sign final states~\cite{ss_note}.}
%\label{fig:limit_dilep}
%\end{center}
%\end{figure}
%
%\begin{figure}[htb]
%\begin{center}
%\begin{minipage}{0.3\textwidth}
%\epsfig{file=singlelep_bprime_limit.eps,width=0.8\textwidth}
%\end{minipage}
%\begin{minipage}{0.33\textwidth}
%\epsfig{file=singlelep_tprime_limit.eps,width=1.0\textwidth}
%\end{minipage}
%\caption{\textit{Left:} Limit on $b'$ mass in single lepton final states~\cite{singlep_bprime_note}.
% \textit{Right:} Limit on $t'$ mass in single lepton final states~\cite{singlep_tprime_note}.}
%\label{fig:limit_singlep}
%\end{center}
%\end{figure}
\begin{figure}[htb]
\begin{center}
%\begin{minipage}{0.32\textwidth}
\epsfig{file=singlelep_bprime_bins.eps,width=0.32\textwidth}
%\end{minipage}
%\begin{minipage}{0.32\textwidth}
\epsfig{file=singlelep_bprime_limit.eps,width=0.32\textwidth}
%\end{minipage}
\caption{\textit{Left:} Discriminant. %~\cite{singlep_bprime_note}.
\textit{Right:} Limit on $b'$ mass in single lepton final states~\cite{singlep_bprime_note}.}
\label{fig:limit_singlep}
\end{center}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%
%% use this format to include a LaTeX table into your paper
%%
%\begin{table}[b]
%\begin{center}
%\begin{tabular}{l|ccc}
%Patient & Initial level($\mu$g/cc) & w. Magnet &
%w. Magnet and Sound \\ \hline
% Guglielmo B. & 0.12 & 0.10 & 0.001 \\
% Ferrando di N. & 0.15 & 0.11 & $< 0.0005$ \\ \hline
%\end{tabular}
%\caption{Blood cyanide levels for the two patients.}
%\label{tab:blood}
%\end{center}
%\end{table}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{thebibliography}{99}
%%
%% bibliographic items can be constructed using the LaTeX format in SPIRES:
%% see http://www.slac.stanford.edu/spires/hep/latex.html
%% SPIRES will also supply the CITATION line information; please include it.
%%
\bibitem{atlas}
ATLAS Collaboration, 2008 JINST 3 S08003
%\bibitem{lhc}
%L. Evans and P. Bryant, 2008 JINST 3 S08001
\bibitem{Holdom:2006mr}
B.~Holdom, JHEP 0608 (2006) 076
%\bibitem{Kribs:2007nz}
%G.~D. Kribs et al., %, T.~Plehn, M.~Spannowsky, and T.~M. Tait,
%Phys.Rev. {\bf D76} (2007) 075016
\bibitem{Hung:1997zj}
P.~Hung, Phys. Rev. Lett. 80 (1998) 3000--3003
\bibitem{Hou:2008xd}
W.-S. Hou, Chin. J. Phys. 47 (2009) 134
%\bibitem{Djouadi:2012ae}
%Djouadi, Abdelhak and A.~Lenz, arXiv:1204.1252
%\bibitem{Eberhardt:2012sb}
%O.~Eberhardt, G.~Herbert, H.~Lacker, A.~Lenz and A.~Menzel,
%arXiv:1204.3872
%\bibitem{Eberhardt:2012ck}
%O.~Eberhardt, A.~Lenz, A.~Menzel, U.~Nierste, M.~Wiebusch,
%arXiv:1207.0438
%\bibitem{Gerhold:2010wv}
%P.~Gerhold, K.~Jansen, J.~Kallarackal, arXiv:1011.1648
\bibitem{singlep_bprime_note}
ATLAS Collaboration, Phys. Rev. Lett. 109 (2012) 032001
%arXiv:1202.6540
%%CITATION = PWASA,13,1564;%%
\bibitem{ss_note}
ATLAS Collaboration, JHEP 1204 (2012) 069
%%CITATION = PWASA,13,1564;%%
\bibitem{singlep_tprime_note}
ATLAS Collaboration, Phys. Rev. Lett. 108 (2012) 261802.
%arXiv:1202.3076
%%CITATION = PWASA,13,1564;%%
\bibitem{os_note}
ATLAS Collaboration, Phys. Rev. D86 (2012) 012007
%arXiv:1202.3389
%%CITATION = PWASA,13,1564;%%
\end{thebibliography}
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\setlength{\parskip}{0.3cm}\setlength{\parindent}{0.0cm}
\bigskip\bigskip {\Large {\bf Discussion}} \bigskip}
\def\speaker#1{{\bf #1:}\ }
\def\endDiscussion{}
%\Discussion
%\endDiscussion
\end{document}
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\[(\tfrac{1}{2}p-\ell+\tfrac{1}{2})\xi B_{2\ell-1}+\left((2\ell+1)^{2}-\eta%
\right)B_{2\ell+1}+(\tfrac{1}{2}p+\ell+\tfrac{3}{2})\xi B_{2\ell+3}=0,\]
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\begin{document}
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\toccontents
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\subsection*{הוראות כלליות לכל המסלולים:}
\begin{itemize}
\item{} מומלץ ללמוד בסמסטר א, שנה א את הקורס \href{/teaching/generic_courses/proof-writing-workshop}{סדנה בכתיבת הוכחות}. קורס זה הינו קורס בחירה שניתן ללמוד רק בסמסטר א שנה א (פרט לאישורים חריגים של ועדת הוראה). קורס זה אינו נחשב קורס בחירה רגיל במסגרת חובות הבחירה של המסלולים השונים.
\item{} חובה להשלים את דרישות הלימודים באנגלית (2 נק״ז או מבחן פטור), עד תם שנה ב`.
\item{} חובה להירשם לקורסים הבאים בסמסטר א, שנה א:
\begin{itemize}
\item{} \href{/teaching/generic_courses/library-orientation}{הכרת הספרייה}
\item{} \href{/teaching/generic_courses/prevention-sexual-harrasment}{לומדה להכרת החוק למניעת הטרדה מינית}
\end{itemize}
\item{} חובה ללמוד במקביל לקורס \href{/teaching/generic_courses/probability}{הסתברות} את הקורסים \href{/teaching/generic_courses/intro-analysis}{מבוא לאנליזה} ו\href{/teaching/generic_courses/geometric-infi-1}{חשבון אינפיניטסימלי גיאומטרי 1}. לחלופין, ניתן ללמוד את הקורס בהסתברות לאחר שנלמדו שני הקורסים הנ``ל.
\item{} הקורס \href{/teaching/generic_courses/probability}{הסתברות} יינתן בשפה האנגלית, בהתאם להנחיית האוניברסיטה.
\item{} ישנן דרישות נוספות לשם המשך לתואר שני, המסומנות בטבלאות הקורסים. על תלמידים אשר שוקלים המשך לתואר שני לקחת זאת בחשבון בתכנון מערכת השעות.
\end{itemize}
\section*{סיכום הדרישות במסלולים השונים}
\global\tabulinesep=1mm
{
\begin{tabu} to\linewidth{X|X|X|X|X|X|X|X|X|X}
\hline
\BTC{דרישת מסלול}
& \BTC{מתמטיקה כללית}
& \BTC{מתמטיקה מחלקה ראשית עם חטיבה}
& \BTC{חטיבה במתמטיקה}
& \BTC{מסלול דו-מחלקתי במתמטיקה ומדעי המחשב}
& \BTC{מסלול דו-מחלקתי במתמטיקה ופיסיקה}
& \BTC{מסלול דו-מחלקתי במתמטיקה ומדעי הרוח}
& \BTC{מתמטיקה והנדסת תעשייה וניהול}
& \BTC{מתמטיקה והנדסת חשמל ומחשבים}
& \BTC{מתמטיקה והנדסת מכונות}
\\
\hline
\hline
\BTC{לימודי חובה במתמטיקה}
& \TC{61.0 נק``ז}
& \TC{61.0 נק``ז}
& \TC{20.0 נק``ז}
& \TC{49.0 נק``ז}
& \TC{56.0 נק``ז}
& \TC{45.0 נק``ז}
& \TC{65.0 נק``ז}
& \TC{73.0 נק``ז}
& \TC{69.0 נק``ז}
\\
\hline
\BTC{חובת בחירה באנליזה}
& \TC{קורס אחד לפחות}
& \TC{קורס אחד לפחות}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{קורס אחד לפחות}
& \TC{}
& \TC{}
\\
\hline
\BTC{חובת בחירה באלגברה}
& \TC{קורס אחד לפחות}
& \TC{קורס אחד לפחות}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{קורס אחד לפחות}
& \TC{קורס אחד לפחות}
& \TC{קורס אחד לפחות}
\\
\hline
\BTC{בחירה במתמטיקה}
& \TC{לפחות 24.0 ולכל היותר 50.0 נק``ז}
& \TC{בדיוק 18.0 נק``ז}
& \TC{לפחות 8.0 נק``ז}
& \TC{לפחות 2 קורסים, לכל היותר 26.0 נק``ז}
& \TC{לפחות 8.0 נק``ז}
& \TC{לפחות 5 קורסים, לפחות 20.0 נק``ז}
& \TC{לפחות 8.0 ולכל היותר 13.0 נק``ז}
& \TC{לפחות 8.0 ולכל היותר 13.0 נק``ז}
& \TC{בדיוק 17.0 נק``ז}
\\
\hline
\BTC{לימודי חובה במדעי המחשב}
& \TC{3.0 נק``ז}
& \TC{3.0 נק``ז}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
\\
\hline
\BTC{לימודי השלמה למדעים}
& \TC{לכל היותר 20.0 נק``ז}
& \TC{}
& \TC{}
& \TC{לכל היותר 8.0 נק``ז}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
\\
\hline
\BTC{לימודי בחירה כלליים}
& \TC{לכל היותר 6.0 נק``ז}
& \TC{לכל היותר 4.0 נק``ז}
& \TC{}
& \TC{לכל היותר 6.0 נק``ז}
& \TC{}
& \TC{לפחות 2.0 ולכל היותר 3.0 נק``ז}
& \TC{לכל היותר 3.5 נק``ז}
& \TC{לכל היותר 2.0 נק``ז}
& \TC{לכל היותר 4.0 נק``ז}
\\
\hline
\BTC{לימודי חובה כלליים}
& \TC{2.0 נק``ז}
& \TC{2.0 נק``ז}
& \TC{2.0 נק``ז}
& \TC{2.0 נק``ז}
& \TC{2.0 נק``ז}
& \TC{2.0 נק``ז}
& \TC{2.0 נק``ז}
& \TC{2.0 נק``ז}
& \TC{2.0 נק``ז}
\\
\hline
\BTC{חטיבה חיצונית}
& \TC{}
& \TC{בדיוק 28.0 נק``ז}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
\\
\hline
\BTC{חובת-בחירה במתמטיקה}
& \TC{}
& \TC{}
& \TC{}
& \TC{לפחות 2 קורסים}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
\\
\hline
\BTC{לימודי חובה במחלקה האחרת}
& \TC{}
& \TC{}
& \TC{}
& \TC{35.5 נק``ז}
& \TC{לפחות 55.0 נק``ז}
& \TC{לפחות 54.0 נק``ז}
& \TC{לפחות 119.5 ולכל היותר 123.5 נק``ז}
& \TC{לפחות 119.0 ולכל היותר 123.0 נק``ז}
& \TC{בדיוק 114.5 נק``ז}
\\
\hline
\BTC{לימודי בחירה במחלקה האחרת}
& \TC{}
& \TC{}
& \TC{}
& \TC{קורס אחד לפחות}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
\\
\hline
\BTC{לימודי חובה בפיסיקה}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{}
& \TC{4.0 נק``ז}
& \TC{}
\\
\hline
\BTC{נק``ז כולל}
& \TC{124.0}
& \TC{124.0}
& \TC{124.0}
& \TC{124.0}
& \TC{124.0}
& \TC{124.0}
& \TC{210.0}
& \TC{210.0}
& \TC{210.0}
\\
\hline
\end{tabu}
}
\restoregeometry
\chapter*{מסלולי לימוד -- תואר חד-חוגי}
\section{מתמטיקה כללית}
\subsection*{תיאור מסלול הלימוד}
המגמה במתמטיקה כללית מעניקה לתלמיד רקע רחב ומעמיק במגוון התחומים המרכזיים של המתמטיקה. הלימודים כוללים מקצועות חובה בסיסיים במתמטיקה, ואפשרויות רחבות של מקצועות בחירה לפי הכוון המועדף על התלמיד; במיוחד ניתן לבנות תוכנית בכיוון המתמטיקה השימושית.
\subsection*{דרישות המסלול}
\subsubsection{לימודי חובה במתמטיקה}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:basic-mandatory-courses}},
\textbf{\nameref{tab:basic-mandatory-courses}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
1011 &
חשבון אינפינטסימלי 1 &
4/2 &
5.0 &
סתיו &
\\
\tabucline-
1021 &
חשבון אינפינטסימלי 2 &
4/2 &
5.0 &
אביב &
חשבון אינפינטסימלי 1\\
\tabucline-
1031 &
חשבון אינפינטסימלי גאומטרי 1 &
4/0 &
4.0 &
סתיו &
חשבון אינפינטסימלי 2, אלגברה לינארית 2\\
\tabucline-
1051 &
מבוא לאנליזה &
4/0 &
4.0 &
סתיו &
חשבון אינפינטסימלי 2, אלגברה לינארית 2\\
\tabucline-
1211 &
אלגברה לינארית 1 &
4/2 &
5.0 &
סתיו &
\\
\tabucline-
1221 &
אלגברה לינארית 2 &
4/2 &
5.0 &
אביב &
אלגברה לינארית 1\\
\tabucline-
7031 &
מבנים אלגבריים &
4/0 &
4.0 &
סתיו &
אלגברה לינארית 2\\
\tabucline-
8001 &
הסתברות &
4/0 &
4.0 &
סתיו &
חשבון אינפינטסימלי 2\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{36.0}\\
\tabucline-
\end{tabu}
\caption[קורסי חובה בסיסיים]{קורסי חובה בסיסיים. חובה ללמוד במקביל לקורס \href{/teaching/generic_courses/probability}{הסתברות} (קורס ניתן בשפה האנגלית) את הקורסים \href{/teaching/generic_courses/intro_analysis}{מבוא לאנליזה} ו\href{/teaching/generic_courses/infi1}{חשבון אינפיניטסימלי גיאומטרי 1}. לחלופין ניתן ללמוד את הקורס בהסתברות לאחר שנלמדו שני הקורסים הנ``ל.\label{tab:basic-mandatory-courses}}
\end{table}
\item %
\emph{טבלה~\ref{tab:advanced-mandatory-math-courses}},
\textbf{\nameref{tab:advanced-mandatory-math-courses}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
0061 &
משוואות דיפרנציאליות רגילות &
4/0 &
4.0 &
אביב &
מבוא לאנליזה\\
\tabucline-
0091 &
מבוא לטופולוגיה &
4/0 &
4.0 &
אביב &
מבוא לאנליזה\\
\tabucline-
0251 &
תורת הפונקציות המרוכבות &
4/0 &
4.0 &
אביב &
חשבון אינפינטסימלי גאומטרי 1, מבוא לאנליזה\\
\tabucline-
1041 &
חשבון אינפינטסימלי גאומטרי 2 &
4/0 &
4.0 &
אביב &
חשבון אינפינטסימלי גאומטרי 1\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{16.0}\\
\tabucline-
\end{tabu}
\caption[קורסי חובה מתקדמים במתמטיקה]{קורסי חובה מתקדמים במתמטיקה. \label{tab:advanced-mandatory-math-courses}}
\end{table}
\item %
\emph{טבלה~\ref{tab:additional-core-math-courses}},
\textbf{\nameref{tab:additional-core-math-courses}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
0171 &
מבוא לתורת הקבוצות &
4/0 &
4.0 &
אביב &
מתמטיקה בדידה\\
\tabucline-
2201 &
מתמטיקה בדידה &
4/2 &
5.0 &
סתיו &
\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{9.0}\\
\tabucline-
\end{tabu}
\caption[קורסי ליבה נוספים במתמטיקה]{קורסי ליבה נוספים במתמטיקה. \label{tab:additional-core-math-courses}}
\end{table}
\end{itemize}
\subsubsection{חובת בחירה באנליזה}
קורס אחד לפחות מתוך %
\emph{טבלה~\ref{tab:analysis-mandatory-elective}},
\textbf{\nameref{tab:analysis-mandatory-elective}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
0081 &
יסודות תורת המידה &
4/0 &
4.0 &
סתיו &
מבוא לאנליזה\\
\tabucline-
0291 &
התמרות אינטגרליות ומשוואות דיפרנציאליות חלקיות &
4/0 &
4.0 &
סתיו &
חשבון אינפינטסימלי גאומטרי 1, מבוא לאנליזה\\
\tabucline-
\end{tabu}
\caption[חובת בחירה באנליזה]{חובת בחירה באנליזה. \label{tab:analysis-mandatory-elective}}
\end{table}
\subsubsection{חובת בחירה באלגברה}
קורס אחד לפחות מתוך %
\emph{טבלה~\ref{tab:algebra-mandatory-elective}},
\textbf{\nameref{tab:algebra-mandatory-elective}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
6031 &
תורת המספרים &
4/0 &
4.0 &
סתיו &
חשבון אינפינטסימלי 1, אלגברה לינארית 1\\
\tabucline-
7041 &
תורת השדות ותורת גלואה &
4/0 &
4.0 &
אביב &
מבנים אלגבריים\\
\tabucline-
7071 &
מבוא לאלגברה קומוטטיבית &
4/0 &
4.0 &
סתיו &
מבנים אלגבריים\\
\tabucline-
\end{tabu}
\caption[חובת בחירה באלגברה]{חובת בחירה באלגברה. \label{tab:algebra-mandatory-elective}}
\end{table}
\subsubsection{בחירה במתמטיקה}
קורסים מתקדמים הניתנים על-ידי המחלקה. הקורסים מתפרסמים ב\href{/teaching/term}{דף המחלקה}, לפני תחילת כל סמסטר
\subsubsection{לימודי חובה במדעי המחשב}
יש לקחת את כל הקורסים מתוך %
\emph{טבלה~\ref{tab:mandatory-computer-science-courses}},
\textbf{\nameref{tab:mandatory-computer-science-courses}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
202.1.9041 &
מבוא לתכנות עם פייתון &
2/2 &
3.0 &
סתיו &
\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{3.0}\\
\tabucline-
\end{tabu}
\caption[קורסי חובה במדעי המחשב]{קורסי חובה במדעי המחשב. במקרים חריגים ניתן ללמוד במקום קורס תכנות אחר באישור ועדת ההוראה\label{tab:mandatory-computer-science-courses}}
\end{table}
\subsubsection{לימודי השלמה למדעים}
קורסי בחירה ממחלקות בפקולטה למדעי הטבע או באחת המחלקות להנדסה אשר מקיימות תואר כפול עם המחלקה למתמטיקה, או ביחידה לקידום המקצועיות בחינוך. רשימת הקורסים המוצעים תפורסם במחלקות הרלוונטיות או במערכת שעות האוניברסיטאית.
רישום לקורסים בסעיף זה יהיה באישור ועדת ההוראה של המחלקה למתמטיקה, ועל בסיס מקום פנוי. ככלל, למעט מקרים חריגים, ועדת ההוראה למתמטיקה תאשר קורסים בתנאים הבאים:
\begin{itemize}
\item{} אין חפיפה משמעותית בין הקורס המבוקש לבין קורס אשר ניתן ע``י המחלקה למתמטיקה.
\item{} אין חפיפה משמעותית בין הקורס המבוקש לבין קורס אחר בתכנית הלימודים של התלמיד/ה.
\item{} לא יאושרו קורסי שירות אשר ניתנים על ידי מחלקה אחת לתלמידי מחלקה אחרת, למעט אם ישנה הצדקה מיוחדת לכך.
\item{} לא יאושר רישום לקורסים במסגרת זו במהלך סמסטר א` של שנה א`.
\end{itemize}
למען הסר ספק, אין חובה לבחור בקורסים מקטגוריה זו
\subsubsection{לימודי בחירה כלליים}
קורסים כלשהם מכלל המופיעים בשנתון
\subsubsection{לימודי חובה כלליים}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:mandatory-english-courses}},
\textbf{\nameref{tab:mandatory-english-courses}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
153.1.5051 &
אנגלית מתקדמים 2 טבע &
4/0 &
2.0 &
סתיו, אביב &
\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{2.0}\\
\tabucline-
\end{tabu}
\caption[קורסי חובה באנגלית]{קורסי חובה באנגלית. \label{tab:mandatory-english-courses}}
\end{table}
\item %
\emph{טבלה~\ref{tab:mandatory-general-courses}},
\textbf{\nameref{tab:mandatory-general-courses}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
299.1.1121 &
הדרכה בספריה &
0/0 &
0.0 &
סתיו, אביב &
\\
\tabucline-
900.5.5001 &
לומדה להכרת החוק והנהלים למניעת הטרדה מינית &
0/0 &
0.0 &
סתיו, אביב &
\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{0.0}\\
\tabucline-
\end{tabu}
\caption[קורסי חובה כלליים]{קורסי חובה כלליים. \label{tab:mandatory-general-courses}}
\end{table}
\end{itemize}
\newpage
\subsection*{גרף תלויות}
\begin{tikzpicture}[remember picture, overlay]
\node [shift={(0cm,-1.5cm)}] at (current page) {%
\tikzstyle{every node}=[text width=70pt,align=center]
\begin{tikzpicture}[>=latex,line join=bevel,]
%%
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\node (GenericCourseDecorator-134) at (246.0bp,451.83bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי גאומטרי 1}}};
\node (GenericCourseDecorator-133) at (246.0bp,327.83bp) [draw,ellipse] {\footnotesize{\RL{מבוא לאנליזה}}};
\node (GenericCourseDecorator-8) at (246.0bp,541.83bp) [draw,ellipse] {\footnotesize{\RL{הסתברות}}};
\node (GenericCourseDecorator-13) at (79.0bp,485.83bp) [draw,ellipse] {\footnotesize{\RL{תורת הפונקציות המרוכבות}}};
\node (GenericCourseDecorator-135) at (79.0bp,574.83bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי גאומטרי 2}}};
\node (GenericCourseDecorator-113) at (79.0bp,406.83bp) [draw,ellipse] {\footnotesize{\RL{התמרות אינטגרליות ומשוואות דיפרנציאליות חלקיות}}};
\node (GenericCourseDecorator-136) at (79.0bp,327.83bp) [draw,ellipse] {\footnotesize{\RL{משוואות דיפרנציאליות רגילות}}};
\node (GenericCourseDecorator-12) at (79.0bp,249.83bp) [draw,ellipse] {\footnotesize{\RL{מבוא לטופולוגיה}}};
\node (GenericCourseDecorator-4) at (79.0bp,170.83bp) [draw,ellipse] {\footnotesize{\RL{יסודות תורת המידה}}};
\node (GenericCourseDecorator-176) at (372.0bp,299.83bp) [draw,ellipse] {\footnotesize{\RL{אלגברה לינארית 2}}};
\node (GenericCourseDecorator-1) at (246.0bp,92.833bp) [draw,ellipse] {\footnotesize{\RL{מבנים אלגבריים}}};
\node (GenericCourseDecorator-17) at (79.0bp,92.833bp) [draw,ellipse] {\footnotesize{\RL{תורת השדות ותורת גלואה}}};
\node (GenericCourseDecorator-92) at (79.0bp,13.833bp) [draw,ellipse] {\footnotesize{\RL{מבוא לאלגברה קומוטטיבית}}};
\node (GenericCourseDecorator-177) at (485.0bp,322.83bp) [draw,ellipse] {\footnotesize{\RL{אלגברה לינארית 1}}};
\node (GenericCourseDecorator-102) at (372.0bp,569.83bp) [draw,ellipse] {\footnotesize{\RL{מבוא לתורת הקבוצות}}};
\node (GenericCourseDecorator-56) at (485.0bp,569.83bp) [draw,ellipse] {\footnotesize{\RL{מתמטיקה בדידה}}};
\draw [->] (GenericCourseDecorator-131) ..controls (446.59bp,453.98bp) and (422.01bp,462.61bp) .. (GenericCourseDecorator-132);
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\draw [->] (GenericCourseDecorator-132) ..controls (327.63bp,470.04bp) and (305.66bp,465.08bp) .. (GenericCourseDecorator-134);
\draw [->] (GenericCourseDecorator-132) ..controls (343.27bp,445.84bp) and (282.36bp,371.17bp) .. (GenericCourseDecorator-133);
\draw [->] (GenericCourseDecorator-132) ..controls (332.37bp,499.15bp) and (288.74bp,520.96bp) .. (GenericCourseDecorator-8);
\draw [->] (GenericCourseDecorator-134) ..controls (183.83bp,464.44bp) and (140.49bp,473.37bp) .. (GenericCourseDecorator-13);
\draw [->] (GenericCourseDecorator-134) ..controls (205.99bp,480.91bp) and (134.74bp,534.03bp) .. (GenericCourseDecorator-135);
\draw [->] (GenericCourseDecorator-134) ..controls (188.21bp,436.34bp) and (138.87bp,422.89bp) .. (GenericCourseDecorator-113);
\draw [->] (GenericCourseDecorator-133) ..controls (227.97bp,347.6bp) and (191.44bp,389.26bp) .. (158.0bp,420.83bp) .. controls (137.0bp,440.66bp) and (110.93bp,461.73bp) .. (GenericCourseDecorator-13);
\draw [->] (GenericCourseDecorator-133) ..controls (208.65bp,345.24bp) and (135.23bp,380.39bp) .. (GenericCourseDecorator-113);
\draw [->] (GenericCourseDecorator-133) ..controls (199.94bp,327.83bp) and (166.53bp,327.83bp) .. (GenericCourseDecorator-136);
\draw [->] (GenericCourseDecorator-133) ..controls (208.53bp,310.59bp) and (134.53bp,275.61bp) .. (GenericCourseDecorator-12);
\draw [->] (GenericCourseDecorator-133) ..controls (227.93bp,308.31bp) and (191.33bp,267.14bp) .. (158.0bp,235.83bp) .. controls (136.95bp,216.06bp) and (110.89bp,194.97bp) .. (GenericCourseDecorator-4);
\draw [->] (GenericCourseDecorator-176) ..controls (344.03bp,332.9bp) and (286.76bp,403.09bp) .. (GenericCourseDecorator-134);
\draw [->] (GenericCourseDecorator-176) ..controls (325.48bp,310.11bp) and (294.51bp,317.1bp) .. (GenericCourseDecorator-133);
\draw [->] (GenericCourseDecorator-176) ..controls (346.94bp,259.66bp) and (278.94bp,146.15bp) .. (GenericCourseDecorator-1);
\draw [->] (GenericCourseDecorator-1) ..controls (194.7bp,92.833bp) and (159.13bp,92.833bp) .. (GenericCourseDecorator-17);
\draw [->] (GenericCourseDecorator-1) ..controls (208.65bp,75.43bp) and (135.23bp,40.279bp) .. (GenericCourseDecorator-92);
\draw [->] (GenericCourseDecorator-177) ..controls (450.52bp,343.04bp) and (411.59bp,366.54bp) .. (GenericCourseDecorator-16);
\draw [->] (GenericCourseDecorator-177) ..controls (444.06bp,314.56bp) and (424.67bp,310.54bp) .. (GenericCourseDecorator-176);
\draw [->] (GenericCourseDecorator-56) ..controls (446.25bp,569.83bp) and (430.71bp,569.83bp) .. (GenericCourseDecorator-102);
%
\end{tikzpicture}
};
\end{tikzpicture}
\newpage
\subsection*{תכנית לימודים מומלצת}
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה ראשונה--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
1011 &
חשבון אינפינטסימלי 1 &
5\\
\tabucline-
202.1.9041 &
מבוא לתכנות עם פייתון &
3\\
\tabucline-
299.1.1121 &
הדרכה בספריה &
0\\
\tabucline-
900.5.5001 &
לומדה להכרת החוק והנהלים למניעת הטרדה מינית &
0\\
\tabucline-
2201 &
מתמטיקה בדידה &
5\\
\tabucline-
2241 &
סדנא בכתיבת הוכחות &
1\\
\tabucline-
153.1.5041 &
אנגלית מתקדמים 1 (טבע) &
0\\
\tabucline-
1211 &
אלגברה לינארית 1 &
5\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{19}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה ראשונה--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
0171 &
מבוא לתורת הקבוצות &
4\\
\tabucline-
1021 &
חשבון אינפינטסימלי 2 &
5\\
\tabucline-
153.1.5051 &
אנגלית מתקדמים 2 טבע &
2\\
\tabucline-
1221 &
אלגברה לינארית 2 &
5\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה מחוץ למחלקה} &
\LR{4.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{20}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שניה--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
1031 &
חשבון אינפינטסימלי גאומטרי 1 &
4\\
\tabucline-
1051 &
מבוא לאנליזה &
4\\
\tabucline-
7031 &
מבנים אלגבריים &
4\\
\tabucline-
8001 &
הסתברות &
4\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה} &
\LR{6.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{22}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שניה--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
1041 &
חשבון אינפינטסימלי גאומטרי 2 &
4\\
\tabucline-
0091 &
מבוא לטופולוגיה &
4\\
\tabucline-
0061 &
משוואות דיפרנציאליות רגילות &
4\\
\tabucline-
0251 &
תורת הפונקציות המרוכבות &
4\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה} &
\LR{5.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{21}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שלישית--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
0081 &
יסודות תורת המידה &
4\\
\tabucline-
6031 &
תורת המספרים &
4\\
\tabucline-
6061 &
לוגיקה &
4\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה} &
\LR{10.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{22}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שלישית--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
7041 &
תורת השדות ותורת גלואה &
4\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה} &
\LR{16.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{20}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\clearpage
\section{מתמטיקה מחלקה ראשית עם חטיבה}
\subsection*{תיאור מסלול הלימוד}
בתוכנית לימודים של ראשי + חטיבה התלמיד לומד את רוב הקורסים במחלקה הראשית וקורסים נוספים של חטיבה במחלקה אחרת המציעה חטיבה על פי בחירתו. בסיום הלימודים התלמיד מקבל תואר עפ``י המחלקה הראשית.
המחלקה למתמטיקה מציעה תוכנית לימודים של מתמטיקה כמחלקה ראשית וכן חטיבת לימודים במתמטיקה. בוגרי מתמטיקה במחלקה ראשית הזאת יכולים להתקבל לתואר שני במחלקה למתמטיקה ללא השלמות (בתנאי שציוניהם עומדים בדרישות הסף). לימודים בתוכנית מתמטיקה ראשית מקנים תעודת תואר B.Sc במתמטיקה.
במקרה של חפיפה חלקית בין טבלאות מקצועות החובה של המחלקות, על הסטודנט להשלים את יתרת הנקודות מתוך מקצועות הבחירה במחלקה הראשית או החטיבה, בכפוף לאישור יו``ר ועדת הוראה של המחלקה הראשית.
במחלקות בהן ניתנים קורסי שירות של המחלקה למתמטיקה על התלמיד ללמוד את קורסי המתמטיקה עפ``י תכנית הלימודים של מתמטיקה כללית, כפי שיקבע על ידי ועדת ההוראה של המחלקה למתמטיקה. לתלמידים הלומדים חטיבה במתמטיקה מומלץ ללמוד קורסים מתקדמים של המחלקה למתמטיקה במקום קורסי השירות המקבילים בתחום המתמטיקה הניתנים במחלקה הראשית.
\subsection*{דרישות המסלול}
\subsubsection{לימודי חובה במתמטיקה}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:basic-mandatory-courses}},
\textbf{\nameref{tab:basic-mandatory-courses}}
\item %
\emph{טבלה~\ref{tab:advanced-mandatory-math-courses}},
\textbf{\nameref{tab:advanced-mandatory-math-courses}}
\item %
\emph{טבלה~\ref{tab:additional-core-math-courses}},
\textbf{\nameref{tab:additional-core-math-courses}}
\end{itemize}
\subsubsection{חובת בחירה באנליזה}
קורס אחד לפחות מתוך %
\emph{טבלה~\ref{tab:analysis-mandatory-elective}},
\textbf{\nameref{tab:analysis-mandatory-elective}}
\subsubsection{חובת בחירה באלגברה}
קורס אחד לפחות מתוך %
\emph{טבלה~\ref{tab:algebra-mandatory-elective}},
\textbf{\nameref{tab:algebra-mandatory-elective}}
\subsubsection{בחירה במתמטיקה}
קורסים מתקדמים הניתנים על-ידי המחלקה. הקורסים מתפרסמים ב\href{/teaching/term}{דף המחלקה}, לפני תחילת כל סמסטר
\subsubsection{לימודי חובה במדעי המחשב}
יש לקחת את כל הקורסים מתוך %
\emph{טבלה~\ref{tab:mandatory-computer-science-courses}},
\textbf{\nameref{tab:mandatory-computer-science-courses}}
\subsubsection{חטיבה חיצונית}
על הסטודנט ללמוד חטיבה מובנית ממחלקה אחרת
\subsubsection{לימודי חובה כלליים}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:mandatory-english-courses}},
\textbf{\nameref{tab:mandatory-english-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-general-courses}},
\textbf{\nameref{tab:mandatory-general-courses}}
\end{itemize}
\subsubsection{לימודי בחירה כלליים}
הקורסים ייבחרו על ידי הסטודנט מתוך מכלול הקורסים המוצעים באוניברסיטה
\newpage
\subsection*{גרף תלויות}
\begin{tikzpicture}[remember picture, overlay]
\node [shift={(0cm,-1.5cm)}] at (current page) {%
\tikzstyle{every node}=[text width=70pt,align=center]
\begin{tikzpicture}[>=latex,line join=bevel,]
%%
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\node (GenericCourseDecorator-132) at (372.0bp,479.83bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי 2}}};
\node (GenericCourseDecorator-16) at (372.0bp,389.83bp) [draw,ellipse] {\footnotesize{\RL{תורת המספרים}}};
\node (GenericCourseDecorator-134) at (246.0bp,451.83bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי גאומטרי 1}}};
\node (GenericCourseDecorator-133) at (246.0bp,327.83bp) [draw,ellipse] {\footnotesize{\RL{מבוא לאנליזה}}};
\node (GenericCourseDecorator-8) at (246.0bp,541.83bp) [draw,ellipse] {\footnotesize{\RL{הסתברות}}};
\node (GenericCourseDecorator-13) at (79.0bp,485.83bp) [draw,ellipse] {\footnotesize{\RL{תורת הפונקציות המרוכבות}}};
\node (GenericCourseDecorator-135) at (79.0bp,574.83bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי גאומטרי 2}}};
\node (GenericCourseDecorator-113) at (79.0bp,406.83bp) [draw,ellipse] {\footnotesize{\RL{התמרות אינטגרליות ומשוואות דיפרנציאליות חלקיות}}};
\node (GenericCourseDecorator-136) at (79.0bp,327.83bp) [draw,ellipse] {\footnotesize{\RL{משוואות דיפרנציאליות רגילות}}};
\node (GenericCourseDecorator-12) at (79.0bp,249.83bp) [draw,ellipse] {\footnotesize{\RL{מבוא לטופולוגיה}}};
\node (GenericCourseDecorator-4) at (79.0bp,170.83bp) [draw,ellipse] {\footnotesize{\RL{יסודות תורת המידה}}};
\node (GenericCourseDecorator-176) at (372.0bp,299.83bp) [draw,ellipse] {\footnotesize{\RL{אלגברה לינארית 2}}};
\node (GenericCourseDecorator-1) at (246.0bp,92.833bp) [draw,ellipse] {\footnotesize{\RL{מבנים אלגבריים}}};
\node (GenericCourseDecorator-17) at (79.0bp,92.833bp) [draw,ellipse] {\footnotesize{\RL{תורת השדות ותורת גלואה}}};
\node (GenericCourseDecorator-92) at (79.0bp,13.833bp) [draw,ellipse] {\footnotesize{\RL{מבוא לאלגברה קומוטטיבית}}};
\node (GenericCourseDecorator-177) at (485.0bp,322.83bp) [draw,ellipse] {\footnotesize{\RL{אלגברה לינארית 1}}};
\node (GenericCourseDecorator-102) at (372.0bp,569.83bp) [draw,ellipse] {\footnotesize{\RL{מבוא לתורת הקבוצות}}};
\node (GenericCourseDecorator-56) at (485.0bp,569.83bp) [draw,ellipse] {\footnotesize{\RL{מתמטיקה בדידה}}};
\draw [->] (GenericCourseDecorator-131) ..controls (446.59bp,453.98bp) and (422.01bp,462.61bp) .. (GenericCourseDecorator-132);
\draw [->] (GenericCourseDecorator-131) ..controls (446.92bp,423.8bp) and (413.11bp,408.26bp) .. (GenericCourseDecorator-16);
\draw [->] (GenericCourseDecorator-132) ..controls (327.63bp,470.04bp) and (305.66bp,465.08bp) .. (GenericCourseDecorator-134);
\draw [->] (GenericCourseDecorator-132) ..controls (343.27bp,445.84bp) and (282.36bp,371.17bp) .. (GenericCourseDecorator-133);
\draw [->] (GenericCourseDecorator-132) ..controls (332.37bp,499.15bp) and (288.74bp,520.96bp) .. (GenericCourseDecorator-8);
\draw [->] (GenericCourseDecorator-134) ..controls (183.83bp,464.44bp) and (140.49bp,473.37bp) .. (GenericCourseDecorator-13);
\draw [->] (GenericCourseDecorator-134) ..controls (205.99bp,480.91bp) and (134.74bp,534.03bp) .. (GenericCourseDecorator-135);
\draw [->] (GenericCourseDecorator-134) ..controls (188.21bp,436.34bp) and (138.87bp,422.89bp) .. (GenericCourseDecorator-113);
\draw [->] (GenericCourseDecorator-133) ..controls (227.97bp,347.6bp) and (191.44bp,389.26bp) .. (158.0bp,420.83bp) .. controls (137.0bp,440.66bp) and (110.93bp,461.73bp) .. (GenericCourseDecorator-13);
\draw [->] (GenericCourseDecorator-133) ..controls (208.65bp,345.24bp) and (135.23bp,380.39bp) .. (GenericCourseDecorator-113);
\draw [->] (GenericCourseDecorator-133) ..controls (199.94bp,327.83bp) and (166.53bp,327.83bp) .. (GenericCourseDecorator-136);
\draw [->] (GenericCourseDecorator-133) ..controls (208.53bp,310.59bp) and (134.53bp,275.61bp) .. (GenericCourseDecorator-12);
\draw [->] (GenericCourseDecorator-133) ..controls (227.93bp,308.31bp) and (191.33bp,267.14bp) .. (158.0bp,235.83bp) .. controls (136.95bp,216.06bp) and (110.89bp,194.97bp) .. (GenericCourseDecorator-4);
\draw [->] (GenericCourseDecorator-176) ..controls (344.03bp,332.9bp) and (286.76bp,403.09bp) .. (GenericCourseDecorator-134);
\draw [->] (GenericCourseDecorator-176) ..controls (325.48bp,310.11bp) and (294.51bp,317.1bp) .. (GenericCourseDecorator-133);
\draw [->] (GenericCourseDecorator-176) ..controls (346.94bp,259.66bp) and (278.94bp,146.15bp) .. (GenericCourseDecorator-1);
\draw [->] (GenericCourseDecorator-1) ..controls (194.7bp,92.833bp) and (159.13bp,92.833bp) .. (GenericCourseDecorator-17);
\draw [->] (GenericCourseDecorator-1) ..controls (208.65bp,75.43bp) and (135.23bp,40.279bp) .. (GenericCourseDecorator-92);
\draw [->] (GenericCourseDecorator-177) ..controls (450.52bp,343.04bp) and (411.59bp,366.54bp) .. (GenericCourseDecorator-16);
\draw [->] (GenericCourseDecorator-177) ..controls (444.06bp,314.56bp) and (424.67bp,310.54bp) .. (GenericCourseDecorator-176);
\draw [->] (GenericCourseDecorator-56) ..controls (446.25bp,569.83bp) and (430.71bp,569.83bp) .. (GenericCourseDecorator-102);
%
\end{tikzpicture}
};
\end{tikzpicture}
\newpage
\subsection*{תכנית לימודים מומלצת}
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה ראשונה--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
1011 &
חשבון אינפינטסימלי 1 &
5\\
\tabucline-
7011 &
אלגברה 1 למדעי המחשב &
5\\
\tabucline-
202.1.9041 &
מבוא לתכנות עם פייתון &
3\\
\tabucline-
299.1.1121 &
הדרכה בספריה &
0\\
\tabucline-
900.5.5001 &
לומדה להכרת החוק והנהלים למניעת הטרדה מינית &
0\\
\tabucline-
2201 &
מתמטיקה בדידה &
5\\
\tabucline-
2241 &
סדנא בכתיבת הוכחות &
1\\
\tabucline-
153.1.5041 &
אנגלית מתקדמים 1 (טבע) &
0\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{19}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה ראשונה--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
0171 &
מבוא לתורת הקבוצות &
4\\
\tabucline-
1021 &
חשבון אינפינטסימלי 2 &
5\\
\tabucline-
7021 &
אלגברה 2 למדעי המחשב &
5\\
\tabucline-
153.1.5051 &
אנגלית מתקדמים 2 טבע &
2\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{16}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שניה--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
1031 &
חשבון אינפינטסימלי גאומטרי 1 &
4\\
\tabucline-
1051 &
מבוא לאנליזה &
4\\
\tabucline-
7031 &
מבנים אלגבריים &
4\\
\tabucline-
8001 &
הסתברות &
4\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה} &
\LR{6.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{22}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שניה--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
1041 &
חשבון אינפינטסימלי גאומטרי 2 &
4\\
\tabucline-
0091 &
מבוא לטופולוגיה &
4\\
\tabucline-
0061 &
משוואות דיפרנציאליות רגילות &
4\\
\tabucline-
0251 &
תורת הפונקציות המרוכבות &
4\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה} &
\LR{6.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{22}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שלישית--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
0081 &
יסודות תורת המידה &
4\\
\tabucline-
6031 &
תורת המספרים &
4\\
\tabucline-
6061 &
לוגיקה &
4\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה} &
\LR{12.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{24}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שלישית--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
7041 &
תורת השדות ותורת גלואה &
4\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה} &
\LR{18.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{22}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\clearpage
\section{חטיבה במתמטיקה}
\subsection*{דרישות המסלול}
\subsubsection{לימודי חובה במתמטיקה}
יש לקחת את כל הקורסים מתוך %
\emph{טבלה~\ref{tab:external-mandatory-courses}},
\textbf{\nameref{tab:external-mandatory-courses}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
1011 &
חשבון אינפינטסימלי 1 &
4/2 &
5.0 &
סתיו &
\\
\tabucline-
1021 &
חשבון אינפינטסימלי 2 &
4/2 &
5.0 &
אביב &
חשבון אינפינטסימלי 1\\
\tabucline-
1211 &
אלגברה לינארית 1 &
4/2 &
5.0 &
סתיו &
\\
\tabucline-
1221 &
אלגברה לינארית 2 &
4/2 &
5.0 &
אביב &
אלגברה לינארית 1\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{20.0}\\
\tabucline-
\end{tabu}
\caption[קורסי חובה לחיצוניים]{קורסי חובה לחיצוניים. \label{tab:external-mandatory-courses}}
\end{table}
\subsubsection{לימודי חובה כלליים}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:mandatory-english-courses}},
\textbf{\nameref{tab:mandatory-english-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-general-courses}},
\textbf{\nameref{tab:mandatory-general-courses}}
\end{itemize}
\subsubsection{בחירה במתמטיקה}
קורסים מתקדמים הניתנים על-ידי המחלקה. הקורסים מתפרסמים ב\href{/teaching/term}{דף המחלקה}, לפני תחילת כל סמסטר
\newpage
\subsection*{גרף תלויות}
\begin{tikzpicture}[remember picture, overlay]
\node [shift={(0cm,-1.5cm)}] at (current page) {%
\tikzstyle{every node}=[text width=70pt,align=center]
\begin{tikzpicture}[>=latex,line join=bevel,]
%%
\node (GenericCourseDecorator-131) at (402.0bp,94.154bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי 1}}};
\node (GenericCourseDecorator-132) at (102.0bp,94.154bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי 2}}};
\node (GenericCourseDecorator-177) at (402.0bp,482.15bp) [draw,ellipse] {\footnotesize{\RL{אלגברה לינארית 1}}};
\node (GenericCourseDecorator-176) at (102.0bp,482.15bp) [draw,ellipse] {\footnotesize{\RL{אלגברה לינארית 2}}};
\draw [->] (GenericCourseDecorator-131) ..controls (309.33bp,94.154bp) and (210.23bp,94.154bp) .. (GenericCourseDecorator-132);
\draw [->] (GenericCourseDecorator-177) ..controls (315.41bp,482.15bp) and (205.45bp,482.15bp) .. (GenericCourseDecorator-176);
%
\end{tikzpicture}
};
\end{tikzpicture}
\newpage
\clearpage
\chapter*{מסלולי לימוד -- תואר דו-חוגי}
\section{מסלול דו-מחלקתי במתמטיקה ומדעי המחשב}
\subsection*{תיאור מסלול הלימוד}
תוכנית הלימודים היא באחריות משותפת של המחלקה למתמטיקה והמחלקה למדעי המחשב.
\subsection*{דרישות המסלול}
\subsubsection{לימודי חובה במתמטיקה}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:basic-mandatory-courses}},
\textbf{\nameref{tab:basic-mandatory-courses}}
\item %
\emph{טבלה~\ref{tab:additional-core-math-courses}},
\textbf{\nameref{tab:additional-core-math-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-math-for-cs}},
\textbf{\nameref{tab:mandatory-math-for-cs}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
0121 &
תורת הקירובים &
4/0 &
4.0 &
אביב &
מבוא לאנליזה\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{4.0}\\
\tabucline-
\end{tabu}
\caption[מתמטיקה שימושית]{מתמטיקה שימושית. \label{tab:mandatory-math-for-cs}}
\end{table}
\end{itemize}
\subsubsection{חובת-בחירה במתמטיקה}
לפחות 2 קורסים מתוך %
\emph{טבלה~\ref{tab:analysis-mandatory-elective}},
\textbf{\nameref{tab:analysis-mandatory-elective}}
%
\emph{טבלה~\ref{tab:advanced-mandatory-math-courses}},
\textbf{\nameref{tab:advanced-mandatory-math-courses}}
%
\emph{טבלה~\ref{tab:algebra-mandatory-elective}},
\textbf{\nameref{tab:algebra-mandatory-elective}}
\subsubsection{בחירה במתמטיקה}
קורסים מתקדמים הניתנים על-ידי המחלקה. הקורסים מתפרסמים ב\href{/teaching/term}{דף המחלקה}, לפני תחילת כל סמסטר. לתלמידים המתעניינים בתיאוריה של מדעי המחשב מומלץ לבחור בקורסים בתחום הקומבינטוריקה והלוגיקה.
\subsubsection{לימודי חובה במחלקה האחרת}
יש לקחת את כל הקורסים מתוך %
\emph{טבלה~\ref{tab:mandatory-cs-courses}},
\textbf{\nameref{tab:mandatory-cs-courses}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
202.1.1011 &
מבוא למדעי המחשב &
4/2 &
5.0 &
סתיו &
\\
\tabucline-
202.1.1031 &
מבני נתונים &
4/2 &
5.0 &
אביב &
מבוא למדעי המחשב\\
\tabucline-
202.1.2011 &
מודלים חישוביים &
4/2 &
5.0 &
אביב &
מתמטיקה בדידה, מבוא ללוגיקה ותורת הקבוצות, תכנון אלגוריתמים\\
\tabucline-
202.1.2031 &
תכנות מערכות &
4/2 &
5.0 &
סתיו &
מבני נתונים\\
\tabucline-
202.1.2041 &
תכנון אלגוריתמים &
4/2 &
5.0 &
סתיו, אביב &
מבני נתונים, מתמטיקה בדידה\\
\tabucline-
202.1.2051 &
עקרונות שפות תכנות &
4/2 &
5.0 &
אביב &
תכנות מערכות\\
\tabucline-
202.1.2081 &
מעבדה מורחבת בתכנות מערכות &
0/3 &
2.0 &
סתיו &
תכנות מערכות\\
\tabucline-
361.1.3131 &
מבוא להנדסת חשמל 2 &
3/1 &
3.5 &
סתיו &
\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{35.5}\\
\tabucline-
\end{tabu}
\caption[קורסי חובה במדעי המחשב]{קורסי חובה במדעי המחשב. \label{tab:mandatory-cs-courses}}
\end{table}
\subsubsection{לימודי בחירה במחלקה האחרת}
קורס אחד לפחות מתוך %
\emph{טבלה~\ref{tab:mandatory-elective-cs}},
\textbf{\nameref{tab:mandatory-elective-cs}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
202.1.2061 &
שיטות אופטימיזציה ויישומיהן &
3/1 &
3.5 &
אביב &
חשבון אינפי 2, מבוא למדעי המחשב, אלגברה לינארית 2\\
\tabucline-
202.1.3101 &
מבוא ללמידה חישובית &
4/2 &
5.0 &
&
מודלים חישוביים, הסתברות\\
\tabucline-
202.1.3201 &
מבוא למודלים גרפיים ולמידה עמוקה &
4/1 &
4.5 &
אביב &
חשבון אינפינטסימלי גאומטרי 1, הסתברות\\
\tabucline-
202.2.1111 &
סיבוכיות חישוב &
4/0 &
4.0 &
סתיו &
תכנון אלגוריתמים, תכנות מערכות, מודלים חישוביים\\
\tabucline-
361.1.3301 &
מבוא למחשבים למדעי המחשב &
3/1 &
3.5 &
אביב &
מבוא להנדסת חשמל 2\\
\tabucline-
\end{tabu}
\caption[חובת בחירה למדעי המחשב]{חובת בחירה למדעי המחשב. \label{tab:mandatory-elective-cs}}
\end{table}
\subsubsection{לימודי בחירה חופשית}
הקורסים ייבחרו על ידי הסטודנט מתוך מכלול הקורסים המוצעים באוניברסיטה
\subsubsection{לימודי השלמה למדעים}
ניתן ללמוד רק בקורסים של המחלקות למדעי הטבע או המחלקה להנדסת חשמל אשר מיועדים לתלמידי המחלקות הללו, ולא בקורסי שירות
\subsubsection{לימודי חובה כלליים}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:mandatory-english-courses}},
\textbf{\nameref{tab:mandatory-english-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-general-courses}},
\textbf{\nameref{tab:mandatory-general-courses}}
\end{itemize}
\newpage
\subsection*{גרף תלויות}
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\newpage
\subsection*{תכנית לימודים מומלצת}
על התלמידים לדאוג לכלול שני קורסי חובת-בחירה במתמטיקה וקורס חובת-בחירה במדעי המחשב מהרשימות המתאימות באחד משלושת הסמסטרים האחרונים. מאחר שיש שוני במספר נקודות הזכות שכל קורס מקנה, באחריות התלמידים לתכנן ולוודא שהם מגיעים ל-124 נק``ז. קיימת גמישות בהיקף נקודות הזכות האפשרי בקטגוריות של בחירה כללית ובבחירה ממחלקות אחרות במדעי הטבע, ואין צורך להגיע למקסימום האפשרי; באחריות התלמידים לאתר קורסים בהיקף מתאים אשר משלים את מספר נקודות הזכות הנדרש.
\bigskip
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה ראשונה--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
153.1.5041 &
אנגלית מתקדמים 1 (טבע) &
0\\
\tabucline-
1011 &
חשבון אינפינטסימלי 1 &
5\\
\tabucline-
2201 &
מתמטיקה בדידה &
5\\
\tabucline-
2241 &
סדנא בכתיבת הוכחות &
1\\
\tabucline-
202.1.1011 &
מבוא למדעי המחשב &
5\\
\tabucline-
299.1.1121 &
הדרכה בספריה &
0\\
\tabucline-
900.5.5001 &
לומדה להכרת החוק והנהלים למניעת הטרדה מינית &
0\\
\tabucline-
1211 &
אלגברה לינארית 1 &
5\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{21}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה ראשונה--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
153.1.5051 &
אנגלית מתקדמים 2 טבע &
2\\
\tabucline-
0171 &
מבוא לתורת הקבוצות &
4\\
\tabucline-
1021 &
חשבון אינפינטסימלי 2 &
5\\
\tabucline-
202.1.1031 &
מבני נתונים &
5\\
\tabucline-
1221 &
אלגברה לינארית 2 &
5\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{21}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שניה--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
1051 &
מבוא לאנליזה &
4\\
\tabucline-
8001 &
הסתברות &
4\\
\tabucline-
202.1.2041 &
תכנון אלגוריתמים &
5\\
\tabucline-
1031 &
חשבון אינפינטסימלי גאומטרי 1 &
4\\
\tabucline-
7031 &
מבנים אלגבריים &
4\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{21}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שניה--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
0121 &
תורת הקירובים &
4\\
\tabucline-
202.1.2011 &
מודלים חישוביים &
5\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה במתמטיקה} &
\LR{12.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{21}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שלישית--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
202.1.2031 &
תכנות מערכות &
5\\
\tabucline-
361.1.3131 &
מבוא להנדסת חשמל 2 &
3.5\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה} &
\LR{11.5}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{20}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שלישית--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
202.1.2071 &
מעבדה בתכנות מערכות &
2\\
\tabucline-
202.1.2051 &
עקרונות שפות תכנות &
5\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה} &
\LR{13.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{20}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\clearpage
\section{מסלול דו-מחלקתי במתמטיקה ופיסיקה}
\subsection*{תיאור מסלול הלימוד}
המתמטיקה והפיסיקה התפתחו במקביל ויש ביניהם קשרי גומלין והפריה הדדית. לפיכך, מדובר בשילוב לימודים טבעי ומקובל. התכנית הדו-מחלקתית לתואר ראשון במתמטיקה ובפיסיקה מיועדת לתלמידים אשר מעוניינים לשלב בין התחומים תוך מתן משקל דומה ללימודים בכל אחת מהמחלקות. ניתן להמשיך מתכנית הלימודים לתואר שני הן במתמטיקה והן בפיסיקה, אך כדי להימנע מהצורך בהשלמות, יש לשים לב לבחור בקורסים מתאימים במסגרת קורסי הבחירה לקראת סוף התואר.
תלמידים המעוניינים לשים דגש רב יותר על לימודי המתמטיקה יכולים ללמוד לתואר ראשי במתמטיקה עם חטיבה בפיסיקה, ואילו תלמידי פיסיקה המעוניינים רק ללמוד כמה מקורסי היסוד במתמטיקה ברמה גבוהה יותר מהנדרש בלימודי הפיסיקה יכולים לבחור באפשרות של לימודי פיסיקה כמחלקה ראשית עם חטיבה במתמטיקה.
\subsection*{דרישות המסלול}
\subsubsection{לימודי חובה במתמטיקה}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:basic-mandatory-courses}},
\textbf{\nameref{tab:basic-mandatory-courses}}
\item %
\emph{טבלה~\ref{tab:advanced-mandatory-math-courses}},
\textbf{\nameref{tab:advanced-mandatory-math-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-math-for-physics}},
\textbf{\nameref{tab:mandatory-math-for-physics}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
0291 &
התמרות אינטגרליות ומשוואות דיפרנציאליות חלקיות &
4/0 &
4.0 &
סתיו &
חשבון אינפינטסימלי גאומטרי 1, מבוא לאנליזה\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{4.0}\\
\tabucline-
\end{tabu}
\caption[מתמטיקה חובה לפיסיקה]{מתמטיקה חובה לפיסיקה. \label{tab:mandatory-math-for-physics}}
\end{table}
\end{itemize}
\subsubsection{בחירה במתמטיקה}
קורסים מתקדמים הניתנים על-ידי המחלקה. הקורסים מתפרסמים ב\href{/teaching/term}{דף המחלקה}, לפני תחילת כל סמסטר
\subsubsection{לימודי חובה במחלקה האחרת}
כמפורט בשנתון המחלקה לפיסיקה
\subsubsection{לימודי חובה כלליים}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:mandatory-english-courses}},
\textbf{\nameref{tab:mandatory-english-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-general-courses}},
\textbf{\nameref{tab:mandatory-general-courses}}
\end{itemize}
\newpage
\subsection*{גרף תלויות}
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%%
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\node (GenericCourseDecorator-8) at (246.0bp,560.94bp) [draw,ellipse] {\footnotesize{\RL{הסתברות}}};
\node (GenericCourseDecorator-13) at (79.0bp,282.94bp) [draw,ellipse] {\footnotesize{\RL{תורת הפונקציות המרוכבות}}};
\node (GenericCourseDecorator-135) at (79.0bp,560.94bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי גאומטרי 2}}};
\node (GenericCourseDecorator-113) at (79.0bp,411.94bp) [draw,ellipse] {\footnotesize{\RL{התמרות אינטגרליות ומשוואות דיפרנציאליות חלקיות}}};
\node (GenericCourseDecorator-136) at (79.0bp,22.942bp) [draw,ellipse] {\footnotesize{\RL{משוואות דיפרנציאליות רגילות}}};
\node (GenericCourseDecorator-12) at (79.0bp,152.94bp) [draw,ellipse] {\footnotesize{\RL{מבוא לטופולוגיה}}};
\node (GenericCourseDecorator-176) at (372.0bp,254.94bp) [draw,ellipse] {\footnotesize{\RL{אלגברה לינארית 2}}};
\node (GenericCourseDecorator-1) at (246.0bp,133.94bp) [draw,ellipse] {\footnotesize{\RL{מבנים אלגבריים}}};
\node (GenericCourseDecorator-177) at (485.0bp,254.94bp) [draw,ellipse] {\footnotesize{\RL{אלגברה לינארית 1}}};
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\draw [->] (GenericCourseDecorator-132) ..controls (342.63bp,449.89bp) and (282.59bp,519.62bp) .. (GenericCourseDecorator-8);
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\draw [->] (GenericCourseDecorator-134) ..controls (185.89bp,411.94bp) and (177.14bp,411.94bp) .. (GenericCourseDecorator-113);
\draw [->] (GenericCourseDecorator-133) ..controls (198.4bp,269.31bp) and (152.97bp,274.54bp) .. (GenericCourseDecorator-13);
\draw [->] (GenericCourseDecorator-133) ..controls (216.51bp,289.49bp) and (127.55bp,369.29bp) .. (GenericCourseDecorator-113);
\draw [->] (GenericCourseDecorator-133) ..controls (221.59bp,229.75bp) and (120.59bp,82.223bp) .. (GenericCourseDecorator-136);
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%
\end{tikzpicture}
};
\end{tikzpicture}
\newpage
\subsection*{תכנית לימודים מומלצת}
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה ראשונה--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
153.1.5041 &
אנגלית מתקדמים 1 (טבע) &
0\\
\tabucline-
1011 &
חשבון אינפינטסימלי 1 &
5\\
\tabucline-
1211 &
אלגברה לינארית 1 &
5\\
\tabucline-
2241 &
סדנא בכתיבת הוכחות &
1\\
\tabucline-
203.1.1141 &
עקרונות מתימטיים בפיסיקה 1 &
4\\
\tabucline-
299.1.1121 &
הדרכה בספריה &
0\\
\tabucline-
900.5.5001 &
לומדה להכרת החוק והנהלים למניעת הטרדה מינית &
0\\
\tabucline-
203.1.1181 &
תגליות ואתגרים בפיסיקה מודרנית &
0.5\\
\tabucline-
203.1.1281 &
מכניקה קלסית 1 &
5\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{20.5}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה ראשונה--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
153.1.5051 &
אנגלית מתקדמים 2 טבע &
2\\
\tabucline-
1021 &
חשבון אינפינטסימלי 2 &
5\\
\tabucline-
1221 &
אלגברה לינארית 2 &
5\\
\tabucline-
203.1.1623 &
מעבדה בפיסיקה א לפיסיקאים &
1.5\\
\tabucline-
203.1.2371 &
תורת החשמל 1 &
5\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה מחוץ למחלקה} &
\LR{2.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{20.5}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שניה--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
1031 &
חשבון אינפינטסימלי גאומטרי 1 &
4\\
\tabucline-
1051 &
מבוא לאנליזה &
4\\
\tabucline-
8001 &
הסתברות &
4\\
\tabucline-
203.1.2121 &
פיסיקה 3 &
4\\
\tabucline-
203.1.2281 &
מכניקה קלסית 2 &
4\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{20}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שניה--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
0061 &
משוואות דיפרנציאליות רגילות &
4\\
\tabucline-
0251 &
תורת הפונקציות המרוכבות &
4\\
\tabucline-
1041 &
חשבון אינפינטסימלי גאומטרי 2 &
4\\
\tabucline-
203.1.2381 &
תורת החשמל 2 &
5\\
\tabucline-
203.1.3141 &
תורת הקוונטים 1 &
4.5\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{21.5}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שלישית--א}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
0291 &
התמרות אינטגרליות ומשוואות דיפרנציאליות חלקיות &
4\\
\tabucline-
7031 &
מבנים אלגבריים &
4\\
\tabucline-
203.1.2161 &
תרמודינמיקה ומכניקה סטטיסטית 1 &
3.5\\
\tabucline-
203.1.2551 &
מעבדה בפיסיקה ג לפיסיקאים &
0\\
\tabucline-
203.1.2553 &
מעבדה בפיסיקה 2 לתלמידי פיסיקה &
2\\
\tabucline-
203.1.3241 &
תורת הקוונטים 2 &
5\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה במתמטיקה} &
\LR{4.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{22.5}\\
\tabucline-
\end{tabu}
\end{minipage}
\hfill
\noindent\begin{minipage}[t]{.48\linewidth}\small
\subsection*{\centering{שנה שלישית--ב}}
\begin{tabu} to\textwidth{|X|X|X|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{נק"ז}\\
\tabucline-
0091 &
מבוא לטופולוגיה &
4\\
\tabucline-
203.1.2261 &
תרמודינמיקה ומכניקה סטטיסטית 2 &
3.5\\
\tabucline-
203.1.2291 &
סמינר על נושאי מחקר במחלקה &
0.5\\
\tabucline-
203.1.3161 &
בטיחות במעבדה ג' &
\\
\tabucline-
203.1.3162 &
מעבדה ג1 לניסויים מתקדמים בפיסיקה &
2\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה בפיסיקה} &
\LR{5.0}\\
\tabucline-
\multicolumn2{|c|}{קורסי בחירה במתמטיקה} &
\LR{4.0}\\
\tabucline-
\multicolumn2{|c|}{נק"ז כולל} &
\LR{19}\\
\tabucline-
\end{tabu}
\end{minipage}
\bigskip
\clearpage
\section{מסלול דו-מחלקתי במתמטיקה ומדעי הרוח}
\subsection*{תיאור מסלול הלימוד}
התואר מקנה תעודה משולבת B.Sc. במתמטיקה ו- B.A. במדעי הרוח והחברה
\subsection*{דרישות המסלול}
\subsubsection{לימודי חובה במתמטיקה}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:basic-mandatory-courses}},
\textbf{\nameref{tab:basic-mandatory-courses}}
\item %
\emph{טבלה~\ref{tab:additional-core-math-courses}},
\textbf{\nameref{tab:additional-core-math-courses}}
\end{itemize}
\subsubsection{בחירה במתמטיקה}
בחירה ממבחר הקורסים המוצעים על-ידי המחלקה למתמטיקה
\subsubsection{לימודי חובה במחלקה האחרת}
בהתאם להוראות השנתון המתאים
\subsubsection{לימודי בחירה כלליים}
\subsubsection{לימודי חובה כלליים}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:mandatory-english-courses}},
\textbf{\nameref{tab:mandatory-english-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-general-courses}},
\textbf{\nameref{tab:mandatory-general-courses}}
\end{itemize}
\newpage
\subsection*{גרף תלויות}
\begin{tikzpicture}[remember picture, overlay]
\node [shift={(0cm,-1.5cm)}] at (current page) {%
\tikzstyle{every node}=[text width=70pt,align=center]
\begin{tikzpicture}[>=latex,line join=bevel,]
%%
\node (GenericCourseDecorator-131) at (445.0bp,368.77bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי 1}}};
\node (GenericCourseDecorator-132) at (272.0bp,368.77bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי 2}}};
\node (GenericCourseDecorator-134) at (79.0bp,363.77bp) [draw,ellipse] {\footnotesize{\RL{חשבון אינפינטסימלי גאומטרי 1}}};
\node (GenericCourseDecorator-133) at (79.0bp,184.77bp) [draw,ellipse] {\footnotesize{\RL{מבוא לאנליזה}}};
\node (GenericCourseDecorator-8) at (79.0bp,542.77bp) [draw,ellipse] {\footnotesize{\RL{הסתברות}}};
\node (GenericCourseDecorator-176) at (272.0bp,173.77bp) [draw,ellipse] {\footnotesize{\RL{אלגברה לינארית 2}}};
\node (GenericCourseDecorator-1) at (79.0bp,27.767bp) [draw,ellipse] {\footnotesize{\RL{מבנים אלגבריים}}};
\node (GenericCourseDecorator-177) at (445.0bp,173.77bp) [draw,ellipse] {\footnotesize{\RL{אלגברה לינארית 1}}};
\node (GenericCourseDecorator-102) at (272.0bp,547.77bp) [draw,ellipse] {\footnotesize{\RL{מבוא לתורת הקבוצות}}};
\node (GenericCourseDecorator-56) at (445.0bp,547.77bp) [draw,ellipse] {\footnotesize{\RL{מתמטיקה בדידה}}};
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\draw [->] (GenericCourseDecorator-132) ..controls (207.19bp,367.1bp) and (170.97bp,366.15bp) .. (GenericCourseDecorator-134);
\draw [->] (GenericCourseDecorator-132) ..controls (231.62bp,330.83bp) and (128.46bp,231.45bp) .. (GenericCourseDecorator-133);
\draw [->] (GenericCourseDecorator-132) ..controls (230.6bp,405.57bp) and (128.93bp,498.19bp) .. (GenericCourseDecorator-8);
\draw [->] (GenericCourseDecorator-176) ..controls (233.73bp,210.84bp) and (135.34bp,308.71bp) .. (GenericCourseDecorator-134);
\draw [->] (GenericCourseDecorator-176) ..controls (204.92bp,177.57bp) and (148.4bp,180.83bp) .. (GenericCourseDecorator-133);
\draw [->] (GenericCourseDecorator-176) ..controls (228.3bp,141.13bp) and (131.42bp,67.072bp) .. (GenericCourseDecorator-1);
\draw [->] (GenericCourseDecorator-177) ..controls (384.76bp,173.77bp) and (344.84bp,173.77bp) .. (GenericCourseDecorator-176);
\draw [->] (GenericCourseDecorator-56) ..controls (391.86bp,547.77bp) and (348.74bp,547.77bp) .. (GenericCourseDecorator-102);
%
\end{tikzpicture}
};
\end{tikzpicture}
\newpage
\clearpage
\chapter*{מסלולי לימוד -- תואר כפול}
\section{מתמטיקה והנדסת תעשייה וניהול}
\subsection*{תיאור מסלול הלימוד}
זוהי תכנית שמקנה תואר ''בוגר`` B.Sc במתמטיקה, ותואר ''מוסמך למדעים`` B.Sc (מהנדס) בהנדסת תעשייה וניהול.
משך הלימודים בתכנית הוא 8 סמסטרים לפחות (התוכנית המומלצת הינה ל- 9 סמסטרים). התכנית מיועדת לסטודנטים מצטיינים. יוכלו להתקבל לתכנית סטודנטים עם סכם הנדסה 530 ומעלה העומדים בתנאי הקבלה של שתי המחלקות. סטודנטים שאינם עומדים בתנאים אלה יוכלו להתקבל לתוכנית, במקרים מיוחדים, בהסכמת שתי המחלקות. כמו כן סטודנט יוכל להתקבל לתוכנית על תנאי ואם יעמוד בתנאים של סוף שנה א` יתקבל לתוכנית.
סטודנטים בתכנית יידרשו לעמוד בתנאים הבאים בתום שנה א` ללימודיהם בתכנית:
\begin{enumerate}
\item{} לעבור בהצלחה את כל מקצועות החובה של שנה א` (כמפורט בהמשך).
\item{} ממוצע כל הקורסים 80 לפחות.
\item{} ממוצע כל הקורסים במתמטיקה והקורסים במדעי המחשב 75 לפחות.
\item{} ציון כל קורס במתמטיקה 65 לפחות.
\end{enumerate}
הסטודנט יירשם לתכנית המשולבת במדור רישום. הודעה על קבלה לתכנית המשולבת תשלח ע``י מדור רישום לשתי המחלקות. על הסטודנט המתקבל לתכנית המשולבת לבצע רישום לקורסים בשתי המחלקות.
בוגרי התכנית בציון מתאים יוכלו להתקבל לתואר שני במחלקה להנדסת תעשייה וניהול או במחלקה למתמטיקה.
לפרטים נוספים על \href{http://in.bgu.ac.il/engn/Documents/YearBooks/2018/IemMath2018.pdf}{תכנית הלימודים}
\subsection*{דרישות המסלול}
\subsubsection{לימודי חובה במתמטיקה}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:basic-mandatory-courses}},
\textbf{\nameref{tab:basic-mandatory-courses}}
\item %
\emph{טבלה~\ref{tab:advanced-mandatory-math-courses}},
\textbf{\nameref{tab:advanced-mandatory-math-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-math-for-cs}},
\textbf{\nameref{tab:mandatory-math-for-cs}}
\item %
\emph{טבלה~\ref{tab:additional-core-math-courses}},
\textbf{\nameref{tab:additional-core-math-courses}}
\end{itemize}
\subsubsection{חובת בחירה באלגברה}
קורס אחד לפחות מתוך %
\emph{טבלה~\ref{tab:algebra-mandatory-elective}},
\textbf{\nameref{tab:algebra-mandatory-elective}}
\subsubsection{חובת בחירה באנליזה}
קורס אחד לפחות מתוך %
\emph{טבלה~\ref{tab:analysis-mandatory-elective}},
\textbf{\nameref{tab:analysis-mandatory-elective}}
\subsubsection{בחירה במתמטיקה}
\subsubsection{לימודי חובה במחלקה האחרת}
\subsubsection{לימודי בחירה כלליים}
\subsubsection{לימודי חובה כלליים}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:mandatory-english-courses}},
\textbf{\nameref{tab:mandatory-english-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-general-courses}},
\textbf{\nameref{tab:mandatory-general-courses}}
\end{itemize}
\newpage
\subsection*{גרף תלויות}
\begin{tikzpicture}[remember picture, overlay]
\node [shift={(0cm,-1.5cm)}] at (current page) {%
\tikzstyle{every node}=[text width=70pt,align=center]
\begin{tikzpicture}[>=latex,line join=bevel,]
%%
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\newpage
\clearpage
\section{מתמטיקה והנדסת חשמל ומחשבים}
\subsection*{תיאור מסלול הלימוד}
התוכנית המשולבת הנדסת חשמל ומחשבים - מתמטיקה המוצעת מיועדת להכשיר תלמידים ''דו-לשוניים`` בשני התחומים. לבוגרי התוכנית יוענק תואר כפול: בוגר B.Sc בהנדסת חשמל ומחשבים ובוגר B.Sc במתמטיקה. לפיכך, בוגר התוכנית יוכל (ויעודד) להמשיך ללא השלמות לתואר שני במחלקה להנדסת חשמל ומחשבים או במחלקה למתמטיקה, לפי בחירתו.
התוכנית מיועדת לתלמידים שיעמדו בתנאי הקבלה לתואר ראשון של כל אחת מהמחלקות –מתמטיקה והנדסת חשמל ומחשבים. לכל תלמיד יותאם מנחה אישי שילווה אותו מקבלתו לתוכנית לכל אורך הלימודים. התלמידים ישתייכו לשתי המחלקות. בכל שלב תלמיד יוכל לפרוש מהתוכנית ולהמשיך לתואר יחיד לפי בחירתו, במידה ומצבו האקדמי יהיה תקין (לאחר אישור ועדת הוראה של המחלקה המתאימה), תוך השלמת החובות הרגילות של המחלקה המתאימה.
על מנת להמשיך בתוכנית על התלמיד לשמור על ממוצע שנתי מעל 85 (למעט אישור חריג של האחראים על התוכנית בשתי המחלקות)
לפרטים נוספים על \href{http://in.bgu.ac.il/engn/Documents/YearBooks/2018/EceMath2018.pdf}{תכנית הלימודים}
\subsection*{דרישות המסלול}
\subsubsection{לימודי חובה במתמטיקה}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:advanced-mandatory-math-courses}},
\textbf{\nameref{tab:advanced-mandatory-math-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-math-for-cs}},
\textbf{\nameref{tab:mandatory-math-for-cs}}
\item %
\emph{טבלה~\ref{tab:additional-core-math-courses}},
\textbf{\nameref{tab:additional-core-math-courses}}
\item %
\emph{טבלה~\ref{tab:basic-mandatory-courses}},
\textbf{\nameref{tab:basic-mandatory-courses}}
\item %
\emph{טבלה~\ref{tab:analysis-mandatory-elective}},
\textbf{\nameref{tab:analysis-mandatory-elective}}
\end{itemize}
\subsubsection{חובת בחירה באלגברה}
קורס אחד לפחות מתוך %
\emph{טבלה~\ref{tab:algebra-mandatory-elective}},
\textbf{\nameref{tab:algebra-mandatory-elective}}
\subsubsection{לימודי בחירה במתמטיקה}
\subsubsection{לימודי חובה בפיסיקה}
יש לקחת את כל הקורסים מתוך %
\emph{טבלה~\ref{tab:mandatory-physics-courses}},
\textbf{\nameref{tab:mandatory-physics-courses}}
\begin{table}[pbht]
\begin{tabu} to\textwidth{|X[r]|X[r]|X[r]|X[c]|X[c]|X[r]|}
\tabucline-
\textbf{מספר} &
\textbf{שם הקורס} &
\textbf{\footnotesize{שעות}} &
\textbf{\scriptsize{נק"ז}} &
\textbf{\scriptsize{סמסטרים}} &
\textbf{קורסי קדם}\\
\tabucline-
203.1.1141 &
עקרונות מתימטיים בפיסיקה 1 &
3/2 &
4.0 &
סתיו &
\\
\tabucline-
\multicolumn3{|c}{נק"ז כולל:} & \multicolumn3{r|}{4.0}\\
\tabucline-
\end{tabu}
\caption[קורסי חובה בפיסיקה]{קורסי חובה בפיסיקה. \label{tab:mandatory-physics-courses}}
\end{table}
\subsubsection{לימודי חובה במחלקה האחרת}
\subsubsection{לימודי בחירה כלליים}
\subsubsection{לימודי חובה כלליים}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:mandatory-english-courses}},
\textbf{\nameref{tab:mandatory-english-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-general-courses}},
\textbf{\nameref{tab:mandatory-general-courses}}
\end{itemize}
\newpage
\subsection*{גרף תלויות}
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\section{מתמטיקה והנדסת מכונות}
\subsection*{תיאור מסלול הלימוד}
זוהי תכנית לתלמידים מצטיינים המקנה תואר כפול : B.Sc במתמטיקה ו-B.Sc בהנדסת מכונות. משך התוכנית ארבע וחצי שנים. יתקבלו תלמידים העומדים בתנאי הקבלה של המחלקה למתמטיקה והנמצאים ב-20\% העליונים של הנרשמים להנדסת מכונות. על התלמידים לשמור על ממוצע שנתי שלא יפחת מ-80.
לפרטים נוספים על \href{http://in.bgu.ac.il/engn/Documents/YearBooks/2018/MeMath2018.pdf}{תכנית הלימודים}
\subsection*{דרישות המסלול}
\subsubsection{לימודי חובה במתמטיקה}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:basic-mandatory-courses}},
\textbf{\nameref{tab:basic-mandatory-courses}}
\item %
\emph{טבלה~\ref{tab:additional-core-math-courses}},
\textbf{\nameref{tab:additional-core-math-courses}}
\item %
\emph{טבלה~\ref{tab:advanced-mandatory-math-courses}},
\textbf{\nameref{tab:advanced-mandatory-math-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-math-for-physics}},
\textbf{\nameref{tab:mandatory-math-for-physics}}
\item %
\emph{טבלה~\ref{tab:mandatory-math-for-cs}},
\textbf{\nameref{tab:mandatory-math-for-cs}}
\end{itemize}
\subsubsection{חובת בחירה באלגברה}
קורס אחד לפחות מתוך %
\emph{טבלה~\ref{tab:algebra-mandatory-elective}},
\textbf{\nameref{tab:algebra-mandatory-elective}}
\subsubsection{לימודי חובה במחלקה האחרת}
\subsubsection{לימודי בחירה כלליים}
\subsubsection{לימודי חובה כלליים}
יש לקחת את כל הקורסים המופיעים בטבלאות הבאות
\begin{itemize}
\item %
\emph{טבלה~\ref{tab:mandatory-english-courses}},
\textbf{\nameref{tab:mandatory-english-courses}}
\item %
\emph{טבלה~\ref{tab:mandatory-general-courses}},
\textbf{\nameref{tab:mandatory-general-courses}}
\end{itemize}
\subsubsection{בחירה במתמטיקה}
\newpage
\subsection*{גרף תלויות}
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\chapter*{מסלולי לימוד -- תכנית מיוחדת}
\section{תכנית אשלים}
\subsection*{תיאור מסלול הלימוד}
מטרת התוכנית היא לחשוף את הסטודנטים המצטיינים ביותר לתחומי מדעי הטבע. לתוכנית
יתקבלו מועמדים מצטיינים לתואר ראשון עם נתוני קבלה גבוהים ביותר. סטודנטים
ותיקים שיצטיינו בלימודיהם יוכלו להצטרף החל משנה שניה.
למצטייני ''אשלים`` תיבנה מסגרת לימודים שתאפשר להם למצות את יכולותיהם הגבוהות.
תכנית הלימודים תכלול מערכת לימודים עיקרית במחלקה למתמטיקה בשילוב של קורסים
מכלל הקורסים המוצעים באוניברסיטה (באישור מנחה אישי והמחלקה נותנת הקורס).
בנוסף יתקיימו מפגשים עם נוכחות חובה לתלמידי ''אשלים``, כתנאי להשתתפותם בתוכנית.
פרטים נוספים ניתן למצוא ב\href{http://in.bgu.ac.il/welcome/Pages/degree_1/Ashalim_Program_degree_1.aspx}{אתר הפקולטה למדעי הטבע}.
\paragraph*{תוכנית קורסים במגמת מדעי המוח בתוכנית ''אשלים``}
משנת הלימודים תשע``ז הסטודנטים בתוכנית ''אשלים`` יוכלו ללמוד במסגרת הלימודים
שלהם לתואר ראשון במגמת מדעי המוח. התוכנית דורשת מהתלמידים להשלים לפחות חמישה
קורסים (אפשרי יותר) מרשימת הקורסים הבאה. הקורסים הספציפיים שהתלמידים ילמדו
ייקבעו בהתייעצות עם היועץ המחלקתי של התוכנית.
מדובר בשלושים הנקודות שתלמידי אשלים יכולים לבחור. מגמת מדעי המוח תופיע
בגיליון הציונים של כל מי שילמד מקבץ של חמישה קורסים של המגמה. התואר שירשם
בתעודת הבוגר יהיה התואר של המחלקה אליה משתייך התלמיד. בפרטי תעודת התואר ירשם
רק שם המחלקה שאליה משויך התלמיד: מתמטיקה, מדעי המחשב, פיסיקה, כימיה, מדעי
החיים, או מדעי הגיאולוגיה והסביבה.
התוכנית בנויה בצורה המותאמת לתוכנית הלימודים המחלקתית של התלמיד ומאפשרת הבנת
תהליכים מרמת תא העצב הבודד עד לרמת המוח השלם.
\subparagraph*{קורסים במגמת מדעי המוח}
\begin{enumerate}
\item{} על התלמידים ללמוד לפחות חמישה קורסים מהרשימות הבאות.
\item{} יש ללמוד לפני את קורסי הקדם.
\item{} קורסי הקדם של הקורסים ברשימה לא יחשבו כחלק מחמשת הקורסים.
\item{} על התלמידים ללמוד לא יותר משני קורסים מרשימה ב` במסגרת המגמה.
\item{} קורסים שנמצאים בתוכנית החובה המחלקתית לא יחשבו כקורסים במגמה.
\end{enumerate}
\clearpage
\section{תכנית דקלים}
\subsection*{תיאור מסלול הלימוד}
התכנית מיועדת לתלמידים מצטיינים, הלומדים במחלקה למתמטיקה ונמצאים בשנת הלימודים השנייה או השלישית לתואר הראשון.
התכנית מאפשרת לתלמידים מצטיינים, המתכוונים להמשיך ללמוד לתארים מתקדמים, להיחשף לפעילות מחקרית.
פרטים נוספים ב\href{http://in.bgu.ac.il/welcome/Pages/General_BGU/Dekalim-program.aspx}{אתר הפקולטה למדעי הטבע}.
תלמידים המשתתפים בתוכנית ''דקלים`` ילמדו במסגרת תכנית הלימודים את הקורס ''התנסות
מחקרית``, במהלך כל סמסטר לימודים, במסגרת קורסי בחירה במחלקה:
\begin{enumerate}
\item{} 201-17771 התנסות מחקרית 1
\item{} 201-17772 התנסות מחקרית 2
\item{} 201-17773 התנסות מחקרית 3
\item{} 201-17774 התנסות מחקרית 4
\end{enumerate}
\clearpage
\appendix
\chapter{קורסים מתקדמים לתואר ראשון בשנה אקדמית 2021–2022}
\section*{סמסטר סתיו}
\begin{itemize}
\item \emph{מבנים אלגבריים}, פרופ` איליה טיומקין
\item \emph{יסודות תורת המידה}, פרופ` אילן הירשברג
\item \emph{לוגיקה}, פרופ` אסף חסון
\item \emph{הסתברות}, פרופ` אריאל ידין
\item \emph{תורת המספרים}, פרופ` פיודור פקוביץ
\item \emph{מבוא לאלגברה קומוטטיבית}, פרופ` איתן סייג
\item \emph{התמרות אינטגרליות ומשוואות דיפרנציאליות חלקיות}, פרופ` ארקדי פוליאקובסקי
\item \emph{מבוא לאנליזה}, ד``ר יזהר אופנהיים
\item \emph{חשבון אינפינטסימלי גאומטרי 1}, ד``ר דניאל מרקייביץ`
\item \emph{מבוא לתורת החבורות הגיאומטרית}, פרופ` יאיר גלזנר
\end{itemize}
\end{document}
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\begin{document}
\def\RunningAuthor{Tatyana V. Kabanova}
\firstPage{2350}
\articleType{Original Article}
\receivedDate{05 Dec 2019}
\acceptedDate{07 Jan 2020}
\revisedDate{03 Jan 2020}
\journalVolume{2020, 11}
\journalIssue{2}
\journalDoi{ijrps.v11i2.2212}
\copyrightYear{2020}
\def\authorCount{7}
\def\affCount{1}
\def\journalTitle{International Journal of Research in Pharmaceutical Sciences}
\title{\textbf{The Influence of Fermented Mare's Milk Processing Under Pressure of Gaseous Nitrogen on the Quality of Koumiss}}
\author{Tatyana~V.~Kabanova\textsuperscript{*},
Mariya~V.~Dolgorukova,
Sergey~I.~Okhotnikov,
Elena~V.~Tsaregorodtseva,
Sergey~Yu.~Smolentsev,
Ekaterina~A.~Savinkova,
Andrey~V.~Onegov~\\[5pt]{Mari State University, Lenin Square 1, Yoshkar-Ola city, 424000, Russia}}
\begin{abstract}
The urgency of the problem of increasing the nutritional qualities of mare's milk is that, it has versatile uses. Many people traditionally prefer koumiss made of milk with high fat content, as it has good taste, mouth-watering appearance, high calorie content, helps to increase liver weight. In many patients, including tuberculosis, hypotrophic and malnourished due to a long and serious illness, the latter aspect is an important part of treatment. In the production of raw materials for baby and diet food, first of all, it is necessary to achieve an increase in the protein and fat content of mares, as well as to stabilize the content of valuable components in milk during lactation. Therefore, the search for methods for correcting the composition of dairy raw materials is relevant to industry issues. This article presents the results on the research of influence of Hydro mechanical processing on the physicochemical and microbiological properties of mare's milk. Milk processing treatment by gaseous nitrogen was carried out at laboratory-patented system. This method has not had a negative impact on its physical and chemical characteristics, but significantly increased the bactericidal life. The introduction of pure cultures of koumiss leaven in processed and not processed raw milk has reduced the acidity build-up to the optimal value allowing increase in the life of koumiss.
\end{abstract}\def\keywordstitle{Keywords}
\begin{keywords}quality,\newline milk,\newline processing,\newline mares
\end{keywords}
\twocolumn[ \maketitle {\printKwdAbsBox}]
\makeatletter\textsuperscript{*}Corresponding Author\par Name:\ Tatyana~V.~Kabanova~\\ Phone:\ 89033146175~\\ Email:\ [email protected]
\par\vspace*{-11pt}\hrulefill\par{\fontsize{12}{14}\selectfont ISSN: 0975-7538}\par%
\textsc{DOI:}\ \href{https://doi.org/10.26452/\@journalDoi}{\textcolor{blue}{\underline{\smash{https://doi.org/10.26452/\@journalDoi}}}}\par%
\vspace*{-11pt}\hrulefill\\{\fontsize{9.12}{10.12}\selectfont Production and Hosted by}\par{\fontsize{12}{14}\selectfont Pharmascope.org}\par%
\vspace*{-7pt}{\fontsize{9.12}{10.12}\selectfont\textcopyright\ \@copyrightYear\ $|$ All rights reserved.}\par%
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\section{Introduction}
Bacterial contamination of milk is one of the primary indicators that helps to determine its quality, to choose the direction of its processing and tolerantly vary the parameters of the technological process. In addition, this ratio significantly affects the safety and quality of finished products. Traditional and widely used ways to suppress the microflora of milk is to pasteurize. As we know, the pasteurisation destroys vegetative forms of microorganisms, including pathogens. However, the effect of milk disposal by heat causes its undesirable changes that generally reduces biological value of products. This is especially true for milk of mare's and unlike the cow's milk being dominated by albumin and globulin with the thermal stability is much lower than casein. Heat pasteurisation method nowadays is not the only one. Laws of modern life encourage the search for optimal solutions at a lower cost of various types of resource to increase the efficiency of any processes. For example, in literature sources there are descriptions of methods of milk disinfection based on the effects of vacuum ultraviolet radiation, ultrasound and others \unskip~\citep{683561:16167944,683561:16167943}.
One of the new non-traditional ways of processing milk is treating it by gaseous nitrogen pressure followed by a dramatic collapse to atmospheric pressure. One of the most common and useful products is mare's milk Koumiss. Koumiss production is currently regulated by the GOST r 52974-2008 "Koumiss. Technical conditions". Implementation of Koumiss in long term usually significantly changes its organoleptic and its biochemical properties. Kumis is produced from natural fermented mare's milk by stewing clean cultures of Bulgarian and acidophilic sticks, as well as yeast, fermented with milk sugar. The whole technological process and ultimately the quality of the product will depend on the quality of leaven and matching the appropriate level of microflora, as well as the quantity and composition of the initial microflora in milk. Lack of heat treatment in milk does not guarantee the purity of the specified type of fermentation, so the search for new, not connected with high-temperature processing of milk, the methods of partial or complete destruction of the primary mare's milk microflora is urgent. One of such methods may be the method that will handle the pressure of mare's milk by gaseous nitrogen (DGA), followed by its sudden drop.
\section{Materials and Methods}
Milk processing treatment by gaseous nitrogen was carried out at laboratory-patented system \unskip~\citep{683561:16167946}. The suppression of microflora occurs as a result of the destruction of microorganisms with a sharp drop in pressure. The cells of microorganisms have gas bubbles, which, when pressure drops sharply increases its volume, bursting the shell cell or stretching it that lead to disruption of cell structures, resulting in reduced overall level of bacterial contamination of milk \unskip~\citep{683561:16167945,683561:16167947,683561:16167948}.
Based on the results of earlier research about the effect of cow's milk processing treatment by gaseous nitrogen, we as the Department of Technology of meat and dairy products, carried out the processing of milk mares treatment by the following way. Chilled to a temperature of 4 \ensuremath{\pm} 2 .1\text{\textordmasculine}C milk was exposed to gaseous nitrogen pressure p = 1.2 MPa for 5 min followed by sudden drop. To help to control it, chilled raw milk from the same batch was used. In general, this kind of treatment had no impact on changing colour of milk. In both cases, the milk was white with a bluish tint. However, the consistency of mare's milk, handled by the pressure of gaseous nitrogen with its sharp drop has become more homogenous due to the removal of the original gas, which gave raw milk slightly bubbled texture. In both cases, there were no flakes and protein sediment. In our view, the taste of mares' milk became more saturated sweet that most likely may be explained by the occurrence of milk gas freeing. Thus, in general, mare's milk processing by the pressure of gaseous nitrogen with consequent dramatic drop did not have a negative impact on organoleptic indicators with some gas-freeing, reduced fuzzy features and the appearance of more saturated sweet taste.
\section{Results and Discussion}
The research has shown that the basic physical and chemical characteristics of fermented mare's milk before and after processing by gaseous nitrogen pressure with sharp drop indicate some change in the initial indicators (Table~\ref{tw-638ac14a878f}).
\begin{table*}[!htbp]
\caption{\boldmath {The effect of mare's milk processing by pressure of gaseous nitrogen on its physicochemical indicators} }
\label{tw-638ac14a878f}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.5293\linewidth-2\tabcolsep}p{\dimexpr.29009999999999998\linewidth-2\tabcolsep}p{\dimexpr.18060000000000002\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Indicators & Raw milk \mbox{}\protect\newline Unprocessed & Raw milk \mbox{}\protect\newline Processed\\
\tblmidrule
Density, kg/m \ensuremath{^3} &
1030 &
1032\\
Titrated acidity, \ensuremath{^\circ} &
6,5 &
5,5\\
Mass fraction of fat, \% &
0,94 &
0,99\\
Mass fraction of proteins, \% &
2,05 &
2,09\\
Mass fraction of dry non-fat milk solids, \% &
8,33 &
8,43\\
The content of vitamin C in milligrams, mg/l &
121,5 &
117,8\\
\tblbottomrule
\end{tabulary}\par
\end{table*}
\bgroup
\fixFloatSize{images/7fa3c985-48ae-43c8-8b8f-9a26097d6045-upicture1.png}
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\centering \makeatletter\IfFileExists{images/7fa3c985-48ae-43c8-8b8f-9a26097d6045-upicture1.png}{\includegraphics{images/7fa3c985-48ae-43c8-8b8f-9a26097d6045-upicture1.png}}{}
\makeatother
\caption{\boldmath {Increase of titratable acidity of processed and unprocessed fermented mare's milk.}}
\label{f-6bf14bfded0a}
\end{figure*}
\egroup
\bgroup
\fixFloatSize{images/bcb97ad2-40bd-4a36-a66e-5b9b1a1cc87c-upicture2.png}
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\makeatother
\caption{\boldmath {Change the titratable acidity of Koumiss (capacity 0.50 l)}}
\label{f-45b6304fd5e8}
\end{figure*}
\egroup
\bgroup
\fixFloatSize{images/48664f5b-072b-4e38-8498-6c164f4cd187-upicture3.png}
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\centering \makeatletter\IfFileExists{images/48664f5b-072b-4e38-8498-6c164f4cd187-upicture3.png}{\includegraphics{images/48664f5b-072b-4e38-8498-6c164f4cd187-upicture3.png}}{}
\makeatother
\caption{\boldmath {Change the titratable acidity of Koumiss (capacity 0.25 l)}}
\label{f-3bd01f8dfffe}
\end{figure*}
\egroup
After processing mare's milk by gaseous nitrogen pressure, milk density remained almost without changes (before-and-after processing 1030 1032 kg/m\ensuremath{^{3}}). Fat and protein content increased by one hundredth of a percent, which made before processing-0.94 and 2.05\% and after processing is 0.99 and 2.09\% respectively. Mass fraction of dry non-fat milk solids increased from 8.33 to 8.43\%.
In our view, these changes can be explained as follows: when processing mare's milk by gaseous nitrogen pressure in case of sudden drop, part of gases evaporates from milk, which increases its weight.
After processing by gaseous nitrogen pressure titratable acidity of mare's milk has changed more dramatically (before and after processing respectively 6.5 and 5.5\ensuremath{^\circ} t) this can also be associated with the partial removal of gases (O\ensuremath{_{2}} and CO\ensuremath{_{2}}). [5] Nitrogen has a small coefficient of solubility in liquids; it can therefore be assumed that, because of the gas pressure and its dramatic drop, there is degassing of mare's milk happening.
The data obtained is consistent with our research on cow's milk, where there was a reduction in the acidity of milk at 1-2\ensuremath{^\circ} t at handling pressure of gaseous nitrogen with sharp its drop \unskip~\citep{683561:16167949,683561:16167950}.
In our view, reducing acidity after mare's milk processing by pressure of nitrogen gas with its sharp drop is a positive factor, because with 8 times milking mares there is the need for storage and keeping milk till the beginning of its fermentation. This is especially important during the summer period at higher temperatures.
The studies of the fermented mare's milk before and after processing of the content with vitamin C, proves to have its high levels in dry milk (121.5 mg/l), with some tendency to its decline after processing (117.8 mg/l) that generally is not essential.
Thus, the composition of mare's milk after it being processed by gaseous nitrogen is undergoing some changes, i.e. a slight increase in the core components of the composition, in addition to vitamin C, and a decline in titratable acidity of milk at 1 \text{\textordmasculine}T.
The question of increasing the duration of bactericidal phase is of great practical significance, because it gives the option to preserve fermented mare's milk refrigerated, if there is a need for storage in small quantities \unskip~\citep{683561:16167952,683561:16167951}.
To some extent, the bactericidal properties of mare's milk can be judged by ascend of its acidity in cooling and storage. In this case, the length of time will be equal to the bactericidal phase increasing acidity on 1\ensuremath{^\circ} T.
The following data shown in Figure~\ref{f-6bf14bfded0a} has been obtained in chilling mare's milk to 4 \ensuremath{\pm} 2 .1\text{\textordmasculine}C in our studies.
During the study lasting 16 hours, we found that the titratable acidity of unprocessed fermented mare's milk increased from 6.5\ensuremath{^\circ} to 10\ensuremath{^\circ} T. Changing acidity on 1\ensuremath{^\circ} T was noted during 12-hour research. In a processed mare's milk, the titrated acidity increased from 5.5\ensuremath{^\circ} t to 6.5\ensuremath{^\circ} T during 16-hour observation.
Thus, mare's milk processing by pressure of gaseous nitrogen with the following sharp drop produces a positive effect on the increase of acidity, and increases the duration of bactericidal phase by approximately 4 hours. Research of impact of mare's milk processing by pressure of gaseous nitrogen showed that sharp pressure difference also reduces the amount of microflora (KMAFAnM, yeast and mold) compared to raw milk.
The laboratory of the Department of Technology of meat and dairy products manufactured experimental batch of Koumiss processed from mare's milk by the pressure of gaseous nitrogen followed by the sharp drop and raw milk, in accordance with the requirements of GOST R 52974-2008 "Koumiss. Technical conditions". With a view to working leaven, delivered with the current enterprise CJSC ``Semenovsky'' it was produced about 8 litres of Koumiss from each batch, and after fermentation it was packaged in glass bottles with a capacity of 0.75; 0.50; 0.25 litres sealed and sent to the maturation of at t =4 \ensuremath{\pm} 2.1 \text{\textordmasculine}C.
Data on change in titratable acidity as the primary indicator of quality of Koumiss, within 5 days refer to Figures~\ref{f-45b6304fd5e8} and~\ref{f-3bd01f8dfffe}.
These data indicate notable differences in accumulation of titratable acidity of Koumiss processed and raw fermented mare's milk. So, the acidity of the drink in a standard container (0.5 litres) of raw and processed mare's milk accordingly: one-day-89 and 88\ensuremath{^\circ} T; two-day-94 and 91\ensuremath{^\circ} T; three-day-108 and 94\ensuremath{^\circ} T; four-day-115 and 99\ensuremath{^\circ} T; five day-120 and 105\ensuremath{^\circ} T.
\section{Conclusions}
Comparing the results with the requirements of GOST r 52974-2008 "Koumiss. Technical conditions'', five day Koumiss processed from mare's milk at a temperature of 4 \ensuremath{\pm} 2 .1\text{\textordmasculine}C by gaseous nitrogen pressure p = 1.2 MPa for 5 min followed by sudden drop is equal to a three-day one from raw milk that gives an opportunity to distribute it for a longer time, especially if shipment to other regions required. Therefore, this method of mare's milk processing can have a significant increase in economic efficiency because of price difference in other regions and production in general. In addition, we observed that the volume of packaging also had a significant impact on titratable acidity. When matured, Koumiss in packaging 0.25 litres of titratable acidity build-up occurred less intensively-in control and experimental batch 109 respectively and 101\ensuremath{^\circ} T, that can also be used to increase the duration of produced product.
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\module{Scientific notation}{01}{AP05}{1 period}
\prereq{\begin{itemize}
\item Positive and negative powers of ten.
\end{itemize}}
\object{\begin{itemize}
\item Discover the scientific notation, the E notation used by the calculators and the engineering notation.
\item Switch from one notation to another.
\end{itemize}}
\mater{\begin{itemize}
\item Fact sheets about the three notations (36 copies).
\item Test paper with 16numbers (36 copies).
\item Matching cards with different notations of the same numbers (20 cards).
\end{itemize}}
\modpart{Matching game}{10 mins}
Students are handed out cards with numbers written in ordinary decimal notation, scien-
tific notation, E notation or engineering notation. They commit their numbers to memory
and mingle to find the students with equal numbers.
\modpart{Team work}{20 mins}
Working in the same teams as in the first part on a lesson handed-out by the teacher, students have to undersand the
three notations and how to switch from one to another.
\modpart{Test}{Remaining time}
Still working in teams, students have to convert numbers from one notation to the others, including the ordinary
notation.
\pagebreak
\newcommand{\E}{\textrm{E}}
\moddocdis{Scientific notation}{01}{AP05}{Lesson}
\defi{Scientific notation}{A decimal number $x$ is said to be in scientific notation if there exist two numbers $k$ and $n$
such as $$x=a \times 10^n,$$
where $a$ is a decimal number such that $1\leq |a|<10$, and $n$ is an integer (positive or negative).}
{\sl Examples :}
\begin{itemize}
\itemb $158$ ; $-45.8$ and $3.7$ are not in scientific notation.
\itemb $-2.58749 \times 10^5$ is in scientific notation, as $1\leq 2.58749 < 10$ and $5$ is an integer.
\itemb $7.451 \times 10^{-4}$ is the scientific notation of the number $0.0007451$.
\itemb $12 \times 10^{-8}$ is not a scientific notation because $12$ is greater than $10$.
\itemb $0.587 \times 10^2$ is not a scientific notation because $0.587$ is lower than $1$.
\end{itemize}
\bigskip
\defi{E notation}{An alternative notation used by calculators is the ``E'' notation. It's basically the scientific
notation, with the letter E standing for $\times 10$\^{}.}
{\sl Examples :}
\begin{itemize}
\itemb The E notation of the number $-2.58749 \times 10^5$ is $-2.58749\E5$.
\itemb The E notation of the number $0.0007451$ is $7.451\E-4$.
\end{itemize}
\bigskip
\defi{Engineering notation}{Engineering notation differs from normalized scientific notation in that the exponent $n$ is
restricted to multiples of $3$. Consequently, the absolute value of $a$ is in the range $1\leq |a|<1000$, rather than
$1\leq |a|<10$.}
{\sl Examples :}
\begin{itemize}
\itemb The engineering notation of the number $-2.58749 \times 10^5$ is $-258.749 \times 10^3$.
\itemb The engineering notation of the number $0.0007451$ is $745.1 \times 10^{-6}$.
\end{itemize}
\pagebreak
\moddocdis{Scientific notation}{01}{AP05}{Test}
\begin{center}
\begin{tabularx}{16cm}{c|X|>{\columncolor{lightgray}} c|X|>{\columncolor{lightgray}} c|X|>{\columncolor{lightgray}} c|X|>{\columncolor{lightgray}} c}
&\centering Ordinary &&\centering Scientific &&\centering Calculator &&\centering Engineering & \\ \hline
1 &\centering $45$ && && && & \rule{0pt}{20pt}\\[5pt] \hline
2 & &&\centering $-3.8405 \times 10^5$ && && & \rule{0pt}{20pt}\\[5pt] \hline
3 & && &&\centering $1.15\E-2$ && & \rule{0pt}{20pt}\\[5pt] \hline
4 & && && &&\centering $458.7 \times 10^0$ & \rule{0pt}{20pt}\\[5pt] \hline
5 &\centering $-4879$ && && && & \rule{0pt}{20pt}\\[5pt] \hline
6 & &&\centering $7.7484 \times 10^{-4}$ && && & \rule{0pt}{20pt}\\[5pt] \hline
7 & && &&\centering $2.951\E0$ && & \rule{0pt}{20pt}\\[5pt] \hline
8 & && && &&\centering $55.0002 \times 10^{-3}$ & \rule{0pt}{20pt}\\[5pt] \hline
9 &\centering $0.258$ && && && & \rule{0pt}{20pt}\\[5pt] \hline
10 & &&\centering $-3.7 \times 10^{-4}$ && && & \rule{0pt}{20pt}\\[5pt] \hline
11 & && &&\centering $2.3548792\E7$ && & \rule{0pt}{20pt}\\[5pt] \hline
12 & && && &&\centering $-80 \times 10^{-6}$ & \rule{0pt}{20pt}\\[5pt] \hline
13 &\centering $100000$ && && && & \rule{0pt}{20pt}\\[5pt] \hline
14 & &&\centering $1 \times 10^{-5}$ && && & \rule{0pt}{20pt}\\[5pt] \hline
15 & && &&\centering $4.01\E4$ && & \rule{0pt}{20pt}\\[5pt] \hline
16 & && && &&\centering $510 \times 10^{-6}$ & \rule{0pt}{20pt}\\[5pt] \hline
\end{tabularx}
\end{center}
\pagebreak
\moddocdis{Scientific notation}{01}{AP05}{Correction}
\begin{center}
\begin{tabularx}{16cm}{c|X|>{\columncolor{lightgray}} c|X|>{\columncolor{lightgray}} c|X|>{\columncolor{lightgray}} c|X|>{\columncolor{lightgray}} c}
&\centering Ordinary &&\centering Scientific &&\centering Calculator &&\centering Engineering & \\ \hline
1 &\centering $45$ && \centering $4.5\times10^1$&& \centering $4.5\E1$ && \centering $45\times10^0$& \rule{0pt}{20pt}\\[5pt] \hline
2 &\centering $-384050$ &&\centering $-3.8405 \times 10^5$ && \centering $-3.8405\E5$&& $-384.05 \times 10^3$& \rule{0pt}{20pt}\\[5pt] \hline
3 &\centering $0.0115$ &&\centering $1.15\times 10^{-2}$ &&\centering $1.15\E-2$ &&\centering $11.5\times 10^{-3}$&\rule{0pt}{20pt}\\[5pt] \hline
4 &\centering $458.7$ &&\centering $4.587 \times 10^{2}$ &&\centering $4.587 \E2$ &&\centering $458.7(\times 10^0)$ & \rule{0pt}{20pt}\\[5pt] \hline
5 &\centering $-4879$ &&\centering $-4.879 \times 10^{3}$ &&\centering $-4.879 \E3$ &&\centering $-4.879 \times 10^{3}$ & \rule{0pt}{20pt}\\[5pt] \hline
6 &\centering $0.00077484$ &&\centering $7.7484 \times 10^{-4}$ &&\centering $7.7484 \E-4$ && \centering $774.84 \times 10^{-6}$& \rule{0pt}{20pt}\\[5pt] \hline
7 & \centering $2.951$&&\centering $2.951\times 10^{0}$ &&\centering $2.951\E0$ &&\centering $2.951\times 10^{0}$ & \rule{0pt}{20pt}\\[5pt] \hline
8 & \centering $0.0550002$&&\centering $5.50002 \times 10^{-2}$ &&\centering $5.50002\E-2$ &&\centering $55.0002 \times 10^{-3}$ & \rule{0pt}{20pt}\\[5pt] \hline
9 &\centering $0.258$ &&\centering $2.58\times 10^{-1}$ && \centering $2.58\E-1$&& \centering $258\times 10^{-3}$& \rule{0pt}{20pt}\\[5pt] \hline
10 &\centering $-0.00037$ &&\centering $-3.7 \times 10^{-4}$ &&\centering $-3.7 \E-4$ && \centering $-370 \times 10^{-6}$& \rule{0pt}{20pt}\\[5pt] \hline
11 &\centering$23548792$ &&\centering $2.3548792\times 10^{7}$ &&\centering $2.3548792\E7$ &&\centering$23.548792\times 10^{6}$ & \rule{0pt}{20pt}\\[5pt] \hline
12 &\centering $-0.00008$ &&\centering$-8 \times 10^{-5}$ &&\centering $-8 \E-5$&&\centering $-80 \times 10^{-6}$ & \rule{0pt}{20pt}\\[5pt] \hline
13 &\centering $100000$ && \centering $(1\times)10^{5}$&& \centering $1\E5$ && \centering $100\times10^{3}$ & \rule{0pt}{20pt}\\[5pt] \hline
14 &\centering $0.00001$ &&\centering $1 \times 10^{-5}$ &&\centering$1 \E-5$ &&\centering $10 \times 10^{-6}$ & \rule{0pt}{20pt}\\[5pt] \hline
15 &\centering $40100$ &&\centering $4.01 \times 10^{4}$ &&\centering $4.01\E4$ &&\centering $40.1 \times 10^{3}$ & \rule{0pt}{20pt}\\[5pt] \hline
16 &\centering $0.00051$ &&\centering $5.1 \times 10^{-4}$ &&\centering $5.1\E-4$ &&\centering $510 \times 10^{-6}$ & \rule{0pt}{20pt}\\[5pt] \hline
\end{tabularx}
\end{center}
\pagebreak
\moddoc{Matching cards with various notations of the same numbers}
\begin{center}
\includegraphics[scale=0.8]{Euro.s01AP05.eps}
\end{center}
\end{document}
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\usepackage{graphicx}
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\usepackage{latexsym}
\usepackage{textcomp}
\usepackage{longtable}
\usepackage{tabulary}
\usepackage{booktabs,array,multirow}
\usepackage{amsfonts,amsmath,amssymb}
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\usepackage[utf8]{inputenc}
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\usepackage{float}
\begin{document}
\title{LEFT-RIGHT QUANTUM DERIVATIVES AND DEFINITE INTEGRALS}
\author[1]{Mehmet Kunt}%
\author[2]{Abdul Baidar}%
\author[3]{Zeynep ŞANLI}%
\affil[1]{Karadeniz Technical University}%
\affil[2]{Kabul University}%
\affil[3]{Karadeniz Teknik Universitesi}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
In this work, the concepts of quantum derivative and quantum integral
are renamed to be the left quantum derivative and the left
de\ldots{}nite quantum integral. Symmetrically to the left, a new
quantum derivative (the right) and de\ldots{}nite quantum integral (the
right) are de\ldots{}ned. Some properties of these new concepts are
investigated and as well as according todo these new concepts some
inaccuracies in quantum integral inequalities are corrected. Moreover,
some new quantum Hermite-Hadamard type inequalities are established.%
\end{abstract}%
\sloppy
\textbf{Hosted file}
\verb`M_Kunt_, A_ W_ Baidar_and_Z_\selectlanguage{polish}Ş\selectlanguage{english}anl\selectlanguage{polish}ı.\selectlanguage{english}pdf` available at \url{https://authorea.com/users/346089/articles/472188-left-right-quantum-derivatives-and-definite-integrals}
\selectlanguage{english}
\FloatBarrier
\end{document}
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%%
%% The LaTeX Companion, 2ed (second printing August 2004)
%%
%% Example A-3-6 on page 874.
%%
%% Copyright (C) 2004 Frank Mittelbach, Michel Goossens,
%% Johannes Braams, David Carlisle, and Chris Rowley
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%% It may be distributed and/or modified under the conditions
%% of the LaTeX Project Public License, either version 1.3
%% of this license or (at your option) any later version.
%%
%% See http://www.latex-project.org/lppl.txt for details.
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\newcommand{\IX}[2][R]{\texttt{#2}%
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we define item \IX[D]{AAAA}
\ldots{} we reference item \IX{AAAA}
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%
% Modified 21.8.1995 from Norbert Schwartz's original dcstdedt.tex and
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% [email protected]
%
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\documentclass[11pt]{article}
\usepackage{amsmath}
\usepackage{german}
% \usepackage{program}
\usepackage{epsfig}
\usepackage{theorem}
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% ------------------------------------------------------------
% Spezielle Def. fuer das Skript
% ------------------------------------------------------------
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% ------------------------------------------------------------
% theorems
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\begin{document}
\begin{center}
{\bf \large Vorlesung: Logik lebdender Systeme} \\
Peter Dittrich, JCB \& FSU Jena \\
started: 14. Oktober 2003, compiled: \today \\
\vspace{1cm}
\end{center}
%%% Local Variables:
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%%% End:
\parskip=0pt plus 3pt
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\section{Fragen}
Die Fragen sind nat"urlich nicht vollst"andig.
Sie d"unnen nach hinten naturgem"a"s aus.
Sie geben aber hoffentlich einen Eindruck, wie
die Fragen bei einem Scheingespr"ach oder einer m"undlichen
Pr"ufung aussehen k"onnten.
\subsection{Grundlegende Fragen}
\subsubsection{Systemtheorie}
Diese Fragen waren nicht direkt Inhalt der
Vorlesung. Die Themen sind aber z.T. angerissen worden,
von grundlegender Bedeutung, und deshalb auch Teil
von Pr"ufungen (falls es die entsprechende PO erlaubt).
\begin{enumerate}
\item Erkl"are folgende Begriffe: System, Modell, Selbstorganisation,
Emergenz, Systemstruktur, dissipatives System, abgeschlossenes Systeme,
offenes System.
\item Wie unterscheidet sich ein abgeschlossenes System von einem
geschlossenen
\item Unterscheide Regelung und Steuerung. Wird die K"orpertemperatur geregelt
oder gesteuert? Steuert oder regelt man ein Auto wenn man es f"ahrt?
\item Was versteht man thermodynamisch unter Entropie?
\item Was kann eine positive oder negative R"uckkopplung bewirken?
\end{enumerate}
So, jetzt geht's aber los mit Fragen direkt zur Vorlesung.
\subsubsection{Leben}
\begin{enumerate}
\item Was versteht man unter Leben?
\item Wie lassen sich die unterschiedlichen charakterisierenden
Eigenschaften des Lebens ordnen?
\item Auf welcher Zeitskaler sind welche Eigenschaften wichtig?
\item Erkl"are den Unterschied zwischen ``das Leben" und "es lebt!''.
\item Was sind die wichtigsten Eigenschaften eines lebenden Systems.
Unterscheide dabei die Eigenschaften, die nur f"ur biologische Systeme gelten
von
solchen Eigenschaften, die auch f"ur ``lebendige'' technische, sozial, und
kulturelle Systeme gelten m"ussen.
\end{enumerate}
\subsubsection{K"unstliches Leben - Artificial Life}
\begin{enumerate}
\item Was versteht man unter Artificial Life (AL)?
Und was sind die grunds"atzlichen Ideen und Annahmen, die sich
hinter diesem Begriff verbergen?
\item Auf welche Art der Erkenntnis zielt AL im Vergleich zur Biologie ab?
\item Was versteht man unter ``weak AL'' und ``strong AL''?
Nenne Beispiele.
\item Welche Eigenschaften des nat"urlichen Lebens versucht man
in AL Systemen abzubilden oder nachzubilden ?
\item Ordne die in der Vorlesung vorgestellten Modelle bez"uglich
ihrer Zugeh"origkeit zur ``weak'' bzw. ``strong'' AL ein.
\item Macht die Unterscheidung zwischen ``weak'' und ``strong'' AL Sinn?
Nenne Argumente, die daf"ur und welche, die dagegen sprechen.
\item Welche Anwendungsm"oglichkeiten bieten AL Systeme.
\item Was ist das prim"are Ziel der AL Forschung?
\item Nenne jeweils ein offenes (aber l"osbares) Problem einer
Organisationsebene (z.B. Entstehung, Evolution, Sozialisation) des Lebens.
(vgl. vorletzte Vorlesung)
\item Ordne die 12 offenen Probleme (Bedau et al., Open Problems in Artificial
Life, ...,) bez"uglich ihrer Schwierigkeit und bez"uglich ihrer
wissenschaftlichen, "okonomischen, und gesellschaftlichen Bedeutung.
\item Wann (ca.) hat sich der Begriff ``Artificial Life'' als
Bezeichnung f"ur die in der Vorlesung pr"asentierten Inhalte herausgebildet?
\item Nenne (mind. drei) Arbeiten, die vor dieser Zeit entstanden sind und die
man als AL Forschung interpretieren kann.
\item Beschreibe einen typischen Weg der Erkenntnisgewinnung mit ``AL
Methoden''.
\item Wie kann man ein AL Modell qualitativ validieren?
\item Wie kann man ein AL Modell quantitativ validieren?
\item Wie kann man ``reale Daten'' beim Entwurf von AL Systemen oder AL
Modellen verwenden? Nenne ein Beispiel.
\item Welche Beziehung besteht zwischen AL und der Systembiologie?
\item Erl"autere Grey Walters elektronische Schildkr"ote.
Was kann man daraus "uber die Logik lebender Systeme lernen?
\item Menschen haben schon immer lebende System nachzubilden versucht.
Welchen Schritt haben aber Kleen, Church, Turing, G"odel, Post und andere
Anfang des 20. Jahrhunderts gemacht?
\end{enumerate}
\subsection{Replikation}
\begin{enumerate}
\item Nenne die wichtigsten Prinzipien der Selbstreplikation technischer
und nat"urlicher Systeme.
\item Nenne Prinzipien der Selbstreplikation von Automaten und
Computerprogrammen. Welche kommen am ehesten in der Natur vor?
\item Beschreibe (mit Zeichnung!) den zur Selbstreplikation
f"ahigen Automaten nach von Neumann.
Was kann dieser Automat aus"serdem?
\item Kann man diesen Automaten auf einem heutigen PC simulieren?
(ja)
\item Kann man einen vollen Replikationszyklus dieses Automaten auf einem
heutigen PC simulieren? (nein)
\item Wie verl"auft eine solche Simulation typischerweise?
Welcher Trick wird zur Effizienzsteigerung eingesetzt? (nicht alle Zellen
"anderen ihren Zustand bei einem Zeitschritt)
\item Beschreibe die Entwicklung der Forschung im Bereich
selbstreplizierender Strukturen in zellul"aren Automaten.
Welche Prinzipien kamen dazu? (z.B. template basierte Replikation)
\item Welches Prinzip der Selbstreplikation wird in Assemblerautomaten
(z.B. Tierra oder Avida) verwendet?
\end{enumerate}
\subsection{Selbstorganisation und Edge of Chaos}
\begin{enumerate}
\item Was versteht man unter Selbstorganisation?
Mache dies anhand eines zellul"aren Automaten (CA) deutlich.
\item Welche Formen der Selbstorganisation gibt es? Wonach
kann man Selbstorganisationsvorg"ange klassifizieren?
\item Nenne Beispiele in denen die Selbstorganisation von
der Perspektive eines Beobachter abh"angig und unabh"angig ist.
\item Beschreibe eine Arbeit (CA oder boolesches Netz) die
das Konzept vom Rand des Chaos (engl. edge of chaos) illustriert.
\item Warum ist dieser Rand gerade so interessant f"ur das Leben?
\item Nenne die drei Komponenten, die f"ur die Informationsverabeitung
bedeutsam sind. (Speicherung, Transport, Verrechnung)
\item Wie zeigen sich die Komponenten in zellul"aren Automaten?
\item Wie ist Langtons Lambda-Parameter definiert?
\item Wie kann man Entropie und gegenseitige Entropie in CAs messen?
\item Wie kann man die Komplexit"at eines CA messen?
\item Was versteht man unter einem Attraktor, einer Trajektorie, und
einer transierten Phase.
\item Was ist ein asynchroner CA?
\item Wie "andert sich das Verhalten (bspw. vom Game of Life), wenn
man die Zellen asynchron aktualisiert?
\end{enumerate}
\subsection{Informationstheorie}
Literatur: Adami
\begin{enumerate}
\item Was versteht die Informationstheorie unter Information?
Was ist das Ziel der Informationstheorie? Wie wird Information gemessen?
\item Warum ist die Informationstheorie f"ur AL interessant?
\item Warum ist die Informationstheorie f"ur die Biologie interessant?
\item Leite die Entropie nach Shannon her. (haben wir 2007 leider nicht gemacht)
\item Wie kann man die Kapazit"at eines "Ubertragungskanals bestimmen.
\item Illustriere anhand der Replikation von DNA die Anwendung der
Informationstheorie. Wo ist Empf"anger, Sender und der Kanal?
\item Erl"autere die wichtigsten Konzepte der Informationstheorie
(s. Kapitel in Adami 19??).
\item Welche Zusammenh"ange bestehen zwischen
Entropie, gemeinsame Entropie, bedingte Entropie, gegenseitige Entropie.
\end{enumerate}
\subsection{Selbstorganisierte Kritikalit''at (SOC) und Power Law}
\begin{enumerate}
\item Was versteht man unter einem ``Power Law''? Wo findet man es?
\item Erl"autere den Unterschied zwischen polinomiellen und einer
exponentiellen Verteilung anhand zweier Beispiele.
\item Welche Mechanismen f"uhren zu einer polonmiellen Verteilung
(Power Law)?
\item Welcher Zusammenhang besteht zwischen SOC und Power Law?
\item Was versteht man unter ``skalenfrei''?
\item Nenne Beispiele f"ur SOC, Power Law und Skalenfreiheit aus der
Biologie.
\item Erl"autere die in der Vorlesung vorgestellten Modelle.
\end{enumerate}
\subsection{Random Boolean Networks (RBN)}
\begin{enumerate}
\item Erkl"are anschaulich was ein RBN ist. Wie kann man es
formal darstellen?
\item Was bedeuten die ber"uhmten Parameter $N$ und $K$?
Und wie h"angt das Verhalten von ihnen ab?
\item Was ist ein Attraktor in einem RBN? Hat jedes RBN einen
Attraktor?
\item Wie unterscheidet sich ein Attraktor eines RBNs von dem
einer gew"ohnlichen Differntialgleichung $\dot x(t) = f(x(t))$ ?
\item Wie kann man anschaulich die Attraktoren und Einzugsgebiete von RBNs
darstellen? (vgl. Wuensche)
\item Geht das auch f"ur zellul"are Automanten?
\item Hat jedes RBN einen Attraktor?
\item Welche Beziehung besteht zwischen den Attraktoren eines RBNs und
den Zelltypen eines Organismus?
\item Wie kann man den Rand des Chaos mit Hilfe eines RBNs zeigen?
Welches sind die ``Ordnungsparameter'', die den Grad der
Ordnung beschreiben.
\end{enumerate}
\subsection{K"unstliche Chemie (AC)}
\begin{enumerate}
\item Was ist eine k"unstliche Chemie?
\item Nenne Beispiele f"ur Molek"ule, Reaktionen und Techniken die
Dynamik zu realisieren.
\item Erl"autere anhand von Beispielen den Unterschied zwischen
expliziter und impliziter Repr"asentation der Menge der Molek"ule
(und Reaktionen).
\item Was versteht man unter ``Struktur-Funktions-Abbildung''?
\item Was versteht man unter syntaktischer und semantischer Abgeschlossenheit?
\item Welche Ziele werden mit den in der Vorlesung vorgestellten ACs
verfolgt? Nenne ein paar wichtige Ergebnisse.
\item Erkl"are die in der Vorlesung vorgestellten KCs (z.B. Fontana's
Alchemie, Jain/Krishna, Hyperzyklus).
\item Nenne ein Beispiel (aus der Literatur) f"ur eine physikalische KC und
erl"autere kurz eines mit ihr erzielten Ergebnisse.
\end{enumerate}
\subsection{Assembler Automaten}
\begin{enumerate}
\item Beschreibe die grobe Historie der Assemblerautomaten.
(Core Wars, Core World, Tierra, Avida, Amoeba (Pargellis))
Worin unterscheiden sich diese Systeme?
\item Nenne wichtige Ergebnisse, die mit diesen Systemen erzielt worden sind.
\item Welche Besonderheiten (``Tricks'') werden bei Tierra eingesetzt
(im Vergleich zu Core World) und warum? (vgl. Evolvierbarkeit)
\item Mit welchem Trick wird bei Avida eine Verl"angerung der
transienten evolution"aren Phase erreicht? (zus"atzliches Fitnesskriterium)
\item Was versteht man unter ``open ended evolution''? In wieweit wird das
in den Assemblerautomaten erreicht?
\item Wie beobachtet man das dynamische Verhalten eines Assemblerautomaten?
\item Was versteht man unter einer Fitnesslandschaft?
\end{enumerate}
\subsection{Form und Funktion}
\begin{enumerate}
\item Warum ist es schwierig gleichzeitig Form und Funktion zu
evolvieren? Welche Schwierigkeiten treten bei einem solchen Vorhaben
typischerweise auf?
\item Wie w"urde man heute vorgehen?
\item Welche Strukturen lassen sich mit Hilfe eines L-Systems leicht
und welche nur schwer nachahmen?
\item Wie funktioniert ein parametrisiertes L-System? Nenne ein
biologisches Beispiel f"ur sein Anwendung.
\item Skizziere, wie man L-Systeme evolvieren kann? (statische
Fitness, Koevolution oder Simulatio einer ``"okologischen'' Umgebung).
\item Erl"autere die Arbeit von Karl Sims.
\item Welchen Trick hat Sims angewendet damit sich leicht Bewegung
entwickeln kann?
\end{enumerate}
\subsection{Evolution"are Spieltheorie}
\begin{enumerate}
\item Wie ist der grunds"atzliche Ansatz, den die
evolution"are Spieltheorie verfolgt?
\item Erl"autere anhand des Gefangenendilemmas das Nashgleichgewicht?
\item Gibt es ein Spiel ohne Nashgleichgewicht?
\item Wie ver"andert sich das Gefangenendilemma im Falle von
zuf"alligen St"orungen? Berechne den Payoff von Tit-For-Tat f"ur ein
unendlich langes Spiel.
\item Was versteht man unter eine evolution"ar stabilen Strategie? Und
wie erkennt man sie?
\item Beschreibe die Arbeit von C. Lindgren (in der Vorl. vorgestellt).
\item Haben wir es hier mit ``open-ended evolution'' zu tun?
(fast, aber nicht ganz, warum fast? warum nicht ganz?)
\item Wie wird eine Spielstrategie repr"asentiert? Und wie wird sie
mutiert ?
\item Was bewirkt eine Verdoppelung des Genoms?
\end{enumerate}
\subsection{Emergenz und Evolution der Sprache}
(nicht SS 2007, daf"ur SOC im Detail)
\begin{enumerate}
\item Was versteht man unter einem Sprachspiel?
\item Beschreibe das ``nameing game''?
\item Welche Struktur besitzt eine System, dass eine ``Sprache'' lernt?
Wie arbeiten diese Komponenten zusammen?
\item Beschreibe die in der Vorlesung vorgestellten Arbeiten von Steels.
\end{enumerate}
\subsection{Kunst}
(nicht SS 2007, daf"ur SOC im Detail)
\begin{enumerate}
\item Fasse die Ans"atze zusammen, bei denen AL in der Kunst und Musik
eingesetzt wird.
\end{enumerate}
\subsection{Weiterf"uhrende Fragen}
\begin{enumerate}
\item Wie konstruiert man ein AL System, das die Evolution von
Form und Funktion (a la Karl Sims) mit einem k"unstlichen Genom
(incl. Expressionsdynamik) verbindet. Warum sollte man sowas tun?
\item Welche Beziehung besteht zwischen AL und aktuellen
gesellschaftlichen Fragestellungen?
\item Welche Verantwortung hat eine AL ForscherInn?
\end{enumerate}
%% \bibliographystyle{chicago}
%% \bibliography{bib/ACbib.bib,bib/all.bib,bib/gp-bibliography,local}
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\bibitem{Lucas1892} Lucas, F. A. (1892). Up to date. \textit{The Auk: Ornithological Advances}, \textit{9}(4), 395.
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\itemti{Cornerstone mathematics: designing digital technology for teacher adaptation and scaling.}
\itemso{ZDM, Int. J. Math. Educ. 45, No. 7, 1057-1070 (2013).}
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Summary: We report the results of a design-based research project in England that embeds digital technology. The research followed from two phases in the USA: (1) a design phase that used dynamic representations to foster conceptual understanding of hard-to-teach mathematical ideas, and (2) a research phase that measured the efficacy of the resulting technology-based curriculum units as implemented in Texas schools. The goal of the third phase in England was initially to ``scale up'' the US approach. We determined, however, that the materials had to be re-designed for adaptability by English teachers. We report how the features of the innovation -- particularly its technological infrastructure -- could be leveraged, not only to achieve positive learning outcomes, but also to lay the foundations for change in pedagogy and learning at scale. We identify an emergent framework of design affordances for teacher adaptability that are particularly salient when technology is a critical element.
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\bibitem{Krieg2017} Krieg, H. (2017). \textit{{\"O}kologisch-{\"o}konomische Systemanalyse}. Forschungsergebnisse aus der Bauphysik, 28. Doctoral thesis, Fraunhofer Verlag, Stuttgart.
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\frontmatter
The Project Gutenberg EBook of The Grey Book, by Johan M. Snoek
This eBook is for the use of anyone anywhere at no cost and with
almost no restrictions whatsoever. You may copy it, give it away or
re-use it under the terms of the Project Gutenberg License included
with this eBook or online at www.gutenberg.net
** This is a COPYRIGHTED Project Gutenberg eBook, Details Below **
** Please follow the copyright guidelines in this file. **
Title: The Grey Book
Author: Johan M. Snoek
Release Date: January 23, 2005 [EBook \#14764]
Language: English
*** START OF THIS PROJECT GUTENBERG EBOOK THE GREY BOOK ***
Copyright (C) 1969 Johan M. Snoek.
Produced by the nephew of the author.
Transcriber's Note:
(Gutenberg preparation by Ge J. Snoek 2004: [email protected]
The original printed paper book pages are marked as \textit{xxx} right
aligned, (because lots of pages are referenced: omitting page nrs
troubles comfortable searching, while footnotes are marked/numbered
between square [123] hooks.)
JOHAN M. SNOEK
THE GREY BOOK
A COLLECTION OF PROTESTS
AGAINST ANTI-SEMITISM AND PERSECUTION OF JEWS
ISSUED BY
NON-ROMAN CATHOLIC CHURCHES AND CHURCH LEADERS
DURING HITLERS RULE
INTRODUCTION BY URIEL TAL
Van Gorcum \& Comp. N.V. dr. H.J. Prakke \& H.M.G. Prakke---Assen, 1969
CONTENTS
INTRODUCTION (by Uriel Tal)
Part I
\begin{quotation}
1 PROBLEMS OF EVALUATION
2 FACTORS LEADING TO PUBLIC PROTESTS
3 RESULTS
4 HELP TO CHRISTIANS OF JEWISH ORIGIN
5 ``MERCY-BAPTISMS''\end{quotation}
Part II
\begin{verbatim}
6 HISTORICAL EVENTS
7 GERMANY
8 THE NETHERLANDS
9 BELGIUM
10 FRANCE
11 SWITZERLAND
12 DENMARK
13 SWEDEN
14 HUNGARY
15 RUMANIA
16 GREAT BRITAIN AND IRELAND
17 THE UNITED STATES
18 INTERNATIONAL ORGANIZATIONS OF CHURCHES\end{verbatim}
Part III
\begin{verbatim}
19 HISTORICAL EVENTS, 1939-1945
20 GERMANY
21 NORWAY
22 THE NETHERLANDS
23 FRANCE
24 YUGOSLAVIA
25 GREECE
26 DENMARK
27 SLOVAKIA
28 RUMANIA
29 BULGARIA
30 HUNGARY
31 SWITZERLAND
32 SWEDEN
33 GREAT BRITAIN
34 THE UNITED STATES
35 THE WORLD COUNCIL OF CHURCHES
36 TERRITORIES IN WHICH THE CHURCHES REMAINED SILENT
37 IN CONCLUSION
APPENDIX I
APPENDIX II
BIBLIOGRAPHY
PERIODICALS AND REPORTS\end{verbatim}
\textit{Introduction} (by Uriel Tal)
The protests of the non-Roman Catholic Churches against the persecution
and extermination of the Jews during the Nazi period, carefully compiled
and amply documented in this volume, possess a significance that is not
confined to the history of Christian-Jewish relations. They constitute an
important chapter in the history of Christianity itself in that they reveal
the deeper aspects of the Church's antagonism to the anti-religious and
hence anti-Christian character of Nazi anti-semitism.
The well-attested facts presented to us in this volume are a clear
confirmation of the Church's reputation of Nazi doctrines, not only when
these doctrines were directed against the Jews but, first and foremost,
when they threatened the very existence of the Church itself, both as a
system of theological doctrines and beliefs and as an historical institution.
The Church regarded freedom, freedom of man as well as its own, as an
inalienable right rooted in the nature of man as a rational being created
in God's image. Hence, when the Church was deprived at the right of
self-determination, it felt its very existence endangered, and it was then
that it recognized the full symbolic import of Jewish persecution. This
view was plainly set forth at the beginning of the persecution of the Jews
by the Nazi-regime in Holland, by D. J. Slotemaker de Bruine, Protestant
pastor and Minister of State, who declared:
``...Freedom of the spirit is our life-blood. By that I mean freedom in
questions of the spirit, freedom of conscience, freedom of the Church,
freedom of instruction, freedom of the Word of God, freedom to bear
witness...'' [1]
\textit{}
\textit{
In the light of this statement it is obvious that the Church was provoked
to raise its voice in protest chiefly because the Nazis appropriated the
messianic structure of religion which they exploited to their own ideological
and political ends.
This was made clear already in the early days of the Third Reich by ``Die
Geistlichen Mitglieder der Vorlaufigen Leitung der Evangelischen Kirche'' who,
in a memorandum (Denkschrift) addressed to the Fuehrer (May 1936), accuse
Hitler of pursuing a policy that is not only directed against the Church but
which is designed ``to de-Christianize the German people'' (das deutsche Volk
zu entchristlichen), quoting, among other things, the words of
Reichsorganisationsleiter Dr. Robert Ley: }
``The Party lays total claim to the soul of the German people...and hence
we demand the last German, whether Protestant or Catholic...'' [2]
To those Church circles that raised their voices in protest this totalitarian
structure of the Nazi regime presented a double threat to the very existence
of the Church. First, the pseudo-religious and pseudomessianic character of
Nazism was calculated to weaken the Church from within and to mislead the
Christian community, especially its youth. It became increasingly clear to
these circles that the Nazi racial doctrine --- which Hitler and also the
``Deutsche Christen'' had called positive Christianity in their first
formulation as early as 5 May 1932 --- constituted a kind of additional
gospel of messianic redemption that ostensibly strengthened Christianity
as an institution and as a religion of revelation. Secondly, this pseudo-messianic and pseudo-religious authority that the Nazi regime arrogated to
itself was able by means of its repressive measures to curtail the influence
of the Church and even to reduce it to silence. This danger was perceived at
an early date by the ``Bekenntnissynode der Deutschen Evangelischen Kirche'' in
its Botschaft (Part I, par 2, 5) adopted by the Conference held in Berlin-Dahlem 19-20 October 1934, which stated:
\textit{II}
``The National Church that the Reich's bishop has in view under the slogan:
One State --- one People --- one Church, simply means that the Gospel is no
longer valid for the German Evangelical Church and that the mission of the
Church is delivered to the powers of this world.... The introduction of the
Fuehrer principle into the Church and the demand of unconditional obedience
based upon this principle are contrary to the Word of Scripture and bind the
officials of the Church to the Church regiment instead of to Christ... [3]
Towards the end of the period that is dealt with in the sources collected in
this volume, in the year 1943, we also meet with a clear expression of the
Church's opposition to this pseudo-religious and pseudo-messianic character
of Nazism in the ``Pastoral concerning National Socialist Philosophy'' that
was sent in Holland:
... to parochial church councillors to give them the necessary basis for
their opposition in the struggle against National Socialist ideology, and
especially against the intangible, but all the more dangerous religious
ideas and expressions of National Socialism which will exercise an influence
even after the war.''
In its penetrating analysis of the totalitarian character of Nazism this
Pastoral observes:
``...It is not surprising that National Socialism has the power to become the
religion of the masses, and its assemblies to take the form of a kind of
popular worship in which a great deal of latent religious emotion is
released.... In carrying out its ministry the Church must therefore make
its work in this connection even more definite in character, and must tell
its members very clearly and resolutely that what is at stake here is the
first commandment: Thou shalt have no other gods besides me...!'' [4]
\textit{III}
This pseudo-religious and pseudo-messianic character of Nazism was by no
means accidental or the product of mass hysteria induced by some skilful
propagandists. It was rather an ideological structure that was consciously
given definite patterns and developed within a conceptual system in accordance
with its own laws of logic. In this development the traditional theological
concepts of Christianity were retained but given an altogether different
meaning. Values that had previously been regarded as relative in the culture
of Christianity and of the West now became absolute; and values that had
formerly been considered absolute, being interpreted as metaphorical or
visionary, became relative. Phenomena with an imminent historical essence
were lifted to a meta-historical plane. Means were converted to ends, and
ends were endowed with absolute authority in so far as they sanctified
the means.
In this manner the fundamental concepts of religion were not invalidated
nor the integrative functions served by these concepts impaired, such as
those cohesive factors that hold together the social structure and ensure
its normal functioning. The Nazis retained these concepts and their functions
as a legitimate part of their racial theory and, after depriving them of
their authentic historical content, turned them into political expedients
to be used in their attack against humanism, religion and Christian values.
Basic theological concepts such as God, redemption, sin and revelation were
now used as anthropological and political concepts. God became man, but not
in the theological Christian sense of the incarnation of the Word:
``...and the Word became flesh and dwelt among us...'' (John 1. 14) or in
the Pauline conception of the incarnation of God in Christ in whom ``the
whole fullness of deity dwells bodily'' (Colossians 2. 9).
In the new conception God becomes man in a political sense as a member of
the Aryan race whose highest representative on earth is the Fuehrer.
This change in the essential meaning of the concepts God-man is, from the
standpoint of cognition, effected by converting the relative into the
absolute and, from the standpoint of theology, by transferring the Pauline
conception (Ephesians 4. 24; Colossians 3. 10) from the plane of metaphysics
and eschatology to that of nationality rind politics.
\textit{IV}
It was this radical change from Christian doctrines to pagan myths that
aroused the Churches to express their protest against Nazism, and also
against the persecution of the Jews, in the above Pastoral of the year 1943:
``And there is now a return to the worship of life and power by accepting and
exalting the old Adam as the original and eternal \textit{man}. There is an attempt at
self-salvation --- the old Adam is not crucified with Christ (Rom. 6. 6) but
by his very own inmost strength achieves a new life and a heightened
vitality...'' [5]
Similarly, the theological concepts of sin and redemption were transferred
to a legal category of administrative regulations that demanded outer
conformity and inner obedience. The traditional conception of sin and
redemption that was common to all currents of Christian thought held that
man's redemption, and hence eschatological existence, depends on his faith:
``the righteousness of God through faith in Jesus Christ... since all have
sinned and... they are justified by grace... through the redemption which
is in Christ Jesus..."(Rom. 3. 22-24). In the totalitarian Nazi regime the
concepts sin and redemption were used as means by the State or the Party
to convert man into a loyal subject whose allegiance is assured by his
constant fear not only of violating some concrete ordinance or governmental
decree but simply of just deviating from the official ideology. The Christian
belief that man could be saved through faith in the forgiveness of Jesus who
died for his sins, ``so that the sinful body might be destroyed, and we might
no longer be enslaved to sin'' (Rom. 6.6), was transferred from the theological
to the secular, political plane. Even the comforting assurance of the believer
that his sins shall be forgiven and that he shall be found worthy of the
purifying influences of grace could now be gained only by the individual's
complete identification with the State, the Party and the superior Aryan race.
An instructive illustration of this shift from theology to ideology is to be
found in the circular letters (Rundschreiben) and in the speeches of the
Reichsorganisationsleiter Dr. Robert Ley, for example in his words of
26. June 1935:
``Strength through joy (Kraft durch Freude) is the embodiment of National
Socialism.
$<$V$>$
Over against sin we put discipline, over against penitence pride! Over
against the weak and their infirmities we put strength... " [6]
This doctrine was not mere Aryan propaganda; it became an integral part of
school studies and was systematically inculcated into the minds of the young.
The following is an example of a dictation given in 1934 to the third grade
of an elementary school:
``Just as Jesus redeemed mankind from sin and hell, so did Hitler rescue the
German people from destruction. Jesus and Hitler were persecuted; but whereas
Jesus was crucified, Hitler rose to be Chancellor... Jesus worked for
heaven, Hitler for the German soil...'' [7]
This same pattern of reversing meanings was also applied by the totalitarian
Nazi regime to the basic concepts of western culture. Nationalism as an
historical phenomenon of a people with a common language and culture and
with the consciousness of a common destiny was raised to a mythical,
meta-historical plane. The essence of national unity was discovered to
reside in race and soil; the cultural and spiritual creations of the nation
were attributed to man's biological resources. Similarly, the State became
an end in itself, an ideal meta-historical entity that was identical with
the national spirit. [8]
This view was critically described by the Dutch Church as follows:
``... The whole cult of National Socialism finds its most powerful
manifestation in a State which claims to support, lead and fill in the
material and spiritual, educational, cultural and religious spheres,
the whole life of its subjects. Not only does the State order the life of
the individual, but it takes a creative part in it. It becomes the founder
of the true religion and the dispenser of the true philosophy; it furnishes
the data for knowledge...'' [9]
\textit{VI}
Mythical nationality in the totalitarian regime thus developed a monolithic
structure which functioned as the only ontological framework in which the
individual may acquire his own identity, his selfknowledge and understanding.
While in a different, non-totalitarian civilization man establishes his inner
freedom by means of intellectual autonomy, the Nazi regime made the actual
biological belonging to the Aryan race into the ultimate condition for the
self-realization of Man.
Hence one who could not belong to the Aryan race, the prototype of whom was
the Jew, was doomed to be completely alienated, deprived not only of all
rights, but of the very justification to exist. It was this reversal of the
status of the individual which prepared the ground for subsequent developments
against which the Church protested, such as forced labour, the repression
of independent thought, the indoctrination of the young by the State and
their estrangement from their parents, teachers and preachers. An example
of this tendency towards the total dehumanization of the individual, as
reflected in the persecution of the Jews, and that provoked the Church to
protest, was the decree authorizing sterilization. The stand of the Church
in this matter was stated in the ``Letter on the Question of Sterilization''
that was sent in May 1943 by the Protestant and Catholic Churches in Holland
to the officials of the Reich and in which, among other things, we find the
following:
``...In the last few weeks the sterilization of the so-called mixed marriages
has begun. But God, who created heaven and earth and whose commandments are
for all men, and to whom even your Excellency will have to give account one
day, has said to mankind: `Be fruitful and multiply' (Gen. 1. 18).
Sterilization is a physical and spiritual mutilation directly at variance
with God's commandment that we shall not dishonour, hate, wound or kill our
neighbours. Sterilization constitutes a violation of the divine commandment
as well as of human rights. It is the last consequence of an anti-Christian
racial doctrine that destroys nations, and of a boundless self-exaltation.
It represents a view of the world and of life which undermines true
Christian human life, rendering it ultimately impossible... [10]
\textit{VII}
The fact that the protest of the Church against the persecution and
annihilation of the Jews was an inseparable part of its general protest
against the inhuman and anti-Christian character of modern anti-semitism
places the documents collected in this volume in a broad historical context.
These documents offer ample evidence of the Church's opposition to an
historical phenomenon rooted long before the Nazis came to power, hence
also prior to the rise of modern anti-semitism. The protest of the Church
was fundamentally directed against those pagan and mythological elements
that had crept into Christianity itself in the course of its historical
development among the heathen.
To many of the fathers of modern anti-semitism, which is the racial and
political Anti-semitism that arose towards the end of the 19th century and
reached its highest stage during the Third Reich, the rejection of Judaism
was tantamount to the rejection of religion in general.
This view goes back to Feuerbach's anthropological criticism of religion,
to the young Hegelians (Max Stirner, Bruno Bauer) and to the early Romantics
who longed to return to the primitive forms of a religion called
``vorchristliches Germanenthum''. [11] Modern anti-semitism was influenced by
these streams of thought through Nietzsche's concept of the `Antichrist',
although Nietzsche himself kept aloof from the more vulgar manifestations
of political anti-semitism of his day. In him the anthropological view
reaches its culmination --- God, who is nothing more than the deified form
of man [12] is finally overthrown by Dionysian man who found courage to
assert his instinctive life and abjure the gross and enslaving notions of
Christianity that men
\textit{VIII}
are equal and can be redeemed by faith, the gospel of the downtrodden and
everything that creeps on earth. [13] These views, inimical to religion and
to Christianity, were already being expounded with great vigour towards the
end of the 19th century. Christian doctrine was accused of perverting man's
instinctive life, vitiating his natural enthusiasm, inflaming his ego,
invading his private life over which it declares its dominance only to
enslave human nature, to weaken and alienate man, by imposing upon him
``un-natural'' restraint such as the anguish of his conscience.
Wilhelm Marr, one of the early fathers of modem racial and political
Anti-semitism and the man who during the late 70's coined the term
'anti-semitism'[14] included in the rejection of Judaism his critique of
Christianity as early as the year 1862. In a polemical work called
``Der Christenspiegel von anti-Marr'' by Moritz Freystadt, a member of
the ``Society for History and Theology'' in Leipzig, written in answer
to Marr's ``Judenspiegel'', the author interprets Marr's rejection of
Judaism as a rejection of monotheism, based on his anthropological view
of God as a subjective product of our conscious life --- an antireligious
analysis Marr evidently borrowed from Voltaire, Feuerbach and Bruno
Bauer. [15]
With Marr's intensification of anti-Jewish propaganda inspired by the new
racial anti-semitism we find increased criticism of Christianity both as a
system of beliefs and as an institution. In one of his popular books
``Religioese Streifzuege eines Philosophischen Touristen'' (1876) Marr,
relying on theories propounded by Voltaire and Feuerbach, observes
that from the atheistic point of view it is evident:
\textit{IX}
``that Christianity, in its dogmas and precepts, is like every religion, a
malady of human consciousness. The philosopher explains... every religion
as a product of man's conscious life and relegates to the sphere of phantasm
the so-called `revelations' of which all people boast depending on the state
of their culture...'' [16]
Most additional factors in the rejection of Judaism, Marr continues, go
beyond the attack directed against Christianity as a system of beliefs and
superstitions that demoralizes man and corrupts his nature. Anti-semitism is
not only called to combat religion and Christianity; its chief aim is to
save the German nation and the whole world from Jewish domination and from
the moral depredation of the Jewish race. Christianity is not yet fully
cognizant of the gravity of the problem, and it deceives itself when it
thinks that baptism or conversion is a gratuitous deliverance from native
corruption, for the Jew's aberrations are not religious but biological
and hence incorrigible. The Jewish question, Marr concludes, is a racial
question for the infidelity of the Jew is essentially biological, and
hence Christianity is in no position to save the world from the perils
of the Semitic-Jewish race. [17] \textit{X}
We here encounter a primary distinction between the doctrines of racial
anti-semitism and those of the Christian Heilsgeschichte, a contradiction
that awoke the Church to the dangers of Nazism when, in 1933, it opposed
the ``Arierparagraph''. This racial law rejected the notion that the Jews
could still hope for redemption, and for a renewed status of election,
assured them in the New Testament (Rom. 9-11) on condition that they
acknowledge their error and accept the redeeming truth of Christianity.
Even in the early years of racial anti-semitism, in the seventies and
eighties of the last century, we already find this inner contradiction
between a racial theory that regards Jews as the ontological embodiment
of an ineradicable evil and the views of the Heilgeschichte that believes
this evil to be remedial if only the Jews could be persuaded that salvation
comes from the Savior who was sent first of all to the Jews themselves, and
who atoned for the sins of all mankind.
It is this inner tension between the recalcitrance of the Jew and the
incorrigibility of Judaism that refuses to acknowledge Jesus as the Messiah,
already conspicuous in the change that took place in Luther's attitude to
the Jews between 1523 and 1543, which charactarizes the theological and
political attitude of Adolf Stoecker, court preacher in the Bismarck era
and one of the leading figures of modern anti-semitism. Until recently
historians concentrated much on his importance in preparing the ground
for racial and political anti-semitism. It is true that without his powerful
influence during the last decades of the 19th century the rise of modern
political anti-semitism would be incomprehensible. A more balanced approach
has been taken lately, as may be seen in the instructive study by Walter
Holsten on the part played by Stoecker in the rise of modern anti-semitism.
The author shows that many phases of Stoecker's anti-semitism had their roots
in the conservative tradition of Lutheranism and at the same time were
opposed to the anti-Christian tendencies of racial anti-semitism. [18]
\textit{XI}
The early phases of Stoecker's activity already reveal the ambivalent
nature of his attitude to the Jews and to Judaism, an ambivalency that
characterized the anti-Christian elements in antisemitic ``Christian''
ideology throughout the days of the Third Reich. In his speeches after
the political defeat of his Christian Social Labor Party in the summer
of 1878, Stoecker insisted on making a distinction between the anti-Jewish
attitude that arises in conjunction with or flows from Christianity and
the antisemitic attitude which at the same time also impugns Christian
ethics. In his well-known antisemitic speech as early as 19.9.1879 Stoecker
warns his listeners:
``We can already detect here and there a hatred directed against the Jews
that is contrary to the Gospels''. [19]
Even in his most violent speeches against the Jews Stoecker did not draw
the extreme biological consequences of his racial theories and continued
to maintain that conversion was the only authentic solution to the Jewish
question that would complete the universal mission of Christianity and that
only baptism could save the Jews from their ignominious belief in the
validity of the halacha after the coming of Jesus. The salvation promised
to the Jew then is to be saved from his Judaism. The final redemption,
however, will not raise the Jews above the nations of the world, as
promised in the Old Testament, but this position of eminence and election
will pass, or actually has already passed, from the Jews not just to the
Christians but to Christian Germany. The redemption promised to the Jews
is thus to be attained by way of the baptismal font at the entrance to
the Church:
\textit{XII}
``All Israel will be saved when the fullness of the heathen shall have come
to an end. This was Paul's promise to his beloved people --- final salvation
and not a future glory that will raise Israel above the other nations as
proclaimed in the Old Testament... and every believing Christian knows well
what a rejoicing there will be in the Kingdom of God when the people of the
Old Testament finally acknowledge their sin against Christ and repent. This
event will be hailed by all Christendom and by the angelic hosts with paeans
of praise, and it will be turned by the Church in the End of Days into glory
and renown when Israel will bring to it its uncommon religious talents and
intellectual gifts...'' [20]
The inner tension between the theological view that sees the solution of the
Jewish question in the liquidation of Judaism and the racial view that sees
it in the liquidation of the Jews is clearly expressed in an address delivered
by Stoecker on 8.2.1882 about the danger to the German Reich from Jews in
public life, in which he states:
``We regard the Jewish question not as a religious nor indeed as a racial
question. Although it is at bottom both of these, it appears in its external
form as a social-ethical question, and is treated by us as such. No people
can tolerate the preponderance of an alien spirit without degenerating and
being destroyed? We would not solve the Jewish question radically by force,
but gradually in a spirit of peace and amity... We must keep the wounds
open until they are healed...'' [21]
Although Stoecker himself was opposed to the use of force, modern political
anti-semitism, which was to no small degree influenced by him, did not shrink
from advocating violence in its hostility to Judaism, to religion and finally
to Christianity.
A significant contribution in this direction was made by the Darwinian
racial doctrines of Eugen Duehring and his antisemitic disciples. Whereas
Marr had formulated the anti-religious meaning of modern anti-semitism in
ominous terms of the Jewish domination of Europe and especially Germany,
D\"{u}hring adopted a so-called constructive approach by suggesting an alternative
to religion and religious culture, namely, race. In his antisemitic writings
after 1880 Judaism serves as the prototype of religion in general, including
Christianity. \textit{XIII}
The primary aim of this anti-Christian anti-semitism was for Duehring the
struggle against Jews and Judaism, and this also entailed the struggle
against the monotheistic religions and all forces that suppressed what he
called ``the instinct of the free, natural life.'' In his anti-religious
book ``Wert des Lebens'' (1877), and especially in the third edition issued
four years later, he points out that Christianity as a monotheistic religion
is opposed to life and that all religious systems are nothing but pathological
maladies (ein Stueck weltgeschichtliche Krankheitslehre des Geistes).
Christianity is thus not interested ``in ennobling man, but rather in
suppressing his natural instincts'' as is evident, for example, in the
``paradox Christian doctrine'' of the crucifixion of the flesh. [22] Hence,
it is absurd and hopeless to conduct the struggle against the Jews with
Christian theological concepts borrowed from Judaism, and those Christians
who attach importance to this only deceive themselves for it is plain that:
``...their anti-semitism lacks the primary truth, namely, that Christianity
itself is Semitism, a truth... that must serve as the terminus a quo for
all genuine anti-Hebraism...'' [23]
As long as the Christians fail to disavow their Jewish source and their
Jewishness they themselves will be tainted by its anti-natural influence.
But since Christianity is inextricably bound to its Jewish origins, and
even the New Testament is nothing but ``a racially Jewish tradition'' (eine
rassenjuedische Ueberlieferung), the only hope for struggling humanity is
to throw off once for all this humiliating yoke, meaning the religious
heritage of Jews and Christians alike.
The liberation from the Jewish-Christian heritage, on the one hand, and
the strengthening of the Nordic German race on the other cannot be achieved
through the process of education or civilization but only by means of racial
purity which will cleanse man of religious depravities and restore the vital
sources of his instinctive life. Christianity is inadequate for this struggle
since it is itself ineradicably debased by its complicity with Judaism:
\textit{XIV}
``Those who would cling to Christian tradition are in no position to combat
Judaism effectively. ...An understanding Christian cannot be a serious
antisemite... The Nordic gods are rooted in nature itself, and no millennial
diversion can eradicate them... We here see a vivid phantasy in operation
that is incomparably loftier than the Jewish slave-imagination...'' [24]
This basic thesis that racial anti-semitism must also be directed against
Christianity continued to be elaborated from the end of the 19th century
onwards by Theodor Fritsch as well as in a number of journals:
the Antisemititche Correspondenz, which in 1888 became the official organ
of the D.A.P. under the name of Deutsche-Soziale Blaetter, the
Antisemiten-Katechismus which was later called Handbuch zur judenfrage
and, in the early years of the present century, the influential journal
Hammer. The general tendency of this movement was directed against
Christianity as an ecclesiastical institution, sometimes chiefly against
the Catholic Church which was suspected of ``ultramontanist'' sympathies
for a foreign ecclesiastical power. Christianity was also opposed as a
system of beliefs and practices that tended to debilitate the German
Aryan race in its struggle for existence.
Finally, Christianity was opposed because of its Jewish origins which
deteriorate the whole human race by elevating spirit over body, rational
thought over the wisdom of the senses, abstract ideas over direct and
spontaneous experience, and the discursive intellect over the vital emotions.
In the course of this debate the antisemitic movement displayed a readiness
to reconcile itself to the continued existence of Christianity on condition
that it subsitute the biological values of the Aryan race for its Jewish
origins, as was recommended by the idealogues who made Jesus a member of
the Aryan race --- Julius Langbehn, Max Bewer, Houston Stewart Chamberlain,
Leopold Werner, and the German Christians in the days of the Third Reich. [25]
\textit{XV}
We find the same line of thought pursued by the followers of Duehring,
such as Prof. Paul Foerster, as well as in those circles connected with
the antisemitic journals, such as Heimdall, Freideutschland, Staatsburger
Zeitung, also some of the functionaries connected with the imperialist
Der Alldeutscher Verband, such as Friedrich Lange, the author of the
anti-Christian Reines Deutschtum (1893), and numerous writers, historians,
orientalists, scientists and students influenced by anthropology, materialism
and Darwinism. A popular exposition that reveals the national and Romantic
roots of this ideology appeared in the Hammer (Oct. 1908), and reads in
part as follows:
``What shall we do with a Christ whose kingdom is not of this world? A
Bluecher, a Gneisenau, a Koerner, an Arndt can always be useful for Germany,
but not a Christ. The God who was called upon at Leuthen, Leipzig and Sedan
was not the God of love, nor the God of Abraham. Christ comforts the lowly,
the weak and the sick. We too are sorry for these poor folk and try to
alleviate their condition; but they are of no use to us and to our future.
They only degrade that which we deem to be the highest good --- the German
character. Strength, health, the joy of life are what we need. The kingdom
of Heaven can be left to the lowly and the wretched, as long as we possess
the earth. Give the Bible to the sick and the lonely, the shut-ins and the
scholars who wear their faces on their backs!...'' [26]
Similarly, the antisemitic propagandist, Dr. Ernst Wachler, writes in the
same journal (Jan. 1911):
``Away with the stones and tales, the doctrines and precepts of Jews as well as
of Christians!... Not only the free-thinkers, but our basic Aryan instincts
demand: the Church with all its trappings must be done away with...'' [27]
\textit{XVI + (whole page XVII is footnote)}
The available historical sources, including the documents collected in this
volume, clearly indicate that the protests of the Church against the
persecution of the Jews, with its human and ethical concern for their fate,
were an inseparable part of a more comprehensive opposition directed against
the pseudo-messianic and hence anti-Christian character of Nazism. Seen in
this context, the protest of the Church gives rise to a number of historical
and theological questions that require further study. The questions that
arise fall into three groups.
A. To what extent did the secularizing tendencies of the last century, the
rationalistic attacks on religion, the Romantic philosophies, pagan mythology,
Darwinism and the anthropological critique of religion, contribute to the
anti-Christian character of modern anti-semitism?
How did the process of secularization influence the teachings and art of
Richard Wagner, the Christian mythology of Houston St. Chamberlain, Julius
Langbehn, Ernst Bergmann and the movement of the ``German Christians'', or
the ``Mythus'' of Alfred Rosenberg? Can modem historiography support the
psychoanalytical Freudian explanation of anti-Christian anti-Semitism in
terms of a revival of vestigial pagan elements which were latent in
Christianity itself, and which consequently revolted against the ethical
Judaic basis of Christianity and against the Jews who were now made
responsible for all that disturbed the Christian conscience?
From the vast literature that has grown up around these problems [28] we see
that side by side with the all-pervasive secularization of life there were
also historical and theological factors embedded in Christianity which
later turned against Christianity itself. Through further study, we might
find in the history of Christianity traditions \textit{XVIII}
that originated in the barbarism of the pagan world, turned anti-Christian
by that very paganism, then continued as anti-Jewish attitudes and policies
on the part of the Christian world --- and finally culminating dialectically
into a destructive force that was directed not only against Judaism, but
through Judaism against Humanity and hence also against Christianity. One
of these powerful anti-Jewish elements which rooted in Christianity, and after
having been secularized became an effective means used by totalitarianism
against the Jews as well as against the Jews as the symbol of non-conformism,
as the embodiment of the human quest for a free existence, for the right to be
different and yet to be, is the very concept of Collective Guilt. Its origin
is the idea of guilt for the crucifixion of God who took on Flesh (Matt. 27. 25;
I Thess. 2. 15), a guilt which lies as a heavy yoke on the shoulders of all
the Jews till the end of the days.
It was applied to social life by various Church Synods (such as Elvira in 306,
Clermont in 535, Orleans in 538, the Lateran Councils of 1179 and 1215) with
their succession of repressive measures and harassments directed against the
Jews. It culminated under the influence of blood libels in the late Middle
Ages (Andreas of Ryn p. 1462, Simon of Trient 1475) [29], and in Modern
Times (Tisza-Esslar, Korfu, Xanten, Polna, Konitz) --- down to the days of
the Third Reich. By using the very pattern of a Collective Guilt, the
Christian projected on to the Jew the frailties common to all human beings.
This mechanism enabled the Christian to see his own weakness reflected in
the Jew so that by persecuting the Jew, moreover by exterminating him, the
Christian could obliterate his own image as a sinner, and cleanse his
conscience from the burden of guilt.
These patterns of thought and conduct, these models of generalization,
projection and prejudice that originally were established by Christianity
with respect to the Jews --- to what extent were they now employed by the
Nazi regime against Humanity, as well as against the Church itself
whenever the racial antisemites attacked its ethical Judaic basis?
\textit{XIX}
B. The second group of questions concerns the problems as to whether the
survival of the Jews on the one hand, and their ultimate Christianization
on the other, are both indispensable to Christianity.
Since the promise made to the Jews in the Old Testament (Gen. 22 .18;
II Sam. 7. 12; Isaiah 7.14), will be fulfilled or perhaps superseded by
those of the New Testament (Rom. 9-11) when the Jews return in penitence
and acknowledge Jesus as the Messiah, it seems that the Christian concern
for the fate of the Jews, even in the days of the holocaust, is unavoidably
accompanied by an interest in his salvation. Alas, his salvation is
conceived by the Christian in terms that are unacceptable for the Jew as
long as he wishes to adhere to Judaism as a religion, a people and an
unfulfilled eschatology.
As we study the documents before us in their total historical context
including parts not directly relevant to the very protest and therefore
not printed in this volume, we are impressed with the following fact;
while the Church raised its voice against the persecution of the Jews
out of human motives, as well as in the hope of thereby strengthening
its own members, the traditional, dogmatic concept of the Jew continued
to be dominant. According to this view the persecution of the Jews
constitutes an error, not only for reasons of humanity, but mainly
because persecution prevents the Jew from seeking redemption among his
persecutors. It prevents the Jew from turning to Jesus as the Messiah
and from seeking in the New Testament that salvation which not only is
promised him, but without which Christianity itself is doomed to remain
unfulfilled. From the theological point of view regarding the right of
Judaism to exist, the Church in its protest against the Nazis reverted
to the original attitude of Luther, as expressed in ``Das Jesus Christus
eyn geborener Jude sey'' of 1523. When Luther protested against the
anti-Jewish policy of the Church, claiming that the Church treated the Jews
``als waren es hunde'', and that under such circumstances he himself would:
``...ehe eyn saw geworden denn eyn Christe'', this very protest was also not
based on an acknowledgment of the right of Judaism to exist as an independent,
autonomous religion. The motive that inspired this protest was the hope that
Christianity would mitigate the persecution of the Jews and apply to them
instead the Christian Commandment of love and tolerance, as written by Luther:
\textit{xx}
``...Ob etliche halsstarrig sind was ligt daran? Sind wyr doch auch nicht alle
gute Christen...''. In that case, and only in that case, Christians might be
hopeful that the Jews would return in penitence and believe in the salvation
brought to them by their own Messiah.
Against this historical background [30] it seems that even during the Holocaust,
Christianity continued to identify the Jew not in his own, authentic, terms,
but according to the classical traditions. The Jew is one who persists in the
impenitent rejection of Christ, but must be saved, for it is the Jew who has
to complete the eschatological process of the Heilsgeschichte. Therefore Jews,
and especially converts, have to be rescued from racial discrimination.
Moreover, since Judaism continues to be an integral part of Christianity,
the very notion of the Jews as a race can have no basis whatsoever in
Christian theology. [31]
This has been stated as early as September 1933 by the theological faculty
of the University of Marburg in its statement against the ``Arierparagraph''.
Similar statements were issued by theologians such as Rudolf Bultmann and
the members of the Bekenntnis der Vaeter und die bekennende Gemeinde (Betheler
Bekenntnis), 1934 [32]. Thus, even at the height of Nazi persecution and in
times of the extermination of the Jews, the Church would not acknowledge
Judaism as a religion in its own right and on its own terms, but insisted
that a Jew who became a Christian was merely fulfilling his predestined role;
such a Jew did not leave his faith, he returned to his true faith.
It is most symptomatic and instructive to note that in the controversy
between Heinrich Vogel, one of the leaders in the protests against the
persecution of Jews and the author of the ``65 Theses of Protest'' (March 1933)
and Friedrich Gebhart, a spokesman of the ``German Christians'' and the author
of the ``Reply to the 65 Theses'' (May 1933), \textit{XXI}
both sides, despite their theological and political contradictions adhere to
the same traditional Christian view that the Jews are in a state of rejection
(Verwerfung). One view holds that the Jew can abrogate his old covenant with
Jehovah and step over to the side of the Redeemer; the other holds that the
derelict Jew is beyond salvation and the redeeming influences of the Church,
that Ueberzeuging cannot overcome Zeugung, that the Vollendung of Judaism in
Christ should be turned into the Endloesung of Jewish existence. Both, however,
despite the far reaching differences and contradictions between them, deny the
Jew the right to live on his own terms and according to his own autonomy.
This approach to the Jewish question on the part of those who protested
against the persecution of the Jews was not confined to the Bekennende Kirche
in Germany. Even the Dutch Church, in the early forties, did not deviate from
its theological tradition. A typical illustration is to be found in the
Pastoral Letter written by the General Synod of the Dutch Reformed Church
(Sept. 1941), a document that will go down in history as a striking witness
to spiritual integrity and moral courage. Although the Letter emphasizes the
fact that the New Testament is dependent on the Old Testament (Deut. 6, 4-5;
Mark 12, 29-31) and that the love of one's fellowman also applies to the
treatment of the Jew (Lev. 19, 18; Matt. 22, 39), it defines Judaism as a
religion that is destined to disappear by being redeemed through and in
Christianity. Again, this is in keeping with Christian tradition which holds
that the metaphysical status of election and the promise of eschatological
salvation as given to the Jews in the Old Testament are fulfilled in Jesus
who is ``...the fulfilment of all God's promises to Israel, the true king of
this nation sent by God...'' [33]
Hence, the document continues, having rejected Jesus as the redeemer,
the Jews are still sunk in sin: ``...Israel did not recognize Him, but
rejected Him... In this way they hardened their hearts against the grace
of God... They are no longer Israel in the original sense of the lord,
they are `Jews' now. A Jew is a man of Israel who rejects Jesus Christ,
and thus is to us a sign of human hostility to the Gospel...'' [34]
The Church that protested Jewish persecution by the Nazis with such courage
and religious conviction still finds it indispensable to advocate conversion
as the only solution to the problem of Jewish stubborn existence, an
existence which equals infidelity: \textit{XXII}
``...The true destiny of the Jewish people lies in its Conversion to Christ,
by joining the Christian Church. The Jew remains a Jew in the bitter sense
which this word has for him first and foremost; the Jew cannot free himself
from himself, as long as he does not come to Christ...'' [35]
Are there any pronouncements of the Church that offer a Christian-Jewish
relationship other than that of conversion? [36] Is there a possibility that
the Church may acknowledge the inherent right of self-determination for
the Jew, so that he could retain his identity and not seek to ``free himself
from himself?'' This ``bitter sense'' of the Jew the Church spoke about even
when protesting against Nazism, is it indigenous to Judaism or rather the
result of the social and political conditions in a Christian world?
Similar questions arise when we read the documents in Appendix I which do
not deal with the period of the Third Reich but with the period after the
Second World War. In these documents we find a number of explicit statements
by eminent Christian theologians condemning anti-semitism. But even here we
find no acknowledgment of the right of Judaism to exist on its own terms.
Nor do we find such acknowledgment in the special declaration of a group of
theologians, during the Second Assembly of the World Council of Churches
which convened in Evanston in 1954, entitled ``Hope of Israel'' [37]. In this
declaration a systematic attempt is made to renew relations with \textit{XXIII}
Judaism since"... to be a member of the Christian Church is to be involved
with the Jews... and the people of the New Covenant cannot be separated
from the people of the Old Covenant...'' [38]
Jews, however, are still regarded as candidates for salvation on Christian
terms, so that even in this enlightened document --- a document which was
composed years after the wholesale extermination of the Jews by the Nazis ---
theologians find no other solution but ``...to hope for the conversion of the
Jewish people...'' [39].
Moreover, when these circles in the Church desire for reasons of conscience
and remorse to express ``...the grievous guilt of the Christian people towards
the Jews throughout the history of the church...'', they find no better way to
express their deep sorrow than to revert to the ``Findings of the pre-Evanston
Conference of the American committee on the Christian Approach to the Jews''
(Lake Geneva, Wisconsin, August 8-11, 1954) which states that '... the Church
cannot rest until the title of Christ to the Kingdom is recognized by His own
People according to the flesh...''' [40].
Another typical example of this attitude is the proclamation of the Joint
Committee of the World Council of Churches and the International Missionary
Council, after its Consultation at the Ecumenical Institute, Bossey, Sept.
12-18, 1956. An attempt was made to elevate the missionary activities of the
Church, to seek the salvation of the Jews by the power of the spirit only:
``...Our hope for the Jews does not mean that we can calculate the time or
define the nature of the coming of Christ in his Kingdom... We may find a
further warning against too precise speculation with regard to the Second
Coming of Christ...'' (Ch. III, par. c, d.). [41] In conclusion, however, the
Joint Committee could not help adding a declaration which for the Jew makes
any authentic dialogue meaningless if not impossible: ``... The Jewish people
will not find their true destiny until they return and acknowledge Jesus as
Christ and Lord'' (Cf. Chap. IV, 6) [42]. \textit{XXIV}
C. The third group of questions that arise from reading the documents and
require careful study, deal with the actual situation as it existed during
the Nazi regime. Were the protests of the Church effective, in rescuing Jews
and then in strengthening the spirit of resistance, or even the religious
feelings among Christians? Were the protests raised at the right time and
under the proper circumstances, to mitigate the persecution or to postpone
the annihilation of the Jews? Was the protest the most useful means of
rescuing Jews, or would it have been more helpful for the Church to keep
quiet so that it could devote itself more to actual underground activities
- but, then, could the Church keep quiet? Was the Church, in its protest,
ready to endanger its members as well as their relatives for the sake of
an effective anti-Nazi struggle, or did the protest function as a Catharsis,
relieving the members of the Church from the burden of moral responsibility
towards the persecuted? Did the protests create a new, perhaps even a
revolutionary non-conformist stand of the Church over against political
power? How was the protest of the Church related to the concept of obedience
to the existing regime, as expressed in Paul's Letter to the Romans Ch. 13,
and in Luther's ``Von weltlicher Obrigkeit wie weit man ihr Gehorsam schuldig
ist'' 1523? Finally, what was the reaction of the Jews who were persecuted,
and especially of those Jews who lived in free countries and who might have
been expected to exert themselves to save their brethren? Did they endanger
their personal safety to rescue their fellow-Jews and display a deeper sense
of responsibility towards them than the Church?
This collection of sources, by concentrating on only one aspect of the
entire interrelationship between Christianity and Judaism during
the period of the holocaust may confuse the reader in thinking that the
Protest was the prime characteristic and policy of the Church regarding
anti-semitism, the persecution of the Jews and their extermination. The
author of this book, the Rev. Johan M. Snoek, is correct in bringing to
our attention that the Protest must be viewed as one and only one aspect
of the position of the Church and of the Christian world as a whole
during the Nazi regime. A collection of sources on the Protest of the
Church does not preclude the fact that there existed other positions
among Christians; the position of cooperation with antisemites, whether
it was active or passive, direct or indirect, with knowledge of without,
whether voluntary or \textit{XXV}
through coercion. This volume does not attempt to research the entire
and definite historical and theological position of the Church during
the Holocaust. Its purpose is to bring light upon one aspect, which until
now has not been sufficiently investigated. By having gathered these
documents, and by having placed them before us in their historical and
geo-political order, a major contribution has been made towards a more
balanced and varied understanding of this period.
Uriel Tal
The Hebrew University, Jerusalem
\textit{XXVI}
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\gutchapter{PREFACE}
Much has been published as to whether the Pope remained silent during the persecution of the Jews in Europe, primarily as a reaction to Rolf Hochhuth's play ``The Representative'' (Der Stellvertreter in German or in Dutch ``Plaatsbekleder"). Not so much, however, has been published about the attitude of the non-Roman Catholic Churches. When there is a vacuum in our knowledge, it is an excellent breeding place for myths. We should fill a gap therefore as well as possible.
There exist certain myths, which die hard. Many people still believe that it was Richard III who murdered the princes in the Tower, though this has been shown to be false. The Dutch people for instance did not behave as courageously during the Second World War as is generally believed, but the myth seems to be firmly established; just as is the story that the King of Denmark walked through the streets of Copenhagen wearing the yellow badge in protest against the German measures concerning the Jews.
One should not lightly dismiss the existence of such historical untruths on the assumption that there is always a grain of truth in every myth. Sometimes a myth is completely false, as in the case of murdered princes. Moreover, this type of myth is sometimes very harmful. We need only remind ourselves of the infamous ritual-murder myth, suggesting that the Jews used the blood of a Christian child for ritual purposes. \textit{1}
It is undeniable that throughout the ages many Christians took an active part in the persecution of Jews. [43] This fact has been officially and repeatedly admitted by Christian bodies. Some of the statements in this documentation unequivocally plead guilty in this respect. Small wonder, then, that many Christians, as well as Jews, honestly believe that ``there was a complete and terrible silence on the part of the Church'' [44]. In the process of creation of anti-Jewish myths, there is a tendency to generalize: ``The Jews have ...'' We like to think in general terms because stereotypes are so easy, whilst it costs us much more mental effort to discriminate. Let us not commit the same offence against logic as the anti-Semites have and let us remember that it is just as fallacious to talk about ``the Churches'' as about ``the Jews''.
It is important for many reasons not to overrate the positive things the Churches did and said. It is also important, again for many reasons, not to belittle them. We certainly must denounce acts of anti-Semitism, even when outstanding leaders of the Church were the perpetrators, but this remains a negative. We must also mention the positive, which is more encouraging. I believe this is one of the underlying intentions of ``Yad Vashem's'' competent Department in trying to seek out and honour the ``righteous of all Nations'': non-Jews who helped Jews at the risk of their own lives. [45] It seems far too early to come to a definite evaluation of many aspects of the holocaust. Far be it from me, to claim that I can say the last word about that one aspect under discussion here: the attitude \textit{2} of the non-Roman Catholic Churches. I can and must try to be objective, but I cannot be detached, as probably none of our generation can: we were all involved, in one way or another. [46] But I am convinced that our generation can and must do the groundwork. It must collect the material that may otherwise be completely lost or forgotten, and investigate it before even more people, who were personally involved, have passed away.
Collecting these documents was like trying to make a jigsaw puzzle from which many pieces are missing, the difference being that in this case one often does not even know that something is missing. However, the lack of other pieces is known. [47] As regards my own country (the Netherlands), I am fairly sure that the collection of documents is well-nigh complete. Some statements issued by Churches were published in Bulgarian or Slovak, \textit{etc}., but not in English. Even such documents as were available in English were not generally known. Most of the material in this book had to be translated from Hebrew, French, German, Dutch, Norwegian, Swedish, Danish, Slovak, Hungarian, Bulgarian or Greek. Initially, I sent a circular letter to the heads of Churches in Eastern Europe asking for information and I received some replies, though not many. Some replies stated that no documents were available because everything had been destroyed during the war. This seems quite possible, and perhaps we must give the Churches in question the benefit of the doubt. Moreover, it was not advisable, for security reasons, to keep certain documents. Thus, for instance, all documents of the World Council of Churches and its preceding organizations, which might incriminate Christian leaders in Germany, were destroyed when, in 1940, it was feared that Germany would invade Switzerland. [48]
\textit{3}
Yet, some Churches, which probably could have sent material, and which in some cases as, for example, the Churches in Bulgaria and Greece, had a good record of resistance against anti-Semitism, failed to do so. It would appear that Church archives are sometimes the safest place in the world for documents not to be found. The Library and Archives of ``Yad Vashem'', in Jerusalem, had much material. I was also able to spend some days in the Wiener Library, in London, and in the Library of the World Council of Churches, at Geneva. I could never have succeeded in finding the material and having it translated without the help of many interested friends, Jews as well as Christians, to whom I am deeply indebted. It would be difficult to mention all their names, but I should like at least to express here my indebtedness to the late Director of ``Yad Vashem'', Dr. Arjeh Leon Kubovy, of blessed memory; and to Dr. Shaul Esh, of blessed memory, who made valuable suggestions for the chapters on Germany. I am also especially indebted to Dr. J. Robinson, of New York, and Prof. Dr. C. Augustijn, of Amsterdam, who read the manuscript and suggested many improvements. Of course the responsibility for any eventual mistakes solely rests on me. I am deeply grateful for all the kind help rendered to me by the Chief Librarian of ``Yad Vashem'', Miss Ora Alcalay, and her assistants.
Most of the chapters in Part III (During the War) have some particulars about anti-Semitic measures taken by the Germans: I wanted to give some historical background for the statements issued by Churches. For the background of statements issued in the different countries before the Second World War, the historical survey and the chapter on Germany in part II should be consulted. One can never have too much knowledge of the situation and background in the countries concerned, if one is to see facts clearly in their historical context and interpret them correctly. Thus, more publications are mentioned in the notes for further study. Some figures concerning the membership of Churches are given in Appendix II, though they are of limited value. Many territorial changes took place in Central and Eastern Europe. Some Churches count as members all who were baptized, whether they ever attended services or not; others count \textit{4} ``communicants''; the Baptists do not count the children. But one will at least acquire a conception of the numerical strength of a certain Church.
An investigation into the question whether the non-Roman Catholic Churches kept silent, must necessarily have certain limitations. Firstly, no statement issued by a Church under the authority of the Pope are recorded in this book, with the exception, of course, of joint statements issued by Protestants and Roman Catholics, as was the case in the Netherlands. Thus I have recorded nothing from the Polish Greek Catholic Metropolitan Sheptitsky, or from the Maronite Patriarch of Syria, Mgr. Arida. [49] Secondly, this investigation is not concerned with the acts of individual Christians, unless they were leaders of the Church and clearly spoke in the name of their Church. [50] Thirdly, I have not recorded the contents of protests issued solely against the treatment of Christians of Jewish origin. It was certainly the duty of the Churches to do all in their power to protect those Christians, but this is not my subject. I am interested in what manner the Churches acted or failed to act on behalf of the Jews in general.
This book is first of all an attempt to draw up an inventory, rather than to draw up the balance-sheet. However, the fact that I have often had the privilege of lecturing on the subject to Jewish, Christian or mixed audiences, always followed by brisk discussion, encourages me to feel that I have correctly understood some of the problems and questions which arise.
\begin{quotation}
The Introduction arrived only just in time to be printed. I am particularly\end{quotation}
grateful to Dr. Uriel Tal for his penetrating comment and questions. It stands to reason that our views need not agree in every detail, but \textit{5} Christians should know that such questions as are raised in the Introduction are asked by many Jews. It is of the utmost importance for Jewish-Christian relations to discuss them as frankly as Dr. Tal did.
\textit{6}
I
GENERAL CONSIDERATIONS
1 PROBLEMS OF EVALUATION
Commentators on the attitude of the Churches in certain lands frequently contradict one another. Some Christians, such as Rev. Niemoeller [51] and Rev. Buskes [52] for instance, pass a severe verdict on the Churches and include themselves also. It seems to me that at least one Jewish commentator gives too positive a picture about the attitude of the population in his country, Greece.[53] He may, consciously or unconsciously, have tried not to embarrass the people amongst whom he still lived when he wrote his book. But also the opinion of a Christian that ``the hundreds of thousands of Jews that escaped the doom decreed for them owed their survival more to the rescue activities of individuals and private groups, above all the Churches, than to governmental resistance policy'' [54], seems to me too favourable.
It must be difficult for Jews who know of anti-Semitic actions perpetrated by Church leaders throughout the centuries, and who personally suffered and lost their relatives in the holocaust, to believe that not merely a few ``righteous of all Nations'' but also Churches publicly and unequivocally spoke out against Hitler's murderous anti-Semitism. On the other hand, Christians are in danger of trying to whitewash the Church and ignoring the many instances when the Church failed. We all tend to forget our failures and to remember our victories.
Some commentators tend to forget how the actual situation was in those days. Indeed, it is difficult even for people who themselves lived through it, to project themselves back into the time when Hitler seemed all-powerful. Moreover, we now have the benefit of living after the events, and thus we know many facts, which were not generally known in those days. \textit{9}
It seems unbelievable now, but in the summer of 1940, when some people somewhere in the Netherlands formed a resistance group, their leader stated that the British would not liberate us before Christmas 1940, and everybody present felt sorely disappointed. This kind of unwarranted optimism was fostered by many people throughout the war, and thus they underestimated the danger to the Jews and believed that, if German action against them could be delayed by some kind of compromise, much, and perhaps all, would be won. Many people in occupied Europe, in Great Britain and in the United States thought, that the information about the gas-chambers was ``atrocity propaganda''. The President of the Federal Council of Churches of Christ in the U.S.A. stated, on May 1, 1943: ``What is happening to the Jews on the Continent of Europe is so horrible that we are in danger of assuming that it is exaggerated'' [55]. We quote the following from ``Unity in Dispersion'':
``The undertaking was so staggering that, until the revelation about the Maidanek camp, a majority of the people in the United States as well as in England dismissed the facts of extermination as 'atrocity mongering'... It should be conceded, as extenuating circumstances, that never before in history had states descended to such depths of bad faith, deceit, and treachery as did Germany and some of her satellites in their resolve to murder. In 1942, tens of thousands of Polish Jews volunteered for cunningly disguised `resettlement' and agricultural work in the territories recently conquered by the Germans in the East, and thus entered of their own accord on a road at the end of which destruction awaited them.'' [56]
The Germans tried to deceive the victims about their aims as well as the people amongst whom these victims lived, and they succeeded in this to a considerable extent. [57] \textit{10} They had, in occupied Europe, all the instruments of mass communication, such as press and radio, at their disposal. All these and other factors are mentioned in ``Unity in Dispersion'' [58] in order to explain to some extent ``the failure of organized Jewry to halt or even to slow down the most terrible catastrophe in Jewish history''. Much of it is, mutatis mutandis, also applicable to ``organized Christianity''.
On the other hand, when the true facts became known, there was danger mentioned by the Archbishop of Canterbury: ``It is one of the most terrible consequences of war that the sensitiveness of people tends to become hardened... There is a great moral danger in the paralysis of feeling that is liable to be brought about.'' [59]
We now are in danger of forgetting that so many other problems burdened people in those days. The British people were fighting their life-and-death struggle against the Third Reich, but were free. In the occupied countries, many young people were sent to Germany for compulsory labour; food was rationed and became more and more scarce. People went out in the night to cut wood illegally as there was hardly any fuel.
One cannot understand what happened in occupied Europe without remembering these things; neither can one understand, without realising the power of human egoism and the will to survive. No one who has never really been hungry, nor has been deprived of his liberty, can understand what it meant in practice to ``love one's neighbour'' during the Second World War. The persecution of the Jews was not the only challenge confronting the Churches in those days, though we only now can perceive better that it was the most important one. The list of steps taken by the Churches in the Netherlands shows the type of problems which faced the Churches: intercession in church services for the Queen; arrest of pastors; suppression of the Church press; compulsory labour for youth; requisition of church bells; deportation of labourers to Germany; closing down of the Bible Society; ban on Church conferences; death sentences: plea for mercy; deportation of students, and national-socialist education in Christian schools. [60] \textit{11}
We tend now to underestimate the power of the Hitlerite terror. It has been said that all the Dutch should have blocked the railways with their own bodies, thus preventing the deportation of the Jews, because Hitler could not have murdered the entire Dutch population. I do not doubt that he could have and he would have done precisely that. [61]
It is not surprising then that many lay members of the Church and Church leaders were afraid, and therefore failed to fulfil their duties. Gerstein said, in Rolf Hochhuth's play: ``A Christian in these days cannot survive if he is truly Christian''. [62] Dr. Banning said: ``If the Church had fully exercised the obedience of faith, no pastor or priest would have come out alive. [63]
But the greatness of the risks matched the appalling need to help: the Germans committed genocide. Whenever the Church remained silent in view of the holocaust, it was guilty. ``Nevertheless a crime of such magnitude falls in no small measure to the responsibility of those witnesses who never cried out against it --- whatever the reason for their silence.'' [64] Therefore, all the considerations mentioned above cannot exempt Churches, Christians or non-Christians, though they can help us to be fairer in our judgment.
One is sometimes in danger of becoming irritated by people who did not stand the test themselves, and yet claim to know exactly what should have been said and done. There recently appeared a book [65] in which the author sharply criticizes much what was done, or was not done, during the German occupation of the Netherlands. \textit{12} He himself took a very active part in the struggle. Perhaps that is the reason why his criticism is not without compassion, and that it is to a large extent self-criticism. In order to understand how difficult it was to risk one's life or even freedom on behalf of others, one had to have been in it oneself.
I, who am now living in Israel, have sometimes, when lecturing on the subject, invited my audience to imagine for a moment that (God forbid!) some foreign power should occupy the land of Israel, say in the year 1980; and that this foreign power should deport many Jews for compulsory labour abroad, and also ration all food supplies, but that the Jewish part of the population should not risk their lives when complying with the demands of the enemy; that, however, the Christian minority in Israel should be deported and exterminated; that they should be deprived of their ration cards, that their identity cards should be stamped with a C, and that they must wear a yellow badge in the form of a cross, in order to distinguish them as Christians.
I then asked the question: ``would you be willing, in such a situation, to hide my wife, one of my children or me, who all look very ``Aryan'', though you knew that, as in every community, you were in danger of being betrayed and in even greater danger of being given away by careless talk of other people? Or would you, if you were the Chief Rabbi, be prepared to denounce the anti-Christian measures publicly and unequivocally?''
2 FACTORS LEADING TO PUBLIC PROTESTS
There were many factors that led Churches to protest publicly. One of them is mentioned by the Executive Council of the Federal Council of Churches in the U.S.A. in 1941:
``No true Christian Can be anti-Semitic in thought, word or deed without being untrue to his own Christian heritance.'' [66]
But how often true Christians were untrue... \textit{13}
The National Council of the Reformed Church in France made a similar statement, in September, 1942, declaring:
``A Christian Church would lose its soul and the reason for its existence, were it not to maintain... the Divine law above human contingencies.'' [67]
The Bible (the Old as well as the New Testament) was frequently cited in the protests. This may appear strange to people who only knew that the New Testament was used as a source of anti-Semitic influence. The same applies, by the way, to the Old Testament. [68] In my opinion, this use is quite indefensible. We list some of the texts cited in the protests:
``Open thy mouth for the dumb in the cause of all such as are appointed to destruction. Open thy mouth, judge righteously, and plead the cause of the poor and the needy. (Proverbs 31, 8-9).
Indirect reference, particularly in Switzerland and Germany, was made to Ezekiel 33, when the Church's office as Watchman is mentioned.
``When I bring the sword upon a land, if the people of the land take a man of their coasts, and set him for their watchman: if when he seeth the sword come upon the land, he blow the trumpet and warn the people; then whosoever heareth the sound of the trumpet, and taketh not warning; if the sword come, and take him away, his blood shall be upon his own head... But if the watchman see the sword come, and blow not the trumpet, and the people be not warned; if the sword come, and take any person from among them, he is taken away in his iniquity; but his blood will I require at the watchman's hand. So thou, O son of man, I have set thee a watchman unto the house of Israel...'' (Ezekiel 33, 2-4, 6-7). ``With what measure ye mete, it shall be measured to you again.'' (Matthew 7, 1). ``Verily I say unto you, Inasmuch as ye have done it unto one of the least of these my brethren, ye have done it unto me. (Matthew 25, 40). ``We ought to obey God rather than man.'' (Acts 5, 29). ``...and (God) hath made of one blood all the nations of men...'' (Acts 17, 26). ``There is neither Jew nor Greek...: for ye are all one in Christ Jesus. (Galatians 3, 28). \textit{14}
In addition to this, the parable of the good Samaritan (Luke 10, 30 --- 37) was quoted. It was frequently pointed out, though the wordings differ, that Jesus was born a Jew.
With regard to the Churches in the Netherlands, it has been stated that ``the moral implications of Christian doctrine motivated the resistance of the Churches''. [69] Such a statement seems to me to oversimplify matters. I believe that the Christian doctrine (or rather: the teachings of the Bible) demanded the resistance of the Churches, but it is always possible to find convenient excuses to escape a challenge, as for example the opinion that the Church should not interfere in political matters.
I once tried to convince a devout Protestant (he was an elder of the Church) that he should hide a Jewish child, by reminding him that one day he would have to give account of his deeds to the Supreme Judge. The man, who certainly could have hidden that child (he had a large farm) flatly refused, not because he denied that he would have to give account of his deeds, but because he was afraid, --- too afraid to hide the child. I pointed out to him that he should rather fear God and not man, but my words simply had no effect.
Christian teaching did not work in this case, though that does not mean that it did not work in other cases. Chief Rabbi Safran spoke to the Rumanian Patriarch Nicodemus of the terrible responsibility he was taking upon his conscience in the eyes of the Supreme Judge [70], and in this case it worked, though there were probably other motivations as well.
Everybody's decisions are also motivated by the principles to which he adheres, and thus a Christian's decisions are influenced by Christian principles, though it must be admitted that mostly there are many other influences and motivations, probably more than the person who makes a decision, realizes.
\textit{15}
The whole matter of the attitude of the Churches during the war was once discussed at a conference, and one of the speakers began by expressing as his opinion that Hitler and Eichmann were Christians, but later on he said that Mr. Johannes Bogaard, one of the ``righteous of all Nations'' who saved many Jews and whose father, brother and son were murdered by the Germans, was ``just a courageous Dutchman''.
I happen to know Mr. Bogaard very well and I am convinced that he acted as he did during the war, primarily because he is a committed Christian. Of course this does not alter the fact that many Christians did not do very much, if anything, on behalf of their neighbours, the Jews; nor should it be denied that many non-Christians did do what they could, out of national, socialist, humanist or communist convictions.
The same applies to the attitudes of a community. A member of a left wing kibbutz stated his views very clearly to me, and I know that many people hold views similar to his:
``Allow me to express my position which is based on dialectical materialism. The Protestant Churches were active everywhere according to the local circumstances, first of all according to the nature of the people amongst whom they lived. The Churches did not act in a vacuum. For instance, in the countries of Western-Europe, such as Holland, Norway and Denmark, where the `final solution' met with the resistance of all sections of the population, the courageous stand of those nations found its vehement expression in the attitude of the different Churches. The non-Roman Catholic Churches merely reflected the opinion and reactions of the people.''
It seems to me that there is more than a grain of truth in such a view and certainly no Church ever acted in a vacuum. Much in the protests issued by Churches in countries such as Bulgaria and Greece, points to nationalist rather than to spiritual-Christian considerations. Reading and analysing the contents of the statements may be of some help when assessing the motivations of Christians and groups of Christians who resisted the persecution of Jews. \textit{16}
If, however, one indeed believes that everything can be explained by the influences of local circumstances \textit{etc}., one should be consistent and stop holding Churches responsible for acts of anti-Semitism committed by Churches or by people professing to be Christians throughout the ages, for in such a case they were also ``merely reflecting the opinion and reactions of the people amongst whom they lived''. In the case of such a rigid determinism, it seems difficult to hold anyone anywhere responsible for his acts and decisions.
In my opinion we are all influenced by the people amongst whom we live, by social circumstances and by many other factors. We are all subject to a kind of mimicry, but that does not necessarily mean that we are just chameleons and nothing else. Churches are certainly influenced, just like any other group of people, by circumstances and surroundings, but they on their part influence these circumstances and surroundings. There is interplay of factors.
Similar to the opinion mentioned above is the viewpoint that Churches always tend to support the Establishment. The United States and Great Britain were at war with Germany, and the Churches participated in the crusade against the enemy. The same applies to Churches in occupied Europe, even when their own Government was in exile. I think that the Old Testament already gives us many examples of organized religion supporting the Establishment, but it also gives us some instances when religious leaders (the prophets!) refused to do so. [71]
It is doubtful whether the British Government was pleased with the Church's protest against the pogroms of the ``Crystal Night'', just after the Munich agreement. [72] The Archbishop of Canterbury's speech in the House of Lords and the Bishop of Chichester's letters to The Times, in 1943, must have embarrassed political leaders who were of the opinion that the main object was to win the war, and that attempts to rescue Jews were of less importance. [73]
The Swiss Churches could hardly be accused of supporting the Establishment, when they protested against the decision of the Swiss Government to return refugees to Nazi Germany who had illegally entered Switzerland. [74] Similar examples can be given regarding the United States, Sweden and other lands. The little that was said by the ``Confessing Church'' in Germany on behalf of the Jews was certainly not in support of the Establishment. \textit{17} A Church must try to be the conscience of nation and Government, even though this may mean that its leaders have to speak out against the seeming interests of their nation. Churches frequently failed to do so, but we should refrain from generalizing.
Whenever Churches were conscious of belonging to a worldwide fellowship, this contributed to their making a stand against anti-Semitism.
Church leaders in the Netherlands followed the struggle of the ``Confessing Church'' in Germany, and were on the alert when they were challenged themselves. The Church in Sweden was moved to protest by the statement issued by the Church of Norway. Church leaders in Hungary realized, when they did not carry their protest before the Hungarian public, that this course would ``incur... the reproach and accusation of the leading bodies of the Christian Churches'' and stated that, if their intervention proved ineffective, they would be obliged ``to testify before the congregations of our Church and the Protestants of the world that we did not suppress the message of God''. [75] Many of the Church leaders who took a clear stand, knew one another personally. [76] In view of the attempts of the Germans to deceive world opinion as to their ultimate aims concerning the Jews, and in view of the tendency to dismiss reports about what was going on as ``atrocity propaganda'', the importance of the information given by the World Council of Churches through its Press Service and by other means can hardly be overestimated. The need to combine efforts and thus break through denominational barriers in order to come to a joint stand, was understood in some countries. In the Netherlands, Protestants and Roman Catholics began a new chapter in their relationship by protesting together. In France and Hungary there was consultation between Roman Catholics and Protestants, but it is to be regretted that they did not achieve a common front. \textit{18}
Sometimes there existed close contact between Christian and Jewish leaders, as for example in the United States, in Great Britain, in Bulgaria and between the leaders of the World Council of Churches and the World Jewish Congress, in Geneva. Thus, again, information about what was going on was communicated and action could be co-ordinated.
The negative implication is also clear: whenever a spirit of particularism, provincialism and isolationism was strong in a Church, it did not fulfil its duty toward the persecuted Jews.
3 RESULTS
In order to ascertain the practical effects that could be expected from steps taken by the Churches, the political and geographical position of the countries concerned, as well as the time factor, must be born in mind. Where there was a national Government, as was the case in Slovakia, Hungary, Rumania and Bulgaria, protests had a better chance of some success than in countries under direct Nazi control. Yet even then what Jeno Levai stated about Hungary was sometimes true:
``The Church was not in a position to promise or to threaten. Thus, in spite of their very best intentions, they could obtain only very little. Naturally this little meant life to the persons concerned.'' [77]
Typical were the differences between the Scandinavian countries: Sweden was neutral; Finland was an ally of the Germans; Denmark was occupied but it had its own King and was officially not even in a state of war with Germany; Norway's King had fled and the infamous Quisling had become Prime Minister. \textit{19}
Geography also played an important role. The Jews in the Netherlands were in a deadly trap; Hungary was, at least for some time, a place of refuge for Jews in the neighbouring countries; Jews in Denmark and Norway had a chance to flee to Sweden and the Jews in France and Italy to Switzerland, in so far as that country was willing to admit them. The time persecution began was a vital factor. The earlier it started, the smaller the chance of saving at least some lives. It should be noted that these three factors were utterly unfavourable in Germany. It is difficult to assess the range of influence of any Church. Figures have been given about membership in Appendix II, but one must remember that many Churches have a high percentage of nominal members who, perhaps since their baptism, never attended a church service. Therefore it can be misleading to read that there were forty-five million Protestants in Germany, or, that 96,2 per cent of the population of Norway are members of the State Church. Only 5 per cent of the members of the Norwegian Church regularly attend Sunday services. In many other countries the situation is similar.
Many people who were not church goers may never even have known about the protests of the Church, and this is especially true of occupied Europe in those days, for there the Church could only speak from the pulpits, not through press and radio. Moreover, many nominal Christians are influenced by other outlooks on life, rather than by the Christian faith. However, when press and radio were silenced and the Church alone could voice an open and public protest, it met with the response of many people who were outside the fold. Church services were better attended than in times of peace. The former editor-in-chief of the Dutch communist daily De Waarheid relates that he went to a church service in those days:
That church meant something to us in those black days, were it only to listen to the prayer of a man, who dared make a public address on behalf of the people tortured in the concentration camps. [78]
\textit{20}
I myself belong to the persons who, in those days, found their way back to the fold, attracted as we were by the Church's spiritual resistance to the Nazis.
When attempting to assess the practical results of steps taken by Churches or Church leaders on behalf of the Jews, we distinguish between countries under German occupation, countries under a satellite government, neutral countries, and countries that were at war with Germany. In countries under German occupation, efforts made by the Churches had hardly any direct practical result for the Jews in general. Personal intervention did not help or, at best, could only cause some delay in the deportations. The only step that had some effect on the Germans (as we now know!) was the issuing of a public protest.
Again it was evident, that the German authorities did not fear or have any step taken by the Churches as much as their protests which were read from the pulpits. Letters of protest they could throw in the dustbin or file away. They could listen to oral protests without taking them to heart. But they tried in every way to prevent public protests (in those days the only form of public protest), fearing their effect upon the people."[79]
The most effective protests were those, which clearly encouraged the faithful to help the Jews. Others called for non-cooperation with the Germans, and this had at least some result.
Six Roman Catholic police-agents at Utrecht informed their chief on February 24, 1943, that on the grounds of a pastoral letter read in their church on February 21, they would have to refuse if ordered to arrest Jews. Their chief threatened to dismiss them without pension and said that ``those who do not announce their intended refusal and yet have the impudence to carry it out will be considered saboteurs, with all the serious consequences. The Germans immediately tried to arrest these agents but they had gone into hiding. The Germans then arrested their wives and children.'' [80]
\textit{21}
Generally speaking, the positive indirect effect of public protests was, that it counteracted the attempts of the Germans to separate and isolate the Jews from the non-Jewish population, in order to break their will to resist deportation and annihilation. [81] It is impossible to count the lives saved through the activities of the Churches in the occupied territories. I agree with the opinion of Dr. Visser 't Hooft:
``So far we have only spoken of public protests. But were these protests implemented by deeds? The answer is that they were, though by no means as generally as ought to have been the case. The full story of Christian assistance to the Jews in their hour of great need will never be fully told, for in many cases individuals acted quietly and behind the scenes.'' [82]
In the countries under a satellite government, actions undertaken by the Churches were of some and sometimes even of much avail. [83] Concerning the neutral countries, the steps and protests of the Churches in Switzerland contributed to the relaxation of measures against the refugees [84], and in Sweden the Lutheran Archbishop encouraged his government to broadcast its willingness to take in the Jews of Denmark. [85] It is difficult to assess how far the protests of the Churches in countries that were at war with Germany had a practical effect. [86] They apparently helped to combat anti-Semitic influences in these countries (the same applies to protests issued in the countries mentioned above) and they contributed towards ``breaking the wall of silence.''
\textit{22}
``The world wide public, overburdened with the issues and the incidents of a world conflict fraught with the gravest consequences, was not receptive to reports which it was ready to dismiss as propaganda tales; besides, the facts were hidden from it, not withstanding persistent endeavours by the (World Jewish) Congress to keep it informed. A wall of secrecy concealed the terrible tragedy... The main difficulty was how to convince public opinion and induce the Allied Governments to act. The battles of World War II raged fiercely on three continents, the onslaught of barbarity was nowhere decisively checked, the democratic nations feverishly tried to overcome their unprepared ness for a conflict of such dimensions. The Governments in Exile were chiefly concerned with the sufferings of their nations as a whole.'' [87]
The pressure exerted by Jewish and Christian leaders on their Governments did not, however, result in effective rescue activities being undertaken by these Governments.
It has been suggested that the protests from the Churches mostly came too late, and thus fell flat. This is partly true. The Protestant leaders in Hungary did speak out very late, and Bishop Wurm of Wurttemberg sent his letters when there only remained a chance of doing something for the ``privileged'' Jews.
On the other hand, Churches or Church leaders in Belgium, France, Switzerland, Sweden, Great Britain and the United States began to protest in 1933. The Churches in the Netherlands protested at the very beginning of the German attacks on the Dutch Jews, in 1940. The Church of Denmark had prepared a public protest before the deportations started.
It is, however, necessary to keep the dates of protests in mind, in order to arrive at a fair evaluation of the moral courage which such protests required. After Hitler's defeat at Stalingrad, at the end of 1942, and the defeat of Rommel at El-Alamein, it became more and more clear that Germany would lose the war. The measure of success is in itself no yardstick for the moral value of a deed. One can hardly say that Church leaders in Rumania behaved better than Church leaders in, for instance, the Netherlands, because the former, contrary to the latter, actually succeeded in saving many lives. \textit{23}
To this it must be added, however, that the seeming absence of any chance of success could not be an excuse for maintaining silence or for doing nothing against the terror of the Nazis. Prince William the Silent is said to have stated that it is not necessary to hope in order to try, nor to succeed in order to persevere.
4 HELP TO CHRISTIANS OF JEWISH ORIGIN
Apart from the 500,000 Jews who registered as members of their community in 1933, there were some 50,000 Jews in Germany who no longer belonged to the Jewish community. Though born as Jews, they had been baptized. In addition, some 210,000 people had at least one Jewish parent, and another 80,000 one Jewish grandparent; thus a total of some 340,000 people in Germany were, in addition to the ``full Jews'', affected by racial legislation. [88]
Until the end of the year 1938, Christian leaders and Churches tended to stress the necessity of helping Christian refugees of Jewish origin, rather than calling for help for Jews in general. A notable exception to this rule was the Appeal of the Ecumenical Council for Life and Work, in 1933, to help ``Jews, Christians of Jewish origin and political refugees''. [89] During the war, Churches in countries such as Bulgaria, Hungary and the Netherlands, instituted steps to protect their members of Jewish origin. It can hardly be denied that it was the right as well as the duty of the Churches to do so, but more than once the Churches were tempted to try and save their own members while neglecting the Jews in general. The announcement read from the pulpits of the Hungarian Protestant churches, on July 16, 1944, is significant:
``The Bishops... wish to inform the congregations that in connection with the Jewish question, and particularly in the case of baptised Jews, they have repeatedly intervened with the competent Government authorities...'' [90]
\textit{24}
A comment on the ``Confessing Church'' in Germany is:
``The Church took up the cudgels for the baptized Jews and that meant to the average churchgoer that the unbaptized Jew, \textit{i.e}. the Jew as such, was left to the devil.'' [91]
Church leaders in the Netherlands regarded the issue as a temptation:
``Great dangers and temptations threatened continually. From the German side came the voice of the tempter: `do not protest; only negotiate'. 'Do not speak on behalf of the Jews any more; then we shall be lenient to the Christians of Jewish origin.'... It is a great miracle that, in general, the Church recognized these voices as coming from the tempter, and boldly rejected the temptation.'' [92]
That these questions were very difficult indeed, becomes clear from the following comment of Herzberg:
``The baptized Jews [in the Netherlands], who were able to save their lives, owed this exclusively to the resistance of the Churches, a resistance which was especially impressive because of the principles by which it was motivated.'' [93]
Quite different, however, is the sharp verdict of Presser:
``And the Churches (in the Netherlands)? With what hesitation did they begin their resistance? How many were there, unfortunately, who were resigned to the fatal decrees of the occupying power, even appealing to texts in the Bible, and actually helping to carry out the decrees. How many times did they stand up only on behalf of baptized Jews and not on behalf of others.'' [94] \textit{25}
It makes a difference, whether Churches on their own initiative stressed the importance they attached to the fate of Christians of Jewish origin, or were forced into a compromise by the tactics of the Germans. The latter was the case when the DUTCH REFORMED CHURCH failed to read the telegram of protest publicly in the church services. [95] The Protestant Christians of Jewish origin in the Netherlands indeed survived. We should not pass judgment lightly and we must realize that we now have the benefit of being after the events, Church leaders in those days were not always aware of the fact that the Germans, who offered not to deport the Christians of Jewish origin, were not just making a concession, but were also providing themselves with a means of exerting further pressure on the Churches. It is most regrettable that on several occasions certain Churches interpreted the saying ``charity begins at home'' as they did.
5 ``MERCY-BAPTISMS''
Christian clergymen in many lands were prepared to baptize Jews if the ceremony of baptism meant that lives could be saved. The following is related of the Lutheran Church in Slovakia:
``Many Jews who tried to escape persecution sought rescue by giving up their religion and by requesting to be received into the Evangelical Church, for the Catholic Church did not receive them. [96] The Evangelical Church did not refuse them, which was an act of courage in those days, but enabled them to become members of the Church... Here some examples follow: \textit{26} In 1940, 20 persons, most of them adults, became Christians in Bratislava. For the year 1941 the number was 83; for the first half of the year 1942: 47 persons; for the second half: 7.828 persons were admitted in 1943; only 2 in 1944. In Horne Zelenice (near Hlohovec), 169 persons became Christians in 1942; 39 in 1943; in 1945 only one. In Frencin 120 persons; in Kochanovce (near Treucin) 45; in Banska Bystrica, 202 persons became Christians in 1942. This help aroused the anger of the rulers, of the Gestapo and of the Hlinka Guard. They began to arrest Evangelical Christians and pastors. 9 pastors were sent to the concentration camps in Germany. Joseph Bucko, minister at Martine, perished in the camp.'' [97]
It is reported that in Bulgaria,
``... Ministers of various Christian denominations engaged in mass 'mercy baptisms'; several of them were removed from office because of this (one of these ministers, with a community of about 200 souls, managed to baptize 200 additional persons between January 1 and September 1, 1940). High dignitaries of the Bulgarian Orthodox Church declared that 'conversion to Christianity' and `formal baptism' were two different acts, the first of which necessarily preceded the second, sometimes by a considerable period; because the law spoke of conversion and not of baptism having to have taken place before September 1, 1940, Jews baptized later could also be saved if the minister declared that they had expressed their will to adopt Christianity before that date. Many courts accepted this reasoning. In this way, a number of baptized Jews and offspring of mixed marriages escaped the provision of the law.'' [98]
The following is quoted from the testimony of Richard Simantov: \textit{27}
``... It must be admitted that, with a few exceptions, all the Christian religious institutions [in Bulgaria], as also their clergy, behaved with sympathy towards the Jewish victims of the anti-Jewish legislation. When issuing the required legal documents to the Christian Jew, the clerk of the court or the judge himself interrogated the priest, whether he had indeed carried out all the religious formalities, and how long the teaching of the catechism had lasted for the person of Jewish origin concerned. The priest would always reply in the affirmative and would declare that the man had received instruction for 3, 4 or 5 months, and that he regularly attended church services \textit{etc}., although often these documents, which were issued by the Church, were given only in exchange for a payment, without the ceremony having been performed...'' [99]
We have the following particulars about Greece :
``Many tried to evade the racial laws through baptism. More than 500 Jews embraced the Orthodox religion; some scores preferred to become Catholics. it was clear that it was not out of conviction that these Jews entered into the Church. It was well-known, that only the desire to escape persecution moved them to seek refuge in the shadow of the cross. Out of compassion, the priests did not hesitate to accept the new converts. They were on friendly terms with them in different ways. Out of noble feelings and not in order to receive a reward, the priests also distributed baptismal certificates to Jews who had never attended a church service...'' [100]
The biographer of the Archbishop of Athens, Damaskinos, relates:
``Later on, when the persecutions started affecting the Jews of Athens, the Archbishop decided on the following measures. He summoned the Director General of the Administrative Services of the Community of Athens, Mr. P. Haldezos, and said to him: ``I have made the sign of the cross and have spoken to God, and have decided to save as many Jews as I can, even though I run a great risk. I am going to baptize them, and you must give certificates enabling them to obtain the identity cards of Christian Greeks. Mr. Haldezos agreed to this. With the help of a Municipal official, they opened a register wherein they registered 560 Jews as Christians, all of whom were saved. There was no treachery.'' [101]\textit{28}
Rev. J.J. Buskes discussed the considerations, which led clergymen in the Netherlands to provide Jews with false certificates of Baptism:
``We are well aware that many pastors had conscientious objections to giving forged baptismal certificates. But, thank God, there were other ministers who had conscientious objections about not doing so. Such a certificate was, of course, false. But the man who wrote it out and gave it to a Jew, did service to the truth and helped his neighbour. The one, however, who would not write it and thus refused help to a Jew, served falsehood and failed the Jew. There is a truth which is like a lie and there is a lie which is like the truth. God commanded us to lie in the service of the truth. Not the end, but the obedience to God's commandment (to love our neighbour as ourselves) justified the means. Thus the humble and scrupulous Dr. Oorthuis wrote in a pamphlet of the underground movement: even forged passports can be safe-conducts from the Lord, and stolen ration cards be gifts of mercy from God, which we accept with Thanksgiving.'' [102]
Many people may feel horrified when reading the views of Rev. Buskes. The same author stated in another publication:
``If I can save a man whose life is threatened by a scoundrel by saying to that scoundrel that two and two make five, I shall say so to him, in obedience to the ninth commandment. In such a case I am even prepared to declare that two and two make ten.'' [103]
A personal friend of mine, who is a devout Christian, took the oath declaring that a child in his house was not Jewish but his own child born out of wedlock. He saved the child. People who are horrified at such behaviour, probably never lived under German occupation. At any rate, they should remember St. Paul's saying: ``Owe no man any thing, but to love one another: for he that loveth another hath fulfilled the law''. [104]
In my opinion, it was morally permissible and even laudable to baptize Jews in those days in order to save their lives, as long as it was mutually understood that this was in order to deceive the persecutors and that the baptism in fact was invalid. \textit{29}
II
BEFORE THE WAR
6 HISTORICAL EVENTS
a. Hitler's Rise to Power --- the Nuremberg Laws. (Jun., 1933-Sept., 1935)
President Hindenburg entrusted Hitler with the Chancellorship on January 30, 1933. The Reichstag fire, on February 27, was followed by a wave of arrests. The ``Ordinance for the Protection of the People and the State'', issued on February 28, suspended the sections of the Constitution which guaranteed individual and civil rights. The ``Enabling Act'' (March 23) stripped Parliament of its power and handed it over to the Reich Cabinet. Laws enacted by the Cabinet were to be drafted by the Chancellor (Hitler) and might deviate from the Constitution.
On April 1, Jewish shops throughout Germany were boycotted. Jewish civil servants were dismissed on April 7. On the same day the exclusion of ``non-Aryan'' lawyers was ordered. According to a decree of April 22, no Jewish physicians were allowed to work for sick funds anymore. At the end of April another decree restricted the admission of Jewish children and students to schools and universities. In the following months Jews were excluded from working in the fields of art, music, literature and journalism. The ``Law on revocation of naturalizations and deprivation of German citizenship'' (July, 14) robbed Jews, who had been naturalized before or had been born outside Germany, from their citizenship. In January, 1934, it was decreed that Jews could no longer be members of the Labour Front. When President Hindenburg died, on August 2, 1934, Hitler became President and Supreme Commander of the Army. On May 21, 1935, it was decreed that only ``Aryans'' could serve in the army.
It is estimated that 37,000 Jews emigrated from Germany in 1933; in 1934, the number was 23,000, whilst 21,000 Jews left Germany in 1935. [105] \textit{33}
b. The Nuremberg Laws --- Crystal Night. (Sept., 1935-Nov., 1938)
On September 15, 1935, two fundamental laws were adopted by the Reichstag meeting at Nuremberg. One, the ``Law Respecting Reich Citizenship'', decreed that only a national of German or kindred blood, who proved by his conduct that he was willing and likely to serve the German people and Reich faithfully, could be a citizen. The second, the ``Law for the Protection of German Blood and Honour'', specifically referred to the Jews and singled them out as undesirable aliens, impure of blood and dangerous to the honour and security of the German people. There followed seven paragraphs, the first of which dealt with the prohibition of marriages between Jews and nationals of German or kindred blood. Paragraph three prohibited Jews to employ in domestic service female nationals of German or kindred blood, under the age of forty-five years.
On April 26, 1938, it was decreed that all Jewish assets in excess of 5,000 marks should be registered. On June 15, 1938, about 1,500 Jews were arrested and deported to a concentration camp. On July 25, it was decreed that Jewish physicians were no longer permitted to treat non-Jewish patients. In the same month, Jews had to apply for special identity cards. On August 17, 1938, the first name ``Israel'' for Jewish men and ``Sara'' for Jewish women was made compulsory in addition to their own names. In October, all passports of Jews were stamped with the letter J. Austria had been incorporated into the Third Reich, on March 13, 1938. The German anti-Jewish laws were also enforced in Austria, where about 180,000 Jews were living. It is estimated that 25,000 Jews emigrated from Germany in 1936, and 23,000 in 1937. [106]
In March, 1938, President Roosevelt invited thirty-three governments to join in a co-operative effort to aid the emigration of refugees from Germany and Austria. On July 6, 1938, the Intergovernmental Conference met at Evian, France. Nearly all the delegates expressed their sympathy for the refugees but were very careful not to assume any obligations on behalf of their Governments. \textit{34}
It is important to keep some of the major political events of those days in mind. Italy attacked Ethiopia on October 3, 1935. In May, 1936, the Ethiopian emperor went into exile into Great Britain. On March 2, 1936, the Rhineland was remilitarized. On November 25, 1936, the anti-Comintern Pact with Japan was signed. On July 16, 1936, civil war in Spain broke out. On September 30, 1938, the Munich agreement was signed by Hitler, Chamberlain, Mussolini and Daladier. As a result, Sudetenland was occupied by Germany. Poland and Hungary also occupied part of Czechoslovakia.
c. Crystal Night --- the Outbreak of the War. (Nov., 1938-Summer, 1939)
On October 28, 1938, 15,000-17,000 Jews of Polish origin were rounded up and expelled. On November 7, a seventeen-year-old Jewish boy, Herschel Grynspan, whose parents had been among the people expelled to Poland, shot Ernst vom Rath, a minor Nazi official in the Paris Embassy. He died two days later. This was the pretext for unleashing a pogrom that has entered history under the name Crystal Night: 7,500 Jewish shops were looted and windows of shops and houses were smashed; many synagogues were burned; more than 26,000 Jews were arrested, many of whom were sent to concentration camps; at least 91 were killed. [107]
On November 12, the Jews in Germany were ordered to pay a collective fine of thousand million Reichsmark. On November 15, Jewish children were dismissed from German schools. Jews were prohibited from visiting theatres, cinemas, concert halls, museums and public baths. On December 8, a decree was issued expelling Jews from the universities.
At the beginning of January, 1939, the ``Aryanisation'' of Jewish enterprises began. Since January 17, 1939, Jews were forbidden to be employed in the professions of dentist, pharmacist and veterinary surgeon. On January 30, 1939, Hitler publicly declared that the Jewish race in Europe would be annihilated if war broke out. \textit{35} Hitler annexed Czechia, on March 15, 1939. Slovakia became ``independent''. In April, 1939, Mussolini occupied Albania.
From the beginning of 1938 until October 1, 1941 (when further emigration was forbidden) an estimated 170,000 Jews left Germany. [108] Jews in Germany at the beginning of the Hitler regime, numbered 499,682. The 1939 census, registered within the borders of the pre-Hitler Reich, amounted to no more than 213,930. [109]
7 GERMANY
The vast majority of the Protestants of Germany belonged to one of the 28 Landeskirchen (Lutheran, Reformed or Uniate), of which the largest was the Church of the Old Prussian Union, with 18 million members. The Landeskirchen were independent members of the German Evangelical Church Union, founded in 1922. In all, there were forty-five million Germans who were, nominally at least, members of the Protestant Church. In 1932, members of the Church who supported Hitler had founded the ``German Christians' Faith Movement''. These ``GERMAN CHRISTIANS'' demanded the creation of one Protestant Church, the application of the Fuehrer principle in Church affairs, the introduction of racialism within the Church, the ``Germanization'' of Christianity (the ``Aryan Jesus"!) and the elimination of ``Jewish influence'' from teaching, liturgy and preaching. In 1933, some 3000 pastors belonged to this group. Church elections took place on July 23, 1933. On the eve of the elections, Hitler made an unexpected radio appeal asking the electorate to vote ``GERMAN CHRISTIANS''. They won a decisive victory.
On September 21, 1933, Rev. Martin Niemoeller and others created the ``Pastors' Emergency League'', which opposed the ``GERMAN CHRISTIANS''. In the beginning, Niemoeller's group was definitely in the minority. By December, 1933, its membership had grown to 6,000. Between the two groups, a majority tried to remain neutral while more or less sympathizing with the group of Niemoeller, but in practice obeying Hitler's orders without open protest.
\textit{36}
After a protege of Hitler, Ludwig Mueller, had been elected as Reich Bishop under pressure of the Government, Niemoeller's opposition group constituted the ``CONFESSING CHURCH'' which declared itself to be the legitimate Protestant Church of Germany and set up a provisional Church government. [110] The ``GERMAN CHRISTIANS'' had, in the meantime, gained control in several Landeskirchen, sometimes with the active help of the national-socialist party. In April, 1933, the Landeskirche of Thuringia required of its clergy a formal oath of allegiance to Hitler; the ``Thuringian Christians'' wanted to give this symbol of unconditional obedience to Hitler as a birthday present. There was a division in other Landeskirchen, as for instance in the largest: the Church of the Old-Prussian Union. In the summer of 1933, a law had been issued forbidding the appointment of pastors or Church officers of ``non-Aryan descent'' and ordering the dismissal of such pastors and Church officers. [111] In its session on Sept. 5, 1933, the Synod of the Old Prussian Union accepted this law; the opposition party protested and, when this was of no avail, left the meeting. Later on the opposition organized the ``CONFESSING Synod of the Evangelical Church of the Old-Prussian Union''. \textit{37}
It is not, as has been stated in the Preface, my intention to record the contents of statements issued by Churches or Church leaders on behalf of Christians of Jewish origin. It is of importance, however, to know to what extent the ``GERMAN CHRISTIANS'' supported discrimination against these members of the Church, and, also, to know that the CONFESSING CHURCH defended them. Thus I mention the more important statements, which were issued, without recording their full contents. [112] There was sharp controversy and much discussion as to whether the anti-Jewish laws should be applied within the Church. The following persons and institutions protested against such a measure: the Theological Faculty of the University of Marburg (Sept. 19, 1933); the Theological Faculty of the University of Erlangen (Sept. 25, 1933); Rev. Martin Niemoeller (Nov. 2, 1933), and Prof. Rudolf Bultmann (Dec., 1933). On the other hand, the ``GERMAN CHRISTIANS'' declared at the beginning of April, 1933, that only those who were ``of pure German blood'' should be admitted to the ministry. On May 26, 1932, they had already decided to consider Missionary work amongst the Jews as a great danger ``as it is the entrance gate for foreign blood into our national body''. The example of the Synod of the Old-Prussian Union (see above) was followed by other Landeskirchen, as for instance in Saxony, Thuringia and Braunschweig: ministers of Jewish origin were to be dismissed. The Church in Saxony even voted, on Dec. 10, 1933, to accept the principles of blood and race, and that only those who according to the laws of the State were compatriots should be members of the national Church! \textit{38}
The decision of the Church of Saxony was publicly rejected by the Theological Faculty of the University of Leipzig, and by the Pastors' Society of the Rhine. The majority of the Theological Faculty of the University of Berlin, however, supported the racialism of the Saxonians. This all happened in the years 1933-1934. In those days, it certainly needed courage to stand up publicly for the rights of Christians of Jewish origin in the Church. It should be noted, however, that the publications mentioned above did not publicly oppose discrimination against the Jews in general, nor even discrimination against Christians of Jewish origin outside the Church.
In March, 1935, the CONFESSING Synod of the Evangelical Church of the Old-Prussian Union sent a ``Word to the Congregations'', which was read from the pulpits. We quote the following:
``We believe that our nation is threatened by a mortal danger. This danger lies in a new religion.... in it, racial and nationalistic ideology becomes supreme. Blood and race, nationality, honour and freedom become its idols. ... Whoever substitutes blood, race and nationality as the creator and source of authority instead of God, undermines the state.'' [113]
The Government struck back with arrests. 500 pastors were imprisoned.
* * *
After the notorious Laws of Nuremberg had been promulgated, only individuals in the CONFESSING Church pleaded for the issue of a public declaration. Dietrich Bonhoeffer said : ``Only the man who loudly cries out on behalf of the Jews, is at liberty to sing the Gregorian chants''. [114]
The ``Council of Brethren'' of the CONFESSING Church stated, in a declaration in defense of the right to baptize Jews, in September, 1935: ``We only say the necessary minimum (alas, perhaps even not the minimum) concerning things about which we are not allowed to keep silent...'' [115] \textit{39}
The Provisional Church Council of the CONFESSING Church sent a Memorandum to Hitler, in May, 1936. We quote the following from it:
``... When blood, race, nationality and honour are thus raised to the rank of qualities that guarantee eternity, the Evangelical Christian is bound by the first commandment to reject that assumption. When the Aryan human being is glorified, God's word bears witness to the sinfulness of all men. When in the framework of the National-Socialist ideology, anti-Semitism is forced on the Christian obliging him to hate the Jews, he has nonetheless the divine command to love his neighbour...'' [116]
The Memorandum, which was published in the foreign press without the consent of the CONFESSING Church, resulted in the arrest of Dr. Weissler who worked in the office of the Provisional Church Council. He perished in a concentration camp. [117]
On June 23, 1937, several members of the Reich Brethren Council were arrested, and on July 1, 1937, Rev. Martin Niemoeller also. He remained a prisoner until the end of the war. The office of the Provisional Church Council was closed by the authorities, and thus the CONFESSING Church was to a large extent forced into underground resistance.
* * *
No public protest was voiced after the Crystal Night pogroms. In September, 1938, an office for helping persecuted Jews, but mainly Christians of Jewish origin, was opened under the direction of Rev. Grueber. Rev. Grueber also contacted the Jewish and Catholic relief-organizations. Repeated journeys to Switzerland, the Netherlands and Great Britain were made to find places for Jewish refugees. \textit{40}
At the end of 1940, Rev. Grueber was arrested and the office in Berlin was closed. The branches in Heidelberg, under Rev. Maas; in Breslau, under Vikarin Staritz; and in Kassel, continued to function, though under the pressure of fierce hostility. On the initiative of Rev. Werner Sylten, Grueber's deputy, an attempt was made to continue the work of the Berlin office on a smaller scale. Conversations with the Evangelical Church Council of Berlin took place; negotiations with the Gestapo were held. This eventually led to the arrest of Rev. Sylten, who perished in the concentration camp of Dachau, at the end of 1942. Only a few of the 35 members of Grueber's office, most of them of Jewish origin, lived to see the end of the war. Most of them died in the gas chambers. [118]
In Dec., 1938, the Kirchentag of the CONFESSING Church stated:
``... We again face the fact that many servants of the Church are being hampered in the execution of their ministry and are being expelled from their offices. In the hour of threatening war some fulfilled the duty of the Church, doing penance for the whole nation and beseeching forgiveness and deliverance from God's judgment. Thereupon, they were charged with high-treason. in view of what happened to the Jews others earnestly preached the Ten Commandments and were persecuted for it...'' [119]
The Thuringian Church, followed by Mecklenburg, Anhalt and Sachsen (all directed by ``GERMAN CHRISTIANS'') promulgated (February, 1939) a law which eliminated Jews from membership in their Churches. \textit{41}
In April, 1939, the infamous declaration of Godesberg was published. It accepted National-Socialism and stated that ``the Christian faith is in irreconcilable opposition to Judaism''. The declaration was accepted by the leaders of 11 Landeskirchen in which the ``GERMAN CHRISTIANS'' were the ruling party. The CONFESSING Reich Brethren Council sharply opposed the Godesberg declaration in a statement issued on April 13, 1939. One day later, it also opposed the law of the Thuringian Church (see above) which denied permission to Christians of Jewish origin to be members of the Church. The Reich Brethren Council stated:
``... The men responsible for these laws thereby show themselves to be enemies of the Cross of Christ. They cannot exclude anybody from the Church of Christ. They have, however, separated themselves from the holy Christian Church, by the promulgation of these laws...'' [120]
The fundamental difference between ``GERMAN CHRISTIANS'' and the CONFESSING CHURCH is obvious: the former completely identified themselves with national-socialist racialism, the latter repudiated it verbally but showed weakness of action. One feared that, by an all-out intervention on behalf of all non-Aryans, the theological protest against the separation of Christian non-Aryans from the community of the Church would be politically misinterpreted, and that thus the intervention on behalf of them would become even more difficult. [121]
That the CONFESSING Church hardly spoke out at all was not the worst fact; it seems infinitely worse that the so-called ``GERMAN CHRISTIANS'' supported Hitler and his racialism. One may agree with the words of the German Lutheran pastors in England: ``It is not for us who now live in safety to criticise those who, under fire, have done their utmost not to bow to Baal''. The fact remains, however, that so many did bow to Baal. [122]
8 THE NETHERLANDS \textit{42}
Before the second world war, no Church in the Netherlands publicly protested against German anti-Semitism, as distinct from Churches in Great Britain, France, Sweden, the United States \textit{etc}. The following reasons for this can be given: 1. There was little co-operation between the Protestant Churches. 2. The Churches did not speak out publicly on any subject. 3. The spiritual life of many Churches was at a low ebb. 4. Many people were afraid of endangering Holland's precious neutrality and
its economic interests with Germany.
5. Many Christians considered National-Socialism a bulwark against Communism.
[123]
The exceptions to the rule were provided by inter-denominational Church bodies. In April, 1933, the Dutch Council of the ``World Alliance for International Friendship through the Churches'' adopted and published the following motion:
``The Dutch Council of the World Alliance for International Friendship through the Churches, aware of its duty to promote friendly relations among the nations, and convinced that the anti-Jewish measures taken and carried out in Germany must be regarded as a manifestation of racial hatred which considerably prejudices such an understanding, requests the International Executive Committee to define publicly its position with regard to these measures and, subsequently, to do everything in its power in accordance with the aims and principles of the Alliance, to disperse the tension and indignation which these measures have provoked in the Netherlands as well as in the entire civilized world, and to work towards the establishment of those relations which, according to the principles of the Christian conscience, ought to exist among the different races.'' [124]
This appeal to the International Executive Committee was successful. [125] The same Council also sent a letter to the ``Permanent General Committee of the Dutch Israelite Community'', informing them that they had heard with a sense of shame and distress of the treatment of the Jews by the German government on grounds of racial hatred. The Council expressed its conviction ``that this hatred is contrary to the Christian conscience'' and quoted the letter sent to the International Committee. [126] \textit{43}
In May, 1933, a Manifesto was published, signed by many individual Dutchmen, denouncing anti-Semitism. [127] In the same month, Christians of Jewish origin turned to the ``Synodal Committee of the DUTCH REFORMED CHURCH'', requesting that on one particular Sunday the Jewish question should be the main theme of the sermon. The Committee replied that ``they were convinced that it is the duty of the Church to pay attention to Israel and pray for it, but that in the present circumstances it would not be wise to set apart a special Sunday for this purpose''. [128] On May 23, 1933, a public meeting of protest was held. Amongst other speakers was the Rev. J.J. Buskes, who later became one of the leaders of Church resistance during the war. He then spoke ``as a member of a Christian Church''. Dr. W. Banning also protested against the Nazi terror, ``in the name of Socialism and of the Gospel''. [129]
* * *
On September 19, 1935, a meeting of protest was held at Amsterdam. There were three Protestant speakers, one of them, Rev. J.J. Buskes. [130] In 1936, the Synod of the Reformed Churches in the Netherlands declared that members of the Church who were members of the Dutch National-Socialist Party, must be advised to terminate their membership of the party. If they would not heed this admonition, they must be barred from participating in Holy Communion. This measure was maintained throughout the war. The report to the Synod on the N.S.B. (National-Socialist Movement of the Netherlands) says:
``Even though the N.S.B. rejects idolization of the race, the manner in which it stresses in its Program the unity of the Aryan race [131] shows, that it is not blameless in this respect.'' [132] \textit{44}
A Protestant Committee for help to Protestant refugees of Jewish origin was formed on May 5, 1936.
* * *
In 1938, 7,000 Jewish refugees were admitted into the Netherlands. The Government was of the opinion that Holland could not bear too heavy a strain on the labour market. The Protestant Prime Minister declared: ``If an unlimited stream of foreign Jews were admitted, public opinion regarding the Jews will take an unfavourable turn''. [133] Thus the border was closed and Jews who had ``illegally'' entered into Holland were sent back to Germany, unless they could prove that their life was in danger there. Of course it is easy to be wise after the event, and in those days it was not yet clear to everybody that the life of all Jews in Germany was in mortal danger. The fact remains that these inhuman measures were taken by a Government of which most of the members were professing Christians. And no Church protested. Prof. D. Cohen states:
``Our Committee [for help to Jewish refugees] had clashed vigorously with the Government on this point, notwithstanding our good relations and good co-operation with it. However, we had public opinion with us.'' [134]
The last part of his statement is doubtful, at least regarding a large section of the Protestant press. [135]
A national collection was held on December 3, 1938, and recommended by the Synodal Committee of the DUTCH REFORMED CHURCH:
``The Committee, concerned about the bitter sufferings resulting from the persecution of the Jews, considers it to be the duty of the Church to practise Christian mercy. It urgently recommends that all local churches should take up a special collection, on behalf of the victims of this persecution, so that their suffering may be alleviated.'' [136] \textit{45}
Here help to the persecuted Jews in general was recommended, not just to Christians of Jewish origin. In November, 1938, the Executive of the Dutch Ecumenical Council turned to the World Council of Churches, Geneva, requesting it to organize immediate action on behalf of the German Jews. [137]
9 BELGIUM
The Protestant Churches in Belgium are minority Churches, together comprising less than half a percent of the population. The following statements are all from the year 1933. To the best of my knowledge no other statements were issued after this year. On April 4, 1933, the Federation of Protestant Churches of Belgium sent the following letter to Dr. Kapler, the President of the Protestant Federation of Germany:
``The Federation of Protestant Churches of Belgium has directed us to send a fraternal message to the Protestant Federation of Germany. We would ask you, Mr. President to accept it in the same Christian spirit, and to do us the honour of transmitting it to your Executive Council. We are much distressed by the events of recent weeks during which the German Jewish population has been subjected to discriminatory measures; the situation threatens to deteriorate even further. Our German co-religionists, imbued with a sense of justice, must certainly be equally distressed by these excesses. It certainly cannot be pleasing to them that, in most countries, spontaneous public opinion has espoused the cause of German Jewry. We would therefore ask you, Mr. President, if it would not be possible for the Federation of German Evangelical Churches itself to intervene, discreetly as they may deem fit, on behalf of the German Jews so that they may be reinstated in all their rights of citizenship. Would it not be a great triumph for the spirit of tolerance, which is certainly a Protestant attribute? Would it not mean a re-establishment, in the eyes of the world, of that reputation which your country has enjoyed for so long, of being a highly cultured country? May one not say that German Jews have, up till now, been much attached to their country; that they have added to its distinction in the field of science, art and literature. \textit{46} In short, that they are known for their adherence to the principles of freedom of conscience? Inspired as we are by purely Christian and humane sentiments, we have no doubt that you will accept the above message in the spirit of grace.''
Yours faithfully,
Henri Anet, Secretary; A. Rey, President. [138]
This letter was certainly not lacking in courtesy and we get the impression that it was written in a spirit of moderate optimism. Apparently it was some months later that the President of the Synod of the Evangelical Protestant Churches of Belgium sent the following letter to the Chief Rabbi of Belgium:
``Time has passed since, during the first explosion of hate throughout Germany, it might be supposed that a period of calm would follow. But according to accounts in the press, it seems that a general and lasting exclusion of all Jewish intellectuals cold-bloodedly continues. This illegal and cruel oppression of a highly respectable minority shows that the new Germany is descending into a mental attitude fit only for the Middle Ages. The destruction of such an out-grown mentality had been, until now, the noblest work and the most imperishable glory of the new spirit of the last four centuries.'' [139]
Even more outspoken was the address of Rev. Schijns, the President of the Federation of Protestant Churches, at a Meeting of Protest in Bruxelles, on April 6, 1933:
``You have heard the lay protests against anti-Semitic persecutions in Germany. You have heard the Catholic protest. May I be permitted to speak on behalf of the Protestant Churches of Belgium. It is true that the voice of Christ, who clearly proclaimed the inviolable rights and imperative demands of justice, has not always been listened to over the centuries; on many occasions Christians themselves have had recourse to violence; I cannot forget that in the 16th century my ancestors, the Huguenots, and the Beggars, [140] also suffered cruel persecution... Nevertheless, thanks to a clearer understanding of the demands of the Gospel, as well as to the progressive evolution of the lay conscience, we had become sincerely convinced that henceforth violence, which was unanimously condemned by public opinion, is morally inconceivable. Yet now we discover that violence has been `honourably' reinstated, so that even today it is still attacking innocent victims.'' \textit{47} We never supposed that, in our times, any person, on religious grounds, could be accused of a political offence! Yet, now we hear that in Germany a religion (the Jewish religion) is being formally and coldly proscribed, by the civil authorities. This inhuman attitude, inspired by a narrow, sectarian nationalism, stands in absolute contradiction to the Gospel: it is a monstrous heresy, which cannot but dwarf all other crimes. The ancient Jewish law contains the following beautiful maxim: 'Thou shalt love the Lord thy God with all thy heart, and with all thy soul, and with all thy might'. It is therefore with all my heart, with all my soul and with all my might that I deliver here, in the name of my Protestant co-religionists, a message of vigorous and profound sympathy for all innocent victims of violence. The sufferings of today, like those in the past, tragically illustrate the struggle of brute force against the forces of the spirit. But just as moral strength has triumphed in ages past, we are sure that to-day also, by virtue of an eternal law, victory lies with the powers of the spirit!'' [141]
10 FRANCE
Though a small minority, numbering altogether not more than 800,000 souls, the spiritual sons of the Huguenots early and unequivocally protested against the persecution of Jews. They themselves had been persecuted.
Rev. Marc Boegner, President of the Protestant Federation of France, sent the following letter to the Chief Rabbi of France, in 1933:
``The Council of the Protestant Federation of France which reassembled to-day, for the first time since the beginning of the period of the great sufferings of your coreligionists in Germany, has asked me to assure you that the Protestants of France whole-heartedly associate themselves with the indignation of their Jewish compatriots and with the distress of the victims of such base fanaticism. The spiritual sons of the Huguenots are stirred with emotion and sympathy whenever a religious minority is persecuted. They are well aware how much Christianity, and in particular the Reformed Churches, owe to the prophets who paved the way for the Gospel, and feel afflicted by the blows descending upon their Jewish brothers. May God help your sorely tried co-religionists to find in Him their strength and consolation, as did their frequently persecuted ancestors. May He impart to you, and to the Jews of France, the secret of soothing pain and reviving hope. \textit{48}
I wish to reassure you, that we are certain, that all our Churches will unite, during the Holy Week, in fervent intercession on behalf of the Jews of Germany.'' [142]
In the same year the following letter was sent by Rev. Cleisz, Honorary President of the Consistory of the Reformed Churches of Lorraine, to the Chief Rabbi of Nancy:
``You will hardly be surprised to find me among those who energetically protest against the wave of anti-Semitism in Germany, which has cast so many Jewish families in distress. I abhor fanaticism, whatever its source, and am dismayed to observe in the middle of the twentieth century such an excess of folly. Therefore I join whole-heartedly with those who protest against such a tyranny. I wish to assure you of my deep compassion for so many human beings overcome by grief...'' [143]
Rev. Wilfred Monod sent the following letter to the French Committee for the Protection of Persecuted Jewish Intellectuals:
``Allow me to express my feelings of relief at the thought that France is offering hospitality to Jews escaping from the darkness of a new Mediaevalism. Although Jews were crushed by the great Empires of the West; later becoming the vassals of the anti-Semitic Kings of Egypt and Syria; politically annihilated by the Romans; hated by the Moslems; persecuted by the Church; held in public disdain; treated as a stateless and homeless people even in the twentieth century, and sometimes deprived of their civil rights in the countries in which they were dispersed; the Jews have not disappeared as did the Phoenicians or the people of Nineveh. Without territory, without government, without currency, without flag, Abraham's race has kept itself alive. What marvellous obstinacy! What supernatural tenacity! In spite of all this, Judaism has given the human race that mysterious Book which maintains alive on this earth the inextinguishable flame of a universal, international ideal, the world-embracing ideal of human catholicity. Israel has bequeathed to men the Bible, Jesus Christ, and the Messianic vision of the Kingdom of God... On 29th August, 1914, up in the Vosges, one of our Catholic soldiers, mortally wounded, asked for a crucifix, and it was the Jewish chaplain who brought him this venerable symbol, some minutes before he himself gave up his soul in the arms of a Jesuit priest. This happened on a Saturday, the holy day of the Jewish Sabbath. Welcome to the representatives of the wandering nation! On French soil they will find a place to rest their head.'' [144] \textit{49}
On April 17, 1933, a Protest Meeting was held at Lille. Rev. Bosc was the Protestant spokesman, speaking in his ``triple capacity as a human being, a Frenchman and a Christian''. We quote the following:
``... Finally, to protest against the persecutions and victimisations of the Jews is a task in harmony with the spirit of Jesus Christ, and here I thank Monsieur l'abbe who has just sounded forth a note of profound truth. Everyone of us knows that the spirit of Jesus Christ is the spirit of peace, the spirit of justice, and more than that: the spirit of brotherhood and of love. It is the spirit which to-day imbues all moral and social systems in the world, so that Jesus Christ is acknowledged as the unrivalled ruler not only by Christianity as a whole but also by all mankind... The spirit of Jesus Christ which, Ladies and Gentlemen, means the spirit out of which are woven the dreams we have of a better future for mankind, the dreams we dream when, surrounded by all sorts of iniquities and by all kinds of ugliness, we nevertheless look towards some glorious dawn! The spirit of Jesus means that spirit which will triumph because it is the living truth. It is in my triple capacity as human being, Frenchman and Christian that I fully pledge my entire, conscious support to this movement of truth in its efforts to infuse a little justice and kindness into mankind, against the attempt to lead humanity back to the night and the iniquities of the Middle Ages, from which it began to emerge.'' [144]
* * *
On November 20, 1935, a Meeting of Protest was held in the Hall of Chopin, Paris. Rev. Marc Boegner, President of the Protestant Federation of France, said the following:
``... Since I am here representing both Christian and Protestant France, I should say that in the light of what is going on in Germany --- whether it be the persecutions of Jews or of Christians --- it is impossible for us not to add our most energetic protests to those you have heard so far.
What Christianity Owes to Judaism
``Christianity, as has been indicated by President Reynaud, is essentially a universal creed. Once one believes in Christ, whatever one's denomination may be, it is impossible not to subscribe fully to the words of that Jew of olden times St. Paul, the apostle, who having plumbed the depths of Christ's thought, exclaimed: 'There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female; for ye are all one in Christ Jesus'. (Gal. 3, 28). This is the basic tenet on which, since July 1933, all preaching in the Churches of Germany has been practically proscribed. \textit{50}
First I wish to state that what has shocked and appalled Christian conscience, what has provoked protests from one end of the Christian world to the other? Protests which will certainly be reiterated and increased --- is precisely the fact that this new gospel of racialism already has been applied to the Jews, and seems to have reached its culmination point in the Nuremberg Decrees. One cannot know whether even worse may not happen later on. I have met many Jews who had been driven out of Germany since the Hitler revolution, and when I went to Germany as recently as this year, on two occasions while travelling through a large part of Germany, I could not but feel intensely moved on seeing, at the entrance of villages and towns, large signboards forbidding access to the Jews; and on many trees along the roads, posters full of insults against them. Christian as I am, and knowing what Christianity owes to Judaism, I know that the Church of Jesus Christ is the daughter of what it calls the ancient Church of Israel. The Protestant in me knows what the Gospel owes to those prophets who, beginning eight centuries before Jesus Christ, have presaged the universalism which the religion of Christ would later proclaim throughout the world. Did not Isaiah welcome the day when all nations would flow unto the mountain of the Lord? And others after him, such as Jeremiah, did they not show their people, the only people ever elected, the road by means of which they were to bring to others the revelation which had been bestowed upon them, so that all nations might come to know the true God? The Gospel is the heritage and fulfilment of that great hope of the prophets. It is impossible for a Christian, when he sees the infamous crusades conducted against Judaism, not to be among those who declare that they are unable to forget what they owe to the Jewish people. We are among those who remember all this with deep gratitude. We believe that this gratitude, in view of the suffering of this people who are being crucified once again, ought to be shown in acts of sympathy and solidarity.
Racialism inside the Christian Churches
``The gospel of racialism of which you have just been told does not rear its ugly head solely outside the Church, but also inside the Christian Churches. Since July 1933, under the pretext of rallying the whole of Germany round the doctrine of racial superiority --- you have seen the outcome of such teaching --- and of purity of blood, they have begun to persecute those who are not 100 percent Aryan, even inside the churches. I was in Berlin in July, 1933, and there, where every wall might have been equipped with a hidden microphone, I met one of the most representative personalities of the Evangelical Church of Prussia. He informed me of what had been happening during the last few weeks. He said that he had felt compelled to resign from the high position he had occupied in the German Church, even though his resignation would mean a considerable financial sacrifice. He and those who were thinking and acting like him were now unable to speak, to write, to telephone, or to do anything whatsoever. \textit{51} ``But,'' said he, ``how could I have agreed to go and tell the young Evangelical pastors whom I ordained two or three years ago, that they are not fit to preach the Gospel or to carry out the duties of their ministry, simply because they have a Jewish grandmother or grandfather? This problem of the non-Aryam has since then caused much anguish to many men who are pastors or simply beadles. The new gospel has made its appearance in the Church, propagated, preached and spread by groups calling themselves the `Deutsch-Christliche' or the German Christians. It is necessary, they claim, to expel from the churches and from all church posts, in every denomination, those men who are not of absolutely pure Aryan blood for three or four generations back.
The Church Resists
``This has resulted in unbearably painful conflicts. It should be acknowledged that tremendous pressure was exerted by the State authorities as well as by the pressure of the opinion which increasingly tends to assert in religious circles that Adolf Hitler was the man through whom Germany was able to re-establish herself. in spite of all this, however, there have been instances of Catholic and Protestant consciences refusing to bow and submit. Resistance was organized in the Catholic Church, where the warning bell to the conscience of Christians was rung by that admirable man, the Cardinal Archbishop of Munich. In the Protestant Church, the great voice of the theologian, Kar1 Barth, and the voices of many others, have been raised to rally Christian consciences to their call. A completely new Confessing Church has sprung up comprising more than half of the pastors in Germany, quite apart from those who are still hesitating, because they must make a living, and, therefore, ask themselves what will happen to-morrow. About a thousand pastors have placed themselves behind Bishop Mueller, the `German Christian', whom you, Mr. Paul Reynaud, have just mentioned.
The ``New Gospel''
``It is not only through the persecution of the non-Aryans that the desire has arisen amongst many Germam to preach a new gospel, but because of a claim to meet Christ on a new basis, particularly on the basis of the glorification of the German race and blood. The most extraordinary statements have been made. Paganism has asserted itself on the fringes of the Church and its influence gradually has pervaded it. An effort even has been made in certain churches by pastors imbued with the spirit of national-socialism, to have the Old Testament --- containing the magnificent history of the Jewish people and I even would say, of God's great acts toward the Jewish race --- banned and barred from religious instruction. Included also is that moving page in the first book of your Bible, and ours --- note this, any Jews who may be listening to me --- where we are told that Abraham went so far as to be ready to sacrifice Isaac, his only son, to God!
``Never will the Churches Agree...'' \textit{52}
``Subsequently there have been attempts to make peace with the Churches, and the papers during the past few weeks have brought us news of 'peace feelers' offered to the Churches. Negotiations were envisaged both with the Catholic Church and with the Evangelical Churches. But they never will induce either true Catholics or Protestants, to put as a Gospel source, an affirmation of the superiority of the German race over the others, nor a denial of anybody's right to belong to the Church of Jesus Christ. They may again start their persecutions, and I think they will. They may chase pastors and priests from their churches and send them to concentration camps. They may resort to petty annoyance and to persecution; however, I am absolutely convinced that the Christian conscience has been aroused. Perhaps this experience was necessary to awaken it out of a certain stupor? The Christian conscience will absolutely oppose the events which have succeeded each other which such rapidity over the past few years and any attempt which may be made to persuade the Churches in any way to insert into the Gospel (which desires that all men should be considered the children of God and be reconciled in universal brotherhood) an addition which asserts that some shall rank first and others may be excluded. Never will the Churches agree that the Gospel of love, symbolized by the two arms of Christ extended on the cross, will be replaced by a gospel of race and blood. I am convinced that by affirming our sympathy with all in Germany who are being persecuted for their views, and with all in the Christian Church who make efforts to resist (as I have tried to show) the determined attempt to lead them onto the ground of racial discrimination, we are helping them in their resistance. We are helping them to discover that there is a Christian, as well as, a merely secular public opinion, throughout the entire world, which is aware of all that this resistance implies in the way of present sacrifice, and perhaps of still more suffering in the future. Let us therefore be among those who by word and example give evidence of that sympathy and solidarity. Let us unite here, as Mr. Paul Reynaud has asked us to do, without distinction of religious, philosophic or even political convictions, in protest against the besmirching of justice and the dignity of man.'' [145]
* * *
The Council of the Protestant Federation of France, in its session of November 29, 1938, unanimously adopted the following Resolution:
``The Council of the Protestant Federation of France, reassembled for the first time since a terrible crime has provided a pretext for new persecutions against the Jews, feels itself to be the mouthpiece of all the Churches which it represents in our country, in making a solemn Protest against a similar outburst of violence and cruelty. \textit{53}
The Christian Churches will betray the message entrusted to them, if they do not unreservedly condemn racial doctrines which are contrary to the teaching of Christ and the apostles; and if they do not express their utmost disapproval of the barbaric methods by which such doctrines are practised...'' [146]
In the light of ``the serious problem confronting the authorities by the arrival on French territory of numerous foreigners who had been expelled from their own country by persecution'', the Council of the Protestant Federation in France instructed ``all Protestant Frenchmen'' as follows:
1. To aid the Government --- in determined resistance to any suggestion of violence, wherever it may come from and in whatever manner it expresses itself --- to solve so complex a problem in a quiet atmosphere and with respect for human dignity. 2. To contribute as much as possible, by their gifts and by their co-ordinated initiative, for the relief of the terrible distress which they are witnessing and which makes its appeal to them. The Council draws their attention to the existence of a French Committee for Protestant Refugees, Aryans and non-Aryans, which is now functioning and to which financial contributions can be sent... [147]
11 SWITZERLAND
The Protestant Churches of Switzerland are cantonal Churches, distinct and independent from one another. In most of the cantonal Churches, the legislative body is the Synod and the executive organ the Synodal Council. The Federation of the Protestant Churches of Switzerland at first consisted only of National Churches, but it soon admitted the Free Evangelical Churches, the Methodist Church and the ``Evangelische Gemeinschaft''. The Federation has 2,888,122 baptized members. At the beginning of April, 1933, the following Declaration, signed by 21 Protestant ministers, was addressed to ``various Protestant Ecclesiastical groups in French-speaking Switzerland'':
``Moved by the present situation of the German Jews, and unable to understand how the authorities, otherwise attentive to moral values, can ignore the right of freedom of conscience, and of work, as well as security to every human being, we, the undersigned, think that the time has come to draw the attention of Christians to the serious implications in an attitude which is the very negation of the evangelical spirit; a spirit which is synonymous with love, freedom and mutual assistance. We expect the Churches to raise their voices in order to claim for the Jews the same degree of justice, which it is their duty to demand for every oppressed minority.'' [148] \textit{54}
On May 31, 1933, the Synod of the Free Evangelical Church of the Canton Vaud sent the following letter to the President of the Council of the Federation of Protestant Churches of Switzerland:
``We beg to bring to your attention the fact that the Synod of the Free Evangelical Church of the Canton of Vaud, at its annual meeting at Lausanne, unanimously resolved upon the following Declaration, which we now submit to use as you see fit. ``Moved by the news which has reached us from Germany concerning the numerous and regrettable restraints imposed upon the freedom of conscience, and, in particular, concerning the ill-treatment of the Jewish population of that country; ``and with the conviction that the Gospel of Jesus Christ constitutes an affirmation of freedom and love among the races of mankind; the Synod of the Free Evangelical Church of the Canton of Vaud, assembled at Lausanne, unites itself with all protests raised in favour of freedom of conscience and respect for the Jews of Germany.'' [149]
In September, 1933, the Protestant Churches of Geneva published the following Declaration:
``Events shocking and hurtful to a sense of justice are mounting in Germany and have repercussions here. Men are persecuted for their opinions. Dismissed, boycotted, ostracized, they are suffering as in the days when neither freedom of thought nor of conscience were tolerated. The mere fact of belonging to the Jewish race, even if only by descent, frequently incurs implacable treatment. These actions have given rise to protests in numerous countries and in the most varied circles. Here too, our Christian conscience has been roused. It would be dangerous to consider ourselves better than others. Intolerance and injustice have their roots in our own soil. We must be on our guard. Several papers make appeals for violence. The seeds of discord are being sown among our people. Anti-Semitism, which until now has been foreign to us, now finds its advocates among us. Members of our Churches, also, forgetting that the same blood flows in all mankind, and that, before God our Father, we are all brothers, have been swayed by the passions of these times. Let us not permit a spirit incompatible with the teachings of Jesus Christ to take root in our country.'' The National Protestant Church of Geneva; the Free Evangelical Church of Geneva; Evangelical Christian Association; the Committee for Popular Evangelism; the Council of the Methodist Church. [150] \textit{55}
It is striking that the declarations and resolutions issued in Switzerland, so many times mention the danger of anti-Semitic influences within the country itself, and sometimes within the Church. [151]
* * *
On November 14, 1938, the Church Council of Canton Zurich addressed the following public letter ``To the Reformed People of Zurich'':
``In indignation and horror we recently have witnessed, in the state neighbouring us to the north, that Jew baiting has erupted and, in its dimensions, surpassed the severest atrocities yet experienced. We feel in spirit united with all our brothers and sisters in the neighbouring country who, whatever their attitude toward Jewry may be, deeply deplore such injustice, yet they must keep silent on the subject. We must not be silent. We must consider it a Christian obligation to cry out against it, not only within our church walls but to the world at large. It is a terrible injustice to exterminate, by all conceivable means, a nation which possesses, as does every nation, the right to exist. It fills us with deep humiliation and shame to discover in a country living for centuries under the influence of the gospel and of Luther, that sentiments of passionate hatred can break out and boil over against a small racial and religious minority, and that all humane and Christian feelings be suffocated. It plainly shows us, to our horror, what human hearts are capable of when racial hatred and blind raving passion win the upper hand, drowning the voice of justice, mercy and goodness. Can we Swiss suppose that we are immune against such frenzy? But are not the same dark powers active within our own people, openly at times and sometimes secretly, confusing conscience; stirring passions; igniting racial hatred? It pains us that consideration for so many unemployed citizens in our own nation prevents us from offering a protecting asylum to the suffering refugees, who, like wild game, are chased from country to country. \textit{56} At least let us do for them all that is in our power! When in the next few days a general collection is made for the benefit of these refugees, among whom are not a few who, although Jewish by birth, are of the Christian faith and thus a part of the Evangelical Church, let us open our hearts and hands and express loving-kindness towards these remorselessly persecuted people. Let us close our hearts to all feelings of unchristian racial and religious hatred. Neither hate, slander, oppression nor violence, but Jesus Christ's love alone is capable of bringing longed for peace to restless humanity. But above all, let us pray to the Almighty that He will protect all those who are persecuted, and that He will save our Swiss people from the disgrace of an anti-Jewish campaign and deliver us, and all nations, from the forces of violence and injustice, and bring His Kingdom of justice, love and peace.'' [152]
Again (as in 1933) the danger of anti-Semitic influences within Switzerland was mentioned. The letter also gave as an excuse for not admitting more refugees, that there were ``so many unemployed citizens in our own nation''. The same motive had led other Governments --- as for instance the Dutch Government --- to issue decrees restricting immigration.
The members of the Ministers Union of Geneva wrote a letter to the Chief Rabbi of the City of Geneva in which they expressed their deep sympathy with the persecuted Jews. This letter, together with the declaration of the Church Council of Zurich (see above) was read at a service, held in the synagogue on a Sunday and not, as usual, on a Saturday. This postponement was in order that the prayers of that day could be united with those of all the Christian Churches in Switzerland for the persecuted Jews. [153]
In December, 1938, the Synod of the Canton of Bern issued the following Declaration:
``The Synod of the Evangelical Reformed Church of Canton Bern declares, that it views the merciless persecution of Jews and Fellow-Christians stemming from Jewry, as an expression of a spirit which has nothing in common with the spirit of Jesus Christ. It calls upon all members of our Church to intercede on behalf of the persecuted, especially our persecuted brothers; to stand up for them on every occasion; and to oppose any further attempt to poison the soul of our people with the spirit of racial hatred.'' [154]
* * * \textit{57}
12 DENMARK
Leading Danish theologians --- three professors and one lecturer of the Copenhagen University [155] and the Bishop of Copenhagen, Fuglsang-Damgaard --- published a declaration on January 10, 1936, denouncing an anti-Semitic brochure, ``The Christian Church according to the concept of the peoples of the North'', based on the ``Protocols of the Elders of Zion''. Professor Frederik Torm related the history of this forgery in an informative article. The matter drew attention, even in Germany, where the ``Volkische Beobachter'' in its edition of January 14, 1936 reported the story as told by its correspondent in Copenhagen under the caption ``Danish theologians grow nervous'' and with the subtitle: ``The Jewish question arises in Denmark''. The report of the former German Envoy, Richthofen, dated January 13, 1936, shows the same attitude, considering the article of the theologians as an act of defence against ``the ever increasing understanding of the Jewish question in Germany among the Danish public''. [156]
In the autumn of 1938, Bishop Fuglsang-Damgaard said in his sermon at the opening of a new church, Lundehuskirken, that it was with deep pain that the Christian community had heard about the persecution of the Jews in Germany, which had reached a culminating-point in those days. 149 pastors of Copenhagen supported these words by a public statement and pronounced their ``deep sympathy with our Jewish countrymen on account of the sufferings which at this time befall their brethren and which must fill every Christian with horror''. \textit{58} Dr. Fuglsang-Damgaard asked the pastors to pray for the suffering Jews in the services the following Sunday, and he himself declared at a service in Helligkors Church, that we must pray to God ``to protect our people against the poisonous pestilence of anti-Semitism, hatred of the Jews and persecution of the Jews. Our Lord and Saviour Jesus Christ was David's Son after the flesh, and those who love Him cannot hate His people''. [157]
13 SWEDEN
The Swedish Ecumenical Council sent the following letter, dated April 3, 1933, to the German Evangelical Church Council in Berlin:
``The Swedish Ecumenical Council, a representation of different Swedish Church communities, sincerely regrets the existing conditions in Germany and the boycott of German goods abroad, and is deeply concerned by the anti-Semitic action in your country, such as has been expressed in official statements and actions. We hope and pray that, with God's help, it will be possible for the German Evangelical Churches actively to stress the genuinely Christian principles, which you upheld in your appeal before the latest elections. ``Be not overcome by evil, but overcome evil with good.'' As Christian brothers, we are anxious to be in communication with you in this matter and further hear your views. In sincere communion in the faith, for the Swedish Ecumenical Council: Arch-bishop Erling Eidem, Chairman. [158]
The Appeal of the German Evangelical Church Council to which this letter referred, was published on March 3, 1933, just before the elections for the Reichstag. Unfortunately, we do not know whether any reply was received by the Swedish Ecumenical Council.
In 1933, 64 prominent Protestant Church leaders also published an ``Appeal to Swedish Christianity'', warning against anti-Semitic influences in Sweden:
``Action against the Jews in Germany seems to work as a stimulant --- and no small one --- for the anti-Semitism which exists in certain Swedish circles. Many of us may have been prone to consider this movement in our country as insignificant, and not worth combating. But the matter is more serious than that. If sufficiently great spiritual strength is not mobilized against this fanatical and shortsighted nationalism, it is difficult to foresee the result. \textit{59} The undersigned regard it as their duty to express the worry and anger with which this anti-Semitic movement has filled them, and to appeal to Swedish Christianity of all denominations to fight against racial hatred, stressing Christ's valuation of man and his brother-love. \textit{59} Already from a general and cultural viewpoint, anti-Semitism is an expression of ingratitude and shortsightedness. No less in our country, citizens of Jewish descent, have contributed in all fields to such a degree that, if all trace of what they have done were erased from the Swedish civilization, to-day, it would be much poorer. But first, anti-Semitism must be condemned from a Christian-religious viewpoint. Here too one can, rightly, speak of a debt of gratitude. The prophets and psalms of Israel also belong to our holy heritage. And in spite of all wild racial hypotheses, Jesus Christ is a son of Israel and a perfecter of these prophets' work. However, it is not only, and not first and foremost, the gratitude for a spiritual inheritance which urges Christian people to take their stand against anti-Jewish activity. They would be denying their Master if they did not do so. For in Him all racial differences are overcome, in the divine love, which has taken form in Him, we are all each other's brothers, no matter to which nation or race we belong. Whosoever professes himself a follower of Christ, yet lets himself be seized by nationalistic presumption, of which anti-Semitism is one of the most repellant expressions, must realize that any action designed to attach a stamp of inferiority on members of the Jewish people or deprive them of full civil rights, is in absolute opposition to the spirit and teaching of Jesus. The gravity of the situation has impelled us to make public this declaration, which is also an appeal to Swedish Christianity to oppose unmitigatedly a propaganda which is becoming louder and more aggressive anti-Jewish, and the mentality of violence from which it stems. Time must not be lost. Freedom of speech is not yet stifled. The gospel of Truth and Love may still sound its voice.'' [159]
At a meeting of the Stockholm Pastors' Society, held in 1934, Professor Nygren of Lund opened the discussion on the subject: ``What is the reason for the struggle within the German Church?'' The Pastors' Society unanimously decided to publish in the press their agreement with the fundamental viewpoints expressed in Prof. Nygren's address. The Society's Resolution reads as follows:
``The furious struggle now taking place within the German Church is not on a personal question, a question of rights or a question of organization. Nor is it a struggle for or against the National-Socialistic State or for or against the liberalistic freedom ideal. \textit{60} The struggle concerns Christianity itself, its existence or non-existence. What is happening in Germany to-day is nothing more or less than the appearance of a new religion, beside and in contrast to Christianity --- a religion based on `Blut und Boden', on racial idealism and racial egoism. This has to some extent thrown Christians and non-Christians into jail. From a deeper viewpoint, the difference between `German Christians' and the heathen `German Faith Movement', therefore, becomes surprisingly small. If we observe the deepest tendency, of which, in general, the followers of these movements are quite unconscious, it can even be said that, for the former group, it is a question of the new religion in Christian guise; for the latter, the same religion in Germanic guise. The extraordinary danger is that the present Church management has not the least understanding of the reason for the struggle. It believes that it is fighting for the sake of Christianity and does not realize that it has slipped into a new racial religion. True, it often stresses that the Bible and the Confession should be left 'unas-sailed', but the tone of the voice itself reveals that it is on something else that one subsists. Out of the abundance of the heart the mouth speaks. The real pathos first appears when one can talk of `Blut und Boden', `Blut und Rasse', `Blut und Ehre'. The god one really worships is the idol of one's own people. But in the German Church there are men --- and fortunately these are not few --- who understand what is at stake; what this new religion has to offer the people, from a Christian viewpoint, is nothing less than idolatry. One creates a new god in one's own image, the image of `the German Man'. The Christians who see this must, through their faithfulness to the Gospel, be forced out into the struggle. Because of this they find themselves in tragic conflict; for there is so much in the new state to which, in their hearts, they say `yes', and with joy. But when they fight this new heathen spirit that has penetrated the Church and seized the power in it, they are stamped as enemies of the state by the uncomprehending Church management. The point has been reached, where those who do not want to give up their Christian faith are attacked by the German Church management: with external means of power, the secret state police, removals from office and suspensions. We, Evangelical Christians of a kindred people, have seen with grief and concern that the German Church management through such activities has tarnished the Christian name. With the deepest sympathy we follow the oppressed Christians' brave and joyfully self-sacrificing struggle, in defence of Evangelical Christianity, not only in Germany but also the world over.'' [160]
\textit{61}
The Resolution contains points that to-day are obvious to us, but in those days they undoubtedly enlightened many ignorant people. Much that has been said by the Lutheran Church leaders of Sweden, already in the first years of Hitler's regime, shows a deep theological insight into the nature of anti-Semitism. Few Churches in other lands showed this insight at so early a date. This fact should prevent us from over-simplifying the answers to the question, as to how far certain of Luther's views about the Jewish people influenced the Lutheran Churches in the twentieth century.
* * *
The following statement, signed by Erling Eidem, Archbishop of Uppsala, and 25 other Church leaders was issued by the Swedish Ecumenical Council, in autumn 1938:
``A storm of violence and cruelty goes through the world. The Jewish people are severely hit by this. Their horrible fate must awake in Christian minds strong indignation, as well as deep sympathy for the victims. To belong to the Jewish race is becoming equivalent to being stateless within that portion of humanity which calls itself Christian. This brings shame upon the Christian name.
Anti-Semitic Propaganda in Sweden
``In our country, too, anti-Semitic propaganda is prosecuted, even though it may, in some respects, avoid publicity and, especially under the pressure of recent occurrences, has met with deserved resistance. More than others, Christians here must be on their guard. No racial differences exist in the Christian evaluation of man. Love of Christ forbids branding any person inferior. Persecution of the people of Israel on the one hand requests Christ's congregation to fight against violence and injustice, and preventive action on the other. The Swedish Ecumenical Council, representing the ecumenical world organizations as well as the larger Swedish Church communities, hereby begs to remind you of our Christian responsibility in this matter. We must not forget that we too, bear a measure of guilt for this evil power that has arisen through loveless ness and injustice in the world. We appeal to all who, in their capacity as pastor, congregation head or preacher, are responsible for the creation of public opinion in such circles as come under Christian influence, to resist the spirit of mercilessness and injustice in the anti-semitic propaganda, by all ways and means available in each community. It seems especially important to us to try to prevent its poison penetrating the minds of the Young. Not only religious instruction in the schools can give an opportunity for this, but also instruction in Sunday schools, confirmation classes and Bible classes. A few congregational evenings could be used to throw light upon the plight of the Jewish people and to stress our Christian responsibility towards them. The un-Christian element in all racial hatred could at times be stressed in the sermon. All discussion of politics naturally must be banned from such Christian instruction and preaching. \textit{62}
Aid of Refugees
``Where the feeling of responsibility has been awakened, it must be transformed into action. This can be done by gifts to the relief organizations among the banished, which also have branches in our country. In co-operation with other organizations, the Swedish Ecumenical Council's Refugee Committee seeks to aid refugees both within and outside our country's borders, particularly Christians of non-Aryan descent. The money already collected is now almost spent, but the need for help is still very great. Gifts for this activity can be deposited under the name ``Help for Refugees'' on the Swedish Ecumenical Council's postal current account No. 80710, Stockholm. Recently, the Council's Refugee Committee, the Deacon Board's Social Committee and the Swedish Israel Mission have started other aid activities, such as accommodating children of Jewish refugees, preferably Jewish-Christian, in Swedish homes for a shorter or longer period, and trying to find places farmers' homes for about a year for Jewish-Christian youth, particularly male, who need re-education for later emigration to countries which have declared themselves willing to receive them. Information of such homes as well as financial contributions will be gratefully received by Pastor B. Pernow, Idungatan 4, Stockholm, postal current account No. 125545.
Intercession
``At this period, with the mentality of violence penetrating minds more and more, it is important not to neglect the possibilities we still have to make Christ's mind and Christ's thoughts heard regarding the relation between man and man, between people and people. Scarcely at any other point has this task seemed clearer and more demanding than as it concerns the Western peoples' conduct towards Israel. May Christ's love in our hearts light a flame of concern for a people who were the Lord's own, the people of the Prophets and the Apostles. May Christ's love make us burning and persistent in our intercession for those who suffer persecution, as well as, for those who persecute. May they receive the grace to repent. May Christ's love make us firm against all hatred, drive out all fear, and make our hands ready for service. Brethren, in the name of Christ we beg you to receive this appeal in a brotherly spirit.'' [161]
It is difficult to understand how ``all discussions of politics'' can be banned from Christian instruction and teaching, as the statement demands, whilst at the same time resisting ``the spirit of mercilessness...''. \textit{63} In this same statement, support was requested for the Refugee Committee, which sought ``to aid refugees... particularly Christians of non-Aryan descent''. We have seen the same trend in Churches in other countries. However, the appeal of the Bishops of Sweden, also in 1938, pleaded for aid to Jewish children and youth in general. This ``Appeal for Help to Jewish Refugees'' was signed by Archbishop Eidem and 12 other Church leaders:
``With deep sorrow and sincere sympathy, we have witnessed the terrible sufferings to which the Jewish people, not least during recent months, have been exposed spiritually as well as physically. The question of the Jewish people has become a question for all mankind. No one can escape responsibility any longer. Our consciences shaken by the suffering of innocent people will not rest until peace and refuge has been provided for the Jewish people. Each one of us must be on his guard against contamination by the plague of racial hatred; we must not betray the Christian commandment of love to every suffering neighbour. May we willingly do our Samaritan service in aiding mercy. The duty and possibility nearest to us is to support Jewish refugees who have had to relinquish home and property. We must hurry to help provide a refuge and a new future for innocent children and youth. Various collections in this respect have already begun. We hereby wish to stress that collections for Jewish children and youth are being mediated by the Swedish Church's Deacon Board. Contributions should be sent to ``Deacon Board, Help for Jewish children, Stockholm 7, Postal Cheque Account No. 155650'.'' [162]
14 HUNGARY
The first anti-Jewish Law, restricting the economic activities of Jews, was enacted in 1938. The representatives of the Churches in the Hungarian Upper House, amongst whom was the Protestant Bishop Ravasz, voted for the passage of this law. [163] \textit{64}
``The only amendment the representatives of the Churches wished to be introduced was that certain modifications should be included for the benefit of the baptized Jews. Apart from that, they took the view that once the Bill had become law 'it would be possible to avoid emphasis being laid on the Jewish question and thus to allay anti-Semitism'. This attitude turned out to be a fatal mistake. It was the stone that started the landslide, and it is all the more regrettable that the Christian Churches lent this Bill their support.'' [164]
Rabbi Fabian Hershkovits (former Chief Rabbi of Budapest, now living in Tel-Aviv, Israel) had the following to say:
``Bishop Ravasz was certainly not an anti-Semite. After the war, in 1947, he was the President of the Council of Christians and Jews of which I also was a member. He and his friends intended, by supporting the anti-Jewish law in 1938, to guard the national Hungarian interest. He did not understand that Europe, after Hitler had come to power, had become a powder-magazine; one should not light a match in a powder-magazine; that was Bishop Ravasz's historical mistake.'' [165]
The fact remains that Protestant Bishops supported an anti-Semitic Law. If this was an error of judgment, it certainly was a fatal error.
In 1939, the Hungarian government introduced a bill for the enactment of the second anti-Jewish Law. The measures included drastic curtailments of personal rights. The representatives of the Churches ``stood solidly against the passage of the bill'' but ultimately ``refrained from voting down the Teleki government,'' that is to say they did not vote against the passage of the Law but tried ``to incorporate such provisions in the law as would insure the greatest possible benefits for particular Jewish categories, the first among these being the Jewish converts to Christianity''. [166] Hilberg comments:
\begin{quotation}
``In waging the struggle for the baptized Jews in the first place, the church\end{quotation}
had implicitly declined to take up the struggle for Jewry as a whole. In insisting that the definition exclude Christians, the church in effect stated the condition upon which it would accept a definition that set aside a group of people for destruction.'' [167]
\textit{65}
15 RUMANIA
We hardly found any statement against anti-Semitism issued by one of the Orthodox Church leaders in Eastern Europe, before the second world war. Rumania was notorious for the strong anti-Semitic influences in that country. The following Declaration, issued on April 15, 1933, by Mgr. Pimem, Metropolitan of Moldavia and Suceava, is the more striking:
``We now are in the Holy Week and for a time we must forget petty affairs and acts of men. Nevertheless I wish to state one thing, namely, that I do not approve of the actions and policies of the Nazis with respect to the Jews of Germany, just as I disapprove of the anti-Christian campaign carried out in Russia. I desire peace for the entire world and on the occasion of this Holy Feast I express my wishes for the health and progress of our people. We should follow but one course: the way of Christ, for only thus can we be led to salvation.'' [168]
16 GREAT BRITAIN AND IRELAND
Many leaders of the Churches in Great Britain publicly protested against the first anti-Jewish measures in Germany. Most of the protests were made by the leaders of the Church of England, though some made by other Churches are also recorded. The Church of England, however, certainly had the widest range of influence in England. I have not recorded all protests that were made. [169] \textit{66}
Already in 1933 the protests were clear and unequivocal, though the Church leaders seemed to be afraid of offending the German Government. The Archbishop of Canterbury said in the House of Lords, on March 30, 1933, in reply to statements made by Lord Reading:
``I feel that it would be a decided omission on my part, were I not to state publicly, in the name of the worthiest citizens of our country, whom I represent here, that I entirely agree with the words just spoken by the Right Hon. Lord Reading, words which touched us all. I sincerely hope that His Majesty's Government will, as I know it hopes to do, be able to assure us that it is doing its utmost to express to the Jewish community the sympathy of this country and of all Christian subjects, --- not least of those amongst us who have a feeling of sincere friendship for the German nation.'' [170]
The Archbishop himself apparently belonged to ``those amongst us who have a feeling of sincere friendship for the German nation''. The Bishop of Ripon addressed the following Message to the International League combating Anti-Semitism and Racialism, on May 1, 1933:
``Most gladly do I avail myself of this opportunity of expressing my sympathy with you and the International League in your struggle against anti-Semitism, on the occasion of the distressing situation created in Germany by the new form of government. It seems almost incredible that such things should happen in the 20th century, and above all in a country like Germany. The leaders of this country, --- of the Church as well as of the State, --- have not left the German government in doubt as to the feelings aroused in us by its policy of cruelty and suicide.'' [171]
On May 5, 1933, the Archbishop of York issued the following Message:
``Racial persecution is an insult to civilization and culture. It is our duty to endeavour to understand the cause and the character of the Nazi revolution in Germany, which has gained the support of a large number of the best citizens of the country. But although it generally happens that understanding produces sympathy, the persecution of Jews, Pacifists and others, such as has so far disgraced the conquests of the Revolution, cannot but alienate all sympathies. It is highly important that the government and leaders of the German nation should realize how great the animosity is which these acts provoke among the best British citizens. \textit{67} Whatever excuses may be made for deeds of violence committed in the course of a revolution, no condemnation can be too severe for the persecution and the organized terror, which undeniably are typical aspects of the recent revolution.'' [172]
No doubt the Nazi revolution in Germany had gained the support of a large number of citizens of that country. That the Archbishop believed that they belonged to the best citizens of Germany, is typical of the atmosphere that reigned in those days. Fortunately, however, ``the best British citizens were provoked by the persecution''.
On May 15, 1933, a Meeting of Protest was held in Birmingham. The Bishop of Birmingham presented the following Resolution:
``This meeting of Christian citizens of Birmingham who are anxious to promote friendly international relations, expresses its profound conviction that the discriminating measures adopted against the Jewish race, both in Germany and elsewhere, are contrary to the spirit and the principles of Christianity. It urges Christian men and women everywhere to exert their influence in order to do away with racial and national prejudice.'' [173]
The resolution was adopted at the close of the Meeting.
On May 31, 1933, the Archbishop of Canterbury addressed a Meeting of Anglican Clergy at Westminster. The English Primate appealed to the German nation:
``to give up, without delay, the racial discrimination which is now being practised. The true strength of a nation and the respect owing to it by other nations lies in the impartial administration of justice to all those who live in its territory''. [174]
On June 27, 1933, the Archbishop of Canterbury addressed a Meeting of Protest, held at Queen's Hall, London:
``We all know that at this very time while we are gathered here in an atmosphere of peace and security, the members of the Jewish community in Germany are being expelled from all public employment, from the posts which they had obtained in virtue of their qualifications, in law, in medicine, and at the universities, and that they are even excluded from concert halls, where music was always considered to be the language common to all mankind. They are being progressively deprived --- even when permission is given to practise their profession or their trade --- of every chance of earning a living... \textit{68} I think with particular indignation of what I have heard concerning the treatment inflicted on Jewish children, who are set apart in schools, separated from other children as though they were unclean. Think of the effect this must produce on such children in whom the feeling is inculcated from their tenderest years that they are not worthy to mix with other Germans! And then picture to yourselves the effect this is bound to have on non-Jewish German children, who are thus taught from their earliest days to despise and look down upon other children. When injustice prevails to such an extent, it is impossible here or in any part of the civilized world, that men for whom justice is a part of the heritage they desire to keep intact should remain silent. They must needs speak, were it only to ease their own conscience.'' [175]
The Archbishop showed a remarkable insight when he expressed his particular indignation about the separation of Jewish children in schools from other children. That was at a time when many Christians and Jews tended to underestimate the malevolent intentions of the rulers of the Third Reich.
Representatives of all religious creeds, responding to an appeal of the United Council of Christian Churches in Ireland (now renamed the Irish Council of Churches) voted for the following Resolution, on the occasion of a public Meeting of Protest, held at Belfast, in May 1933:
``We have met here in order to express our deepest regret that millions of law-abiding citizens who are not guilty of any crime or of any criminal intentions, should have been accused, persecuted and placed beyond the pale of the law, for the sole reason that they belong to the race which was, after all, the source of our European religion, and to which the founder of Christianity belonged. The meeting is horrified at the thought of the sufferings endured and the consequences, which are bound to ensue for Europe and the whole world. The history of the human race, of these islands, and of Ireland herself presents countless examples of the disastrous effects that persecution has had for us, not to mention the repercussion elsewhere. We know the obstacles that intolerance placed in the way of our national development, the harm it has done, the wounds it has inflicted, the hatred it has caused to accumulate in the course of centuries; hatred by which the minds of men are poisoned long after the actual grievances have disappeared. For this reason we deplore this new seed of death, the dire results of which we foresee, not only for Germany, but also for the whole of Europe."[176] \textit{69}
The Church of Scotland is by far the largest Church in Scotland. The General Assembly of the Church of Scotland is the final authority of that Church. It is convened annually in May and attended by about 700 ministers and 700 elders, delegated by the presbyteries of the Church. The following statement was issued by the General Assembly, in May, 1933:
``The General Assembly rejoice that, in this country, the longstanding traditions of friendliness and goodwill to the Jewish people continue to be maintained; they deplore the growth of anti-Semitism in many lands to-day, and, in particular, its recent intensified manifestations in Germany; and they respectfully appeal to the sister German Churches to secure, through their influence with their fellow countrymen and governing authorities, that, notwithstanding the inevitable unsettlement of revolutionary conditions, the suffering of the innocent shall cease, and justice and charity towards all shall prevail.'' [177]
The Church of Scotland apparently was optimistic about the ``influence of the sister German Churches with their fellow-countrymen and governing authorities''. We, who now live after the events, are not astonished that the General Assembly lamented, in 1937, that, ``the protesting voice of the Christian Church has been so barren of results''. [178] The General Assembly of the Church of Scotland was the only ecclesiastical authority, which as far as I know, spoke out against anti-Semitism year after year. The contents of the statements show that it was not an automatic affair, for the changing character of the situation was reflected in these protests. In May, 1934, the following Statement was adopted:
``The General Assembly of the Church of Scotland, in light of the present world situation as concerns the Jewish race, place on record the following expression of their view and convictions. \textit{70} Remembering the age-long sufferings of the Jewish people, their homelessness a nation which has lasted for centuries, the persecutions, injustices and hardships they have endured, from Governments, Churches and individuals; in view also of the present fresh outbreaks of anti-Semitic fanaticism manifested in many lands, the General Assembly offer to the Jewish people their heartfelt sympathy with them in their almost intolerable wrongs. The General Assembly of the Church of Scotland desire to assure the entire Jewish world that ill-treatment of the Jews on account of their race or religion is to them abhorrent; that in their judgment it is a denial of the first principles laid down by the great Founder of the Christian Faith, who places love and kindness to all as fundamental laws of His Kingdom; and that it is their firm belief that any Church which claims to be animated by the spirit of Jesus Christ and which nevertheless acts with intolerance towards members of the Jewish race, is thereby denying the elementary doctrines of the Christian Faith. The General Assembly acknowledge with gratitude to God the great contributions to human knowledge which the Jewish race has made in many realms; in a special degree they express their debt to the Jewish people for the scrupulous care with which they preserved the early documents of Holy Scripture for the ultimate benefit of all nations, which for centuries have nourished the piety of myriads who thereby have learned of the grace of Almighty God. The General Assembly would, in conclusion, again express their sense of the profound significance of the fact that the One whom they rejoice to believe in as the divine Saviour of the world came, according to the flesh, of the Jewish race, and they feel that this thought imparts to the Hebrew nation a special and peculiar position in world history, rendering it a duty on the part of all who love the Lord Jesus Christ to love also the race from which He sprang.'' [179]
It was then moved and resolved that the Assembly send to the Chief Rabbi a message of sympathy. The statement issued in May, 1935, is as follows:
``The General Assembly renew their protest against the anti-Semitic spirit which still prevails in many countries, express their sympathy with the Jews in their sufferings, and urge their faithful people to a greater earnestness in commending the Gospel as the one sure basis of fellowship and peace among all men.'' [180]
Not all statements and protests issued over this period in Great Britain and Ireland can be recorded here, but we mention in conclusion two statements issued by Churches, not yet mentioned. \textit{71}
In April 1933, the following Message was sent by Dr. Scott Midgett, President of the United Methodist Church, to a meeting at the White-chapel Art Gallery:
``All the different branches of the Christian Churches share the Jewish Communities' horror of all deeds of violence against citizens, and especially of such outbursts of violence against any race or class of society. I feel convinced that I am interpreting the feeling of the Methodist Church in stressing our hope that measures will instantly be taken in Germany in order to prevent a recurrence of explosions of this nature in the future.'' [181]
In 1934, the ``Report to the General Assembly of the Presbyterian Church in England'' stated:
``There has unhappily appeared in various parts of the world, notably in Germany, a recondescense of that irrational and wholly unchristian spirit of anti-semitism, which from time to time has disgraced European civilisation. A number of its victims have arrived in our country, and the Archbishops of Great Britain have issued a moving appeal for their relief. But we must do something more. To quote the News Sheet issued by the International Committee for the Christian Approach to the Jews: ``We must play the part of the Good Samaritan''. But that is only one of our objectives. Wise Christian statesmanship demands that in addition to our relief activities, we must also endeavour to eliminate the causes, which create anti-semitism and its victims. Those who are in a position to know, maintain that the outbreak in Germany is sure to spread to other lands. Indeed it has already begun to do so. We know of attempts to foster the spirit in our own country. And there are so-called Christians who attempt to justify it. But note the fact that anti-semitism is essentially anti-Christian. No conscious anti-Semite can do homage to Christ, the Jew.''
The Assembly adopted the following Resolution:
``The Assembly regrets the spirit of anti-semitism now prevalent in Germany and other parts of Europe, and urges its faithful people so to act towards all Jews as to allay the spread of this spirit.'' [182] \textit{72}
* * *
On November 20, 1935, the Bishop of Chichester (Dr. George Bell) moved a resolution in the Church Assembly. [183] The Archbishop of Canterbury (Dr. Lang) had to leave to officiate at the christening of His Majesty's grandson. He asked the Archbishop of York to take his place in the chair. Without a word of explanation, however, his absence might be misunderstood.
``Speaking simply for himself, he felt bound to say that he did most strongly protest against the persecution of the Jews... He was sure that the continuation of the present modes of persecution must seriously affect the good will with which the people of this country desired to regard the German nation.''
The Bishop of Chichester then moved: ``That this Assembly desires to express its sympathy with the Jewish people and those of Jewish origin in the sufferings which are being endured by many of their number in Germany, and trusts that Christian people in this and other countries will exert their influence to make it plain to the rulers of Germany that the continuance of their present policy will arouse widespread indignation and prove a grave obstacle to the promotion of confidence and good will between Germany and other nations''.
He said he moved the resolution with great reluctance, as one who had a profound admiration for Germany, as one who had many friends in that country, and desired the closest co-operation and the firmest mutual understanding between Germany and Great Britain. He was compelled to move his resolution because, as a human being, he saw a wrong done to humanity in one great area of German life and action. As a friend of Germany he saw the hoped-for friendship between two kindred countries tumbling into ruin through the prosecution of a policy against a section of its population, which was unworthy of a great civilized nation. He appealed to the rulers of Germany to desist from a course which shocked Christian opinion in this country in a way to which the nearest analogy was the oppression of the Jews in Russia by the Tsarist Government exactly 30 years ago. The hardships suffered by baptized persons of Jewish origin made a peculiar claim upon their Christian sympathy and compassion. There were two points of attack: \textit{73} the casting out of the Jews from all cultural and professional life, together with the precariousness of their position in business, and the defamation of the Jews throughout Germany. The Nuremberg laws passed last September were supposed to give protection and security within limits to the Jews, yet suffering of individuals increased and the personal attacks grew bolder. No doubt they saw in The Times not so many weeks back that prayer was asked in all German synagogues for protection for the Jews against slander, with the result that the Chief Rabbi suffered imprisonment for one day and other Rabbis suffered punishment. He was sure that great masses of German people themselves abhorred the policy of persecution. They, too, must feel as we felt, that it was a great scar across the fair fame of Germany.
The Bishop of Southwark (Dr. Parsons), in seconding the resolution, said they had hoped that the days of the Ghetto had passed for ever. Now the Jewish people in Germany apparently were being forced back into conditions which reminded them all too vividly of the Ghetto. Their whole position, if it could not be compared with that of slaves, could be compared with that of helots. An article in The Times had described the whole policy as a ``cold pogrom''.
Mr. S. Carlile Davis, the German Vice-Consul at Plymouth, in opposing the resolution, said that every member of the Assembly would agree that they should all express sympathy with those who suffered from persecution, envy, hatred, malice, or any uncharitableness... The Jewish question, so far as it affected Germany, was purely a race question, and it was nothing new in Germany. It was not for us to dictate to any people how they should handle a race question... The Bishop of Durham (Dr. Henson) submitted that they had in the resolution brought before them by the Bishop of Chichester one of those matters which required from them as a great representative Assembly of Christian men a clear pronouncement of their convictions. One thing which they ought to emphasize was the solidarity of civilization... The Jews were just as mixed a race as the Germans --- they could hardly be more. This nonsense about race --- as if there were some poison in the ancestry of Judaism which must be guarded against --- was sheer hallucination and nonsense. We knew in this country that the Jews could be as prominent in good citizenship as any other section of His Majesty's subjects. We, who were the children of Christendom, could not exclude from our minds the vastness of the obligations under which we stood to the Jewish people. Our Divine Lord, according to the flesh, was a Jew. His Apostles were all Jews. The Sacred Book, which we used was a Jewish Book. It was preposterous, base and almost incredibly mean that we, the children of Christendom, should turn on the ancient children of God, to whom religiously, spiritually and morally we owed almost everything we value. \textit{74} ``The least we can do,'' Dr. Henson concluded, ``is to make it clear from our hearts that we loathe and detest this attitude which is obtaining in Germany, and protest against the continuance of this brutal oppression of a small minority of Jewish citizens in Germany.'' (Loud and continued cheers.)... Mr. G.F. Lefroy (Exeter), in opposing the resolution, said that Parliament itself would not dream of passing it. He moved, as an amendment, that only the first portion of the resolution should be moved, confining it to the words ``That the Assembly desires to express its sympathy with the Jewish people and those of Jewish origin in the sufferings which are being endured by many of their numbers in Germany''.
On being put to the vote, Mr. Carlile Davis's motion for the previous question and Mr. Lefroy's amendment were rejected by very large majorities. The Bishop of Chichester's motion was then carried, with few dissentients. [184]
Some of the Bishop of Chichester's words mentioned above could create misunderstanding, for instance, that he ``had a profound admiration for Germany''. Dr. Bell's record regarding the fight against anti-Semitism (as well as in many other respects) is outstanding. [185] One should note the policy of deception practised by the Germans: ``The Nuremberg laws passed last September were supposed to give protection and security within limits to the Jews...''. [186] That seems incredible, and yet it provided a pretext for people who wanted to do nothing. In the discussion on the above mentioned resolution, one Mr. Lefroy, in opposing the resolution, said: ``Parliament itself would not dream of passing it. Therefore, why should the Assembly pass it?'' Apparently it escaped the attention of Mr. Lefroy that a Church Assembly is not a Parliament, and that a Church body often can and ought to say things publicly, even though a Parliament is not prepared to do so, or perhaps for that very reason. However, the Bishop of Durham's speech, in the same meeting of the Church Assembly, is an outstanding example of how a Christian leader could and should speak.
\textit{75}
The Chief Rabbi, Dr. J.H. Hertz, wrote to the Bishop of Chichester:
``Your words will come as a ray of hope to hundreds of thousands whose annihilation seems to have been decided upon by the Nazi rulers.'' [187]
At a meeting of the London Diocesan Conference [188] held in Central Hall, Westminster, in 1936, the following resolution was submitted for discussion by permission of the Bishop of London:
``This Conference, while fully aware of the difficulties that must arise from the presence in certain districts of large populations of people of other religious beliefs and social habits, asserts that the Jew and the Christian are equal children of God, and therefore calls upon all Christians to stand firm against any and every attempt to arouse anti-Semitic feeling for political or any other needs.'' [189]
The Bishop of Chichester was very active in promoting help for Christians of Jewish origin. [190] This subject is, however, beyond the scope of this book. In the summer session of the Church Assembly, in June 1938, Dr. Bell pleaded that the needs of Jews and Christians alike should be remembered. ``The Bishop of Chichester moved: That this Assembly records its deep distress at the sufferings endured by `non-Aryan' Christians, as well as by members of the Jewish race, in Germany and Austria, and urges that not only should everything possible be done by Government aid to assist their emigration into other countries but also that Christians everywhere should express their fellowship with their suffering brethren by material gifts as well as by personal sympathy and by prayer.''
He said he did not want to speak of political matters in a country with which they desired to be friends, nor to attack the leadership of the great German State. He asked the Assembly not to make any protest against a system, but to record its deep distress at the suffering of Christians and Jews... \textit{76} What could members do? First of all they must not forget it, but let it be printed on their memory and never rest while the distress was unhealed. They must remember the needs of Jews and Christians alike. It was wrong to separate the Jews and leave the Jews to the Jews and the Christians to the Christians. They both made a deep appeal by their sufferings to all humanity and above all to the Christian Church.'... First of all they could pray for the sufferers; prayer from the heart availed and was a great bond of fellowship. Next they could feel deeply for and with them until something was done. Thirdly there was material help... He asked for their (the Assembly's) help and for the help of their constituents all over England and he asked for the awakening of conscience. They would not forget and he could not forget that their Master was a Jew, a non-Aryan. They thought in their hearts that if they saw their Master in sorrow they would wish to help him, but it was right to remember the parable that their Master uttered of judgment and what He said when He rebuked certain disciples: 'For I was an hungered and ye gave me no meat: I was thirsty and ye gave me no drink: I was a stranger and ye took me not in: naked and ye clothed me not: sick and in prison and ye visited me not.' When the disciples in defending themselves asked what he meant, the Master added: 'Verily I say unto you, in as much as ye did it not to one of the least of these, ye did it not to me'. He was convinced that their attitude in England and in the Church of England to the needs of those suffering non-Aryan Christians and members of the Jewish race was the test of their attitude to their Master himself. It was because of that that he felt so deeply and that he asked them to give their prayers and sympathy and their material help.
The motion was carried. [191]
The Bishop of Chichester followed this move with a plea for more vigorous Government action in his maiden speech in the House of Lords, on July 27, 1938. He began with a strong condemnation of the Nazi persecution:
``I cannot understand --- and I know many Germans --- how our own kinsmen of the German race can lower themselves to such a level of dishonour and cowardice as to attack defenceless people in the way that the National-Socialists have attacked the non-Aryans. \textit{77}
He then pleaded with the Government to follow up the initiative of President Roosevelt by increasing its facilities for training younger refugees in Great Britain, by providing greater scope for settlement in the Colonies, and by persuading the Dominions to open their doors more widely. The Under-Secretary of State for Foreign Affairs assured him that the Government would do what it could. But Dr. Bell remarked a few weeks later in his Diocesan Gazette:
``It is almost as hard to understand the seeming apathy with which the fate of the Jews and the non-Aryan Christians is being regarded by the people of the British Empire... These non-Aryans can no longer be called `refugees' for they have as yet no country of refuge. We emphasize the responsibility of the British Empire in this connection, because the British Colonies and the British Dominions cover the larger part of the whole available globe. It seems to us impossible, both on the grounds of charity and on the grounds of statesmanship, that the doors can remain forever shut.'' [192]
Resolutions adopted by the Presbyterian Church of England exposed the danger of anti-Semitism existing in England in those days. In 1937, the General Assembly stated:
``The Assembly notes with concern the attempts which have been made to create racial antipathy against the Jews, with whom the Assembly expresses its sympathy. The Assembly expresses its conviction, that in a nation professing Christianity, no discrimination on grounds of race must be recognised. The Assembly urges that the freedom accorded by law in this country to citizens of any faith to live in peace and pursue their lawful callings shall be specially safeguarded. The Assembly resolves to send a copy of this resolution to the Board of Deputies of British Jews, and to the Home Secretary.'' [193]
In May, 1938, the General Assembly adopted the following Resolution:
``The Assembly urges its faithful people to encourage every effort to overcome the evil spirit of anti-Semitism which thing we hate.'' \textit{78}
There was hesitancy in the minds of some about the word `hate', when the Convener moved this resolution, but the Assembly overwhelmingly approved of it. [194]
The General Assembly of the Church of Scotland certainly did not mince words. It declared in 1936:
``The General Assembly learn with profound regret that the past year has brought no alleviation of the sufferings caused to the Jewish people by the inhuman political, social and economic persecutions prevalent in Central and Eastern Europe. They protest against the religious intolerance, the narrow nationalism and race-pride on which anti-semitic hatreds are based. They call on the Christian people of Scotland, in loyalty to the law of Christ and their own high traditions of liberty and toleration, to rid their minds of all narrow anti-Jewish prejudice, and to broaden out their obedience to the Gospel ever commanding peace and goodwill to all men. The General Assembly again commend to the liberality of their faithful people appeals made on behalf of refugee Jews from Germany and other lands, specially remembering the Christians of Jewish race who are involved in the terrors of persecution.'' [195]
In 1937, the General Assembly declared:
``The General Assembly renew in Christ's name their condemnation of the unabated brutality still being dealt to the Jewish minorities in Central and Eastern Europe, and lament that the protesting voice of the Christian Church has been so barren of result. They deprecate the attempts in certain parts of England to create antipathy against the Jews.'' [196]
The statement adopted in May 1938, reads as follows:
``The General Assembly renew their protest against the virulence and cruelty of the attacks still being directed against helpless Jewish minorities in Central and Eastern Europe, and they affirm that no Church can be truly Christian and anti-semitic at one and the same time.'' [197]
* * *
The first reaction to the horrors of the ``Crystal Night'' pogroms was a letter of the Archbishop of Canterbury to ``The Times'':
``I believe that I speak for the Christian people of this country in giving immediate expression to the feelings of indignation with which we have read of the deeds of cruelty and destruction which were perpetrated last Thursday in Germany and Austria. \textit{79} Whatever provocation may have been given by the deplorable act of a single irresponsible Jewish youth, reprisals on such a scale, so fierce, cruel and vindictive, cannot possibly be justified. A sinister significance is added to them by the fact that the police seem either to have acquiesced in them or to have been powerless to restrain them. it is most distasteful to write these words just when there is in this country a general desire to be on friendly terms with the German nation. But there are times when the mere instincts of humanity make silence impossible. Would that the rulers of the Reich could realize that such excesses of hatred and malice put upon the friendship which we are ready to offer them an almost intolerable strain. I trust that in our churches on Sunday and thereafter remembrance may be made in our prayers of those who have suffered this fresh onset of persecution and whose future seems to be so dark and hopeless.'' [198]
The Archbishop's letter expressed ``feelings of indignation'', but also reflected the spirit of appeasement: the British Prime Minister Neville Chamberlain had signed the Munich agreement with Hitler, only six weeks before.
On November 16, 1938, during the Autumn Session of the Church Assembly, the Bishop of Chichester pleaded that help should be given to Christian refugees of Jewish origin. In January 1939, he was to urge ``to aid the entire mass of non-Aryans''. Now the tendency still was to stress the help to Christians of Jewish origin, not to the Jews in general. There was one notable exception, in which Jews and Christians jointly took action, without asking themselves whether the persons to be helped were Jews or Christians. Lord Gorell was asked by the Archbishop of Canterbury to be joint Chairman (with Lord Samuel) of the ``Movement for the Care of Children from Germany'', in February 1938. This movement succeeded in bringing over 9,354 children from Germany to England. Roughly nine-tenths were Jewish, and one-tenth Christian children.
\textit{80}
``Where a Jewish child was received in a Christian home --- which occurred frequently --- it was prescribed by the Movement, and accepted by the foster-parents, that there should be no attempt to proselytise. The nearest Rabbi, or Jewish teacher, was put in touch with the child, and if personal contact was not possible, instruction was arranged by correspondence. The last transports of the children from Germany reached England a few days after the outbreak of the war.'' [201]
A Joint Statement was issued by British Church leaders, in April 1939:
``In making the following statement, we, the undersigned, --- the Archbishop of York; Dr. Jas. Black, Moderator of the Church of Scotland; the Bishop of Edinburgh; Dr. S.M. Berry, Congregational Union of England and Wales and Federal Council of Free Churches; the Rev. M.E. Aubrey, Baptist Union of Great Britain and Ireland, --- feel that we are giving expression to the convictions of a large number of Christians in Great Britain: 1. We believe that the following is an essential and basic principle of all
true civilization: Religious freedom, freedom of opinion and action in
accordance with religious beliefs, provided that social order is in no
way endangered thereby; legal equality for all, independently of social
position or race...'' [202]
In November, 1938, the Moderator of the Church of Scotland wrote a letter to the Chief Rabbi of the British Empire, who replied as follows:
London, 24th Nov. 1938/5699. Dear Dr. Black, ``I am indeed touched by your letter of the 18th inst. conveying to me on behalf of the General Assembly of the Church of Scotland, the deep horror of the suffering inflicted on the Jewish people throughout Europe. In the agony through which hundreds of thousands of my coreligionists are now passing, it is fortifying to read your strong repudiation of all persecution as unchristian, inhuman and pagan; and to learn that the love of God, love of fellow-man, and love of freedom rule with undiminished strength in little, but great Scotland. I should be glad if you would kindly convey to the General Assembly the deep felt thanks of my community for their kind expression of Christian sympathy with the suffering of Israel.
The General Assembly commented: \textit{81}
``It is now the duty of the Church to contrive that the wave of sympathy shall not ebb, but, while it is on the flow, shall be turned into the only channel, which, as we believe, reaches the heart of the Jewish problem. The immediate duty, however, is to direct sympathy towards practical and generous action with regard to the gigantic Refugee problem which confronts the free peoples of the world...'' [203]
The following statement was issued by the Conference of the Methodist Church in Ireland, in June 1939:
``The Conference notes with grave concern the growth of anti-Semitism in Europe and America, and expresses its profound conviction that this tendency is directly contrary to the spirit of Christianity. It views with horror the treatment now being meted out to men, women and children in Germany on purely racial grounds, and regards with apprehension the possibility of the spread of such policy to other countries. It commands to the sacrificial sympathy of the Church, the efforts being made on behalf of non-Aryan Refugees both in Eire and in Northern Ireland, and suggests that they offer a most effective method of bearing Christian testimony against the terrible divisions of the present hour.'' [204]
17 THE UNITED STATES
Protestant Churches in America have protested against racial discrimination in general. We only record, however, the resolutions and statements, which expressly denounced anti-Semitism.
On March 22, 1933, American Christian clergymen and laymen appealed to the German people to put an end to the persecution of Jews. They urged preachers throughout the United States to rally their congregations on the following Sunday for a united stand against Hitlerism. The summons to the Churches was sponsored by the Interfaith Committee and signed by Bishop Manning (Episcopalian), Mr. Al Smith, the former Governor of New York State (a Roman Catholic), and others equally prominent. [205] \textit{82}
On March 28, 1933, a mass meeting was held in New York, Madison Square Garden, attended by 20,000 persons, as a protest against anti-Semitic activities in Germany. 38,000 swarmed round the building to hear the voice of speakers brought to them through amplifiers. The meeting followed a day of fasting and prayer with similar protests being staged in 300 other cities. Former Governor Alfred Smith, Bishop William T. Manning, and Senator Robert F. Wagner were among the speakers. [206]
On May 26, 1933, a Manifesto signed by 1200 Protestant ministers from 42 States of the United States and Canada was published:
``We Christian ministers are greatly distressed at the situation of our Jewish brethren in Germany. In order to leave no room for doubt as to our feelings on this subject, we consider it an imperative duty to raise our voices in indignant and sorrowful protest against the pitiless persecution to which the Jews are subjected under Hitler's rule. We realize full well that there are religious and racial prejudices in America, against which we have repeatedly protested and for this very reason we all the more deeply deplore the retrogression which has supervened in Germany where so much had been achieved while we in America were still fighting for human rights. For many weeks we have waited, refusing to believe all the reports concerning a State policy against the Jews. But now that we possess the irrefutable testimony of facts, we can no longer remain silent. Hitler had long vowed implacable hatred against the Jews. One of the fundamental Nazi doctrines is that Jews are poisonous germs in German blood and must therefore be treated as a scourge. Hitler's followers now apply this doctrine. They systematically pursue a `Cold Pogrom' of inconceivable cruelty against our Jewish brethren, dismissing them from important positions they had occupied, depriving them of civil and economic rights, and deliberately condemning those who survive to a life without legal protection, --- as outcasts, threatening them with massacre should they make the slightest protest. We are convinced that the efforts made by Nazis to humiliate an entire section of the human family, are liable to cast the civilized world back into the clutches of mediaeval barbarism. We deplore the consequences which may ensue for the Jews and also for Christianity which tolerates this barbarous persecution, and, more particularly, for Germany herself. We are convinced that in thus protesting against Hitler's cruel anti-Semitism we are acting as sincere friends of the German nation.'' [207] \textit{83}
Speaking of their ``Jewish brethren in Germany'', those 1200 Protestant ministers apparently had in mind the Jews of Germany in general, not just the Christians of Jewish origin.
* * *
The next statement to be recorded in this chapter was issued by the Federal Council of the Churches of Christ in America. This organization represented the great majority of American Protestants. The total membership of Churches affiliated with it was, in 1941-1942: 25,551.560. The Executive Committee of the Federal Council published the following statement in November 1935:
``At a recent meeting of protest against the treatment at present inflicted on Jews in Germany, the Assembly of the Church of England expressed the hope that other Christian bodies would join in this protest. We feel constrained to do so. We are members of churches which have numerous and close bonds of union with the German church. We recognise our indebtedness to the great German preachers and teachers of Christianity, who have done so much to enrich our common heritage from the days of Luther to the present day. After the last war we protested strongly against the limitations to which Germany was subjected by the Treaty of Versailles and made constant efforts for their suppression. For this very reason we consider it our duty to speak equally freely now that Germany is pursuing a policy, which threatens her with moral isolation. We protest against this policy because the treatment of the Jews is unworthy of a great nation. To treat a considerable part of the population as being essentially inferior for racial reasons only, and to impose restrictions on the normal life of persons whose families have lived in Germany for generations, and who have rendered eminent services in the realms of education, art, and government, is to violate the codes of honour and good faith which are the common property of civilized nations. But our reason for protesting goes far deeper. We protest against this policy because the philosophy on which it is based is a heathen philosophy. Founded on a religious interpretation of race, the actual treatment inflicted on the Jews raises far greater problems than any former persecutions of Jews and other minorities, which were founded on political and incidental considerations. It is an attempt of a tribal heathen movement, based on race, blood, and soil, to separate Christianity from its historical origin and a Christian nation from its religious past. All the different branches of the Christian Church are, therefore, in duty bound to protest, not only in the name of the human brotherhood, but also in the name of our Christian faith. [208] \textit{84}
The meeting of protest mentioned at the beginning of this statement was held on November 20, 1935. [209] The response of the Federal Council came very promptly indeed. International contacts between Churches were a factor the importance of which can hardly be overestimated.
Dr. Charles S. Macfarland, the then General Secretary of the Federal Council, had had a personal interview with Hitler in the autumn of 1933. Before accepting Hitler's invitation to call, he was warned that no one was even permitted to mention the Jewish issue to him. Dr. Macfarland, however, had made it clear that he was not going there to discuss Tennyson or Browning and that he would have to be permitted to choose his own subjects. Word came that ``His Excellency desired me to talk freely with him''. Dr. Macfarland relates:
``I told Herr Hitler that, in my judgment, the German Evangelical Church could not and would not yield itself to his polito-social theory, including his so-called Aryan laws, and that if it did, it would not only cut itself off from the Christian churches of the world, but would cease to be Christian...'' [209]
Dr. Macfarland followed up this conversation by correspondence. In one letter he wrote that the near complete hostility of the American people was deeply ethical in nature and could be modified only by two processes:
1. ``A constructive measure of justice in dealing with the Jews in Germany,
stopping all continuation of the boycott, conferring with leading Jews of
high character, and, while still recognizing the social problem involved,
endeavouring to secure needed readjustments by friendly measures and,
above all, restoring neighbourly good feeling between Jewish rabbis and
Christian pastors and among Jews and non-Jews who live side by side...
I also hope that, by a final settlement of the Jewish problem which will do
full justice, this barrier between the German people and the peoples of the
world may be removed.'' [210] \textit{85}
Apparently Hitler did not underestimate the influence of the American Churches: he replied to Dr. Macfarland's letters, stating that he wished ``to promote the unity of the Church'', that he accepted one of these letters ``in the same spirit in which it was written'' and that he thanked Dr. Macfarland for his ``candid and sympathetic appeal''. [211] On June 2, 1937, however, Dr. Macfarland published an open letter to Hitler, from which we quote the following:
``You especially demarcated the church's ``confession'' as a sacred ground on which the State could not and would not intrude, and I handed you a memorandum calling attention to the fact that by that confession the church was supernatural, supernational and superracial and that the so-called 'Aryan paragraph' cut right across the confession; that if the church accepted it, it would make a breach between the church in Germany and the 'positive Christianity' for which you declared you stood. As previously mentioned, you replied to later correspondence that you accepted my appeal `in the spirit in which it was given'. That appeal was for a constructive measure of justice in dealing with the Jews in Germany, stopping all continuation of the boycott, conferring with leading Jews of high character and, while still recognizing the social problem involved, endeavouring to secure needed readjustments by friendly measures and, above all, restoring neighbourly good feeling between Jewish rabbis and Christian pastors and among Jews and non-Jews who live side by side'. And I added: 'I hope that this barrier between the German people and the peoples of the world may be removed'... What now are the results of my continued study and how do they appear in the light of your earnest assurances?... Instead of doing justice to the Jews, you have permitted them to be harassed and despoiled. Your treatment of them has been ruthless, without the slightest appearance of mercy, even reminding one of the infamous edict of Herod in stretching the hand of violence to the littlest child. Your attitude toward the little handful of Jews in Germany and your so-called Aryan and Nordic ideas have had no little effect in confusing members of the Evangelical Church, so that, in this way, you divided instead of fulfilling 'the desire you expressed to me of uniting the church. You undermined the most basic ideal of Christianity, on which unity alone could be secured... I have been reading a paper called Der Stuermer. Not only does it explicitly teach and urge hate-hate-hate, but does it in forms whose viciousness never would be believed by one who had not seen it. The language in this paper is too vile for repetition, and its falsehoods are obvious to any ordinarily informed person who knows Germany. The best that one can say of the illustrations is to hope that they emanate from a disordered, rather than a depraved mind...'' [212] \textit{86}
I think that, if Dr. Macfarland had been a citizen of my country (the Netherlands), legal proceedings might have been instituted against him in those days, for ``public offence to the Head of a friendly State''.
The Home Missions Council, early in December 1937, issued a special Christmas message concerning Jewish and Christian relations which it addressed to all Christians of North America. We quote the following from this message:
``As Christians of the United States and Canada we desire to express to those Jews who are the victims of injustice and abuse our sincere sympathy, and we emphatically declare that such conduct is utterly alien to the teaching and spirit of the faith we profess and an affront to all our ideals of civil liberty and justice.'' [213]
* * *
The Executive Committee of the Federal Council of Churches of Christ in America proposed to set aside November 20, 1938, as ``the occasion when prayer will be sought in the United States for refugees, both Christian and Jewish''. [214] The officials of both the Roman Catholic Church and Jewish Organizations, following the example set by the Federal Council, designated the same date for a period of prayer and intercession. The Governors of about a score of States issued statements or proclamations urging citizens to repair to their places of worship on that day for united prayer for the suffering. The day of prayer was widely observed in all parts of the country and in all the churches. [215] The Executive of the Federal Council had issued ``an appeal to all church people to respond generously to the efforts for the relief of refugees as carried on by the American Committee for Christian German Refugees and also by the Catholic and Jewish organizations''. [216]
When the first reports of the new measures of oppression and persecution of the Jews in Germany appeared in the press, the Federal Council's office invited outstanding Christians, both ministers and laymen, to express their views and give wide publicity to them. \textit{87}
Among the lay voices, which were most widely heard across the nation was that of Honourable Herbert Hoover, who, in a message telegraphed to the Federal Council, gave expression to the sympathy of all thoughtful Christian people. A statement of Dr. Edgar De Witt Jones of Detroit, President of the Federal Council, was also quoted in all parts of the country. [217]
On the evening of November 13, 1938, the Federal Council of Churches sponsored a national broadcast over the Columbia Broadcasting System in which Christian sympathy was again expressed and carried to every part of the nation. There also was a national broadcast under the auspices of the National Conference of Jews and Christians, on November 20, 1938. [218]
On January 9, 1939, a petition on behalf of German refugee children was left for President Roosevelt at the White House by a deputation of clergymen. The petition was signed by leaders of the Catholic and Protestant Churches. It read as follows:
``The American people has made clear its reaction to the oppression of all minority groups, religious and racial, throughout Germany. It has been especially moved by the plight of the children. Every heart has been touched, and the nation has spoken out its sorrow and dismay through the voices of its statesmen, teachers and religious leaders. Americans have felt that protest, however vigorous and sympathy, however deep, are not enough, and that these must translate themselves into such action as shall justify faith. We have been stirred by the knowledge that Holland and England have opened their doors and their homes to many of these children. We conceive it to be our duty, in the name of the American tradition and the religious spirit common to our nation to urge the people, by its Congress and Executive, to express sympathy through special treatment of the young, robbed of country, homes and parents. A heartening token of the mood of America is to be found in the fact that thousands of Americans of all faiths have made known their eagerness to take these young children into their homes, without burden or obligation to the State. Working within and under the laws of Congress, through special enactment if necessary, the nation can offer sanctuary to a part of these children by united expression of its will to help. \textit{88} To us it seems that the duty of Americans in dealing with the youthful victims of a regime which punishes innocent and tender children as if they were offenders, is to remember the admonition of Him who said, 'Suffer little children to come unto me'. And in that spirit we call on all Americans to join together without regard to race, religion or creed in offering refuge to children as a token of our sympathy and as a symbol of our faith in the ideals of human brotherhood.'' [220]
Senator Robert F. Wagner, attempting to implement the clergymen's proposal, introduced a resolution in the Senate. Known as the Child Refugee Bill, it proposed that a maximum of ten thousand children under the age of fourteen be admitted in 1939, and a similar number in 1940. Their entry would be considered apart from and in addition to the regular German quota. [221] The Executive of the Federal Council supported the Bill:
``In the extraordinary circumstances which have created the problem of Jewish and Christian refugees from Germany, we feel that it is not enough to call upon other nations to help or to voice our protests but some such practical step as the one here contemplated is imperative and will do much to facilitate a larger approach to the problem of which it is but one part.'' [222]
On July 1, 1939, the proposed Bill was modified: the twenty thousand childrens' visas would be issued against the German quota, not in excess of them. Senator Wagner, realizing that the twenty thousand children's visas might become twenty thousand death warrants for adults they would replace, withdrew his proposal. [223] \textit{89}
In March 1939, the Federal Council urged the United States to continue to provide asylum for refugees of other countries in the face of any legislative proposals to suspend immigration or curtail existing quotas. Declaring that the Churches were deeply concerned with the refugee problem and that ``as Christians we have responsibility for suffering human beings as children of our common Father wherever they may be'', the Council said:
``We, therefore, urge our government to maintain its historic policy of friendliness to refugees. We oppose legislative proposals, which would suspend immigration at this time or curtail the established quotas.''
In its objection to any change in the immigration policy the Council pointed out that refugees ``would be consumers as well as producers'' and added:
``However, even if they were not an economic asset as well as a liability, we would still have a Christian responsibility to them.'' [224]
In April 1939, the National Council of the Protestant Episcopal Church issued the following Resolution on behalf of aid to refugees:
``In view of the persecution of minorities now taking place in Europe, we, as Christians and members of the National Council of the Protestant Episcopal Church, and in keeping with the traditional spirit of our country, reaffirm our conviction that the United States should continue to show its spint of generosity and hospitality in opening its doors to afflicted people. We commend the program, as prepared by the Episcopal Committee on German Refugees, to the interest and support of all members of the Church, reminding ourselves of our Lord's admonition: 'in as much as ye have done it unto the least of these my brethren, ye have done it unto me'.''
The program prepared by the Committee on German Refugees called for co-operation with local refugee committees in helping to obtain employment, in placing children in homes and in obtaining affidavits of support for individual immigrants. [225] To the best of my knowledge, there is no other country in which Churches and Church leaders in those days so unequivocally demanded asylum for the refugees.
So far we have recorded actions and statements on behalf of the refugees only. The following statements also denounced anti-Semitism in Germany and/or in the United States. \textit{90}
The Presbyterian Church in the U.S.A. stated, at the end of 1938:
``... We are deeply shocked at the continuance of persecutions based on race in Germany, Austria, Rumania and other nations. We sympathize with our Jewish brethren in the United States, many of whose relatives are the innocent victims of fanatical hatred abroad. We commend the National Conference of Jews and Christians for all its labour to the end that race murders and race discriminations shall not happen here...'' [226]
In its Bulletin (February, 1939) the Federal Council of the Churches of Christ published the following article:
The Christian Attitude towards Anti-Semitism
Every thoughtful Christian must gratefully acknowledge his spiritual indebtedness to the Hebrews. We Christians have inherited the ethical and religious insights of Israel. We hold them with a difference --- at one point with a momentous difference --- but we can never forget that the historic roots of our faith are in the Hebrew people. From Israel we inherit the Ten Commandments, which are still our basic moral standards. From Israel we inherit the priceless treasure of the Psalms, which are an essential part of Christian worship around the world. From Israel we inherit the vision of social justice which has come to us through Amos and Isaiah and Micah. From Israel we inherit even our own unique Christian classic, the New Testament, nearly all of which (if not all) was written by Jews. A Christian who faces the modem world must also be conscious of a present spiritual kinship with his Jewish neighbours to whom their religious heritage is still a vital force. That kinship is grounded in our common faith in the ultimate spiritual foundations of the universe. Over against those who adhere to a materialistic philosophy of life and a mechanistic conception of human destiny, we recognize ourselves as at one with the Jews in the first sublime affirmation of the Pentateuch: `In the beginning God'. Over against current disillusionment and despair Christian and Hebrew stand together in their belief in the one Holy God Who is the Creator of all and whose righteous will gives meaning and direction to life. A Christian who knows anything of history must also speak a word of confession. For he cannot help recalling how grievously the Jewish people have suffered at the hands of men who called themselves Christians. The record of the treatment of Jews in Europe through long centuries is one which Christians of to-day view with penitence and sorrow. One has also regretfully to admit that the day of cruel treatment of the Jews by some who call themselves Christians is not yet a thing of the past. Even in our own country there are misguided groups which circulate statements that spread a poison of mistrust and hate which is antithetical to the true genius both of America and of the Christian religion. Anti-Semitism is inherently un-Christian, contrary to the plain teaching and spirit of our Lord, and it can be asserted with confidence that an intolerant attitude towards the Jews is opposed by the great body of American Christians... \textit{91} But everything which has happened since shows that what started as a movement against the Jews turns out to be a movement against Christianity also... [227]
In May 1939, the Commissioners of the General Assembly of the Presbyterian Church in the U.S.A. stated:
``... We confess the sins of our country in this respect. We condemn the attacks on Jews and Christians and other minority groups throughout the world. We would be lacking in a sense of common morality and decency if we did not express our strong disapproval of such an outrageous assault by any government upon an innocent and defenceless people. We urge our government to continue its efforts to make generous arrangement for the settlement of refugees, so continuing our national tradition of being an asylum for the oppressed of all the nations.'' [228]
The General Synod of the Reformed Church in America, attended by 200 pastors and delegates, adopted (June, 1939) the social welfare report which said in part:
``The failure of the Church to recognize the Jew has behind it a record of misunderstanding, intolerance and spiritual malpractice that has been unequaled in dealing with any other people. Even America is not free from the blight of anti-Jewish prejudice. Both Jew and Gentile are responsible for existing conditions and both must co-operate for their betterment. Christians must rebuke all anti-Semitism... Third, in reference to the refugee problem, a linking up of our efforts and agencies with all others in more adequately caring for those who are so greatly in need. Fourth, a wholehearted endorsement of the legislation permitting 10,000 children (refugees) to be received each year for two years.'' [229]
18 INTERNATIONAL ORGANIZATIONS OF CHURCHES
The World Alliance for International Friendship through the Churches was constituted at Constance, in 1914, at the eve of the first world war. Its supreme body, the International Council, was composed of some 145 members appointed by the various National Councils. \textit{92} In some countries, especially on the continent of Europe, the National Councils worked in close relation with ecclesiastical authorities; in most areas, however, they remained entirely independent agencies, based on the personal adhesion of their members. The Executive Council of the ``World Alliance'', at its meeting in Sofia, 1933, unanimously adopted the following Resolution:
``... We especially deplore the fact that the State measures against the Jews in Germany have had such an effect on public opinion that in some circles the Jewish race is considered a race of inferior status. We protest against the resolution of the Prussian General Synod and other Synods [230] which apply the Aryan paragraph of the State to the Church, putting serious disabilities upon ministers and church officers who by chance of birth are non-Aryans, which we believe to be a denial of the explicit teaching and spirit of the Gospel of Jesus Christ.'' [231]
The International Council of the ``World Alliance'', at its meeting in Chamby (August, 1935), adopted the following Resolution:
``In view of the pitiable situation of refugees and stateless persons in Europe, having regard to the policy of expulsion which is being pursued by the majority of the European States, to the inadequacy of the measures for providing refugees with valid identification papers and residence and labour permits, and recognising the fact that a turn for the better cannot be attained by legislation undertaken by individual States but only on the basis of international agreements, the World Alliance most warmly welcomes the initiative taken by the Norwegian Government which, in the spirit of Fritjof Nansen, has proposed to place the situation of the refugees upon the agenda of the next plenary assembly of the League of Nations. It expresses the hope that in this way it will be possible to secure for refugees and stateless persons a minimum of individual rights and, by the setting up of a central organisation for refugees, within the framework of the League of Nations, to provide a basis for the settlement of the problem. In order to make this resolution effective, the World Alliance resolves: a. to bring the text of this resolution of the Norwegian Government to the
knowledge of the General Secretary of the League of Nations and of all
States members of the League of Nations;
b. to request the Churches and organisations affiliated to the World
Alliance in the different countries to make representations to their
governments in the spirit of the resolution before the next meeting of
the League of Nations in order to obtain the support of these governments
for the Norwegian initiative.'' [232] \textit{93}
Another International Organization of Churches, more influential than the ``World Alliance'', was the Ecumenical Council for Life and Work, which had its first world conference in 1925, in Stockholm, and its second in 1937, in Oxford. Its purpose was ``to stimulate Christian action in society''. Its President, Dr. George Bell (Bishop of Chichester) wrote a letter to Dr. Kapler, President of the Federation of Protestant Churches in Germany, dated May 17, 1933:
``... We do not wish to enter into political questions, nor indeed is it our business to do so. At the same time it would not be fair to disguise from our friends in Germany that certain recent events, especially the action taken against the Jews, have caused and continue to cause us anxiety and distress; and we feel that we ought to share our concern with you here...'' [233]
The annual meeting of Life and Work was held at Novi Sad, in Yugoslavia, on 9-12 September, 1933. A German delegation under the leadership of Dr. Heckel, who supported Hitler's policy, was present at the meeting. The minutes record that representatives of other Churches had expressed grave anxiety over the severe action taken against people of Jewish origin. [234] Bishop Bell proposed that, in addition to this, he should write a letter to the leaders of the German Church. This proposal was adopted unanimously. Only Dr. Heckel abstained from voting. Bishop Bell wrote this letter to the German Reich Bishop Mueller, on October 23, 1933. He referred to two features, which were gravely disturbing to the Christian conscience, namely, the adoption of the Aryan Paragraph by the Prussian Church Synod [235] and certain other Synods, and the forcible suppression of minority opinion. Mueller's reply of 8 December was intended to be reassuring. The enactment of the Aryan Paragraph had been stopped, and he hoped for an opportunity when they might discuss together the problems of race, the state, and international order. [236] The Executive Committee of the Ecumenical Council of Life and Work at Novi Sad issued the following ``Appeal on Behalf of German Refugees'' in November, 1933: \textit{94}
``A new appeal is hereby addressed to Christians, at this Christmastide. It is an appeal to help those who are suffering because there is no place for them in Germany: Jews, Christians of Jewish origin and political refugees. They are dispersed in Palestine and in different lands of Europe. They are in a deplorable situation and a great number of them are destitute... The gifts of the Churches will constitute a welcome proof of that truly ecumenical and Christian spirit which, beyond all differences of race and class, regards every man as a brother.'' George Cicestr, President of the Ecumenical Council for Life and Work; Germanos, Archbishop of Thyatira, Co-President; W.A. Brown, President of the Administrative Committee; Waldemar Ammundsen, Interim President of the European Section; Wilfred Monod, Vice President. [237]
* * *
The International Missionary Council was organized in 1921, to co-ordinate missionary work throughout the world. Its ``Committee on the Christian Approach to the Jews'' met at Vienna, 28 June-2 July, 1937. A report of the Subcommittee on Anti-Semitism and the Church was submitted, and adopted in the following form:
``We desire to record our conviction that in contemporary anti-Semitism we face an extraordinary menace against which all Christians must be warned. All forms of hatred and persecution must be deplored by Christians, and their victims must be succoured; but there exists to-day a type of racial anti-Semitic propaganda inspired by hatred of everything springing from Jewish sources; and this creates more crucial issues for Christianity than ordinary outbursts of race feeling. Christian Churches must be warned that they cannot be silent in the presence of this propaganda, still less connive at or participate in the extension of its errors and falsehoods, without betraying Christ, undermining the basis of the Church, and incurring the most severe judgment of God. The Christian Church must let no doubt about this attitude prevail in the eyes of the world. Realizing that enmity to the Jews has now become a cloak for the forces of anti-Christ, and conceals hatred for Christ and His Gospel, the Christian Church must reject anti-Semitism with complete conviction. \textit{95} To realize its true nature and to vindicate its right to the title of the `Body of Christ', the Church must preach the Gospel and open its fellowship to men of all race, including the Jews. Our mission to the Jews cannot consistently be carried out without at the same time combating anti-Semitism among Christians, and giving more tangible evidence than has been given of our sympathy with Jews and Hebrew Christians in their present distress. Anti-Semitism can and should be combated systematically: 1. By suitable literature, capable of influencing specially wide classes,
also by sustained treatment in Christian Reviews and newspapers.
2. By occasional conversations, discussions, and lectures, on the destiny
and the hope of the people of Israel.
3. By sincere and friendly discussion between Jews and Christians. 4. By the realization among Christians of the treasures committed to them
(Christianizing of Christians).'' [238]
The same Committee submitted the following resolution to the Oxford Conference, in 1937:
``The International Committee on the Christian Approach to the Jews desires to lay before the Oxford Conference on Church and Community and State the problem of Anti-Semitism. The fact of Anti-Semitism is proved, by the ample material in the possession of the Committee to be of growing importance and menace in the world. It constitutes one of the principal denials of modern life of the Christian doctrine of man. It is an attack upon the unity of the Una Sancta, it is even a denial of the person of Christ Himself. It has been largely instrumental in aggravating existing economic and social strains until they have become intolerable. The human misery created, maintained and at the same time concealed by the influence of Anti-Semitism is difficult to estimate. Graver, however, than the volume of human misery is the poisoning of the spirit, the drying up of sympathy and the warping of judgment caused by the influence of Anti-Semitism, especially among the young. Deepest of all is the denial which Anti-Semitism offers to the Unity of the Church, and to the meaning of the Person of Christ Himself. The Committee would further ask the Conference to consider the terrible fact that this problem is not, like many on the Conference will consider, that of an influence external to the Christian Church with which it must make its account, but also of an evil within the Church. Anti-Semitism antedates Christianity and it is not suggested that it is a purely Christian phenomenon, but it is aided by false Christian teaching and it results in the appalling situation, present in several countries where Christian Churches are reluctant, or frankly refuse, to receive a Jewish convert. \textit{96} It is plain that where racial and physical conditions of church membership override the conversion of heart and will, the Christian religion has ceased to exist except in a vain form. But this devitalising influence is present within the Church, not only in one country but in many, and far more widely than is suspected. The Committee therefore invites the Oxford Conference to do two things: in the first place, realizing that the Conference can make its voice heard widely among the Churches of all lands it begs the Conference to speak out clearly on the dangers of Anti-Semitism to the Church itself and to recognize openly the total impossibility of a Church tainted with this form of racial absolutism bearing any valid witness to the word of God in the world. Secondly, it asks that in any provision that is made after the Conference for international Christian study of the great problems that confront the Church in the modem world, attention shall be given to this problem of Anti-Semitism. The International Committee which has already collected a certain amount of information on the subject would gladly co-operate in such a study.'' [239]
The Oxford Conference (July, 1937), organized by ``Life and Work'', was an event of major importance. [240] The 425 regular members of the Conference included 300 delegates officially appointed by the Churches, representing 120 communions in forty countries, and constituting a cross-section of Christendom, with the exception of the Roman Catholic Church; only some personal observers from that Communion were present by invitation. Not less than 300 delegates came from the United States and the British Common-wealth. The Orthodox Churches and the Lesser Eastern Churches were represented by some two score dignitaries and scholars. This delegation represented the Ecumenical Patriarchate, the Patriarchates of Alexandria and Antioch, the Churches of Cyprus, Greece, Rumania, Yugoslavia, Bulgaria, Poland, the Russian Church in Exile, the Coptic Orthodox Church, the Armenian Church, and the Church of the Assyrians.
\textit{97}
German Church leaders had taken a prominent part in the preparations for the Conference, but the German secret police had seized the passports of leading members of the Confessing Church, including those of Dibelius and Niemoeller, who had been chosen as delegates to Oxford. On July 1, 1937, before the Conference opened, Niemoeller was arrested. Other delegates of the Confessing Church who still retained their passports decided that, unless all the representatives of the Confessing Church were allowed to attend, none of them would come, thereby demonstrating their unity. [241]
The German authorities must have realized that the absence of the leaders of the Confessing Church would make a bad impression on world opinion, but apparently they were also aware that the position of Niemoeller and his friends would have been strengthened, if they had been able to attend the Conference.
The Oxford Conference sent a ``Message to the Churches of Christ throughout the World''. We quote the following:
``The Christian sees distinctions of race as part of God's purpose to enrich mankind with a diversity of gifts. Against racial pride or race-antagonism the Church must set its face implacably as rebellion against God. Especially in its own life and worship there can be no place for barriers because of race or colour. Similarly the Christian accepts national communities as part of God's purpose to enrich and diversify human life. Every man is called of God to serve his fellows in the community to which he belongs. But national egotism tending to the suppression of other nationalities or of minorities is, no less than individual egotism, a sin against the Creator of all peoples and races. The deification of nation, race, or class, or of political or cultural ideals, is idolatry, and can only lead to increasing division and disaster.'' [242]
We also quote the following from the Oxford Conference's ``Longer Report on Church and Community'':
``Each of the races of mankind has been blessed by God with distinctive and unique gifts. Each has made, and seems destined to continue to make, distinctive and unique contributions to the enrichment of mankind. All share alike in the love, the concern and the compassion of God. Therefore, for a Christian there can be no such a thing as despising another race or a member of another race. Moreover, when God chose to reveal Himself in human form, the Word became flesh in One of a race, then as now, widely despised... \textit{98} Against racial pride, racial hatreds and persecutions, and the exploitation of other races in all their forms, the Church is called by God to set its face implacably and to utter its word unequivocally, both within and without its own borders. There is a special need at this time that the Church throughout the world should bring every resource at its command against the sin of anti-Semitism... The recrudescence of pitiless cruelty, hatred, and race-discrimination in the modern world (including most notably anti-Semitism) is one of the major signs of its social disintegration. To these must be brought not only the weak rebuke of words but the powerful rebuke of deeds. For the Church has been called into existence by God not only for itself but for the world; and only by going out of itself in the work of Christ can it find unity in itself.'' [243]
An immense effort was made, notably in the Anglo-Saxon world, to bring home the message of the Conference to the rank and file of the Churches. The message was referred to by Church leaders when the fight against anti-Semitism intensified as, for instance, by the 170 ministers in the city of New York, 1941, [244] and Rev. Bertrand in France, in his circular letter of June 11, 1942. [245] Many Church leaders who were present at the Oxford Conference were to denounce anti-Semitism vehemently and publicly, during the Second World War. We mention: Dr. Visser 't Hooft, the General Secretary; the Archbishop of York (Dr. Temple); the Bishop of Chichester (Dr. Bell); Archbishop Eidem, of Sweden; Bishop Fuglsang-Damgaard, of Denmark; Archbishop Stephan, of Bulgaria; Dr. Samuel Osusky, Czechoslovakia; the Rev. Marc Boegner, France; Prof. Emil Brunner, Switzerland; and Dr. Samuel McCrea Cavert, the United States.
Another statement to be recorded in this chapter was adopted by the World Alliance for International Friendship through the Churches, on its meeting at Larvik (Norway), in August, 1938:
``The Council appeals to its members to do all they can to awaken public opinion in their own countries to the great evils involved in the systematic ostracism and persecution now being directed against the Jewish race and against thousands of Christians who have kinship with the Jews. Whilst acknowledging the weakness, hesitancy and failure of Christians in this matter, it is appalled by the growth of racial and religious intolerance throughout the world. \textit{99} It holds it to be a total denial of faith in the fatherhood of God and the brotherhood of men as revealed in Jesus Christ and it calls upon all Christians to unite their efforts so that in a distracted and divided world Christ may be made manifest 'Who is our peace. Who made both one and hath broken down the middle wall of partition between us'.'' [246]
* * *
In 1938, two great Ecumenical Movements --- Faith and Order and Life and Work --- associated together in forming a Provisional Committee of the World Council of Churches (in process of formation). The World Council of Churches was officially constituted in Amsterdam, in 1948.
On November 16, 1938, Dr. Visser 't Hooft, General Secretary of the World Council of Churches, H. L. Henriod, General Secretary of the World Alliance for International Friendship through the Churches and Adolf Keller, Director of Inter-Church Aid sent the following letter to the member Churches:
``At the moment when the terrible persecution of the Jewish population in Germany and in other Central European countries has come to a violent climax, it is our duty to remind ourselves of the stand which we have taken as an ecumenical movement against anti-Semitism in all its forms. The World Alliance at the meeting of its Executive in Sofia in 1933 and at its recent Assembly at Larvik in August 1938, and the Conference on Church, Community and State at Oxford in 1937 have unequivocally expressed the Christian attitude on this point and called upon the Churches to help those who suffer from racial persecution. We suggest that at this time all Churches should take immediate action based on these statements. The most practical action would seem to be: 1. Corporate prayers of intercession. 2. An approach to the Governments of the various countries requesting that
they should act immediately.
a. in order to allow a larger percentage of non-Aryan refugees to enter
provisionally or definitely into the country concerned;
b. to further without delay the plan proposed by the Evian Conference [247]
for securing a permanent settlement of a large number of actual and
potential non-Aryan refugees.
3. Undertake as a Church the responsibility of the maintenance of some
non-Aryan and Christian families and particularly of at least one
non-Aryan pastor or theological student. \textit{100}
We put ourselves at your disposal for further information on any of these projects.'' [248]
We know that Church leaders in the United States made the requested ``approach to the Government''. [249]
The International Missionary Council held a large international conference at Tambaram, Madras, in December, 1938. It reiterated the Vienna (1937) statement of the International Committee on the Christian Approach to the Jews on anti-Semitism [250], expressed ``its deep concern about the increasingly tragic plight of the Jews'', and urged ``that this constitutes a claim of first importance on the Christian Church''. It recommended:
1. That prayer should be regularly made in Christian Churches, and
particularly on Good Friday and the Jewish Day of Atonement, for all Jews
and non-Aryans who are suffering persecution.
2. That individuals, Churches and Christian Councils in countries suitable
for the reception of immigrants should use their influence, wherever
possible, to secure an open door for refugees.
3. That Christian people in all countries should make a special effort to
welcome and help such of their refugee brethren as arrive in their country.
4. That an appeal be made in all churches for help for recognized refugee
funds...'' [251]
In January 1939, at the First ordinary session of the Provisional Committee of the World Council of Churches, the Bishop of Chichester proposed that the Council create a special department to deal with refugee problems.
``He felt that the time had come to aid the entire mass of non-Aryans. He meant not only the non-Aryan members of the Church but also the others, albeit there being a special responsibility towards members of the Christian Church. Soon afterwards Dr. Adolf Freudenberg was appointed the first secretary of this new Department for Aid to Refugees.'' [252]
\textit{101}
III
DURING THE WAR
19 HISTORICAL EVENTS, 1939-1945
\begin{verbatim}
1939 Sept. 1 Germany attacks Poland. Sept. 3-4 Great Britain and France declare war upon Germany. Sept. 17 Russia invades Poland. Nov. 30 Russia attacks Finland.\end{verbatim}
\begin{quotation}
1940\end{quotation}
March 20 Finland accepts peace with Russia.
Apr. 9 Germany occupies Denmark and attacks Norway.
May 10 Germany attacks the Netherlands and Belgium.
May 14 Capitulation of the Netherlands.
May 28 Capitulation of Belgium.
May 20-June 4: Evacuation of the British expedition force at Dunkirk. June 10 Italy attacks France. June 24 France concludes armistice with the Axis.
Aug.-Nov. The ``Battle of Britain'': Hitler tries to subdue Great Britain by
air raids.
Oct. 28 Italy attacks Greece.
Dec. 7-11 Victory of Great Britain in North Africa.
\begin{verbatim}
1941 April Victory of Germany in North Africa. April 6 Germany attacks Yugoslavia and Greece. April 13 Belgrade occupied. April 27 Athens occupied. May 10 Rudolf Hess flies to Scotland. June 22 Germany invades Russia. Dec. 6 Russian counter offensive; Germany fails to take Moscow. Dec. 7 Japan attacks Pearl Harbour. Dec. 11 Hitler declares war upon the United States.\end{verbatim}
1942
Jan. 20 The Wannsee Conference on the ``Final Solution of the Jewish
Question'' in Europe.
June Rommel defeats Great Britain in North Africa and captures El
Alamein.
Aug. 23 Germany's sixth army reaches the Volga near Stalingrad. Nov. 2 Montgomery breaks through at El Alamein. Nov. 8 Allied forces land in Morocco and Algeria.
Nov. 11 Germany seizes the unoccupied zone of France.
Nov. 19 Russia launches its counter offensive near Stalingrad.
1943
Jan. 3 End of the Battle of Stalingrad.
April 19-May 16 Warsaw Ghetto uprising.
July 10 Allied forces land on Sicily.
Sept. 3 Allied forces land in Southern Italy.
Dec. The Soviet armies approach the Polish and Rumanian frontiers.
\begin{verbatim}
1944 June 6 The beginning of the Invasion. July 20 Attempt on Hitler's life. August Russia conquers Rumania. Aug. 25 Liberation of Paris. Sept. 3 Liberation of Brussels.\end{verbatim}
\begin{quotation}
1945\end{quotation}
Jan. 17 Russia captures Warsaw.
Feb. 13 Russia captures Budapest.
Apr. 30 Hitler commits suicide.
May 2 Capitulation of Berlin.
May 7 Unconditional surrender of Germany.
\textit{106}
20 GERMANY
Deportations from Austria and the Protectorate (Bohemia-Moravia) began in the winter of 1939/1940. On February 12, 1940, Jews were deported from Stettin. On July 31, 1941, Heydrich was charged by Goering with the preparation and execution of the ``Final Solution''. On October 14, 1941, the systematic deportation of the Jews from the Reich began. On January 20, 1942, the Wannsee-Conference on the ``Final Solution of the Jewish Question in Europe'' was held. From July, 1942, the selections for the gas chambers took place in concentration camps such as Auschwitz. On June 19, 1943, Goebbels declared Berlin to be free of Jews. An estimated number of 3,000 Jews, however, succeeded in surviving ``illegally'', until the end of the war.
It is, as has been explained before, not my intention to record the help rendered by individuals to Jews. There are, however, indications that organized help to Jews did not wholly stop with the closure of Rev. Grueber's office. [253] Mr. Krakauer relates how he and his wife were helped and hidden during the last years of the war. [254] Not less than 34 ministers of the CONFESSING Church were involved in the rescue of these two people. They all had them in their homes for some days or longer, as staying at the same place for too long a period was too dangerous. It appears that there existed a kind of organization of pastors who passed on persecuted Jewish people from one manse to another. The book also shows how difficult it was in those days, to help and hide people who had no identity cards and no ration cards. Mr. Krakauer stated: ``On May 20, 1945, I had the opportunity to speak with Landesbischof D. Wurm, the highest prelate of the country (of Wurttemberg), and to thank him for the fact that by his attitude he had made it possible for his pastors to interest themselves actually on our behalf''. [255] Some Church leaders did not speak out publicly, or, only spoke when it was too late; the reason may just have been that they were afraid to accept the personal risk involved. We know of Bishop Wurm's protests, which came late, even too late to do any good for the Jews in general. \textit{107} I do not know very much about his ``attitude'' in the time before he took official action. The fact that Mr. Krakauer felt that he should thank the Bishop, throws an important sidelight on the dilemma which Church leaders sometimes had to face. If they spoke out publicly against the persecutions, they did not only risk their own freedom and life, but they also risked the lives of the persecuted Jews whom they secretly tried to save. Mr. Krakauer's story should certainly be read by anyone who is interested in the attitude of Protestants in Hitler's Germany toward the Jews.
No public statement whatsoever against anti-Semitism was issued by the CONFESSING Church in Germany, or by any of its leaders, from the end of 1938 until 1943. In April, 1943, a letter was sent by a group of Christian laymen to the Lutheran Bishop of Bavaria. The Bishop asked for at least two signatures to enable him to raise the matter officially, but no one was willing to sign. However, the letter had an indirect influence because Bishop Wurm of Wurttemberg read it, and then sent two letters to the German Government.
Letter of a Group of Christian Laymen:
``As Christians we no longer can tolerate that the Church in Germany should keep silent in regard to the persecution of the Jews. in Churches where the true Gospel is preached, all members are equally responsible for supporting such preaching. We are therefore aware that we also, are equally guilty for the Church's failure in this matter. The inclusion of the so-called `privileged' Jews in this persecution is the next threat: the dissolving of marriages which are valid according to God's law, should cause the Church to protest, in faithfulness to the World of God, against this violation of the fifth, sixth, seventh, eighth, ninth and tenth commandments, thus, at last, doing what it should have done long ago. What moves us is the simple commandment to love one's neighbour, as expounded by Jesus in the Parable of the Good Samaritan. Here He explicitly precluded any limitation of our love only to members of our own faith, race or nation. At this time every `non-Aryan' in Germany, whether Jew or Christian, 'has fallen among murderers'; we are challenged as to whether we will act towards him as did the priest and the Levite, or as the Good Samaritan?
The Duty of the Church \textit{108}
No `Jewish Question' can release us from this decision. Rather should the Church declare that the Jewish question is primarily an evangelical question and not a political one. The politically unusual, and unique existence and character of the Jews is, according to the Holy Scriptures, based on the fact that God has chosen this people as the instrument for His revelation. The Church, just as the first apostles after the crucifixion, must tirelessly testify to the Jews: 'Unto you first, God, having raised up his Son Jesus, sent him to bless you, in turning away every one of you from his iniquities' (Acts of the Apostles 3, 26). This testimony of the Church will only seem worthy of belief to Israel, if the Church is also concerned about the Jews who 'have fallen among murderers'. The Church must especially resist `Christian' anti-Semitism within its borders, which excuses the actions of the non-Christian world against the Jews, as well as, the inactivity of the Church in this matter, by saying that a `deserved' curse lies upon Israel. Let us not forget the apostle's exhortation to us Gentile Christians: 'Be not high-minded, but fear: For if God spared not the natural branches, take heed lest he also spare not thee' (Romans 11, 20, 21). The Church must testify to the State about the purpose of Israel in the plan of salvation, thus actively resisting every attempt, to `solve' the Jewish question, according to a man-made political gospel, which brings about the annihilation of the Jews. This is an attempt to fight against God and his first commandment. The Church must confess that she, as the true Israel, is united with Jewry by indissoluble ties, both in her guilt and in her right to the promises of God. She must not try any more to remain in safety while Israel is attacked. Rather, she must testify that by the attack on Israel, the Church and her Lord Jesus are also being attacked.
God remains faithful to his Covenant.
The parable of the Good Samaritan reveals the kind of example which should be given by the Church, in regard to the Jewish question. The phenomenal history of the Jews, in which the prophecy has been fulfilled: 'they shall be a curse, and an astonishment, and a hissing, and a reproach, among all the nations' (Jeremiah 29, 18), proclaims to the whole world that the God who gave the first commandment, by his dealings with Israel has manifested to the nations his sovereignty. The Church must explain this phenomenon. She also must, by her faithful testimony, make certain that the authorities are not able to avoid the challenge by obliterating the phenomenon of the Jews. She must therefore proclaim the message of God, who brought both Israel and the Church `out of the land of Egypt, out of the house of bondage' (Exodus 20, 2). Notwithstanding all the unfaithfulness of those He has elected among both Jews and Gentiles, He remains faithful to his Covenant. The Church thus proclaims to the authorities that only by faith in Jesus Christ can they be delivered from their demonic political `gospel', which they in their obsession wish to realize, being unrestrained by the law of God. The Church, therefore, must proclaim the commandments concerning our neighbour to the authorities in connection with their attitude to Israel, but also the first commandment concerning their attitude towards God. For the rulers can only exercise their powers rightly by upholding the law rightly, in obedience to the first commandment. \textit{109}
A Public Protest Demanded
The protest of the Church against the persecution of the Jews in Germany thus becomes a specially important example of the witness she is charged to give against all violations of the ten commandments by any power. The Church must warn the State, in the name of God, not with political arguments, as has happened occasionally, that it must 'not oppress the stranger, the fatherless and the widow' (Jeremiah 7, 6). She must remind the State of its duty to maintain public justice in an orderly, legal system based on humane laws; of the commandment to execute punishment in righteousness; of its duty to protect the oppressed and to respect certain basic rights of its citizens, \textit{etc}. This witness of the Church must be made publicly, either through preaching or by means of a special pronouncement of the Bishop in his function as Shepherd and Watchman. Only thus can the Church fulfil her duty towards all who, either in a legislative or in an executive capacity, participate in this persecution. Also the conscience of the stricken Jews and the Christian community, which is tempted to deny its faith, must be instructed. So far the Church in Germany cannot be said to have made such a witness, for nothing that she has said in public has done justice to her responsibility to preach the truth in this respect. [256]
It is significant that the authors of this letter claimed that as Christians they no longer could tolerate that the Church in Germany should keep silent regarding the persecution of the Jews; that all members of the Church are equally responsible for supporting such preaching (of the true Gospel) and that the protest of the Church must be made publicly. Yet, they themselves refused to sign their own letter.
On January 28 1943, Bishop D. Wurm of Wurttemberg sent a letter to a ``Senior State Official'' (Ministerial Director Dr. Dill, of the Ministry of Interior). We quote the following:
``... Apart from these matters, ecclesiastical in the limited sense of the word, I would like to raise another delicate and difficult, but unfortunately, unavoidable point. Wide circles, and not only those in the Confessing Church, are unhappy at the manner in which the war against other races and nations is conducted. \textit{110} From soldiers on home leave we learn how Jews and Poles are systematically murdered in the occupied territories. Also those who objected to Jewish predominance in public life (even at a time when the entire press was in favour of the Jews), cannot assume that one nation is entitled to exterminate another through measures applied to individuals irrespective of their personal blame. The putting to death of people without any trial, solely on the basis of their belonging to a different nationality, or on account of their diseased health, clearly contradicts the divine commandments, and therefore also every concept of justice and humanity which is indispensable in a civilised nation. There can be no blessing on such an attitude. It leads one to consider the fact that from the time these measures were adopted, the German forces have not been as successful as they were at the beginning of the war. Many Germans see in these occurrences not only a disaster but also a sign of guilt, which will bring its own vengeance. Their moral burden would be lightened, if a courageous and noble-minded decision were taken by the Government, which would cleanse the besmirched shield of honour of the German nation. The Evangelical Church has not publicly protested before, to avoid embarrassing the German nation in the eyes of foreign countries. But now that new and great sacrifices are being demanded of the German people, it should also be granted relief from its moral burdens.'' [257]
On July 16, 1943, Bishop Wurm sent a letter to all the Members of the Government, in which he pleaded for the ``so-called privileged non-Aryans''. We quote the following:
``... In the name of God, and for the sake of the German nation, we urgently request that the responsible leaders of the Reich stop the persecution and the annihilation of so many men and women, which under German domination is being carried out without any judicial sentence. Now that non-Aryans under German domination have to a great extent been removed, it is much to be feared that individuals, the so-called privileged non-Aryans, who until now were spared, are now in danger of being treated likewise. In particular we emphatically protest against those measures which threaten to dissolve legal marriages and thus penalize the children born out of these marriages. These aims are, like other actions of annihilation taken against non-Aryans, in flat contradiction to God's commandment, and they violate the foundation of all Western existence and human values in general...'' [258]
\textit{111}
On December 20, 1943, another letter was sent by Bishop Wurm, to the Chief of the Reich Chancellery, Lammers:
``... Not because of any philosemitic sympathies but solely from religious and ethical considerations, I must declare, in accordance with the opinion of all positive Christian circles in Germany, that we as Christians consider the policy of annihilation of the Jews as a terrible injustice, fatal to the German people. Killing without military necessity and without trial is contrary to God's commandments, even though it is ordered by the Goverment. Just as every conscious transgression of God's commandments, it will recoil sooner or later on its perpetrators. Our people in many respects is experiencing sufferings which it has to bear from the air-attacks of the enemy, as if in retribution for what was inflicted upon the Jews...'' [259]
A Public Protest, issued not by one Church leader but by the CONFESSING Synod of the Evangelical Church of the Old-Prussian Union, was the ``Interpretation of the Fifth Commandment'':
14. ``The sword is given to the State only that it may execute criminals and for the destruction of enemies in war-time. What it does beyond that, it does arbitrarily and to its own detriment. When life is taken for other reasons than those mentioned, men's confidence in one another is undermined and thus the unity of the people is destroyed. The divine world order knows no such terms as `to expunge', `to liquidate' or `valueless life' with regard to human beings. To slay human beings simply because they are related to criminals, because they are old or mentally afflicted, or because they belong to a different race, is not the use of the sword sanctioned by the Scripture... 17. In our time, especially, elderly people are more than ever before dependent on our help. The same is the case with the incurably ill, the weak-minded and the mentally diseased. We must also not forget those who receive no support - or almost no support --- from public funds. In such matters the Christian is not concerned with public opinion. His neighbour is always the one who is helpless and who especially needs him, and he makes no distinction between races, nations or religions. \textit{112} God alone has authority over human life. All life is sacred to him, even that of the people of Israel. Israel has indeed rejected the Christ of God, but neither as human beings nor as Christians are we called upon to pass sentence on their unbelief...'' [260]
The publication of the ``Interpretation of the Fifth Commandment'' was an act of courage but one shudders to read the opinion that ``Israel has indeed rejected the Christ of God''. It was only after the war that the Kirchentag (1961) declared: ``Jews and Christians are insolubly linked with each other: ...God hath not cast away his people, which He foreknew''. [261] Such declarations were lacking at the time when they were most necessary.
Several leaders of the CONFESSING Church have severely criticized their Church, and themselves. Rev. Martin Niemoeller, who himself was imprisoned from 1937 until the end of the war, stated:
``Nobody wants to take the responsibility for the guilt. Nobody admits to guilt but instead points to his neighbour. Yet the guilt exists, there is no doubt about it. Even if there were no other guilt than that of 6,000,000 clay urns; the ashes of burnt Jews from all over Europe. This guilt weighs heavily on the German people, on the German name, and on all Christendom. These things happened in our world and in our name... I regard myself as guilty as any SS man.'' [262]
Rev. Grueber, who himself suffered in a concentration camp because of his help rendered to Jews, said:
``In a few meetings of the Confessing Church a call to protest was given. But protests were made by the few, in comparison with the millions who co-operated or kept silent, who, at best, played the ostrich or clenched their fists in their pockets.'' [263]
\textit{113}
The following is the opinion of Dr. Freudenberg, who was the Director of the World Council of Churches' Secretariat for Refugees, during the war:
``The attitude of the Christians, also of the adherents of the Confessing Church, towards the national-socialist persecution of the Jews, shows great weakness and uncertainty. The anti-Semitic outcry of the environment made a greater impression than the word of Jesus Christ, the Son of David... But even the apparently feeble witness of the Church demanded great confessional courage in the situation of that time. One wrestled to give many a witness, and one suffered when the right word at the right time was not given... It certainly is not accidental that even the Confessing Church, though offering determined resistance against the introduction of the Arierparagraph within the Church, only very hesitatingly made its stand against the anti-Semitic laws and the persecution of the Jews in the State... The fact that the policy of the State towards the Jews ultimately is the policy of the Church and that persecution of the Jews is persecution of Christ, was not acknowledged in time, and when finally it was made, it was far from adequate. Moreover, this policy was effectively veiled by the national-socialist methods of camouflage. At the beginning of the regime one simply could not believe that the rulers relentlessly pursued a plan for the annihilation of the Jews and the elimination of the Christian Church from public life... If we want to evaluate the documents correctly, we must always consider Hitler's incomprehensible terrorization in the Reich. It may disappoint us that the matter was not raised more often and more forcibly. We should, however, bear in mind under which circumstances speaking or keeping silent took place. We should keep in mind that only now, after all the atrocities have become known, has it become customary to make a categorical condemnation of national-socialism. But this phenomenon was, in general, judged quite differently, that is to say, much more positively, not only by the Germans but everywhere in the world, at the time when (some of) these documents were issued.'' [264]
The Evangelical Church in Germany herself, after the war, pleaded guilty, unequivocally and repeatedly. [265] The verdict seems obvious: even the Protestant group in Germany which resisted Hitler, totally failed when they should have stood up in the defence of the Jews. After all this has been said, however, something should be added. 1. The CONFESSING Church in Germany did speak out against anti-Semitism
in 1936, and, indirectly, also in 1935 and 1938, when already this meant
martyrdom. Churches in other lands, for instance in the Netherlands, did
not speak out in those days. Many Churches outside Germany denounced anti-
Semitism long before 1940, but it cost them little, if anything. \textit{114}
2. The CONFESSING Church, when speaking on behalf of the Jews, spoke against
its own Government and seemingly against national interests. Church
leaders in countries occupied by the Germans also risked their lives when
denouncing German anti-Semitism, but they spoke against the national enemy.
Public declarations of Church leaders in Germany were used by foreign
propaganda media against the Third Reich. [266]
Fortunately, this served to open the eyes of many blind people outside
Germany, but it certainly made things even more difficult for Church
leaders in Germany: many of their compatriots regarded the issue of such
declarations as an act of high-treason.
3. Guenter Lewy, discussing the attitude of the Roman-Catholic Church in
Germany, states:
``The concern of the Gentile populations of these countries (France, the Netherlands and Belgium) for their Jewish fellow citizens was undoubtedly one of the key factors behind the bold public protests of the French, Dutch and Belgian bishops --- just as the absence of such solicitude in Germany goes a long way toward explaining the apathy of their German counterparts.'' [267]
This is also applicable to the leaders of the CONFESSING Church.
\textit{115}
THE OCCUPIED COUNTRIES
21 NORWAY
Only 1,700 Jews were living in Norway. In October 1940, the Jews were barred from certain professions. In June 1942, registration was ordered and in October confiscation of Jewish property was decreed. The Jews received identity cards stamped with the letter J; at the same time, arrests of Jews began. On October 25, 1942, all male Jews of sixteen and over were arrested and interned. On November 25, the women and children were seized. 770 Jews, including 100 refugees from Central Europe, were deported by boat to Stettin and thence to Auschwitz. The majority of Norwegian Jews (930) were smuggled to Sweden. [268]
The Constitution of Norway proclaims: ``The Evangelical-Lutheran religion shall remain the official religion of the State''. The majority of Government ministers must be members of the Church of Norway. Quisling had received the title of Minister-president on February 1, 1942. The Bishops of the Church of Norway decided unanimously, on February 24, 1942, to ``cease administrative co-operation with a State which practices violence against the Church'', although maintaining the right to exercise the spiritual vocation given them by ordination at the Lord's altar. On April 9, 1942, the Quisling authorities imprisoned Bishop Berggrav and four other Church leaders. Later on Bishop Berggrav returned from the concentration camp in which he was held, but remained under house arrest.
\textit{116}
On November 11, 1942, the (Lutheran) Bishops of Norway sent a letter of Protest to the Minister President Quisling. This Protest was also signed by the Baptists, the Methodist Church, the Norwegian Mission Association, the Norwegian Mission Alliance, the Sunday School Union and the Salvation Army. Following is the text of the Protest:
``The Minister President's law, announced October 27, 1942, regarding the confiscation of property belonging to Jews have been received by our people with great sorrow, and was deepened by the decree that all Jewish men over 15 years of age were to be arrested. When now we appeal to the Minister President, it is not to defend whatever wrongs Jews may have committed; if they have committed crimes they should be tried, judged and punished according to Norwegian law, just as all other citizens. But those who have committed no crime should enjoy the protection of our country's justice. For 91 years Jews have had a legal right to reside and to earn a livelihood in our country. Now they are being deprived of their property without warning; men were being arrested and thus prevented from providing for their property-less wives and children. This not only conflicts with the Christian commandment to `love thy neighbour', but with the most elemental of legal rights. Jews have not been charged with transgression of the country's laws, much less convicted of such transgressions by judicial procedure. Nevertheless, they are being punished as severely as the worst criminals are punished. They are being punished because of their racial background, wholly and solely because they are Jews. This disaffirmation by the authorities of the Jews' worth as human beings is in sharp conflict with the Word of God which from cover to cover proclaims all racial groups to be of one blood. See particularly Acts 17, 26. There are few references where God's Word speaks more plainly than here. God does not differentiate among people. Romans 2, 11. There is neither Jew nor Greek. Galatians 3, 28. There is no difference. Romans 3, 22. Above else: When God through incarnation became man, He allowed Himself to be born in a Jewish home of a Jewish mother. Thus, according to God's Word, all people have, in the first instance, the same human worth and thereby the same human rights. Our state authorities are by law obliged to respect this basic view. Paragraph 2 of the Constitution states that the Evangelical Lutheran religion will remain the religion of the State. That is to say, the State cannot enact any law or decree which is in conflict with the Christian faith or the Church's Confession. When now we appeal to the authorities in this matter we do so because of the deepest dictates of conscience. To remain silent about this legalized injustice against the Jews, would render ourselves co-guilty in this injustice. If we are to be true to God's Word and to the Church's Confession we must speak out. \textit{117} Regarding worldly authority, our Confession states that it has nothing to do with the soul but that it shall 'protect the bodies and corporal things against obvious injustice, and keep the people in check in order to maintain civic peace and order'. (Augustana, Article 28). This corresponds with God's Word which says the authority is of God and established by him, not as a terror to good works, but to the evil. Romans 13, 3. If the worldly authority becomes a terror to good works, that is, to the one who does not transgress against the country's laws, then it is the Church's God-given duty as the conscience of the State to object. The Church, namely, has God's call and full authority to proclaim God's law and God's gospel. Therefore it cannot remain silent when God's commandments are being trampled underfoot. One of Christianity's basic values now is being violated: the commandments of God which are fundamental to all society, namely law and justice. One cannot dismiss the Church with a charge that it is mixing into politics. The apostles courageously spoke to the authorities of their day and said: `We ought to obey God rather than men'. Acts 5, 29. Luther says: 'The Church does not interfere in worldly matters when it warns the authority to be obedient to the highest authority, which is God'. By the right of our calling we therefore warn our people to desist from injustice, violence and hatred. He who lives in hatred and encourages evil invokes God's judgment upon himself. The Minister President has on several occasions emphasized that Nasjonal Samling, according to its program, will safeguard the basic values of Christianity. To-day one of these values is in danger. If it is to be protected, it must be protected soon. We have mentioned it before, but re-emphasize it now in closing: This appeal of ours has nothing to do with politics. Before worldly authority we maintain that obedience in all temporal matters which God's Word demands.'' [269]
The close relationship between Church and State in Norway is reflected in the protest: ``The State cannot enact any law or decree which is in conflict with the Christian faith or the Church's Confession''.
Important is the reference to Luther; the attitude of the Lutheran Churches in Germany has been explained by recalling Luther's conception of the two dominions through which God rules this world: the spiritual one, or the Church, and the secular one, or the ``worldly authorities''. The people, according to Luther, have not the right to resist the authorities; only princes have. \textit{118}
The Lutheran Church of Norway, however, quoted the Confession (Augustana) and Luther, in order to stress that it was ``the Church's God-given duty as the conscience of the State to object'' (to the anti-Semitic measures).
The letter of Protest won response throughout the country. It was read in the churches on the 6th and 13th December, 1942. It was also noted outside the borders of the country. The Swedish newspapers quoted it in full. The Swedish Lutheran Bishops referred to it in a pastoral letter which they issued at the beginning of December. [270] In radio London the Protest was quoted in full. ``Breaking the wall of silence'' did not help much, if at all, the Jews of Norway; but it warned people in Sweden and Denmark, so that they were on their guard when the Germans tried to apply their `final solution' to the Jewish community in Denmark.
Naturally enough, the attitude taken up by the Christians earned them fresh attacks from the Quisling followers. On December 30, 1942, the Trondheim paper Adresseavisen concluded an editorial on the 'detrimental Jewish influence' in Norway with these words:
``... But now all this is forgotten. On Boxing day the Norwegian clergy read a new pastoral letter from the pulpits, glorifying the Jews and their activities, sighing and lamenting because the chosen race of Israel is not allowed to pursue its activities among the Norwegian people as before, but must be held responsible for its actions.'' [271]
Nevertheless, in a New Year's message for 1943, which was read from the pulpits throughout the country, the Provisional Church Council boldly declared that it would continue to fight Nazism to the end. The Council called upon the congregations to pray for imprisoned clergymen and persecuted Jews. It added:
\begin{quotation}
``The appeal which the Norwegian Church and the Christian people recently\end{quotation}
sent to the Minister President on account of the atrocious treatment of the Jews, has not yet been answered. In this case we have clearly seen what may happen when God's words concerning the worth of man and love are being trampled underfoot.'' [272]
\textit{119}
22 THE NETHERLANDS
a. The Preliminary Phase
On May 14, 1940, the Dutch army surrendered to the Germans. Seyss-Inquart was appointed Reich Commissioner to the Netherlands; Rauter was Chief of Police and Security; General Christiansen was head of the military administration. The political situation in the Netherlands was better than in occupied Poland and Bohemia, but worse than that in most of the other occupied countries, such as Denmark. The Queen and the Cabinet were in exile. The German rulers in the Netherlands were ruthless and efficient. In October, 1940, the first anti-Jewish decrees were promulgated. In November, Jews were dismissed from public posts. On January 10, 1941, the decree ordering registration of the Jews was signed. On February 9, 1941, the first raid on the Jewish quarter of Amsterdam was made. On February 25, 1941, a general protest-strike was declared in Amsterdam which paralyzed transport and industry, spreading to other districts. It was suppressed by force within three days. In May, 1941, the Jews were banned from parks and places of public amusement. In July, 1941, identity cards of Jews were stamped with the letter J. Between January and April of 1942, thousands of Jews were deported to labour camps. After May, 1942, the Jews had to wear the yellow star.' [273] There are people who believe that the record of Dutch resistance against National-Socialism is outstanding and that the majority of the population was engaged in rescue activities on behalf of the Jews. To those who believe this, the reading of Dr. J. Presser's book ``Destruction'' must be a shattering experience.
On June 20, 1940, the Synodal Committee of the DUTCH REFORMED CHURCH [274] invited seven other Protestant Churches to a consultation. \textit{120}
The Churches invited were: The Reformed Churches in the Netherlands, the Christian Reformed Church, the Re-united Reformed Churches, the Evangelical Lutheran Church, the Re-united Evangelical Lutheran Church, the Brotherhood of Remonstrants and the Society of Mennonites. Representatives of these Churches convened for the first time on June 25, 1940. A ``Council of Churches'' was established, and later on, became known as the ``Inter-Church Consultation''. [275] Most of the public protests were issued by this Council.
Particularly at the beginning, the attitude of several members of the ``Council of Churches'' showed a lack of determination. One of the factors that led the Council, as well as the Churches themselves, to a more determined attitude, was the influence of the ``Circle of Lunteren''. This group, consisting of ministers belonging to different Churches but mainly to the DUTCH REFORMED CHURCH, had followed the plight of the Confessing Church in Germany with deep sympathy; many of them were influenced by the clear stand and the teachings of Prof. Karl Barth. [276] The ``Circle of Lunteren'' secretly met for the first time in the village of Lunteren, on August 22, 1940. A letter was sent to the Synodal Committee of the DUTCH REFORMED CHURCH, urging the Church to give clear advice to the local churches and to the nation at large, especially regarding increasing anti-Semitic propaganda. [276] The reply of the Synodal Committee, however, was both reserved and evasive. [277]
The ``Circle of Lunteren'' also published clandestine brochures; 50,000 copies of the brochure ``Almost too late'' were distributed. It was written by Rev. J. Koopmans. He spoke of the danger of following new Messiahs, instead of the Messiah who came ``not from our race, but from the much hated Jewish race''. \textit{121} He especially mentioned the fact that people in official posts were commanded to sign a document stating that they were ``Aryan'', and that the vast majority of those concerned had signed it, perhaps not even realizing its implications for the Jews. [278] Rev. Koopmans pointed out that it was a grave mistake to sign the document, and since many people had already signed it, indeed it was ``almost too late''. Therefore quick action should be taken if it would not be too late altogether. Everyone should explicitly declare that he would not take part in the expulsion of the Jews from public life. The pamphlet closed with the words:
``Dutchmen, it is almost too late, but still not too late! It is still not too late to return to the Christian faith and to a clear conscience. It is still not too late to stand up for our Jewish compatriots, for the sake of mercy and on the grounds of Holy Scripture. It is still not too late to show the Germans that their wickedness has not overcome everything, but that there are people who are determined not to be robbed in this way of their Christian faith and their clear conscience.'' [279]
Someone was caught distributing this brochure; he was sentenced by a German judge to one and a half year imprisonment. [280] Another clandestine pamphlet was published by the ``Circle of Lunteren'': ``What we believe and what we do not believe''. It was written in the summer of 1941 and widely distributed. We quote the following:
``Therefore we believe that he who stands up against Israel, stands up against the God of Israel... Therefore we believe anti-Semitism to be something much more serious than an inhuman racial theory. We believe it to be one of the most stubborn and most deadly forms of rebellion against the holy and merciful God whose name we confess.'' [281]
\textit{122}
On October 24, 1940, the Protestant Churches sent a letter to the Reich Commissioner for occupied Holland, protesting against the discriminatory regulations against Jewish officials. The letter reads as follows:
``We, the undersigned, representing the following Protestant Churches in questions regarding the relations between the Church and the civil authority: The DUTCH REFORMED CHURCH; the Reformed Churches in the Netherlands; the Christian Reformed Church; the Re-united Reformed Church; the Brotherhood of Remonstrants; the Society of Mennonites, feel impelled to appeal to your Excellency in view of the regulations recently issued forbidding the appointment or promotion in the Netherlands of officials or other persons of Jewish blood. In our view the spirit of these regulations, which bear in a special way upon important spiritual questions, is contradictory to Christian mercy. Moreover, these regulations also effect members of the Church itself insofar as they have adopted the Christian faith in recent generations and who have been received as perfect equals into the Churches, as is expressly demanded by the Holy Scripture (Rom. 10, 12; Gal. 3, 28). Finally, the Churches are deeply concerned since this affects the people from whom came the Saviour of the world, and for whom all Christians intercede that they may recognize in Him their Lord and King. For these reasons we urgently appeal to your Excellency to induce the authorities to abolish the said regulations. Moreover, we refer to your Excellency's solemn promise to respect our national character and to refrain from enforcing on us any ideology alien to us.'' [283]
As the Boards of both the Lutheran Churches refused to associate their Churches with this protest, it was only submitted on behalf of six of the eight Protestant Churches. The text was made public in an abbreviated form on Sunday, October 27, in most of the churches. However, the Reformed Churches in the Netherlands and the Christian Reformed Church did not make the protest public to their congregations. Therefore Prof. H. H. Kuyper, who was the representative of the Reformed Churches in the Netherlands, was sharply criticized and some of the other members of the Council refused to co-operate with him further. He then resigned on account of his ``deafness'', and another was appointed in his place. [284]
\textit{123}
On January 10, 1941, the decree ordering registration of the Jews was signed by Seyss-Inquart. On February 9, 1941, a general protest-strike was declared in Amsterdam which paralyzed transport and industry, spreading to other districts. It was suppressed by force in three days.
The next protest of the Churches was a letter, dated March 5, 1941, and sent to the Assembly of General Secretaries (an Assembly which, in the absence of the Ministers of State, represented the supreme Dutch authority in the Netherlands). The Evangelical Lutheran Church also signed this protest; thus seven Protestant Churches participated in this action. Here follows the text:
``The Churches are deeply distressed about the development of events, which is becoming increasingly clear. The proclamation of the Word of God entrusted to the Church charges us with the express duty to make its stand for right and justice, truth and love. It must raise its voice when these values are threatened or attacked in public life. The fact that these values are being seriously threatened cannot be denied by anyone who observes the present situation of our nation. Clear symptoms of this state of affairs which not only weighs as a heavy burden on the conscience of our fellow citizens but is also, according to the deep conviction of the Church, contrary to the Word of God, are incidents in the public street and the treatment to which the Jewish part of the Dutch population is being increasingly subjected. There is growing insecurity in the administration of justice and a continual attack on the freedom indispensable to the fulfilment of Christian duties. For this reason the Churches deem it their duty to request the Assembly most urgently to employ all means at its disposal to ensure that also at this time, justice, truth and mercy may be guiding principles of Government action. The Churches humbly consider it their bounden duty to influence the lives of the people as to inculcate in them these spiritual values. We trust that you will be prepared to pass on the word of the Churches as expressed in this document in any way you deem expedient to those who, in the present period of occupation, bear the ultimate responsibility for the course of events in our country. We fully realize the extremely difficult task which faces the Assembly at this juncture, and we pray God that He may give it His light and His help.'' [285]
\textit{124}
The Churches intended to inform all the congregations of the nature and contents of this letter by a short announcement from the pulpits. The necessary circulars had been prepared in time for the reading of the declaration on Sunday, March 23, 1941. But on March 20, the secretary of the Synod of the DUTCH REFORMED CHURCH and the Chairman of the ``Council of Churches'' were arrested. The authorities were sure that the reading of the declaration would become the signal for an insurrection and that the Churches would be responsible for a disturbance of public order. When it was shown that this was a misunderstanding, the two representatives of the Churches were released. To show that the Churches had not intended political action, the pastors who could still be contacted were asked not to read the letter from the pulpit. Thus it was only read in those towns and villages which did not receive the counter-order until too late. [286]
On March 23, 1941, a Pastoral Letter of the General Synod of the Reformed Churches in the Netherlands was read from the pulpits. We cite the following:
``In our time the notion is advanced with ever increasing emphasis that it is not personal relationship to God's Name but belonging to a certain people or race which determines the meaning of a person's life and which divides mankind into distinct divisions. You will always be able to give the right answer to this doctrine (which has already been accepted by many) if you are faithful to the Holy Scripture. In repudiation of this doctrine the Church should not present its own ideas but only convey the powerful Word of God. You have already shared the anxiety which has filled the hearts of so many of our compatriots in recent months. This is a matter of course because, as the Church of Christ well knows from the Gospels, it was in the course of the history of the Jewish people that Christ was born. Therefore the fact of belonging to a special race must never limit our love towards our neighbour, nor the mercy that we owe him.'' [287]
On January 5, 1942, delegates of the Protestant and Roman Catholic Churches together applied to the General Secretary of the Ministry of Justice for an interview with the Reich Commissioner, Seyss-Inquart. \textit{125}
This was the first time in Dutch history that the Protestant and Catholic Churches acted together and signed a document of protest. Moreover, this was a unique proceeding in occupied Europe and considerably increased the impact of the protests. The National-Socialist daily ``Volk en Vaderland'' commented:
``What God has been unable to achieve for centuries, the Jewish star has achieved. Churches which were never able to unite for the greater glory of God, now conduct a united action.'' [288]
An interview was arranged for February 17, 1942. Two delegates of the Protestant Churches and one from the Roman Catholic Church submitted a translation into German of the Memorandum to the Reich Commissioner which had previously been given to the General Secretary of the Ministry of Justice, who had already passed on a copy to Seyss-Inquart. In handing over the document the delegates declared that they were speaking in the name of the entire Christian Church of the Netherlands. We quote the following:
``Then the treatment of people of Jewish origin must be mentioned. At the moment the Churches do not offer judgment on anti-Semitism which, incidentally, they reject utterly on Christian grounds; nor do they wish to initiate a discussion on the political measures taken against the Jews in general. They wish to confine themselves to the fact that a large number of Jews were arrested in the course of the year 1941 and deported, and that since then an alarmingly large number of official announcements of death among these deportees has been received. The Churches would be neglecting their elementary duty if they did not insist that the authorities should put an end to these measures. This is a duty of Christian mercy.'' [289]
Prof. Aalders, one of the spokesmen, then gave an oral explanation of the Memorandum. In his reply the Reich Commissioner said:
``... In our treatment of the Jews there can be no talk of mercy; only, at best, of justice. The Jewish problem will be solved by the Germans and no distinction will be made between Jews and Jews...'' [290] \textit{126}
The results of the interview were negative. Shortly afterwards, Prof. Aalders was arrested. The Churches intended to inform all the congregations of the interview from their pulpits. The German security service, however, threatened heavy punishment, if this intention were carried out. The DUTCH REFORMED CHURCH protested against this in a letter sent to Seyss-Inquart, dated March 17, 1942. [291] Moreover, a short message was read from the pulpits on April 19, 1942:
``... The Church has protested against the lawlessness and cruelty to which those of Jewish faith in our nation are being subjected and against the attempt to enforce a national-socialist philosophy of life which stands in direct contradiction to the Gospel...'' [292]
A full report was sent to all local Church councils, at the same time.
A decree, which initially did not seem so dangerous, actually resulted from the desire to isolate the Jews from other Dutchmen in order to exterminate them more easily. It was the regulation to place a notice ``Forbidden to Jews'' on public gardens, public baths and cinemas. At the beginning of 1942 it was ordered that such a notice must be placed on all public buildings. The Churches refused to obey this order:
``It is absolutely forbidden to place the notice on any church building or on premises used by the Church. On a building with Christian purposes the notice in question cannot be permitted as a matter of principle, because it would be a denial of the Gospel.''
In some church buildings concerts were held, which required placing the notice. But the advice of the leaders of the Church was, that in such cases the concerts must be cancelled. The advice to sports clubs which were compelled to display the notice was: ``For reasons of principle there is no other way but to stop the activities''. [293] Many ministers of religion were fined or imprisoned because of their refusal to display this notice. \textit{127}
b. Mass Deportation
Mass deportations of Jews began in June, 1942. The Jews were assembled in Westerbork camp; trains to the extermination camps in Poland left every week. The last large-scale deportations were in the spring and summer of 1943. In January, 1941, there were 160,000 Jews in the Netherlands, of whom 138,000 were Dutch citizens, and 22,000 foreign Jews. At least 104,000 of them were murdered.
After the systematic rounding up of Jews had started in Amsterdam, the representative of the Remonstrant Fraternity proposed to the Council of Churches, to turn the ``New Church'', in the centre of Amsterdam, into a house of refuge for persecuted Jews, and that attired in their robes of office the ministers of the different Churches should occupy the entrances of the church and stand or fall with the Jews in the church.
The proposal was not accepted. The majority of the Council believed that it would be a sublime but useless gesture which might well cause a bloodbath and at the very least an acceleration of deportations. [294] The Council decided, however, to send a telegram of protest to Seyss-Inquart, to General Christiansen, and to the two German General-Commissioners Rauter and Schmidt. The telegram read as follows:
``Dismayed by the measures that have been taken against the Jews in the Netherlands by excluding them from participation in the normal life of the community, the undersigned Churches have now learnt, with horror, of the new measures whereby men, women and children, as well as whole families, are being deported to Germany or countries now subservient to it. The suffering which this brings to tens of thousands, the recognition that these measures offend the deepest moral sense of the Dutch people, the opposition to God's laws of justice and mercy, all this forces us to address to you the most urgent plea not to implement these measures. Moreover, as far as Christians of Jewish origin are concerned this plea is strengthened by the fact that they have been debarred by this decree from participation in the life of the church.'' [295] \textit{128}
Thereupon the Germans offered a concession. They declared their readiness not to deport Christians of Jewish origin. On the other hand, they made it clear that the sending of the telegram of protest had better not be made public during church services. This was accepted by the General Synod of the DUTCH REFORMED CHURCH. The Synod considered that ``among decent people one party does not publish any document if the other party objects''. Another important argument was the fear that all that had been gained in favour of the Christians of Jewish origin might be lost. [296] None of the other Protestant Churches followed the example of the DUTCH REFORMED CHURCH, nor did the Catholic Bishops. [297] The Germans took their revenge: all Roman Catholics of Jewish origin (amongst whom was the philosopher Edith Stein) were deported, on July 26, 1942, and perished, while most of the Protestants of Jewish origin survived. On September 24, 1942, Rauter wrote to Himmler:
``... Since my last report the Catholics among the Christian Jews have been deported because the five Bishops, with Archbishop de Jong of Utrecht at their head, did not abide by our original agreements. The Protestant Jews are still here, and attempts to break through the united front presented by the Catholic and Protestant Churches have indeed been successful. Archbishop de Jong declared at a Conference of Bishops that he would never again form a united front with the Calvinists and other Protestants. The storm of protest raised by the Churches when the evacuation began has thus been greatly undermined and has now subsided...'' [298] \textit{129}
Rev. H.C. Touw, the historian of the resistance of the DUTCH REFORMED CHURCH, asked the questions:
``Did the Synod take the right decision? Or did it succumb to a satanic temptation? Was it unfaithful to its Lord in order to save the lives of its own members?'' [299]
The question of choosing between ``quiet diplomacy'' and public protest now seems to be easy: negotiations with the devil are senseless. We should not forget, however, that Church leaders who issued a public protest not only took considerable personal risks, but also took upon themselves the responsibility for endangering the freedom and life of others. Noteworthy is the opinion of a group of Christians of Jewish origin who addressed themselves to the Synod of the DUTCH REFORMED CHURCH:
``Be assured that --- if the proclamation of the Word of God (concerning the persecution of Jews) needs to be more clearly emphasized at this time --- those among us who truly belong to the Lord are willing to be deported to Poland, confidently trusting in the lord.'' [300]
In the summer of 1942, regular contact was established between Protestants in Holland and Dr. Visser 't Hooft, general secretary of the World Council of Churches, Geneva. Couriers brought copies of protests of the Churches (and much other information) in microfilm to Geneva, Dr. Visser 't Hooft sent the microfilms to the Dutch Government in London.
The Churches expressed themselves again in a protest which was sent to Seyss-Inquart on February 17, 1943, and which was read from the pulpits in all the churches. We quote from this protest the following:
``The Churches would be culpable if they failed to point out to the authorities the sins they committed in the execution of their authority, and if they failed to warn them of God's judgment. The Churches have already drawn your attention to the increasing lawlessness, the persecution unto death of Jewish compatriots... \textit{130} But it is also the duty of the Churches to preach this Word of God: 'We ought to obey God rather than men'. This commandment is the touchstone in all conflicts of conscience, also in those that arise out of the recently taken steps. Because of God's justice, no one may participate in unjust actions since thereby he would become equally guilty of injustice.'' [301]
It was important that this protest was read out in all the local churches for it frequently happened that Dutch police agents were ordered to arrest Jews and others. The Churches thus warned the faithful that ``no one may participate in unjust actions''. [302]
c. The ``privileged categories''; the ``other God''
In spring 1943, after nearly all Jewish families had been deported, the occupying authorities confronted Jews in mixed marriage with the alternative of being deported or sterilized. We quote below the protest of the Churches. It was sent on May 19, 1943, and signed by the delegates of the nine Protestant and the Roman Catholic Churches, while the Bishop of the Old Catholic Church sent a letter of adherence to the protest, a month later.
``Following on the many happenings in the years of occupation which have forced the Christian Churches of the Netherlands to complain to your Excellency --- especially in the matter of Jewish citizens of our county --- something so frightful is now being perpetrated that we cannot but address a word to your Excellency in the name of our Lord. We have already protested about several acts committed by the occupation authorities, which are in absolute contradiction to the spiritual principles of our people --- a people and its Government which, from the very beginning, have at least endeavoured to live under God's Word. In the last few weeks the sterilisation of so-called mixed married has begun. But God who created heaven and earth and whose commandments are for all men, to whom even your Excellency will have to give account one day, has said to mankind: `Be fruitful and multiply' (Gen. 1, 28). Sterilisation is a physical and spiritual mutilation directly at variance with God's commandment that we shall not dishonour, hate, wound, or kill our neighbours. \textit{131} Sterilisation constitutes a violation of the divine commandment as well as of human rights. It is the latest consequence of an anti-Christian racial doctrine which destroys nations, and of a boundless self-exaltation. It represents a view of the world and of life which undermines true Christian human life, rendering it ultimately impossible. At the present time your Excellency is de facto the highest political authority in the Netherlands; you have been entrusted with the task of maintaining law and order in this country --- entrusted not only by the leader of the German Reich but also by the inscrutable will of the God whom the Church proclaims here on earth. The commandments of this God and Judge of all the earth apply to you as much as to anybody else and all the more in view of your high position. It is for this reason that the Christian Churches of the Netherlands say to your Excellency in the name of God and of His Word: It is your Excellency's duty to stop this shameful practice of sterilisation. We have no illusion. We are well aware of the fact that we can hardly expect your Excellency to listen to the voice of the Church, which is the voice of the Gospel, which is God's voice. But things that cannot be expected of men, may be hoped for in the Christian faith. The living God has the power to incline even the heart of your Excellency to repentance and obedience. For that we pray God, both for the benefit of your Excellency and of our suffering people.'' [303]
This time again no official reply was received from Seyss-Inquart. However, he communicated by a verbal message that all cases which had occurred up till then, were dealt with on a voluntary basis and furthermore, that he had transferred the matter to General-Commissioner Rauter to deal with. Thus the Churches were advised to send any further protest to Rauter. The Churches turned again to Seyss-Inquart in their letter of June 24, 1943, in which was written, amongst other things:
``The Churches must, irrespective of the question of who is charged with a particular matter, consider your Excellency as ultimately responsible for everything that has happened, and is happening, in our country during the years of occupation.'' [304]
The letters had no practical effect. Many hundreds of Jews of mixed marriages were forced to undergo sterilisation; some, by using bribery or appealing to patriotic physicians, were able to arrange sham operations or get certificates of exemption. [305] \textit{132} German racial policy encouraged the partners of ``mixed marriages'' to divorce the ``non-Aryan'' spouses. By a nominal formality, a partner could part from the one to whom he was legally married. The reaction of the Churches to this is laid down in their letter to Seyss-Inquart of October 14, 1943, which ran as follows:
``Time and again the Christian Churches in the Netherlands have approached your Excellency in matters concerning the Jews of our country, who long have been settled in the Netherlands, and who have been integrated into the life of our people. Your Excellency decided not to listen to the urgent words of warning from the Churches. Most of our Jewish compatriots who, until now enjoyed a limited liberty, have been deported. For them as well as for the very small group which yet remains, we appeal urgently to your Excellency, to prevent deportation and allow them privileged treatment in the Netherlands. Further, the Churches are seriously alarmed by indications that the German administration is again paying particular attention to the so-called mixed-marriages, with the aim of bringing about divorce, at least in a number of these marriages. This aim may, as happened in the case of sterilisation, be made to appear more harmless by a pretension that each divorce is a voluntary one. As before, the Churches beg emphatically to stress to your Excellency that this way of dissolution of marriage may not be followed. The Lord Jesus says, and He does not say that to His Church alone, but to the whole world, and thus also to your Excellency: 'What therefore God hath joined together, let no man put asunder' (Matthew 19, 6). Therefore the Churches urgently appeal to your Excellency to let these small groups which are at present under consideration for the clauses of exemption, share also in the possibility recently opened for some of them, \textit{i.e}., to be exempted from the restrictions that are in force for Jews. Commotion and indignation cannot diminish if actions are continued which injure the Dutch people in their deepest religious and moral convictions.'' [306]
In the autumn of 1943 a pastoral letter was sent to parochial church councillors of the DUTCH REFORMED CHURCH, to give them the necessary basis for their opposition in the struggle against national-socialist ideology. After sections on ``Another God'' and ``Another Morality'', there follows the section on ``Anti-Semitism''. We quote the following from this section:
\textit{133}
``This `other god' and this `other morality' is clearly recognizable in deliberate anti-Semitism. That the people of Israel should be hated and persecuted with fanatical passion and systematically annihilated with malice aforethought, is a phenomenon which has never before appeared in history in this form; for in the last resort there are no strategic, economic or cultural reasons to be adduced for this; the basis of anti-Semitism lies deeper, and this the Church should clearly perceive. The boundless and unrestrained hatred of the Jews comes from natural aversion to the `Jewish God' and the `Jewish Bible'. This outrage, this blasphemy, spread as it has in many written tracts and his been made into the spiritual nourishment of millions (of course under a regime where the state and the state alone is responsible, and intends to make itself responsible for the guidance of the people, and where public utterances and printed statements can thus never be attributed to the whim of private persons or groups as is the case under a democratic regime), must be an absolutely clear indication to the Christian Church that Faith, itself, is being attacked in its deepest foundations. The Church must not overlook the fact that in this respect, too, its members urgently need guidance based on the Scriptures. There are still members of the Church who, while detesting the systematic annihilation of our Jewish fellow-men and fellow-citizens, yet justify their aversion to the Jews by adducing the judgment of God.'' [307]
d. Some Comments and Evaluations
It is to the honour of the Churches in the Netherlands, that they already protested against one of the first steps taken against the Jews, in October, 1940. It is regrettable that sometimes the Churches chose to ask for ``mercy'' on behalf of the Jews instead of demanding the maintenance of justice. It is even more regrettable that the Churches never publicly exhorted their members, to actively help and hide Jews. Much in the declarations and protests issued, however, shows a deep Biblical insight, in contrast to protests of Churches in other countries in which the national-socialist terminology often was used, or national reasons were stressed rather than the Biblical viewpoint. There have been many comments on the attitude of the Churches in the Netherlands, and we quote some of them below.
\textit{134}
Dr. W.A. Visser 't Hooft, general Secretary of the World Council of Churches:
``These documents must be read carefully. They are precious, for those who composed them and also those who read them from the pulpit were in great danger; they risked much when giving their witness.'' [308]
Rev. H.C. Touw, the historian of the resistance of the DUTCH REFORMED CHURCH:
``The Church's struggle on behalf of the Jews was a struggle of mixed failure and success. Nevertheless this struggle was the most moving, the most dramatic, and the most persistent part of the resistance of the Dutch Church.'' ``Just as too many kept silent in the pulpits, certainly too few took persecuted persons into their houses. Many felt that the Synod had failed to give sufficient guidance in this respect. It did not issue any exhortations, nor did it find any way by which to quicken the conscience of the people. This must be considered a great, collective guilt. Here there is no reason whatsoever for Christian self-glorification, but there is every reason to be ashamed.'' [309]
H. Wielek: ``In April 1942, important declarations showing dignity and courage were proclaimed from the pulpits of the churches. The activity of the Church did not slacken. The pastors evinced personal courage; even without Synodal exhortation they understood how to act. Their sermons did not lack clarity, particularly in regard to the persecution of the Jews and their persecutors. Many pastors had to pay for their courageous attitude by a term in a concentration camp.'' [310]
W. Warmbrunn: ``The attempt of the churches to caution the Germans in their actions, especially with respect to the persecutions of the Jews, could not be effective, since the course of action in major matters of this kind was determined by the Reich leadership.'' [311] ``It appears to this writer that groups that excelled in effective resistance were voluntary organizations independent of state control that were conveyers of religious or ethical norms. The moral implications of Christian doctrine motivated the resistance of the Churches.'' [312]
Rev. J.J. Buskes: ``Why did I let myself be seduced? Yes, indeed, seduced into making compromises. Why did I not say: `Thus speaks the Lord'? \textit{135} It is a painful matter also for others of whom it is said (as of myself) that they have behaved excellently. For it depends on the standard by which one judges.'' [313]
Message of the DUTCH REFORMED CHURCH to the Church in Germany, March 9, 1946:
``...We publicly confess before God and the world, that in this struggle we have not been sufficiently faithful, nor willing to accept suffering gladly and courageously.'' [314]
23 FRANCE
The armistice was signed on June 22, 1940. It was stipulated that 3/5 of the French territory would be occupied by the Germans. In the unoccupied zone a nominally independent regime was established. Marshall Petain became President; Laval was Vice-president until April, 1942, when he was succeeded by Admiral Darlan. Delegate for the occupied zone was Ambassador Brinon. In November, 1942, the Germans occupied Vichy France. Thus we have inserted this chapter under ``Occupied Countries'', not under ``Satellite Countries''. It should be noted, however, that the Vichy Government maintained diplomatic relations with the outside world and that it had at least a certain freedom of action in its own territory, until November, 1942. Laval was in a position to bargain for the French Jews by sacrificing the foreign Jews in France. [315]
a. The Preliminary Phase \textit{136}
At the end of 1939 the Jewish population of France had reached a total of about 270,000. After May, 1940, more than 40,000 Jews streamed into France from Holland, Belgium and Luxembourg. [316] The number of Jews deported from France is estimated to be approximately 80,000 persons. [317] According to Tenenbaum, the number was 100,000 out of a total pre-war Jewish population of some 350,000. ``This relatively favourable result in comparison with the other countries is due primarily to the determined attitude of the French people with regard to their Jewish neighbours.'' [318] Chief Rabbi Kaplan shows us the other side of the picture:
``I do not forget, when recalling these dreadful crimes, that priests, pastors, men and women of all confessions and philosophical doctrines and of all classes, exposed themselves to the greatest dangers in order to come to the rescue of the persecuted Jews. Here I wish to mention particularly, the energetic and courageous protests issued by the eminent leaders of French Catholicism and Protestantism. Nonetheless the undeniable fact remains, that Christian ethical education --- inculcated over a long succession of generations --- has not prevented the majority of the people of a nation claiming to be Christian, from becoming more or less responsible for the abominable Hitlerite persecution.'' [319]
Many factors played their part. It was easier to go into hiding in France than, for instance, in the Netherlands. The attitude of the Italians who held part of occupied France was an important factor: they either found excuses for their non-cooperation with the Germans or just refused. France was the first country to be liberated: the invasion started on June 6, 1944.
On September 27, 1940, the decree for compulsory registration of Jews was promulgated in the occupied zone, including the marking of Jewish stores with the star of David. A few days later --- October 4, 1940 --- the Vichy French Council of Ministers decreed the Statute des Juifs which disfranchised the Jews in all France. On March 29, 1941, a ``Department for Jewish Affairs'' was created by the Vichy Government. In May, 1941, 3,600 Polish Jews were rounded up in Paris. In August, there was another raid. The victims were placed in three camps (Drancy, Pithiviers and Beaune la Rolande). On June 2, 1941, Jewish registration was made compulsory in both zones. On November 29, 1941, the Vichy regime decreed that all Jewish organizations were to be dissolved.
\textit{137}
The Protestants in France are a small minority, numbering altogether not more than 800,000 souls. France is, to my knowledge, the only country where a small minority group of Protestants publicly protested against the persecutions. Poliakov stated one of the reasons:
``It must also be remembered that the French Protestants are themselves a minority and have known centuries of persecution --- such trials, when they are surmounted, sharpen one's sensitivity to injustice.'' [320]
Another positive factor was the fact that the President of the Protestant Federation of France, Rev. Marc Boegner, was also one of the three Vice-chairmen of the Provisional Council of the World Council of Churches. He had many international contacts. This fact gave an additional impact to the protests. Rev. Boegner did not only speak in the name of the French Protestants, but also informed Marshal Petain ``of the deep emotion felt in Swiss, Swedish and United States Churches''. [321]
Rev. Boegner relates that he first stayed in Vichy at the end of July, 1940. A ``very highly placed personality'' told him: ``The Jews have done so much damage to the country that they need collective punishment''. He himself realized then ``where we are going to be dragged and what would be the responsibility of the Churches''. [322]
The establishment of the Department for Jewish Affairs, in March, 1941, aggravated the situation. German pressure on the Vichy Government became stronger. Rev. Boegner spoke of this to Admiral Darlan, who tried to calm him by saying that ``it primarily was a matter of saving the French Jews''. A high police officer sought to persuade him that this was a government matter which was no business of the Churches. [323] \textit{138}
In Lyon, where the National Council of the Reformed Church had convened before the end of 1940, Rev. Bertrand informed Rev. Boegner that the Council of the Protestant Federation wanted a written protest without delay. It was agreed, however, that Rev. Boegner should continue with his oral interventions for some time longer. But when the National Council of the Reformed Church reconvened in March, 1941, it was unanimously resolved that the position of the Reformed Church should be set down in writing without delay. It was on these instruction that Rev. Boegner wrote two letters. The first was sent to the Chief Rabbi of France, on March 26, 1941:
``The National Council of the Reformed Church of France has just convened for the first time since the law of October 3rd, 1940, came into force. It has instructed me to express to you the grief we all feel at the introduction of racial legislation in our country, and at the trials and innumerable injustices which it has brought upon the French Jews. There are some among us who have thought that the State has been faced with a great problem as a result of the extensive immigration of a large number of foreigners --- Jews and non-Jews --- and by hasty and unjustifiable naturalisations, but they have always expressed the conviction that this problem should be handled with the respect due to human beings; with strict adherence to State undertakings; and in accordance with the demands of justice which France has always championed. They are all the more distressed because of the rigorous enforcement of a law which, applying exclusively to Jews, makes no distinction between Jews who have been Frenchmen for many generations, in many cases for centuries, and between those who received their citizenship only yesterday. Our Church which has in the past known all the sufferings of persecution, harbours feelings of warmest sympathy for your communities whose freedom of worship in certain places has already been restricted and whose faithful members have so suddenly been afflicted with misfortune. It has already taken steps --- which it will not fail to pursue vigorously - for the necessary repeal of the law.'' [324]
This letter shows hesitation: it considers the ``extensive immigration of a large number of foreigners'' as a problem and creates the impression that the French Protestants cared less for the Jews who had ``received their citizenship only yesterday'' than for the Jews who had been Frenchmen for many generations. \textit{139}
The same applies to the letter sent to Admiral Darlan, also on March 26, 1941:
``We have just convened at Nimes, for the first time since the enforcement of the Law of October 3rd, 1940, concerning the status of the Jews. On the eve of our meeting we learned from a notice in the press, of your intention to set up an office for Jewish Affairs. We consider it our duty to inform you in the name of the Reformed Church of France, comprising the vast majority of French Protestants, of our feeling on this painful question. We in no way disregard the seriousness of the problem which the State has to face in view of the recent, large immigration of a great number of foreigners, many of them of Jewish origin; and in view of hasty unjustifiable naturalisations. We are convinced that this problem ought, and can be, resolved with due respect to individual people and due care for the justice, of which France has always desired to be a champion. We also know that under the present circumstances strong pressure is undoubtedly being exerted on the government of France in order to force its decision to pass anti-Jewish laws. We are nonetheless deeply distressed, as Frenchmen and as Christians, by a law which introduces the principle of racial discrimination into our legislation, the strict enforcement of which entails severe trials and tragic injustices for the French Jews. Especially, do we protest against the principle of racial discrimination, because it has caused the State to break its formal undertakings on behalf of men and women, the vast majority of whom have served it loyally and disinterestedly. We are assured that the Law of October 3rd, 1940, is not a law of religious persecution. But if freedom of worship really remains untouched, for Jews as for Catholics and Protestants, why then is it, in fact, already being barred or threatened in certain places? The fact is, that a religious minority is being wronged. Our Church which has known all the sufferings of persecution, will fail in its primary mission if it does not raise its voice on behalf of this minority. We know that by setting up an office for Jewish Affairs, you sincerely wish to do whatever is in your power, to avoid even greater hardship from befalling the French Jews. We believe we may give you our assurance that the Christian denominations will give their unreserved approval to your effort, the difficulty of which they are well aware of. At the same time, however, we would ask you most earnestly to take even further measures, and as from now, to amend the law imposed on the French Jews, so that, on the one hand, further injustices may be prevented, and on the other hand, the disastrous impression made on a large part of the civilised world by the law of last October, may be removed. The defeat suffered in the war, the painful consequences of which we are now experiencing, constitutes a further reason why France should seek to safeguard those values which, in the moral sphere, have gained it the respect and affection of Christian nations.'' [325] \textit{140}
Admiral Darlan did not reply to this letter in writing. He told Rev. Boegner that he wanted to discuss the matter with him. Rev. Boegner relates:
``In May (1941) I had a long meeting with him. He informed me that a new draft law was being studied, certain provisions of which would seem very severe to us, but there were others which would attenuate their effect. His sole care was to save those Jews who had been established in France for several generations. Regarding the others, who had recently immigrated, his one wish was that they should leave the country.'' [326]
On May 29, 1942, it was decreed that every Jew who had reached the age of six must wear the yellow star. The Council of the Protestant Federation, under the chairmanship of Rev. Bertrand (in the occupied zone) decided to express the feelings of the Churches in the occupied zone directly to the Chief of State, Marshal Petain. Their letter read as follows:
``The Council of the Protestant Federation of France, assembled in Paris, takes the liberty of addressing itself with respectful confidence to the French Chief of State to express to him the painful impression made upon its affiliated Churches by the new measures taken by the Occupation Authorities with respect to the Jews. The decree of May 29th, compelling our compatriots of the Jewish race to wear a distinctive badge, has in fact deeply moved thousands of Protestants in the occupied zone. Our President, Rev. Marc Boegner, has already had the honour of informing you, as well as Admiral Darlan of the Fleet, who is Vice President of the Council of Ministers, of the unanimous desire of the Protestants of France that the solution of the Jewish question, the importance of which none of us can fail to recognize, shall be found in a spirit of justice and understanding. Yet at present we are faced with a measure which far from contributing to the proper solution of this problem, seems to aggravate it further. Socially and economically unworkable, it is designed to inflict uncalled for humiliation on Frenchmen, many of whom have shed their blood fighting under our Rag, by pretending to set them apart form the rest of the nation. It exposes six year old children to mischievous behaviour, easily liable to occur in the disturbed atmosphere prevailing among the population. Finally, it compels converts to Catholicism or Protestantism to wear before other men, the visible sign of being Jewish, whereas, before God, they have the honour to be acknowledged as Christians. \textit{141} The Churches of Christ also cannot keep silent in view of the undeserved suffering imposed on Frenchmen, and sometimes on Christians, which ignores their dignity as men and as believers. The Council of the Protestant Federation has therefore instructed me to convey to you our feelings of distress. It hopes that you may consider it as a sign of confidence and respect that it submits this expression of pain and distress to the heart of a great soldier who is the Chief of State of France.'' [327]
The letter was handed over to Marshal Petain by Rev. Boegner. The subsequent conversation left him with the same impression as that on his previous meeting with the Chief of State: deep emotion, complete impotence. In a circular letter dated June 11, 1942, Rev. Bertrand informed the pastors in the occupied zone that the Council of the Federation had instructed him to write to Marshal Petain. After having quoted part of his letter to Marshal Petain, he reminded his colleagues that ``the spiritual value of such interventions depends on careful avoidance of any allusion to political events or worldly ideologies, and on strict adherence to the sphere of thought and of Christian action alone''. Rev. Bertrand added:
``In particular the Ecumenical (Oxford) Conference of 1937 affirms that `all men are by birthright children of God.' 'Therefore, for a Christian there can be no such thing as despising another race or a member of another race.' `All races share alike in the concern of God.' 'The sin of man asserts itself in racial pride, racial hatred and persecutions, and in the exploitation of other races. The Church is called upon by God to express itself unequivocally on this subject.'' [328]
Perhaps more important than the protests sent to the French Government, was a Message issued by the National Synod of the Reformed Church of France, in May, 1942, which was read out publicly in all the local churches. This Message included the following passage:
``The Church has been commanded by God to resist the attack of every doctrine and every ideology, every threat and every promise which seeks to assail the message of the Bible, both Old and New Testaments. \textit{142} It must proclaim absolute sovereignty of God, who creates His own people For Himself by calling to Him men of every race, nation and language, in spite of the rights and privileges to which men may deem themselves to have a claim. It knows that all men were created equal, equal in perdition and equal in salvation, and that God's justice demands that every man shall be respected.'' [329]
b. Mass Deportations
On July 16, 1942, mass raids struck the stateless Jews living in Paris. In two days 12,884 of them, including 4,051 children, were rounded up by the French police. [330] Thereupon, the President of the Protestant Federation in the occupied zone, Rev. Bertrand, sent the following letter to Mr. de Brinon, General delegate of the French Government to the Occupation authorities:
``When the German authorities made it incumbent upon the Jews living in the occupied zone to wear a distinctive badge, the Council of the Protestant Federation of France submitted a letter to the French Chief of State which was well received by him and of which I enclose a copy. One would have thought that now the anti-Jewish laws have reached their climax with this humiliating measure designed to place the Jews apart from the rest of the nation and to single them out for the kind of malevolence, systematically meted out to them since the beginning of the occupation. However, the month of July has seen an increase of personal violence on a scale never before attained; and we have noted among the general population of Paris a feeling of distress and disapproval which the present generation undoubtedly will never forget. The Churches of Jesus Christ to whom God has entrusted the message of peace, love, and mutual respect among men, cannot keep silent in view of events which for many years have threatened any possibility of a normal relationship between two great nations. Because Frenchmen at present have no means of making their opinions and feelings known, it should not be inferred that they are indifferent onlookers at the extermination of a whole race, and at the undeserved martyrdom of its women and children. The men who profess to be working towards closer relations between the conqueror and the nations over which he exercises his authority, surely should be able to make the occupying forces understand that declarations of good will during these years cannot efface the effect of the cruelties we have witnessed. \textit{143} A Christian Church would be failing in its vocation were it to let the seeds of hatred be sown in this fashion without raising its voice in the name of Him who gave His life to shatter all barriers between men. I leave it to Your Excellency to judge whether the appeal I have made to you to-day should be brought to the notice of the occupying authorities, and whether the voices of Christians, who are solely concerned with seeking to alleviate suffering and hatred, ought to be ignored, rather than those of men who know no other response to violence than that of hatred. Before concluding this letter I wish expressly to state that the message to Marshal Petain was the only subject of the deliberations of the Council of the Protestant Federation, which has just ended its sessions and it is collectively responsible for it. With regard to the present letter, I take upon myself full responsibility for it, not only before the Church and the French nation but also --- eventually --- before the German authorities.'' [331]
Rev. Boegner relates: ``Events succeeded one another precipitately. After the occupied zone came the turn of the so-called `free zone'. We saw a new wave of horror unleashed in camp, town and village. Our chaplains, together with the `Cimade' [332] and the parish pastors, in the face of tremendous suffering, accomplished a task of Christian love which was a powerful testimony to Jesus Christ. I supported their efforts to the best of my ability. But renewed appeals became necessary. I thought that at this tragic juncture the Catholic Church and the Protestant Churches should at least unite in making their appeals. I spoke of this to Cardinal Gerlier on August 13th. It was agreed that each of us should write an urgent letter to Marshal Petain. Mine was sent on August 20th.'' [333] The letter read as follows:
``When you did me the honour of receiving me on June 27th, I placed in your hands a letter whereby the Council of the Protestant Federation of France entrusted to your soldier's heart the pain and agitation caused in the Protestant Churches by measures taken in the occupied zone against the Jews, and those Christians whom the law has marked as Jews. \textit{144} To-day it is my regrettable duty to write to you in the name of the same Council in order to express the unspeakable sorrow felt in our Church, in face of new measures ordered by the French Government and directed against the foreign Jews (baptised and unbaptised), and the ways and means of their execution. No Frenchman can remain unmoved in view of the events occurring since August 2nd, in concentration and internment camps. As is known, the reply is that France is only returning to Germany those Jews whom the latter had sent in autumn 1940. In truth, however, man and women who for political and religious reasons fled to France, and who know the terrible fate awaiting them, are now being deported or facing immediate deportation to Germany. Christianity has hitherto inspired nations, and especially France, with respect for the hallowed right of sanctuary. The Christian Churches, irrespective of their different confessions, would be disloyal to their original calling if they did not raise a protest against the abandonment of this principle. I am forced to add that in several places these `deliveries' have occurred under such inhuman conditions that they shock the most hardened consciences, and brought tears to the eyes of witnesses: herded together in goods trucks, without the slightest hygienic precautions, foreigners intended for deportation were treated like cattle. The Quakers, who were doing the utmost possible for those who suffer in our country, were refused permission to feed the deportees at Lyons. The Israelite Consistorium was not allowed to give them foodstuffs. Respect for the human personality which you intend to maintain in the Constitution and which you want to grant to France has often been trodden underfoot. Here, also, the Churches see themselves obliged to protest against such a grave misunderstanding of undeniable duties. The Council of the Protestant Federation appeals to your high authority to order the introduction of absolutely different methods in the treatment of foreigners of the Jewish race, whether baptized or not, whose deportation has been admitted. The tenacious fidelity of France, especially during the tragic days which it has lived through in the past two years, towards its traditions of human generosity and noble-mindedness, remains one of the main grounds of respect which certain nations still have for us. As Vice President of the World Council of Churches which includes all great Christian Churches, with the exception of the Roman Catholic Church, I am compelled to inform you of the deep emotion felt in Swiss, Swedish, and American Churches, in face of the events now occurring in France, and with which the entire world is acquainted. I beg you to dictate the indispensable measures in order that France may not inflict upon herself a moral defeat of unfathomable weight.'' [334]
Some days later, the letter was broadcast over the American and British radio, and subsequently reproduced in the foreign press. \textit{145} The deportations continued. By September 1, 1942, the Vichy authorities had handed over 5,000 Jews to the Germans and another 7,100 had been arrested. [335] On August 27, 1942, Rev. Boegner sent the following letter to the Chief of the Government, Laval:
``Authorized to speak on behalf of the Protestant Churches of the entire world, many of which have already asked for my intervention, and aware of the events of the past few days, I beg to urge you to give me your assurance that in no event shall foreigners be convicted in their own countries for political reasons, and those who have sought refuge in France, for similar reasons, be expelled to the occupied zone.'' [336]
He then had an interview with Laval, who said that foreign Jews must be handed over to the Germans in order to save the French Jews. ``Would you agree that we save their children?'', asked Rev. Boegner ``The children must remain with their parents'', was the reply. Laval then asked: ``What would you do with the children?'' Rev. Boegner. answered: ``French families will adopt them''. Laval retorted: ``NO, not one must remain in France''. Rev. Boegner than had an interview with the Charge d'Affaires of the United States, who promised him to cable to Washington, to be authorized to tell Laval that the United States would accept the children of deported parents. [337]
As the Council of the Federation of Protestant Churches in France could not be convened, Rev. Boegner then urgently called a gathering of the National Council of the Reformed Church. It addressed to the faithful the following Message, dated September 22, 1942, which was read from nearly all the pulpits: \textit{146}
``The National Council of the Reformed Church of France, being convened for the first time since the application of measures against the Jews, among whom are many Christians, was informed of the demarches which its President had made, in writing and verbally, to the highest State authorities in the name of the Federation of French Protestants. The Council associated itself fully with the President. Without ignoring or belittling the extreme complexity of the situation with which the authorities of our country are faced and more than ever determined to exercise loyally --- among the people --- the spiritual vocation to which God has called her; although composed of people faithful to the old principle of abstaining from any intrusion into the sphere of politics, the Reformed Church of France cannot keep silent in face of the suffering of thousands of human beings who have received asylum on our soil. A Christian Church would lose its soul and the very reason for its existence, were it not to maintain --- for the safeguard of the whole nation in the midst of which God has placed it --- the Divine law above human contingencies. That Divine law does not permit families created by God to be broken up, children to be separated from their mothers, the right of human beings to asylum and pity to be disregarded; nor respect for human rights to be trodden upon, nor defenceless beings to be delivered to a tragic fate. Whatever the problems may be which are beyond the scope of the Church and which the Church is not called upon to resolve, it is its duty to assert that they shall not be resolved by means which contravene the law of God. The Gospel commands us to consider all men, without exception, as our brothers, for whom our Saviour has died on the cross... How can the Church ever forget that it was among the people from whom the Jews are physically descended, that the Saviour of the world was born? And how can it be anything but profoundly grieved --- as a Church which must affirm the unity of the body of Christ --- by measures which also effect non-Aryan Christians, who are members of our Protestant parishes? In the face of these painful facts the Church feels compelled to make heard the cry of its Christian conscience, and to implore, in the name of God, those who exercise authority in the world, not to aid to the natural horrors of war --- in itself a violation of Christ's commandments --- still worse violations which will in the most fearful manner hinder reconciliation between the nations, in a repentant and peaceful world, submissive to God. It calls upon the faithful to incline toward the distressed and the suffering with the compassion of the good Samaritan, and to intercede ceaselessly with God on their behalf, for He alone can deliver us from evil by the grace He has revealed in Jesus Christ.'' [338] \textit{147}
Everybody knowing the parable of the Good Samaritan [339] must have fully understood that the last sentence of this message was a call to practical and effective acts of rescue, on behalf of those who had fallen ``among thieves'' and murderers.
No public protests were issued by the French Protestant Churches after that of September 22, 1942. On November 11, 1942, the Germans seized unoccupied France. The demarcation line had disappeared. The deportations continued.
c. Practical Help
It is difficult to assess the practical results of public messages such as the one mentioned above. They certainly made more impact than protests sent by Churches to the authorities. S. Lattes is of the following opinion:
``Also, as might have been expected, when the first anti-Semitic measures were taken by the Germans and the Vichy government, many authoritative voices, Catholic and Protestant, were raised in demonstration of their sympathy towards the Jews... These written manifestos had hardly any practical effect, but they were a display of true courage and by their distribution exercised a deep influence on the conscience of the French. They also afforded moral encouragement to the Jewish victims.'' [340]
L. Poliakov gives the following account of the results of the public appeal, made by Rev. Boegner in the name of his Church, and he also gives an interesting analysis of what moved the ordinary Protestant to help the Jews:
``A picturesque little town of 2,000, Chambon-sur-Lignon lies at the foot of Mont-Lisieux, in the centre of a little plateau almost exclusively inhabited by Huguenots. The word Huguenot immediately calls to mind the thousands of victims of persecution who, escaping from France in the 17th century, settled in Prussia, the Netherlands and the United States. One section, however, instead of leaving their country, fled to the savage region of Velay. Protected by practically impassable ravines, they hid in the woods, and remained faithful to their religion. Only in the 19th century were they able to resume their religious worship openly. This period of persecution has made them deeply pious, melancholic and austere; they are suspicious of any authority but unquestioningly follow their pastors. It is here that they have preserved almost intact the customs and virtues of the past centuries. \textit{148} immediately after the terrible raids of July 1942, Pastor Boegner, President of the Federation of Protestant Churches of France, issued an appeal to all this followers, asking them to do everything in their power to help the Jews. The appeal was heeded. Nearly every Sunday the pastors of Chambon, Mazet and Fay-Le-Froid, exhorted their congregations to renewed efforts. The country-people never tried to evade their responsibility. The persecutions which their own grand-parents had suffered were still alive in their memory. They provided food and lodging for the persecuted; in certain small hamlets in the area there was not a single farm which did not give shelter to a Jewish family... On the evening, at the hotel May, I witnessed a spectacle typical of the whole region of Chambon: a social worker arrived with several children whose parents had either been deported or were in hiding in Marseille. They huddled together in fear, in a corner of the room. A couple of country people first came in. 'We should like a little girl of eight or ten,' explained the woman. Little Miriam is called. `Would you like to go with this uncle and auntie?' Intimidated, the little girl does not answer, but she was muffled up in blankets and carried into the sledge; and so she left for a home where, until the end of the war, she would live a simple, healthy life with temporary foster parents. And as if by sleight of hand, all the other children were taken care of in the same way.'' [341]
Perhaps France was the only occupied country where an official Protestant organization rendered direct and practical help to the persecuted Jews. The Cimade [342] was a Protestant Youth organization which sent teams of young Protestants into the camps, in order to render relief to the internees. Miss Madeleine Barot, general secretary of the Cimade, states: ``All racialism is inadmissable from the Christian point of view. It was necessary to give tangible signs of this conviction, to alert public opinion, to protest to the responsible authorities, to mobilize the forces of \textit{149} Protestantism, and, above all things, to help those who suffered most.'' [343] The first relief team was installed in the camp of Curs. It was partly justified to the police by the presence of a number of baptized internees, who were registered as Protestants. ``Our work was labelled as 'Protestant assistance', which was of a great help, though we ourselves did not even consider for one moment restricting our help to the Protestants.'' [344] Thanks to the financial support of the Ecumenical Committee for Aid to Refugees, Geneva, the number of rations to be distributed in Gurs could be increased. [345] In 1941, teams were also placed in the caps at Rivesaltes, Brens, le Recebedou, and Nexon. In the spring of 1942, the Cimade opened four houses (at Chambon-sur-Lignon, Tarascon, le Tarn and Marseilles) for the accommodation of old or sick people and women with little children, who were permitted to leave the concentration camp if an authorized organization took charge of them. The Swedish Church and the World Council of Churches rendered financial aid. [346]
The leaders of the Cimade permanently kept in touch with the Rev. Marc Boegner so that he, when he intervened with the Vichy Government, could make proposals which corresponded with the actual situation in the camps. [347]
After mass deportations had begun, the members of the Cimade became more and more involved in ``illegal'' activities. The Secretariat of the Cimade at Nimes provided false identity cards. ``We set up a record by once producing fifty identity cards in one night.'' [348] Several members of the Cimade were active as guides, bringing refugees through the mountains to safety in Switzerland. ``According to my estimations, we helped to evacuate about four hundred persons, from August, 1942, until December, 1943.'' [349] After the Swiss Government had ordered that refugees who had illegally entered into Switzerland be returned to France [350] the Rev. M. Boegner obtained in Berne the agreement that non-Aryans coming from France for whom he had given personal guarantee, would be admitted. [351] \textit{150}
24 YUGOSLAVIA
On April 5, 1941, Yugoslavia concluded a treaty of friendship with Moscow, and within hours Belgrade was bombed by the German air force. Yugoslavia was dismembered by the Nazis. The north-eastern part, the Backa basin, with 20,000 Jews, came under Hungarian annexation. Old Serbia, where 12,000 Jews lived, came under German occupation. In Croatia, with 21,000 Jews, a puppet regime was established. The Bulgarian-annexed territory of Yugoslavia (Serbian Macedonia) contained between 7,000 to 8,000 Jews. Before the war, Yugoslavia harboured some 70,000 Jews. Fifty-five thousand of them were murdered. [352]
The greatest non-Roman Catholic Church in Yugoslavia is the Serbian Orthodox Church. Much smaller Churches are: the Reformed Christian Church of Yugoslavia and the Slovak Evangelical Church of the Augsburg Confession in Yugoslavia. None of these Churches replied to my circular letter. The persecution of Orthodox Serbs matched the persecution of Jews, both in cruelty and fanaticism. [353] I hardly found any material about the attitude of the Churches in Yugoslavia; only the following quotations can be mentioned:
``High Orthodox and Catholic circles were unanimous in condemning anti-Jewish propaganda. Early in 1940, the Serbian Patriarch Gavrilo, while visiting a synagogue near Belgrade, deplored religious persecution, and the official Catholic organ die Donau condemned racialism. In October, the Patriarch of Sarajevo expressed to representatives of the Jewish community his sympathy for their sufferings.'' [354] ``At the end of May (1943), some Jews who were still living in Zagreb under the protection of the Archbishop, were seized one night and deported, before the churchman could intervene to save them.'' [355]
\textit{151}
``Contrary to what we know about the attitude of the Catholic and Protestant Churches on the Jewish question, we have only meagre knowledge of the aid and comfort rendered by the Orthodox Churches in Nazi-subjugated Europe. [356] Nazi persecution of the Orthodox faith was not checked by the minor hesitation the Nazis showed in their dealings with the other Christian denominations. A few enlightening examples of a deeply humane attitude in some of the conquered countries rend the mist surrounding the tragedy into which these unhappy lands were thrust. Thus it is known that the heads of the Yugoslavian Orthodox Church bravely protested against the atrocities perpetrated on the Jews and exhorted priests and people to abstain from participating in the outrage of Nazis and Ustasa (Croation Fascists) alike.'' [357]
25 GREECE
a. Salonika
Greece was overrun by the Germans on April 6, 1941; the armistice was signed on April 23, 1941. There were three separate occupation zones: Italy was assigned the territory comprising ``old Greece'', with Athens as capital, and the Ionian islands; Bulgaria occupied Western Thrace and Greek Eastern Macedonia; Germany had a narrow belt of Eastern Thrace bordering on Turkey, along with the Salonika harbour and the island of Crete. A puppet government, seated in Athens, functioned in both Italian and German zones. About 13,000 Jews lived in the Italian zone, but the number of Jewish inhabitants in German dominated territory was over 55,000. In March, 1943, the Jews of Salonika were put in a concentration camp. From the middle of March, through May, deportation trains rolled from Salonika to Auschwitz. About 46,000 Jews were deported. [358]
Friedman is of the opinion that ``the Greek Orthodox Church, always a power in the political life of the country, used its considerable influence to oppose anti-Jewish laws, and, later, to help rescue the victims. The humblest papas of remote villages as well as the highest dignitaries of the Church enlisted in the crusade to help Jews''. [359] \textit{152}
It is doubtful, however, whether any Church in any country had a ``considerable influence'' with the German occupying forces. The Church did not, and probably could not, prevent the extermination of the great majority of the Jews of Greece.
At the end of February, 1943, two lawyers turned to Genadius, Bishop of Salonika, and submitted to him a Memorandum concerning the danger threatening the Jews. Bishop Genadius immediately went to Dr. Merten, who was in charge of all civilian affairs in Salonika, and protested, in the name of his Christian faith, against the preparations for the transports. Replying hypocritically, Dr. Merten stated: ``I expected this step of yours, but all your efforts are in vain, for the orders are official and no intervention can change them''. [360] Mr. Moissis, a Jewish lawyer in Athens, commented:
``The attitude of Genadios, Bishop of Salonika, was excellent. He submitted a vehement protest to the military commander of the Macedonian capital who had issued the order of deportation, in March, 1943, in which Bishop Genadios characterized the order as inhuman and anti-Christian. During the deportations, he secretly received Chief Rabbi Koretz and other representatives of the Jewish community, and it was at his residence that the meeting took place of Rabbi Koretz and the Greek Prime Minister, John Rallis, who had come to Salonika especially, and solely, in order to save the Jewish population.'' [361]
As soon as the measures against the Jews started, desperate appeals were addressed to Damaskinos, Archbishop of Athens and Primate of all Greece, by the Jews of Salonika, begging him to mediate with the representatives of the Reich in order to prevent their extermination. [362] Greek delegations went to see the Archbishop asking him to intervene. Archbishop Damaskinos, who shared the feelings of his followers, asked to see Altenburg, the representative of the Reich. He expressed to him the anguish of the Greek people at his inhuman and anti-Christian measure, and asked for his intervention to stop persecution. \textit{153} Altenburg replied that the Jewish question was of capital importance to National Socialism; that it was dealt with by the central administration and that, consequently, he, personally, could do nothing on behalf of the Jews of Greece. Actually, he shared the opinion that this measure should be taken, and should be applied to Jews throughout Greece. In spite of all protests, Jews of Greek nationality should be forced to go to Poland, while those of other nationalities should be returned to their countries of origin. The Archbishop asked: ``Why should Jews of Greece, who are of Spanish nationality, go to Spain, and those of Italian nationality to Italy, whereas, Jews of Greek nationality should be sent to Poland rather than be allowed to stay in Greece?'' Annoyed by this question, Altenburg refused to answer, except to say that Jews of Greek nationality were sent to Poland `to work'. ``If they are sent to Poland `to work', 'the Archbishop asked, why are women, children and aged people also sent?'' ``Because it is cruel to separate the families; if they are united they will have a better life'', the representative of the Reich replied. Another strong appeal to the German representative, based on the claims of a humane and Christian civilization, was made by the Archbishop. Altenburg vaguely replied that he would try to ease the strictness of the measure.
The extermination of the Jews of Salonika, however, continued unabated; the anguish of the Greeks increased. Greek organizations from all the towns sent appeals to the Archbishop of Athens who received an incessant stream of protestations and appeals from the Jewish organizations of Larissa, Chalkis, Volos and Verria, declaring their solidarity with the Jews of Salonika. The Archbishop decided again, to convey this general concern to the German authorities. He invited the representatives of the chief intellectual Institutions and of the scientific and professional organizations in the Archbishopric, to join with him. Under the auspices of the Church, they addressed a strong protest to the Greek Prime Minister, and to the representative of the Reich. The memorandum sent to the Prime Minister was as follows: \textit{154}
Athens, March 23, 1943.
Mr. Constantine Logotheropoulos,
Prime Minister,
In Town.
The Greek people have recently learned, with great surprise and grief, that
the German military occupation forces in Salonika have begun the gradual
expulsion of Jews living in Greece, and that the first groups of displaced
Jews are already en route to Poland.
The grief of the Greek people is even deeper because:
1. According to the spirit of the armistice terms all Greek citizens were to
be treated equally by the occupation forces, irrespective of religion and
race.
2. Greek Jews not only have been valuable contributors to the financial
progress of the country, they generally have been loyal and have shown full
understanding of their duties as Greek citizens. They have shared in the
common sacrifices on behalf of their Greek mother country, being among the
first to join in the struggle of the Greek nation to defend its historical
rights.
3. The well-known loyalty of the Jews living in Greece already rules out any
claim that they participated in actions likely to endanger the security of
the Military Forces of Occupation.
4. In the conscience of the Nation, the children of our common Mother Greece
are regarded as being an integral part of the Nation, entitled to enjoy
all the privileges of the national community, independently of any religious
or dogmatic differences.
5. Our holy religion repudiates any racial or religious distinctions,
supremacy or inferiority, stating that `there is neither Jew nor Greek'
(Gal. 3, 28), and condemns every tendency to create distinctions on
grounds of racial or religious differences.
6. The sharing of a common fate, both in days of glory and in periods of
national disaster, has produced unbreakable bonds between all Greek
citizens of every race.
We are well aware of the deep opposition between the new Germany and the
Jews, nor do we intend do defend or criticize international Jewry and its
activities in the sphere of the political and financial problems of the world.
We are only interested in, and concerned with, the lives of 60,000
fellow-citizens.
We deeply appreciate their noble feelings, brotherly disposition,
progressiveness, economic activities, and, above all, their incontestable
love for their country during the long periods we have lived together.
As a proof of this last statement, we point to the great number of Greek-
Jewish sacrifices offered, without complaint or hesitation, on the altar
of duty for our common homeland.
We are sure that the Government and the people of Greece are agreed on
this matter. We are confident that you have already taken the necessary
steps to plead with the Occupation Forces, to defer this painful measure
of the expulsion of Jews living in Greece. We are hopeful that you already
have pointed out to the highest authorities that such treatment of the
Greek Jews --- cruel in comparison with what happened to the Jews of other
nationalities --- makes this measure even more unjust, and thus morally
inadmissible. \textit{155}
If they pretend that these measures are taken for security reasons, an
adequate solution should be possible. Preventive measures could be taken,
such as the confinement of the males only (except aged men and children) in
a place in the country, under the supervision of the Occupation Forces.
Thus, security will be protected even against imaginary dangers, and the
Jews of Greece will not suffer the adversities of the expulsion. The Greek
people will be ready, if asked, to give their full guarantee for a measure
taken on behalf of their brothers in distress.
We hope the Occupation Forces will understand the senselessness of the
persecution of Greek Jews, who are considered the most peaceful, loyal
and productive elements in our country.
If, however, the Germans insist, against every hope, on their policy of
expulsion, we think that the Government, as the holder of the remaining
political power in our country, should take a firm stand against these
actions. It should be made clear that full responsibility for this injustice
will lie with the foreigners.
Let no one forget that all acts committed during this difficult period, even
those committed against our will and beyond our power, will one day be
examined by our Nation; it will ascertain the responsibility of everyone.
On that day of National judgment, the moral responsibility of those in
authority, who have failed to express by some courageous gesture the
unanimous anguish and protest of the Nation against all actions which are
derogatory to our unity and pride, such as the expulsion of the Jews, will
weigh heavily. [363]
\begin{quotation}
Yours Truly, Damaskinos, Archbishop of Athens and Primate of all Greece.\end{quotation}
The memorandum was signed by the president of the Greek Academy; the rectors of the University and the Polytechnic Institute; the chairman of the Association of writers, painters and artists; lawyers, surgeons, industrialists, and chambers of commerce.
It should be noted that the memorandum mentions six reasons why the Jews should not be deported; only one of them is strictly religious; four reasons stress that the Jews were loyal citizens of Greece and that they belonged to the nation. \textit{156} The Archbishop and his friends did not intend ``to defend or criticize international Jewry and its activities in the sphere of the political and financial problems of the world''. It is not clear whether they really meant this or tried to appeal to the mind of the addressee. At all events, the remark is regrettable.
Another memorandum was sent to the Representative of the Reich.
It read as follows:
Athens, March 24, 1943.
To His Excellency the Representative of the Reich for Greece,
Mr. Guenther Altenburg,
In Town.
Excellency, The undersigned are not seeking at present to interfere in any way in the questions of general tactics of the German forces in our country or elsewhere, but simply to submit certain views, regarding a question which is keeping the entire Greek population in suspense and anxiety; we are sure that you will examine these views in a spirit of benevolence and understanding. They concern the persecution of the Greek Jews of Salonika, who have long been legally under the jurisdiction of our country. Not only have they never given occasion for complaint, but on the contrary, they have always offered proof of earnest and sincere collaboration. In critical times, their acts of self-sacrifice and self-abnegation were apparent. We must add that the above mentioned Jews have never acted against our interests, even in the smallest matters; on the contrary, they have always felt a sense of responsibility towards the Greek majority. Most of them belong to the poorer classes. It should be noted that Greek Jews have quite a different mentality to that of the Jews living in Germany and have no knowledge whatsoever of the language of Poland where they are being sent to live. in addition to the above facts, we wish to add that during the long course of our history, ever since the era of Alexander the Great and his descendants, and through all the centuries of Greek Orthodoxy down to the present time, our relations with the Jewish people have always been harmonious. We believe therefore that, in your high office as ruler of our country during the present war, you will not hesitate to accept our present request and decide, even if provisionally, to suspend the expulsion of Greek Jews from Greece until the Jewish question can be examined in the light of a special and detailed investigation. Our present request is based upon the recent historical fact, that during the surrender of Salonika and, later, that of the whole of Greece, among the clauses of the protocol, the following is included: 'The Occupation forces promise to protect the life, the honour and the properties of the population'. Certainly this clause implies, that no persecution would be made against Greek subjects, on the account of religion and race, and that consequently the theory relating to racial or religious discrimination would not be applied in Greece. \textit{157} This was further confirmed later by a clear declaration made by General Tsolakoglou, to whom the Occupation Forces had entrusted the Presidency of this country, and who stated explicitly: ``There is no Jewish question in Greece and there never will be.'' ``All Greeks occupied in peaceful work may rest assured that their honour, life and property are under protection of the Occupation Forces and of the Government. Excellency, some days ago the Berlin radio transmitted an article of a German reporter, which was a real hymn to the traditional quality of hospitality of the Greek people in all occasions, even in the cases of supposed enemies. What must be the anguish of these people, who have been infused by thousand years of Christianity and its message of love of one's neighbour, when they see their brothers tom away from their homeland. Especially, when, for many years they have embraced it with unlimited confidence and a spirit of irreproachable solidarity towards us. Excellency, in the name of the lofty ideas of the Greek spirit, and of the culture of your country, both of which have so powerfully influenced the whole world, we beg that the expulsion of our Jewish fellow-citizens be halted as soon as possible. We assure you that the whole Greek nation will sincerely appreciate a gesture of such historic importance.
Damaskinos.
Archbishop of Athens and
Primate of All Greece.
(This Memorandum was also signed by the leading citizens who had signed the Memorandum sent to the Prime Minister).
There are some dubious remarks in this Memorandum: ``It should be noted that Greek Jews have quite a different mentality to that of the Jews living in Germany'', and ``In the name of the lofty ideas of the Greek spirit and of the culture of your country (Germany)''. That does not alter the fact that much in the Memorandum is to be lauded.
Archbishop Damaskinos did not cease his activities. He again saw Altenburg asking for his intervention. Following the formation of the new Government of John Rallis, he briefed the new Prime Minister and asked him to discuss fully the question with the commander-in-chief of East-Europe, Marshal Loehr. At the same time he took the following steps: \textit{158}
a. He requested the President of the International Red Cross in Greece to ask the Governments of the European countries, to interest themselves on behalf of the Jews of Greece, considering that their expulsion to Poland would mean total extermination. b. He negotiated with the International Red Cross to supply food for the kitchen established for the Jews of Salonika who had been put into a concentration camp. He then asked the Greek Government to furnish the necessary technical means. In fact, the kitchen started operating immediately. The Ministry of Social Welfare undertook its organization and the International Red Cross provided large supplies of food. c. He undertook, secretly, to send to Salonika the contribution of the Jews of Athens to the Jews of Salonika. Their contributions were sent by the Archbishop to Genadios, the Bishop of Salonika. Thus far the biographer of Archbishop Damaskinos.
Comments on the attitude of Church leaders and lower clergy are favourable:
``Monks, regardless of the great dangers or considerations of religion or faith, hid persecuted families and rendered secret but effective help to multitudes of unfortunate people, who could no longer subsist without employment, and thus had to leave their hiding place and give themselves up to the Germans.'' [364] ``The heads of the Orthodox Church in Greece defied the Nazi edicts and exhorted their faithful followers to shun anti-Semitic slogans and outrages. It is reported that in May 1943 alone, six hundred Greek priests were arrested and lodged in concentration camps because they refused to obey a Nazi order to preach anti-Jewish sermons. Much help and Jewish rescue work go to the credit of the Greek Orthodox clergy.'' [365]
What happened in Salonika enables us to realize that the attitude of Church leaders frequently had a very limited influence on the population, even in Greece. Dr. Nathan Eck, the editor of the revised edition in Hebrew of the book of Michael Molho and Joseph Nehama, has the following to say about the situation in Salonika:
``... The attitude of the non-Jewish population in Salonica to their Jewish neighbours was not very friendly. \textit{159} Many of them were former residents of Turkey who, in 1922, were transferred to Greece on an exchange basis, and their economic and social status was similar to that of the Jews. As a result of their feelings of hatred and competition, it was not easy to find anyone among the non-Jews who would agree to endanger his life and the life of his family in order to hide Jews in his home... The authors Molho-Nehama are wary of casting aspersions and blame on the general non-Jewish population but remain satisfied with mere hints. Here and there, there is a short remark which outweighs a host of express statements. For example, the following remark: 'It is likely that local factors (in Salonica) were active in the implementation of the deportations in order to get rid of competitors who proved a burden to them in their commercial life' (Part II, p. 11). Indeed, as the authors point out, only seventy Jews, most of them married to non-Jews, succeeded in finding hiding places in Salonica...'' [366]
Another comment:
``The great bulk of the population, while not indifferent, played the role of an interested if shocked spectator. However, this situation began to change after Archbishop Theophilos Damaskinos, who later became a regent, intervened forcefully on behalf of the Jews threatened with deportation. The Archbishop's vigorous protest about the action contemplated against the small Jewish population of Greece created a stir throughout the country.'' [367]
The attitude of the non-Jewish population in Salonika, where most of the Jews were living, was lamentable. Such information should prevent us from accepting stereotypes such as ``the Greek --- or the Dutch, or the French --- population has done everything to save the Jews''.
b Athens and Southern Greece
Following the Italian armistice, the Germans took over the administration of Athens and other parts of Southern Greece. General Stroop, the ``Conqueror of the Warsaw Ghetto'', arrived in Athens on September 10, 1943, and took over the function of Higher SS leader. \textit{160}
On October 3, 1943, the Jews were ordered to register. The seizure of the Jews on the Greek mainland was to be completed in three days, from March 23-25. Jews living on the Greek islands were deported in June and Jule, 1944. More than sixty thousand Jews out of the 79,950 who had been living in Greece, were deported. [368] The following is quoted from ``The Destruction of Greek Jewry, 1941-1944'':
``... On Tuesday, September 21, 1943, Athens' Chief Rabbi, Elia Barzilai, was ordered to submit to the German authorities a list containing the names and addresses of all Jews living in Athens... A delegation led by Rabbi Barzilai paid a visit to the Archbishop who declared that, to his deep regret, he did not see how he could do anything on behalf of the Jews, despite his willingness to help them. The only alternative left was to go into hiding, or disappear, the Archbishop said. When the Rabbi requested permission for the Jews to hide in the churches, the Archbishop replied: 'Willingly, but it is a mistake to think that there you will be safe. They will not hesitate to seize you. However, I could, with the help of the English, arrange a transfer to the Middle-East for those Jews who are prepared to go...''' [369]
At the instigation of Archbishop Damaskinos, priests preached in the churches that Jews should be aided. He also intervened with the German authorities so that children younger than 14, as well as, persons married to parties of the Greek Orthodox faith, should be exempted from the strict anti-Jewish regulations. [370]
According to Moissis, the fact that more than 10,000 Jews saved themselves was largely due to the efforts of the Orthodox Church under Archbishop Damaskinos. A few days proceeding the German attempt to corral the Jewish population, the Church issued a circular to all priests, parishes and convents, exhorting them to lend succour and safety to the victims of Nazi barbarism. [371] \textit{161}
I have not succeeded in retrieving a copy of this circular, nor was Mr. Moissis able to give any additional information. He confirmed to me that Archbishop Damaskinos had done much for the rescue of the Jews:
``Archbishop Damaskinos knew my place of refuge, in the neighbourhood of Athens, and sent me provisions every month. He did the same for other Jews ...whose hiding place he knew.'' [372]
It seems unlikely that a circular letter was issued: a copy might easily have fallen into the hands of the persecutors. In those days one did not put such a message in writing but it was passed on orally.
26 DENMARK
a. The Time of Moderation
Germany occupied Denmark on April 9, 1940. The position of Denmark under the German occupation was unique in many respects: the King had remained; the Danish Government continued to function until August, 1943; the Germans were interested in keeping things as quiet as possible and granted to Denmark a certain independence in internal affairs, and the attempt to deport and exterminate the Jews of Denmark started relatively late: September, 1943. A total of 7,700 Jews were living in Denmark, a number of them refugees from Germany and elsewhere. \textit{162}
In December 1941, participants in a conference of Danish pastors [373] considered the possibility of presenting a petition to Parliament demanding that all members of Parliament should vote against any racial legislation. But the proposal was withdrawn as it was considered undesirable to focus to much public attention on the question. [374] The same question was discussed at another conference of pastors which met in the provincial town of Askov. One of the participants wrote to Rabbi Friediger:
``... For us it is not just a question of the Jews and their rights; for the Danes this first of all must be the question of the right of a small nation to exist, particularly as this is also a question of our whole national attitude and the basis of democracy: equality and human dignity.'' [375]
Frederik Torm, a professor of theology at the University of Copenhagen, brought about a common decision of the theological faculty and of the students, declaring that, should persecution of the Jews begin, they would voice their opposition vigorously and publicly. This internal decision was put into practice, in October, 1943. [376]
The Church Press in Denmark could publicly denounce anti-Semitism at a time when the Press in other countries had long since been completely silenced. The Rev. Johannes Nordentoft, in one of his articles, called for an active war against the anti-Semitic propaganda of the Nazi press. He pointed out that ``those who remain silent or disapprove by merely shrugging their shoulders become accomplices''. [377] An article in the Church gazette of Sonderbourg, edited by Dean Halfdan Hoegsbro, stated: \textit{163}
``Hatred of the Jews is prompted by the demand for a scapegoat... We will not lend our support to the introduction of anti-Jewish laws; Jew hatred is an infectious disease, to which the innate sense of justice of the Danish people will not permit them to succumb. It is a disease that we shall cast out from our midst. Shame upon us if we ever allow ourselves to fall victim to it.'' [378]
The Skydebjerg-Aavup Church Gazette, comparing the anti-Jewish drive to that of medieval times, wrote:
``Our Danish minds will not let themselves become infected by this disease... Anti-Jewish legislation is tantamount to lawlessness, and if we forsake justice, then we will be submitted to a degradation worse than war and suppression.'' [379]
In January, 1943, the Bishop of Copenhagen, Dr. Fuglsang-Damgaard, publicly warned against racial hatred. [380] The pro-Nazi press frequently attacked ``the Church's dogged opposition to attempts to initiate anti-Jewish restrictions''. [381]
The first occasion on which the Danish Bishops approached the authorities en bloc to protest on behalf of the Church of Denmark, was when they addressed a protest to the Minister of Justice containing the following paragraph:
``... We draw to your attention the feeling of protest which is spreading in the Church of Denmark. This feeling of protest is due, above all, to the way in which justice is administered in these days. Men are being arrested without the public being given any information about how the arrested persons are treated in prison. Anti-Semitic propaganda is being artificially incited. At the same time pastors receive warnings from the Government that they must not comment on the persecution of the Jews...'' [382] \textit{164}
b. The Deportation Attempt; the Protest
In the summer of 1943, disturbances occurred in several provincial towns. The Germans took reprisals and the people reacted to this by proclaiming strikes. A German ultimatum was rejected by the Danish Government. Thereupon martial law was proclaimed on August 29, 1943. Dr. Werner Best, the German envoy in Copenhagen, received full powers as Reich pleni-potentiary. The Danish Government had resigned. The day to day affairs of its ministries remained in the hands of the permanent Department directors; the director of the Danish Foreign Ministry, Nils Svenningsen, became the chief spokesman of the administration. The Germans now planned the deportation of the Jews in one night, October 1-2. On September 28, however, a German in Copenhagen, Duckwitz, revealed this to Danish friends of his, H.C. Hansen and H. Hedtoft, who warned Henriques, the president of the Jewish community. On the morning of September 29, the Jewish congregations which met in their synagogues for the services of the Jewish New Year were warned. The raids took place as planned. In the night of October 1-2, 202 Jews were captured in Copenhagen and 82 elsewhere in Denmark. About 200 others were arrested later on, most of them caught in flight. The great majority, however, succeeded in hiding themselves. The Swedish Government had publicly expressed its willingness to admit the Danish Jews into Sweden. 7,220 Jews were secretly moved to the beaches and then ferried by Danish fisherman to safety. [383]
At the end of August, 1943, the Bishop of Copenhagen, Dr. Fuglsang-Damgaard, asked for an interview with the Director of the Foreign Ministry, who declared that the Jewish question had not been raised. Nobody had been arrested because of race or religion. When the Director had asked Dr. Best about this matter, he had answered: ``The question has not been broached at all''. [384] \textit{165} Dr. Fuglsang-Damgaard reported this in a letter to the pastors of his diocese, dated September 4, 1943, adding that later developments would be followed attentively. ``From our experience with the German habit of breaking promises, it was not thought wise to take Dr. Best's words too seriously. Unfortunately however, his words perhaps did set our minds too much at rest.'' [385] The Churches, however, made necessary preparation in case persecution of the Jews would begin. Bishop Fuglsang-Damgaard convened with pastors belonging to the unofficial Pastors' Organisation P.U.F. [386] and asked them to prepare a draft for a public protest, to be read out from the pulpits. It was ready a short time later. The Bishop suggested some changes but there was hardly time to make them as events developed rapidly. On September 17, 1943, some Jewish houses in Copenhagen were raided. Bishop Fuglsang-Damgaard thus had another interview with the director of the Foreign Ministry, Svenningsen. In a letter to the Bishops, dated September 23, he informed them that:
``... the raid did not indicate that they (the Germans) would raise the Jewish question, but that it was connected with a suspicion of certain persons. Thereafter I asked the Director of the Department to inform the German authorities that their raising of the Jewish question would be met by a joint protest by the Church and the Bishops. The Director promised to inform the Germans of this...''
Bishop Fuglsang-Damgaard relates:
``The Jewish community was in a very difficult situation. The chief-rabbi, Dr. Friediger was interned in the camp of Horsercad, just as the time of the great feasts of the year was approaching. We did what we could to obtain his release, so that at least he could lead the services during the feasts. At the beginning of the fateful week (during the night of the first October) I paid a visit to the chairman of the Jewish community organisation, the advocate of the Supreme Court, C.B. Henriques. \textit{166} I shall never forget it. I came to express our heartfelt fellowship with his community and to say that we were remembering the Jews in our prayers, not the least in those days when they celebrated their great feasts, and also in order to assure him that we would do what we could to help the interned to get their liberty again...'' [387] ``On 29th September, about 10 o'clock, the chairman of the Jewish community organisation, Advocate Henriques, came to me and told me that it was almost sure that the Jewish question would now be raised. There existed an order from Hitler himself to raise it. The ships for the deportation were said to be in the harbour. I went at once to the Department of Religious Affairs and asked for an interview with the Director of the Department who, however, at that time did not know anything about such imminent action. Immediately after this I went to the Department of Foreign Affairs and obtained an interview with the Director. He told me that, according to information he had received from different sources, there could be no doubt that the situation was very serious. A meeting of the Directors of the Departments was to be held on the question at two o'clock...I returned to the Ministry of Religious Affairs, in order to tell Mr. Thomsen, the Director of the Department, how serious the situation was, asking him to present a protest to the meeting and to inform the German authorities also about the contents of the protest.'' [388]
Bishop Fuglsang-Damgaard then returned to his residence. The protest was written in the presence of his assistants in the office. He signed it on behalf of the Bishops.
``We were conscious that this was a decisive moment. We expected at the time, that the signature would cost me both my office and my freedom. The protest was sent by a messenger to the Director of the Department to whom personally it was handed. I went to the Dean in order to arrange with him the things to be done if I should be arrested.'' [389]
All the Bishops received the protest by express letter, with a request for their immediate support and with the appendix:
``In case persecution of the Jews should begin, this Protest must be read in the churches, and I propose that the pastors commence the reading with the following sentence: 'On the 29th September of this year the Bishops sent to the leading German authorities, through the Directors of the Departments, a letter with the following contents:...''' \textit{167}
On Saturday, October 2, 1943, theological students despatched the Protest to all the manses in Bishop Fuglsang-Damgaard's diocese. On that same evening, the Bishop again was advised by the Ministry of Religious Affairs and the Ministry of Foreign Affairs, to consider the consequences. ``But there was nothing to reconsider. The matter had to be completed.'' [390]
The Protest
``Wherever persecutions are undertaken for racial or religious reasons against the Jews, it is the duty of the Christian Church to raise a protest against it for the following reasons: 1. Because we shall never be able to forget that the Lord of the Church, Jesus Christ, was born in Bethlehem, of the Virgin Mary into Israel, the people of His possession, according to the promise of God. The history of the Jewish people up to the birth of Christ includes the preparation for the salvation which God has prepared in Christ for all men. This is also expressed in the fact that the Old Testament is a part of our Bible. 2. Because a persecution of the Jews is irreconcilable with the humanitarian concept of love of neighbours which follows from the message which the Church of Jesus Christ is commissioned to proclaim. With Christ there is no respect of persons, and He has taught us that every man is precious in the eyes of God. 'There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Christ Jesus.' (Gal. 3, 28). 3. Because it contradicts the sense of justice, inherent during centuries in our Danish civilisation and which lives in the Danish people. In accordance with the above principles, all Danish citizens have equal rights and duties before the law and freedom of religion assured to them by the constitution. We understand by freedom of religion the right to exercise our faith in God according to vocation and conscience, in such a way that race and religion can never be in themselves a reason for depriving a man of his rights, freedom or property. Despite different religious views we shall therefore struggle to ensure the continued guarantee to our Jewish brothers and sisters of the same freedom which we ourselves treasure more than life. The leaders of the Danish Church are conscious of our responsibility to be law-abiding citizens; we do not needlessly revolt against those who exercise the functions of authority over us; but at the same time, we are obliged by our conscience to maintain the law and to protest against any violation of human rights. Therefore, we desire to declare unambiguously our allegiance to the word that we must obey God rather than man.''
On Behalf of the Bishops:
Fuglsang-Damgaard. [391]
\textit{168}
What strikes us is that the Public Protest stressed the special relationship existing between Christians and Jews, while the second point of the protest states that ``every man is precious in the eyes of God''. The text mentioned (which also was quoted by many other Churches in different lands) seems more applicable to the position of members of the Church who are of Jewish origin ("There is neither Jew nor Greek,... for ye are all one in Christ Jesus"). However, Christians of Jewish origin were not mentioned in the Protest at all. This in itself was certainly fortunate, for reasons discussed in ch. 4. Finally, the letter of Protest states that ``we must obey God rather than man''. It must have been clear to every church goer that, in fact, the Bishops were summoning him to active resistance against the German measures.
In one of the churches in Copenhagen the Bishop began his sermon on that particular Sunday by telling what had happened and unequivocally expressing his own view. Finally, when the protest was read out to the congregation as a Pastoral Letter of the Church leaders, all those who were present stood up in order to express their approval. [392] A Danish Lutheran pastor informed me that whenever the Danish Bishops issue a public declaration, the faithful consider two questions: 1 Is what the Bishops say right? 2 What gave them the right to speak on my behalf? When, therefore, the congregation stood up when Bishop Fuglsang-Damgaard read out the protest, this can be seen as expressing the congregation's opinion that he had rightly spoken on their behalf. No Bishop nor pastor, to the best of my knowledge, directly suffered or was even arrested because of the public protest.
In conclusion of this paragraph we record Bishop Fuglsang-Damgaard's comment on the situation after the Church had given its testimony:
``The protest had been made and it was not repeated. A repetition would have meant a weakening of it. Furthermore, it would not have been of any use. That was clear to everyone who knew the situation. What had now to be done was to bring help to those compatriots who were deported, persecuted or in hiding. \textit{169} The whole Danish population understood this and all circles in our country came together to render this help. This was a time when there was no rest by day or by night; when it happened that a man in the street would come and give one 10,000 kroner; when a code-language developed in order to keep the mutual contact alive; when one felt an unspeakable happiness and gratitude whenever somebody was saved.'' [393]
c. After the Rescue Operation
Bishop Fuglsang-Damgaard and other Church leaders also contributed to the sending of gift parcels to the Jews who had been deported. On November 29, 1943, the Bishops jointly addressed Dr. Best through the Ministry of Foreign Affairs in order to gain his support for this work. The appeal read as follows:
``It is with deep sorrow and disappointment that we perceive through developing circumstances, that our appeal to the German authorities over the Jewish question has not born fruit. But our interest in, and deep sympathy with, our deported countrymen is undiminished, and as there now seems to be a possibility that we can send support and aid in the form of food from this country, we wish to suggest to the Danish Church communities that they should send help to the interned Jews, in the form of gift parcels, through the Red Cross. In our relationship with the community, we know that the Christian conscience of our people and their conception of justice has suffered a painful wound, and how deep a need they feel to help. We would therefore be grateful to the Director of the Department of Foreign Affairs if he would inform Dr. Best of our attitude and point out to him that support from competent German representatives towards a good solution of this question would be met with deep satisfaction within Church circles, the members of whom would, through this Christian and humane activity, find a way to express their deep concern over this matter.'' [394]
The Ministry of Foreign Affairs, the Ministry of Social Affairs, the Red Cross and the pastors of Copenhagen acted unanimously in this large-scale assistance to the Danish Jews in Theresienstadt, an action which was successful beyond all expectation. Of the 475 Jews who were deported to Theresienstadt, all returned with the exception of 53 who had died. \textit{170}
In December, 1943, Bishop Malmstrom prayed for the Jews in a broadcast religious service. Thereupon the German authorities demanded the right to make a preliminary censorship of broadcast services. Bishop Fuglsang-Damgaard then sent a statement through the Foreign Ministry to the German authorities, in which he stated that if censorship was introduced, neither the Sunday services nor the morning devotions would continue to be broadcast, and that the reason for this measure would be made public from all pulpits. A week later, the Bishop was informed by the German authorities that ``the incident was due to a misunderstanding''. [395]
In February, 1944, the Bishops sent a letter to their congregations in which they requested prayer ``for God's ancient chosen people, trusting that God will help where we see no way to do so.'' [396] \textit{171}
The crucial question, whether the Church was influenced by general public opinion or whether it was the other way round, has been discussed in ch. 2.
THE SATELLITE COUNTRIES
27 SLOVAKIA
On the eve of the German invasion of Czechoslovakia, on March 14, 1939, Slovakia declared its independence, and on March 23, the agreement of German protection was signed. Following the first Vienna award on November 2, 1938, parts of former Slovakian territory with about 40,000 Jews were annexed by Hungary, together with parts of Sub Carpathian Ruthenia. After the occupation of all of Carpatho-Ruthenia containing 100,000 Jews, by Hungary, about 90,000 Jews remained in ``independent'' Slovakia. A Catholic priest (Dr. Josef Tiso) was head of the Slovakian State. On April 18, 1939, the first anti-Jewish decree was enacted. A special Department for Jewish Affairs was opened in the Ministry of Interior. It co-operated with the Hlinka Guard.
The Council of the Evangelical (Lutheran) Pastors' Union decided, in its session of November 21, 1939, to send a Memorandum to the President and the Government of Slovakia, regarding the Hlinka Youth organization and the Hlinka Guard. We quote the following:
``We, as Evangelical Christians and as citizens, cannot agree with the following facts: the annulment of individual rights and freedom of certain people; the taking of steps against the Jews without legal basis, by means of violence, for instance, that the men of the Hlinka Guard, during the night, dragged Jews --- women, mothers and children --- out of their beds and transported them to concentration camps; illegally imposing of fines \textit{etc}.; transgressions which are performed though they are contrary to the law and to Christian ethics.'' [397]
\textit{172}
The first deportation train left Slovakia on March 26, 1942. In August 1942, the Jewish population had been reduced to 25,000. On August 23, 1944, a rebellion broke out which was ruthlessly quelled. In the autumn of 1944, 13,500 of the remainder of Slovakian Jewry were deported. In the whole of Slovakia there remained not more than about 4,000 to 5,000 Jews. [398]
The Convent of (Lutheran) Bishops, under the Chairmanship of Dr. Vladimir Cobrda and Dr. Samuel Stefan Osusky, decided to issue a Pastoral letter about the ``Jewish Question'', on May 20, 1942. We quote the following:
``... The Evangelical (Lutheran) Church neither can nor wishes to interfere in the executive power of the competent government departments, whose duty it is to solve the problems. The Church, however, is convinced that it is possible and thus also necessary to solve this problem in a just, humane and Christian way, according to the Christian principles which are based on the eternal laws of God and the teaching of Christ. According to this teaching, all men are endowed with the right to live, to earn a honourable livelihood, and the right to family-life. It also protects the honour of the Jews as human beings, so that not one of them should feel deprived because of his national, religious or racial attachments. The racial law however, which some people champion, is contrary to the Christian faith, which accepts the biblical message that God is the Creator of all things and of all mankind, 'from whom every family in heaven and on earth takes its name' (Ephesians 3, 14). 'He is the head, and on him the whole body depends. Bonded and knit together by every constituent joint, the whole frame grows through the due activity of each part, and builds itself up in love' (Ephesians 4, 16). To our sorrow we have been compelled to witness deeds which cannot be justified. They are contrary to human feelings, to justice and to the law of God; they are in no way related to love. Such things could not happen, if all would honour the declaration broadcast by the Ministry of Interior, that no harm would be done to the Jews, that they would be treated in a humane and Christian way, and that they should just have to work as the other citizens. \textit{173} The Church cannot reconcile itself to these deeds which we have witnessed in many places. The Church cannot but express its sorrow about them and reject them. If members of the Evangelical Church participated in these deeds, they must be severely condemned for this...'' [399]
``The Times'' of August 11, 1942, commented on this pastoral letter as follows:
``The Slovak Lutheran Church, under the leadership of the Bishops Dr. Cobrda and Dr. Osusky, has taken the lead in the fight against Nazism in Slovakia. From the pulpits of all Protestant Churches in Slovakia a pastoral letter was read on May 31. In this the bishops condemned an `immature political ideology' modelled on Nazi and Fascist lines and emphasized loyalty to the Gospel of Christ. They also condemned the anti-Jewish policy and defended the right of the Church, to baptize proselytes from Judaism on religious grounds. The pastoral letter, the first of its kind in this part of Europe, has caused a profound sensation in central and south-eastern Europe (particularly in Hungary, where a substantial Protestant congregation exists). Nazi circles in Slovakia are particularly aggrieved since the bishops in question are considered as leading authorities in Church matters, even outside Slovakia... Roughly one sixth of the Slovak population are Protestants.''
We have discussed the matter of the so-called ``mercy-baptisms'' in chapter 5. Suffice it here to mention that pastors in Slovakia were in peril of their life if they dared to baptize Jews, during the second world war.
28 RUMANIA
In June, 1940, the Russians took back Bess Arabia and occupied Northern Bucovina. In August, Hungary carved out for itself Northern Transylvania and the Bulgarians occupied Southern Dobrudja. On September 5, General Ion Antonescu took over the government as Conducator of Rumania, and on October 7, 1940, German troops arrived in Rumania. At the beginning of 1941, the Fascist Iron Guard tried to overthrow General Antonescu. The revolt was crushed, but members of the Iron Guard had murdered hundreds of Jews in Bucharest. \textit{174} In June, 1941, Germany invaded Russia; Rumania reconquered Bucovina and Bess Arabia. On July 29, 1941, Rumanian soldiers murdered at least 4,000 Jews in Jassy. The Rumanians deported an estimated 185,000 Jews from Dorohoi, Bucovina and Bess Arabia to Transnistria, in the Soviet Ukraine. By May, 1942, about two-thirds of these Jews had died. [400]
Strong anti-Semitic influences were manifest in the Rumanian Orthodox Church. On August 18, 1937, Patriarch Miron Cristea had issued a statement calling upon the Rumanian nation ``to fight the Jewish parasites''. [401]
Chief Rabbi Dr. Safran relates his frantic efforts to try to avert the deportation of the Jews in the districts of Dorohoi, Bucovina and Bessarabia. It was decided that he should approach the head of the Orthodox Church, the old Patriarch Nicodemus.
``... During the dramatic conversation I had with the Patriarch, who was rather indifferent at the beginning pretending that it was all the affair of the government, he changed his attitude in view of my growing emotion which I was unable to hide from him. I spoke of the terrible responsibility he was taking upon his conscience in the eyes of the Supreme Judge, and ended by throwing myself at the feet of his pontifical seat. Deeply moved, the Patriarch lifted me up and promised to do his best. On taking my leave of him I sensed that he intended to ask for the support of the Queen-mother.'' [402]
Chief Rabbi Safran immediately took steps to get in touch with King Michael and the Queen-mother Helena to prepare them for a possible appeal from the Patriarch Nicodemus.
``The Patriarch, on his part, first sought unsuccessfully, to intervene with Antonescu; and then addressed himself to the King and the Queen-mother.
\textit{175}
The Queen-mother suggested that Baron Manfred von Killinger, the German ambassador, should be invited to the palace for a meal during which a last appeal should be attempted. In the course of this dinner the Queen-mother spoke fervently on behalf of the innocent victims, but he, in the presence of the King and the Patriarch, responded with an obstinate, brutal refusal.''
These interventions of the Queen-mother and the Patriarch (who unfortunately was to disappoint Dr. Safran later on) nevertheless helped to make it possible for the rest of the Jewish population of Czernovitz to stay in the Bucovinian capital. [403]
Chief Rabbi Safran then heard of the arrival of the Metropolitan of the Bucovina, Tot Simedrea, in Bucharest, whose anti-Semitic feelings were known. Nevertheless Dr. Safran called on him.
``Contrary to my expectations, Mgr. Simedrea revealed an understanding attitude. He told me of the feelings aroused in him by the sight of the Jews of Czernovitz being deported to the ghetto, during which he had seen a Rumanian soldier carrying a sick old Jewish women on his shoulders. He also had heard the heart-rendering cries of Jewish mental patients who formed part of this tragic convoy. The Metropolitan effectively intervened with the Government of Bucharest and on his return to Czernovitz exerted pressure on the Governor-General of the Bucovina. These, together with other similar appeals, brought to an end the deportation of Jews from the capital of this province.'' [404]
In the summer of 1942, pressure was exerted on Antonescu by the Germans, to order the deportation of all Jews of Rumania. The Germans obtained the consent of the Rumanian Government for this. Trains were already prepared for the deportation. Then a delegation of the Jewish communities of South Transylvania informed Dr. Safran that all technical steps for the operation had just been taken in their province. Appeals to the authorities had been in vain. Dr. Safran relates:
``One sole course remained to be tried --- an appeal to Metropolitan Balan, head of the Orthodox Church of Transylvania, well-known both for his anti-Semitism and for the great influence he had with leading figures in the government, and with Marshal Antonescu in particular. \textit{176} Following a brief consultation we gave up the original idea of my proceeding to Sibiu, for fear of arousing the attention of the Gestapo and the Centre for Jewish Affairs. I accordingly adopted a most daring course. Using the services of an intermediary, I begged the Metropolitan to come to Bucharest.''
In the meantime, Metropolitan Balan had come to the capital and informed Dr. Safran by telephone that he would be waiting for him at the house of General Vaitoianu with whom he was staying.
``Our meeting took place in an extremely tense atmosphere. I assumed an accusing tone which could only have been inspired by despair. [405] The Metropolitan walked up and down the room without saying a word. Finally he took up the telephone and called Marshal Antonescu with whom he asked for an urgent interview. The Marshal was reported to be busy, but they agreed to have lunch together. In the meantime I communicated to Mgr. Balan the news that for several weeks the authorities in Bucharest had been deporting not only Jews, condemned without trial, of not having reported for compulsory labour, but also their parents and children. The Metropolitan immediately telephoned the Vice-Premier, Minister Michael Antonescu, and told him what he just had learned. The Minister promised to look into the matter. As a result, after a few days there were no more deportations from Bucharest. I accompanied the Metropolitan to his car which was to take him to the Dictator, pleading with him to use all the means in his power to obtain a favourable decision. My prayers followed him after he had left... Three hours later the sonorous voice of the Metropolitan told me over the telephone that the Marshal had given in. The Jews of South Transylvania had been saved.'' [406]
There are other countries in which Church leaders courageously and whole heartedly stood up for the Jews and yet their interventions seldom had any result at all. In Rumania, however, the intervention of the Orthodox leaders seems to have been quite successful. It is typical of Rumania that no public protests were issued. Church leaders personally intervened. These interventions took place only after Chief Rabbi Safran had implored the Orthodox leaders to come to the rescue of the Jews.
\textit{177}
It is difficult to ascertain what exactly moved these apparently reluctant saviours to take action. The change of heart with Patriarch Nicodemus seems to have come after Dr. Safran had spoken ``of the terrible responsibility he was taking upon his conscience in the eyes of the Supreme Judge''. Metropolitan Simedrea told Dr. Safran ``of the feelings aroused in him by the sight of the Jews of Czernovitz being deported to the ghetto ``. According to a report of Matatias Carp, there was in 1940 a Jewish population in Rumania of approximately 760,000, of whom 400,000 were massacred. ``Among the victims, 250,000 lie on the conscience of the Rumanian Fascist Government directly. [407]
There are two other non-Roman Catholic Churches of some importance in Rumania. The Reformed Church of Rumania is the Church of the Hungarian national minority. I have not been able to find any particulars about the attitude of this Church regarding anti-Semitism. The Evangelical (Lutheran) Church of the Augsburg Confession is mainly the Church of the German immigrants.
In spring 1942, the National Consistory of this Church decided, on a motion of Bishop Staedel, that their Church would join the ``Institute for Research into the Jewish influence upon German Church life'', founded in Eisenach (Germany). A study group was formed, which, in close contact with the Eisenach Institute, sought ``to make the results of its scientific work fruitful for the life and future form of the Lutheran Church in Rumania''. At the first conference of the study group, at the beginning of March, 1942, the following statement by Bishop Staedel was accepted as the guiding principle for the work as a whole:
``We are deeply convinced that at this time of national revival, we are making it extremely difficult for a German to come to Jesus Christ if we present him with a continuous and detailed treatment of the Old Testament. In the two hours every week, which are meant to be devoted to bringing the message of the Saviour to the German in his national character and community, we have absolutely no room for the national and messianic history of the Israelite-Jewish people. \textit{178} Therefore we advocate the elimination of the Old Testament so far as possible from the religious life of the Germans, and thus from the Lutheran religious instruction.'' [408]
A statement made by the leader of the ``scientific work'' of the study group described the motives underlying its work as follows:
``The decisive impulse has come to us from outside, from the political life of the German people. In the national-socialist revolution, however, this nation has confessed to a year-long guilt, for having failed to guard its God-given torch of the Nordic Aryan vision of life, allowing it to flicker and die out under the influence of foreign, especially Jewish, intrigues. Now this light will once again burn for the nation in all its purity. What wonder then that people are now coming to the Church, demanding of it the same confession of guilt, even more insistently because the Church has taken the Bible of the Jews into its own canon of Scripture. Thus it has consciously held open at least an aperture through which an essentially foreign spirit could infiltrate into our national life. [409]
Bishop D. Friedrich Mueller, the present head of this Church, replied to my circular letter and stated:
``As soon as the alliance (between Germany and Rumania) came into force, the fascist government of Rumania promulgated a law by which a 'German community in Rumania' was constituted. Par. 4 of this law granted to the leader of this community the right, to issue decrees compulsory upon the Rumanian citizens of German descent. Thus supported, the `leader of the community' succeeded in compelling Dr. Victor Glondys, the Bishop of the Evangelical Lutheran Church in Rumania, to resign, whereupon he appointed his political associate, Wilhelm Staedel, as the head of the Church. Even Staedel did not give in to him completely but tried to follow the policy of the `German Christians'. [410] In a admonition to my congregation I made a stand against both attempts. This led to several actions of persecution... By secret consultations we could win about 80 per cent of our pastors for resistance and a clear Christian preaching, based on the Old as well as the New Testament... I do not know of any case in which members of my Church co-operated in the persecution of the Jews. Unfortunately there are no documentary proofs of this, because of the atmosphere of the time. During the fascist dictatorship in Rumania censorship existed, which prevented publication of statements on behalf of the Jews. \textit{179} I could not, for instance, publish my warning mentioned above nor send it by mail. Copies of it had to be passed on from hand to hand. Similarly, as a precaution, I had to destroy my archives during the persecution. I myself no longer have a copy.'' [411]
I requested Dr. Safran to comment on this letter. He replied:
``Concerning the attitude of the Evangelical Lutheran Church in Rumania towards my co-religionists in distress during the period of Nazi oppression, I must tell you that we did not receive any help or comfort from this Church in our terrible suffering, not even a token of human compassion.
In 1942, in order to request his intervention on our behalf, I intended to go to the Metropolitan of the Orthodox Rumanian Church, Mgr. Balan, whose residence was in Sibiu, where also was the Centre of the Evangelical Church. I was warned, however, that the members of this Church living in Sibiu were capable of betraying me to the Gestapo --- with which they maintained direct relations --- in order to prevent me from approaching Mgr. Balan.'' [412]
The letter from Bishop Mueller seems to suggest that there existed a kind of ``Confessing Church'' in Rumania. If this name is correct for the group mentioned by him, it should be added that the existence of this ``Confessing Church in Rumania'' was not, contrary to what can be said about the Confessing Church in Germany, a very manifest phenomenon. Apparently its existence was not manifest to Dr. Safran. Bishop Staedel ``tried to follow the policy of the German Christians''. He certainly matched them in anti-Semitic heresies.
29 BULGARIA
a. The Preliminary Phase
Bulgaria was part ally, part satellite of Germany. In September, 1940, it acquired southern Dobrudja from Roumania. In March, 1941, the German army was admitted to Bulgaria. The Germans took Macedonia from Yugoslavia, Thrace from Greece, and handed them over to Bulgaria. \textit{180} The number of Jews in Bulgaria at the end of 1939 amounted to 50,000. Approximately 15,000 more were added to the Bulgarian power sphere in the newly won territories. [413]
The ``Law for the Protection of the Nation'; containing provisions for the definition, expropriation and concentration of the Jews, was adopted by a majority of the Parliament at the end of December 1, 1940, and promulgated on January 21, 1941. In August, 1942, wearing the yellow star was made compulsory for the entire Jewish population. At the same time Belev was appointed as Bulgarian Commissioner for Jewish Affairs.
On November 15, 1940, the ``Holy Synod of the Bulgarian Church'' sent a letter of Protest to the Prime Minister (Filov), with a copy to the Speaker of the Parliament. The letter was signed by the Deputy Chairman of the Holy Synod, Metropolitan Neophyte. It read as follows:
``The Bulgarian Church has always kept a faithful and watchful eye on the destiny of the Bulgarian people throughout its existence. She has always had an unbroken link with its destiny, and shared in its wishes and longings, its joy and sorrow, its pain, its misfortunes and ideals. This concern of the Bulgarian Orthodox Church for the Bulgarian people was strongest in days of trial and danger. In such days she did everything in her power to prevent the nation from making big mistakes, as it was capable of doing, and to protect it from the dangers and calamities that threatened it. And whenever the warning voice of the national Church was heeded our people was kept from major disasters. On the other hand, when it ignored the warning voice of the Church, our people underwent danger and suffering. The Bulgarian Church follows with great satisfaction the efforts of our people and those of the Bulgarian authorities to protect the people and the fatherland from dangers that lie in wait for them from different quarters. Therefore, now too, the national Church is very glad to note that the Government is preparing a `Law for the Protection of the Nation', to protect our people and everything Bulgarian from such dangers. The Church considers it her duty, however, precisely for the benefit of the nation, to draw the attention of the competent authorities to several defects in the proposed law, which could have bad consequences, and which also touch the Church as a divine institution, whose duty it is to watch over all her spiritual children and cause the will of God to rule in the cause of righteousness and mercy among human beings and the nations... \textit{181} Let no account be taken of laws against the Jews as a national minority, but let purposeful steps be taken against all the real dangers to the spiritual, cultural, economical, public and political life of the Bulgarian people, from whatever direction these dangers come.'' [414]
It is typical of this letter that most of its contents could also have been written by any anti-fascist political, party, instead of by a Church.
Early in 1941, it became known that the ``Law for the Protection of the Nation'' was going to be ratified. Metropolitan Stephan then called for a plenary session of the Holy Synod of the Bulgarian Church, which passed a resolution agreeing to send a letter of protest to the Prime Minister and to the Minister of Foreign Affairs, in which it was pointed out that:
``... The principle of racialism which is the basic idea on which the above mentioned law is founded, has no justification from the point of view of the teachings of Jesus... The principle of racialism which encourages persecution and denies the rights of people, merely because of their race, in this case the Jewish race, has no justification, and therefore one cannot base the 'Law for the Protection of the Nation' on the principle of racialism. One cannot turn the `Law for the Protection of the Nation' into a means of oppression and persecution of the Jewish minority in the land.'' [415]
On September 9, 1942, the Metropolitan of Sofia, Stephan, preached a sermon, probably in preparation of the ``Feast of the Exaltation of the Honourable and Life-giving Cross''. This feast of the Orthodox Church falls on September 14. The Metropolitan declared that:
``... God had punished the Jews for the crucifixion of Jesus in that He had expelled them from their country and had not given them a country of their own. And thus, God had determined the destiny of the Jews. \textit{182} However, men had no right to exercise cruelty towards the Jews and to persecute them. Especially Christians ought to see their brothers in Jews who had accepted the Christian religion and to support them in every possible way. He stressed several times in his sermon that truly it is in God's hands to punish twice and three times, but it is forbidden for Christians to do such a thing.'' [416]
Apparently there existed a brand of ``theological'' anti-Semitism in the Church of Bulgaria. Fortunately, it is difficult to state that ``God had punished the Jews ... and had not given them a country of their own'', since, in 1948, the State of Israel came into being. Perhaps we may consider it an encouraging fact that people who held such views of ``theological'' anti-Semitism, nevertheless have such an excellent record when practical help to the persecuted was proved necessary. This consideration, however, should not be used to exempt Church leaders from their duty to educate the faithful in a more Biblical and thus more humane spirit than that of Metropolitan Stephan's sermon, in 1942.
b. The Attempt to Deport the Jews
In January, 1943, Eichmann's representative Dannecker arrived in Bulgaria. On February 22, 1943, he concluded a written agreement with the Bulgarian Commissioner for Jewish Affairs, Belev, which provided for the deportation of 8,000 Jews from Macedonia, 6,000 from Thrace and 6,000 from Old Bulgaria. In March, deportations from the occupied Greek and Yugoslavian territories started. 11,363 Jews were deported from these regions. [417]
There were personal interventions by Church leaders, and an official Protest from the Synod of the Bulgarian Orthodox Church was issued, on behalf of the Bulgarian Jews who were threatened with deportation. \textit{183}
Abraham Alphasy, who was then Head of the Jewish Community of Sofia, relates:
``... At that time I went, as the Chairman of the Jewish Congregation, to Metropolitan Stephan, a man with a highly-developed sense of justice, who was a faithful friend of the Jews. When I informed him about the preparations to deport the group of Jews to Germany and requested his intervention, he asked me from whom I had received this information. I replied that it was from a reliable source but for obvious reasons I could not reveal it. Then he immediately dressed and went to the palace of King Boris. The King, who guessed for what reason the Metropolitan had come, sent a message informing him that he was ill and could not receive him. The Metropolitan intimated, as he himself told me, that he would not leave the palace before he had seen the King. Finally, the King was compelled to receive him. The Metropolitan requested him to cancel the order to deliver the Jews to the Germans. The Metropolitan told him that, in the event that they would assault the Jews in order to send them to Germany, he would give instructions to open the gates of the churches and monasteries. They would give the Jews shelter. `In this situation the King was compelled to promise to do as requested,' the Metropolitan told me...'' [418]
We quote the following from the testimony of Joseph Geron, who served as head of the Jewish Community in Sofia, and afterwards became the Chairman of the Union of Jewish Congregations in Bulgaria:
``... Continuing, the witness gave details about united action with the head of the Church in Sofia, Metropolitan Stephan, by whom he was received three times. Dr. Kalmi, one of the leaders of Jewry, kept in touch with the general secretary of the Holy Synod, the body authorized to direct religious affairs in Bulgaria. Thanks to these contacts a meeting between the King and representatives of the Church took place concerning the rescue of the Jews... During his first meeting with the head of the Church in Sofia, the Metropolitan Stephan, he had said to him among other things: 'Cannot the Bulgarian Church do something similar to what the Catholic Church and the Pope himself are doing for the Jews, with an action for their rescue?' To this Stephan answered that the Bulgarian Church would follow the example of the Catholic Church and would do, and allow to be done, everything possible on behalf of the Jews...'' [419] \textit{184}
In March, 1943, Metropolitan Stephan called for a plenary session of the Holy Synod which was held April 2, 1943. He informed all the Metropolitans of the danger that was threatening Bulgarian Jewry. The Metropolitans unanimously decided to send a letter of Protest to the Prime Minister, Filov, and to the Minister of the Interior and of Religions. The letter read as follows:
The Law for the Protection of the Nation
``The idea of passing a Law for the Protection of the Nation which would annul dangers to our people and our state, on which the national, spiritual and moral unity of the Bulgarian people is founded, was accepted by our Holy Orthodox Church, which is the eternal guardian of the destiny of the Bulgarian people, and which knows better than others, from bitter historical experience, what it would mean to our people to be divided by false religious, national and economic teaching, and to be exploited by any minority. The need to restrain such disintegrating political and religious-sectarian ideas, has always existed in our country, as it also exists now. To-day, too, when the new destiny of our people is being decided, it is more than ever necessary to limit, with the help of the law, disintegrating factors in our land and, to harness them to the building of the healthy spiritual powers of our people and, to guarantee economic opportunities for every Bulgarian. However, already when this Law for the Protection of the Nation was made, the Holy Synod of our Church gave warning and begged that it should not be only based on the existing foundations and concepts, because in that case it would not meet the great objective standing before it: to safeguard against disintegrating influences and, to unite the Bulgarian people in a spiritual entity. The Law for the Protection of the Nation was created with the express purpose of limiting the Jewish minority; the main concept of the law is based on racialism. At that time the Holy Synod informed the Government, that the principle of racialism cannot be justified from the point of view of the Christian doctrine, being contrary to the fundamental message of the Christian Church, in which all who believe in Jesus Christ are men and women of equal worth. 'There is neither Jew nor Greek, there is neither bond nor free, there is neither male nor female: for ye are all one in Christ Jesus' (Gal. 3, 28). The principle of racialism, according to which certain members of the community can be persecuted, restricted and deprived of their rights only because they belong to a certain race, in this case the Jewish race, cannot be justified from the standpoint of Christian ethics. Therefore the Church emphatically demands that the Law for the Protection of the Nation shall not be based mainly on the principles of racialism, but on those of spiritual wholeness and the protection of our people, so that it may safeguard them from those disintegrating influences which affect spiritual and religious values, and also from economic financial exploitation. \textit{185} They did not listen to the voice of our Holy Synod. We now see, that the Law for the Protection of the Nation, nearly two years after its promulgation, instead of meeting its great task of safeguarding the Nation from damaging and disintegrating influences, and uniting its creative, healthy, spiritual and economic powers into a spiritual and moral unity, has turned into a means of restricting and persecuting the Jewish minority in our country.''
Christians of Jewish Origin
``Many times our Holy Synod has requested in writing the honourable Government, from the promulgation of the Law until to-day, to ease the restricting passages of the Law against Christians of Jewish origin, and against the Jews in general. Until now both the written requests and the interventions of the Holy Synod have remained unanswered. Neither has any alleviation in the fate of the Jewish minority been granted. The Christians of Jewish origin are still forced to wear the star with the six points, the symbol of the Jewish religion, and they pay taxes to the Jewish consistory; in fact this is a gross profanation of our holy Orthodox religion, in as much as they have been baptized and received into the Church, some of them long before there was any word at all about the Law for the Protection of the Nation. In spite of our repeated requests to exempt them --- what insults they have to bear as Christians --- there has been no alleviation whatsoever.''
The Jewish Minority
``Neither has there been any easing in the situation of the Jewish minority as a whole. Quite the contrary, restrictions are increasing daily. It has gone so far that these citizens of our country are deprived of the most elementary rights, and the Department for Jewish Affairs is free to do with them as it wishes; sending them to camps and deporting them from the country. Our people, with soul and conscience, mind and conviction, cannot tolerate injustice, cruelty and violence against anybody. It cannot accept what is being done now to the Jewish minority. Its human and Christian conscience is perplexed. The Holy Synod has also received requests from different quarters --- from leading citizens who are outstanding Bulgarians, from well-known businessmen who love their fatherland, from Bulgarian mothers --- to demand righteous and a humane treatment of the Jewish minority in the country. The Holy Synod of the Bulgarian Church cannot ignore its divine command and its holy duty. It must, according to the teaching of the Gospel concerning love of one's neighbour, raise a compassionate and defending voice in aid of the suffering sand wronged people; it must beg, guide and convince, so that the measures in general against the Jews may cease or at least be eased. God's law, which transcends all human laws, definitely obliges us not to be indifferent in the face of the sufferings of innocent people, of whatever race. The majority of our people also place their relationship with the suffering Jewish minority on this biblical and humane foundation. \textit{186} Understandably our Holy Synod, as we have already stressed in another letter to the honourable Government, does not deny the right of the Bulgarian authorities to preserve the security of the State and to take all steps to safeguard this security; to persecute, to restrict, to punish. But the Holy Synod is charged with the divine duty to remind the Government that these steps must be taken with justice and in a humane spirit, so that they may surely attain their aim and be effective and lasting for the protection of the State. Until now, a historical line of justice and integrity has been the sure means for the protection of our people and our State. On these eternal foundations we also base our national and righteous demands, side along our hopes. The Bulgarian people as a whole has always, until now, been just and tolerant. Our nation, although it has suffered more than all the nations, does not love, nor tolerate, violence and cruelty We have this name and by it we are known amongst the other nations. We have realized our national aspirations, precisely because we knew they were just; and we wanted justice, both for ourselves and for others. May we Bulgarians, who have longed so much for a fair and decent attitude towards ourselves, now forsake our strongest weapon? The Bulgarian Orthodox Church fears that, if we destroy the eternal foundation --- the right to live as free men and the divine commandment to be just --- there no longer will be left to us, as a small people, any other strong support for our existence. The Bulgarian State must, therefore, abide by these truths, and apply them to all its subjects, who are guiltless (except for the fact that they were born in Bulgaria, but not of Bulgarian parents). A divine command and divine justice cannot be disregarded. The Holy Synod, meeting in the special session of April 2, of this year, has decided --- after considering its deep concern for the honour and future of the Bulgarian people, and its responsibility before God --- to inform you that the Bulgarian Orthodox Church, as a national and divine institution, cannot agree to principles such as racialism, in which it is possible to foment hatred and to indulge in violence and cruelty. It cannot accept the principle that any race be deprived of the human right to live, since this right is in accordance with the fundamental principles of Christian religion and morality. The Bulgarian Orthodox Church is of the opinion that she cannot deny help and protection to the persecuted and oppressed. If she were to refuse such help, she would be unfaithful to herself. In this case our Holy Church was asked for help, by the Jews as well as by Christian Bulgarians, in order to improve the fate of the Jews in general. The Church does not deny and even especially stresses the duty and the right of the honourable Government to take the necessary steps to protect the people and the State from all dangers. However, she must stress the duty of the State to abide by the principles of justice and the Christian Gospel.''
Three Requests
``In consideration, therefore, the Holy Synod has decided to request you
urgently: \textit{187}
1. Not to deprive the Christians of Jewish origin and the Jews of our country
in general of the elementary rights of human beings and of citizens; not to
deprive them of the right to live in the country and of the possibility to
work and to live as human beings.
2. The restricting decrees regarding the Jews must be both eased and not be
enforced too strictly.
3. To cancel the unjustifiable obligation whereby Christians of Jewish origin
wear both the Christian cross and the Jewish star, and whereby they pay taxes
to the Jewish community.
The Bulgarian Church considers herself especially obliged to raise her voice for
the protection of the Christians of Jewish origin, who have cut themselves off
from the Jewish community and who have been received into the bosom of the
Bulgarian Church.
She cannot accept that these Christians wear the symbol of the Jewish religion
and that they pay taxes to the Jewish religious community, or that they be
deported from their fatherland. In this case the Church cannot help but recall
to mind the words of our Lord: 'and with what measure ye mete, it shall be
measured to you again' (Matthew 7, 2), and with concern raise her voice in
warning. We pray that God's blessing may be upon you, and fervently praying in
the name of Jesus, we remain,...'' [420]
Unfortunately, this Protest complained that ``Christians of Jewish origin are still forced to wear the star with the six points'', stating that ``this is a gross profanation of our holy Orthodox religion''. It also defended the Jews in general, stating that ``the principle of racialism cannot be justified from the point of view of the Christian doctrine, being contrary to the fundamental message of the Christian Church.'' However, much of the argumentation was still national, rather than religious. Typical is the expression: ``The Holy Synod,... after considering its deep concern for the honour and future of the Bulgarian people, and its responsibility before God... (in that order!). Absence of sound theology as regards the position of the Jewish people, combined with national considerations, is especially dangerous when one considers that (contrary to the case in Bulgaria) the national interest does not require to stand up for the persecuted Jews. \textit{188}
After meetings had been held between the Metropolitan Stephan and Jewish businessmen, it was deemed essential to bring about a meeting between King Boris and the Holy Synod. However, the meeting did not take place immediately. This moved the Metropolitan of Vidin, Neophyte, the Chairman of the Holy Synod, to appeal to members of his flock (and intentionally, wide publicity was given to this letter) expressing opposition to the anti-Jewish measures. The Metropolitan Stephan, for his part, preached in the churches of Sofia, condemning the anti-Semitic policy of the Government and thus defending the Jews of his town.
``In that period, nobody in Bulgaria could compare with the higher clergy in courage. As a result of this outcry, the Government was compelled to arrange an audience between the King, the Cabinet and the higher clergy. The meeting took place on April 15, 1943, in the royal palace in Sofia. King Boris, the Metropolitan Stephan, Neophyte, Kyril, the Prime Minister Filov and others participated in the discussions in which the clergy defended the Jews with great courage.'' [421]
In May, 1943, the Commissioner for Jewish Affairs, Belev, submitted to King Boris two alternate plans: one for the deportation of all Bulgarian Jews to Poland, the other for their evacuation to the country. The King chose the latter. The expulsion order was published on May 25. [422] The Jews expelled from the cities were housed with Jewish families in the country and in schools. They were never deported from Bulgaria. On May 23, instructions concerning the deportations from Sofia began to be received by the Jews. Rabbis Daniel Ben Zion and Dr. Hanael, together with the lawyer Adolf Chaymov and Mr. Menachem Moshonov, decided to go to Metropolitan Stephan, who had called for them, in order to beg his intervention for the cancellation of the deportation decision. Mr. Moshonow relates: \textit{189}
``... We went to the Metropolitan at 8,30 a.m. He wanted to know what we were doing and we told him everything in detail. He received us early and apparently was greatly concerned about our situation, because he was still in his dressing gown. After he had listened to us, he calmed us and promised to continue to do everything in his power to prevent the deportation of the Jews from the country. Metropolitan Stephan added that at the ceremony in honour of the feast of the saints Kyril and Methodius, which was taking place on that same day, May 24, 1943, he would meet the King and would speak to him again about that same matter. He seemed to be very moved and full of hope. He stressed anew that at one of his last meetings with the King, the latter, in the presence of the ecclesiastical high official Kyril, had specifically stated that the Jews would not be deported from Bulgaria. When we parted from the Metropolitan, he reassured us saying: 'Go and calm your brethren, tell them from me that the King has promised, and a King's word is not reversed'.'' [423]
Contrary to the situation in Rumania, the Church leaders in Bulgaria could indeed claim to express the feelings of ``the majority of our people'', [424] when they stood up for the Jews. Moreover, the great majority of the Bulgarians belonged to the Orthodox Church. Seldom, however, can a Church leader afford to address his King as Metropolitan Stephan addressed King Boris, in the telephone conversation which is related by Solomon Mashiach. His visits to Metropolitan Stephan probably took place on May 25 and 26, 1943.
\textit{190}
``I went to the residence of prelate Stephan. He gave me a kind welcome and ordered that we should not be disturbed. He locked the door and I began to tell him our troubles. After he had listened to me with emotion and attention for nearly half an hour, he said: 'This I cannot permit as long as I live. There are many among the Jewish people who have rescued Bulgaria; they sacrificed much on behalf of the nation. I shall speak with the King immediately. I wish you to hear our conversation.' The prelate took the telephone and was connected with the King. After an exchange of words of no interest as far as the Jews are concerned, the prelate said: 'Boris, my son, I am not at all satisfied about you. One hears lately of many things done to our Israelite brethren. Think very hard [425]; it is unworthy of you and of the Bulgarian people.' The King asked: `But what --- what did you hear and from whom?' 'Things have come to my knowledge which I would rather not believe. They are a disgrace and shame to you and to the Bulgarian people. I cannot explain them to you by telephone. If you wish, come to me, or I shall come to you at once, to see with my own eyes what your reaction will be.' The King began to stammer and to excuse himself, saying that he could not meet Stephan on that day. He then made an appointment with him for the next day. I whispered to prelate Stephan: `That will be too late'. Then the prelate said to him: `Boris, let it not be too late. Pull yourself together, my son.' 'It will not be too late, I promise you. To-morrow we shall see one another.' Thus ended the first conversation. Prelate Stephan said to me: 'Come to-morrow morning, between 9 and 10. He is trying to give me the slip but I shall not permit him to bring such a disgrace, even if I would lose my head...''
``Next morning I again went to the Metropolitan Stephan to hear the outcome. He immediately took the telephone and was connected with the palace. The King's Councillor (Dr. Neshev, if I am not mistaken) replied. He said that the King had been urgently called away and had not intimated where he was going or when he would return. Metropolitan Stephan got very angry and said: 'Tell me where the little king is, you milksop. Tell others that you do not know, but beware if you continue to be stubborn'. Dr. Neshev apologized, saying he had been instructed not to reveal that the King was in his palace Krichim. He begged the Metropolitan not to divulge this information, as it would cause him trouble. Metropolitan Stephan promised to say nothing, but he asked Dr. Neshev: 'Did he expressly instruct you that you should not even reveal his whereabouts to me'? Dr. Neshev replied in the affirmative.''
\textit{191}
``Metropolitan Stephan was very angry and said to me that the King would regret his deed very much. 'At one time I saved his father's head and to him I gave the throne; now this is his reward to me.' In a great perturbed state of mind he took the telephone and spoke to the King in Krichim palace. I heard, word for word, the following: 'Boris, you forget yourself. You elude me and hide. You know that for me there are no secrets under the sun. You know that at one time I saved your father's head and your throne. But it is doubtful whether I, after these acts of yours, shall be able to save your head. Give the matter serious thought and uproot this demoniac influence from your heart.' He then put down the receiver. Afterwards the telephone began to ring. I said to Metropolitan Stephan: `They are calling you'. He replied: 'I know; he wants to speak to me over the telephone but I shall not answer him unless he will come personally to apologize to me. You will see that he will not dare to cause you evil.'...'' [426]
The last recorded activity of Metropolitan Stephan on behalf of the Jews is a telegram sent to the King, in which was written:
'Do not persecute, so that you may not be persecuted. With what measure ye mete, it shall be measured to you again. I know, Boris, that from heaven God will keep watch over your actions.''' [427]
Another outstanding leader of the Bulgarian Orthodox Church who intervened on behalf of the Jews was the Metropolitan of Plovdiv, Kyril. Belev had ordered the internment of the ``influential Jews'' in several cities. On March 10, 1943, some of the Jews in Plovdiv were arrested. Early in the morning of that day, Kyril sent a telegram to the King and called upon the representatives of the regime, to inform the government that from that very moment, because of the action against the Jews, he (Kyril) had ceased to be a loyal citizen and would act according to the dictates of his conscience. [428] \textit{192}
Leviev relates another incident in which Metropolitan Kyril was the hero:
``It is fitting to bear in mind, as a token of the personal courage of Metropolitan Kyril, the date of May 20, 1944. Early in the morning, when it was still dark, he was awakened by a Jewish boy, who had been sent to inform the Metropolitan that during that night large groups of Jews had again been arrested. The Metropolitan went to the Jewish quarter, where many Jews were gathered in a square; Rabbi Samuel, who was wholly dedicated to his people, was at their head, giving them courage. The appearance of the Metropolitan was received by the Jews with relief and hope. The Metropolitan immediately went to the district office where he only found Kolev, the deputy district officer, of whom he demanded particulars about the extent and the meaning of the arrests. Kyril draw his attention to the dangerous consequences which might result from the confusion, created in the mind of the public, and who were not likely to remain inactive in the face of renewed injustice and violence. It was explained to him that about 2,000 Jews had been arrested because a group of five Jewish youngsters had joined the underground movement. The Metropolitan demanded that they set the arrested Jews free; otherwise great public disturbances would occur. After having obtained a promise in this respect, he went to the police station, where the arrested people were held, and encouraged them. He met with the police commander and with his assistants, and pointed out to them that the entire public was following with attention the fate of the arrested Jews. The arrested people were set free at the end of that day.'' [429]
It appears that an important factor influencing Church leaders in Bulgaria to act was their genuine concern. Thus they were easily accessible whenever their help was needed. It seems to be a small feature in the over-all picture, but it is significant: Kyril got up early in the morning when it was still dark and rushed to the rescue of the arrested Jews in Plovdiv; Stephan received Jewish leaders when he was still in his dressing gown. Jewish leaders in Israel as well as Jews in Bulgaria who now live under a Communist government, have expressed their appreciation of the help rendered by the Church in Bulgaria. We quote the following from the testimony of Joseph Geron, who served as head of the Jewish community in Sofia, and afterwards became the Chairman of the Union of Jewish Congregations in Bulgaria: \textit{193}
``... The witness stressed the fact that the Bulgarian Church, on many occasions and at different periods, revealed understanding and sympathy for the Jews, and took important actions for their rescue... Concerning the Bulgarian Church, her attitude to the Jews was always very correct, but during the events which accompanied Jewish life under the rule of Prime Minister Filov, the Church revealed an attitude of open sympathy, and exercised strong moral pressure on all the decisive factors in Jewish affairs... What then were the factors that, directly and indirectly, helped in the rescue of the Jews of Bulgaria? One may answer that there were collective and individual factors. Among the former, the Orthodox Bulgarian Church, with its leaders Stephan, Neophyte and Paisly, take the first place...'' [430]
Of course, in Bulgaria just as in other countries there were many factors helping to influence the outcome. King Boris and the Cabinet were in a position to withstand German pressure to some extent if they wished so. The victories of the Soviet armies made their mark on the minds of the people. It appears, however, that the activities of the leaders of the Orthodox Church were an important contribution to the positive outcome. All the Jews of Bulgaria survived. Yet, there remains one nagging question: did the Orthodox Church of Bulgaria try to render any aid to the more than 11,000 Jews who were deported from the Greek and Yugoslavian territories occupied by Bulgarian troops? It seems that they did not, but perhaps there was no time to intervene. [431]
30 HUNGARY
a. The Preliminary Phase
In November, 1938, Hungary annexed some Slovakian districts and part of Sub Carpathian Ruthenia. In March, 1939, the remainder of the latter territory was annexed. In August, 1940, Hungary occupied Northern Transylvania. In April, 1941, part of Yugoslavia was occupied. In its enlarged state, Hungary had a Jewish population of 750,000 within its borders. [432] \textit{194}
On June 22, 1941, Germany invaded Russia and the Hungarians joined forces with the Germans. On August 8, the third anti-Jewish law was enacted. [433] This law defined who was to be considered a Jew, according to the well-known principles of the Nuremberg laws.
``Bishop Ravasz, the leading speaker of the representatives of the Reformed Church, after having delivered his address of refusal, read a solemn declaration signed by all the Bishops, and by four general elders, in which the signatories protested against the passage of the law and disclaimed all responsibility for its passage.'' [434]
I have tried to obtain a copy of Bishop Ravasz's address of refusal and of the declaration mentioned above. Dr. Elek Mathe, of the Reformed Church of Hungary, replied to my request:
``Unfortunately there is no available copy of the address referred to in your letter...; even less, newspaper cuttings, for the simple reason that at that time the daily press was under strict government control and the text of such an address could not be printed. [435]
In the summer of 1941, the Hungarian government ordered an inquiry into the citizenship of all the Jewish residents of Northern Hungary. 11,000 Jews unable to give satisfactory proof of their citizenship were deported to Galicia, where a systematic extermination was carried out by the German troops.
``Baroness Edith Weisz called on Bishop Ravasz, and asked for his intervention. The Bishop requested an audience with the Regent, and appearing before him, informed him of the situation and asked that the Minister of Interior be instructed to give due regard to humanitarian viewpoints. \textit{195} Bishop Ravasz then called on Francis Kereszres-Fischer, Minister of the Interior, who himself later on was carried away by the Germans, warning him that after the conclusion of the war an account would have to be given before world Protestantism, of the fate meted out to the Jews. He requested the adoption of such measures as would enable him to appear before any foreign Church body in future days, with a clear conscience regarding these matters. An end was put to all abuses and the lives of many persons were saved.'' [436]
Bishop Ravasz thus tried to do something on behalf of non-Hungarian Jews, this in contrast to the tendency of those in other lands who rendered resistance only when Jews of their own nationality were deported.
From March, 1942, to March, 1944, Kallay was Prime Minister. His Cabinet withstood German pressure to deport the Jews.
b. Mass Deportations
On March 17, 1944, Regent Horthy was ``invited'' to a conference with Hitler, who informed him of the imminent occupation of Hungary by German troops. Horthy had to agree to Kallay's dismissal. The aerodromes of Budapest were seized by a German task force. A new Government was appointed under Sztojay. The Arrow Cross leader, Laszlo Baky, was appointed Undersecretary of State in the Ministry of Interior, and Laszlo Endre Administrative Under-secretary and expert on Jewish affairs. Veesenmayer was appointed as Ambassador to Hungary and as Plenipotentiary of the German Reich. Eichmann came to Budapest at the end of March. On March 29, 1944, it was decreed that all Jews must wear the yellow star. Concentration of all the Jews took place at a rapid pace. In May, the first deportation trains left for Auschwitz. At the end of June, 381,661 Jews had been deported. On July 9, 1944, the total number was 437,402. The evacuation of the Jews of Budapest was planned for July. \textit{196} Concerted pressure was exerted on Regent Horthy to stop the deportations. Switzerland and Sweden made urgent requests. The Turkish and Spanish governments also intervened. The Papal nuncio was, according to Sztojay, calling ``several times'' a day. On July 6, Sztojay informed Veesenmayer that the Regent had given the order that the deportations should stop. In fact, the stoppage occurred in the middle of July and it lasted until October.
On April 3, 1944, Bishop Laszlo Ravasz addressed a letter of protest to the Minister of the Interior. In this document Bishop Ravasz did not object to the stigmatization of the Jews, but to the regulations that required members of the Reformed Church to wear the star of David. [437] At the same time he called on Ambrozy, the Regent's chef de cabinet, and asked to be granted an audience with the Regent. He was informed that ``the Regent regards himself a prisoner and will not receive anyone''. Subsequently Bishop Ravasz called on the Minister of Interior, who asked him to return at 7 p.m. the same day.
``Jaross, who kept the Bishop waiting till 8 p.m., agreed, after a heated argument, to exempt certain Church dignitaries and persons of Jewish origin who had contracted mixed marriages.'' [438]
On April 6, the General Assembly of the Reformed Church addressed a petition to the Prime Minister, urging him to be mindful of the claims of humanitarian thinking, and demanding the extension of granting exemptions.
``All the activities carried on by the Churches in these days, centred around the Jewish question. However, as the government was but a mere tool in the hands of the Nazi regime, expressly antagonistic toward the Churches, it paid little regard to the action of the Churches. The results reached accordingly were rather meagre. Yet, meagre as they were, they meant the saving of many lives. The most important result was the exemption secured for members of mixed marriages. This one measure alone meant exemption from the wearing of the yellow star and its fearful consequences for several thousand families \textit{197} The Churches already at that time demanded the formation of a body authorised to grant exemptions whenever individual merits made the granting of the same justified. This demand, although not granted in its original form, led later on the recognition of the Regent's right to grant exemption, through which channel some twenty thousand persons were given exemption. [439]
Under the influence of the news reaching the capital, Bishop Ravasz asked the Regent --- a Protestant --- for an audience, which was granted on April 12, 1944.
``He entreated the Regent to abstain from any action in connection with the Jewish question, for which at some future date he might have to bear the responsibility, pointing out that the blame for cruelties, should these occur, would be laid at his door and that he would render himself liable to trial for same. `The desperadoes,' said the Bishop, 'will not fail to make an attempt to have their own accounts paid out of the moral capital of others'. Horthy reassured the Bishop.'' [440]
A few days later, however, Baron Zsigmond Perenyi, President of the Upper House, called upon Bishop Ravasz and informed him of his sad experiences in Northern Hungary. On April 28, Bishop Ravasz was again received in audience by the Regent, to whom he passed on the information gathered by Perenyi. Horthy's answer was as follows:
``Only a few hundred thousands Jews were scheduled to leave the country with the labour battalions. No harm will befall them, not a hair of their heads will be touched. They will enjoy the same treatment as the nearly hundred thousand Hungarian labourers employed abroad?...''
The Regent admitted that complaints had been received from Nyiregyhaza, whereupon he had sent for the Minister of Interior, Jaross, and had asked for an investigation to be instituted. Jaross had charged his two Under-Secretaries with the investigation, and had since reported that a stop had been put to the scandalous treatment. [441]
On May 9, 1944, Bishop Ravasz called on Prime Minister Sztojay and protested against the atrocities committed against the Jews. \textit{198}
``He presented the petition of the Reformed Assembly referring to the horrors which occurred during the concentration of the Jews at Marosvisihely, Kolozsvir, Kassa and Nagybanya.
The Prime Minister seemed to have been informed about the situation and declared that he condemned the brutalities, stating that he had given instructions for the separation to be carried out drastically, but humanely. ``The Jews are a race'', he said, ``and thus the regulation of the Jewish problem is not a question of religion, but of race''. [442]
On May 17, 1944, the Assembly of the Reformed Church sent a letter to Prime Minister Sztojay in which two matters were emphasized. First, it recalled the promises which the Prime Minister had made regarding amelioration of the cruel measures and, second, it protested against the segregation of the Jews which had already begun.
``We are compelled to declare that we most resolutely disapprove the segregation of persons classified as Jews. We are of the opinion that the measures adopted by Christian Society in times past in this direction, must not be repeated... The second thing which we have to mention is as yet an anxious presentiment. Signs are not lacking to show that, besides segregation, the deportation of the Jews beyond the country's boundaries is also in preparation. We have to call your Excellency's attention to the tragic developments which mark the conclusion of Jewish deportations in other countries, and we beg your Excellency to do all that can be done in order to impede such happenings and to avert responsibility for such acts from the Royal Government and from the whole nation.'' [443]
Bishop Ravasz then tried to join forces with the Roman Catholic Church and informed the Chairman of the Holy Cross Society (which was charged with the protection and care of Catholics of Jewish origin) of his willingness to make the introductory steps for a united action. On June 15, 1944, he sent a letter to the Primate, Justinian Serkdi, saying that he had already prepared the draft for a memorandum (of which a copy was included) to be sent to the Government, as ``a final earnest warning'' before the Churches should ``voice their solemn declaration in protest, in the presence of the country and the world''. No reply from Primate Seredi, however, was received. [444] \textit{199}
As nothing could be learned from the press, the authorities of the Church sent a young pastor to Kassa, the largest ghetto. He returned from his trip of inspection, reporting that, notwithstanding the protestations and promises of the Prime Minister, the deportation of Jews had begun. Therefore pastors were dispatched to carry the text of the memorandum that had been prepared, to the nine Bishops, in order to obtain their consent. After they had signed it, a deputation presented it to the Prime Minister, on June 23, 1944. Its text was as follows:
``In our memorandum of May 19 we mentioned, with foreboding, that there was a possibility of the deportation of the Hungarian Jews to an unknown destination. Since then information has reached us, according to which Jews have been crossing the frontier in sealed wagons day after day, disappearing from our sight, bound for an unknown destination. Each of these wagons contained about 70 to 80 persons of different sex, age and social standing, of both Israelite and Christian faith. The persons deported, as well as, their relatives are convinced that this journey is leading to final destruction. The solution of the Jewish question is a political task. We now are not dealing with politics. The execution of this solution is a great work of administration. We are not experts on that. But the moment the solution of the Jewish question challenges the eternal laws of God, we are in duty bound to raise our voice, condemning, but at the same time imploring, the head of the responsible Government. We cannot act otherwise. We have been commanded by God to preach His eternal Gospel, to give evidence of the unalterable laws of His moral order for this generation, whether people like it or not. Although humble and sinful men, we, in the bondage of faith and obedience to this heavenly command, possess the right to give evidence of the Word of God and to condemn every action which outrages human dignity, justice or charity, and which loads upon the head of our people the horrible responsibility of innocently shed blood. As Bishops of the two Protestant Churches we protest against devout members of our congregations being punished merely for being considered Jews from a racial point of view. They are being punished for a Jewish mentality from which they, and in many cases their ancestors, have solemnly disconnected themselves. Their lives, as regards Christian spirit and morality, are not considered in the least. \textit{200} Finally we, as Hungarians and as clergymen, repeatedly implore Your Excellency to put an end to the cruelties, even disapproved of by yourself, and to enforce the declaration made by a prominent member of your Cabinet, protesting against the very idea of a senseless and cruel destruction of the Jews. We do not wish to aggravate your Excellency's political position; we even wish to promote the solution of the great task you have taken upon yourself. For this reason, for the time being, we do not carry our protest before the Hungarian public, although this course will incur for us the reproach and accusation of the leading bodies of the Christian Churches. Should, however, our intervention prove ineffective, we will be obliged to testify before the congregations of our Church and the Protestants of the world, that we did not suppress the message of God. As a last attempt we appeal, through the kindness of your Hungarian heart and the Christian feelings of your Excellency, to the leniency of the Royal Hungarian Government. We desire that this, the most painful manifestation in our whole history hitherto, should become the case of the Government.'' [445]
The Bishops were afraid of ``the reproach and accusation of the leading bodies of the Christian Churches'', in the event of remaining silent. Apparently the anticipated verdict of Churches in other countries, and of the World Council of Churches, was one of the factors which urged action. The Churches made a clear stand in this memorandum, though certainly not everything in it is of prophetic power.
The deputation submitting this memorandum, which in its way was a kind of ultimatum, supported it by word of mouth. Prime Minister Sztojay answered bluntly:
``The two Secretaries of State of the Ministry of Interior have reported that, except in certain cases, no atrocities have been committed. Germany has need of man power, and the Jews are being taken there for labour purposes.''
In their reply, the members of the Protestant deputation pointed out to the Prime Minister that deported babies, pregnant women and old people were certainly of no use for that purpose. Sztojay's answer to this was that the authorities did not want the Jews working abroad to feel anxious about the families they had left behind, nor the families to worry as to the fate of the deported Jewish men. \textit{201}
The deputation proposed that the Churches be permitted to care for children whose parents had been selected for such ``labour purposes''. The Prime Minister consented to this request, but asked that it be submitted in writing. The deputation immediately composed and handed over the written request. It was never answered. [446]
A Confidential Report sent by the Hungarian Protestants to the Secretary of the World Council of Churches, Geneva, states:
``... Not a single line on all this has been published, either in the ecclesiastical or in the daily press; for when the first Memorandum was personally handed over, the Government made it a condition that the whole intervention be handled with the utmost discretion and no press-comments whatsoever were to be made: in this case the Government were willing on their part to do everything possible; if press-comments were made it would appear as though the Government, considering the echoes in the press, had taken alleviating \textit{i.e}. modifying measures in their sudden fear. The date to answer the last Memorandum expires to-morrow and if no adequate essential measures are taken by then, uniform sermons and a strongly worded pastoral letter will be read from every pulpit on next Sunday.'' [447]
No answer came to the Memorandum. Thus Bishops Ravasz and Kapi decided to address an open declaration to the congregations and to the Protestant Christianity of the world:
``We request all our brethren, the pastors belonging to our communions, that they read the following message to the congregations at the conclusion of next Sunday's morning service: To all congregations of the Hungarian Reformed Church and of the Hungarian Lutheran Church, Grace unto you and peace from God our Father, and the Lord Jesus Christ. Brethren in Christ! The undersigned Bishops of the Hungarian Reformed Church and of the Hungarian Lutheran Church address you and inform you, in the presence of God, of the steps taken before the Royal Hungarian Government in the name of the Protestant Churches. \textit{202} We notify the congregations that the two Protestant Churches, after several proposals both by word of mouth and in writing, on June 21st presented to the Royal Hungarian Prime Minister a solemn memorandum of request and protest. This memorandum pointed out the more than regrettable events accompanying the concentration and deportation of Jews, whether Christian or not. After having stated that the solution of the Jewish question violates eternal Divine laws, the memorandum continued its proposals as follows: 'We have been commanded by God to preach His eternal Gospel, to give evidence of the unalterable laws of His moral order in this generation, whether people like it or not. Although humble and sinful men, we, in the bondage of faith and obedience to this heavenly command, possess the right to give evidence of the Word of God and to condemn every action which outrages human dignity, justice or charity, and which loads upon the head of our people the horrible responsibility of innocently shed blood.' At the same time we beseeched the Royal Hungarian Government to put an end to the cruelties which were also condemned by members of the Cabinet, and to enforce those declarations that protested against the very suggestion of the destruction of the Jews, while at the same time they issued orders that the Jews should be treated humanely. We were deeply afflicted when we were forced to admit that our entreaties had been in vain. We, the Bishops of the two Protestant Churches, considered it to be our duty to inform our faithful, as well as, every member of our congregation and the universal community of Christ's Holy Church of these events. We summon the congregations to repentance and the entire Hungarian nation to penitence under the mighty hand of God. Pray to Him and beseech Him to turn His mercy and His supporting Grace towards our Hungarian nation.''
Your loving brethren in Christ:
the last Sunday in June, 1944. [448]
The proclamation was lithographed and, as a necessary precaution, put into differently coloured envelopes. It was intended to post it to the two thousand clergymen in the country from different provincial post offices. At this juncture, the Minister of Religion and Education sent word by telephone that a pastoral letter of the Primate, addressed to the Bishops and priests of the Roman Catholic Church, had been intercepted and that the Government wished to have a conference with the Churches. On July 11, 1944, the Minister visited Bishop Ravasz, who was ill.
``The Minister declared that the Prime Minister had promised the abolition of atrocities, the cessation of further deportations and that the isolation of the Jews would be carried out in a humane manner. `That,' he said, 'was the agreement with the Catholic Church'. \textit{203} He could not at that time produce it in writing, but that was the text and essential contents of the agreement with the Catholics. At great length he persuaded and threatened the sick Bishop that if they persisted in protesting in public, the Church would be `overwhelmed', or the Government might resign, thereby paving the way for the coming into power of the Arrow Cross Party. If, however, they came to an agreement, the lives of 250,000 Budapest Jews would be saved.'' [449]
Bishop Ravasz believed the Minister's statement regarding the agreement with the Primate, but he insisted that the clergy should, at any event, be allowed to read out a short note in the Reformed Churches. This note was immediately drafted. It reads as follows:
``Reverend Pastor! We request that you read the following announcement at the conclusion of next Sunday's morning service [July, 16]: 'The Bishops of the Reformed Church of Hungary and the Evangelical (Lutheran) Church of Hungary wish to inform the congregations that in connection with the Jewish question, and particularly in the case of baptised Jews, they have repeatedly intervened with the competent Government authorities. Their endeavours in this respect are continuing'.'' [450]
It is difficult to answer the question whether it was the right decision to cancel the public proclamation of the pastoral letter form all the pulpits. Dr. Mathe wrote to me:
``... This circular [the pastoral letter] reached all the pastors, and they undoubtedly communicated its contents to most of their parishioners.'' [451]
The deportations were stopped. The Archbishop of Canterbury addressed ``the Christian people of Hungary'' through the B.B.C., in July, 1944. He begged them ``to do your utmost, even taking great personal risks, in order to save some if you can''. [452] This appeal may have had more direct, practical effects than the short note that was read out from the pulpits.
\textit{204}
c. The Terror at the End
On October 15, 1944, Regent Horthy was arrested by the Germans and the new Nazi-dominated regime of Szalasi was installed. On October 20, 22,000 Jewish men were rounded up. By the end of October, 35,000 Jewish men and women had been seized. The majority of them were marched off to Austria, without food. All who fainted and fell, were killed on the spot. The 160,000 Jews who had remained in Budapest were herded into a ghetto where they were exposed to raids by German and Hungarian Nazis, and to the bombardments of the Russian guns. On December 13, 1944, the Russians stormed Budapest. On January 18, 1945, the ghetto was liberated by the Red Army. The fighting for the Buda citadel continued until February 13, 1945.
After Szalasi's reign of terror had begun, Bishop Ravasz intervened in the name of the Protestant Churches. He demanded the fulfilment of five points. Three of them were in connection with the Jews:
... c. Humane methods in the treatment of Jews. Revocation of the order which,
in cases of mixed marriages, empowered the non-Jewish party to obtain a
divorce, and declared as Jewish the party that failed to comply with this
regulation.
d. The cessation of the deportations. e. Security for the lives of the Jews.
On November 24, 1944, the Deputy Prime Minister replied in the name of the Government. He informed then that Szalasi had succeeded in obtaining the Fuehrer's permission to grant the following points:
``No alteration to be made in the legal status of mixed marriages, the Jews to be separated from the rest of the population of Budapest, and the labour service companies to be directed towards the German frontier, because it was to be feared that they might commit atrocities in the case of a Russian occupation. When carrying out these measures, however, the principle of humanity would be respected.'' [453]
\textit{205}
On November 26, 1944, Bishop Ravasz again wrote to the Roman Catholic Primate proposing united action. ``The Primate, tired and very ill, replied that he had already intervened with Szalasi and that he did not feel like repeating the intervention in the company of others.'' [454]
On Dec. 1, 1944, the Bishops of the Reformed and Lutheran Churches presented a note to the so-called ``Leader of the Nation''.
``It follows from the prophetic office of Christ's Church that the servant of the Church should always raise his voice when men's acts gravely violate God's laws'', wrote Bishop Ravasz. The letter stressed, that ``the treatment meted out mocks God's eternal laws which prescribe humane treatment even toward one's enemies, and brings down God's anger on the head of the nation. This treatment casts a dark blot on the name of the Magyar nation which, for a thousand years, had been known to the world for its generosity and chivalry.'' [455]
A pastoral letter issued in December, 1944, called on the pastors to pray at the services for ``the scattered flock of Israel, the homeless and the persecuted.'' [456]
\begin{quotation}
On May 9, 1946, the Hungarian Reformed Church declared that ``in deep humility\end{quotation}
she confesses her guilt and offence against God's honour... She had not laboured in time to warn the people and the rulers, when they embarked on a course contrary to God's laws, and she had not strongly taken her stand on the side of the innocent persecuted human beings.'' [457]
\textit{206}
THE NEUTRAL COUNTRIES
31 SWITZERLAND
a. Press Censorship
Switzerland remained neutral throughout the second world war, but it was surrounded by the Axis powers and to a great extent economically dependent on them. The Swiss Government tried to avoid offending the Germans, and thus the press was forbidden to make foreign propaganda or to publish stories about atrocities committed by the warring parties, ``of which the objective correctness could not be verified''.
Even in June, 1943, the press censorship issued the following order: ``There recently appeared several articles about Jews and Polish clergy, without mentioning their source of information. It is understandable that our conscience should be moved by all such inhuman treatment, but yet we must strictly obey the instructions of the Press emergency law, which stipulates that it is our duty to suppress rumours and foreign propaganda.'' [458]
Thus censorship imposed silence on the press concerning reports of ``bloody murders of hostages and persecution of Jews''. The first time, however, that, to the best of my knowledge, Church leaders in Switzerland spoke out about the persecution of the Jews during the second world war, they did so in a protest against censorship of the Press.
On October 27, 1941, the following Petition was presented by the ``Social Study Committee of the Swiss Reformed Pastors Union'' to the Swiss Federation of Churches: \textit{207}
1. We take the liberty of drawing the attention of the Swiss Federation of
Churches, which is the spokesman of the Swiss Churches to the Federal
Authorities, to the alarming position of the Evangelical Reformed Church.
2. The press-censor has repeatedly taken severe measures against men who,
as representatives of the Evangelical Church, have raised their voices
to inform public opinion.
These measures have aroused deep and widespread concern in many circles
to whose notice they have come, despite the ban on the publication of
such matters, and have led to the opinion that the Evangelical Church is
no longer allowed to pronounce the truth entrusted to its care...
3. We especially bear in mind the silence imposed on us by our censorship
concerning the injustice of the bloody murders of hostages and the
persecution of Jews. When mentioning this subject, we should certainly
not shout about it from the roof tops, but under no circumstances should
our sense of justice and injustice be blunted within our national
conscience. Otherwise we shall invoke God's heavy punishment on our country.
Therefore, we take exception to the reproach levied at us by some, that such
intrepid talk of injustice by a foreign nation, is a misjudgement.
4. The apprehension we bring to your notice particularly gains alarming weight
by the fact that we, as Evangelical and democratic citizens of Switzerland,
have to look on while un-Christian and undemocratic ideologies and deeds
cross our borders unhindered in the form of many foreign newspapers and
illustrated periodicals, which are thus able to exert their influence on
young and old. Does not this give rise to the impression that our
highest authorities do not sufficiently recognize the danger of a moral
and spiritual capitulation on our part, or consider it to be of only
secondary importance?
We therefore request the Federation of Churches:
a. That it remonstrate with the highest responsible authorities of our
country and draw their attention to the deep concern and alarm which
these measures by our censorship have aroused in large circles of our
Evangelical Church.
b. That it publicly voice its opinion on the matter and unequivocally make
known its stand, with the full weight of its authority.
c. That it emphatically take a stand on behalf of all persons in our Church,
whose freedom of speech is endangered or impaired, and that it encourage
our Church authorities and Synods to make use of their divinely authorized
right of freedom of speech. [459]
On November 17, 1941, a conference of the ``Swiss Protestant Relief Society for the Confessing Church in Germany'' was held at Wipkingen near Zurich. It was attended by 300 churchmen from all parts of Switzerland. A Resolution regarding the Censorship of ecclesiastical publications was presented and unanimously adopted. We quote from this resolution the following:
\textit{208}
``The undersigned Reformed Swiss Pastors have taken note of the following facts: That the Department for Press and Radio of the army has imposed preliminary censorship on the periodical Neue Wege, and thus has prevented its further publication; ... that the same office has demanded of the Swiss press that they refrain henceforth from taking any stand on the execution of hostages by a foreign power;... They herewith publicly protest against these measures, as they are concessions to the spirit and methods of a policy incompatible with the Reformed Confession and pernicious to the Swiss Federation. They herewith publicly declare that they are determined to continue to fulfil their duty, to declare the truth to our people, the suppression of which is attempted by these measures.''
One hundred pastors signed this ``Protest and Declaration'', which was submitted to the Federal Government and to the Army Commanders. [460] The same Conference adopted the following Resolution on ``The Jewish Problem'':
``The Conference meeting to-day at Zurich-Wipkingen sends to the Committee of the Swiss Protestant Church Federation the urgent request that it should take action so that all the Reformed Churches in Switzerland may make a public statement on the Jewish problem. Not only the most recent deportations of the Jews, whose number and character are particularly frightful, but also certain announcements which have appeared even in the Swiss press, make it a duty for the Church, for the sake of its own members, to proclaim before the whole world: 1. That the Church, to which the Gospel of the mercy of God is entrusted, calls
its members to pray for the suffering Jewish people and to do everything they
can to alleviate this suffering.
2. That the Church, to which the message of the creation of man in the image of
God is entrusted, condemns as a revolt against the will of God as Creator the
violence which is done to the image of God in persecuting a race and
humiliating it.
3. That the Church, to which the message of the Revelation of God in the
people of Israel is entrusted, knows itself, as the Church of Jesus Christ,
to be bound up in a special way with the fate of the Jewish nation. Because
`salvation comes of the Jews' (St. John 4, 22), anti-Semitism is
incompatible with membership in the Christian Church.'' [461]
\textit{209}
On August 30, 1942, a meeting was held of the ``Young Church'', attended by about 6,000 young people. Supreme Court Justice Dr. M. Wolff, who in his capacity as President of the Synod conveyed the greetings of the Church of Zurich, declared:
``Switzerland is in extremely danger. One speaks of a new order in Europe, but this order is characterised by attacks on other nations; by the murdering of hostages and the persecution of Jews. This new order means a denial of the Christian faith... The best contribution the Church of Switzerland can render to-day is, to be a true Church, faithfully proclaiming the word of God. Its freedom to preach must therefore be preserved unrestrictedly. We shall be grateful for a State Church, so long as the State recognizes the Church's right to exercise its function as Watchman. Zwingli has unequivocally insisted on this. Unfortunately, the Church's function as Watchman is now being threatened by press censorship. The State must not demand that the Church should refrain from clearly distinguishing between right and wrong. The Church must now fight for its right to raise its voice against the rejection of poor refugees; in a clear ``Yes'' to Jesus Christ, and in a clear ``No'' to the dark powers of this world.'' [462]
On October 28, 1942, Dr. Wollf said in his Opening Address to the Synod of Zurich:
``... It is therefore not surprising that, when a clearer profession resulted through the awakening of the Church in recent years, its freedom of speech was often denied by political coercion, and the Church was told that it had no right to interfere in Government matters. ... The Gospel knows nothing about neutrality of opinion and nothing of a policy of false silence, advocated currently by higher circles.'' [463]
b. Anti-Semitism within and outside Switzerland
In several declarations anti-Semitism within, as well side, Switzerland was denounced unequivocally, but the name of Germany was seldom mentioned. In May, 1942, the Synodal Council of Bern published the following Declaration:
\textit{210}
``Deeply concerned by the fact that hatred of the Jews is being stirred up both openly and secretly also in our country, the Council of the Evangelical Federation of Churches has requested the cantonal Church authorities to use their good offices so that our attitude as Evangelical Christians towards the Jewish question be maintained against all such plots. Our attitude towards Jewry is not based on economical or racial problems. It is not even a matter of conducting oneself humanely and decently; the question has a far deeper significance and only can be understood correctly and answered in the light of biblical teaching. Therefore above all it is essential, that we reach a Christian understanding of the Jewish question; only then shall we be able to overcome, on the basis of a deeper understanding, the common prejudices and slogans; and especially, the latent disparaging attitude towards the Jews. Wherever anti-Jewish attitudes appear within a congregation, we must not remain passive; we have an obligation to emphasize the Evangelical stand on this matter, and to admonish and counsel. Above all, we should not slacken in our intercession on behalf of the people of Israel.'' [464]
In June, 1942, a similar Declaration was issued by the Council of Pastors in Geneva:
``Our Church cannot keep silent in face of anti-Semitic propaganda which is in danger of becoming stronger in our own country. At a time when the Jews elsewhere are the victims of plunder and persecution, the Church must define her spiritual position. 1. The Apostolic message which declares that there is no longer Jew nor Greek
in Jesus Christ forbids us to make any distinction in the community of the
baptized. A Jew attached to the Christian Church by his conversion and
baptism is a member of it on exactly the same basis as every other faithful
Christian.
2. Christendom has denied the spirit of her Lord every time she has maltreated
or persecuted the descendants of those for whose pardon Jesus prayed to the
Father. Our obligation is to deal with all men in justice and charity on
the grounds that they are indeed our brothers.
3. The race from which came the prophets and the apostles, and to which Jesus
Christ belonged, deserves our respect. We owe Jews a debt of gratitude,
and if Christians pray to God for the conversion of the chosen people they
must also implore divine mercy for persecuted Israel; they must sympathize
with the grief which they are suffering; they must suffer in sympathy the
injustices which Jews once more are suffering.
Strong in the convictions of our Evangelical faith we invite members of our Churches to resist all efforts to introduce in our country anti-Semitic racialism which is condemned by the spirit of our Master and by all teaching which is derived from the Holy Scriptures. [465]
\textit{211}
On September 30, 1942, the Assembly of the Swiss Pastors' Union, meeting at Liestal, adopted the following Resolution:
``We confess on the basis of the Holy Scripture that the hope of the Church through the grace and faithfulness of God in Jesus Christ is indissolubly bound up with the hope for the Jews. We therefore declare that all anti-Semitism is irreconcilable with confession of Jesus Christ. It is the holy duty of every Christian to help the tortured Jews by intercession and active love.'' [466]
That warnings against anti-Semitic influences in Switzerland were not superfluous, was shown in the Report concerning the Fund Drive for Aid to Refugees, which was held in October-November, 1942. The Report stated:
``... The reasons for the diverse reactions to our fundraising appeals in the German and the French-speaking parts of Switzerland, are of a complex nature. From an inquiry made by the cantonal committees we learn that the press in general took a stand against aid to refugees. An article by Pierre Grellet, the Bern correspondent of the 'Gazette de Lausanne', published in November, had a distinct undercurrent of anti-Semitic feeling, characteristic of his attitude. There were also other expressions of anti-Semitism in the press. In contrast to the German-speaking press which protested against the turning away of refugees from our borders, this action triggered no particular reactions in the French-speaking press. [467]
c The Admission of Refugees
Like the Government of the Netherlands, the Swiss Government had already closed the Swiss borders before the war. \textit{212} The Church Council of Canton Zurich stated, in 1938: ``It pains us that consideration for so many unemployed citizens in our own nation prevents us from offering a protecting asylum to the suffering refugees, who, like wild game, are chased from country to country.'' [468] In the summer of 1942, mass raids took place in France and many Jewish refugees tried to find asylum and safety in Switzerland. They often crossed the French-Swiss frontier ``illegally''. On August 13, 1942, the border police were instructed to send back civilian refugees from France who had entered into Switzerland illegally, with the exception of political refugees. ``Refugees for racial reasons only, for instance Jews,'' were not considered political refugees. [469] The Federation of the Protestant Churches as well as other organizations turned to the Federal authorities. [470] Their protests were not ineffectual. On August 23, Federal Councillor von Steiger ordered that in special cases rejection should be waved. On August 24, a meeting was convened with the ``Swiss Central Office for Refugee Aid'', where all the Institutions for refugee aid were represented. The ``Central Office'' informed the press of the result of this partly tumultuous meeting on the same day: ``Foreign refugees, who had entered Switzerland before 13th August, 1942, and register with the police, will be sent back only if, after careful investigation, they must for important reasons be considered undesirable.'' [471] \textit{213}
On August 30, 1942, at the meeting of the ``Young Church'' which has been mentioned before [472], Rev. W. Luthi said:
``Sin separates us from God. What has happened in the case of the refugee problem comes under the same heading. Even though we understand that events may be motivated by political considerations, our conscience is burdened by such events in three ways. First, because the rejection of the poorest of the refugees was not an act of humanity. Second, because any claim to humanitarianism becomes hypocritical. And third, because it was an act of ingratitude towards God, who has so graciously protected our own country. Now we may well fear that, after what has happened, God will no longer be for us, but against us.''
The morning session ended with words of greetings by the Rev. Hans Roduner, who thanked the authorities for their consent ``to revoke the painful measures in force against the refugees''. He called upon the Young Church to make great sacrifices for the refugees and ensure the support of fifty of them. The reply of Federal Councillor von Steiger, who spoke in the afternoon, was typical of the Government point of view:
``Of course the Federal Councillor would like to help all the refugees. However, when thousands of victims of a shipwreck cry out for help, the one in command of a small and fully occupied lifeboat, that is limited in capacity and provisions, must seem heartless if he cannot take them all into his boat. Nevertheless, it is humane to give warning against false hopes, and at least try to save those already aboard. As regards the measures adopted concerning the refugee problem, Federal Councillor von Steiger is prepared to accept full responsibility.'' [473]
Since September 26, 1942, the following categories of refugees were admitted: a. Obviously ill persons and pregnant women. b. Refugees over 65 years old; married couples if at least one of them was
over 65 years. \textit{214}
c. Children under the age of 16 travelling alone. d. Parents with children under 16 years. e. Refugees who claimed and could prove that they had close relatives in
Switzerland or, otherwise, close relations with Switzerland (Residence for
a long time).
However, French Jews without exception had to be deported ``as they were in no danger in their own country''. In doubtful cases (when it was not clear whether a refugee came under one of the categories mentioned, or when deportation appeared to be exceptionally severe) the Police Department had to be contacted by telephone. It was ascertained that 3,800 persons had entered Switzerland illegally during September. [474]
On October 28, 1942, in his opening address to the Synod of Zurich, Dr. Wollf said:
``... The dominant spirit, in no way identical with the sentiments of the people, has become despondent and even pitiable. Its exponents, who can be found not only in the Federal Council but also in the Parliament, pay homage to the opinion that expediency, craftiness and a so-called realistic policy are greater importance to our salvation than the spirit of the Gospel and of freedom and of truth. The misery of the dominant spirit has become evident in recent months in the shameless treatment of the refugees. We must not pass over in silence the disgrace and shame we have brought upon ourselves when, because of cold political calculations, we returned to misery and threat of death, those refugees who believed they had found within our borders a refuge from danger... It is not the beauties of our country nor our safe existence, which make Switzerland worthy of our defence and devotion, but the fact that it is the centre of freedom and justice. The Declaration of the Federal Council and the three coalition parties, contained no sign of their having grasped the challenge of the hour. [475] In contrast to this, it may be said that the Reformed Church, and, in particular, the Executive Committee of the Swiss Federation of Evangelical Churches, has in no uncertain manner fought for recognition of the demands of our Christian conscience. These have found their most impressive formulation by the President of our Federation of Churches: \textit{215} 'God, through His commandments in the Old and New Testament, has placed us unequivocally on the side of the weak, the oppressed and the destitute, no matter what their race or nationality. Confession of faith in Jesus Christ is, for the Christian, almost always also related to recognition of our responsibility to our suffering brothers. The least of His brethren to-day are the oppressed refugees in their physical and mental distress. Christ will either find us on their side or on the side of His persecutors.' The mitigations, now granted by the authorities, may be accepted as revoking their heartless orders. But this is not a lasting or definite solution. The fight for an honourable and humane conduct must continue. Protests alone will not suffice. Indignation is shallow if it is unaccompanied by the will to act. The members of the Church, as well as its critics, justly demand that it put up a determined stand on behalf of the outcasts. Generous contributions to the Refugee fund, and willingness to accept refugees in our homes, must now furnish proof that our nation wishes its ancient Christian traditions to be upheld. Each one of us should do his part to atone for our guilt in this matter. Injustice, force and inhumanity triumph around our borders. These terrible events can no longer allow us only to consider expediency. The only truly realistic policy is the one which accepts God as the highest Reality, and considers Him more important than all calculations of worldly wisdom, which only lead us astray.'' [476]
In the months October and November, 1942, a general collection for the Aid to Refugees was held. Because of political considerations on the part of the authorities concerned, the planned 5-minute broadcasts could not take place. Nevertheless, the General Management of the broadcasting services agreed to broadcast short appeals under the slogan ``Contribute towards an Asylum for the Homeless''. [477] The ``Swiss Central Office for Aid to Refugees'' stated:
``If, however, the result of the fundraising is disappointing, all is lost. Not only will the organizations have no more money, but our opponents who even now are urging the complete closing of our borders, will then say to the Federal Government: 'Close the doors, let nobody in. The Swiss people do not want them...''' [478]
Many Church leaders publicly recommended this collection. Prof. Karl Bart did so in the following words:
\textit{216}
``There are reasons for and against aid to refugees as currently suggested to us Swiss. The reasons for are: The Christian reason. 'In as much as ye have done it unto one of these least, ye have done it unto me.' The refugees are our concern: not because they are valuable or agreeable human beings, but because in all the world they are to-day the lowest and the most miserable people, and as such they, with their inseparable companion the Saviour, knock on our door. They are our concern, not in spite of their being Jews but just because they are Jews, and as such are the Saviour's brethren in the flesh. (I suggest that this first reason is the strongest and may well be the one decisive and effective reason in this matter). The Swiss reason. The refugees (whether they are aware of it or not) do us a great honour, in looking upon our country and seeking it out as the last refuge of justice and mercy. Many of the great and dreadful things which occurred in our time will be forgotten. After centuries, however, it will still be asked, whether Switzerland proved true to its name as the free Switzerland in these days, or renounced it. The question whether the Lest that we Swiss are capable of and have, can be preserved throughout the present crisis, will be decided only by opening our hearts and hands to these refugees, or by turning our backs upon them. The Humane reason. We see in these refugees the fate we have miraculously been spared. It is quite true that we also are not too well off to-day. It is, however, equally true that we are well enough off to be in a condition exactly opposite to these unfortunate fellow-creatures: well-fed and even rich. Can we bear this, without wanting to help them to the best of our ability? Would it not be disgraceful, even to let our lips suggest any reasons at all against offering such aid?'' [479]
In December, 1942, 1,595 refugees were admitted and 330 sent back. At the end of December, the number of immigrants and refugees amounted to 16,200. Of the refugees, 8,467 had entered Switzerland illegally between August 1, 1942, and Dec. 31, 1942. This development led the Department of Police to propose to the Federal Council that new decrees, more stringent than the preceding ones, be issued for the whole of the Swiss border. Apparently the order of Sept. 26, 1942, that being a Jew was no reason for admittance, mostly was not observed. [480] \textit{217} The decree of Dec. 29, 1942, ordered that foreigners arrested whilst crossing the border or in the region of the border (up till 10 kilometres) must be turned back immediately. Exemption would be granted to the categories a, b and c mentioned in the decree of Sept. 26, 1942. [481] ``Further, parents with children not over six years old; or if at least one of their children is not older than six; refugees who can prove that they have a spouse, parents or children in Switzerland; or when at least one of a married couple has been born in Switzerland. [482] The ``Report of the Swiss Protestant Relief Society'' comments:
``We are grateful that a Delegation of the Federation of Churches also remonstrated with the Federal Government in the matter. No substantial amendments to the decree were obtained, but in practice the attitude of the authorities was more obliging than the wording of the decree leads one to assume. The possibilities of providing asylum, and the readiness of the authorities to grant it, are in no small measure dependent on the willingness of the Swiss people to make sacrifices for the refugees. We therefore emphatically insisted that the congregations of the Evangelical Church should take upon themselves the financial responsibility for the upkeep of as many refugees as possible, and so to fulfil towards individual refugees Christ's commandment of love. [483]
In the first seven months of 1943, 1,821 refugees were sent back and 4,733 admitted. ``Its is impossible to determine, how many Jews were among those admitted; apparently they made up the vast majority.'' [484] On May 9, 1943, the Synod of the Evangelical-Reformed Church of the City of Basel adopted the following Resolution:
``The Synod, deeply concerned by the information received regarding instruction given by the authorities to the border guards and the dreadful horrors still being undergone by refugees wanting to cross our borders, charges the Church council to urge the Executive Committee of the Federation of Churches to remonstrate afresh with the responsible authorities on behalf of the refugees according to the Church's responsibility to be a Protector, and desires that the congregation, through the `Church Messenger', be kept suitably informed of the Synod's negotiations concerning the refugee and asylum problem. [485]
\textit{218}
In October, 1943, the Church Council of Zurich addressed the following message ``To the Reformed People of Zurich'':
``... We are able only through rumours to gain a vague impression of the dreadful reality. And because it is beyond the powers of our imagination, we are in danger of closing our hearts and trying to suppress any awareness of the fact that daily, hourly, indeed every single moment, thousands suffer, bleed, starve, despair, die. We also let ourselves be misled by a falsely understood neutrality, which freezes our feelings towards the distress of foreigners, or causes in us a moral apathy towards injustice and inhumanity, sometimes even making us adopt the catchwords and evil slogans of anti-Semitism and racial hatred, and persuading us to accept ideals which are hostile to the Gospel of love to God and to ones neighbour... All humane people are haunted by descriptions of the sufferings to which members of the Jewish people have been exposed during these past four years of war, this following centuries of being slandered, ridiculed, beaten and persecuted throughout the Christian era! Expelled from home and work, forcibly separated, children tom from the arms of their mothers, mothers from the arms of their children, anew they are uprooted just when they had supposed they had found a protecting refuge. They have been tossed towards an uncertain destiny, which all too often only spelled destruction, misery, starvation, beatings, despair and death. Indeed, no other nation has been so overwhelmed by storms of persecution and deluged by sufferings, as has been the people of Israel. Who as a Christian, or as a Swiss, can fail to be oppressed by the distress of the Jewish people, or to be confronted by questions unsolvable by the words guilt and atonement, because we have certainly sufficient cause to ask questions about our own guilt in this matter and to apply to ourselves Christ's word: `Except ye repent, ye shall all likewise perish.' Such an admission can result only in one thing: the emergence of a deep sympathy and a desire to help wherever and however we can, to grant refuge to the homeless, to shelter the exposed, to feed the hungry, to clothe the naked, to visit the imprisoned, to support the weak, to comfort the mourning; in accordance with the example of the good Samaritan and the teaching and promise of our Master: 'In as much as ye have done it unto one of the least of these my brethren, ye have done it unto me'...'' [486]
In November, 1943, the ``Social Study Committee of the Swiss Union of Reformed Pastors'' published the following Statement:
\textit{219}
``... With shame and sorrow we see this purposeful turning away from Christ in a monstrous effort to exterminate entire races and peoples. The Christian conscience cries out against this. We therefore appeal to all those in responsible positions in the world, to save what still may be saved. We demand that the Swiss Government which maintains diplomatic relations with all governments in the world, devise with them and with the International Red Cross, a plan of rescue. In the name of Jesus we demand that our authorities put a stop to the driving back of refugees to their death, until final measures are taken, and to grant them a safe, Christian asylum. Our thanks go to the people of Switzerland for their cordial hospitality, even though it is hampered by authority!...'' [487]
In my opinion this is the sharpest protest against the official refugee policy of the Swiss government that was ever published during the second world war.
At the end of 1943, it was ordered not to send Jewish refugees back if they objected. Thus Jews who fled Italy after its occupation by the Germans, were to be admitted; however, in the case of a real `run' one might have to stop admitting them for some time. [488] It is, in my opinion, undeniable that the protests of the Churches and Church'' leaders contributed to alleviating the measures against the refugees and their ultimate cancellation in practice. In the meantime, unspeakable sufferings had been inflicted on refugees who had been sent back and fell into the hands of their mortal enemies.
d. Aid to Refugees
We already mentioned some of the activities of the ``Swiss Protestant Relief Society for the Confessing Church in Germany'', [489] for instance the Annual Conference held on November 17, 1941, and its participation in the general collection for Aid to Refugees, held during October-November, 1942. In order to show the spirit in which this refugee work was done, we record the following letter which was sent by the Executive Council of the Society to the Swiss-Israelite Union of Congregations, on June 22, 1943: \textit{220}
``You have ordered a call to an Assembly of Mourning, for next Sunday, 27th June, 1943. You will then recall the horrible decrees to which Jews in Europe are subjected, and the unspeakable hardship and oppression under which people nowadays suffer and die. Together with you we are deeply shocked at the mass murder that has engulfed European Jewry. Only with dread and horror can one read of the number deported from Germany, France, the Netherlands, Rumania and Greece. We fight against allowing suffering to become a familiar routine, and against blunting of concern on the part of our people of Switzerland at such distress. To us these dry figures represent human beings, who have lived, suffered and died. Their mass graves and their ashes will, till the coming Day of the Lord, be a shocking accusation against a Europe which forgot God. As Christians we cannot let the Assembly of Mourning of the Swiss-Israelite Union of Congregations pass without a cordial word of sympathy and participation. Deeply moved, we shall join our thoughts with yours in intercession. We know that each murder and every act of violence is rooted in the godless thinking of godless minds. The unkind word and the unappreciative gesture are signs of poisoning of the minds. The fact that this poisoning could assume such terrible proportions in `Christian' Europe, where especially the Jewish people are victimised, shames us and gives us cause for severe self-accusation. So little have we Christians understood Jesus Christ and so far apart from him have we lived, that godless thinking was able to create this insane racial hatred and merciless cruelty in our midst, raging as a demon against the Jews. On your day of mourning we join hands with you in sympathy and sorrow. At the same time we confess our guilt before God and mankind. We regret every word of contempt, we Christians ever uttered against Jews. We regret that we have shamed Jesus Christ by our self-righteousness and our hardness of heart. We regret that we Christians were not more loyal to our Master and thus failed courageously to struggle, in time, against every expression of anti-Semitism. On this day of your mourning we implore the Almighty for his mercy, for the sake of Jesus Christ, with the publican's prayer of penitence: 'God be merciful to us sinners'. On your day of mourning we pray that God's mercy may be upon you, and the deep consolation of His promise from the precious Old Testament, which also has comforted us again and again: 'And I will cause the captivity of Judah and the captivity of Israel to return, and will build them, as at first. And I will cleanse them from all their iniquity, whereby they have sinned against me. And it shall be to me a name of joy, a praise and an honour before all the nations of the earth, which shall hear all the good that I do unto them: and they shall fear and tremble for all the goodness and for all the prosperity hat I procure to it' (Jeremiah 33, 7-9). \textit{221} 'The Lord thath sent me to bring good tidings, to heal the broken-hearted, to proclaim liberty to the captives and the opening of the prison to them that are bound; to proclaim the acceptable year of the Lord, and the day of vengeance of our God; to comfort all that mourn' (Isaiah 61, 1-2). 'Yea, though I walk through the valley of the shadow of death, I fear no evil; for thou art with me; thy rod and thy staff they comfort me' (Psalm 23, 4). 'For the mountains shall depart, and the hills be removed; but my kindness shall not depart from thee, neither shall the covenant of my peace be removed, saith the Lord that hath mercy on thee' (Isaiah 54, 10).'' [490]
Strong powers in Switzerland objected to the admission of refugees. Therefore the ``Protestant Relief Society'' undertook action in two different but interdependent fields: influencing public opinion, and rendering practical aid. Books and pamphlets were published and distributed. [491] Rev. Paul Vogt was appointed `Refugee pastor', and was later joined by two other ministers. They launched the ``Place of Refuge Operation'', [492] asking members of the Church to provide places in their homes to Jewish refugees who were unable to work: pregnant women, mothers with little children; people ill, invalided or old. Another way to help for the local churches was to pay the maintenance (120 Franc per month) of a refugee being cared for in one of the houses of the homes of the Society. [493]
``Help was not just rendered to Protestant refugees; the majority of them were Jewish... We are convinced that we may not exploit the difficult situation of our proteges by trying to convert them. Rather, we respect the religious conviction of the Jews, whose care has been entrusted to us. \textit{222} Therefore two Refugee homes were opened for observant Jews;... one accommodating 35 refugees, the other 26. Plans for a third refugee home were prepared. In order to reunite married couples and families, houses were rented in which a total of 111 persons were accommodated. Up till the end of 1943, 348 persons were helped and places for another 219 persons were in preparation.'' [494]
On October 1, 1944, 868 refugees who were unable to work were accommodated by the Protestant `Place of Refuge Operation'. 739 of them were Jewish, 115 Protestant, 8 Catholic (mixed marriage) and 6 without religion. One hundred and seventy-nine places, especially for children, were reported to the Committee for Aid to Children. [495]
Far be it for us to belittle the efforts of Rev. Paul Vogt and others, who did what they could. Yet the number of refugees who were helped is small in relation to the terrible need that existed. Moreover, in Switzerland, people did not risk their lives or freedom by taking in a Jewish refugee, as happened in many other countries.
e. The Deportation of the Hungarian Jews
On July 4, 1944, the following circular letter was sent by Prof. Karl Barth, Prof. Emil Brunner, Dr. W.A. Visser 't Hooft and Rev. Paul Vogt to pastors in Switzerland:
``We send to you, enclosed, two messages from Hungary and a covering letter dated June 19, 1944, which came from reliable sources and reached Switzerland through diplomatic channels. The messages have shocked us deeply. Out of a sense of responsibility we feel it our duty to convey these messages to you. We do not doubt that you will read them and let them circulate within your own group. They are also known to the competent authorities.'' [496]
There followed a wave of public protests. We quote some of them. [497] \textit{223}
On July 9, 1946, the Church Council of the Canton Zurich urged that the following message be read from every pulpit:
``The present day truly has revealed enough frightful things, but in the last weeks one piece of news has reached us which far exceeds anything that we have heard for years. Reliable witnesses inform us of terrible persecution of the Jews in Hungary. In a few weeks between three and four hundred thousand people have been sacrificed, and who knows how many more there will be. Many are dying of exhaustion or hunger, but the majority meet their death by gas. In one single place, at Birkenau, four crematoria are in use, in which every day six thousand people can be gassed and burned and incinerated. Hitherto Hungary had more than a million Jews. A number of towns already have been cleared of Jews. Persecution is said to be impending in the capital, if it has not already begun. We do not know what can have induced the government to take these dreadful measures and at whose door the responsibility for this dreadful deed must lie. What can we do? It is not for us to pass self-righteous judgment on the acts of other peoples, for we are not guilt-free. It does not lie in our power to order the cessation of atrocities. The Swiss Protestant Church Federation addressed an urgent request to the Federal Council and to the International Red Cross that they would do everything possible to rescue the Jews still living in Hungary. We invite congregations to make solemn intercession for all those who must tread this dark road simply because they belong to another race. Let us also pray for our sister-Evangelical Church in Hungary, that strength and guidance be granted to her at a time when her people is taking upon itself such a terrible load of guilt. Let the word of the Psalmist be our prayer: 'Keep not Thou silence, O God: hold not Thy peace, and be not still, O God'.'' [498]
The Council of the Evangelical Church of Canton Glarus, on July 12, issued a circular letter to all local ministers, drawing attention to the horrible reports of the extermination of Hungarian Jewry and stressing that, in all probability, reality would prove even more horrible than those reports implied. After having mentioned the appeals made by the Federation of Churches to the Federal Council and the International Red Cross, the circular letter continued:
\textit{224}
``We fully realise that at present we are incapable of stopping the demonic powers by any human means. Only God can do that, and we invoke His aid. We request you to inform your congregation of these horrors in a fitting manner, and in your prayers to intercede with God on behalf of those that are threatened. Finally, the Church Council hereby issues a call to leave all vengeance and retaliation to Him who has proclaimed: 'Vengeance is mine, saith the Lord'.'' [499]
At about the same time, the following Proclamation was issued by the Council of the Synod of Bern:
``Added to the untold miseries that the Jews have had already to suffer in various countries, the terrible mass murders of the Jews in Hungary which, according to reliable information were carried out recently, surpass all imagination and defy any description. The inhuman removal of people of all ages who, solely on account of their racial origin, no longer are considered fit to exist, is a monstrosity unparalleled in history, as well as a grievous sin and guilt before God. As Christians who have received light and life, salvation and mercy from Jesus Christ, we feel a painful indignation in the face of such methods of extermination. We declare that such destruction of our fellow-men was conceived by a spirit and will which came from below, and which will bring a curse and doom on humanity. A deep sympathy unites us with the countless victims. We thank our brethren of the faith in Hungary for their courage in standing up, in time, against these monstrosities in spite of great difficulties, and we urge them to continue to do everything in their power to stop these horrible mass murders. We call on Christians in our own country to fight all hatred and thirst for revenge among people of different origin and race, and to resist all prejudices and offensive slogans wherever they may appear. Let us not tire of intercession on behalf of the ancient people of the Covenant, of Israel.'' [500]
In August, 1944, the following circular letter was published by the Church Council of Canton Graubunden:
``Ecclesiastical and other proclamations and directives draw our attention to the fate of Hungarian Jewry. No pen is able to describe, no soul can sense, and no Christian mind can imagine what is being done to these unfortunate people. The human mind is powerless to grasp the horrors, day by day enacted with cool determination and limitless hate. Shocked cries, objections and protests, to those in authority have remained ineffective. Brotherly help to those threatened by death is not possible. \textit{225} Only one thing remains to the Christian, of which he cannot be deprived: prayer. Dear brethren, prepare the people, before offering your public prayers, by referring to the sufferings of the Jewish people of whom the Son of God was born. Tell openly from the pulpits how many hundreds of thousands are being exterminated in gas chambers and crematoriums, while further multitudes tremble in desperation, because the hour is at hand when they too will be herded like cattle, deported and slaughtered. Tell how human dignity is degraded, how man's rights are trampled upon, so that all Christian feelings have received a deadly wound. Remember in your prayers at church the unfortunate who are persecuted, urging all members of your congregations likewise to remember them in their prayers at home in solitude. The prayer of the merciful heart availeth much. Through it God's presence may become real in distant gas chambers; consolation and indestructible faith may shine in their horror-stricken eyes and upon their deadly-pale faces. Such prayer may also have the power to reawaken petrified consciences, and to paralyse the hands engaged in deadly tasks. The prayer should be said in repentance over unbrotherly words also spoken in our country about Jews now and in the past...'' [501]
32 SWEDEN
The political situation of Sweden was comparable to that of Switzerland: each country tried to preserve its neutrality, was afraid of a German attack, and sometimes gave in to German demands. Yet Sweden expressed its willingness to receive all the Jews from Denmark, whilst Switzerland closed its borders. We should, however, bear in mind that to Switzerland, owing to its geographical position, the challenge of the refugee problem was much greater than to Sweden. The Swedish Church denounced the persecution of the Jews more sharply than did the Swiss Churches. Apparently Swedish Church leaders were not afraid of offending Germany. But in Sweden there was no press censorship, as was the case in Switzerland.
On November 29, 1942, Manfred Bjorkquist was consecrated first Bishop of Stockholm. Along with the Bishops of the Swedish Church, there were also present representatives of the Church of Denmark and the Church of Finland. The Quisling Minister for Church Affairs in Norway sent an indignant letter to Archbishop Eidem, because he had not been invited to send a representative to the consecration. Dagens Nyheter, commenting on this report, wrote: \textit{226}
``What happened in Norway recently is sufficient explanation, if it is confirmed that Sweden's Archbishop did not reply to the letter. Archbishop Eidem's warning at Lutzen on November 6th against national self-sufficiency and arrogance provides an adequate answer. When now for the first time we see these things happening near at hand we are aghast at this self-sufficiency and arrogance; this complete contempt for human values.'' [502]
The last sentence refers to the deportation of the Jews of Norway.
On the first Sunday in Advent, 1942, the following Proclamation was issued by the Swedish Bishops:
``Hatred blinds and hardens. Hatred leads to destruction. Hatred is the most frightful and monstrous of the dark powers which now are dominating an unhappy earth. Jesus Christ condemns hatred in all its forms without exception. His words and deeds, His life and death, all mean an absolute judgment upon hatred. Whatever stands in contradiction with the royal command of love, which is the sum of the will of God, is sin, sin against the living God. Men may trample upon the commandments of the All-Highest. But God Almighty lives. And whoever turns away from Him has deserted the springs of life and is walking in the way of death. If we really want to be Christians, we must in all seriousness take up the fight against hatred, against all hatred. We must be strictest towards ourselves, so that we may not leave the smallest room in our heart for the evil spirit of hatred. So far as our voice reaches, we must, each in his own circle, stand up for love in word and deed, and fight hatred and the deeds of hatred. With horror and dismay we have learned in the last two days how an un-Christian racial hatred, which has spread over many lands in the world like a mortal pestilence, has now expressed itself in shocking acts of violence in our immediate neighbourhood, on our Scandinavian peninsula. Human beings are being subjected to the greatest sufferings, not because they have been legally convicted of misdeeds --- they have not even been accused of such things by regular legal procedure --- but solely because they belong by descent to a certain race. \textit{227} We have been deeply moved to hear the courageous Christian admonitions which our oppressed Norwegian sister-Church has directed to those in power in their country, not to rebel against the clear Word of God by doing deeds of violence in blind racial hatred. Everything that lies in our power to assist the poor people affected by this hatred is being done. That is our elementary duty as Christians and as human beings. But even if we cannot do much to help the unfortunate, we can and must bear them and their needs upon our hearts. We Bishops of the Swedish Church call all our fellow-Christians in Sweden, in the Name of God, to include these our tortured brethren of the race of Israel in our faithful and constant intercessions, and to make daily prayers to our Father in Heaven for the many who are suffering violence and disaster at this time.'' [503]
At a service of intercession in Goteborg Cathedral on the first Sunday in Advent, Dean Nysted said:
``Everything we have heard of the nameless sufferings of the Jewish people in past times dwindles to nothing in comparison with the fate that has overtaken them in recent years. We have read with disgust of the slave hunts of former times and the cargoes of slaves which were carried like cattle to America. Who could have dreamt anything so frightful as that such a ship would sail along our coasts last week, laden with men, women and children, who have no other fate to expect than that of the slaves or cattle for slaughter, and that not because of any crime of which they have been convicted but because they are of Jewish descent. The Church of Sweden must not keep silent when such a thing happens at our frontiers. If we were to keep silent, the stones would cry out. We are shocked to the depths of our hearts when we think of the sufferings of these unhappy people. We tremble at the dragon's teeth of hatred which are senselessly being sown... What harvest must grow from such seed? We stand powerless. What is being prepared for the Jews who have remained in Norway? Can our authorities do anything to save them? We implore them to consider this question seriously and without delay.'' [504]
In a broadcast sermon, Bishop Aukn of Strangnas commented upon the events of the time:
``Violence is triumphing, and the commandments which form the bases of our human common life are remorselessly being trampled upon. Every day brings new pictures of horror. Recently we received the news that the frightful plague of racial persecution has descended upon our Scandinavia... There are probably no limits to the depths to which people who are blinded by hatred may sink. \textit{228} But at the same time a wonderful thing is happening: in the midst of this darkness we are witnessing a bold and firm steadfastness which remains unmoved even when it leads to persecution and martyrdom. Such events have opened the eyes of many people who were subject to the prejudice that we have only to reckon with material factors and the resources of outward force. They bear witness to the power of the Holy Spirit, to the power of Christ, which works in secret and is unconquerable. If we in our Swedish Church are able to begin the new Church year as a free Church in a free country, that lays upon us a great responsibility: to stand up in unshakable faithfulness for the holiness of the laws of God, when the most elementary demands of justice are trampled upon.'' [505]
At a Meeting of Protest, held in Stockholm on the same Sunday, Dr. Natanael Beskow said:
``Here we are not concerned with neutrality or politics, but with humanity or inhumanity. Nothing of that kind must ever happen in Sweden. Indifference in face of a crime is in itself a crime.''
The meeting passed the following Resolution:
``In the name of Christianity and democracy, humanity and justice, we protest against the mass deportations of Jewish citizens from our nearest neighbour country, not for crimes committed but because of their race. We do this for the sake or our Northern community, but we are angry and distressed that Northern men have been able to commit this deed of shame. We protest in the name of international law, for without security in law all human order collapses, whether it be called old or new.'' [506]
Svenska Morgonbladet reported that it had received expressions of sorrow and sympathy from the leaders of various Church congregations. Bishop John Cullberg said at Strangnas:
``After what happened earlier in Norway, the latest telegrams about the persecution of Jews are not surprising. But we are profoundly shocked. The Norwegian Church has, through its statements, already interpreted the Christian conscience's protest against these atrocities. It must be loudly proclaimed that we in Sweden support this protest. \textit{229} With bleeding hearts, we think of the martyrs. And what should we say of their tormentors? All we can say is: 'Father, forgive them, for they know not what they do'.'' [507]
Although leading men within the Swedish Free Churches already at an early stage had separately expressed their feelings in the press regarding the persecution of the Jews in Norway, the Free Churches' Co-operation Committee wished to emphasize their mutual standpoint:
``God is the Father of all, and all men are called to receive the advantage of the adoption of sons, independent of race and birth. Racial persecution is thus a sin and a rebellion against God. The Jew is our neighbour, and we wish to love him as ourselves. Facing what is happening in Norway, we feel grief and distress. We are onlookers at a situation where our neighbour is being treated as something sub-human. We cannot remain silent witnesses to this We wish that our deeds could bring help, to undo what has been done. Our hope is that God will turn evil to good. We wish to join in the appeal of the Bishops of the Swedish Church, in the name of God, for intercessions for our tortured brethren of the race of Israel, and to make daily prayers to our Father in Heaven for the many who are suffering violence and disaster at this time.'' [508]
Under the subject heading ``Christian Gathering'', a meeting was held on December 6, 1942 at Hedvig's Church at Norrkoping. This meeting was arranged by clergymen. Speakers were Vicar Thysell, Pastor Einitz Genitz and Vicar Knut Ericson. We quote the following from Vicar Thysell's address:
``The information concerning 1,000 Jews driven from their homes, robbed of their property and transferred to Germany to meet a most cruel fate, has shaken us thoroughly and deeply. Those Jews were loyal Norwegian citizens: they had done nothing wrong. They were punished because they were Jews, without trial or verdict. \textit{230} The people of Norway were the first to speak up and protest through their Church. The brave and strong words from Norwegian Church leaders, themselves oppressed and persecuted, have moved us profoundly. Now we, too, must speak. There are occasions when it would be denying truth to remain silent. We bear a special responsibility towards God and humanity when such things are happening around US. We Swedes are best able to represent the world's conscience in this case, and we feel that we also owe our Norwegian brethren a clear and unequivocal declaration on our stand. We also have another responsibility in this case, one that lies even nearer to us: our responsibility towards the Jewish brethren, who belong to our own people. The contamination of anti-Semitism has also reached our own country. Infamous and false propaganda is being spread from plague centres within our own borders. We have hitherto belittled this danger. Now we see to where it is leading. It is time for us to wake up! We must also at this hour think of the mass persecution of Jews which is taking place in other countries. From available information it appears that the anti-Semitic wave is still rising. The threat now also concerns half-Jews. Our taking a stand might seem meaningless to all of these. We cannot stop violence. It may, however, in a secret way, bring a ray of consolation and hope into despairing hearts. We have named our meeting `Christian Gathering'. That our consciences react to the outrage which is happening, is the result of the spiritual values of life which we have received from Christ and the Prophets of Israel --- from the very people who are now being persecuted in so many countries. On those basic values rests our Nordic judicial culture. We pride ourselves on Sweden being a constitutional state. Here no one can be sentenced and punished except on the basis of justice. Here, right is not equal to might. Above the power of the state stand those eternal truths of our relation to God and each other, which have been revealed to us and which, in our consciences, appear as indefeasible values of life. Arnulf Overland says: 'Some things are greater than you. There are mountains with snow. There are dearer things than your life; you shall fight for it'. The dearest thing we have are those values of life that Christ gives us. The persecution of the Jews is not the only proof --- but the most horrible of all --- of a denial of these values of life. We are here to-day to confess our belief in these eternal foundations for human society, which God himself has laid. We believe in God, our Lord Jesus Christ, and our Father, who has called us all, independent of race and all other differences, to receive the adoption of sons and to live in communion with Him and each other. We wish to adhere to this Christian evaluation of man. And we reject as hostile to God and anti-Christian that brutal conception of man, and that contempt of mankind, which forge the acts of violence in anti-Semitism. We regard the brotherhood of humanity as holy, and brotherly action as our goal. We feel it our obligation to act towards our Jewish brethren in accordance with Jesus' rule of life: 'All things whatsoever ye would that men should do to you, do ye even so to them'. \textit{231} Do we seriously mean them to be our confession of faith? Do we dare uphold it, as our Norwegian brethren have done, even if our faith should be tried as gold is tried in fire? Whatever happens, we need not fear, if we follow Jesus Christ, the eternal King. The weapons of iniquity are doomed annihilation. Christ stands on the side of the persecuted. His spirit, the Spirit of Truth, Righteousness and Love, is strongest of all. The day of freedom shall again dawn for the persecuted and oppressed.'' [509]
It is remarkable that earlier deportations of Jews in countries such as Germany, France and the Netherlands, did not prompt the Swedish Church leaders to raise their voices, though the number of deportees was much greater than that of the Jews deported from Norway. It seems likely that what was happening in Western Europe was less known in Sweden than what was happening in Norway. Moreover, human beings generally are more moved by cruelties committed on their doorstep, than by what happens further away. The Proclamation of the Swedish Bishops expressed ``horror and dismay'' because ``an un-Christian racial hatred... has now expressed itself... in our immediate neighbourhood, on our own Scandinavian peninsula''.
As far as we know, the Swedish Church did not issue a Protest against the persecution of the Danish Jews. In fact, events in Denmark took place so rapidly that a Protest would hardly have done any good. The pressure of the Swedish Archbishop (and others) on the Swedish Government to make public their willingness to receive all Danish Jews, was important. It appears that this step, indirectly, saved many lives. Dr. Leni Yahil relates the following:
``The Swedish Foreign Office contacted Richert, the Swedish envoy in Berlin, on the same day, September 29 [1943], and again on the next day, September 30, in order to plan with him the appeal to the German Foreign Office. It was decided that Richert would ask the Germans whether there was a basis to the rumours about an impending deportation of the Jews from Denmark, and that he would stress the fact that such a deportation would cause great indignation in Sweden. Moreover, he was to propose that all Danish Jews be transferred to Sweden and concentrated there in a camp and that the Swedish Government would be responsible that 'they would not be able to undertake any activity that might be harmful to Germany'. \textit{232} It became evident that the Swedes did not intend to take any further action. [Niels] Bohr, Ebbe Munk and their friends, however, were of a different opinion. As we know from entries in Ebbe Munk's diary and from his letters to Christmas Moeller in London, it was the Danish group with the active support of prominent Swedish circles which brought about the publication by the Swedish Government of the appeal to the Germans. On October 2, the day following on the night of the persecution in Denmark, Bohr had an interview with the Foreign Secretary, Guenther. It seems that already on the preceding day the Danes had tried to persuade the Swedes to publish their appeal to the Germans in the hope that such a publication might prevent the deportation. Since this had not been done, Bohr requested the Swedish Foreign Secretary to repeat his appeal to the Germans and to propose to them that the boats on which the Jews were concentrated, be directed to Sweden instead of to Germany. Guenther proposed this to the German Ambassador Thomsen, who called on him at 9 o'clock in the evening on that day. A reply to this proposal was never received. Through Kammerherr von Kruse, the Danish Ambassador in Stockholm, and with the active support of Prof. Stefan Hurvitz, an audience with the King of Sweden was arranged for Bohr, in the afternoon of the same day. During this audience Bohr proposed to the King that the Swedish appeal to Germany be published. The King did not reply, but at the end of the audience the Foreign Secretary was called in. That same evening the Swedish radio broadcasted an announcement about the steps taken by Sweden in Berlin. The announcement stressed that the Swedish Ambassador, on behalf of his Government, had declared that Sweden was willing to receive all the Danish Jews. We know that this announcement encouraged the Jews as well as their Danish helpers to organize the mass escape. In his letter to Christmas Moeller, dated October 12, Munk told that the Swedish Government only agreed to publish the announcement, after the Arch-bishop, professors and other prominent persons had declared that they were prepared to sign an open letter to the Government about the subject.'' [510]
The King of Sweden was present when, in May, 1944, Archbishop Eidem delivered his opening address to the General Assembly, to which 2,600 parish-delegates and guests from all over the country had come. Archbishop Eidem said:
``... Our Christian conscience must keep constantly on the alert in the face of all that is happening in the world around us. Might is not right. Power is not justice. Torture is not permissible in any circumstances. Innocent people must not be made in any way responsible or punished for the acts of others. Houses and entire communities must not be purposely destroyed in order to intimidate or cripple an enemy. \textit{233} People of a particular racial and national group, such as the unhappy people of the Jews, must not be persecuted and martyred because of their membership in that race or national group. All such actions are not only barbarism but sin... It is indeed no wonder that a frightful harvest of hatred and vengefulness is growing from the sowing of such seeds on our poor earth. As Christians we are called to take up the fight against hatred in every shape and form in this world, which now seems to be a free field for unleashed evil forces; and we must conduct this fight first of all in our own hearts, but each man also in the place where he lives. And we must not grow tired or weary in this fight.'' [511]
It would be interesting to know how far the King was influenced by this stand of his Archbishop when, shortly afterwards, he appealed to Regent Horthy on behalf of the Hungarian Jews.
It is my impression that the Church of Sweden also undertook steps on behalf of the Jews about which we know nothing, and perhaps never shall. Concerning two steps, we do know at least something. Firstly, the secretary of the Church of Sweden's Committee for Foreign Affairs, Rev. Johansson, communicated to me: ``It is true that Archbishop Eidem paid a visit to Hitler himself, but no details are officially known''. [512] Secondly, the German Ambassador in Slovakia, Ludin, informed the German Foreign Office in a letter dated January 3, 1945, that the Archbishop of Uppsala had addressed the Slovak Prime Minister (Tiso) with a plea for the transfer of ``the unfortunate Jewish brethren'' to neutral territories. [513] We have, however, not succeeded in retrieving a copy of Archbishop Eidem's letter.
\textit{234}
COUNTRIES AT WAR WITH GERMANY
33 GREAT BRITAIN
a. The First Period
Few voices were publicly raised in England during the years 1940 and 1941. In 1940, the Battle of Britain apparently occupied the national attention so much that people tended to forget everything else. If any statements made by ecclesiastical leaders were issued in 1941 (except the statement of the Church of Schotland, mentioned below), I have failed to find them. The Beckley Social Service Lecture is delivered annually in connection with the Methodist Conference in Great Britain. Its purpose is to review certain major problems in the field of social service from the point of Christian responsibility. In the year 1940 the Rev. W. W. Simpson, now secretary of the Council of Christians and Jews, was invited to deal with the refugee problem and the fight against anti-Semitism. His lecture was published in book-form. [514]
In May 1940, the General Assembly of the Church of Scotland issued the following statement:
``The General Assembly deplore the continued persecution of Jewish minorities in Central Europe, and deeply regret that the situation has worsened in Hungary. The General Assembly warmly appreciate the vigorous protest against the new anti-Jewish legislation made by the Hungarian Reformed Church, and assure the Committee and the missionaries themselves of their sympathy with all endeavours to minister relief and comfort and hope to suffering Jews, so far as it may be in their power to do so.'' [515] \textit{235}
I regret that I have not succeeded in finding any confirmation of the ``vigorous protests'' made by the Hungarian Reformed Church. In May 1941, the Assembly anew expressed:
``their deep sympathy with the Jewish people in their tragic sorrow, and, realising the gravity and intricacy of the problem, approve the settling up of a Sub-Committee to survey the whole situation, and they resolve to appoint six members ad hoc to assist in this survey.'' [516]
This expression of sympathy was repeated in May, 1942, whilst the General Assembly also warned ``their faithful people against the growing menace of anti-Semitism.'' [517]
We record the statements issued by the Presbyterian Church of Ireland during the second world war in this chapter, as most of the members of this Church live in the Northern part of Ireland which is under the sovereignty of Great Britain. In June, 1942, the Assembly of the Presbyterian Church of Ireland issued the following ``Resolution anent the Jewish People'':
``That the following resolution, adopted at a recent conference in connection with the Presbyterian Alliance, be given the warm approval of the General Assembly: 'That this Conference of representatives of the Presbyterian Churches of Great Britain and Ireland, held at Edinburgh on the 28th day of January, 1942, having considered the position of the Jews in the problem of post-war reconstruction, deplores any denial to persons of Jewish descent of the right of equal treatment before the law and of other rights due to their status as ordinary citizens, and urges that all Governments shall take immediate steps to restore to the full status of human dignity such Jewish people as have been deprived of it, and, in particular, that all legislation unjustly diminishing the rights of Jews, as such, shall be repealed at an early date; recognising also that liberty of conscience is an essential part of civil liberty, and that a free exchange of religious convictions is a necessary condition of all understanding between races and nations, the Conference urges on all Governments the recognition of the unfettered right of every individual to free choice in religious faith and to the public profession and preaching of it so long as these rights do not run counter to public law and order. The Conference urges His Majesty's Government, in conjunction with other allied and friendly nations, to provide for some scheme of emigration for Jews who cannot find a home in Europe.'' [518] \textit{236}
b. Mass Massacres. The Fate of the Refugees
On June 26, 1942, Reports of the massacre of Jews in Poland were broadcast by the B.B.C. The Chief Rabbi, Dr. Hertz, based a special Sunday evening broadcast on the reports. On July 8, 1942, the Archbishop of Canterbury inveighed, on the European service of the B.B.C., against ``so terrible a violation of human and Divine law.'' [519] On October 15, 1942, the Bishop of Chichester spoke in the Upper House of the Convocation of Canterbury:
``The torture and the ceaseless and systematic deportation of the Jews form some of the darkest chapters in the tragic history even of that people, and the latest report which has reached this country tells of the deportation, in terrible circumstances, of thousands of Jewish refugees from Vichy France, where they had thought they were safe from the oppressor, to Occupied France and thence to Eastern Galicia, leaving behind them between five thousand and eight thousand children of whom many are now orphans, while large numbers do not know their parents or their own names, and all are waiting for the charity of Britain, or America or Switzerland to give them sanctuary.'' [520]
Also in October, the Archbishop of Canterbury sent the following Message to the Jewish Bulletin:
``The situation of the Jews is unique, and yet has lasted for many centuries. They are a people conscious of close and real unity, and yet they have no motherland. Other people have survived and maintained their identity when there was no national State to which they could be loyal; but there was always a homeland inhabited by the people who remembered their days of independence and hoped for its restoration. For the Jews there has been no such a homeland. Their eyes might turn to Palestine; but though there were Jews among the population there, they did not form the bulk of it. The Jews as a people have been homeless. They have lived among the other peoples of the earth, and they have been loyal citizens of the nations which have made them welcome. But if their hosts turn against them they have no remedy. In earlier periods this has happened from time to time. In our day it has happened on a scale without parallel. Their sufferings are appalling and entirely undeserved. It should be our aim to assist them in all ways in our power; for their need is desperate. \textit{237} But there is more in their claim than a plea for sympathy. One of the tests of a people's civilisation is its capacity to treat well a defined minority. To fail in this is to revert to the ethics of the wolf-pack; and to succeed is the evidence of moral stability. In the case of the Jews our task is the easier because the moral principles which we profess are largely drawn from that sacred literature which we share with them. We should be standing together in loyalty to those principles against all who repudiate or ignore them. Anti-Semitism is evidence of a barbarous outlook and a religious apostasy.'' [521]
In the same month, the Free Church Federal Council sent a letter to the Chief Rabbi, Dr. Hertz, expressing ``the deep feelings of indignation and sympathy with which the Free Churches of this country regard the cruel persecution from which the Jewish race is suffering through the tyranny exercised by the Axis powers''. The message continued:
``We assure you of our continued prayers to Almighty God that its sufferings may speedily be brought to an end, and that all peoples may once again enjoy freedom of worship, preaching and teaching according to conviction without incurring civil disability or penalty in any form.'' [522]
On October 29, 1942, an audience of 10,000 assembled in the Albert Hall to voice their protest against ``the ruthless policy of extermination decreed by the Nazis and their satellites against the Jewish population in all territories under their sway''. The Archbishop of Canterbury was in the chair.
``Speaking about the deportations from France, the Archbishop mentioned the fact that children from two years upwards are now also being deported. 'There is something familiar about that,' he said, 'but when the earlier Nazis massacred the Innocent of Bethlehem it was on those of two years and less that destruction fell; and that in a smaller number.'...
The Archbishop concluded by saying that:
``he was grateful for this opportunity to share in the effort to express our horror at what has been and is being done, our deep sympathy with the sufferers, our claim that our own Government should do whatever is possible for their relief, and our steadfast resolution to do all and bear all that may be necessary to end this affliction.'' \textit{238}
Dr. I. S. Whale, Moderator of the Free Church Federal Council, speaking in the name of the Free Church, declared that anti-Semitism in all its forms was ``an outrage against that sanctity of law which is one of the most precious gifts of ancient Israel to modern Christianity''. Bishop Matthew spoke on behalf of the Roman Catholic Church. The following resolution, moved by the Archbishop of Canterbury, was unanimously adopted:
``This meeting, representative of British public opinion and of the United Nations fighting in the cause of freedom, places on record its profound indignation at the unparallel atrocities which have been and are being committed daily by the German Government and its satellites against the unarmed citizens of countries under the Nazi yoke. It records its horror at the deliberate policy of extermination which the Nazis have declared against the Jews wherever they are to be found, and extends its profound sympathy to the families of the unhappy victims of a systematic terror carried out by wholesale massacre, the murder of innocent hostages, the inhuman separation of children from their parents and other unspeakable cruelties and atrocities. This meeting expresses its heartfelt admiration for the heroism and gallantry of the fighting forces of the United Nations now leading us to victory, and desires to convey its deep sense of gratitude to those people in the occupied territories who, despite the terror, have done so much to help and succour their Jewish fellow-victims.'' [523]
On November 10, 1942, the Archbishop of Canterbury, inaugurating a new Parliamentary session, drew once more the attention to the extermination of the Jews, that ``horror which is going on almost at our door''. Contrasting ``what is still our standard of living'' with the ordeals of the afflicted, ``packed in cattle trucks... sixty in each...given little food'' so that ``on one occasion they all died of starvation'', he inquired ``whether it is thought possible that we may be able to do something to bring relief to these sufferers''. He mentioned as a shining example ``the amazing generosity'' of the Swiss whose ``frontier has been technically closed but actually open'' and suggested that Britain should give aid to the Swiss in support of refugees who can make their way there. He also recommended the granting of visas to those able to reach Britain: \textit{239}
``I hope that we should not in such a case waste our time in considering whether we have done as much or more than other nations for people who are in this kind of distress; the only question which really matters is whether we have done all we can... Again I hope we shall not waste time by considering whether these people fall into the categories drawn up to regulate such matters. Categories are nothing but administrative headings, and can be altered, if we wish, to include some who do not fall under them...'' [524]
The Archbishop of Canterbury again urged the Government, in a letter to ``The Times'' [525], to admit to Britain ``any refugee who might succeed in escaping''.
c. Retribution for the Persecutors; Intercession for the Persecuted
At the beginning of December, 1942, the Archbishop of York delivered a speech in the House of Lords. The Archbishop said:
``Men, women and children are being ruthlessly put to death by massacre, poison, gas, electrocution, or being sent long journeys to unknown destinations in bitterly cold weather without food or drink. Children that die on the way are cast out from the open trucks to the side of the railway. Such is Hitler's new order.''
The Archbishop called upon the Government
``...to state solemnly that when the hour of deliverance comes, retribution will be dealt out not only on the cold-blooded and cowardly brutes who order these massacres, but also on the thousands of underlings who appear joyfully to be carrying them out.'' [526]
The ``Solemn Statement'' requested by the Archbishop of York (and many others) was published on December 17, 1942, simultaneously in London, Washington and Moscow, with the assent and support of all the Allied Governments and of the British Dominions. The text was as follows: \textit{240}
``The attention of the Governments of Belgium, Czechoslovakia, Greece, Luxembourg, the Netherlands, Norway, Poland, the United States of America, the United Kingdom of Great Britain and Northern Ireland, the Union of the Soviet Socialist Republics, and Yugoslavia, and of the French National Committee, has been drawn to numerous reports from Europe that the German authorities, not content with denying to persons of Jewish race in all the territories over which their barbarous rule has been extended the most elementary human rights, are now carrying into effect Hitler's oft repeated intention to exterminate the Jewish people in Europe. From all the occupied countries Jews are being transported, in conditions of appalling horror and brutality, to Eastern Europe. In Poland, which has been made the principal Nazi slaughterhouse, the ghettos established by the German invaders are being systematically emptied of all Jews except a few highly skilled workers required for war industries. None of those taken away are ever heard of again. The able-bodied are slowly worked to death in labour camps. The infirm are left to die of exposure and starvation or are deliberately massacred in mass executions. The number of victims of these bloody cruelties is reckoned in many hundreds of thousands of entirely innocent men, women and children. The above-mentioned Governments and the French National Committee condemn in the strongest possible terms this bestial policy of cold-blooded extermination. They declare that such events can only strengthen the resolve of all freedom-loving peoples to overthrow the barbarous Hilarity tyranny. They reaffirm their solemn resolution to ensure that those responsible for these crimes shall not escape retribution, and to press on with the necessary practical measures to this end.'' [527]
The Bishop of London, Dr. Fisher (later on to be the Archbishop of Canterbury) voiced in the House of Lords ``the whole hearted support for the statement which is forthcoming from Christian circles''. Referring to the appeal made by the Archbishop of York, the Bishop said:
``It would be a satisfaction to the Archbishop and others if it were made clear that retribution will be exacted not only from those who devised and ordered these proceedings, but also in due degree of responsibility from those who carried out joyfully and gladly the orders which were given to them. The deeds were so repugnant to the laws of God and to every human instinct of decency that whoever took a share must receive due retribution for them. He hoped that it would be made clear that we and all our Allies would offer free asylum gladly to all who could escape.''
The Bishop also urged that: \textit{241}
``Neutral countries should be encouraged to grant sanctuary to refugees by a guarantee that for every Jewish refugee from Nazi tyranny they would receive, the United Nations would undertake to share in the cost of maintenance and would make possible the resettlement after the war of refugees in a permanent and abiding home.'' [528]
At the end of January, 1943, the Archbishops of Canterbury, York and Wales issued, ``in the name of the Bishops of the three provinces'', a statement in which they again stressed the two main points in the Bishop of London's speech in the House of Lords in December, 1942, namely: support of the Declaration made by the Allied Governments that ``those responsible for these crimes shall not escape retribution and the demand to provide a sanctuary for the victims. The ``Appeal to the Government'' reads as follows:
``The Bishops of England and Wales have been profoundly stirred by the declaration made in both Houses of Parliament on behalf of His Majesty's Government on December 17th, 1942, describing the barbarous and inhuman treatment to which the Jews are being subjected in German-occupied Europe. They note that the number of victims of this policy of cold-blooded extermination is already reckoned in hundreds of thousands of entirely innocent men, women and children. They note further that the extermination already carried out is part of the carrying into effect of Hitler's oft-repeated intention to exterminate the Jewish people in Europe, which means in effect the extermination of some six million persons in the territories over which Hitler's rule has been extended. The Bishops of England and Wales declare that the sufferings of these millions of Jews and their condemnation, failing immediate rescue, to a cruel and certain death, constitute an appeal to humanity which it is impossible to resist. They believe that it is the duty of civilised nations, whether neutral or Allied, to exert themselves to the utmost possible extent to provide a sanctuary for these victims. They therefore urge the Government of the United Kingdom to give a lead to the world by declaring its readiness, in consultation with the Dominion Governments, to co-operate with the Governments of the United and neutral nations in finding an immediate refuge in territories within the British Empire as well as elsewhere for all persons threatened with massacre who can escape from Axis lands, or for those who have already escaped to neighbouring neutral countries and can make room for other refugees to take their place.'' [529] \textit{242}
That not everyone agreed with the demand for retribution becomes evident from a speech given by the Archbishop of York at a city meeting in Leeds, on March 14, 1943. The Archbishop had been told that he was unchristian in asking for retribution. Objections were evidently made to the Archbishop's request that ``refugees from this horror can find a refuge wherever the British flag flies''. Apparently there was the feeling that there might be spies amongst the refugees; that the territories under the British flag would be flooded by a mass immigration of Jewish refugees, and that this would create insurmountable problems after the war. The Archbishop said the following:
``...The persecution of the Jews is, however, unique in its horror. It has the characteristics which make it stand by itself in the long history of cruelty and tyranny. It is a deliberate policy of extermination directed against, not a nation, but a whole race. Neither their nation, nor their profession, nor their character will save Jews from this sweeping sentence. They are doomed without trial, without crime, without the possibility of defence, simply because they belong to the race from which the prophets came, and of which our Lord and His disciples were members. They are condemned to death to satisfy the blood lust of a cruel and wicked megalomaniac who by fraud and violence now holds the greater part of Europe in his grasp... What can be done? 1. Let the German people know what is being done in their name. 2. Let the German people also be told solemnly and repeatedly that sure
retribution awaits not only the master criminals who have ordered these
horrors, but also their brutal underlings who are carrying them out,
often apparently with zest.
I have been told that I am un-Christian in asking for retribution. Have those who thus criticise never read that the Christ said that rather than a man should offend one of these little ones it were better that a millstone should be hanged about his neck and he be cast into the sea. I ask for this broadcasting of the Allies' determination to punish, in the hope that it may stay the hands of at any rate some of the criminals. Fear is sometimes effective when mercy makes no appeal. 3. We must make it plain that refugees from this horror can find a refuge
wherever the British flag flies. Every precaution will have to be taken
against spies. And the refuge will only be promised for the period of
the terror. Few will be able to reach our shores. But give them this
hope of refuge.
4. Support the Government in the efforts they are now making, with other
allied powers and the neutrals, to help the Jews now in danger and to
provide succour for their refugees. \textit{243}
We must do all we can in the name of Christianity and humanity to save at any rate a remnant from these foul murderers. Victory is the only sure road to their deliverance. The war becomes increasingly a crusade not only to preserve freedom and justice, but also to overthrow and shatter cruelty and tyranny in their most savage and hateful forms.'' [530]
At the end of 1942, a statement was issued by the Archbishop of Canterbury and the Moderator of the Free Church Federal Council, urging that special intercessions be offered in all churches on the first Sunday of the New Year:
``We do not doubt that in all congregations prayer is throughout this time being offered for the Jews of Germany and the occupied countries, who are suffering so terrible an affliction and over whom the threat of extermination is hanging. It is a bitter grief that our nation can do so little to help, but short of victory in the war there is no way in which we can ourselves effect anything comparable with the need, and the massacre goes on day by day. We should be united in constant prayer to Almighty God that this monstrous evil may be checked and the Jews delivered from their tormentors; and as a focus for such united prayer we urge that special intercessions be offered in all churches on the first Sunday of the New Year.'' [531]
Seven ``representative German Lutheran Pastors in England'' commented, in a letter published in ``The Times'', as follows:
``On the first Sunday of the New Year when the Gospel appointed to read in all German Lutheran Churches is the story of the murder of the innocent (St. Matthew 2, 16-18), we ministers of the German Lutheran Church in England feel in duty bound to call our congregations to solemn prayer and intercession for the Jewish people in their unparalleled sufferings. It was the anti-Jewish legislation as applied to the ministry which brought the Lutheran Church in Germany to its first witness against idolatry and barbarism and caused it to become a `Confessing Church'. Some of us wish that the protest then made had been stronger, more general, more frequent; but it is not for us who now live in safety to criticise those who under fire have done their utmost not to bow to Baal. While they are silenced by the terrors of persecution, we know that they would want and expect us to speak on their behalf and in the name of all who confess themselves Christians in Germany. \textit{244} In fellowship with them and in solidarity with the people of whom Christ our Lord was born, in solemn protest and deep repentance we recall the words of the Old Testament: 'Open thy mouth, judge righteously and plead the cause of the poor and needy'. (Prov. 31, 8-9).'' [532]
d. Practical Steps Demanded; the Bermuda Conference
Many times Church leaders in Great Britain demanded that their Government should take practical steps for the rescue of the Jews of Europe. Some of their statements on this subject have already been recorded in the preceding paragraph. In a letter to ``The Times'', the Bishop of Chichester recommended that Germany should be officially requested to let Jews emigrate to neutral countries. [533] In Parliament, an all-party committee of members of both houses was formed to prod the Government into action. Its first meeting, on January 27, 1943, was addressed by the Archbishop of Canterbury. [534] On February 4, 1943, at the annual meeting of the Council of Christians and Jews, the Archbishop of Canterbury referred to ``the deep concern felt by all sections of the British public at the reports of mass extermination of Jews and others at the hand of the Nazis''. He outlined ``the steps which he had taken as one of the Joint Presidents of the Council, and in association with the leaders of the other sections of the Christian community, in the hope of securing some measure of relief to the victims of this persecution.'' [535]
On March 23, 1943, the Archbishop of Canterbury presented the following Resolution to the House of Lords:
``To move to resolve, that, in view of the massacres and starvation of Jews and others in enemy and enemy occupied countries, this House desires to assure His Majesty's Government of its fullest support for immediate measures, on the largest and most generous scale compatible with the requirements of military operations and security, for providing help and temporary asylum to persons in danger of massacre who are able to leave enemy and enemy-occupied countries.'' \textit{245}
The Archbishop said:
``...We are wisely advised not to limit our attention in this connection to the sufferers of any one race, and we must remember that there are citizens of many countries who are subject to just the same kind of monstrous persecution, and even massacre. None the less, there has been a concentration of this fury against the Jews, and it is inevitable that we should give special attention to what is being carried through, and still further plotted against them... ``We are told that the only real solution is rapid victory. No doubt it is true that if we could win the war in the course of a few weeks we could still deliver multitudes of those who are now doomed to death. But we dare not look for such results, and we know that what we can do will be but little in comparison with the need. My whole plea on behalf of those for whom I am speaking is that whether what we can do be large or little it should at least be all we can do.''
The Archbishop then told of the deportation of Jews from Moravia, Germany, Rumania, and Holland, and of the slaughter of Jews in Poland. He continued:
``I believe that part of our difficulty in arousing ourselves and our fellow-countrymen to the degree of indignation that it would seem to merit is the fact that the imagination recoils before it. It is impossible to hold such things at all before the mind. But we are all agreed in this House on the main purpose of this Motion, to offer our utmost support to the Government in all they can do; but with all sympathy for members of His Majesty's Government, I am sure they will forgive some of us who wonder whether quite everything possible has really already been done.''
The Archbishop recalled ``the solemn statement of the United Nations made public on December 17'', and contrasted ``the solemnity of the words then used, and the reception accorded to them, with the very meagre action that had actually followed''.
``It is the delays in the whole matter while these horrors go on daily that make some of us wonder whether it may not be possible to speed up a little. One must admit that some of the arguments hitherto advanced as justifying the comparative inaction seem quite disproportionate to the scale of the evil confronting us.
As reasons for no further action, ``the great part that has been taken by this country and other countries in the relief of the refugees'' was pointed out.
\textit{246}
``That, of course, would be relevant if the people in the other lands were suffering great discomfort or great privation, but when what you are confronted with is wholesale massacre, it seemed to most of us not only irrelevant but grotesquely irrelevant.''
The Secretary of State for the Colonies had given a promise with regard to the admission of Jews to Palestine, on February 3, but on February 24 no attempt to move these persons had yet taken place. The Archbishop made a plea that action should be taken as promptly as possible to carry out the promises given by the Colonial Secretary. He also urged, ``that we should revive the scheme of visas for entry into this country''.
``We want to suggest the granting of blocks of visas to the Consuls in Spain and Portugal and perhaps in Turkey to be used at their discretion. We know of course that the German Government will not give exit permits. What matters is that we should open our doors irrespective of the question whether the German door is open or shut, so that all who can may come... It is of the greatest importance to give relief to those neutral countries because there is at present a steady stream or perhaps more accurately a steady trickle of refugees from France both into Spain and into Switzerland. The numbers that those countries, already suffering a good deal in shortage of food and with their standard of life so far below our own, will be able to receive are of course limited. If we can open the door at the other side and bring away from Spain and Portugal and (if transport is available but probably it would not) from Switzerland and also from Turkey those who are able to make their escape there, we shall render it far more probable that the channels through which that trickle percolates will not be blocked... Then, once more, it is urged, that we should offer help to European neutrals, to encourage them to admit new refugees, in the form of guarantees from the United Nations to relieve them of a stipulated proportion of refugees after the victory, or, if possible, sooner; that we should offer direct financial aid... There is one point I would raise more tentatively... It is that through some neutral power an offer should directly be made to the German Government to receive Jews in territories of the British Empire and, so far as they agree, of the other Allied Nations on a scheme of so many each month. Very likely it would be refused, and then Hitler's guilt would stand out all the more evidently. If the offer were accepted there would of course be difficulties enough, but it would be the business of the Germans to overcome these so far as concerns the conveyance of the refugees to the ports, and efforts could be made to secure help from Sweden and other neutral countries for shipping from the ports... Some of us have wondered how far the possibility has been considered of receiving any considerable number, particularly of children, in Eire and whether the Government of Eire have been consulted about this... \textit{247} ``It is said that there is a danger of Anti-Semitic feeling in this country. No doubt that feeling exists in some degree, and no doubt it could very easily be fanned into flame, but I am quite sure it exists at present only in comparatively small patches. It is very local when it exists at all, and therefore it receives a degree of attention beyond what it deserves. But if the Government were to decide that it was wise and practicable to put in action any of the proposals that I have laid before your Lordships, it would be very easy for the Government, by skilful use of the wireless, to win the sympathy and confidence of the people for their proposals, especially if a large number of those who were brought out were children and were being delivered from almost certain death... The whole matter is so big and other claims are so urgent that we want further to make the proposition that there shall be appointed someone of high standing for whom this should be a primary responsibility... My chief protest is against procrastination of any kind. It was three months ago that the solemn declaration of the United Nations was made and now we are confronted with a proposal for an exploratory Conference at Ottawa. That sounds as if it involves much more delay. It took five weeks from December 17 for our Government to approach the United States, and then six weeks for the Government of the United States to reply, and when they did reply they suggested a meeting of representatives of the Government for preliminary exploration. The Jews are being slaughtered at the rate of tens of thousands a day on many days, but there is a proposal for a preliminary exploration to be made with a view of referring the whole matter after that to the Inter-Governmental Committee on Refugees. My Lords, let us at least urge that when that Conference meets it should not meet for exploration only but for decision. We know that what we can do is small compared with the magnitude of the problem, but we cannot rest so long as there is any sense among us that we are not doing all that might be done. We have discussed the matter on the footing that we are not responsible for this great evil, that the burden lies on others, but it is always true that the obligations of decent men are decided for them by contingencies which they did not themselves create and very largely by action of wicked men. The priest and the Levite in the parable [536] were not in the least responsible for the traveller's wounds as he lay there by the roadside and no doubt they had many other pressing things to attend to, but they stand as the picture of those who are condemned for neglecting the opportunity of showing responsibility. We at this moment have upon us a tremendous responsibility. We stand at the bar of history, of humanity and of God. I beg to move.'' [537] \textit{248}
After the Archbishop of Canterbury had spoken, Lord Rochester spoke ``as a Methodist layman'':
'...No one can preach the Gospel of Jesus Christ and remain indifferent to social institutions which contradict that teaching. Wherever the Churches find practices which are contrary to Christian doctrine, whether they be such diabolical and horrifying practices as these we are more especially considering this afternoon, or others, it is no more than their bounden duty to denounce them... We are concerned with all persecuted minorities, but the Christian necessarily feels an intimate responsibility in regard to the Jews, since Christ `according to the flesh' came out of Israel. Almost every page of the New Testament shows how close was the association between religious Judaism and the first followers of Christ... 'I must needs be that offences come; but woe to that man by whom the offence cometh.' [538] And woe to us if we leave any stone unturned in seeking to aid and succour those of our fellow human beings who are suffering this cruel Nazi stumbling-block of offence. The Nazis have indeed debased themselves even unto hell, but let us remember' the high and lofty One that inhabiteth eternity,' as we recall those words in the 57th chapter of Isaiah: 'Cast ye up, cast ye up, prepare the way, take up the stumbling-block out of the way of my people'. [539] I support the Motion of the most reverend Primate, and I would urge the redoubling of our efforts to succour `one of the least of these', as we recall the latter part of the 25th chapter of St. Matthew.'' [540]
It is remarkable that, contrary to what one might have expected, it was the Archbishop who made the practical suggestions and the ``Methodist layman'' who cited texts from the Bible. It is a pity that one expression in the Archbishop's motion ("immediate measures, on the largest and most generous scale compatible with the requirements of military operations and security'') provided the Government with an excuse to do practically nothing. In order to understand the Archbishop's words, one should, however, try to realize how manifold were ``the requirements of military operations and security'' in those days. \textit{249} Obviously the Archbishop was well-informed about the persecutions on the continent of Europe. He had received (as he himself stated in his speech) reports from the World Jewish Congress, Geneva, and from the Board of Deputies of British Jews. Dr. Riegner, of the World Jewish Congress, sent an aide-memoire to the British Ambassador in Bern ``on behalf of the secretariats of the World Council of Churches and of the Jewish Congress''. The covering letter, dated March 22, 1943, stated: ``We should also appreciate it if His Majesty's Government would see fit to pass on the main contents of this aide-memoire to the Archbishop of Canterbury and the British Section of the World Jewish Congress''. [541] But if the aide-memoire was passed on, it must have come too late for the meeting in the House of Lords. The speech of the Archbishop in the House of Lords deserves careful study. It sheds an important light on the attitude of the Government regarding the Jewish refugees.
The Archbishop mentioned the proposal for an exploratory Conference at Ottawa. The country (Canada) in whose capital the conference was to be held, however, had not been informed, and thus the conference was held at Bermuda, on 19-29 April, 1943. The statement issued at the end of its deliberations merely promised recommendations --- which were not disclosed --- and the setting up of an inter-governmental organization to handle the problem in the future. The verdict on the allied Governments that ``History will record the Bermuda Conference as a monument of moral callousness and inertia'' is not too severe. [542]
The British Council of Churches, made up of the official representatives of the Church of England, the Church of Scotland and the Free Churches, met in London on April 13th and 14th under the presidency of the Archbishop of Canterbury. The following resolution was passed on anti-Semitism: \textit{250}
``The British Council of Churches warmly welcomes the statements made by the leaders of many Christian Churches expressing fellow-feeling with the Jewish people in the trials through which they are passing and the desire to aid them in every practicable way. In particular the Council notes with admiration and thankfulness the statements on this subject which have issued from Christian leaders in enemy-occupied countries. The Council affirms that anti-Semitism of any kind is contrary to natural justice, incompatible with the Christian doctrine of man and a denial of the Gospel. Malicious gossip and irresponsible charges against Jews, no less than active persecution, are incompatible with Christian standards of behaviour. The Council welcomes the decision to hold in Bermuda a Conference in which the British and American Governments will seek jointly to find practical ways of rendering immediate and continuing assistance to Jews and other imperilled people. The Council considers that every possible step ought to be taken to rescue from massacre the Jews in enemy and enemy occupied territories. It is convinced that both Christian and Jewish people in this country would give strong support to a lead from His Majesty's Government in offering sanctuary in Great Britain for a considerable number of children and adults, additional to those received before September, 1939, and would be ready to make sacrifices so as to provide hospitality for them during the war. The Council further asks that the Bermuda Conference will suggest measures for rendering the requisite material assistance for the maintenance of refugees who reach neutral countries, and will give assurance to those countries of readiness to cooperate in plans for post-war settlement of the refugees in other parts of the world.'' [543]
In May, 1943, the General Assembly of the Church of Scotland stated:
``The General Assembly protest anew against the atrocious persecution of the Jews in Nazi-occupied countries, and in the name of Christ condemns the inhumanity and sacrilege of anti-Semitic policy. They warmly approve of the steps taken by the Government to assist refugees, and respectfully urge it to continue and extend its efforts as far as possible. They assure the Jewish people of their deep sympathy in their grievous distress, and earnestly commend them to the prayerful concern and compassion of the Church.'' [544]
The Assembly of the Baptist Union of Great Britain and Ireland passed the following Resolution (also in May, 1943):
``They call upon His Majesty's Government to promote, in concert with the Governments of the United States of America and other associated nations, effective measures for enabling Jews and other victims of German brutality to escape and find refuge. \textit{251} In their view the strong abhorrence and detestation of the persecutors, which are felt throughout the civilised world, and of their purpose of exterminating the Jews, should be followed by energetic action, not only to bring to justice in due course the instigators and perpetrators of the massacres, but to give immediate aid, welcome and asylum in this and other free countries to those in peril, even though some risk to our own country may be involved. To this end they ask that restrictions regarding age, country of origin or means of support should not be put in the way to liberty and safety. They ask the Churches to show and inculcate a friendly and helpful attitude to such refugees, to pray for the deliverance of those who cannot escape beyond the reach of their barbarous enemies, and to resist as un-Christian all tendencies to anti-Semitism. [545]
On June 10, 1943, the General Assembly of the Presbyterian Church in Ireland adopted the following Resolution:
``The General Assembly has learned with great satisfaction that His Majesty's Government is prepared to collaborate with the United States of America in providing asylum for as many victims of German hate as can escape or be rescued from the danger which threatens them, and to consult with the Dominion Governments and the Governments of neutral countries with regard to united action, so that as many of the threatened people as ever possible may be helped. In view of the tremendous urgency of the situation, the General Assembly requests His Majesty's Government to carry out their promises to provide immediate and effective relief for those in such dire peril.'' [546]
Churches and Church leaders had, as quoted so far, expressed their desire and hope that the Government would take practical steps for aiding refugees. The Bishop of Chichester, however, expressed his disappointment in a letter to the Editor of ``The Times'':
``The Foreign Secretary is about to make a statement in the House of Commons on the result of the Bermuda Conference, and the policy of His Majesty's Government with regard to refugees. It will be almost exactly five months after the declaration of December 17, condemning the wholesale massacre of the Jews by the Nazis `in the strongest possible terms'. It is a historic moment in the record of our dealings with the persecuted and the oppressed. It is quite certain that if the British and American Governments were determined to achieve a programme of rescue in some way commensurate with the vastness of the need, they could do it. Nor can there be any doubt about the response which would be given in Britain to a clear lead based on the principles of humanity. \textit{252} There are difficulties. But so far as shipping is concerned, these should be greatly reduced as a result of the victories in North Africa. The need of a big camp to which those now in neutral countries could be sent must be patent to everybody. And the case for a revision of the regulations to allow many more to enter the United Kingdom is overwhelming. The guilt of `this bestial policy of cold-blooded extermination' lies with the Nazis. But can we escape blame if, having it in our power to do something to save the victims, we fail to take the necessary action, and to take it swiftly?'' [547]
A few days later the Bishop of Chichester published the following letter in ``The Times'':
``In the House of Commons on Wednesday Mr. Peake referred to my letter printed in your issue of May 18. His principal charge was 'that the Bishop made no attempt to indicate what was the programme of rescue which he suggested'. He added that he had searched Hansard for the House of Lords ever since December 17, but had failed to find any speech by myself on the subject. I was present at the debate opened by the Archbishop of Canterbury on March 23 and was prepared to speak. But owing to the number of speakers, representing all shades of opinion, on that occasion I, with others, stood down. It is not, however, true to say that I have made no suggestions as to a programme of rescue. In a letter in your columns on December 28, 1942, I referred to the suggestion made by Sir Neill Malcolm in his letter of December 22, and made further suggestions, such as the obtaining of facilities from the protecting Power for the transportation of Nazi victims from Germany and German occupied territories to the nearest frontier, with a view to entry into places of refuge; a guaranteeing to neutral Governments willing to give sanctuary to such victims of an evacuation of as many as possible after the war; and the establishment of reception areas in lands outside Europe. I am also a member of the Parliamentary Committee, and I support the 12-point programme for immediate rescue measures drawn up by the National Committee for Rescue from Nazi Terror, and widely published. I am glad to hear of the extension of categories of individuals eligible for visas, which forms a portion of the first of these points. I entirely agree that a programme of rescue must be a programme of victory. But this is not inconsistent with a determination by the Government to do everything possible for temporary sanctuary. There is a great difference between the spirit of a Government which says, 'We are resolved to do everything in our power, we wish we could do more, but such and such steps shall be taken at once in spite of all the difficulties', and the pessimistic attitude which simply repeats, 'We are filled with burning indignation at the horrors perpetrated by the Nazis against these people. We are determined to punish the guilty when the war is over. But for the present these people are beyond possibility of rescue.'' [548] \textit{253}
On July 28, 1943, the Bishop of Chichester strongly supported the plea for urgent government action in a speech in the House of Lords which was very critical of official policy and action. He contended that:
``...in the matter of the systematic mass murder of the Jews in the Nazi-occupied territories of Europe, which was the reason why the Bermuda Conference was called, there has been a deterioration in the determination to grapple with the problem.''
After quoting earlier promises made on behalf of the Government, he criticized the achievement of this Conference.
``...On April 19-29 the Bermuda Conference took place. It began in a spirit of pessimism. Its official pronouncement at the end said that the delegates `had examined the refugee problem in all its aspects'. The Jews were not mentioned. Agreed confidential recommendations were made which were designed to lead to the relief of a substantial number of refugees of all races and nationalities. Not a word was said about 'temporary asylum'...''
Particularly the Bishop emphasized the obligation to give priority to the persecuted Jews, and the responsibility of both neutral countries and of the Allied Governments to find temporary asylum for Hitler's victims.
``... It is in the face of this systematic murder, especially in the last twelve months, that I and so many others plead with the Government to act in a new way. With the appeal of the stricken people ringing in our ears, we would be false to our tradition if we failed to do everything we can.'' [549]
e. Towards the End
As far as we know, few statements were issued during the last period of the war. Significant was the Archbishop of Canterbury's warning, on Dec. 8, 1943, that ``the sufferings of the Jews be kept in full view of all people so that the spirit of indignation and compassion in them will not die out''. \textit{254}
``It is one of the most terrible consequences of war that the sensitiveness of people tends to become hardened, ``Dr.Temple said. ``We could hardly live these days if we felt the volume of suffering of others in the world as acutely as we felt in peacetime''. ``There is a great moral danger in the paralysis of feeling that is liable to be brought about. It is most important for our own moral health and vigor that we express horror at the persecution of the Jews.'' Dr. Temple said the persecution of Jews on the Continent, and particularly in Poland, ``almost baffles imagination and leaves one horrified at the power of the evil that can show itself in human nature.'' [550]
Another warning came from the General Assembly of the Church of Scotland (May, 1944):
``The General Assembly express their profound sorrow at the lamentable condition of the Jews in Europe, and in the name of Christ renew their reprobation of the inhuman atrocities committed against them. They assure the Jews of their deep concern and sympathy, commend them to the brotherly offices and prayerful compassion of all Christian men and women, and warn the members of the Church of Scotland against the growing danger of anti-Jewish prejudice and propaganda. They respectfully urge the Government to continue to offer every facility to enable refugees to escape from the tyranny and oppression of Nazism.'' [551]
In June, 1944, the Archbishop of Canterbury, presiding at a meeting of the Council of Christians and Jews, denounced the continued persecution and attempted extermination of the Jews by the Germans, whose activities he described as ``one of the most hideous of the elements even in the recent German record''.
Dr. Temple moved a resolution expressing concern at the increasing peril to the Jewish communities involved in the extension of Nazi domination in Central and South Eastern Europe, coupled with satisfaction at the steps taken in North Africa and southern Italy to remove all discriminatory legislation against Jews and other victims of Nazi intolerance. He and many others, he said, had been disappointed that there had not been a greater willingness shown on the part of the authorities to help those who were trying to escape from German-dominated countries... [552] \textit{255}
On July 7, 1944, the Archbishop of Canterbury addressed the following message to Hungary through the B.B.C.:
``I am eager to speak to the Christian people of Hungary, so far as I can do so, because of news sent to me through one of the most reliable of ecclesiastical neutral sources --- and what I hear from that source only confirms what is reported also through other channels. The report is that a wholesale round-up of Hungarian Jews is taking place under orders from the German Government, and that those who are carried off have little chance of survival. According to this report, the Jews are being deported daily. Already the Eastern provinces have been cleared of Jews. Now the process is beginning in the Western districts including the capital. The conditions of travel are such that on arrival many already are dead; others are killed and cremated at Auschwitz. If the Christians of Hungary know the facts I am perfectly confident that they are also doing everything they can to save these doomed people by hiding them and helping them to escape. But it may be that inside Hungary the facts are concealed. It is for this reason that I feel bound to tell you of them, and beg you to do your utmost, even taking great personal risks, in order to save some if you can. Then you will earn in very special degree the words of approval and thanks: `In as much as ye did it unto one of these My brethren ye did unto Me' (Matthew, 25, 40). I speak as a Christian who cannot help to Christians who can. For the honour of our common Christianity I implore you to do your utmost.'' [553]
34 THE UNITED STATES
a. The Time of America's ``Neutrality''
It would have been possible to record the statements in this paragraph under ``The Neutral Countries''. The United States officially entered into the war in December, 1941. Japan attacked Pearl Harbour on Dec. 7 and Hitler declared war upon the United States, on Dec. 11, 1941. Until that time, it was at least pretended that the United States was neutral and the spirit of isolationism was still strong. Before 1942, strong statements against anti-Semitism were issued by Protestant Churches in the U.S.A., especially by the Federal Council of Churches. After Hitler's declaration of war, however, the statements took on an additional clarity: ``Anybody spreading anti-Semitism is helping Hitler just as much as if he were a paid agent of the Reich.'' [554] Anti-Semitism became ``treason against God, treason against the country.'' [555]
\textit{256}
On the evening of December 14, 1939, a mass meeting was held at Madison Square Garden, New York, for the purpose of registering a protest against the treatment of the Jews in Poland and other areas under the Nazi regime. The meeting was attended by 20,000 people. Expressing the sympathy of Christians, Dr. Samuel McCrea Cavert, General Secretary of the Federal Council of Churches, pointed out, that Christians as well as Jews were suffering in Poland and other parts of Europe and that ``Christians have a direct stake in what is happening''. In conclusion, he said:
``Out of the calamity in Europe, there emerges one by-product for which we may be thankful --- the new sense of fellowship between Jew and Christian in America. Nothing so quickly unites men as a cry of desperate human need. I do not believe there has ever been a time when Christian hearts in America beat in such sympathy for their Jewish neighbours. There are differences of religious conviction between Jew and Christian --- at one point a momentous difference --- but we share together the priceless spiritual heritage of Israel. As His Holiness Pope Pius XI truly and nobly said, 'Spiritually we are all Semites'.'' [556]
The United Church of Christ issued the following statement in 1940:
``One of the most disturbing currents in America to-day is anti-Semitism. Under the cover of an attack upon the Jews a covert attack is being made on Christianity. The manipulators of anti- Jewish propaganda are not concerned with the alleged evils they denounce; but they are concerned to destroy the teachings of the Bible --- that God, the Lord and Creator of all men, is a holy God --- and the prophetic morality of the Old Testament. They attack under cover of anti-Semitism God the Lord who is not bound to any nation but is Lord of all nations. They attack justice, righteousness, mercy and the divine command for holiness. They attack the law which Christians and Jews alike acknowledge as God's requirement. Twentieth century anti-Semitism reveals its true character in its demand on the Church to surrender the Old Testament and to deny that the God of Abraham, of Moses and the Prophets is the Father of Jesus Christ. \textit{257} Anti-Semitism is flatly contradictory to the express teaching of St. Paul. In Romans 11, St. Paul reminds the Gentile Christians, just as we need to be reminded today, that Israel is the stem on which Gentile Christians have been grafted. `You owe,' he wrote, 'your position to faith. You should feel awed instead of uplifted.' And again, 'So far as the gospel goes, they (the Jews) are enemies of God, which is to your advantage; but so far as the election goes, they are beloved for their father's sake. For God never goes back upon his gifts and call.' St. Paul discovered in anti-Semitism a pride which needed to be rebuked. `You owe your position to faith'; that means, not something we have by right of possession, not something we can take for granted, not any kind of inherent superiority at all. Faith is the gift of God. Moreover, God has not repudiated Israel. They are still beloved. Anti-Semitism is not only one form of human pride; it is repudiation of the declared purpose of God. We recommend that General Synod declare its condemnation of anti-Semitism and urge upon the members of the Church in the name of Christ the duty to serve in love the brothers of Christ according to the flesh.'' [557]
The Federal Council of Churches of Christ in the United States published the following Resolution, in December, 1940:
``We express as Christians our sympathy with the Jewish people in this hour of calamity for so many of their group in Europe. We deplore the existence of anti-Semitism in America and declare our opposition to it because it is contrary to the spirit and teachings of Christ. We call upon His followers to create Christian attitudes toward the Jews. This should be a matter of primary concern for every Christian Church in every community.'' [558]
On September 19, 1941, the Executive Committee of the Federal Council adopted the following statement:
``On many previous occasions we have expressed our abhorrence of the religious and racial intolerance which afflicts our world today. We have especially emphasized our opposition to unjust and unchristian attacks upon the Jews. In so doing we have been whole-heartedly supported by similar utterances officially made by the highest governing bodies of the great dominations which cooperate in the Federal Council of Churches. \textit{258} Recent evidences of anti-Jewish prejudice in our own country compel us to speak again a word of solemn warning to the nation. Divisiveness on religious or racial grounds is a portentous menace to American democracy. If one group be made the target of attack today, the same spirit of intolerance may be visited on another group to-morrow and the rights and liberties of every group thus be put in jeopardy. We condemn anti-Semitism as un-American. Our nation is a free fellowship of many racial and cultural stocks. It is our historic glory that they have been able to live together in mutual respect, each rejoicing in the rich contribution which the others have made to the common good. Anti-Semitism is an insidious evil which, if allowed to develop, would poison the springs of our national life. Even more strongly we condemn anti-Semitism as un-Christian. As Christians we gratefully acknowledge our ethical and spiritual indebtedness to the people of Israel. No true Christian can be anti-Semitic in thought, word or deed without being untrue to his own Christian inheritance. In behalf of the Christian churches which comprise the Federal Council we voice our renewed determination to unite in combating every tendency to anti-Semitism in our country. We recognize that a special responsibility rests upon us who belong to the numerically strongest group, to be staunch advocates of the rights of minorities.'' [559]
In 1941, the following ``Manifesto to our Brethren and Fellow Citizens of Jewish Race and Blood'' was signed by one hundred and seventy Protestant ministers representing one hundred and sixty-six churches and twenty-four denominations in the City of New York:
``With genuine anguish of heart we behold how in many places across the world today cruel forces of oppression and persecution are being released upon men and women and children of Jewish race and blood. With profound concern we note from time to time within our own beloved nation the manifestation of a spirit of anti-Semitism. The conscience of Protestant Christendom, as recorded at the great ecumenical conference held at Oxford, England, during July of 1937, expressed itself in no uncertain terms when with unanimous voice it affirmed that 'against all racial pride, racial hatred and persecution and the exploitation of other races in all their forms, the church is called by God to set its face implacably and to utter its words unequivocally both within and without its borders. There is a special need at this time that the church throughout the world brings every resource at its command against the sin of anti-Semitism.' With this pronouncement we are in complete accord of heart. Therefore, we would disavow any words or action promoted by the spirit of anti-Semitism, which emanate from sources that purport to be Christian. Such words and actions label themselves unchristian. \textit{259} We call upon our Christian brethren to guard their hearts, their minds, their lips, their hands from emotions, thoughts, words or deeds that partake of `the sin of anti-Semitism'. To that end we command to them the quest for `the fullness of Christ' within their lives. We call upon our fellow citizens to remember that anti-Semitism is a threat to democracy and a denial of the fundamental principles upon which this nation is founded. We extend to our brethren and fellow citizens of Jewish race and blood our solemn assurance that by the constraint of our deepest Christian conviction we shall oppose unceasingly `the sin of anti-Semitism' and we shall strive continuously for the realization of that brotherhood which humanity needs, democracy requires and Christianity demands.'' [560]
b. At War with Germany. Co-operation with Jewish Leaders
The Executive of the Federal Council addressed the following ``Message for Race Relations Sunday'' (Febr. 8, 1942) to its members:
``For all the law is fulfilled in one word even this: thou shalt love thy neighbour as thyself.'' Gal. 5, 14. Let us translate this pattern into a social program. Our pronouncements must now be supported by our practices. Where attacks are made upon Jews or the sinister spirit of anti-Semitism appears, we must protest in the Name of Christ and the Church... Where any racial minority within our borders is exploited or barred from equal opportunity, we Christians must take a stand for the sake of our faith. We must, furthermore, create a genuine fellowship that will prevent the development to such injustice towards any group. Our love for the Church requires that it be pre-eminently the abode of fellowship. The Church, by reason of its origin in the universal Christ, must be a brotherhood of all peoples, remembering that in Him there is neither Jew nor Greek, barbarian, Scythian, bond nor free. Therefore, let every follower of Christ search in his own soul to see if any enemies of brotherhood are lurking there. Let him examine his own daily relationships. Let us all in this awful and creative hour march resolutely forward, not faithless nor fearful, but confident in the future when democracy and brotherhood are one. ``If a man say I love God and hateth his brother whom he hath seen, how can he love God whom he hath not seen.'' 1 John 4, 20. [561] \textit{260}
In September and October, 1942, the General Secretary of the Federal Council, Dr. Samuel McCrea Cavert, visited France and Switzerland. The Director of the World Jewish Congress at Geneva, Dr. Gerhart M. Riegner, stated:
``With regard to our knowledge of the Nazi plan of total extermination of European Jewry, I wish to state that the first report on this plan reached me in the last days of July 1942 and I communicated it to Rabbi Wise in New York and Mr. Silverman in London during the first days of August 1942 (through diplomatic channels). Dr. Wise received the message during the last days of August 1942 and asked Mr. Cavert to use his visit to Geneva at the beginning of September 1942 to find out from us whether deportation really meant extermination. After having spoken to one of us --- I believe to Prof. Guggenheim --- he confirmed this in a cable to the United States.'' [562]
On Dec. 11, 1942, at the great Biennial Assembly of the Federal Council, the following Resolution on Anti-Semitism was adopted:
``The reports which are reaching us concerning the incredible cruelties towards the Jews in Nazi occupied countries, particularly Poland, stir the Christian people of America to the deepest sympathy and indignation. It is impossible to avoid a conclusion that something like a policy of deliberate extermination of the Jews in Europe is being carried out. The violence and inhumanity which Nazi leaders have publicly avowed toward all Jews are apparently now coming to a climax in a virtual massacre. We are resolved to do our full part in establishing conditions in which such treatment of the Jews shall end. The feelings of the Jewish community throughout the world have recently been expressed in a period of mourning, fasting and prayer. We associate ourselves with our Jewish fellow-citizens in their hour of tragic sorrow, and unite our prayers with theirs. We confess our own ineffectiveness in combating the influences which beget anti-Semitism in our own country, and urge our constituencies to intensify their efforts in behalf of friendly relations with the Jews. We urge that all plans for reconstruction in Europe shall include measures designed to secure full justice for the Jews and a safe and respected place for them in western civilisation. For those who, after the war, will have to emigrate from the war-ridden lands of Europe, immigration opportunities should be created in this and other lands. We recommend that the officers of the Federal Council transmit this action to the Jewish leaders in person.'' [563] \textit{261}
On Dec. 31, 1942, the Synagogue Council of America published a New Year message it had addressed to the Rev. Dr. Samuel McCrea Cavert, secretary of the Federal Council of the Churches of Christ in America, and to Mgr. Michael J. Ready, general secretary of the National Catholic Welfare Conference. The message was signed by Dr. Israel Goldstein, president of the Council.
``American Jews,'' the message said, ``share with their Christian brothers the sense of having been privileged to bear burdens not only in answering the call of our nation's defence needs, but also in heeding the call of human needs overseas. ``To the Jews of Hitler-ridden Europe the year 1942 has been the most catastrophic in their tragedy-laden history. Helpless women, aged and children, and defenceless men have been slaughtered wholesale and a whole people has been marked for extermination. Among no other people is such a toll being taken. If the executioner's hand is not soon stayed, all the Jews whom it can reach will perish.'' The message said the greeting was ``preferred to you and to the great body of Christians whom you represent'', and expressed hope for an Allied victory and a just peace in 1943. [564]
On January 6, 1943, the heads of the six Jewish organizations which comprised the Synagogue Council of America, under the chairmanship of Rabbi Israel Goldstein, met in conference with official representatives of the Federal Council of the Churches of Christ in America. The purpose of the meeting was to afford an opportunity to discuss together what the Christian Churches could do to assist the Jews of Europe.
Desiring to express its sympathy in something more than resolutions, the Federal Council arranged for the conference with the Jewish leaders. Several fruitful suggestions emerged as to ways in which the Churches might help to develop stronger support for the needs of refugees from Europe, a measure of relief in the form of food for at least some of the Jews in Europe, and a safe and respected place for Jews in the post-war world. [565]
c. Practical Steps Demanded; the Bermuda Conference \textit{262}
``On March 1, 1943, a great demonstration, one of the largest ever held in the United States, took place in Madison Square Garden at the initiative of the Congress and under the joint auspices of the American Jewish Congress, the American Federation of Labour, the CIO, and the Church Peace Union. Twenty-two thousand people crowded into the great hall, while 15,000 stood outside throughout the evening listening to the proceedings through amplifiers. The demonstration was addressed by Dr. Chaim Weizmann, Dr. Stephen S. Wise, Governor Thomas E. Dewey, Mayor Fiorello H. LaGuardia, Senator Robert F. Wagner, William Green, and others. The British Section transmitted cable messages from the Archbishop of Canterbury and the late Cardinal Hinsley, whose last public utterance it was before his death a week later. The meeting laid down a 12-point program for the rescue of European Jewry prepared by World Jewish Congress experts. The effect was immediate. On the following day, Undersecretary of State Sumner Welles declared that a note had already been sent to Great Britain on February 25 offering the cooperation of the United States in organizing an intergovern-mental meeting for study of methods to save `political refugees' in Europe. The meeting came to be known as the Bermuda Refugee Conference...'' [566]
On March 1, 1943, the Executive Committee of the Federal Council of Churches of Christ in America appealed to the Governments of the United States and Great Britain ``to consider offering financial assistance to Jewish refugees who have escaped to neutral countries from Nazi held territory, and the possible establishment of temporary places of asylum for those evacuated from Europe''.
The committee urged that the proposals be considered at the forthcoming conference in Toronto of representatives of the two governments on the Jewish problem. The suggestion was part of a three-point program calling for a report by the council's department of research and education on the treatment of Jews under the Nazi regime and setting aside May 2 for observance in churches as a ``Day of Compassion'' for the Jews in Europe. The committee's action was a sequel to the adoption at the council's biennial meeting in Cleveland in December of a statement setting forth the organization's determination ``to do our full part in establishing conditions'' in which harsh treatment of Jews should end. The proposals outlined by the committee for consideration of the British and American representatives at Toronto were: \textit{263} ``To offer financial assistance for the support of refugees that neutral governments (for example, Switzerland, or Sweden, Spain, Portugal and Turkey) may receive from areas under Nazi control, as a result either of infiltration across their borders or of negotiations with the Axis powers, with the expectation that, after the war, such refugees would be repatriated in their own countries. ``To provide places of temporary asylum to which refugees whom it may be possible to evacuate from European countries may be removed, these refugees to be supported in camps for the duration of the war, with the understanding that they will then be repatriated in their own country or be provided with permanent homes in other ways.'' At the same time the committee urged Christians throughout the country ``to give their moral support to whatever measures afford promise of rescuing European Jews whose lives are in jeopardy.'' The committee invited all Christians to ``join in united intercession on May 2 for the victims of racial and religious persecution as a special occasion for the expression of Christian sollicitude.'' [567]
The practical steps proposed by the Executive Committee of the Federal Council to the Governments of the United States and Great Britain were similar to the steps proposed by the Archbishop of Canterbury in the House of Lords at about the same time, [568] and to the Aide-memoire sent by the Secretariats of the World Council of Churches and of the World Jewish Congress (Geneva), to the American and British Governments. [569] Not withstanding all this, the Bermuda Conference became ``a monument of moral callousness and inertia''. [570]
d. Different Churches Speaking on Different Occasions
The following is a chronological record of statements made by Churches or Church leaders in the United States from May, 1943, until the end of the second world war.
Henry St. George Tucker, Presiding Bishop of the Protestant Episcopal Church and president of the Federal Council of Churches, in a statement on the observance by the Council of a ``Day of Compassion'' for persecuted European Jews, said that there had been found a ``rising tide of concern among Christians'' over their fate. \textit{264}
Dr. Tucker said it was the first time Christian churches had set aside a specific day for a ``united expression of their sympathy with a suffering and persecuted Jewry''.
``What is happening to the Jews on the Continent of Europe is so horrible that we are in danger of assuming that it is exaggerated,'' he said, and cited a recent survey by the council of evidence that he said indicated that under the Nazis a policy of deliberate extermination of Jews was carried out. ``The survey shows that the actual facts are probably more, rather than less, terrible than the reports,'' he continued. ``The Christian people of America vigorously protest against this brutal and cruel persecution. But protest is not enough.'' Two remedial measures have been set forth by the council: First financial assistance for support of refugees reaching neutral countries from Nazi-occupied areas, and second, provision of temporary asylum to which refugees evacuated from European countries may be removed. [571]
On October 20, 1943, American religious leaders denounced ``the recent acts of terror in Denmark'' and expressed sympathy for the Jews in that country. The Rev. Dr. P.O. Bessel, president of the Augustan Synod, Minneapolis, said that the synod was shocked at the German barbarism in Denmark, but was happy about Sweden's firm stand in offering refuge to the persecuted Jews. The Rev. Dr. Samuel McCrea Cavert, general secretary of the Federal Council, said that ``the American churches have been thrilled by the news that the Danish Church has refused to be cowed into silence in the face of the Nazi attack upon Jews in Denmark''. [572]
The following article in ``The New York Herald Tribune'' shows how strong anti-Semitic influences in the United States were, in 1943:
BISHOP OXNAM ASSAILS BEATING OF JEWISH BOYS
Bishop G. Bromley Oxnam, of the Boston area of the Methodist Church, denounced today the alleged beating of Jewish boys as an apparent expression of incipient Fascism and, in a statement, demanded, ``who is flooding the nation with anti-Semitic literature, and why?'' \textit{265} Declaring that ``the beating of Jewish boys is not the work of hoodlums,'' Bishop Oxnam expressed hope that Jews, Catholics and Protestants could unite ``in demanding that these beatings stop and that steps be taken to discover and destroy the dangerous forces that lie back of them.'' The Bishop's statement followed the placing of charges before Governor Leveratt Saltonstall that Jews had been made the victims of ruffians over a period of months in the Boston area. The Governor, acting upon a petition of which Bishop Oxnam was one of the signers, has appointed five prominent citizens of various faiths to an advisory committee on anti-Semitism. ``The beating of Jewish boys must stop,'' the statement said. ``The beaters must be apprehended and punished. The beating of any boys by gangs is bad enough at any time. The beating of boys of a particular race is worse. But the real menace lies in the apparent fact that these beatings are an expression of incipient Fascism, that they follow a similar pattern, and that, in one case, at least, the beaters wore black shirts. ``Who is flooding the nation with anti-Semitic literature, and why? Who finances these movements? Why is it that the anti-Semitic leaders now under Federal indictment have attacked such religious organizations as the Federal Council of Churches of Christ in America, the Methodist Church and other Protestant religious bodies? Why has Franco, the Fascist dictator of Spain, been extolled? Bishop Oxnam, in an interview with ``The Boston Traveller'', said that the recent outbreaks of racial violence in the Dorchester, Roxbury and Mattapan districts of Boston follow a pattern. ``I was in Germany when these things began there. It is the same pattern in which organized gangs beat up a scapegoat race whether they be Jews or any one else,'' he asserted. He asserted that Fascism is prevalent in Brooklyn now, and predicted that it would show itself in Detroit and sections of the Pacific Coast before long. ``I think Brooklyn, New York and Boston are currently the most difficult centres, however,'' he added. [573]
In Dec., 1943, a Senate resolution proposed the creation of a special commission ``to bring about the rescue of the surviving Jews of Europe''. Eight Protestant leaders sent ``a Christmas Appeal for speedy adoption of the Resolution'' to Vice President Henry A. Wallace, Senate majority and minority leaders and members of the House and Senate committees involved.
Asserting that ``more than 2,000,000 European Jews have been slaughtered by the Nazis, the message added that ``we cannot approach Christmastide without declaring that too many of us have been found wanting in the will to rescue these suffering people.'' \textit{266} ``Let no possible sanctuary be closed, whether in America or elsewhere,'' the appeal said. ``Let each door of refuge be kept open. This is the Christian way.''
The message was signed by Bishop William T. Manning (Protestant Episcopal); Archbishop Athenagoras (Greek Orthodox); Bishop William J. McConnell (Methodist), and others. [574]
On Jan. 15, 1944, fifteen hundred persons attended a rally against anti-Semitism at Carnegie Hall. Dr.Henry Smith Leiper of the Federal Council of Churches of Christ in America, chairman of the meeting, asserted that anti-Semitism was ``treason against God, treason against the country''.
``Anybody spreading such slander,'' he said, ``is helping Hitler just as much as if he were a paid agent of the Reich.'' Dr. Leiper and several others spoke out against what they said was the desire on the part of many to approach the problem of anti-Semitism with too much caution. Dr. Leiper said that exactly this idea prevailed in Germany in 1932, but did not halt the rise of fascism. [575]
The biennial convention of the United Lutheran Church in America adopted, on Oct. 13, 1944, the following Resolution:
``Recognizing that the Jewish problem has been made one of the central elements in the present assault on civilization, the United Lutheran Church in America, viewing with concern the manifestations of a rising tide of anti-Semitism in American life, begs its members to consider their Jewish brethren in the spirit of Luther, who spoke kindly things of them as 'blood brothers of our Lord', to use every available means to assure the Jewish people of their communities of the efforts of our church for the preservation of their rights, and to offer prayers on their behalf.'' [576]
We do not record all the statements issued by Protestant Churches in the United States over the years 1943-1944. The Presbyterian Church in the U.S.A. and the United Presbyterian Church in North America issued a statement in 1943; the American Baptist Convention, the Assembly of the Presbyterian Church in the U.S.A. and the General Synod of the United Church of Christ issued a statement in 1944. Most of these statements condemned anti-Semitic and anti-Negro prejudices.
\textit{267}
e. The Churches in the U.S.A. that kept Silent
Three important Protestant denominations in the United States did not speak out unequivocally against anti-Semitism and the persecution and extermination of the Jews: the Southern Baptist Convention, the Lutheran Church --- Missouri Synod, and the American Lutheran Church. [577] John G. Mager comments:
``... It might have been felt that since a large proportion of the membership of the Synod was of German origin or descent, it would have made for ecclesiastical suicide if the official organ of the Synod made pronouncements against a country to which many were bound by ties of blood, culture and sentiment...'' [578]
It must be borne in mind that the Lutheran Churches in Denmark, Norway, Sweden and Slovakia clearly expressed their horror at German anti-semitism, and they did so under much more difficult circumstances. This should dissuade us from wrong platitudes such as: ``Lutherans tend to be anti-Semitic''.
Recently it has been suggested that ``the causal chain that links Christian belief and faith to secular anti-Semitism begins with orthodoxy --- commitment to a literal interpretation of traditional Christian dogma''. [579] My knowledge of the situation of Churches in America is limited. Therefore I would not venture to suggest that there is a causal chain between the orthodoxy of a Church in America and its failure to denounce anti-Semitism. Moreover, in other countries, like the Netherlands for example, such a connection does not appear to exist. \textit{268} It is noteworthy, however, that the three great Protestant Churches in the United States mentioned above, which failed to issue a clear statement against anti-Semitism, were not members of the Federal Council. Moreover, the Southern Baptist Convention and the Lutheran Church (Missouri Synod) are not members of the World Council of Churches, to this day. These Churches apparently did not feel challenged by the protest issued by the Assembly of the Church of England, in 1935, as was the Federal Council; [580] they did not receive the information provided by the General Secretariat of the World Council of Churches, Geneva, during the war. [581] Ecclesiastical isolationism is very dangerous indeed, especially in a time of crisis. They were probably afraid of watering down their own principles by co-operating with other Churches and this lack of co-operation and communication probably contributed to the fact that they did not fulfil the word of the Bible: ``Open thy mouth for the dumb in the cause of all such as are appointed to destruction. Open thy mouth, judge righteously, and plead the cause of the poor and the needy''. (Prov. 31, 8-9).
35 THE WORLD COUNCIL OF CHURCHES
The war years were the testing time of the World Council. Contacts with Great Britain and the United States were relatively frequent until the end of 1942, when the whole of France was occupied by the Germans. Since it proved impossible to hold fully representative meetings, the Provisional Committee met and continued to meet in three groups --- one in Geneva under the leadership of Dr. Boegner (later of Dr. Koechlin), one in Great Britain under Archbishop Temple, and one in New York under Dr. John R. Mott. \textit{269} The fact that the World Council had offices in New York, London, and Geneva, proved a blessing, for each office had its area of contacts with Churches which the other could not reach. [582] When the second world war broke out, the World Council of Churches was still ``in process of formation'', and it had not as yet an adequate apparatus at its disposal. But the General Secretary, Dr. Visser 't Hooft, and the Director of the Department for Refugees, Dr. Freudenberg, had their contacts with the World Jewish Congress in Geneva, and with Church leaders in Germany and the occupied countries. They could thus pass on valuable information to the Churches in the free world, and stir them to action.
a. Letters Sent to the International Red Cross
On October 29, 1941, Dr. Visser 't Hooft sent the following Memorandum to the President of the Mixed Relief Committee of the International Red Cross, Prof. Dr. Karl Burckhardt:
Memorandum on the Situation in Poland
I. ``We have received some information about the situation in the General government of Poland from a reliable and objective observer who has been travelling there during recent weeks. According to him, there exists a great difference between city and country. In the large cities, especially in Warsaw, the Polish and, to a greater extent the Jewish population, is suffering famine. Typhus is spreading in and outside the ghetto of Warsaw. Our spokesman heard of 2,000 cases in the ghetto alone. The mortality of infants less than three years old is amounting to 26\%%... We know of only one modest relief activity: American Poles have, in co-operation with American Mennonites, the German Red Cross, the Polish and the American Relief Committee (Hoover), organized a soup-kitchen, where they weekly distribute to the distressed population of Warsaw, fish purchased in Danzig for DM. 5,000. This feeding, which is merely a drop in the ocean, reaches Poles as well as Jews. Moreover, a despatch of medicine from the United States is expected to arrive in Lisbon one of these days.
II. The greatest wave of deportations of German Jews and Christians of Jewish origin to Poland has been going on since the middle of October. Seven thousand Jews were deported from Berlin to Litzmannstadt on the nights of October 18/19 and 19/20. 20,000 Jews of the Rhineland are already there, or are en route. 2,000 are to be transported from Prague. Deportations from Vienna have already been going on for some time. A number of Jews from Breslau is believed to be engaged in labour in the Bohemian Riesengebirge. \textit{270} According to our spokesman, the able-bodied men who have been deported to Poland are constructing roads behind the Eastern front whilst the able-bodied women are employed in ammunition factories. In Litzmannstadt hut camps are said to be provided as temporary lodging, but we have no particulars about this. The deportees were allowed to take only a handbag and 10 RM. with them. Sufficient protection against the cold will be out of the question. One may assume that these measures are the beginning of the complete deportation of the Jews and Christians of Jewish origin from the Reich and the Protectorate. This concerns people the majority of whom, owing to their mental powers having been overcharged for many years, will be unfit to cope with these new hard measures.
III. In view of its Christian responsibility the Provisional Ecumenical Council of the Churches cannot heedlessly close its eyes to this misery of the refugees in Poland. As it practically can no longer carry out its own relief work, it feels all the more its duty to intervene with the competent bodies towards quick relief action. The Jewish organizations, generally speaking, are no longer in a position to undertake effective steps on behalf of their co-religionists. The Jewish question touches the centre of the Christian message: neglect of the Church to raise its warning and protective voice here, and do all in its power to help, would be disobeying its God. It is, therefore, the duty of the Christian Churches, and especially of their Ecumenical representative, the Provisional Ecumenical Council, to intervene on behalf of the persecuted.
IV. Therefore the Provisional Ecumenical Council of the Churches appeals to the competent bodies of the Red Cross with the request to pay special attention to the situation in Warthegau and the General government of Poland. We urge that the Red Cross speedily send a delegate, if possible a medical man, to the regions in question. This delegate would have to investigate, especially in the large Polish cities, the most urgent needs of the Polish as well as of the Jewish population, thus ascertaining the medical, sanitary and clothing requirements. Such a survey should include not only the Warthegau (especially Litzmannstadt) but also the region of Lublin where the Jews from Germany, Austria and Bohemia who were deported in the winter of 1939/1940 are said to be living. We hardly know anything about their fate but it is most certainly very critical. The Provisional Ecumenical Council is prepared to request urgent support from its member Churches, especially those in the United States, for a relief action organised by the International Red Cross.'' [583] \textit{271}
Dr. Visser 't Hooft stated in the covering letter that he had also sent a copy to the President of the Red Cross, Dr. Huber, and that he would be grateful for a speedy reply.
On June 3, 1942, the Secretary of the Ecumenical Commission for Refugees, Dr. A. Freudenberg, sent the following letter to the Mixed Relief Committee of the International Red Cross:
``An absolutely reliable correspondent requests us, to communicate to the organizations of the Red Cross the following: 'A serious lack of restoratives, digitalis \textit{etc}. is prevalent in the Jewish ghettos in the East, especially in the camps of Yzbica and Piaski near Lublin, and also in Riga, Wilna, Kowno, Warsaw and Lodz. Many people who had been admitted to the hospitals because of diminishing strength and under-nourishment or other reasons, must now perish there owing to a lack of these restoratives. They could be saved if one could supply them with strengthening food. I have been implored to inform the International Red Cross about this, so that it may render aid wherever possible.' This information, indicating that the deportees and the Polish Jews are suffering terribly from famine, has been confirmed by others As most of them are destitute, numerous cries of distress have reached us both directly and indirectly. Therefore we join in the request of our correspondent, and implore the organizations of the International Red Cross to continue to relieve the fate of these unfortunate people in every possible way.'' [584]
On December 3, 1942, Dr. Visser 't Hooft again wrote to the President of the Mixed Relief Committee of the International Red Cross, Prof. Dr. Karl Burckhardt. The letter reads as follows:
``We refer to our letter of 29th October, 1941, in which we submitted to you a Memorandum concerning the persecution and the misery of the Jews in Poland. Since then the situation has deteriorated in an alarming way. No doubt you have been informed of the mass executions of which the Polish Jews and the Jews in Poland deported from the European countries, are the victims. To the information that has reached other organizations, we can add the contents of a message received from a very distinguished German personality whose reliability we can guarantee. The message informs us that at one place in Poland, 6,000 Jews --- men, women and children --- are being shot every day. These executions are made in three groups, each of 2,000 persons, and this has already been going on for weeks. \textit{272} In our Memorandum of 29th October, 1941, we remarked that the Jewish question touches the centre of the Christian message. Therefore we feel compelled to raise our voice anew on behalf of these people who are being threatened with extermination. We therefore permit ourselves to renew our suggestions of last year, that the International Committee of the Red Cross take urgent steps to send delegates to the areas in question. There is reason to hope that such steps, even if they do not directly have the desired result, would encourage certain circles in Germany to combat the mass executions more energetically. Though from the letters received from Theresienstadt in Bohemia it is not possible to ascertain the real conditions existing in this reception centre, we would be grateful if the requested action could also include that city.'' [585]
The letter mentions ``certain circles in Germany''. These were groups of resistance with which the Secretariat of the World Council of Churches was in contact, especially the ``Kreisau Circle'' and Dietrich Bonhoeffer with his friends. [586]
b. Co-operation with the World Jewish Congress
A unique aspect of the activities of the World Council of Churches regarding the persecution of the Jews was the close co-operation between Dr. Visser 't Hooft and Dr. Freudenberg on the one hand, and the Director of the World Jewish Congress at Geneva, Dr. G. M. Riegner. Dr. Riegner stated: ``... My correspondence with Dr. Freudenberg starts already in November, 1940, and during certain periods we have been in nearly daily contact''. [587] In the same letter to Dr. Visser 't Hooft, Dr. Riegner wrote: \textit{273}
``I remember that you and the World Council have also played an important part in convincing the Swiss authorities of the deadly danger threatening the Jews in all occupied countries and trying to obtain from them a more liberal attitude in admitting refugees. I remember distinctly, though I do not find any trace in writing, that I have put at your disposal several times very detailed information and reports which you have been good enough to communicate on behalf of the World Council of Churches to the Swiss authorities. If I am not mistaken, at least on one occasion you have personally intervened with Federal Councillor von Steiger in such matter.''
Dr. Riegner commented on this point as follows:
``I am still convinced that these interventions of the World Council have been at certain moments of great value. In the course of the discussions which I had during the last year with either Dr. Visser 't Hooft or Dr. Freudenberg, I became convinced that these representations have most probably been made by Dr. Alfons Koechlin, (Base]), the former head of the Protestant Federation of Switzerland and one of the Presidents of the Provisional World Council at that time. Dr. Koechlin, of course, received the material from Dr. Visser 't Hooft and Dr. Freudenberg.'' [588]
Jews and Christians also co-operated together in breaking the wall of silence. The Secretary of the Federal Council of Churches of Christ in the United States, Dr. McCrea Cavert, visited Dr. Visser 't Hooft in Sept., 1942. Dr. Riegner reports about this visit:
``With regard to our knowledge of the Nazi plan of total extermination of European Jewry, I wish to state that the first report on this plan reached me in the last days of July I 942 and I communicated it to Rabbi Wise in New York and Mr. Silverman in London during the first days of August 1942 (through diplomatic channels). Dr. Wise received the message during the last days of August 1942 and asked Mr. Cavert to use his visit to Geneva at the beginning of September 1942 to find out from us whether deportation really meant extermination. After having spoken to us --- I believe to Prof. Guggenheim --- he confirmed this in a cable to the United States.'' [589]
In the same letter to Dr. Visser 't Hooft, Dr. Riegner stated:
``Some of the very forceful speeches by Dr. Bell and other dignitaries of the Anglican Church in the House of Lords were based on reports which we have communicated to them.'' \textit{274}
A telegram was sent by Dr. Visser 't Hooft to the Archbishop of Canterbury and to the Federal Council of Churches in the United States. Its contents were as follows:
15.000 Berlin Jews brought to assembling centres Some hundreds shot. Total evacuation Berlin in execution. Similar news other regions prove extermination campaign at climax. Please back Allied rescue efforts suggest rapid proposals exchange against German civilians and guarantees of re-emigration money food supply enabling European Neutrals to grant transitory asylum. [590]
On March 23, 1944, Dr. Visser 't Hooft and Dr. Freudenberg sent a telegram to the Bishop of Chichester, Dr. Bell:
Most anxious destiny 800,000 Hungarian Jews among whom numerous Christians stop suggest you contact Mr. Silverman World Jewish Congress, I Harley Street W.I. and support suggestions cabled by Riegner to Silverman stop suggest also interest Church of Scotland. [591]
We know of another joint approach made by the Secretariats of the World Council of Churches and the World Jewish Congress. The following Aide-memoire was sent to the Governments of the United States and Great Britain, and to the High Commissioner for Refugees of the League of Nations:
Aide-memoire \textit{275}
The Secretariats of the World Council of Churches and of the World Jewish Congress have taken note with great satisfaction of the aide-memoires exchanged between the Governments of the United States of America and Great Britain on the present situation of refugees in Europe, and of their decision to meet at Ottawa with a view to a preliminary exploration of ways and means for combined action by the representatives of their Governments. [592] Having studied the suggestions and proposals contained in the aide-memoires of the two Governments, the Secretariats of the World Council of Churches and of the World Jewish Congress beg to express their views on the above-mentioned topic. While welcoming most warmly the determination of the Allied Governments to bring help to the persecuted people of all races, nationalities and religions, fleeing from Axis terror, they wish to emphasise that the most urgent and acute problem which requires immediate action, is the situation of the Jewish communities under direct or indirect Nazi control. The Secretariats of the World Council of Churches and of the World Jewish Congress have in their possession most reliable reports indicating that the campaign of deliberate extermination of the Jews organised by the Nazi officials in nearly all countries of Europe under their control, is now at its climax. They therefore beg to call the attention of the Allied Governments to the absolute necessity of organising without delay a rescue action for the persecuted Jewish communities on the following lines: 1. Measures of immediate rescue should have priority over the study of post-war arrangements. 2. The rescue action should enable the neutral States to grant temporary asylum to the Jews who would reach their frontiers. For this purpose a definite guarantee by the Governments of the United States of America and Great Britain, and possibly by other Allied Governments including the British Dominions, should be given to the neutral States, that all refugees entering their territories would be enabled to be repatriated or to re-emigrate as soon as possible after the end of the war. In view of the special characteristics of the Jewish problem, in view of the attitude adopted in the past by many European governments, and furthermore, in view of the present attitude of absolute political neutrality adopted during the hostilities by the neutral countries, it may be stated that the giving of assurance for the prompt repatriation of refugees upon the termination of hostilities, would in the present circumstances not be considered as a sufficient guarantee by the neutral States. Only explicit and comprehensive guarantees of remigration of the refugees, given by the Anglo-Saxon Powers as a reinforcement of any assurances of repatriation which may be given by the Allied Governments in exile, can lead the neutral countries to adopt a more liberal and understanding attitude towards the Jewish refugees. \textit{276} These guarantees should provide for the granting of facilities concerning the supply of food and funds for the maintenance of refugees during their stay in the neutral countries. 3. A scheme for exchange of Jews in Germany and the territories under German control for German civilians in North and South America, Palestine, and other countries, should be pressed forward by all possible means. We should like to stress the fact that the number of nationals of Axis countries living in Allied countries --- particularly in North and South America --- exceeds by far the number of nationals of Allied countries living in Axis countries. We feel that in spite of the great difficulties which we do not underestimate, a workable scheme of exchanging Jews for Germans would constitute an important method of rescuing a considerable number of persecuted people from the countries under Nazi control. In view of the immediate urgency of the situation, the admission of Jews to the scheme of exchange should be granted en bloc to the greatest possible number, as conditions no longer allow time-wasting and in many cases fruitless individual investigations. This scheme might include war-time security measures. Concrete proposals should be submitted without delay to the Governments representing Allied interests in Germany by the Governments of the United States and Great Britain. The International Red Cross Committee may also be approached by the Allied Governments and asked for support in this matter. [593]
Dr. Riegner sent this aide-memoire to the British Ambassador in Switzerland ``on behalf of the Secretariats of the World Council of Churches and of the World Jewish Congress''. Dr. Visser 't Hooft forwarded it to the Ambassador of the United States, requesting in his covering letter, dated March 19, 1943, that the aide-memoire should be forwarded to the American Government, to the Federation of Churches of Christ in the U.S.A., and to the American branch of the World Jewish Congress. He also stated that he had sent a copy to Mr. Allan Welsh Dulles with whom he had ``quite recently had the pleasure of discussing the matter''. Mr.Dulles was the representative of the ``Office of Strategic Services'' of the American Government, at Bern. The sending of this aide-memoire was, I think, the first time in history that an important organization of Churches officially approached Governments, jointly with an important Jewish organization. \textit{277}
c. Aid to Refugees
In 1938, the Provisional Committee of the World Council of Churches was formed. Its first ordinary session took place at Saint-Germain (near Paris), in January, 1939. It was at this meeting that the Bishop of Chichester, George Bell, unequivocally proposed that the Council create a special department to deal with refugee problems. He himself had been a pioneer in this work. He felt that ``the time had come to aid the entire mass of non-Aryans''. He meant not only the non-Aryan members of the Church but also the others, albeit there being a special responsibility towards members of the Christian Church. [594] Soon afterwards Dr. Adolf Freudenberg was appointed the first secretary of this new department for aid to refugees. The Ecumenical Commission for Refugees rendered aid to refugees in the camps of France at the end of 1940. It was also engaged in first aid to the people in the camp of Gurs. Later on, France remained the main field of activities.
``The Christian aid included Christians as well as Jews. There was co-operation with Jewish organizations in many respects. Thus, for instance, the Commission for Refugees could act as the intermediary for financial aid to Jewish families and children who were in hiding in Belgium, Holland, Hungary and other countries.'' [595]
The Churches in three countries rendered financial aid: first and foremost Switzerland, but also Sweden and the United States.
``Switzerland donated Sw. Fr. 77,000 in 1941; the United States donated only Sw. Fr. 10,000 and Sweden Sw. Fr. 6,000. The United States soon realized the importance of the aid to refugees and in the following year the Churches in the United States donated Sw. Fr. 241,000 and later Sw. Fr. 368,000. Obviously they really did understand the significance of this work. I think that this was also due to the fact that Dr. Cavert (the then General Secretary of the Federal Council of the Churches of Christ in the U.S.A.) visited us. Later on I myself went to the United States and was able to explain to them the importance of the matter.'' [596] \textit{278}
When, in the summer of 1942, Lava1 began to hand over the Jews of the unoccupied area of France to the Germans, members of the French Protestant Youth Organization Cimade brought many refugees to the Swiss frontier. Switzerland, however, was not willing to grant asylum to them. The Ecumenical Commission for Refugees, ``closely co-operating with other organizations'', succeeded in assuring the admission of ``many hundreds'' of these refugees. [597] Another endeavour to save lives failed. The Committee had, with the help of American Christians, succeeded in obtaining entrance visas into the United States for 1,000 Jewish children from France, but the occupation of Southern France by the Germans foiled this plan. [598]
Dr. Visser 't Hooft was personally active in an ``illegal'' organization which helped Dutch Jews to pass through France to Switzerland. He helped its leader, Jean Weidner, with money from a collection for this purpose amongst Dutchmen living in Switzerland. [599] The former secretary of the Jewish Committee of Coordination in Switzerland, Mr. H. H. Gans, relates the following incident as regarding to the granting of passports and certificates of citizenship granted by South-American Governments to Jews in French concentration camps:
``...We had declared... that the beneficiaries would not try to use their new `citizenship' after the war. But probably owing to their fear of an invasion of new citizens after the war, some countries dared not postpone the nullification until after the war... The Spanish Ambassador immediately passed on this fatal message (to the Germans) and 300 `South-Americans' were deported from Vitel. The World Congress informed me at night. Consternation was great. \textit{279} I contacted Dr. T. Lewenstein [the then Chief-Rabbi of Zurich and Dr. Visser 't Hooft. Together we sent a telegram to the Queen. There was an immediate reaction: Her Majesty's Ambassador at Buenos Aires was ordered to intervene. Very shortly after this, an entirely favourable result was obtained.'' [600]
Mr. Gans also stated that once he paid a large amount of money on behalf of persons hidden in Holland, through the kind offices of Dr. Visser 't Hooft. From Holland came the confirmation: ``The organization thanks you very much for the money transferred from Switzerland.'' [601] The testimony of Mr. Gans also speaks of the matter of sending gift parcels to the Jews in concentration camps:
``No parcels could have been sent and no other help could have been rendered, if we had not been supported continuously by Dr. Visser 't Hooft, General Secretary of the World Council of Churches, Geneva... His contribution to the Dutch resistance movement will certainly be described by others. Suffice it here to point out the general importance of the presence of such a man in Switzerland, and the fact that busy though he was, he never refused to see me whenever I asked for an interview, and that happened almost every day. No detail of our relief work was unimportant to him.'' [602]
It appears that neither Dr. Visser 't Hooft nor Dr. Freudenberg were formalistic in their activities. They understood, in contrast to so many in and outside occupied Europe, that ``illegal'' acts were, in those special circumstances, morally justified. Thus money was ``illegally'' transmitted to Jews in hiding; and refugees were supported who had entered into Switzerland ``illegally.'' What has been said about Church leaders in Bulgaria, can be applied to Dr. Visser 't Hooft and Dr. Freudenberg: they were gravely concerned, and thus they were available whenever their help was requested.
In June, 1944, the Ecumenical Commission for Aid to Refugees published the following statement:
\textit{280}
The Fate of the Jews in Hungary
``The Ecumenical Commission for Refugees exists in order to give material and spiritual aid to refugees of all faiths. Its main task is therefore to relieve the suffering of the refugees rather than to protest against the treatment meted out to them. But there are situations in which the only aid we can give is in the form of a solemn and public protest. To-day this is the case. Trustworthy reports state that so far some four hundred thousand Hungarian Jews are deported in inhuman conditions and, in so far as they have not died on the way, brought to the camp of Auschwitz in Upper Silesia where, during the past two years, many hundreds of thousands of Jews have been systematically put to death. Christians cannot remain silent before this crime. We appeal to our Hungarian Christian brethren to raise their voice with us to do all they can to stop this horrible sin. We appeal to Christians of all countries to unite in prayer that God may have mercy on the people of Israel.'' [603]
36 TERRITORIES IN WHICH THE CHURCHES REMAINED SILENT
The heading of this Chapter must be regarded with some reservation, firstly because I may have failed to find statements which were issued, and secondly because even the admission by a Church that it did not speak out, cannot always be trusted. In fact, I have in my possession a letter from the official representative of an important Church in Europe, stating that his Church had not publicly protested against the persecution of Jews; yet later on much material was found proving that it had done so. It is notable that the Churches which, as far as we know, kept silent, were minority Churches, with the exception of the Lutheran Church of Finland which was, however, not directly confronted with the challenge of the persecution of the Jews.
a. Austria \textit{281}
On March 12, 1938, German troops entered Austria; it was then absorbed by the German Reich. The Jews in Austria were subjected to all the horrors which the Jews in Germany suffered.
The legend that Austria was the first victim of Hitlerian aggression, to which official endorsement was given by the victorious Allies, is slow to die. In fact, the people in Austria were more national-socialist than in Germany proper: the frenzy with which the ``aggressor'' Hitler was received by the Viennese is proof enough of this. Many of the leaders of the Third Reich were Austrians, as for instance Seyss-Inquart, Kaltenbrunner, Globocnik and Rauter. Hitler himself originally came from Austria.
Little is known about the attitude of the Protestants in Austria with respect to anti-Semitism during the war. [604] In 1966, the General Synod of the Lutheran Church adopted a ``Message to the Congregations on Jews and Christians''. The message stated that:
``...Unfortunately, however, the Christian conscience of our people has not been strong enough to withstand a hatred based on racial differences. This is an alarming sign of the demonic powers of darkness to which we have been exposed and which have not been sufficiently resisted by our Church. Because the Church was entrusted with the Word of reconciliation and the message of peace, its guilt is much greater than that of all other groups. We must acknowledge and confess this guilt. The miracle of God's forgiveness makes our repentance possible...'' [605]
b. Belgium
Professor W. Lutjeharms, who teaches Church history at Brussels, communicated to me why, in his view, the Protestant Churches did not publicly protest against the persecution of the Jews during the war. Part of the reasons he advances are, in my opinion, also applicable to minority Churches in other lands. \textit{282} 1. The Protestants comprise less than half percent of the total population. 2. The Protestants nowhere formed a sufficiently concentrated group among
the population.
3. The Protestants in those days had very few representatives in cultural
and political circles.
4. The Protestant voice was not heard outside its own group before 1940;
hardly at all over the radio and certainly not through daily newspapers.
5. The Protestant Churches represented a distinctly foreign flavour: many
pastors and members were foreigners.
6. An official public protest would neither have impressed the authorities
nor the population. The Protestants could only act effectively on the
personal level. In this respect pastors as well as lay members time and
again risked their lives, to help Jews as much as they could.
There remains the question, why the small Protestant Churches in Belgium undertook official and public steps in 1933, and not, for instance, in the years 1935 and 1938. It is possible that such steps were undertaken, but that they were not sufficiently published, and thus forgotten (Cf. above, point 4).
At least 25,000 Jews were deported from Belgium. Individual Protestants have rescued Jews [606] but these activities are outside the scope of our subject.
c. The Protectorate
Czechoslovakia was deprived of Sudetenland in the Munich pact of September 29, 1938. On March 14, 1939, Slovakia declared its independence. On March 15, 1939, German forces occupied Prague; Czechia as the Protectorate of Bohemia and Moravia became part of the German Reich. \textit{283} An estimated number of 71,000 Jews were deported, and perished. Apparently no Church in Bohemia-Moravia publicly protested. It is true, of course, that there hardly was any address to which they could send a protest, except the Government in Berlin which would probably have paid even less attention than it paid to the protests of the ``Confessing Church'', the members of which were Germans and not Czechs. However, a public protest, read out from the pulpits, could have stirred up the members of the Czech Churches and would have encouraged them to help the Jews. In a letter to me, dated November 12, 1965, it was stated by Dr. Viktor Hhjek, Chairman of the Synod of the Evangelical Church of Czech Brethren:
``Individual members of our Church have tried to help Jewish families in different ways and have indeed helped them. This has always been dangerous, and the persons involved suffered often from the German occupying force. But the pressure of this force was so heavy that it was out of the question to undertake anything publicly and officially.'' [607]
The Synod of the Evangelical Church of the Czech Brethren recognized, in 1945, that ``our Church did not have enough courage or power to withstand the fury of the enemies of Christ directed against the Jews.'' [608]
d. Poland
\textit{284}
The atrocities committed against the Jews in Poland are beyond description. At the end of 1939, 3,300,000 Jews lived in Poland; of these 2,900,000 were murdered. [609] Moreover, most of the Jews arrested by the Germans, in other occupied countries and in the German Reich itself, were deported to Poland and perished there. Thus it was in Poland that the vast majority of the six million was murdered.
There is little to relate about the reactions of the non-Catholic Churches in Poland; there hardly exist such Churches at all. I received two replies to my circular letter; the first is from Dr. Andrzej Wantula, Bishop of the Evangelical Lutheran Church. I quote the following from his letter:
``During the war, our Church was liquidated by the Germans and the majority of the pastors imprisoned, the remainder working in a newly founded German Church. Our Church, therefore, could not carry out any activities. Individual pastors privately have helped the Jews. I myself, in my former parish, have tried to relieve the position of the Jews and partly succeeded in this. These, however, are individual cases, which are outside the scope of your interest.''
The second reply came from the Executive of the small ``Polish-Catholic Church''. [610] I quote the following:
``Our Polish-Catholic Church was exposed to many persecutions, under the National-Socialist domination during the second World War. However, we protested many times, against the persecution of the Jews, also publicly whenever this was possible. In addition to material help, we provided the persecuted Jews with baptismal certificates, enabling them to obtain ration cards and identity cards. In this way they were protected from further persecution. We cannot, unfortunately, send you any proofs, \textit{e.g}. documents, letters or photostats concerning our activities, as all the material was destroyed during the war.''
It is difficult for me to believe that the Polish-Catholic Church has ``protested many times and publicly'', if one is to understand that these protests were made in writing, and officially sent to the German authorities. But perhaps pastors of this Church expressed their protest in their sermons, and if this is so, it was at least something, especially in Poland.
\textit{285}
The activities and attitude of the head of the Greek-Catholic Church in Galicia, the Metropolitan Andrew Sheptitsky, whose Church is united with Rome, is outside the scope of our subject and is thus not related here. [611]
e. Finland
Finland refused to give up her 2,000 Jews. ``We are an honest people,'' declared Witting, the Finnish Foreign Minister. ``We would much rather die with the Jews than give them up.'' [612] I received the following reply to my circular letter:
``...Finland was never actually occupied by the German army, with the exception of the Northern region... Finland remained a sovereign country and it was, as far as I know, the only country within the German sphere of influence where Jews were protected against German claims. It seems to be very difficult to ascertain whether the Church had any direct involvement in this. It must remain, therefore, more or less an academic question, since nothing actually happened, in spite of the hesitation of the Government during some critical days.'' [613]
f. Italy
There are hardly any non-Roman Catholic Churches in Italy. Best-known is the Waldensian Church. The Waldenses themselves have been severely persecuted throughout the centuries. The right of free worship was granted to them by the Constitution of 1848. This ``pre-Reformation Protestant Community'' has 25,000 members. Official declarations against anti-Semitism of such a small minority Church could hardly expected, though the majority of the Waldenses had been strongly anti-fascist. [614]
\textit{286}
g. Russia
The Orthodox Church was the established Church in Russia, until 1917. Under the Communist regime many Church leaders were imprisoned or murdered; many church buildings were closed, some turned into museums. The Constitution of 1936 allows the Church freedom of worship, but not of propaganda. Printing of Bibles was not permitted. Anti-religious propaganda, however, was systematically carried out. In the wake of the German invasion (June, 1941), the Patriarch of Moscow declared himself loyal to the Russian cause and to the Soviet government. Anti-religious measures were relaxed to some degree. As far as we know, no public declaration against anti-Semitism was issued by the Orthodox Church, nor by any of the smaller Christian communities in Russia. [615] It is estimated that 1,500,000 Jews perished in the Nazi-occupied part of Russia.
37 IN CONCLUSION
I have tried to give the answers to some questions related to our subject, but there remain many unanswered questions. It is beyond the scope of this investigation, to analyse the influence of Luther's attitude towards the Jews upon the German Protestants. Suffice it to say, that many anti-Semites quoted from Luther's brochure ``Concerning the Jews and their Lies'' (1542), and not from his earlier: ``Jesus was born a Jew''. (1523) The anti-Jewish sermons of St. Chrysostom, preached at Constantinople at about the turn of the 4th century, are well-known. We have not investigated as to how far these sermons had an influence upon the Eastern Churches in our time.
\textit{287}
Another question: What exactly was the influence of the Lutheran conception of the ``two dominions'' through which God rules this world (the spiritual one, or the Church, and the secular one, or the ``worldly authorities'') on the attitude of the Lutheran Churches towards the persecuted Jews? Why did the Lutheran Churches in Denmark, Norway, Slovakia and Sweden denounce anti-Semitism whilst the record of the Lutheran Churches in America is poor in this respect? The people, according to Luther, have not the right to resist the authorities; only princes have. Was there a notable difference between the Lutheran Churches and the Churches of Calvinist origin regarding their attitude towards the ``ungodly government'' of Hitler in the 20th century, just as such a difference is said to have existed in the 16th century? What about possible differences between continental and Anglo-Saxon Protestants regarding their theological conception of the Jews, between Protestant Churches in the West and Orthodox Churches in Eastern Europe, between non-Roman Catholic Churches and the Church of Rome? How far did the conception of St. Paul about the people of Israel, as expounded in Romans 9-11, encourage the Churches to stand up for the Jews, or how far did the opinion that the Church has ``replaced'' Israel as the people of the Covenant, prevent Churches from taking action? We have hardly touched on practical questions such as the dilemma of whether ``to speak or to save'' ("reden oder retten"). It would be easy to make up a much longer list of unanswered questions, but it is difficult to establish facts even though they happened in our lifetime, and it is even more difficult to interpret them correctly. I can only hope that the documentation provided by this book will stimulate others to further study and investigation.
I hope that I have succeeded to some extent in showing how complicated the situation was, and how careful we ought to be if we try to answer the question, how far Christian leaders and Churches fulfilled or failed to fulfil, the commandment which they profess to consider divine: ``Thou shalt love thy neighbour as thyself''. \textit{288} However, I do not suggest that to understand all is to pardon all. To me, Dr. Visser 't Hooft's conclusion seems to be well-balanced:
``We may conclude this section by pointing out that while many Christians failed in their duty to resist in word and deed the inhuman racialism of National Socialism, there were a not inconsiderable number of Church leaders and simple Church members who rendered a clear witness to the reality of the Christian faith. The Christians who were involved in the struggle know better than anyone how often the Churches and they themselves failed to do what ought have to be done. Thus the Churches in Germany spoke not only for themselves, but for others who had been in a similar situation when after the war they confessed publicly their sense of guilt in this respect.'' [616]
It is difficult to draw conclusions. Mostly generalizations are dangerous. I myself have the impression that public opinion tends to overrate the practical help rendered by individual Christians. Only a minority of professing Christians willingly risked their lives in order to help and save their Jewish neighbours. The Bible condemns such a lack of self-sacrificing love. When, however, human beings judge, particularly if they are people who themselves did not have to undergo the test, they should remember the Jewish saying: ``Judge not thy neighbour until thou art come in his situation.'' [617]
On the other hand, public opinion possibly tends to underestimate the official activities of Churches against anti-Semitism, because they are not generally known. The attitude of the Churches with regard to the persecution of Jews under Hitler's reign of terror was far from uniform. The picture is neither completely black, nor purely white. White and black are mingled. Thus the name chosen for this publication is ``The Grey Book''. The darkness of the holocaust was so great that one can hardly comprehend it. It is understandable that there are people who tend to ignore the lights that were so small, far too small. But ``the greater the darkness, the brighter the light, be it no more than that of a small candle.'' [618]
\textit{289}
In cases where Church leaders or individual Christians did risk their lives, they should remember the words of their Lord: ``Is the master grateful to the servant for carrying out his orders? So with you: when you have carried out your orders, you should say, 'We are servants and deserve no credit; we have only done our duty'."(Luke 17, 9-10). On March 23, 1943, the Archbishop of Canterbury declared in the House of Lords: ``We stand at the bar of history, of humanity and of God''. It is appropriate to conclude this book with the words of Ecclesiastes (12, 13-14): ``Let us hear the conclusion of the whole matter: Fear God, and keep his commandments: for this is the whole duty of man. For God shall bring every work into judgment, with every secret thing, whether it be good, or whether it be evil.''
\textit{290}
\gutchapter{APPENDIX I}
DECLARATIONS AGAINST ANTI-SEMITISM ISSUED
AFTER THE SECOND WORLD WAR
Much has been spoken against anti-Semitism by non-Roman Catholic Churches and Church leaders, after the Second World War. We only mention the most important declarations and statements.
On April 8, 1948, the NATIONAL BRETHREN COUNCIL (Reichsbruderrat) of the Evangelical Church in Germany, meeting at Darmstadt, issued the following ``Message Concerning the Jewish Question''.
``...It may rightly be said that after what has happened, after all that we allowed to happen in silence, we have no authority to speak now. We are distressed about what happened in the past, and about the fact that we did not make any joint statement about it. We have not forgotten that a number of pastors and churches did speak out, and suffered for doing so; we thank God for it, and we thank them. We thank all who, in our own country and abroad, have helped us with old and new insights into the Word of God, and who have taken action by setting up warning signs. Today when retribution is meted out to us for what we did to the Jews, there is increasing danger that we may take refuge from God's Judgment in a new way of anti-Semitism, thus conjuring up all the old evils once again. In this perilous situation and amid this temptation God's Word speaks to us and helps us to find the right attitude to the Jews. It is under pressure of this Word that we speak, because we are filled with anxiety about the future and burdened by the past, and because we feel obliged to express our gratitude to all those individual people who spoke out, took action and suffered doing so... It was a disastrous mistake when the Churches of our time adopted the secular attitude of mere humanity, emancipation and anti-Semitism towards the Jewish question. There was bound to be a bitter retribution for the fact that anti-Semitism rose and flourished not only among the people (who still seemed to be a Christian nation), not only among the intelligentsia, and in governmental and military circles, but also among Christian leaders. \textit{291} And when finally this radical anti-Semitism, based on racial hatred, destroyed our nation and our Churches from within, and released all its brutal force from without, there existed no power to resist it --- because the Churches had forgotten what Israel really is, and no longer loved the Jews. Christian circles washed their hands of all responsibility, justifying themselves by saying that there was a curse on the Jewish people. Christians no longer believed that the promise concerning the Jews still held good; they no longer preached it, nor showed it in their attitude to the Jews. In this way we Christians helped to bring about all the injustice and suffering inflicted upon the Jews in our country. This is what the Word of God teaches us, so that we recognize with shame and grief what a great wrong we have done to Israel, and how deep our guilt is. As a Church we have failed to be the witness of salvation for Israel. Now we have to face the judgments of God which are coming upon us one after the other, so that we may bow beneath the mighty hand of God in sincere repentance, both as a Church and as a nation...'' [619]
On April 27, 1950, the Synod of the Evangelical Church in Germany, meeting at Berlin-Weissensee, issued a ``Message of Guilt towards Israel'', from which we quote the following:
``...We declare that by dereliction of duty and in keeping silent we also are guilty of the crimes committed by people of our nation towards the Jews... We pray all Christians to rid themselves of all anti-Semitism whatsoever, to resist it earnestly where it raises its head again, and to meet Jews and Jewish Christians in a brotherly spirit. We pray the Christian congregations, to care for Jewish cemeteries in their territory if nobody is in charge of them.'' [620]
On January 12, 1960, the Executive of the United Evangelical Church of Germany issued the following Declaration:
``The Executive of the United Evangelical Lutheran Church of Germany most sharply condemns the expressions of anti-Semitism which have stirred the public in the last weeks. Moral condemnation of the crimes committed by Germans against the Jews cannot be evaded, though it is difficult to explain the motives of this wave (of anti-Semitism) which encompasses many lands. The reaction of the public must not be limited to declarations of sympathy towards Jewish fellow citizens but must aim at uncovering their own failures. It is especially important, to break the silence which frequently is maintained here between the older and younger generation, and to help our young people to come to their own clear judgment of the history of the Third Reich and what led up to it.'' [621] \textit{292}
On February 26, 1960, the Synod of the Evangelical Church in Germany, meeting at Berlin-Spandau, published the following Resolution, after the synagogue of Cologne was daubed with swastikas:
``The fact that the honour of our Jewish neighbours has been offended, fills us with horror and shame. We express our solidarity with those who have been offended and insulted... We are guilty towards youth, to whom we have failed to teach and to bear the witness we owe them. It is not surprising, therefore, that the evil spirit increases its influence, again and again, among our youth. However, anew we must realize and attest: the hatred of the Jews which breaks out, again and again, is public godlessness... Therefore, let parents and educators break the widely-spread, painful silence, in our country about co-responsibility for the fate of the Jews. Let them resist everything that seduces the young generation into hatred of the Jews... Therefore, stand up for the payment of reparations. Keep in mind, however, that true repentance is more essential than financial compensation, which only can mean little to people who lost most of their relatives by acts of violence... Therefore, pray for God's peace upon Israel. Pray for the peace of Israel amongst the nations, on the borders of its State and amongst us.'' [622]
During the trial of Eichmann, the Synod of the Evangelical Church in Germany stated (Berlin-Spandau, Febr. 17, 1961):
``...All surviving Germans who at the age of discrimination witnessed the atrocities of the annihilation of the Jews, and even those who helped their Jewish compatriots under oppression, must confess before God, to have become accessories to the deeds by lack of alert and self-denying love...'' [623]
After several years of discussion on the subject of the relationship of the Church to the Jewish people, the 10th German Evangelical Kirchentag, 1961, set aside one work-group to deal specifically with this subject. Work-group VI of the Kirchentag produced the statement which follows, and it was adopted as the official Report of the Assembly: \textit{293}
``Jews and Christians are insolubly linked with each other. The denial of this link brought forth the hostility to Jews within Christendom. It became one of the main causes of the persecution of Jewry. Jesus of Nazareth is betrayed wherever members of the Jewish people among whom he was born are despised as Jews. Every hostility towards Jews is godlessness and leads to self-destruction. The present trial in Jerusalem concerns us all. We Evangelical Christians in Germany recognise that we are involved in it by reason of our guilt. Because of the need for fresh thought and conversion, we call upon the German public to make the following points their own: 1. Parents and educators should break their silence when meeting the young
generation. They should confess their own failure and bring to light the
origins of the crimes so that we all may learn how to face the present
together. In the present world political situation, throwing off our own
failure onto others must threaten not only one section of mankind but all
life.
2. The inhumanity of compulsory systems of command where men can argue that
criminal orders must be obeyed, is calculated to warn us against the
inhuman potentialities of the modern organisation of State and society.
We must be ready to take upon us political responsibility even in spite of
risks. Those who were concerned in the preparation and implementation of
persecution should resign from high office.
3. Where Jews live amongst us, it is our duty to promote their well-being as
best as we can. Likewise everything must be done by us Germans which serves
the reconstruction and peace of the State of Israel and its Arab neighbours.
Compensation claims by victims of racial persecution should be settled with
special urgency and generosity. The material compensation must be matched by
a rebirth of the spirit. In Germany, the so-called Jewish question is today
above all a question concerning the future of the Germans.
4. As against the wrong doctrine preached for centuries that God has cast away
the Jewish people, we once again affirm the word of the Apostle: 'God hath
not cast away His people, which He foreknew' (Romans 11, 2)...'' [624]
On March 13, 1964, the Synod of the Evangelical Church in Germany issued the following Declaration on the trials of Nazi criminals:
``...Only ignorance can speak of `soiling one's own nest' when in fact the cleaning of a badly soiled nest is at stake. Nor is it in any way profitable to try to hide behind the wrongs committed by other nations against members of our people during the war. The mass murder of Jews and other ethnic groups, with which the German name is connected, is not thereby erased... \textit{294} Even the citizen who had no direct share in the crimes, nay, even he who did not know of them, has a share in the guilt because he was indifferent towards the perversion of all moral standards and all notions of right and wrong among our people. Nor can we exempt ourselves and our congregations from this guilt. For where all Christians were called upon to uphold the Gospel entrusted to us, to make public affirmation of the everlasting dominion of God in all spheres of our lives, and, thus armed, to protect the victims of the regime, especially the Jews living amongst us, only few had the insight and the courage to resist. Forced into this humiliating position, we cannot pretend to be unconcerned with the trials now pending, nor can we turn our gaze away from the crimes now being revealed. It was the folly of our entire nation, and the omissions of us Christians, that enabled those crimes to be perpetrated. There is nothing here that can be condoned, and we must resist all temptations to indulge in self-justification. Rather is it our duty to follow the defendants now as they stand before God and His judgment.'' [626]
In 1963, the Evangelical Kirchentag of German speaking Switzerland adopted the following Resolution:
``Israel and the Church belong together. God has chosen them both and formed a Covenant with them --- first with Israel, then with the Church. The Church has been grafted onto the stem of God's People. Hence: if one member suffers, all the other members suffer with it. We confess our guilt with regard to Israel, that we Swiss people did not suffer with the Jews either, thus betraying our fellowship with God's ancient people. Our hope rests in the fact that we know our sins are forgiven. But for us forgiveness must mean active repentance. Let us make atonement through action. As Swiss people and Christians let us recognize the following tasks as our next step, and transform them into action: 1. We urge all Christians, both individually and collectively, to take part in
assisting the Jews and promoting mutual contact and mutual respect.
2. We regard it as our Christian duty to oppose all forms of discrimination
against Jewish people, and we expect the same attitude from all our
fellow-Christians.
3. Short, inexpensive instructive booklets are needed as soon as possible,
so that Christians can remedy their ignorance of Jewish history.
4. We recognize a sub-conscious anti-Semitism here in Switzerland too, with
devastating and far-reaching effects.
We urge the Church to devote more attention to this question. (Parish
evenings, evenings for parents, evenings for mothers, instruction to
religious teachers, training of religious teachers).
5. The intercession for Israel, which exists in most liturgies, should be
made an integral part of the worship-service.'' [627] \textit{295}
On February 6, 1967, the Archbishop of Sweden sent a circular letter to the ministers of the diocese, from which we quote the following:
'On various occasions accusations have been made against the Church of Sweden for anti-Semitism. This is especially directed towards the way in which the passion-history is created. From abroad many appalling examples are known in which `the Jews' are described as a deicide people, as referring not only to the mob in front of the palace of Pilate which wanted Jesus to be crucified but to the whole people and the generations after them. In the Swedish tradition of preaching and teaching this is completely unaccepted. Any feeling of revenge and hatred against Romans and Jews is repudiated and it is emphasized that it is our own sin which has brought Jesus to the cross... The Gospel is against all racial and group discrimination. Each person shall be judged on his own merits according to what he himself is and does. There is no graduation of the value of man; all are called to be children of God and are therefore our brothers and sisters, and Christ died for all. After all, God alone is the Judge, but never we ourselves.'' [628]
On March 18, 1964, the following Statement was issued by the Archbishop of Canterbury:
``It is always wrong when people try to lay blame upon the Jews for the crucifixion of Jesus Christ. In the event the Roman Governor was no less responsible for what happened. The important fact, however, is that the crucifixion was the clash between the Love of God and the sinfulness and selfishness of the whole human race. Those who crucified Christ are in the true mind of the Christian Church representatives of the whole human race, and it is for no one to point a finger of resentment at those who brought Jesus to his death, but rather to see the crucifixion as the divine judgment upon all humanity for choosing the ways of sin rather than the love of God. We all must see ourselves judged by the crucifixion of Christ.'' [629]
The General Assembly of the Church of Scotland adopted a statement against anti-Semitism nearly every year. Here follow some examples.
\textit{296}
1945
``The General Assembly express their renewed sympathy with the Jews in their present circumstances and assure them of the Church's friendly interest in all that pertains to their future welfare and in particular commend the active steps, which are being taken to restrain Anti-Semitism and promote better understanding between Christian and Jew.'' [630]
\begin{quotation}
1947\end{quotation}
``The General Assembly, aware of the growth of anti-Semitic feeling, condemn anew this attitude as un-Christian and contrary to the mind of our Lord; call upon their faithful people to guard against this grave danger; assure the Jewish people of their deep sympathy in the present uncertainties and remember especially the many Jews in the Displaced Persons Camps in Europe still awaiting some scheme for their settlement in other lands.'' [631]
\begin{quotation}
1953\end{quotation}
``The General Assembly view with concern the renewed outbreaks of anti-Semitism in various countries, renew their condemnation of this evil thing and call upon their faithful people to be on the alert to oppose any signs of it in this country.'' [632]
\begin{quotation}
1957\end{quotation}
``The General Assembly express their concern that the threats of annihilation directed against the State of Israel still continue. They express profound sympathy with the State of Israel in the crisis with which she is confronted and earnestly hope that the United Nations will now direct all possible efforts towards a just and lasting settlement between Israel and the Arab States, so that Israel's future will no longer be in jeopardy.'' [633]
\begin{quotation}
1962\end{quotation}
``The General Assembly, in view of the horrors recalled by the Eichmann trial, remind the Church --- especially the youth of the Church --- of the deadly danger of Anti-Semitism, which has in the past so cruelly wounded the brotherhood of the human family.'' [634]
Many Protestant Churches in the United States publicly registered their opposition to anti-Semitism. We only quote the following Resolution, adopted by the National Council of Churches of Christ in the United States of America, on June 5, 1964:
``The General Board of the National Council of the Churches of Christ in the U.S.A., recognizing the ever-present danger of anti-Semitism, renews the call to the Churches and the community to recognize (in the words of the First Assembly of the World Council of Churches) 'anti-Semitism, no matter what its origin, as absolutely irreconcilable with the profession and practice of the Christian faith'. \textit{297} The spiritual heritage of Jews and Christians should draw us to each other in obedience to the one Father and in continuing dialogue; the historic schism in our relations carries with it the need for constant vigilance lest dialogue deteriorate into conflict. We confess that sometimes as Christians we have given away to anti-Semitism. We have even used the events of the Crucifixion to condemn the Jewish people, whereas (in the words of the Third Assembly of the World Council of Churches) 'the historic events which led to the Crucifixion should not be presented as to fasten upon the Jewish people of today responsibilities which belong to our corporate humanity and not to one race or community'. The General Board urges that the members of its constituent communions seek that true dialogue with the religious bodies of the Jewish community through which differences in faith can be explored within the mutual life of the one family of God --- separated, but seeking from God the gift of renewed unity --- knowing that in the meantime God can help us to find our God-given unity in the common service of human need.'' [634]
In 1948, the World Council of Churches held its first Assembly, at Amsterdam. 147 Churches in forty-four countries were represented by 351 official delegates. A report on ``The Christian Approach to the Jews'' was heard by Assembly, and its deliberations were commended to all member Churches ``for their serious consideration and appropriate action''. We quote the following:
Introduction ... We cannot forget that we meet in a land from which 110,000 Jews were taken to be murdered. Nor can we forget that we meet only five years after the extermination of 6 million Jews. To the Jews our God has bound us in a special solidarity linking our destinies together in His design. We call upon all our Churches to make this concern their own as we share with them the results of our too brief wrestling with it."...
3. Barriers to be Overcome ``...We must acknowledge in all humility that too often we have failed to manifest Christian love towards our Jewish neighbours, or even a resolute will for common social justice. We have failed to fight with all our strength the age-old disorder of man which anti-Semitism represents. \textit{298} The Churches in the past have helped to foster an image of the Jews as the sole enemies of Christ, which has contributed to anti-Semitism in the secular world. In many lands virulent anti-Semitism still threatens and in other lands the Jews are subjected to many indignities. We call upon all the Churches we represent to denounce anti-Semitism, no matter what its origin, as absolutely irreconcilable with the profession and practice of the Christian faith. Anti-Semitism is sin against God and man...''. [635]
In 1961, the World Council of Churches held its third Assembly, at New Delhi. 200 Churches were represented by more than a thousand official participants. The following Resolution on Anti-Semitism was adopted:
``The Third Assembly recalls the following words which were addressed to the Churches by the First Assembly of the World Council of Churches in 1948: 'We call upon all the Churches we represent to denounce anti-Semitism, no matter what its origin, as absolutely irreconcilable with the profession and practice of the Christian faith. Anti-Semitism is sin against God and man. Only as we give convincing evidence to our Jewish neighbours that we seek for them the common rights and dignities which God wills for his children, can we come to such a meeting with them as would make it possible to share with them the best which God has given us in Christ.' The Assembly renews this plea in view of the fact that situations continue to exist in which the Jews are subject to discrimination and even persecution. The Assembly urges its member Churches to do all in their power to resist every form of anti-Semitism. In Christian teaching the historic events which led to the Crucifixion should not be so represented as to fasten upon the Jewish people of today responsibi-lities which belong to our corporate humanity and not to one race or community. Jews were the first to accept Jesus and Jews are not the only ones who do not yet recognize him.'' [636]
In 1964, a Consultation on ``The Church and the Jewish People'' under the auspices of the Lutheran World Federation was held at Legumkloster, Denmark. The following statement was adopted:
III. The Church and Anti-Semitism \textit{299}
``Anti-Semitism is an estrangement of man from his fellowmen. As such it stems from human prejudice and is a denial of the dignity and equality of men. But Anti-Semitism is primarily a denial of the image of God in the Jew; it represents a demonic form of rebellion against the God of Abraham, Isaac, and Jacob; and a rejection of Jesus the Jew, directed upon His people. `Christian' anti-Semitism is spiritual suicide. This phenomenon presents a unique question to the Christian Church, especially in light of the long terrible history of Christian culpability for anti-Semitism. No Christian can exempt himself from involvement in this guilt. As Lutherans, we confess our own peculiar guilt, and we lament with shame the responsibility which our Church and her people bear for this sin. We can only ask God's pardon and that of the Jewish people. There is no ultimate defeat of anti-Semitism short of a return to the living God in the power of His grace and through the forgiveness of Jesus Christ our Lord. At the same time, we must pledge ourselves to work in concert with others at practical measures for overcoming manifestations of this evil within and without the Church and for reconciling Christians with Jews. Towards this end, we urge the Lutheran World Federation and its member Churches: 1. To examine their publications for possible anti-Semitic references, and to
remove and oppose false generalisations about Jews. Especially reprehensible
are the notions that Jews, rather than all mankind, are responsible for the
death of Jesus Christ, and that God has for this reason rejected His covenant
people.
Such examination and reformation must also be directed to pastoral practice
and preaching references. This is our simple duty under the commandment
common to Jews and Christians: 'Thou shalt not bear false witness against thy
neighbour'.
2. To oppose and work to prevent all national and international manifestations
of anti-Semitism, and in all our work acknowledge our great debt of gratitude
to those Jewish people who have been instruments of the Holy Spirit in giving
us the Old and New Testaments and in bringing into the world Jesus Christ our
Lord.
3. To call upon our congregations and people to know and to love their Jewish
neighbours as themselves; to fight against discrimination or persecution of
Jews in their communities; to develop mutual understanding; and to make
common cause with the Jewish people in matters of spiritual and social
concern, especially in fostering human rights...'' [637]
An International Conference of Christians and Jews was held at Seelisberg, in 1947, and attended by sixty-five persons from nineteen different countries. They adopted the following ``Address to the Churches'', which became widely known as ``The Ten Points of Seelisberg'': \textit{300}
1. Remember that One God speaks to us all through the Old and the New
Testaments.
2. Remember that Jesus was born of a Jewish mother of the seed of David and
the people of Israel, and that His everlasting love and forgiveness embrace
His own people and the whole world.
3. Remember that the first disciples, the apostles, and the first martyrs were
Jews.
4. Remember that the fundamental commandment of Christianity, to love God and
one's neighbour, proclaimed already in the Old Testament and confirmed by
Jesus, is binding upon both Christians and Jews in all human relationships,
without any exception.
5. Avoid disparaging biblical or post-biblical Judaism with the object of
extolling Christianity.
6. Avoid using the word Jews in the exclusive sense of the enemies of Jesus,
and the words the enemies of Jesus to designate the whole Jewish people.
7. Avoid presenting the Passion in such a way as to bring the odium of the
killing of Jesus upon Jews alone.
In fact, it was not all the Jews who demanded the death of Jesus. It not the
Jews alone who were responsible, for the Cross which saves us all reveals
that it is for the sins of us all that Christ died.
Remind all Christian parents and teachers of the grave responsibility which
they assume, particularly when they present the Passion story in a crude
manner. By so doing they run the risk of implanting an aversion in the
conscious or subconscious minds of their children or hearers, intentionally
or unintentionally. Psychologically speaking, in the case of simple minds,
moved by a passionate love and compassion for the crucified Saviour, the
horror which they feel quite naturally towards the persecutors of Jesus
will easily be turned into an undiscriminating hatred of the Jews of all
times, including those of our days.
8. Avoid referring to the scriptural curses, or the cry of a raging mob: His
blood be upon us and upon our children, without remembering that this cry
should not count against the infinitely more weighty words of our Lord:
Father, forgive them, for they do not know what they do.
9. Avoid promoting the superstitious notion that the Jewish people is
reprobate, accursed, reserved for a destiny of suffering.
10. Avoid speaking of the Jews as if the first members of the Church had not
been Jews. [638]
\textit{301}
\gutchapter{APPENDIX II}
SOME PARTICULARS ABOUT THE CHURCHES MENTIONED [639]
Austria
The Protestant Churches in Austria are minority Churches. The (Lutheran) Evangelical Church of the Augsburg Confession has 406,966 members; the Reformed Church of Austria has 16,078 baptized members.
Belgium
The Protestant Churches in Belgium are minority Churches, together comprising less than half percent of the population. The total number is less than 50,000.
Bulgaria
\begin{quotation}
The Orthodox Church in Bulgaria claims a number of six million members, being\end{quotation}
the vast majority of the population. There is no other Christian community of any numerical importance in Bulgaria.
\textit{302}
Czechoslovakia
The largest non-Roman Catholic Churches in Bohemia and Moravia are: the Evangelical Church of Czech Brethren (295,354 baptized members), the Czechoslovak Church, and the Evangelical Church of the Augsburg Confession in Silesia (48,000 members). There are two Protestant Churches in Slovakia: the Reformed Church of Slovakia (165,000 baptized members) and the Evangelical (Lutheran) Church in Slovakia (520,000 members).
Denmark
The vast majority of the people of Denmark belong to the Lutheran Church, which has 4,104,000 members.
Finland
The vast majority of the population of Finland belongs to the Finnish Evangelical Lutheran Church, which has 4,429,137 members.
France
The Protestants in France are a small minority, numbering altogether not more than 800,000 souls. Members of the Protestant Federation of France are: The Reformed Church of France (375,000), the Reformed Church of Alsace and Lorraine (50,000), the Lutheran Church of Alsace and Lorraine (240,000) and the Evangelical Lutheran Church of France (50,000).
Germany
The vast majority of the Protestants of Germany belonged to one of the 28 Landeskirchen (Lutheran, Reformed or Uniate), of which the largest was the Church of the Old Prussian Union, with 18 million members. In all, there were forty-five million Germans who were, nominally at least, members of the Protestant Church.
Great Britain and Ireland
The main non-Roman Catholic Churches in England are: the Church of England, claiming 2,989,704 members and 15 million adherents (1950); the Methodist Church (775,294 members and 2,2250,000 adherents in 1955); the Congregational Union of England and Wales (451,523 members in 1955); the Baptist Union of Great Britain and Ireland (246,400 members in 1955) and the Presbyterian Church of England, having 70,298 communicants. \textit{303} There are four Free Presbyterian Churches in Scotland, as well as Baptist, Episcopal, Congregational and Methodist Churches. The Church of Scotland is by far the largest Church, having 1,281,559 communicants. The political partition of Ireland did not divide any of the Churches. Most of the non-Roman Catholic Churches were represented in the United Council of Christian Churches in Ireland. The (Episcopalian) Church of Ireland has 400,000 members. The Presbyterian Church in Ireland has 140,395 communicants and 397,500 baptized members. The Methodist Church has approximately 30,000 communicants and 100,000 baptized members.
Greece
The vast majority of the population of Greece belongs to the (Orthodox) Church of Greece, which has an estimated 8,000,000 members.
Hungary
According to the 1941 census, there were in Hungary 9,775,310 Catholics, 2,785,782 Calvinists (Reformed Church of Hungary), and 729,289 Lutherans (Hungarian Evangelical Church).
Italy
The Waldensian Church has 25,000 members. Other non-Roman Catholic communities are the Methodist and Baptist Churches. Their total membership amounts to about 0,19 per cent of the population.
The Netherlands
The DUTCH REFORMED CHURCH has 3,500,000 baptized members. The Reformed Churches in the Netherlands had 640,984 members in 1940. The Evangelical Lutheran Church has 52,587 members. The other Churches mentioned are of about the same size, or smaller.
\textit{304}
Norway
The (Lutheran) Church of Norway has 3,456,687 members, being 96,2 per cent of the population.
Poland
Out of a population of 32,000,000 there are a 130,000 Protestants. 100,000 of them belong to the Evangelical Lutheran Church in Poland. Smaller communities are the Evangelical-Reformed Church (5,000 members); the Baptist Church (2,500 members) and the United Gospel Church (7,500 members).
Rumania
The vast majority of the population of Rumania belongs to the Rumanian Orthodox Church, which has an estimated 11,500,000 members. The Reformed Church of Rumania is the Church of the Hungarian national minority; it has 693,511 baptized members. The Evangelical Church of the Augsburg Confession is mainly the Church of German immigrants; it has 183,399 members.
Russia
Before 1917, the Orthodox Church of Russia claimed a membership of 100,000,000. Estimates about the present situation --- ``perhaps 25-50,000,000'' --- are unreliable. Smaller communities are the Union of Evangelical Christian Baptists of U.S.S.R. and the Lutheran Churches in former Estonia (350,000), Latvia (350,000), and Lithuania (30,000).
Sweden
The vast majority of the population of Sweden belongs to the (Lutheran) Church of Sweden, which claims 7,000,000 members.
Switzerland
The Protestant Churches of Switzerland are cantonal Churches, distinct and independent from one another. In most of the cantonal Churches, the legislative body is the Synod and the executive organ the Synodal Council.
\textit{305}
The Federation of the Protestant Churches of Switzerland at first consisted only of National Churches, but it soon admitted the Free Evangelical Churches, the Methodist Church and the ``Evangelische Gemeinschaft''. The Federation has 2,888,122 baptized members.
The United States
The following are some of the greatest Churches affiliated to the Federal Council of Churches of Christ in America in the year 1942 with their membership for the years ending in 1941-1942.
Northern Baptist Convention 1,538,871
National Baptist Convention 3,911,611
Congregational Christian Churches 1,052,701
Disciples of Christ 1,655,580
Protestant Episcopal Church 1,074,178
United Lutheran Church (consultative) 1,709,290
The Methodist Church 6,640,424
Presbyterian Church in the U.S.A. 1,986,257
The total membership was 25,551,560
The Federal Council of Churches united with 11 other national inter-denominational organizations, to form the National Council of Churches, in 1950. Its 34 member Churches have a total membership of about 42 million persons. The most important Protestant denominations which are not members of the National Council of Churches are: Southern Baptist Convention (present membership 10,770,573); the Lutheran Church-Missouri Synod (present membership 2,692,889); the American Lutheran Church (present membership 2,541,546).
Yugoslavia
The greatest non-Roman Catholic Church is the Serbian Orthodox Church which has about 8,000,000 members. Other Churches are: the Reformed Christian Church of Yugoslavia (30,000 members) and the Slovak Evangelical Church of the Augsburg Confession in Yugoslavia.
The World Council of Churches
\textit{306}
The World Council of Churches is a fellowship of more than 200 Churches of Protestant, Anglican, Orthodox and Old Catholic confessions. It includes in its membership Churches in more than 80 countries. In 1961, the Orthodox Church of Russia also joined the World Council of Churches. A number of large Churches, however, are not World Council members. These include the Roman Catholic Church, the Southern Baptist Convention (U.S.A.), the Lutheran Church --- Missouri Synod, and the majority of Pentecostal Churches. Many of these Churches regularly send observers to the World Council meetings and there were five Roman Catholic observers at the World Council's Third Assembly at New Delhi in 1961.
\textit{307}
\gutchapter{BIBLIOGRAPHY}
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\gutchapter{PERIODICALS AND REPORTS}
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Manchester Guardian.
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The Times, London.
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\gutchapter{FOOTNOTES for The Grey Book by Johan M. Snoek =============================================}
page I
1 The struggle of the Dutch Church for the Maintenance of the Commandments
of God in the Life of the State, Documents collected and edited by
W.A. Visser 't Hooft, London, 1944, p.16 (henceforth: Visser 't Hooft).
page II
2 Kirche im Kampf, Dokumente des Widerstandes und des Aufbaus der Evangelischen
Kirche in Deutschland von 1933 bis 1945, herausgegeben von H. Hermelink,
Tubingen-Stuttgart, 1950, p. 344 ff. On the historiographical use of the term
``political messianism'' in this context, cf. J.L. Talmon, The Unique and
the Universal --- Some Historical Reflections, London 1965, Chap. IV: Mission
and Testimony --- The Universal Significance of Modem Anti-semitism, p. 119 ff.
page III
3 Kirchliches Jahrbuch fur die Evangelische Kirche in Deutschland 1933- 1944,
hrsg. von Joachim Beckmann, Gutersloh 1948, p. 76 f., quoted according to:
Der Nationalsozialismus, Dokumente 1933-1945, herausgegeben, eingeleitet und
dargestellt von Walther Hofer, Frankfurt a/M, 1957 (henceforth: Hofer),
p. 140.
4 Visser 't Hooft, pp. 61, 64-65.
This reference to the first of the Ten Commandments by the Church in its
struggle against the totalitarian and pseudo-messianic character of the
Nazi regime already appeared in:
Wort der Bekenntnissynode der evangelischen Kirche der altpreussischen
Union an die Gemeinden, 4/5 March 1935 in Berlin-Dahlem; par. 1: The first
commandment reads: 'I am the Lord God. Thou shalt have no other gods besides
me. 'We obey this commandment alone having faith in Jesus Christ who was
crucified and resurrected for us. The new religion is a revolt against
the first commandment.'' Cf. Hofer, p. 144.
page V
5 Visser 't Hooft, p. 64.
Page VI
6 Heinrich Schmidt, Apokalyptisches Wetterleuchten, Ein Beitrag der
Evangelischen Kirche zum Kampf im `Dritten Reich', Munchen, 1947, p. 305.
This source also appears in: Friedrich Zipfel, Kirchenkampf in Deutschland
1933-1945 --- Religionsverfolgung und Selbstbehauptung der Kirchen in der
national-sozialistischen Zeit, Berlin I 965, p. 31.
7 Hofer, p. 128.
8 On the origin of the term: ``metapolitics'' cf. Constantin Frantz: ``Offener
Brief an Richard Wagner'', Bayreuther Blaetter, Jahrgang 1, No. 6 (June 1878),
op. 169. Cf.: Peter Viereck, Metapolitics --- The Roots of the Nazi Mind,
N.Y. 1961 (1941), p. 4.
9 Visser 't Hooft, p. 71.
page VII
10 See in this volume pp. 131 --- 132 . Cf. Visser 't Hooft, p. 57.
page VIII
11 These sources of modem anti-semitism have recently been treated in:
Shmuel Ettinger, ``The Critique of Judaism in the Teachings of the 'Young
Hegelians' as one of the Roots of Modem Anti-semitism'', Lecture given at
the Academia scientiarum Israelitica, Jerusalem, 1969 (in press, Hebrew).
Ibid: ``The Roots of Modern Anti-semitism'', (Hebrew) Molad, Jerusalem,
New Series Vol. 11 (xxv), No. (219) Jan.-March, 1969, p. 323 ff.
On the theoretical relationship between theological criticism and racial
theory, cf. Nathan Rotenstreich, Judaism and Jewish Rights, (Hebrew),
Tel-Aviv, 1959, Chaps, 1, 3, 5, 6. Ibid. ``For and against Emancipation:
The Bruno Bauer Controversy'', in Leo Beck Institute, Year Book IV,
London, 1959, p. 3 ff. Cf. also: Eleonore Sterling, Er ist wie Du --- aus
der Fruehgeschichte des Anti-semitismus in Deutschland (1815-1850),
Munchen 1956, 235 pp. For sources on modern anti-semitism in the Critique
of positivistic religion by the deists and rationalists in France,
cf. Arthur Hertzberg, The French Enlightenment and the Jews, N.Y.-Phil.
1968, 420 pp.
12 Ludwig Feuerbach, Das Wesen der Religion, Dreissig Vorlesungen, 1845
(1848), Dritte Vorlesung, Leipzig, 1908, p. 12.
page IX
13 Friedrich Nietzsche, Gesammelte Werke, Gesamtausgabe, Kroner, Leipzig,
Vol. VII, p. 273: ... Das Christenthum ist ein Aufstand alles
Am-Boden-Kriechenden gegen das, was Hoehe hat; das Evangelium der
Niedrigen macht niedrig...''.
14 Alexander Bein, Der moderne Anti-semitismus und seine Bedeutung fur
die Judenfrage, Vierteljahreshefte fur Zeitgeschichte, Stuttgart,
1958, pp. 345/6.
15 Moritz Freystadt, Der Christenspiegel von Anti-Marr, ein offenes
Sendeschreiben an die modernen Judenfeinde, 5e Anlage, Koenigsberg 1863,
pp. 3, 8, 20, 21, 39.
page X
16 Wilhelm Marr, Streifzuege durch das Koncilium von Trient --- Voltaire
frei nach erzaehlt, Hamburg, Otto Meissner Verlag, 1868, pp. 95/6.
In this work Marr emphasizes the fact that the criticism of Christianity
was for the most part directed against the Catholics who were called by
the spokesmen of German nationalism in the period of Bismarck
`ultramontanists.' At the same time he claims: ``We... reject
Christianity as well as Judaism... We reject... all religions...''.
Cf. p. 102.
17 Marr, like most of the fathers of antisemitic ideology, is not consistent
in his antisemitic arguments or in his anti-Christian motives. Different
views are held at different times, and contradictory views are expressed
at the same time. Thus, Marr sometimes does not oppose Christianity but
seems to be a proponent of ``practical Christianity'' with an eye to the
social policy of Bismarck in the 80's of the last century, or as a
proponent of ``Christian-German realism. Thus, we also note an anti-Christian
sentiment directed not so much against Protestantism as against Catholicism.
The Anti-Catholic attitude of the fathers of racial anti-semitism was
part of the national awakening in the days of the Second Reich, an
awakening that was based to a considerable extent on the tradition of
Protestant sovereignty.
Of the many sources of antiCatholic anti-semitism from the first days of
this movement the propaganda of Ottomar Beta is typical, as we find in his
book which he dedicated to Bismarck, ``Juda-Jesuitismus, where, among other
things, he says: ``The arrogant assumption of infallibility of the Jewish
descendents in Rome is nothing more than an ultramontanist firework to
divert the eyes of the Germanic peoples from the more ominous capitalistic
infallibility of their racial brethren in wordly garb... The source appears
in the anti-semitic collection: Antisemiten-Spiegel --- die Antisemiten im
Lichte des Christenthums, des Rechtes und der Moral, Danzig, 1892, (A.S.
further), p. 136.
page XI
18 Walter Holsten, Adolf Stoecker als Symptom seiner Zeit --- Anti-semitismus
in der evangelischen Kirche des 19e Jahrhunderts?
The article appears in: Christen und Juden --- Ihr Gegenueber vom Apostel-
konzil bis heute, herausgegeben von Wolf-Dieter Marsch und Karl Thieme,
Mainz/Goettingen 1961, p. 182 ff.
On this ambivalent character of anti-semitism, cf. the words of A. Stoecker
to the German Kaiser, 25. 9. 1880: ``...Im Ubrigen habe ich in allen meiner
Reden gegen das Judentum offen erklaert, dass ich nicht die Juden angreife,
sondern nur dies frivole, gottlose, wucherische, betruegerische Judenthum,
das in der Tat das Unglueck unseres Volkes ist...''.
This source is found in: Dietrich von Oertzen, Adolf Stoecker --- Lebensbild
und Zeitgeschichte, Berlin 1910, Vo1.1, p. 213. Cf. also: Paul W. Massing,
Vorgeschichte des politischen Anti-semitismus, Frankfurt a/M, 1959,
(P. Massing: further) p. 31.
page XII
19 Adolf Stoecker, Christlich-Sozial; Reden und Aufsaetze, 1885, p. 153 f.
Cf. also P. Massing, p. 238/9, note 64. According to the second edition
of the addresses and works of A. Stoecker of the year 1890, P. Massing
quotes the entire document which concludes with the words: ``... A return
to more of Germanic law and economic life, a return to Christian faith ---
this will be our slogan.''
page XIII
20 R. Seeberg, Reden und Aufsaetze von Adolf Stoecker, Leipzig 1913, p. 141/2.
Cf. also the above essay of Walther Holsten, p. 119.
21 Above, note 19, p. 211.
page XIV
22 Eugen Duhring, Wert des Lebens, 3. Auflage, 1881, p. 5: ``... paradoxe
Lehre von der Umkehrung oder Kreuzigung aller Fleischregungen...''.
23 Eugen Doehring, Die Parteien in der Judenfrage, Separat-Ausgabe von Hefte
7, 8 des ersten Bandes der Schweitznerischen internationalen Monatschrift,
Leipzig 1882, Verlag Theodor Fritsch, p. 403 ff.
page XV
24 A.S. Danzig, 1892, p. 137 fl. Cf. also: these sources in a pamphlet
issued by the ``Verein zur Abwehr des Anti-semitismus'', which also issued
the ``AntisemitenSpiegel. The name of this pamphlet is: ``Antisemitisches
Christenthum und christlicher Anti-semitismus'', Flugblatt No 7, p. 1/2
(Year not given).
25 The anti-intellectual meaning of this doctrine that seeks to relegate the
image of Jesus to the mythology of racial anti-semitism was pointed out by
several writers already during the Second Reich.
Cf. the many publications of the ``Verein zur Abwehr des Anti-semitismus,''
beginning in 1892, and in condensed form: Antisemiten-Spiegel, Berlin-
Frankfurt a/M, 1911 , p. 161 ff.
On the historical background of the anti-intellectual character of racial
doctrines and on the influence of anti-intellectualism on this interpreta-
tion that would transplant Jesus from his Jewish origin and make him a
member of the Aryan race, cf.: Fritz Stern, The Politics of Cultural
Despair --- A study in the Rise of the German Ideology, Berkeley and Los
Angelos, 1961, pp. 41/2: to divorce Christianity and Judaism even at this
late stage would be a recognition of an unambiguous historical truth and
of Jesus' own intent...''. cf. pp. 139, 143, 144, 145, 163, 199.
page XVI + XVII
26 From the journal ``Hammer'', published in: A.S. Berlin-Frankfurt a/M, 1911,
p.201.
27 Ibid. p. 203. The anti-Christian meaning of modem anti-semitism and its
historical sources have recently been noted by Salo W. Baron in: Deutsche
und Juden, Beitraege, \textit{etc}., Frankfurt a/M, 1967,
p. 84/85: ``...It is unmistakable how the resistance against everything
that Judaism and Christianity stand for has increased since the 70's of
the 19th century, and it is no exaggeration to say that this development
prepared the ground for the Nazi assumption of power...''.
I have dealt with this question at length in the last chapter of my book,
cf. UneiTal, Christians and Jews in the Second Reich (1870-1914), Chap. V:
``Christian and anti-Christian Anti-Semitism'', (Hebrew), in Press, The
Magnes Press, The Hebrew University, Jerusalem.
On the anti-Christian elements in racial anti-Semitism, cf. also: Edward
H. Flannery, The Anguish of the Jews, N.Y. London, 1965, p. 180 ff.
See also: Hermann Greive, Theologie und Ideologie --- Katholizismus und
Judentum in Deutschland und Oesterreich (1918-1935), Heidelberg, 1969.
In this book the author stresses that a clear-cut distinction between
Christian anti-semitism or ``kirchlicher Anti-Judaismus'' and racial anti-
semitism or ``Rassenanti-semitismus'' is not warranted. His conclusion is:
``...dass anti-juedische Vorurteile im Katholizismus der diskutierten Periode
auf breitester Basis nachgewiesen werden koennen... Die anfaenglich
vielfach nicht unerhebliche Verschiedenheit in der Judenfeindlichen
Argumentation zumindest der tonangebenden Kreise in Katholizismus und auf
voelkisch-antikirchlicher Seite wich im Laufe der Zeit immer weiter
reichenden Vermittlungstheorien zwischen den sozial, kulturell und religioes
orientierten und den voelkisch-rassischen Anti-semitismus...''; p. 222/223.
A similar conclusion that stresses the direct connection between the
theological and historical anti-Jewishness of Christianity and modem
anti-semitism is reached by: A. Roy Eckardt, Elder and Younger Brothers
-The Encounter of Jews and Christians, N.Y., 1967, Chap. 1, The Enigma, 1,
p. 8: ``There can be little serious doubt that Christendom's traditional
antipathy to the Jews is the major historical root of anti-semitism in the
Western world. Historically speaking, anti-semitism derives from 'the
conflict of the Church and the Synagogue.' Here is the Crime of Christendom.
Such distinguished and authoritative historians as James Parkes and Jules
Isaac have chronicled this fact definitely...''.
This thesis, about the historical continuation between the anti-Jewish
attitude in Christianity and modern anti-semitism rests not only on the
theological attitude of Christianity but also on the legal history of
the persecution of Jews by Christians, beginning in the forth century.
This is treated by: James Parkes, The Conflict of the Church and the
Synagogue --- A Study in the Origins of Anti-semitism, Cleveland, N.Y.,
Phil., 1961, Appendix I, p. 379 ff.
A similar historiographical approach is taken by Raul Hilberg who has
Drawn up a comparative list of Canonical and Nazi .Anti-Jewish Measures,
in his book: The Destruction of the European Jews, Chicago, 1961 , pp. 5-6.
The list also appears in the above mentioned book of A. Roy Eckhardt,
p. 12 --- 13, where he draws the same conclusion, namely, that the Nazis
``...did not discard the past; they built upon it...''. And he adds to
this quotation from Hilberg: ``This fact makes ludicrous any unqualified
claim that the Nazis were the enemies of Christendom.''
page XVIII
28 Cf. the works of E. Flannery, James Parkes, A. Roy Eckhardt, Marsch-Thieme;
cf. above, and also: Karl Thieme, Der religioese Aspekt der
Judenfeindschaft (Judentum und Christentum), in: ``Judentum --- Schicksal,
Wesen und Gegenwart'', hrsg. von Franz Boehm und Walter Dirks, unter
Mitarbeit von Walter Gottschalk, Wiesbaden 1965, Vol. II, p. 603 IT. See
also: ``Judenfeindschaft. Darstellungen und Analysen,'' hrsg. Von Karl
Thieme, Frankfurt a/M. Hamburg, 1963, 326 ff.
Cf. especially the work of: Ernst Ludwig Ehrlich, ``Judenfeindschaft in
Deutschland von der Roemerherrschaft, bis zum Zeitalter der Totalitaet,''
p. 209 ff. Also: W.P. Eckert und E.L. Ehrlich, ``Judenhass --- Schuld
der Christen?'', Essen, 1964, 525 pp.
page XIX
29 Cf. the study of: Willehad Paul Eckert: ``Beatus Simonius --- Aus den
Akten des Trienter Judenprozesses'', in the above collection, note 28,
edited by W.P. Eckert and by E.L. Ehrlich, p. 329 ff; also in the
same collection the work of: Kurt Hruby, Verhangnisvolle Legenden und
ihre Bekaempfung, p. 281 ff.
page XXI
30 Haim Hillel Ben-Sasson, Jewish-Christian Disputation in the Setting of
Humanism and Reformation in the German Empire, H.T.R., 59 (1966), pp.
369-390. Salo W. Baron, Modern Nationalism and Religion, N.Y., Phil.
(1947), 1960, Ch. V.: Protestant individualism, p. 117 ff.
Cf. especially Salo W. Baron, Medieval Heritage and Modern Realities in
Protestant-Jewish lielations, Diogenes Spring 1968, No. 61, p. 32 ff.
31 ``Der Ungekuendigte Bund'', hrsg. von Dietrich Goldschmidt und Hans Joachim
Kraus, Stuttgart, 1962, p. 206.
32 Ibid., p. 218.
page XXII
33 Visser 't Hooft, pp. 35/36.
34 Ibid., p. 36.
page XXIII
35 Ibid.
36 A striking and instructive example to such a different approach has been
given by the Bishops of Denmark, in their protest against the persecution
of the Jews, 3 Oct. 1943. See below in this volume, on p. 168.
Cf. ``The Israel Digest'', X1/22, Jerusalem, 1, 1, 1968, p. 3. A German
translation in Freiburger Rundbrief, Vol. xx, 1968, No. 73-76, Dec. 1968,
pp. 69/70. As to the historical background of this document, cf. Leni Yahil,
Test of a Democracy --- the Rescue of Danish Jewry in World War II, Jerusalem,
1966, pp. 59, 125, 145, 165.
37 World Council of Churches --- Division of Studies, Commission on Faith and
Order in cooperation with the Committee on the Church and the Jewish
People: ``The Relationship of the Church to the Jewish People, Collection
of Statements'', Geneva, July 1964, p. 19 ff.
page XXIV
38 Ibid., p. 22-23
39 Ibid,. P. 22
40 Ibid., p. 23
41 Ibid., p. 26
42 Ibid., p. 27
\begin{verbatim}
Footnotes from Preface ----------------------\end{verbatim}
page 1
43 Much has been published about the subject of ``Christian'' anti-Semitism.
Some literature: Jules Isaac, The Teaching of Contempt (New York, 1965);
James Parkes, Anti-semitism (Quadrangle Books, 1961); Malcolm Hay, Europe
and the Jews, The Pressure of Christendom on the People of Israel for 1900
years (Second printing, Boston, 1961; this book was first published as ``The
Foot of Pride'', in 1950); Karl Thieme (Ed.), Judenfeindschaft (in German;
Fischer Bucherei KG, 1963); E. Flannery, The Anguish of the Jews (New York,
1965).
page 2
44 Thus a Protestant minister from Switzerland who now lives in Israel, in
The Jerusahlem Post, Sept. 27, 1963. The Speaker of the Israeli Parliament,
Mr.Kadish Luz, made a similar statement in the session of the Parliament
on April 21, 1963.
45 The Yad Vashem Martyrs and Heroes Memorial Authority, Jerusalem, was
established by Law in 1953, The Yad Vashem Act assigned to ``Yad Vashem''
the task ``to collect, investigate and publish all evidence regarding the
Catastrophe and its heroic aspects and to inculcate its lesson upon our
people''.
page 3
46 Cf. Dr. Leny Yahil, Historians of the Holocaust; A Plea for a New Approach
(in: The Wiener Library Bulletin, 1967/68, Vol. XXII, pp. 2-5).
47 Dr. Visser 't Hooft and Dr.A.Freudenberg, of the World Council of Churches,
are preparing their memoirs. Rev. Armin Boyens is preparing his thesis which
will have a chapter on ``The Confessing Church and the Jews, 1933-1938''.
48 Cf. Eberhard Bethge, Dietrich Bonhoeffer (Gesammelte Schriften, Munich, 1959;
in German), Vol. I, p. 9 (Introduction).
page 5
49 Cf. ``La Persecution des Juifs en Allemagne: Attitude des Eglises Chretiennes''
(Geneva, 1933; in French), p.25.
50 See for the acts of individuals (Christians and non-Christians): Philip
Friedman, Their Brothers Keepers (New York, 1957); Kurt R. Grossman, Die
unbesungenen Helden (Berlin, 1957; in German); Heinz Leuner, When Compassion
was a Crime (London, 1966); Saul Friedlaender, Kurt Gerstein Ou l'ambiguite
du bien (Tournai, 1967; in French).
\begin{verbatim}
Footnotes Part I ----------------\end{verbatim}
page 9
51 See below, on p. 113.
52 See below on p. 135-136.
53 See below, on p. 160.
54 E.H. Flannery, The Anguish of the Jews (New York,1965), p. 224.
page 10
55 See below, p. 265.
56 ``Unity in Dispersion'', A History of the World Jewish Congress (New York,
1948 pp. 194, 195.Also see: Dr. L. de Jong, Een Sterfgeval in Auschwitz
(Amsterdam, 1967; in Dutch); an English translation will be published in
Yad Vashem Studies, VII (Jerusalem, 1969), pp. 39-55: ``The Netherlands and
Auschwitz.''
57 Cf. Flannery, op. cit., p. 227: ``Criticism of passivity or collaboration
under the Nazis must be tempered by an understanding of the confusion
wrought by the insiduous methods of Nazi propaganda and the paralysis of
wills in the Nazi terror, made all the more effective by its appeal to
patriotic and anti-Communist loyalties. Resistance under the circumstances
should not be entirely assessed from the comfortable perspective of the
postwar era.
page 11
58 ``Unity in Dispersion'', pp. 193-196.
59 See below, p. 255.
60 Cf. H.C. Touw, Het Verzet der Hervormde Kerk ('s Gravenhage, 1946; in
Dutch), 11, p. 388-390.
page 12
61 In February, 1941, a general strike in Amsterdam and other places in the
Netherlands was called as a protest against the deportations of Jews.
The Germans proclaimed martial law and suppressed the strike by force.
They proceeded to deport a total of 430 Amsterdam Jews to the concentration
camp of Mauthausen, where they perished.
Cf. B.A. Sijes, De Februari-Staking ('s-Gravenhage, 1954; in Dutch, with an
English summary), passim.
Also see: H. Knap, Vreemdeling, Bericht de Spartanen (Amsterdam, 1966), p.
III: ``The technical conditions for massive actions of solidarity with the
Jews --- if our people as a whole would have wanted them --- were lacking.''
62 Rolf Hochhuth, The Deputy (New York, 1963; third printing), p. 79.
63 J.J. Buskes, Waar stond de Kerk? (Amsterdam, 1947; in Dutch), p. 93.
64 Francois Mauriac, quoted by Hochhuth, op. cit., p. 6.
65 Knap, op. cit., passim.
page 13
66 See below, p 259.
page 14
67 See beyond, p. 147.
68 Cf., for instance, Exodus 32, 9; Isaiah 1, 2-15; Jeremiah 7, 24-26;
Ezekiel 2, 3; Hosea 4, 7- 8.
page 15
69 Werner Warmbrunn, The Dutch under German Occupation 1940-1945 (London,
1963), p. 279.
70 See below, on p. 175.
page 17
71 Jeremiah, Amos and others. Also see: I Kings 22, 5-28.
72 See below, p. 79 ff.
73 See below, p. 245 ff.
74 See below, p. 212 ff.
page 18
75 See below, on p. 201.
76 See below, on p. 99.
page 19
77 Jenoe Levai, Black Book on the Martyrdom of Hungarian Jewry (Zurich, 1948),
p. 217.
page 20
78 A.J. Koejemans in De Waarheid, Sept. 20, 1945.
page 21
79 Touw, op. cit., I , p. 69
80 Cf. Dr. J. Presser, Ondergang ('s-Gravenhage, 1965; in Dutch), 11, p. 177.
For the protest, which was read from the pulpits in the Roman Catholic and
Protestant Churches, see below, pp. 130 -131.
page 22
81 Cf. H. Leuner, When Compassion was a Crime (London, 1966), p. 13: ``Every
possible means of propaganda and subtle psychology was used to separate
Germans and Jews, to create an unbridgeable gulf between the members of
the Aryan master race and those belonging to the family of `parasites'.
82 W.A. Visser 't Hooft, The Ecumenical Movement and the Racial Problem
(Paris, 1954), p.40.
83 Cf. the chapters 27-30 in this book.
84 Cf. ch. 31 in this book.
85 Cf. ch. 32, p. 233
86 Cf. chapters 33 --- 34 in this book.
page 23
87 ``Unity in Dispersion'', pp. 160-161.
page 24
88 F. Burgdoerfer, ``Die Juden in Deutschland und in der Welt''; in:
``Forschungen zur Judenfrage'' (Hamburg, 1938; in German), pp. 152-199.
89 Cf. ch. 18, p. 95
90 Cf. ch. 30, p. 204
page 25
91 Leuner, op. cit., p. 100.
92 Touw, op. cit., 1, p. 174.
Cf. W. A. Visser 't Hooft, The Struggle of the Dutch Church for the
Maintenance of the Commandments of God in the Life of the State (London,
1944), p. 13: ``When threats were of no avail the Germans attempted to
blackmail the Churches. In this way the Churches were brought into great
conflicts of conscience. Should they give up the open protests so that
this or that group of church-members might be saved? Or should they go
forward, without regard for the consequences that might arise for others?
These are difficult questions that no one can decide on the spur of the
moment or looking at the situation from the outside.''
Cf. also the opinion of a group of Christians of Jewish origin (p. 130 in
this book).
93 Abel J. Herzberg, Kroniek der Jodenvervolging (Arnhem-Amsterdam, 1950;
in Dutch), p. 133.
94 Presser, op. cit., 11, p. 128.
page 26
95 Cf. ch. 22, pp. 129.
96 Cf., however, Pinchas E. Lapide, The Last Three Popes and the Jews (London,
1967), p. 138:
``At that time Archbishop Roncalli, the Apostolic Delegate to Turkey and
Greece, received Mr. Ira Hirschmann, a special emissary for the U.S. War
Refugee Board... sent to interview Yoel Brand in connection with Eichmann's
`blood for goods' deal.
Mr. Hirschmann told Roncalli of the plight of several thousand Jews,
including a number of children slated for deportation and death at Auschwitz.
The Archbishop instantly made available thousands of `baptismal certificates'
for use for the doomed Jews, without conditions, and thousands were saved
from the Nazi furnace.''
page 27
97 Quoted from a letter, dated October 14, 1965, of Dr. Jan Chabada, the
present Generalbischof of the Evangelical Lutheran Church in Slovakia,
to me.
Cf. Dr. L. Rothkirchen, ``Vatican Policy and the `Jewish Problem' in
`Independent' Slovakia (1939-1945)'', in: ``Yad Vashem Studies'' (Jerusalem,
1967), Vol. VI, p. 46: ``...the Protestant clergy... for the most part
supporters of the pro-Czechoslovak line, were prominent for their more
adaptable approach, mainly in the furnishing of certificates of conversion
to Christianity. Some Protestant clergymen have even been charged with
profitmaking motives and with granting hundreds of certificates of
conversion within a day or two.''
Also see: R. Hilberg, The Destruction of the European Jews (Chicago, 1961),
p. 466: ``From the ethnic German organ, the Grenzbote, criticism was more
vociferous. The baptisms were ternied a blasphemy, and the churchmen who
engaged in them were accused of having monetary motives. Two Calvinist
pastors, Puspas and Sedivy, were subsequently arrested, and Sedivy was
accused of having performed not fewer than 717 baptisms.''
98 Peter Meyer (Ed.), The Jews in the Soviet Satellites (Syracuse University
Press, 1953), p. 571.
page 28
99 Yad Vashem Archives, No. 0311 708 (A summary of the evidence by the
interviewer; in Hebrew).
100 Michael Molho and Joseph Nehama, The Destruction of Greek Jewry, 1941-1944
(Jerusalem, 1965; in Hebrew), p. 142.
101 Ilias Venezis, Archbishop Damaskinos (Athens, 1952;) in Greek), ch. 34.
page 29
102 Buskes, op. cit., p. 89.
103 J.J. Buskes, Hoera voor het Leven (Amsterdam, 1963), p. 193.
104 Romans 13, 8.
\begin{verbatim}
Footnotes Part II ------------------\end{verbatim}
page 33
105 Wolfgang Scheffler, Judenverfolgung im Dritten Reich (Berlin-Dahlem,
1960), p. 26.
page 34
106 Ibid., p. 26.
page 35
107 Ibid., pp. 79-80.
page 36
108 Ibid., p.26.
109 Cf. Bruno Blau, The Last Days of German Jewry in the Third Reich (in:
Yivo Annual, vol. VIII, 1953, pp. 197-204).
page 37
110 An immense number of publications appeared about the Church in Germany
during the Third Reich. We mention here: ``Die Evangelische Kirche in
Deutschland und die Judenfrage'', Ausgewaehlte Dokumente aus den Jahren
des Kirchenkampfes 1933 bis 1943 (Geneva, 1945);
G.van Norden, Kirche in der Krise (Dusseldorf, 1963);
W.Jannasch, Deutsche Kirchendokumente (Zurich, 1946);
Heinrich Hermelink, Kirche im Kampf (Stuttgart, 1950);
Guenther Weisenbom, Der lautlose Aufstand (Hamburg, 1953);
Wilhelm Niemoeller, Kampf und Zeugnis der Bekennenden Kirche (Bielefeld,
1948); Wilhelm Niemoeller, Die Evangelische Kirche im Dritten Reich
(Bielefeld, 1956); Renate Maria Heydenreich, Versuch theologischer
Wiedergutmachung; in D. Goldschmidt und H. J. Kraus (Ed.),
Der ungekundigte Bund (Stuttgart-Berlin, 2. Auflage, 1963), pp. 183-283;
Otto Diehn, Bibliographie zur Geschichte des Kirchenkampfes, 1933- 1945
(Gottingen, 1958);
Anton Koch, Vom Widerstand der Kirche (Freiburg, 1947);
Friedrich Zipfel, Kirchenkampf in Deutschland 1933-1945 (Berlin, 1965);
Dr. Joseph Tenenbaum, For the Sake of Historical Balance (in: Yad Vashem
Bulletin, No. 3, Jerusalem, 1958);
Philip Friedman, Was there ``another Germany'' during the Nazi Period? (in:
Yivo Annual of Jewish Social Studies, Vol. x, New York, 1955).
page 38
111 Cf. the ``Gesetz ueber die Rechtsverhaltnisse der Geistlichen und Kirchen-
beamten'' (EKD und Judenfrage, pp. 35 ff.; Heydenreich, op. cit., p. 196).
112 The full contents in: Heydenreich, op. cit. ; EKD und Judenfrage, passim.
page 39
113 Hermelink, op. cit., pp. 250-251. Cf. the article of Kurt Meier,
Kristallnacht und Kirche --- die Haltung der Evangelischen Kirche zur
Judenpolitik des Faschismus (in: Wissenschaftliche Zeitschrift der
Karl-Marx-Universitat Leipzig, 13. Jahrgang, 1964, pp. 91-106), p .99.
114 Heydenreich, op. cit., p. 228.
115 Ibid., p. 230.
page 40
116 Hermelink, op. cit., 351.
117 Meier, op. cit., p. 99. Also see: Wilhelm Niemoeller, Die Bekennende
Kirche sagt Hitler die Wahrheit (Bielefeld, passim). On August 23, 1936,
the Memorandum was published as a ``Proclamation from the Pulpit'' in
an amended form. Cf. Eberhard Bethge, Dietrich Bonhoeffer (Gesammelte
Schriften; Munich, 1958), 11, p. 277 (note).
page 41
118 Cf. ``Die Evangelische Kirche und die Judenfrage'', pp. 180 ff.; H.Grueber,
Wemer Sylten (Berlin, 1956; in German); idem: An der Stechbahn (Berlin,
1960; in German); H.D. Leuner, When Compassion was a Crime (London, 1966).
pp. 114-119.
119 Hermelink, op. cit., p. 461. Cf. Meier, op. cit., p. 100: ``The reason
that no joint protest was issued by the Confessing Church after the
pogrom of November, 1938. was that the Confessing Church as an
organization was under strong pressure at the time. On June 23, 1937,
several members of the Reich Brethren Council were arrested; on July 1,
1937, also Martin Niemoeller.
After that, the Reich Brethren Council was hardly able to act.
page 42
120 Die Evangelische Kirche und die Judenfrage, p. 179.
121 Meier, op. cit., p. 101
page 43
122 Cf. below, p. 244.
123 Cf. H.C. Touw, Het Verzet der Hervormde Kerk (s'Gravenhage, 1946;
in Dutch), pp. 13-34.
124 La Persecution des Juifs en Allemagne, p. 21; ``De Standaard'' (Protestant
Daily in the Netherlands before the war), April 7, 1933.
125 Cf. below, p. 93.
126 ``De Standaard'', April 7, 1933.
page 44
127 Ibid., May 5 and 12, 1933.
128 Ibid., May 16 and 20, 1933.
129 Ibid., May 24 , 1933.
130 The addresses were published in the Brochure ``Vrede over Israel''
(Amsterdam, 1935; in Dutch). Also see: D. Cohen, Zwervend en Dolend
(Haarlem, 1955; in Dutch), pp. 27-28.
131 ``Dietschen stam''.
132 Th.Delleman, Opdat wij niet vergeten (Kampen, 1949; in Dutch), pp. 55-69,
481-489. Action was also taken against members of the ``Christian Democratic
Union'', a party of Christian socialists. Also see: Werner Warmbrunn,
The Dutch under German Occupation 1940-1945 (London, 1963), p. 160.
page 45
133 Ben van Kaam, Opstand der Gezagsgetrouwen (Wageningen, 1966; in
Dutch), p. 16.
134 Cohen, op. cit., p. 56.
135 Cf. for --- at least --- questionable comments in the Protestant Press
on the events in Germany: van Kaam, op. cit., pp. 25-27. Also see the
article ``Van eigen bodem'' (in: ``De Standaard'', Dec. 3, 1938).
136 ``De Standaard'', Nov. 17, 1938.
page 46
137 Ibid., November 14, 1938.
page 47
138 Le IIIe Reich et les Juifs, pp. 191-192.
page 48
139 Ibid., p. 193.
140 The nickname of the Protestants in Belgium and the Netherlands, in the
16th century. It became their name of honour.
141 Le IIIe Reich et les Juifs, pp. 178-179.
page 49
142 Le IIIe Reich et les Juifs, pp. 200-201. La Persecution des Juifs en
Allemagne, p. 6.
143 Le IIIe Reich et les Juifs, p. 201.
page 50
144 Ibid., pp. 202 ff. (2 x used)
page 53
145 ``Pour la dignite humaine'' (Brochure), pp. 48-52.
page 54
146 Archives of the Protestant Federation of France, Paris.
147.Ibid.
page 55
148 ``Journal de Geneve'', April 9, 1933; quoted in: La Persecution des Juifs
en Allemagne, p. 23.
149 La Persecution des Juifs en Allemagne, p. 24.
page 56
150 Ibid., p. 24.
151 Cf. pp. 210-212.
page 57
152 Schweiz. Evang. Pressedienst (E.P.D.), Zurich, Nov. 30, 1938.
153 E.P.D., ibid., p. 2.
154 E.P.D., Dec. 14, 1938.
page 58
155 Professor Aage Bentsen, Docent Flemming Hvidberg, Professor Johannes
Pedersen and Professor Frederik Torm. The declaration was published
in ``Berlingske Tidende''.
156 Dr.Leni Yahil, Test of Democracy, The Rescue of Danish Jewry in World
War II (Jerusalem, 1966; in Hebrew, with a summary in English), pp. 59-60.
page 59
157 Cf. the article of Dr. Fuglsang-Damgaard in: Chr. Refslund --- M. Schmidt
(Ed.),Fem Aar (Copenhagen, 1946; in Danish), II, pp. 100-108.
158 Church Magazine ``Kristen Gemenskap'' (in Swedish), 1933, No. 2.
page 60
159 Ibid., 1933, No. 2.
page 61
160 Ibid., 1934, No. 19.
page 63
161 Ibid., 1938, No. 4.
page 64
162 Ibid.
163 The Hungarian Upper House had 254 members, including 34 representatives
of the Churches. Cf. Albert Bereczky, Hungarian Protestantism and the
Persecution of the Jews (Budapest, 1946), p. 8. Also see: Hendrik Fisch
(Ed.), Kerestzteny egyhazfok felsohazi beszedi a zsidokerdesben (The
Speeches on the Jewish Question by Christian Church Leaders in the Upper
House; Budapest, 1947; in Hungarian); and: ``Schweiz. evang. Pressedienst''
(Zurich), March 27, 1946, pp. 3-6.
page 65
164 Jeno Levai, Black Book on the Martyrdom of Hungarian Jewry (Zurich,
1948), p. 12.
165 Rabbi Hershkovits in my interview with him on March 29, 1966.
166 Cf. Bereczky, op. cit., pp. 9-10.
167 R.Hilberg, The Destruction of the European Jews (Chicago, 1961), p. 514.
Cf. ``Schweiz. evang. Pressedienst'', March 27, 1946, p. 5: ``Special
endeavours were made on behalf of the Protestant Jews, not because they
belonged to the Church but because it was easier to attain something for
them.''
page 66
168 ``Dimineata'', Bucharest, April 15, 1933; quoted in: ``La Persecution des
Juifs en Allemagne'', p. 22.
169 Other protests, statements and declarations:
Speech by the Bishop of Fulham ("The Times'', April 27, 1933).
Speech by the Vicar of Leeds (Manchester Guardian, April 10, 1933).
Speech by the Bishop of Liverpool (Liverpool Post, April 6, 1933).
Speech by the Bishop of Nottingham (Manchester Guardian, May 3, 1933).
Resolution of the Council of the World Evangelical Alliance, British
Section; May 1, 1933
(Dr.A.Freudenberg, the Church and the Jewish Question; Geneva, 1944; p.18).
Resolution of the Baptist Union of Scotland (Glasgow Herald, June 6, 1933).
Message from the Bishop of Durham; ``J'accuse'' (brochure, London), p. 93.
Letter from the Bishop of Chichester to ``The Times'', May 30, 1935.
page 67
170 Freudenberg, op. cit., p. 3
171 Ibid., p. 4.
page 68
172 Ibid., p. 4
173 Ibid., p. 5
174 Ibid., p. 3.
page 69
175 Ibid., p. 5 . Cf. ``Le IIIe Reich et les Juifs'', pp. 218 --- 219. Also see:
``La Persecution des Juifs en Allemagne'', pp. 12-13.
page 70
176 Freudenberg, op. cit., p. 18.
177 Reports to the General Assembly, 1933, p. 709.
178 See p. 79.
page 71
179 Acts, Proceedings and Debates of the General Assembly, 1934, p. 79.
180 Reports to the General Assembly, 1935, p. 772.
page 72
181 Freudenberg, op. cit., p. 15.
182 From the ``Reports and Minutes of the General Assembly of the Presbyterian
Church of England'', 1934.
page 73
183 The Church Assembly of the Church of England usually meets for three
sessions a year. It consists of the three houses of Bishops, clergy
and laity.
At present there are 734 members: 34 Bishops, 344 clergymen and 347
laymen.
page 75
184 ``The Times'', November 21, 1935.
The full report in the Brochure ``The Jews in Germany'', Debate in the
Church Assembly, Nov. 20th, 1935, London.
185 Cf. R.C.D. Jasper, George Bell Bishop of Chichester (London, 1967),
passim.
186 See above, on p. 74.
187 Jasper, op. cit., pp. 137-138.
page 76
188 The Diocesan Conferences meet once a year or at most, twice. They are
the local counterpart of the Church Assembly and consist of two houses,
the Chamber of Clergy and the Chamber of Laity. The Bishop is always the
president of the Diocesan Conference.
189 Freudenberg, op. cit., p. 7.
190 Cf. Jasper, op. cit., pp. 135-163. Also cf.: Norman Bentwich, They Found
Refuge (London, 1956), pp. 43 and 51.
page 77
191 Minutes of the meetings of the Church Assembly (Archives of Church House,
Great Smith Street, Westminster).
page 78
192 Jasper, op. cit., pp. 142-143.
193 ``Reports and Minutes of the General Assembly'', 1937.
194 Freudenberg, op. cit., pp. 12-13.
page 79
195 Reports to the General Assembly, 1936, p. 709.
196 Acts, Proceedings and Debates of the General Assembly, 1937, p. 71.
197 Reports to the General Assembly, 1938, p. 753.
page 80
198 ``The Times'', November 12, 1938.
199 Minutes of the meetings of the Church Assembly (Archives of Church House,
Great Smith Street, Westminster).
200 See p. 101.
page 81
201 Norman Bentwich, They Found Refuge (London, 1956), p. 69.
Also see pp. 78-85.
202 Freudenberg, op. cit., p. 33.
page 82
203 Reports to the General Assembly, 1939, pp. 691-693. See for the full text
of Dr.Black's letter: The Jewish Chronicle, Nov. 25, 1938, p. 26.
204 Minutes of the Conference of the Methodist Church in Ireland,
held in June, 1939.
205 Keesing's Contempary Archives, March 22, 1933; 725-E.
page 83
206 Keesing, March 30, 1933; 735-B.
207 Freudenberg, op. cit., p. 19. Cf. ``La Persecution des Juifs en Allemagne'',
p. 5; and: ``Le IIIe Reich et Les Juifs'', p. 224.
page 84
208 Freudenberg, op. cit., p. 20.
page 85
209 I See above, p. 72 . ff.
210 Dr. Charles S. Macfarland, Across the Years (The Macmillan Co., 1936),
p.168.
211 Ibid., p. 168.
page 86
212 Ibid., p. 169.
213 The New York Times, June 9, 1937; Freudenberg, op. cit., pp. 20 --- 21.
page 87
214 Freudenberg, op. cit., p. 31.
215 Federal Council Bulletin, October 1938, p. 13.
216 Ibid., December 1938, p. 3.
page 88
217 Ibid., October 1938, p. 13.
218 Ibid., December 1938, p. 9
219 Ibid.
page 89
220 The New York Times, Jan. 10, 1939. Among the signers were: Dr. Samuel
McCrea Cavert, Federal Council of Churches; the Rt. Rev. Edwin H.Hughes,
Bishop of Washington area, Methodist Episcopal Church; Bishop Charles
Mead, Methodist Episcopal Church, Kansas City.
221 Cf. Arthur Morse, While Six Million Died (London, 1968), p. 253.
222 The New York Times, Febr. 19, 1939. Cf. Federal Council Bulletin, Febr.,
1939, p. 7.
223 Morse, op. cit., p. 268. Cf. Arieh Tartakower and Kurt R.Grossmann,
The Jewish Refugee (New York, 194.4). p. 90: ``One of the chief arguments
raised against this bill was that the admittance of 20,000 refugee
children to the United States from Germany and the refusal to admit
their parents would be against the laws of God, and therefore would be
an opening wedge for a later request for the admission of about 40,000
adults, the parents of the children in question.''
page 90
224 The New York Times, March 26, 1939.
225 Ibid., April 27, 1939.
page 91
226 National Council of Churches, Department of Information.
page 92
227 Freudenberg, op. cit., p. 51; Federal Council Bulletin, Febr, 1939, p.3 ff.
228 The New York Times, May 31, 1939
229 Ibid., June 13, 1939.
page 93
230 See above, p. 38.
231 Freudenberg, op. cit., p. 22. Cf. ``La Persecution des Juifs en
Allemagne'', p. 27.
The Resolution had been requested by the Dutch Council (see above, p. 43).
232 Freudenberg, op. cit., p. 23.
page 94
233 Jasper, op. cit., p. 101.
234 Minutes, Novi Sad, 1939, pp. 37-38. Cf. Jasper, op. cit., p. 104.
235 See above, p. 38.
236 Jasper, op. cit., p. 105.
page 95
237 ``Le Christianisme Social'' (French Protestant Periodical),
Nov. --- Dec. 1933, p. 606.
page 96
238 Freudenberg, op. cit., p. 27.
page 97
239 Ibid., pp. 28-29.
240 Cf. for the Oxford Conference: ``The Churches Survey Their Task'' (The
Report of the Conference of Oxford, July 1937, on Church, Community,
and State; with and Introduction by J.H.Oldham), London, 1937.
Also see: Ruth Rouse and Stephen Charles Neil (Ed.), A History of the
Ecumenical Movement 1517-1948 (London, 1954) pp. 587-592. The essential
theme of the Oxford Conference, as was stated in the first announcement
of it, was: ``The life and death struggle between the Christian faith
and the secular and pagan tendencies of our time.''
page 98
241 Jasper, op. cit., pp. 221-223.
242 ``The Churches Survey Their Task'', pp. 58-59.
page 99
243 Ibid., pp. 230-238. Cf. pp. 72-73 (the Report on the Church and Race).
244 See p. 259.
245 See p. 142.
page 100
246 Freudenberg, op. cit., p. 24.
247 A conference about the Refugee Problem, called by President Roosevelt,
was held at Evian, in July, 1938. It was attended by representatives
of 32 countries.
page 101
248 Archives of the World Council of Churches, Geneva.
249 See above, on p. 88 . ff.
250 See above, on p. 95.
251 Freudenberg, op. cit., p. 32.
252 ``Conversation entre le Dr.Visser 't Hooft, le Dr.Freudenberg et
le Dr.Barot, concernant les activites Cimade-wcc pendant la guerre''
(Archives of the World Council of Churches, Geneva; in French).
\begin{verbatim}
Footnotes PART III -------------------\end{verbatim}
page 107
253 See for Rev. Grueber's activities: pp. 40-41.
254 Max Krakauer, Lichter im Dunkel (Stuttgart, 1947; in German), passim.
255 Krakauer, op. cit., p. 131.
page 110
256 Hermelink, op. Cit., 651-652.
page 111
257 Ibid., p. 564-565.
Cf. Meier, op. cit., p. 104: ``Apparently Wurm did not protest publicly,
as he wanted to avoid providing amunition to the foreign press and thus
provoking the National-Socialist authorities, which would have blocked
the way of sending petitions in writing to the Government''.
258 Hermelink, op. cit., pp. 654-656.
259 Ibid., pp. 657-658. See for the angry reply of Dr. Lammers: Hermelink,
op. cit., pp. 700-702. The letter was dated March 3, 1944. Bishop Wurm
wrote another letter, dated Febr. 8, 1945, to Reichsstatthalter Murr,
on behalf of the partners in mixed marriages (Hermelink, op. cit., pp.
658-660).
page 113
260 ``Der ungekundigte Bund'', pp. 246-247.
The Message was dated Oct. 17, 1943, and published by ``The New York
Times, on August 4, 1944.
In the Lutheran Churches, the fifth Commandment is: ``Thou shalt not kill.
261 See below on p. 294.
262 Quoted by Friedman, op. cit., p 100.
263 Heinrich Grueber, Dona Nobis Pacem (Berlin, 1956; in German), p. 104.
page 114
264 ``Die Evangelische Kirche und die Judenfrage'', pp. 6, 10, 13 and 14.
265 See below, pp. 291-295.
page 115
266 see above, on pp. 111(note I ) and 113 (note I ) Cf. p. 40 (the fate of
Dr. Weissler).
267 Guenter Lewy, The Catholic Church and Nazi Germany (London, 1964),
pp. 23-294.
page 116
268 J.Tenenbaum, Race and Reich (New York, 1956), p. 292. Also see: Jacob
Robinson, And the Crooked Shall be Made Straight (New York, 1965),
p. 243; Hilberg, op. cit., p. 356.
page 118
269 The original test in: H.C. Christie, Den Norske Kirke I Kamp (Oslo, 1945;
in Norwegian), pp. 267-268. An English translation in: Bjarne Hoye and
Trygve M. Ager, ``The Fight of the Norwegian Church against Nazism'' (New
York, 1943), pp. 146-149.
page 119
270 See pp. 227-228.
271 ``The Spiritual Issues of the War'', No. 167, Jan. 14, 1943.
272 Christie, op. cit., p. 281.
page 120
273 Dr. J. Presser, Ondergang ('s-Gravenhage, 1965; two volumes; in Dutch;
an English edition is in preparation). Also see: Robinson, op. cit., pp.
240-243; Hilberg, op. cit., pp. 365-381; Abel J. Herzberg, Kroniek der
Jodenvervolging (Arnhem-Amsterdam, 1950; in Dutch), passim.
274 We shall write the name of this Church in capitals, in order to prevent
confusion with the ``Reformed Churches in the Netherlands''.
page 121
275 ``Interkerkelijk Overleg.'' Cf. H.C. Touw, Het Verzet der Hervormde Kerk
('s Gravenhage, 1946; in Dutch), 1, pp. 42-43, 138-141.
276 Cf. J.J. Buskes, Waar stond de Kerk? (Amsterdam, 1947; in Dutch),
pp. 77-87. Also see: Touw, op. cit., I, pp. 47, 85 , 373-375; Delleman,
op. cit., pp. 35-39.
277 Touw, op. cit., 11, pp. 259 --- 260.
278 Buskes, op. cit., p. 79.
pag 122
279 A moving exception was N. H. de Graaf. See for his protest: W. A. Visser
't Hooft, The Struggle of the Dutch Church for the Maintenance of the
Commandments of God in the Life of the State (London, 1944), pp. 16- 17.
A few days later Mr. de Graaf was arrested and sent to a concentration
camp. He did not return.
280 The full text in Touw, op. cit., 11, pp. 209-215.
281 Touw, op. cit., I, p. 392.
282 Ibid., 11, pp. 227-232. An English translation of this and most of the
other documents quoted in this chapter, in Visser 't Hooft, op. cit.,
passim.
page 123
283 Visser 't Hooft, op. cit., pp. 23-24.
284 Cf. Touw, op. cit., pp. 132-134; Buskes, op. cit., pp. 62- 63; Delleman,
op. cit., pp. 40, 78- 80.
page 124
285 Touw, op. cit., 11, p. 32; Visser 't Hooft, op. cit., pp. 26-27.
page 125
286 Cf. Buskes, op. cit., pp. 35-36; Touw, op. cit., 1, p. 171; Delleman,
op. cit., pp. 81-83.
287 Delleman, op. cit., pp. 42-44, 512-516.
page 126
288 Quoted in ``Hitler's Ten Year War on the Jews'', p. 244.
289 Visser 't Hooft, op. cit., pp. 42-45; Touw, op. cit., 1, pp. 388-392; 11,
pp. 66- 67; Delleman, pp. 92-100.
290 Ibid.
page 127
291 Touw, op. cit., 11, pp. 78- 83.
292 Visser 't Hooft, op. cit., p. 36; Touw, op. cit., 11, p. 84.
293 Touw, op. cit., I, pp. 395-397.
page 128
294 Buskes, p. 69.
295 Touw, op. cit., 11, p. 101.
page 129
296 Ibid., I, p. 404.
297 Louis de Jong, Jews and non Jews in Nazi-Occupied Holland (in: On the
Track of Tyranny, ed. Max Beloff; London, 1960), pp. 148-149.
Presser is of the opinion that the other Protestant Churches would not
have read out the telegram from their pulpits, if they had known about
the threat, but that they were not warned against doing so (Presser, op.
cit., 1, pp. 260-261). Wielek (in: H. Wielek, De Oorlog die Hitler won,
Amsterdam, 1947; p. 218) is of the same opinion.
The other Churches, however, did know about the threat. Cf. Henberg,
p. 134; Delleman, pp. 155-157; Buskes, p. 50. Also see the version of
the German General Commissioner Schmidt, quoted in Touw, 1, pp. 405-406.
298 Delta, Spring 1965, Vol. VIII/No. 1 (A Review of Arts, Life and Thought
in the Netherlands), pp. 28-29.
page 130
299 Touw, op. cit., 1, p. 173.
300 Ibid., 1, p. 423.
page 131
301 Visser 't Hooft, op. cit., pp. 52-55.
302 For practical results of this protest, see above, on p. 21. Also see:
L. de Jong, De Bezetting (Amsterdam, 1963; in Dutch), 111, pp. 30-31.
page 132
303 Visser 't Hooft, op. cit., pp. 56-58; Touw, op. cit., pp. 150-151.
304 Touw, op. cit., 11, pp. 155-156.
305 ``Delta'', p. 88.
page 133
306 Touw, op. cit., pp. 169-170. Also see: pp. 177-179.
page 134
307 Visser 't Hooft, op. cit., pp. 66-67; Touw, op.
cit., 1, p. 394; 11, pp. 161-169.
page 135
308 Visser 't Hooft, op. cit., p. 7.
309 Touw, op. cit., 1, pp. 371, 434.
310 Wielek, op. cit., p. 216.
311 Werner Warmbrunn, The Dutch under German Occupation 1940-1945
(London, 1963), p. 271.
312 Ibid., p. 279.
page 136
313 J. J. Buskes in the Protestant Weekly ``Hervormd Nederland'', May 1, 1966.
314 Touw, op. cit., 1, p. 660.
315 Hilberg, op. cit., p. 364. Cf. for the situation in France during the war:
Robert Aron, L'Histoire de Vichy (Paris, 1959; in French).
316 Hilberg, op. cit., p. 392.
317 Robinson, op. cit., p. 237.
page 137
318 Tenenbaum, op. cit., p. 280.
319 Chief Rabbi Kaplan in L'Arche, No. 1 10, April 1966, p. 26.
page 138
320 L. Poliakov, Harvest of Hatred (Pocket Edition, 1960), p. 251.
321 Cf. p. 145. Cf. Rev. Boegner's letter to Laval: ``Authorized to speak on
behalf of the Protestant Churches of the entire world, many of which have
already asked my intervention...'' (p. 146).
322 ``Les Eglises Protestantes pendant la guerre et l'occupation; Actes de
l'Assemblee Generale du Protestantisme Francais reunie a Nimes,
du 22 au 26 octobre 1945.'' (Paris, 1946; in French), p. 18.
323 Ibid., p. 23.
page 139
324 Ibid., p. 24.
page 140
325 Ibid., pp. 25-26.
page 141
326 Ibid., p. 26.
page 142
327 Ibid., pp. 27-28. The letter was submitted to Marshal Petain
on June 27, 1942.
328 A copy of the letter is in the Archives of the ``Rijksinstituut voor
Oorlogsdocumentatie'', Amsterdam; file: ``De Zwitserse weg''. For the
Oxford Conference, see pp. 97-99.
page 143
329 Cf. Henri Cadier in: La Chretiente au Creuset de L'epreuve, Vol. 11,
p. 631 (Geneva, 1947; in French).
page 144
330 Cf. Robinson, op. cit., p. 236; Hilberg, op. cit., pp. 407-408.
331 ``Les Eglises Protestantes...'', pp. 28-29.
332 ``Cimade'' (Comite Inter-Mouvement aupres des Evacues), the organization
brought into being during the second world war by the Protestant
Churches and the lay youth movements in France.
333 ``Les Eglises Protestantes...'', pp. 28-29.
page 145
334 Ibid., pp. 30-31.
page 146
335 Hilberg, op. cit., p. 409.
336 ``Les Eglises Protestantes...'', p. 31.
337 Ibid., pp. 33-34.
Cf. p. 279 in this book: ``Another endeavour to save lives failed.
The Ecumenical Committee for Refugees had, with the help of American
Christians, succeeded in obtaining entrance visas into the United
States for 1,000 Jewish children from France, but the occupation of
Southern France by the Germans foiled this plan.'' Also see: Donald A.
Lowrie, The Hunted Children (New York, 1963), pp. 218-228.
page 147
338 ``Les Eglises Protestantes...'', pp. 34-35.
339 Luke 10, 30-37.
page 148
340 Sami Lattes, L'Attitude de L'Eglise en France a L'Egard des Juifs pendant
la persecution (in: Les Juifs en Europe, p. 169).
page 149
341 L. Poliakov, L'Hostellerie des Musiciens. Quoted by David Knout in:
Contribution a L'histoire de la Resistance Juive en France (Paris, 1947),
pp. 107-109.
342 C.I.M.A.D.E.: See above on p. 14 (note 2).
343 Emile C. Fabre (Ed.), Les Clandestins de Dieu (Paris, 1968; in French),
p. 31.
page 150
344 Ibid., p. 31.
345 Ibid., p. 66.
346 Ibid., p. 33.
347 Ibid., p. 35.
348 Ibid., p. 117.
349 Ibid., p. 120.
350 Cf., p. 212 ff.
351 ``Les Clandestins de Dieu,'' p. 27; cf. p. 279. in this book.
page 151
352 Particulars taken from Tenenbaum, op. cit., pp. 301, 339.
353 Cf. Edmond Paris, Genocide in Satellite Croatia, 1941-1945 (Chicago, 1959).
354 Joseph Schechtman in: ``Hitler's Ten-Year War on the Jews'', pp. 99-100.
355 Ibid., p. 108.
page 152
356 See, however, the chapters in this book on Greece, Rumania and Bulgaria.
357 Tenenbaum, op. cit., p. 79.
358 Hilberg, op. cit., p. 42; Tenenbaum, op. cit., pp. 307-308.
359 Philip Friedman, Their Brothers' Keepers (New York, 1957), p. 109.
page 153
360 Michael Molho and Joseph Nehama, The Destruction of Greek Jewry, 1941-1944
(Jerusalem, 1965; in Hebrew), p. 101.
361 Mr. Moissis in a letter to me dated November 2, 1966.
362 The following particulars (unless other sources are mentioned) are taken
from: Ilias Venezis, Archbishop Damaskinos (Athens, 1952; in Greek);
chapter 34.
page 156
363 For the text of this memorandum in French, see: Michael Molho, In Memoriam
(Salonika, 1948), I, pp. 118-120. The text in Hebrew in: Molho-Nehama,
op. cit., pp. 106-107.
page 159
364 Molho-Nehama, op. cit., p. 142.
365 Tenenbaum, op. cit., p. 106.
page 160
366 Nathan Eck, New Light on the Charges Against the Last Chief Rabbi of
Salonica (in: Yad Vashem Bulletin No. 17, Jerusalem; December, 1965),
p. 14.
Cf. the ``Bericht eines aus Athen gefluechteten'' (General Zionist Archives,
Jerusalem, No. 841/44): ``Man muss gestehen, dass die Einwohnerschaft von
Athen sich menschlicher betragen haben als diejenige aus Saloniki.
367 Friedman, op. cit., p. 106.
page 161
368 Molho-Nehama, op. cit., p. 224.
369 Ibid., pp. 135-136. Cf. Friedman, op. cit., p. 107: ``Leaders of the Jewish
community insisted that Rabbi Barzilai take refuge, a notion he rejected
until Archbishop Damaskinos prevailed upon him to change his mind.''
370 ``The Situation of the Jews in Greece'' (Published by the World Jewish
Congress, New York, 1944), p. 7.
371 Asscher Moissis, La situation des Communautes juives en Grece (in: ``Les
Juifs en Europe"), p. 54. Quoted by Tenenbaum, op. cit., p. 310.
page 162
372 Mr Moissis in his letter to me dated November 2, 1966. Another information
from the same source: ``After the Germans had called the Chief Rabbi of
Volos, ordering him to take the necessary steps to facilitate the
deportations, the latter went to the Bishop of Volos asking his advice
and help. The Bishop advised him to abstain from collaboration whatsoever,
and helped him to go into hiding; the majority of the Jewish population
of Volos followed suit and thus was saved from deportation to Auschwitz.''
373 The pastors were followers of Grundtvig (1783-1872), the teachings of
whom in the opinion of Mrs. Yahil had a great influence on the views and
attitude of Christians in Denmark during the German occupation.
Cf. Dr. Leni Yahil, Test of Democracy, the Rescue of Danish Jewry in World
War II (Jerusalem, 1966; in Hebrew, with a summary in English), p. 25.
page 163
374 Yahil, op. cit., p. 33.
\begin{quotation}
Cf. Bishop Fuglsang-Damgaard in: ``Chr. Refslund \& M.Schmidt (Ed.), Fern Aar (Copenhagen, 1946; in Danish, p. 101: ``Our Jews urgently requested [during the first years of the war] that as far as possible there should be silence about the Jewish question''.\end{quotation}
375 Yahil, op. cit., p. 33.
376 Ibid.
377 Ibid., pp. 164-165. The full text of the article in: ``The Spiritual
Issues of the War'', No. 144, August 6, 1942.
page 164
378 ``Christians Protest Persecution'' (Religious News Service, ``The National
Conference of Christians and Jews'', New York), p. 14.
379 Boris Shub (Ed.), Hitler's Ten Year War on the Jews (New York, 1943),
p. 220.
380 Yahil, op. cit., pp. 164-165.
381 ``Christians Protest Persecution'', p. 14.
382 Harald Sandbaek and N. J.Rald (Ed.), Den danske Kirche UNDER BESAETTELSEN
(Copenhagen, 1945; in Danish), pp. 27-28. Cf. Hugh Martin (Ed.), Christian
Counter-Attack (London, 1943), pp. 74-75.
page 165
383 Robinson, op. cit., pp. 243-247; Yahil, op. cit., pp. 158-193.
384 Fuglsang-Damgaard, op. cit., p. 102 ff.
page 166
385 Ibid.
Cf. Yahil, op. cit., p. 148: ``Not only Svenningsen, however, was misled
by Best and believed in his promises that the Jews were not in danger,
but also Bishop Fuglsang-Damgaard came to the congregation, on Sept. 28th,
at 3 p.m., in order to reassure its leaders, telling them that he knew
from very reliable sources that the rumours were without foundation.
386 The Danish resistance movement consisted of ``study circles''. In Copenhagen
such circles were organized on professional lines --- architects, doctors,
clergymen \textit{etc}. The P.U.F. was the ``study circle'' of pastors. Not less than
90\%% of all pastors belonged to it. (Cf. Yahil, op. cit., p.160).
pag 167
387 Fuglsang-Damgaard, op. cit., p. 103.
388 Ibid., pp. 104-105.
389 Ibid., p. 105.
pag 168
390 Ibid., pp. 105-106.
391 Sandbaek and Rald, op. cit., pp. 21-22. Cf. ``Fem Aar'', 11, pp. 141-142.
pag 169
392 Yahil, op. cit., p. 166.
pag 170
393 Fuglsang-Damgaard, op. cit., p. 106
394 Ibid., p. 107.
page 171
395 ``International Christian Press \& Information Service, Geneva, No. 43,
December, 1943.
396 ``Fem Aar'', 11, p. 144.
page 172
397 Dr. Samuel Stefan Osusky, Sluzba Narodu (Bratislava, 1947; in Slovakian),
11, pp. 133, 136.
page 173
398 Cf. for the historical particulars: Hilberg, op. cit., pp. 458-475;
Tenenbaum, op. cit., pp. 318-321; the article of Dr. F. Steiner, ``La
situation des Juifs en Slovaquie'' (in: ``Les Juifs en Europe'',
pp. 216-220); and especially the comprehensive book of Dr.Livia
Rothkirchen, The Destruction of Slovak Jewry (Jerusalem, 1961), passim.
page 174
399 Osusky, op. cit., pp. 230-231.
page 175
400 Cf. for the historical data: Robinson, op. cit., pp. 258-265; Hilberg,
op. cit., pp. 485-509; Tenebaum, op. cit., pp. 312-317; also see the
comprehensive work of Theodore Lavi: ``Roumanian Jewry in World War II''
(in Hebrew; Jerusalem, 1965), especially pp. 11-13.
401 ``Hitler's Ten-Year War on the Jews'', p. 84.
402 Dr.Alexandre Safran, L'oeuvre de sauvetage de la population juive
accomplie pendant l'oppression nazie en Roumanie (in: ``Les Juifs en
Europe''; in French), p. 209. Cf. Lavie, op. cit., pp. 108-110.
page 176
403 Safran, op. cit., p. 209.
404 Ibid., p. 210.
page 177
405 Dr. Safran's secretary, Israel Lebanon, related about this meeting:
``Rabbi Safran quoted texts from the Old Testament and dwelt on the
relationship between Judaism and Christianity, \textit{etc}.'' (Lavie, op. cit.,
p. 110).
406 Safran, op. cit., pp. 211 --- 212.
page 178
407 Matatias Carp, Le martyre des Juifs de Roumanie (in: ``Les Juifs en
Europe"), p. 204.
page 179
408 International Press \& Information Service, Geneva, May 1942, No. 16.
409 Ibid.
410 For the ``German Christians'', see above, on p. 36.
page 180
411 Bishop Mueller's letter to me, dated Dec. 2, 1965.
412 Dr. Safran's letter to me, dated April 18, 1966.
page 181
413 Hilberg, op. cit., pp. 474-475.
page 182
414 Yad Vashem Archives, 013/7-1. On the original there is a marginal
note in handwriting: ``I have read and investigated and shall take into
consideration the contents of the above. 15. 11. 1940.''
415 B. J. Arditi, Les Juifs de Bulgarie sous le regime Nazi 1940-1944
(Tel-Aviv, 1962; in Hebrew), pp. 201-202.
page 183
416 Ibid., p. 92.
417 Robinson, op. cit., p. 258; Hilberg, op. cit., pp. 474-484 . Cf.
``Bulgarian Atrocities In Greek Macedonia and Thrace'', A Report of
Professors of the Universities of Athens and Salonica (Athens, 1945),
p. 47: ``The whole Jewish population of Eastern Macedonia and Western
Thrace, about 9,000 persons, were taken away in the night of the 3rd
of March, 1943... Three or four days later ...they were handed over to
the Germans.''
page 184
418 Yad Vashem Archives, 03/963, pp. 40-43 (in Hebrew).
419 Yad Vashem Archives, 03/1707 (in Hebrew).
page 188
420 Misho Leviev, Nashata Blagodarnost ("Our Gratitude'', in Bulgarian; Sofia,
1945), pp. 81-86. Eleven Metropolitans signed, according to their dioceses.
page 189
421 Arditi, op. cit., p. 202.
422 Hilberg, op. cit., p. 483.
page 190
423 Yad Vashem Archives, No. 03/1707 (in Hebrew). Mr.Moshonov was a goldsmith
who supplied the King's palace with jewelry and had free access to the
palace.
Also see: Ely Barouch, Iz Istoriata na Bulgarskoto Evrejstvo ("From the
History of Bulgarian Jewry'', Tel Aviv, 1960; in Bulgarian), p. 146:
IUGo, 'the Metropolitan began, 'and tell your people that the King
solemnly promised before the Prime Minister and before me, that the Jews
of Bulgaria will not be expelled from the country. Go and transmit to
them this information; reassure them and let they believe in the good
principle that the Lord never forgets his children.
Go in peace.'It was with those words that the Metropolitan Stephan received
the delegation of Rabbi Daniel Ben Zion, Rabbi Asher Hananel and Menachem
Moshonov, on May 24, 1943, when the Jews of Sofia were gathered together
in the synagogue Yutch-Bunar in order to beseech mercy and salvation from
God, as it had become known to them that Bulgarian Jewry would also be
deported to be murdered in Poland. Cf. Arditi, op. cit., pp. 216-217.
424 Cf. p. 186 in this chapter.
page 191
425 Literally: ``bake your head''
page 192
426 Solomon Samuel Mashiach in his article ``Who saved us?'', published in the
newspaper ``Narodno Delo (Sofia, No. 467, July 4, 1958; in Bulgarian).
Quoted by Barouch, op. cit., pp. 147-149.
427 Arditi, op. cit., p. 374. Cf. Matthew 7, 2.
428 Leviev, op. cit., p. 88.
Cf. Arditi, op. cit., p. 289: ``When the Metropolitan Kyril was convinced
that the local authorities and the representatives of the Government were
not inclined to help the Jews, he sent a telegram to the King, in which
he declared that he would cease to be the King's loyal subject and that
he would act as seemed right to him and according to the dictates of
his conscience as a religious man, if the instructions for expulsion
were not cancelled.''
page 193
429 Leviev, op. cit., pp. 88-89.
page 194
430 Testimony of Joseph Geron (in Hebrew); Yad Vashem Archives, No. 03/1707.
431 Cf. above, on p. 183 (note 2).
page 195
432 For the historical data in this chapter, see: Robinson, op. cit.,
pp. 265-269; Hilberg, op. cit., pp. 509-554; Tenenbaum, op. cit.,
pp. 321-332; Livia Rothkirchen, The Attitude of the Vatican and the
Churches in Hungary towards ``The Solution of the Jewish Question''
(in: ``Ha Ummah'' (The Nation), Quarterly, Jerusalem; in Hebrew), 1967,
No. 21, pp. 79-85.
433 For the first and second anti-Jewish law, see above, pp. 64-65.
434 Albert Bereczky, Hungarian Protestantism and the Persecution of the
Jews (Budapest, 1946), p. 10. Also see: Jeno Levai, Black Book on the
Martyrdom of Hungarian Jewry (Zurich, 1948), p. 25.
435 Dr. Mathe's letter to me, dated Aug. 24, 1967.
page 196
436 Bereczky, op. cit., pp. 12 --- 13.
page 197
437 Levai, op. cit., pp. 92- 93; Bereczky, op. cit., p. 14.
438 Ibid.
page 198
439 Bereczky, op. cit., p. 14; cf. Levai, op. cit p 93.
440 Levai, op. cit., p. 117; cf. Bereczky, op. cit., pp. 15-16.
441 Levai, op. cit., p. 117; cf. Bereczky, op. cit., pp. 16-18.
page 199
442 Levai, op. cit., p. 217.
443 Bereczky, op. cit., p. 16.
page 200
444 ibid., pp. 19-21; Levai, op. cit., 217-218.
page 201
445 Bereczky, op. cit., pp. 21 --- 24; Levai, op. cit., 218- 220. Bishop Ravasz
and the Lutheran Bishop Bela Kapi had together prepared the draft. The
Protest was signed by all the Bishops of the Reformed and the Lutheran
Churches (Levai, pp. 218, 220).
page 202
446 Levai, op. cit., pp. 220-221; Bereczky, op. cit., p. 24.
447 Archives of the World Council of Churches, Geneva. The report was dated:
Budapest, June 26, 1944.
page 203
448 Levai, op. cit., pp. 221 --- 222; Bereczky, op. cit., pp. 24- 26.
page 204
449 Levai, op. cit., p. 222; Berezcky, op. cit., pp. 27-28.
450 Levai, op. cit., p. 223; Bereczky, op. cit., p. 28.
451 Dr. Mathe's letter to me, dated Aug. 24, 1967.
452 Cf. p. 256.
page 205
453 Levai, op. cit., pp. 360-361; Bereczky, op. cit., pp. 34-35.
page 206
454 Levai, op. cit., p. 361; cf. Bereczky, op. cit., pp. 35-37.
455 Bereczky, op. cit., p. 37.
456 Ibid.
457 Rothkirchen, ``The Attitude of the Vatican...'', p. 85. Quoted from:
Erno Munkhcsi, Hogyan tortent' Adatok es okmhyok a magyar zsidesAg
Tragediej Ahoz (Budapest, 1947; in Hungarian), p. 146.
page 207
458 For the press censorship in Switzerland during the war, see: Dr. Carl
Ludwig, Die Fluchtlingspolitik der Schweiz seit 1933 bis zur Gegenwart
(Bericht an den Bundesrat zuhanden der eidgenossischen Rate, Zu 7347),
pp. 141, 142, 247, 289.
page 208
459 ``Schweiz. Evang. Pressedienst'' (E.P.D.), Nov. 12, 1941.
page 209
460 E.P.D. , Nov. 19, 1941. Cf. ``International Christian Press \& Information
Service'' (I.C.P.I.S.), Geneva, Nov., 1941. Also see: Alfred A. Hasler,
Das Boot ist voll (Zurich, 1968; second impression, in German),
pp. 131-133.
461 E.P.D., Nov. 19, 1941; cf. I.C.P.I.S., Nov., 1941.
page 210
462 E.P.D., Sept. 2, 1942. Cf. Hasler, op. cit., pp. 147-150.
463 E.P.D., Oct. 29, 1942.
page 211
464 Ibid., May 24, 1942.
page 212
465 Ibid., July 15, 1942. Cf. ``The Spiritual Issues of the War'', Aug. 6, 1942.
466 I.C.P.I.S., NO. 34, Oct. 1942.
467 Cf. the Report: ``Schweiz. Sammlung fur die Fluchtlingshilfe, Oct.-Nov. 1942''
(Erstattet von der Schweiz. Zentralstelle fur Fluchtlingshilfe), p. 15.
page 213
468 Cf. above, on p. 56.
469 Ludwig. op. cit., pp. 204-205.
470 Ibid., p. 209. Cf. Arieh Tartakower and Kurt R. Grossmann, The Jewish
Refugee (New York, 1944), p. 294: ``Thus the Council of the Federation
of Swiss Protestant Churches appealed to the Federal authorities in
August, 1942, urging that the right of asylum be not denied to non-Aryan
refugees who recently arrived in Switzerland, and that liberal methods
be applied to those who may yet come. Again, in September of that year,
when the wave of deportations of Jews from France, Belgium, and Holland
reached its crest, the Swiss National Protestant Church, in a pastoral
letter concerning a nation-wide fast which was read from every pulpit,
declared: '...We forsake our first love if we forget that our country
must remain, as far as possible, a haven of refuge for the persecuted and
refugees. To abandon this role is to betray our spiritual heritage, is 'to
lose our soul in order to gain the world?. In particular, we cannot remain
indifferent to the lot of the people of Israel, in whose midst our Saviour
was born and who are today the object of measures whose cruelty and
iniquity are the shame of our age...?''
471 Ludwig, op. cit., pp. 208-210; cf. E.P.D., August. 26, 1942; Hasler, op.
cit., pp. 138-139.
page 214
472 See above, on p. 210.
473 E.P.D., Sept. 2, 1942. Cf. Hasler, op. cit., pp. 122-125. Also see Ludwig,
op. cit., p. 373: ``In autumn 1942, when we had 10,000-12,000 refugees, it
was declared that the lifeboat was fully occupied and the possibility of
accepting refugees exhausted. At the end of the war Switzerland harboured
115,000 refugees.'' Cf. the reply of Federal Councillor von Steiger (Ludwig,
op. cit., pp. 393-394).
page 215
474 Ludwig, op. cit., pp. 222-224.
475 Cf. Ludwig, op. cit., pp. 214-222.
page 216
476 E.P.D., Oct. 29, 1942.
477 Cf. the Report ``Schweiz. Sammlung fur die Fluchtlingshilfe,
Oct.-Nov. 1942 ``, p.8.
478 ibid., p. 40. Cf. Hasler, op. cit., pp. 186-187. Alsosee: Ludwig, op.
cit., p. 228: ``The result of the collection (about Fr. 1,500,000.-)
organized by the Swiss Central Office for Aid to Refugees and vigorously
supported by the `Young Church', showed that a large proportion of the
Swiss people was moved by the fate of the refugees.
page 217
479 ``Schweiz. Sammlung fur die Fluchtlingshilfe...'', p. 31. For another
statement made by Prof. Karl Bart, see Hasler, op. cit., pp. 129-130.
480 Ludwig, op. cit., pp. 228-229.
page 218
481 See above, on pp. 214 --- 215.
482 Ludwig, op. cit., pp. 229-231.
483 ``Bericht des Schweiz. Kirchl. Hilfskommittee fur Evang. Fluchtlinge uber
das Jahr 1943'', p. 1.
484 Ludwig, op. cit., pp. 245-246.
485 E.P.D., June 16, 1943. Cf. Ludwig, op. cit., p. 245.
page 219
486 Ibid., Oct. 20, 1934.
page 220
487 Ibid., Nov. 11, 1943.
488 Ludwig, op. cit., p. 268.
489 Schweizerisches Evangelisches Hilfswerk fur die Bekennende Kirche
in Deutschland.
page 222
490 E.P.D., June 22, 1943.
491 The following books were published : ``Judennot und Christenglaube''
(Zurich, 1943); ``Soll ich meines Bruders Huter sein?'' (Zurich, 1944);
``Aus Not und Rettung'' (Edited by Paul Vogt, Zurich, 1944). Some of the
brochures published were: ``Das Heil kommt von den Juden'' (Oktober,
1938); ``Thesen zu den Nachkriegsfragen der Fluchtlingshilfe''; ``Vergesst
die evangelische Freiplatzaktion nicht!'' (1944); ``Nicht furchten ist
der Harnisch'' (a circular letter sent monthly to regular supporters of
the Refugee Aid); ``Fluchtlingshilfe als christliche Diakonie'' (by Paul
Vogt, 1944). Rev. Vogt also pleaded the cause of the refugees in sermons
and lectures (Hasler, op. cit., pp. 150-152, 206, 227-229, 301).
492 ``Freiplatzaktion''.
493 ``Bericht des Schweiz. Kirchl. Hilfskomitees fur Evang. Fluchtlinge uber
das Jahr 1943'', passim.
page 223
494 ibid., pp. 6-7.
495 E.P.D. , Oct. 18, 1944.
496 Archives World Council of Churches, Geneva (file CCJP).
497 Also see: E.P.D. , July 5 , 1944: ``Service of Intercession and Mourning
for the persecuted Jews in Hungary''; E.P.D. , July I2, 1944: ``Circular
letter of the Church Council of Thurgau''; ``The Committee of the Synod of
Waadtland, Declaration about the persecution of the Jews''; and E.P.D.,
July 19, 1944: ``A Service of Intercession at Schaffhausen''.
page 224
498 I.C.P.I.C., July 28, 1944. Cf. E.P.D., July 12, 1944. The Press Department
of the German Foreign Office circulated a confidential report ("The Church
Council of Zurich condemns the persecutions of the Jews in Hungary'') dated
July 10, 1944. (Randolph L.Braham, The Destruction of Hungarian Jewry,
New York, 1963; 11, p. 770).
page 225
499 E.P.D. July 19, 1944.
500 Ibid.
page 226
501 Ibid., Aug. 2, 1944.
page 227
502 ``The Spiritual Issues of the War'', December 17, 1942, No. 163.
page 228
503 ``International Christian Press and Information Service'', Dec. 12, 1942,
No. 44-45. Cf. ``Spiritual Issues'', Dec. 12, 1942, No. 162;
and ``The New York Times., Febr. 2, 1943.
504 ``International Christian Press and Information Service'' (I.C.P.I.S.).,
Dec., 1942, No. 44-45; ``Spiritual Issues... ``, Dec. 17, 1942.
page 229
505 I.C.P.I.S., Dec., 1942, No. 44-45.
506 Ibid.
page 230
507 ``Spiritual Issues...'', Dec. 17, 1942. Also see: ``Nordiska Roster mot
Jude-forfoljelse och Vald'' (in Swedish; Documents and Commentaries,
edited by Judisk Tidskrift; Stockholm, 1943), p. 17.
508 ``Dagens Nyheter'', Dec. 5, 1942.
The Free Churches' Co-operation Committee represents the Missionary
Society, the Baptist Church and the Methodist Church in Sweden.
page 232
509 ``Nordiska Roster'', pp. 15-16.
page 233
510 Dr. Leni Yahil, Test of Democracy (Jerusalem, 1966; in Hebrew, with a
summary in English), pp. 228-229.
page 234
511 I.C.P.I.S., May 1944, No. 21.
512 Cf. Eberhard Bethge, Dietrich Bonhoeffer (Muenchen, 1967), p. 415.
513 Livia Rothkirchen, The Destruction of Slovak Jewry (Jerusalem, 1961),
p. XLIX. The full text: ``Ministerprasident Tiso zeigte mich soeben mit
der Bemerkung, `was seine einzelne Leute denken', ein Schreiben des
protestantischen Erzbischofs von Uppsala an den Slovakischen
Staatsprasidenten.
Der Schreiber interveniert `fur die armen judischen Bruder' und bittet,
da der Slovakischen Staat unter den derzeitigen Verhiltnissen keine
humane Behandlung gewahrleisten konne, den in der Slovakei konzentrierten
Juden den Uebertritt auf ein neutrales Gebiet zu ermoglichen.'' (Files of
the German Foreign Ministry, YW/AA-K-327, Inland 11, Geheim, 571-K-2 13007).
page 235
514 William Simpson, Jews and Christians To-day (A Study in Jewish and
Christian Relationships), London, 1940.
515 Reports to the General Assembly, 1940, p. 572.
page 236
516 Reports, 1941, p. 555.
517 Reports, 1942, p. 437.
518 Minutes of the Assembly, Thursday, June 11th, 1942.
page 237
519 Ernest Hearst, The British and the Slaughter of the Jews-(I); in: The
Wiener Library Bulletin, Vol. XXI, No. I , p. 32.
520 Jasper, op. cit., p. 155.
page 238
521 ``The Spiritual Issues of the War'' (Bulletin published by the Religious
Division of the Ministry of Information, London), No. 155, Oct. 22, 1942.
Cf. Freudenberg, op. cit., p. 10.
522 ``The Spiritual Issues...'', No. 155.
page 239
523 ``The Times'', Oct. 30, 1942. This and the following articles from ``The
Times'' are taken from the ``Podro-collection'' (in ``The Jewish Historical
Archives'', Hebrew University, Jerusalem).
page 240
524 Hearst, op. cit., pp. 35-36. The two Archbishops of the Church of England
and twenty-four Bishops (Durham, London, Winchester and the next twenty-one
in order of appointment to a diocese) are members of the House of Lords.
525 Jewish Chronicle, Dec. 11, 1942.
526 ``The Life of Faith'' (Weekly), Dec. 16, 1942.
page 241
527 Keesing's Contempory Archives, Dec. 12-19, 1942 , p. 5506.
page 242
528 ``Spiritual Issues...'', No. 164, Dec. 24, 1924.
529 ``Church of England Newspaper LONDON'', Jan. 29, 1943. A similar statement
was issued by the Executive Council of the World Evangelical Alliance
(Ibid., Febr. 5, 1943). Cf. ``Jewish Chronicle'', Jan. 29, 1943, p 1.
page 244
530 ``Spiritual Issues...'', No. 176, March 18, 1943. Cf. ``The Times'',
March 19, 1943.
531 ``The Times'', Dec. 31, 1942.
page 245
532 ``Spiritual Issues...'', No. 166, Jan. 7, 1943.
533 ``Jewish Chronicle'', Jan. 8, 1943.
534 Ibid., Jan. 29, 1943, p. 5. Cf. p. 10: ``Derby Demands Sanctuary for
Persecuted'', and: ``Mayor of Huddersfield calls Protest Meeting''.
535 ``Jewish Chronicle'', Febr. I2, 1943, pp. 1, 12.
page 248
536 Cf. Luke 10, 30-37 (the Parable of the Good Samaritan).
537 ``Parliamentary Debates House of Lords'', Vol. 126, No. 41, pp. 811-821.
page 249
538 Luke 17, 1.
539 Isaiah 57, 14.
540 ``Parliamentary Debates House of Lords'', Vol. 126, No. 41, pp. 832-841.
page 250
541 A copy of this letter is in the Archives of the World Council
of Churches, Geneva.
542 ``Unity in Dispersion'', pp. 164-165.
See for the Bermuda Conference: ``The Wiener Library Bulletin'', Vol.
xv (1961), No. 3, pp. 44-47. Also see: Morse, op. cit., pp. 43-64;
Tartakower and Grossmann, op. cit., pp. 420-428.
page 251
543 ``Spiritual Issues...'', No. 181, April 22, 1943.
544 Reports to the General Assembly, 1943, p. 338.
page 252
545 ``The Assembly'', May 1943, p. 170.
546 ``Minutes of the Assembly'', Thursday, June 10th, 1943.
page 253
547 ``The Times'', May 18, 1943.
page 254
548 ``The Times'', May 22, 1943.
549 Jasper, op. cit., pp. 156-157.
page 255
550 ``The New York Times'', Dec. 9, 1943.
551 Reports to the General Assembly, 1944, p. 384. The italics are mine.
552 ``The Times'', June 15, 1944.
page 256
553 ``The Spiritual Issues... ``, July 13, 1944. Cf. R.L. Braham,
The Destruction of Hungarian Jewry (New York, 1963), 11, p. 343.
554 See p. 257.
555 Ibid.
556 Federal Council Bulletin, January, 1940.
page 258
557 Department of Information of the National Council of Churches of Christ
in America. The statement was prepared by the Commission on Christian
Social Action and adopted by the following action: ``General Synod approves
Section IV and declares its condemnation of anti-Semitism. It urges upon
the members of the Church in the name of Christ to regard our Jewish
brethren according to the standards of Christian ethics.''
558 The New York Times, Dec. 13, 1940.
page 259
559 Federal Council Bulletin, October, 1941, p. 6.
page 260
560 Freudenberg, op. cit., p. 53.
561 Federal Council Bulletin, Febr. 1942, p. 7.
page 261
562 Dr. Riegner's letter to Dr. Visser 't Hooft, April 14, 1965 (Archives of
the World Council of Churches, Geneva).
563 Federal Council Bulletin, January, 1943. The italics are mine.
page 262
564 The New York Times, Jan. 1, 1943.
565 Federal Council Bulletin, February, 1943.
page 263
566 ``Unity in Dispersion'', a History of the World Jewish Congress (New York,
1948), pp. 162-163. The Archbishop of Canterbury described, in his message
to the meeting, the Nazi extermination of the Jews as ``the most appalling
horror in recorded history'' (Morse, op. cit., p. 47).
page 264
567 The New York Times, March I7, 1943. Cf. Federal Council Bulletin, April,
1943, p. 15.
568 See above, on p. 247.
569 See pp. 276-277.
570 See above, on p. 250.
page 265
571 The New York Times, May 2, 1943. Also see the article ``Day of Compassion
Praised by Rabbis'' (ibid., May 2 , 1943).
572 The New York Times, Oct. 21, 1943.
page 266
573 New York Herald Tribune, Nov. 3, 1943.
page 267
574 The New York Times, Dec. 12, 1943.
575 Ibid., Jan. 16, 1944.
576 New York Herald Tribune, Oct. 14, 1944.
page 268
577 Cf. Alfred Klausner in the monthly ``American Lutheran'', Febr. 1965, p. 16:
...In the course of research through almost all Lutheran publications in the
thirties and forties I have found no direct condemnation of the persecution
of the Jews in Germany. . .''
578 In: ``American Lutheran'', Nov. 1964, pp. 13.
579 Charles Y. Glock and Rodney Stark, Christian Beliefs and Anti-Semitism
(New York, 1966), passim.
page 269
580 See above, p.84.
581 See pp. 274-277
page 270
582 Cf. W.A. Visser 't Hoofd, in: Ruth Rouse and Stephen Charles Neill
(Ed.), A History of the Ecumenical Movement 1517- 1948 (London, 1954),
p. 710 ff.
page 271
583 Archives of the World Council of Churches, Geneva. The original is in German.
page 272
584 Ibid.; the original is in French.
page 273
585 Ibid.; the original is in French. Dr. Visser 't Hooft believes that no
written reply was received from the International Red Cross, but he had
several discussions with Dr. Burckhardt in which the latter told him what
the Red Cross had tried to do unofficially (Communication to me from
Dr. Visser 't Hooft).
586 Communication to me from Dr. Visser 't Hooft.
For the contacts with the ``Kreisau Circle''. see: G.van Roon, Neuordnung im
Widerstand (Munich, 1967), pp. 142, 146, 190, 247, 302, 308-309, 312,
330-332. For the contacts with Dietrich Bonhoeffer, see: Eberhard Bethge,
Dietrich Bonhoeffer (Munich, 1965), pp. 243, 291, 726, 728, 818-819, 824-
835, 848-850, 859, 861, 867, and 1004.
587 Dr. Riegner's letter to Dr. Visser 't Hooft, dated April 14, 1965
(Archives of the World Council of Churches, Geneva).
page 274
588 Dr. Riegner's letter to me, dated Nov. 6, 1967.
589 Dr. Riegner to Dr. Visser 't Hooft, April 14, 1965.
Cf. Arthur D. Morse, While Six Million Died (London, 1968), pp. 3-22.
page 275
590 Archives WCC, Geneva. Marginal note: ``Date? Probably March, 1943''.
591 Archives WCC, Geneva.
The contents of the telegram sent by Dr. Riegner to Mr. Silverman were as
follows: ``Most anxious about destiny Hungarian Jewry the only important
section European Jewry still in existence because of recent political
developments stop suggesting world wide appeal of Anglo-Saxon personalities
non-Jewish and Jewish including chiefs of Protestant Catholic Churches
to Hungarian people warning them not to admit application of policy of
extermination of Jews by German butchers or Hungarian quislings and to
help Jews by all possible means in order to prevent their falling into
hands of Germans stop warning should insist upon fact that attitude
Hungarian people towards Jews will be one of the most important tests
of behaviour which Allied Nations will remember in peace settlement after
war stop similar broadcasts should be made every night in Hungarian
language during the next weeks. Geneva, March 21, 1944.'' (Archives wcc,
Geneva).
page 276
592 For the results --- or rather: the lack of results --- of the Bermuda
Conference, see above, p. 250.
page 277
593 Archives WCC, Geneva.
page 278
594 Cf. ``Conversation entre le Dr. Visser 't Hooft, le Dr. Freudenberg et
le Dr. Barot, concernant les activitCs Cimade-wcc pendant la guerre''
(Geneva, December 14, 1965; mimeographed; in French).
595 Dr. Hans Fraenkel, Die Kirche im Krieg (unpublished manuscript; archives
WCC, Geneva), p. 186.
596 Dr. Visser 't Hooft in ``Conversation...'' (see above, note 1).
page 279
597 Report on Ecumenical Refugee Work since 1939 (Archives WCC, Geneva),
p. 2. Cf. above, on p. 150.
598 OEKUMENISCHER AUSSCHUSS FUR FLUCHTLINGSHILFE, Jahresbericht 1942
(Archives WCC, Geneva), p. 5.
599 Cf. Herbert Ford, Flee the Captor (The Story of the Dutch-Paris
Underground and its compassionate leader John Henry Weidner), Nashville,
1966. For the part played by Dr. Visser 't Hooft, see: pp. 79, 85, 95, 97,
199-201, 208, 225 , 227, 277, 279, 340 and 349.
page 280
600 ``Rapport van de Commissie van Onderzoek inzake het verstrekken van
pakketten door het Rode Kruis en andere instanties aan Nederlandse
politieke gevangenen in het buitenland gedurende de bezettingstijd
alsmede inzake het evacueren van Nederlandse gevangenen kort voor en
na het einde van de oorlog'' (Den Haag, 1947; in Dutch), p. 111.
601 Ibid., p.112.
602 Ibid., p.114-115.
page 281
603 I.C.P.I.S. (Intern. Christian Press and Information Service), Geneva,
No. 26, June 1944.. The statement was also published in ``Jewish News'',
London, July 18, 1944, p. 224; and in ``Basler Nachrichten'', June 29, 1944.
page 282
604 Cf. Karl Stadler, Das einsame Gewissen (Vienna, 1966; in german),
pp. 262-263. See for the persecutions in Austria: Herbert Rosenkranz,
``The Anschluss and the Tragedy of Austrian Jewry 1938-1945''; (in:
Josef Fraenkel (Ed.), The Jews of Austria (London, 1967), pp. 479-546.
605 Reformiertes Kirchenblatt fur Osterreich, March, 1966, p. 4.
page 283
606 Cf. Betty Garfinkels, Les Belges face a la persecution raciale 1940-1944
(Bruxelles, 1965; in French), pp. 74-75, 100. Also see: Fernand Barth,
Presence de l'Eglise (La Belgique sous l'occupation), Geneva, pp. 82, 89.
page 284
607 Cf. the ``Jewish Telegraphic Agency'' (Zurich, July I, 1942): ``A systematic
campaign against the Christian Churches, attacking them for their attitude
towards the Jews, has been launched in the Czech Protectorate by the
`Aryan Society', according to the Prague newspaper `Ceske Slovo'.
It is serious, the paper declares, that the clergy of all churches mostly
keeps silent about the Jews. The reason why the greater part of the clergy
are not opposed to the Jews is that there exist personal and dogmatic
ritual relations between the Church and Jewry. The Christian faith, the
paper demands, must be purged of its Jewish ingredients. Baptisms of Jews
must be declared invalid and the Old Testament must be purged of everything
smuggled into it by Rabbi interpreters.''
608 Archives of the World Council of Churches, Geneva. The statement was
dated Dec. 7, 1945.
page 285
609 Tenenbaum, op. cit., p. 339.
610 The ``Polish-Catholic Church'' does not accept the authority of Rome;
it is a member of the World Council of Churches.
page 286
611 Cf. Philips Friedman, Ukranian-Jewish Relations during the Nazi Occupation
(in: Yivo Annual of Jewish Social Science, New York, 1958/1959, Vol. XII,
pp. 290-294); also see: Philips Friedman, Their Brothers keepers,
pp.133-136.
612 Robinson, op. cit., p. 292.
613 The Rev. Esko Rintala, Secretary of the Archbishop of Finland, in his letter
to me dated Febr. 21, 1966.
614 Some literature: Giovanni Miegge, L'Eglise sous le joug fasciste (Geneva,
1946; in French). Einaudi (Ed.), Lettere do condanati a morte della
Resistenza Italiana (Torino, 1952; in Italian); Prearo, Terra Ribelle
(Torino, 1948; in Italian); Borgna, La Resistenza nel Pinerolese (Pinerolo,
1965; in Italian).
page 287
615 Friedman mentions that, according to Jewish survivors, Ukrainian priests
both rescued and helped Jews. Ukrainian Baptists in Volhynia helped the
Jews and in part also concealed them. Cf. Philip Friedman, Ukrainian-
Jewish Relations During the Nazi Occupation (in: Yivo Annual of Jewish
Social Science, Vol. XII, p. 294).
page 289
616 W.A. Visser 't Hooft, The Ecumenical Movement and the Racial
Problem, p. 40.
617 Ethics of the Fathers, 11, 5.
618 Leuner, op. cit., p. 16.
\begin{verbatim}
Footnotes Appendix I --------------------\end{verbatim}
page 292
619 ``The Relationship of the Church to the Jewish People'' (Geneva, 1964;
mimeographed); pp. 48-52; Cf. Heydenrich, op. cit., pp. 248-254.
620 Heydenreich, op. cit., pp. 256-257. Cf. the comment of Rev. Niemoeller
(Ibid., pp. 257-258).
page 293
621 Frank-Wilkens, Ordnungen und Kundgebungen der Vereinigten Evangelisch-
Lutherischen Kirche Deutschlands (Berlin/Hamburg, 1966; second
imprint), p. 203.
622 Heydenreich, op. cit., pp. 261-262. Prof. D. Gollwitzer criticized
this statement as being far too weak (ibid., pp. 262-264).
623 Heydenreich, op. cit., pp. 264-265.
page 294
624 ``The Relationship of the Church to the Jewish People'', pp. 73-76. Also
see: ``Wiener Library Bulletin'', xv, 1961, No. 3, p. 45.
page 295
625 ``Wiener Library Bulletin'', XVII, 1963, No. 3, p. 39.
626 ``The Relationship of the Church to the Jewish People'', pp. 78- 79.
page 296
627 Quarterly Newsletter from the World Council of Churches' Committee on
the Church and the Jewish People, March 1967, p. 17.
628 The Interpreter (published by the London Diocesan Council for Christian-
Jewish Understanding), August, 1964, p. 2.
page 297
629 ``Reports to the General Assembly'', 1945, p. 389.
630 ``Reports'', 1947, p. 448.
631 ``Reports'', 1953, p. 463.
632 ``Reports'', 1957, p. 528. 633 ``Reports'', 1962, p. 544.
page 298
634 ``The Relationship of the Church to the Jewish People,'' p. 87.
page 299
635 W.A. Visser 't Hooft (Ed.), The First Assembly of the World Council of
Churches (London, 1949), pp. 160-166.
636 W.A. Visser ?t Hooft (Ed.), The Third Assembly of the World Council of
Churches (Second Impression; London, 1962), p. 148. Cf. the interesting
discussion which preceded the adoption of the resolution (pp. 148-150).
page 300
637 ``The Relationship of the Church to the Jewish People'', pp. 83-84.
The total membership of the Churches affiliated to the Lutheran World
Federation is 52,762,379.
page 301
638 ``Reports and Recommendations of the International Conference of
Christians and Jews, Seelisberg, 1947'' (Published by the Intern.
Council of Christians and Jews), pp. 14-16.
In February, 1961, the ``International Consultative Committee of
Organisations for Christian-Jewish Co-operation'' was established.
For the history of the ``International Council of Christians and Jews''
see Rev. W.W. Simpson, Co-operation between Christians and Jews, Its
Possibilities and Limitations; in: Gote Hedenquist (Ed.), The Church
and the Jewish People (London, 1954), pp. 117-142.
page 302
639 The following publications were consulted:
``The World Alliance of Reformed Churches'' (Published by The World
Presbyterian Alliance, Geneva, 1964); ``Lutheran Directory'', Supplement
1966 (Published by the Lutheran World Federation, Geneva);
J. Grundler, Lexikon der Christlichen Kirchen und Sekten (Vienna, 1961;
in German), Vol. 11;
Guy Mayfield, The Church of England (Oxford, 1958);
Stephen Neill, Anglicanism (London, 1958);
Timothy Ware, The Orthodox Church (Pelican Books, 1963);
J. Meyendorff, The Orthodox Church (London, 1962);
Ruth Rouse and Stephen Charles Neill (Ed.), A History of the Ecumenical
Movement 1517-1948 (London, 1954);
Figures about the Churches in the United States were received from the
Department of Information of the National Council of Churches of Christ
in the U.S.A.
***** End of footnotes ******
GJS, dec 2004
\gutchapter{INFORMATION ABOUT THE AUTHOR: }
John Martinus Snoek born 1920, studied theology at the Free University in Amsterdam 1949-1953. Worked and lived with his family 11 years in Israel, (1958-1969) where he served for the Church of Scotland as minister in Tiberias. From 1970-1975 he worked as secretary of the Committee of the Church and the Jewish people with the World Council of Churches in Geneva.
Publications:
1. In English THE GREY BOOK. (1969) see PG e-book \#14764
2. In Dutch: The Dutch Churches en the Jews 1940-1945. (1990)
ISBN 90242 0949 8 NUGI 631
3. In Dutch: Sometimes, One has to show One's Color (1992)
Both the Grey Book and the Dutch Churches 1940-1945 are prepared for Gutenberg eText by his nephew Ge J. Snoek, errors and remarks please mail to: [email protected].
\gutchapter{End of the Project Gutenberg EBook of The Grey Book, by Johan M. Snoek}
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\def\citeauthor{~Archana~G,
~Kusu~Susan~Cyriac,
~Aliya~Kouser~N}
\def\citetitle{Role of COX-2 and ROS in UV-B Induced Skin Carcinogenesis}
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\def\RunningAuthor{Archana G et al.}
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\articleType{Review Article}
\receivedDate{10 May 2021}
\acceptedDate{20 May 2021}
\revisedDate{18 May 2021}
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\copyrightYear{2021}
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\title{Role of COX-2 and ROS in UV-B Induced Skin Carcinogenesis}
\author{Archana~G\textsuperscript{*},
Kusu~Susan~Cyriac,
Aliya~Kouser~N~\\[5pt]{Department of Pharmacology, Karnataka College of Pharmacy, Thirumena Halli, Hegde Nagar Main Road, Bangalore-560064, Karnataka, India}}
\begin{abstract}
Non-melanoma skin tumors are among the cutaneous basal cell and squamous cell tumors (cSCCs), are the most predominant malignancies. While UV-stimulated p53 transformations give off an impression of being an early and essential occasion in the enhancement of skin tumors, another significant provider is the over-expression of cyclooxygenase-2 (COX-2). Oxidative pressures elevate with the moving inflammatory cells, are firmly related with cancer or malignant advancement, and have been demonstrated to be related with beginning, advancement, or movement measures during multistage carcinogenesis. UV generation of COX-2 illustration and Prostaglandin E\ensuremath{_{2}} (PGE\ensuremath{_{2}}) creation is thought to advance skin carcinogenesis, as well as add up to even the most initial stages of UV-instigated skin damage. ROS (Reactive Oxygen Species) - intervened DNA injury assumes participation in the induction of carcinogenesis as well as in dangerous malignant alteration, and it might signify a significant donor in the pathogenesis of human carcinogenesis. The induction of COX-2 expression by acute UV exposure and constitutive up-regulation of COX-2 in UV-induced benign and malignant tumors leads to increased PGE\ensuremath{_{2}} production. ROS present inside the cells which are intracellular signaling cascades perform a function such as secondary messengers where they induce and maintain the oncogenic phenotype of cancer cells; however, cellular senescence and apoptosis can also be induced by ROS, and hence they also, therefore, function as anti-tumorigenic species.
\end{abstract}\def\keywordstitle{Keywords}
\begin{keywords}Cyclooxygenase-2,\newline Reactive Oxygen Species (ROS),\newline Cutaneous Basal Cell and Squamous Cell Tumors (cSCCs),\newline Ultraviolet A \& B,\newline Prostaglandin E2
\end{keywords}
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\makeatletter{\Large$^\ast$}Corresponding Author\par Name:\ Archana~G~\\ Phone:\ 8105909232~\\ Email:\ [email protected]
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\section{Introduction}
Skin is usually a soft layer, outer tissue which is flexible that covers the entire body of any animal. The main 3 functions of the skin are protection, regulation, and sensation. Skin plays a part in shielding the body against many pathogens. Their various other functions include insulation, body temperature maintenance, stimulation, and producing Vitamin D folates. Skin acts as a frontline defense from various exterior factors or exterior stress, which may include various environmental factors that may cause damage to the skin, genetic variations, and contagious substances that, may cause an unwanted impression on the integrity of the skin. Non-melanoma skin tumors are among the cutaneous basal cell and squamous cell tumors (cSCCs), are the most predominant malignancies. Danger factors for sCCSs incorporate high increasing UV exposure, hereditary inclination, persistent swelling, predecessor ionizing radiation, immunosuppression \unskip~\cite{1114320:22298821}. There is significant proof recommending the basic association of oxidative pressure in carcinogenesis. Subjection to bright ultraviolet (UV) light is the major etiologic factor prompting the improvement of the cutaneous squamous, furthermore basal cell carcinomas and is likewise a dangerous factor for melanomas. Ultraviolet radiation-induced DNA injury causes a transient enhancement in p53 protein stable-state levels, which bring about the outflow of the cell cycle inhibitor p21 and impermanent development capture. If cells cannot fix the DNA harm, p53 stimulates apoptosis. As an alternative, if DNA harmed cells go through propagation, a consequence of lasting transformation can be seen. Repetitive UV contact leads to a collection of DNA harm and p53 transformations. While UV-stimulated p53 transformations give off an impression of being an early and essential occasion in the enhancement of skin tumors, another significant provider is the over-expression of cyclooxygenase-2 (COX-2) \unskip~\cite{1114320:22298825}.
Oxidative pressure is an event caused by a difference between the manufacture and gathering of oxygen receptive species (ROS) in cells and tissues and the ability of a biological system to detoxify these responsive items. ROS production surpasses the cell's capacity to metabolize and detoxify them; a state of ``oxidative stress'' arises \unskip~\cite{1114320:22298828}. Normal ingestion of radical scavengers has been appeared to ensure against malignancy or cancer advancement in experimental animal models, as well as in epidemiological investigations in peoples. Oxidative pressure elevates with the moving inflammatory cells, is firmly related to cancer or malignant advancement. Oxidative pressure has been demonstrated to be related to beginning, advancement, or movement measures during multistage carcinogenesis \unskip~\cite{1114320:22298818}. Cancer growth progression is a multi-stage method categorized by the total activity of various changed cell measures including those of replication, angiogenesis, apoptosis, metastasis, etc. These incorporate synthetic chemical substances, x-radiation, ultraviolet radiation, singlet oxygen which are mediated by photosensitizers, and direct electron relocate that doesn't include the contribution of ROS \unskip~\cite{1114320:22298830}.
\textbf{Animals and UV Irradiation}
Hairless SKH-1 mice of around 3\ensuremath{\pm}4 weeks old were bought and were utilized at 8 weeks old. Upon coming, the animals were housed in an environment-controlled quarter (22\ensuremath{\pm}1\ensuremath{^\circ}C and at 50\% humidity) with a 12 hr light and 12 hr dark cycle in yellow fluorescent lights. The mice were permitted free allowance to drinking water and standard feed regimen and were noticed every day during UV irradiation. The UV apparatus comprises 8 FS40 sunlamps, an IL- 1400 radiometer, and an attached UVB photometer. The spectral irradiance for the UV lamps was 280\ensuremath{\pm}400nm, 80\% of which will be in the UVB region and the remaining 20\% will be in the UVA region. The high intensity of the beam source was 297nm [Figure~\ref{f-d0f3ff9f80c8}]. The effluence at 60cm from the dorsal side of the mice was 0.48\ensuremath{\pm} 0.50 mJ/cm\ensuremath{^{2}}/s. The animals were kept in a separate compartment in an open plastic enclosure on a revolving base to annul any distinctions impact across the UV lights.
\bgroup
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\centering \makeatletter\IfFileExists{images/04f61430-2620-4159-a666-ba0081279046-upicture1.png}{\includegraphics{images/04f61430-2620-4159-a666-ba0081279046-upicture1.png}}{}
\makeatother
\caption{\boldmath {Ultraviolet A, B \& C Wave length}}
\label{f-d0f3ff9f80c8}
\end{figure}
\egroup
The animals were provided with the controlled or investigational diets for 6 weeks when they were illuminated with UV radiation 3 times a week, with a starting dose of 90mJ/cm\ensuremath{^{2}} which was enhanced by around 25\% weekly, up to 220mJ/cm\ensuremath{^{2}}. During the tumor study, 30 hairless SKH-1 mice were put on control or medication-containing regime 1 week before the beginning of the UV illumination procedure. The animals were UV illuminated thrice a week by using the starting dose of 90mJ/cm\ensuremath{^{2}} the beginning or 1\ensuremath{^{st}} week, continued by a weekly elevation of 25\% until 275mJ/cm\ensuremath{^{2}} was attained. This procedure promotes skin tumors in hairless SKM-1 mice inside the period of 9 weeks. Every week tumor was tallied after the development of the first tumor and this procedure was continued up to the end of the experimentation period after 25 weeks. The tumor information is conveyed both as assortment (i.e., mean number of tumors per mouse). At the end of the trial period, the widths of the tumors were estimated and the tumors were allotted to either the 1 to 3mm size group or the \textgreater\ 3mm group, and the frequency of tumors in these groups was determined. Arbitrary tumors were then prepared for histological examination or utilized for the separation of RNA and protein \unskip~\cite{1114320:22298830}.
\textbf{UV and Skin Cancer}
Chronic subjection of these animals to UVB illumination prompts the enhancement of the non-malignant epidermal tumors, the majority of which become Squamous cell tumors. Severe subjection of SKH-1 mice to the UVB prompts the manufacture of DNA injury in epidermal cells. Chronic or several UVB contacts leads to an enhancement in epidermal expansion and thickness and can also prompt p53 alteration and/ or allelic loss. This information exhibit that UV-stimulated transformations and alterations in p53 manifestations are early occasions in UV-initiated skin carcinogenesis and are essential in cancer development \unskip~\cite{1114320:22298816}.
Assimilation of UV beam by atoms in the cells brings about the production of reactive oxygen species (ROS), which can cause oxidative DNA injury. DNA damage brings about an elevation in the number of epidermal cells with wild-type p53 articulation. Employment of inflammatory cells into the dermis starts around 4 hrs after the UVB contact and neutrophil penetration stays high for a few days. UV initiation of COX-2 gene illustration seems, by all accounts, to be dangerously dependent on a cyclic AMP response element (CRE).UVB stimulates p38 mitogen-initiated protein kinase (MAPK) that prompts the phosphorylation of CRE binding protein (CREB) and initiating transcription factor-1, which then attach to the CRE site and transactivate the COX-2 gene promoter \unskip~\cite{1114320:22298817}. COX-2 is ordinarily not communicated in the majority of the tissues, yet it is extremely inducible by numerous stimuli, including cytokines, growth development factors, tumor supporters, and UV beam \unskip~\cite{1114320:22298824}. UV stimulation of COX-2 includes p53. DNA injury or ectopically conveyed wild-type p53 can produce COX-2 illustration via the up-guideline of the p53 target gene [Figure~\ref{f-620583417b16}]. COX-2 and PGE2 have been appeared to have antiapoptotic impacts and so the stimulation of COX-2 through the p53/Ras/Raf1/ERK pathway assists with neutralizing genotoxic stress-provoked apoptosis \unskip~\cite{1114320:22298823}.
\bgroup
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\centering \makeatletter\IfFileExists{images/72278e36-5d5a-4885-a396-dd7204dbb99b-upicture2.png}{\includegraphics{images/72278e36-5d5a-4885-a396-dd7204dbb99b-upicture2.png}}{}
\makeatother
\caption{\boldmath {The RAF-MEK-ERK pathway}}
\label{f-620583417b16}
\end{figure}
\egroup
UV generation of COX-2 illustration and PGE\ensuremath{_{2}} creation is thought to advance skin carcinogenesis, as well as add up to even the most initial stages of UV-instigated skin damage. COX-2 specific inhibitor, taken care of in the eating routine hindered UV. The utilization of COX-2 inhibitors to completely determine a part for CO- initiated keratinocyte propagation and elevated UV- initiated apoptosis, particularly in the multiplying basal layer \unskip~\cite{1114320:22298815}. Those COX-2 specific inhibitors have chemopreventive action against UV carcinogenesis just as UV-initiated inflammation and early skin injury COX-2 in carcinogenesis is dangerous in that these inhibitors have likewise been appeared to have COX-2-individual impacts. Genetic methodologies have also been utilized well to show particularly the function of Coxs in UV-initiated carcinogenesis \unskip~\cite{1114320:22298822}. COX-2 over expressing which is sturdily transfected human BCC cells concealed a greater amount of the proangiogenic factors, vascular endothelial development factor, and essential fibroblast development factor, stimulated endothelial cell tube development, and provoked more superior angiogenesis. The COX-2 over expressing BCC cells showed more prominent tumor development.
In addition to p53 modifications, other UV-incited measures are likewise measured as a critical step to the course of skin malignancy progression. COX-2 articulation has been demonstrated to be raised in SCCs, BCCs, and actinic keratoses. In the SKH-1 mouse model, COX-2 is overexpressed in benignant papillomas, hyperplastic skin, and in SCCs as a consequence caused due to the constant or continual exposure to UV radiation. Similar to p53 modifications, the COX-2 constitutive over-expression happens from the initial stage during UV-initiated carcinogenesis \unskip~\cite{1114320:22298814}. COX-2 \ensuremath{/}PGE\ensuremath{_{2}} contains many antiapoptotic properties in various cell types. Along with that, PGE\ensuremath{_{2}} mediates various indications which are known to be involved in the induction of angiogenesis, inflammation, vascular permeability, and vasodilation. All of these various downstream properties of PGE\ensuremath{_{2}} signaling and COX-2 over expression are known to encourage the development of UV-initiated skin carcinogenesis [Figure~\ref{f-46e2bf31fa88}]. Despite the fact, that UVB frequencies are initially liable for UV-initiated DNA injury \unskip~\cite{1114320:22298813}.
\bgroup
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\centering \makeatletter\IfFileExists{images/7fcc1e58-7b18-4b72-b735-1215111a5731-upicture3.png}{\includegraphics{images/7fcc1e58-7b18-4b72-b735-1215111a5731-upicture3.png}}{}
\makeatother
\caption{\boldmath {UV-induced carcinogenesis in the Skin}}
\label{f-46e2bf31fa88}
\end{figure}
\egroup
The electromagnetic energy of UV radiation is consumed by particles inside the cell, and this energy is then moved to atomic oxygen-producing ROS \unskip~\cite{1114320:22298819}. COX-2 illustration can be stimulated by a p53-intervened initiation of the Ras \ensuremath{/} Raf \ensuremath{/}ERK pathway. Genotoxic stress, for example, UVB-initiated DNA injury, initiates p53 articulation, the antiapoptotic movement of COX-2 \ensuremath{/}PGE\ensuremath{_{2}} is avoided to a limited extent the stimulation of apoptosis by DNA injury \ensuremath{/} p53, as treatment of cells with an inhibitor of COX-2 improved genotoxic stress-prompted apoptosis. Ultraviolet-A radiation (UVA), which relates to 90\% of the sun radiation, can enter the skin in a dosage gathering manner, links with the impact of UVB, and encourages dermal collagen degeneration, consequently causing skin irritation, inflammation, and untimely early maturing. UVA can influence different biological tasks including DNA replication, repairing, cell cycle control, and chromatin-modifying \unskip~\cite{1114320:22298827}. Every one of this evidence may add to the enhanced danger of skin malignancy related to UVA exposure.
\textbf{Roles of Reactive Oxygen Species (ROS)} \textbf{in UV Carcinogenesis}
ROS-intervened DNA injury assumes participation in the induction of carcinogenesis as well as in dangerous malignant alteration change and this manner; it might signify a significant donor in the pathogenesis of human carcinogenesis. UV-B prompts DNA injuries altered by ROS. UV-A additionally assumes a part in UV-incited carcinogenesis albeit just UV-B has been viewed as dependable. Skin tumors can be initiated by irradiating mice with UV-B. It has been set up that tumors are the outcomes that have come because of a collection of DNA injuries in critical genetic materials like oncogenes and additionally tumor-suppressing genetic material. UV illumination causes DNA alterations and accordingly is thought to be liable for sunlight prompted skin tumors [Figure~\ref{f-a7032b3b1cdc}].
\bgroup
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\centering \makeatletter\IfFileExists{images/f6b91099-5fcf-4fad-81ba-38d8cf436bc4-upicture4.png}{\includegraphics{images/f6b91099-5fcf-4fad-81ba-38d8cf436bc4-upicture4.png}}{}
\makeatother
\caption{\boldmath {A model for induction of skin cancer by UV}}
\label{f-a7032b3b1cdc}
\end{figure}
\egroup
UVB-initiated H\ensuremath{_{2}}O\ensuremath{_{2}} creation advances the phosphorylation of ERK1/2, JNK, and p38. ERK1/2 initiates development factors and starts the tumor development procedure; phosphorylated JNK activates activator protein-1, engaged with methods like stimulus of tumor aggravation, invasion, metastasis, and angiogenesis; while p38-ERK is engaged with the instruction of NF-\ensuremath{\alpha }B. Surely, in vivo considers have discovered that contact with UVB triggers NF-\ensuremath{\alpha }B/p65 and its movement to the nucleus, in this manner initiating genetic material engaged with various procedures, for example, irritation, inflammation (COX-2 also, iNOS), or the cell cycle \unskip~\cite{1114320:22298811}.
The cancer-causing nature of oxidative pressure is credited to the genotoxicity of ROS in different cell measures. ROS collection is just like an ordinary occurrence in numerous malignancy cells. Such collection can cause direct DNA injury or harm by enhancing a cell's alteration or mutation rate \unskip~\cite{1114320:22298809}. ROS-provoked DNA injuries along with an incapable DNA restoration method are well-recognized injuries similar to human malignancies, for example, on account of breast tumor cell lines and human breast tumor tissue \unskip~\cite{1114320:22298812}. Oxidative pressure is initiated by an irregularity between the creation of reactive oxygen and a biological order's capacity to promptly detoxify the reactive intermediates or effectively fix the resultant injury.
DNA injury assumes a part in the improvement of carcinogenesis [Figure~\ref{f-90d885a2bfd2}]. Activities of ROS should be significant, probably their impacts on p53, cell multiplication, invasive property and even, metastasis. Persistent inflammation inclines to melanoma injury; in any case, the task of ROS in this is probably going to be complicated for the reason that ROS can some of the time produce action as an anti-inflammatory mediator \unskip~\cite{1114320:22298807}.
\bgroup
\fixFloatSize{images/87abe088-5557-40d9-a706-f9e99335a7a7-upicture5.png}
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\centering \makeatletter\IfFileExists{images/87abe088-5557-40d9-a706-f9e99335a7a7-upicture5.png}{\includegraphics{images/87abe088-5557-40d9-a706-f9e99335a7a7-upicture5.png}}{}
\makeatother
\caption{\boldmath {Mechanisms of UV-induced skin carcinogenesis}}
\label{f-90d885a2bfd2}
\end{figure}
\egroup
Hereditary modifications, immune inhibition, and malignant alteration are observable facts associated with the source of disease. Malignancy is generally assumed to emerge from a solo cell that has become "instigated" by mutation of a couple of vital genetic materials, brought about by irregular mistakes in DNA replication or a response of the DNA with free radicals or other chemical substance species. UVB enhances ROS in epidermal cells, and triggers signaling pathways associated with modifying cell development, discrimination, and multiplication and accordingly encouraging the clonal development of tumor cells. Reactive oxygen species are produced through a mixture of occasions and pathways and are known to respond with all segments of the DNA fragment: harming both the purine and pyrimidine bases \unskip~\cite{1114320:22298810}. Lasting alteration of hereditary material which results due to these ``oxidative injury'' events addresses the initial step implicated with mutagenesis, carcinogenesis, and maturing in different malignancy tissues free-radical mediated DNA injury. ROS-initiated DNA injury includes single-or double-stranded DNA breakage, purine, pyrimidine, or deoxyribose changes, and DNA cross-joins. DNA harm can result in capture or enlistment of record, replication mistakes, and genomic precariousness, which are all related to carcinogenesis \unskip~\cite{1114320:22298829}.
\textbf{COX-1 and COX-2 Play Important Roles in PGE\ensuremath{_{2}} Synthesis }
UVB light is a recognized etiologic factor in the growth of Squamous cell carcinoma (SCC). The growth of UVB-initiated SCC happens because of changes at both hereditary and epigenetic levels. Hereditarily, UVB-initiated DNA injury is a serious occurrence at the beginning of SCC \unskip~\cite{1114320:22298806}. Interestingly skin tumor progress and development are frequently reliant upon changes in genetic illustration due to epigenetic occasions. Contact of skin to UV irradiation induces COX-2 illustration, which plays a part in an increase in PGE\ensuremath{_{2}} levels. PGE\ensuremath{_{2}} signaling utilizing its receptors plays a part in UV-initiated inflammation, edema, and keratinocyte propagation and responds to UV-initiated apoptosis. The oxidative nature of the COX enzymatic action may lead to the creation of reactive oxygen species, which in combination with the discharge of reactive oxygen species through the inflammatory cell can also play a role in oxidative DNA injury \unskip~\cite{1114320:22298826}. Chronic inflammation and a pro-oxidant state, together they are joined to COX-2 expression, mutually with UV-initiated p53 alterations are possibly the driving forces following the enhancement of UV-initiated non-melanoma skin tumor. COX inhibitors are efficient in reducing tumor proliferation and occurrence. In usual UV-initiated skin, COX-1 and COX-2 play important roles in PGE\ensuremath{_{2}} synthesis [Figure~\ref{f-42ecff139c77}]. The initiation of COX-2 illustration by acute UV contact and constitutive up-guideline of COX-2 in UV initiated benign and malignant cancers lead to enhanced PGE\ensuremath{_{2}} creation and initiation of EP receptor signaling as a consequence in enhanced epidermal proliferation, initiation of inflammation, angiogenesis, and vascular permeability \unskip~\cite{1114320:22298808}. All these effects of COX-2 up-guidelines along with UV-initiated p53 mutations are probably the driving forces following the carcinogenic procedure induced and elevated by chronic UV exposures.
\bgroup
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\centering \makeatletter\IfFileExists{images/b472fb66-82f5-460e-bc18-526b22a86aa1-upicture6.png}{\includegraphics{images/b472fb66-82f5-460e-bc18-526b22a86aa1-upicture6.png}}{}
\makeatother
\caption{\boldmath {COX-1 and COX-2 play important roles in PGE\ensuremath{_{2}} synthesis}}
\label{f-42ecff139c77}
\end{figure}
\egroup
It has been recognized that oxidative stress plays a significant role in UV-initiated skin carcinogenesis. Inflammatory procedures may initiate DNA alterations in cells utilizing oxidative stress. ROS inside cells behaves as secondary messengers in intracellular signaling cascades which instigates and maintains the oncogenic phenotype of tumor cells. The function of oxidants in the instigation of genetic alterations, it is obvious that ROS provoked cell-signaling pathways those involved in cell development regulatory pathways and hence they are instrumental in the progression of carcinogenesis \unskip~\cite{1114320:22298820}. The commencement of transcription factors includes both MAP-kinase/AP-1 and NF-\ensuremath{\alpha }B pathways that have a straight effect on cell propagation and apoptosis. Thus DNA injury, genetic alterations, and changed genetic illustration, all act as a key player in the progression of carcinogenesis. Cancer is a relatively complex, multi-factorial, and multistage disease with several molecular modifications involved in every phase (which are namely initiation, promotion, and progression) of its enhancement. Oxidative stress has long been known to play a significant role in the human carcinogenesis process.
\section{CONCLUSION}
In UV-induced skin carcinogenesis, the agent nature of pyrimidine photoproducts does have been planned. These days it's been cam upon that inhibition performs a vital role in UV-induced skin carcinogenesis. The global reports are going to be essential to see even if there are any target genes for ROS in different several of carcinogenesis.
\section{Acknowledgment}
I would like to thanks to my guide Kusu Susan Cyriac, Associate professor, Department of Pharmacology, Karnataka College of Pharmacy, Thirumena Halli, Hegde Nagar main road, Bangalore-560064, Karnataka, India.
\textbf{Funding Support}
The authors declare that they have no funding support for this study.
\textbf{Conflict of Interest}
The authors declare no Conflict of interest, financial or otherwise.
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It is argued that quantum traveling of D-particles presents the ``joining-splitting'' processes of field theory Feynman graphs. The amplitudes in $d$ dimensions can be corresponded with those of a $d+2$ dimensional theory in the Light-Cone frame. It is shown that this Light-Cone formulation enables to study processes with arbitrary longitudinal momentum transfers. It is discussed that a massless sector exists which can be taken as the low energy limit. By taking the constant relative distance in the bound-states we find a spectrum for the intermediatory fields.
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\title{Community assembly and metaphylogeography of soil biodiversity: insights
from haplotype-level community DNA metabarcoding within an oceanic
island}
\author[1]{Carmelo Andujar}%
\author[2]{Paula Arribas}%
\author[3]{Heriberto López}%
\author[4]{Yurena Arjona }%
\author[5]{Antonio Pérez-Delgado}%
\author[3]{Pedro Oromí}%
\author[6]{Alfried Vogler}%
\author[5]{Brent Emerson}%
\affil[1]{The Natural History Museum of London}%
\affil[2]{IPNA}%
\affil[3]{Affiliation not available}%
\affil[4]{Real Jardin Botanico}%
\affil[5]{IPNA-CSIC}%
\affil[6]{Imperial College/Natural History Museum of London}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
\let\endcenter\endflushleft
\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
Most of our understanding of island diversity comes from the study of
aboveground systems, while the patterns and processes of diversification
and community assembly for belowground biotas remain poorly understood.
Here we take advantage of a relatively young and dynamic oceanic island
to advance our understanding of eco-evolutionary processes driving
community assembly within soil mesofauna. Using whole organism community
DNA (wocDNA) metabarcoding and the recently developed metaMATE pipeline,
we have generated spatially explicit and reliable haplotype-level DNA
sequence data for soil mesofaunal assemblages sampled across the four
main habitats within the island of Tenerife. Community ecological and
metaphylogeographic analyses have been performed at multiple levels of
genetic similarity, from haplotypes to species and supraspecific
groupings. Broadly consistent patterns of local-scale species richness
across different insular habitats have been found, whereas local insular
richness is lower than in continental settings. Our results reveal an
important role for niche conservatism as a driver of insular community
assembly of soil mesofauna, with only limited evidence for habitat
shifts promoting diversification. Furthermore, support is found for a
fundamental role of habitat in the assembly of soil mesofauna, where
habitat specialism is mainly due to colonisation and the establishment
of preadapted species. Hierarchical patterns of distance decay at the
community level and metaphylogeographical analyses support a pattern of
geographic structuring over limited spatial scales, from the level of
haplotypes through to species and lineages, as expected for taxa with
strong dispersal limitations. Our results demonstrate the potential for
wocDNA metabarcoding to advance our understanding of biodiversity.%
\end{abstract}%
\sloppy
\textbf{Introduction}
Colonisation and speciation, together with extinction, are key processes
contributing to island diversity and core processes within models of
island biogeography (e.g. MacArthur \& Wilson, 1963, 1967; Hubbell,
2001; Rosindell \emph{et al.} , 2011). Most of our understanding of
island diversity, and the mechanisms of diversification and community
assembly on islands, comes from the study of aboveground systems (e.g.,
Gillespie \& Roderick, 2002; Warren \emph{et al.} , 2014; Pati\selectlanguage{ngerman}ño
\emph{et al.} , 2017), while the patterns and processes of importance
for underground biotas remain poorly understood (FAO report, 2020). This
lack of knowledge presents a major limitation to understanding island
biodiversity and dynamics, as patterns and processes are not necessarily
coupled between aboveground and belowground components of ecosystems
(Bardgett \& van der Putten, 2014; Shade et al., 2018)
Soil biodiversity, in particular soil mesofauna (i.e. small-bodied
invertebrates measuring between 0.1 and 2 mm), is globally poorly
understood (Cameron et al., 2018; Decaëns, 2010; White et al., 2020).
Knowledge regarding fundamental biological and ecological traits of soil
mesofauna is absent for most species. For example, dispersal dynamics
within soil fauna remains an open and central question in soil
biodiversity research (Ettema \& Wardle, 2002; Thakur et al., 2019).
Within insular settings, soil faunal diversity is expected to be
strongly influenced by variation among species for dispersal capacity
and niche breadth, as these traits underpin both island colonization and
within island processes of population structure and speciation (Emerson
\& Gillespie, 2008; Gillespie et al., 2012; Kisel \& Barraclough, 2010;
Warren et al., 2014). Thus, insular systems provide an important focus
for the development of a broader understanding of how dispersal and
niche traits shape soil mesofaunal biodiversity.
Arthropod mesofaunal lineages typically exhibit various adaptations to
soil environments, including the reduction of wings, eyes, and legs, and
are thus likely to be limited in their propensity for active dispersal
(Decaëns, 2010; Wardle, 2002). When extrapolated over extended periods
of evolutionary time, such dispersal limitation is consistent with the
high turnover across limited spatial scales and high local endemicity
that has been reported for soil mesofaunal lineages (e.g., Andújar et
al., 2017; Arribas, Andújar, Salces-Castellano, Emerson, \& Vogler,
2021; Cicconardi, Nardi, Emerson, Frati, \& Fanciulli, 2010; Collins,
Hogg, Convey, Barnes, \& McDonald, 2019; Morek, Surmacz, López-López, \&
Michalczyk, 2021). However, it has also been argued that their small
body size and often high local abundances may increase the probability
of passive dispersal and long-distance movement (Ettema \& Wardle, 2002;
Thakur et al., 2019), supporting the ''Everything is everywhere but
environment selects'' hypothesis for soil mesofauna (Fenchel \& Finlay,
2004; Finlay, 2002). In the context of oceanic islands, if passive
dispersal is sufficiently high, island colonisation by soil fauna
lineages should be a recurrent process maintaining species cohesion
between islands and source regions, and panmictic populations at
intra-island scales (Fig. 1A). In contrast, if passive dispersal is
strongly constrained for soil fauna, it is reasonable to assume that
colonization will occur primarily through sporadic events of
long-distance dispersal (i.e. LDD events, Nathan, 2005), and that
geographic speciation, even within islands, will play a more important
role in community assembly (Fig. 1A).
While island colonisation will depend on dispersal capacity, successful
establishment is also reliant upon species-specific traits related to
climatic niche breadth. In general, islands have been proposed to favour
generalist species, either by colonization filters that select for
species with wide niche breadth (ecological tolerance) (Gaston, 2003;
Reaka, 1980) or through lower levels of competition favouring ecological
release following colonisation (Olesen, Eskildsen, \& Venkatasamy,
2002). It has also been demonstrated that climatic gradients within
islands can be characterised by very differentiated invertebrate
communities, comprising species with strong habitat specificity (Lim et
al., 2021). Ecological speciation involving climatic-niche shifts has
been described as an essential process generating diversity within
oceanic island biotas (Gillespie, Roderick, \& Howarth, 2001). However,
recent studies focused on arthropod assemblages have highlighted an
important role for climatic niche conservatism as a driver of community
assembly and diversification within islands (Lim et al., 2021;
Salces-Castellano et al., 2020).
Habitat specialisation and climatic niche conservatism across soil fauna
lineages has been poorly explored. However, previous studies on the
community assembly of soil mesofauna have shown strong evidence for
specialisation to open versus forested vegetation types (Arribas,
Andújar, Salces-Castellano, et al., 2021; Caruso, Taormina, \&
Migliorini, 2012), with further evidence for specialisation among
different forest types (Noguerales et al., 2021). Oceanic islands that
have remained geographically isolated over evolutionary timescales and
present variation in habitat types provide near-ideal conditions to
explore further the relative contribution of generalist and specialist
species composing soil island biotas and the role of habitat-shifts in
the process of diversification within insular settings.
Here we take advantage of a relatively young and dynamic oceanic island
to advance our understanding of eco-evolutionary processes driving
community assembly within soil mesofauna. We achieve this by appling
whole organism community DNA (wocDNA) metabarcoding to soil mesofaunal
communities sampled across the four dominant habitats within the island
of Tenerife. Tenerife is one of the seven principal Canary Islands, an
archipelago within the subtropical region of the North Atlantic Ocean.
The oldest massif of Tenerife emerged approximately 9 Ma, but most of
its 2,034 km\textsuperscript{2} landscape dates back to less than 3 Ma,
with extensive volcanic activity in the last 2 Ma (Ancochea, Maria,
Ibarrola, Cendrero, \& Coello, 1990; Carracedo et al., 2004). Maximum
altitude exceeds 3,000 m, giving rise to an altitudinal-zonal
distribution of main habitat types, strongly mediated by trade winds.
We use spatially explicit and reliable (Andújar et al., 2021)
haplotype-level DNA sequence data for the mtDNA COI gene to conduct
community ecological and metaphylogeographic (Turon, Antich, Palacín,
Præbel, \& Wangensteen, 2019) analyses at multiple levels of genetic
similarity, from the level of haplotypes, through to species and
supraspecific groupings. We estimate local, habitat-level, and
island-level richness, together with measures of local endemicity and
the structuring of community variation across habitats and geographic
distance. We use these data for a joint evaluation of the patterns and
processes driving the diversity and structure of soil mesofauna from the
level of the community down to individual lineages, and address the
following four questions. Is dispersal limitation of soil mesofauna
sufficient to drive geographic structuring of communities and lineage
diversification? How do habitat specificity and habitat shift contribute
to community assembly? What is the relative importance of
spatial\emph{vs} environmental processes as drivers of community
structure and lineage diversification? How do wocDNA diversity estimates
compare with more traditional assessments, and how do they compare to
similar estimates from comparable continental soils?
\textbf{Material and Methods}
\emph{Soil sampling and mesofauna extraction}
Fifty-two sites were sampled across the main habitats of the island of
Tenerife (Canary Islands), including 16 sites on laurel forest, 12 on
thermophilous woodland, 12 on pine forest, and 12 on dry scrubland (Fig.
1B). Distances between sites ranged from a few meters to a maximum of 75
km (Fig. 1B, Table S1). Each site was sampled for: (i) the superficial
soil layer (SUP) by removing one square metre of leaf litter and humus,
and; (ii) the corresponding deep soil layer (DEEP), collected by
extracting 20 litres of soil to a depth of approximately 25-30 cm below
where the superficial layer was collected. SUP and DEEP soil samples
were processed following the flotation--Berlese--flotation protocol
(FBF) of Arribas \emph{et al} . (2016). Briefly, the FBF protocol is
based on the flotation of soil in water, which allows the extraction of
the organic (floating) matter containing the soil mesofauna from soil
samples. Subsequently, the organic portion is placed in a modified
Berlese apparatus to capture specimens alive and preserve them in
absolute ethanol. The last part of the FBF protocol includes additional
flotation and filtering steps of the ethanol-preserved arthropods using
1-mm and 0.45-\selectlanguage{greek}µ\selectlanguage{english}m wire mesh sieves to yield macrofaunal (retained in the
1-mm mesh) and mesofaunal fractions (retained in the 0.45-\selectlanguage{greek}µ\selectlanguage{english}m mesh).
Additional manual sorting was performed to pool together Coleoptera
specimens from both fractions. The remaining macrofauna was stored and
not used for this study. This procedure generates two 'clean' bulk
specimen subsamples for each soil layer, one including all adult and
larval Coleoptera (beeltles) and a second with the smallest mesofauna
typically dominated by Acari (mites) and Collembola (springtails).
\emph{DNA extraction, PCR amplification, and Illumina sequencing}
Bulk specimen subsamples were DNA extracted using the DNeasy Blood and
Tissue Kit (Qiagen) in a Kingfisher Flex robotic system (Thermo
Scientific). The bulk of Coleoptera was extracted non-destructively by
splitting specimens into parts or puncturing the body. The mesofauna
sample was homogenized in 1.5 ml vials with glass pestles. DNA extracts
were quantified using Nanodrop 1000 UV--Vis spectrophotometer (Thermo
Scientific), and the corresponding Coleoptera and mesofaunal subsample
pairs were combined at a ratio of 1:10 in the amount of DNA (in
accordance with the expected species diversity for these two fractions
(Arribas et al., 2021)). The bc3' fragment, corresponding to the 3' 418
bp of the COI barcode region was amplified, using tailed primers
corresponding to the Illumina P5 and P7 sequencing adapters for
subsequent library preparation. Three independent PCR reactions were
performed for each sample, and amplicons were pooled. All information
regarding primers, PCR reagents and conditions is provided in Table S2.
Amplicons were then cleaned using Ampure XP magnetic beads and used as
the template for limited-cycle secondary PCR amplification to add
dual-index barcodes and the Illumina sequencing adapters (Nextera XT
Index Kit; Illumina, San Diego, CA, USA). Metabarcoding libraries were
then sequenced on an Illumina MiSeq sequencer (2 x 300 bp paired-end
reads), dedicating approximately 1\% of a flow cell to each library,
producing paired reads (R1 and R2) with a dual tag combination for each
sample.
\emph{Bioinformatics read processing}
Raw reads were quality checked in Fastqc (Babraham Institute, 2013).
Primers were trimmed using fastx\_trimmer and reads processed in
Trimmomatic (Bolger, Lohse, \& Usadel, 2014) using \emph{TRAILING:20} .
Individual libraries were further processed, implementing several steps
of the Usearch (Edgar, 2013) pipeline: reads were merged
(option\emph{mergepairs} \emph{-fastq\_minovlen 50, -fastq\_maxdiffs 15}
), quality-filtered (\emph{Maxee = 1} ), trimmed to full length
amplicons of 416-420 bp (\emph{-sortbylength} ), dereplicated
(\emph{-fastx\_uniques} ), and denoised and chimera checked
(\emph{-unoise3, -minsize 2} ). Denoised reads from all 104 libraries,
representing putative haplotypes, were then combined and dereplicated to
yield a set of unique sequences across all samples, referred to as
amplicon sequence variants (ASVs from here on; Callahan \emph{et al.} ,
2016). MEGAN V5 (Huson, Auch, Qi, \& Schuster, 2007) with the lowest
common ancestor (LCA) algorithm was used to compute the taxonomic
affinity of each ASV. This classification was based on the result of a
BLAST search (\emph{blastn -outfmt 5 -evalue 0.001} ) against a
reference library including the NCBI \emph{nt} database (Accessed at
June 2018) together with an additional 559 unpublished taxonomically
assigned Iberian sequences of Acari, Collembola, and Coleoptera.
ASVs classified by MEGAN as Acari, Collembola, and Coleoptera were
processed with metaMATE (And\selectlanguage{ngerman}újar et al., 2021). MetaMATE evaluates the
survival of ASVs under alternative filtering procedures based on the
relative abundance of co-distributed ASVs. Briefly, the application of
metaMATE involves a six-step procedure: (i) identification of verified
authentic ASVs (va-ASVs) by 100\% matching against a reference COI
sequence; (ii) identification of ASVs including indels or STOP codons as
verified non-authentic ASVs (vna-ASVs); (iii) generation of a community
table with read-counts (ASV abundance) by sample against the complete
collection of reads (i.e., before the dereplicating and denoising steps)
using Usearch (-search\_exact option); (iv) filtering with a range of
criteria and threshold values; (v) evaluation of the survival of va-ASVs
and vna-ASVs, and (vi) estimation of the predicted number of a-ASVs and
na-ASVs, for every filtering iteration. Filtering parameters can thus be
chosen according to desired stringency for the survival of a-ASVs and
na-ASVs. (see Andújar \emph{et al.} , 2021 for further details)
The following input files were used to run MetaMATE: (i) the set of
unique ASVs (-A option); (ii) a reference dataset (-R) for the
identification of va-ASVs, including all BOLD sequences for Acari,
Collembola, and Coleoptera (downloaded at May 2020) plus 1,011 sequences
from specimens collected at the Iberian Peninsula and the Canary
Islands; (iii) all reads prior to the dereplicating and denoising steps
(-L), and; (4) the specification file including filtering criteria and
parameters to be evaluated (-S) (parameters used: --refmatchlength 350
--refmatchpercent 100 --expectedlength 418). Filtering was explored
using both (i) minimum absolute and minimum percentage abundance by
library and (ii) minimum percentage abundance by library and lineages at
20\% divergence, and all pair combinations of these (See MetaMATE
tutorial for details). Analyses were conducted independently for Acari,
Coleoptera, and Collembola. Filtering parameters were selected for each
taxon to maximize the number of surviving va-ASVs while maintaining the
predicted contribution of na-ASVs to the final dataset to be \selectlanguage{english}[?] 5\%.
Finally, the filtered set of ASVs was further filtered to reduce any
potential cross-contamination problems across samples by removing ASVs
with four or fewer reads from each library. Community tables of fully
filtered haplotypes were then transformed into incidence
(presence/absence) data for further analyses.
\emph{Community richness and structure at multiple thresholds of genetic
similarity}
Filtered ASVs were used to generate a UPGMA tree using F84 corrected
genetic distances, within which haplotypes were grouped into clusters of
genetic similarity at 0.5\%, 1\%, 2\%, 3\%, 5\%, 8\% and 15\% thresholds
for the analysis of \selectlanguage{greek}α \selectlanguage{english}and \selectlanguage{greek}β \selectlanguage{english}diversity from intraspecific haplotype level
(\emph{h} ) variation through to supraspecific lineages. Subsequent
community-level analyses were performed for either a selection of
hierarchical levels (\emph{h} , 3\%, and 15\% clustering) or the
complete set of thresholds.
To test for significant differences in community richness (\selectlanguage{greek}α \selectlanguage{english}diversity)
among different habitats and soil layers for \emph{h} , 3\%, and
15\%-level clusters, repeated-measures ANOVAs were conducted using
habitat and soil layer as grouping factors and sampling site as a
within-subject factor. DEEP and SUP samples were then combined within
each sampling site (n=52), and Kruskal-Wallis rank sum tests were
conducted using habitat as a grouping factor to assess whether \selectlanguage{greek}α
\selectlanguage{english}diversity differed between the communities of each of the four habitats.
Endemicity at the scale of individual sampling sites was also calculated
for \emph{h} , 3\%, and 15\%-level clusters measured as the proportion
of total lineages within a given sampling site that occur exclusively
within that sampling site. Kruskal-Wallis rank sum tests were conducted
to test for differences in community endemicity among the four habitats.
Total observed richness (g \selectlanguage{english}diversity) and accumulation curves (random
method, 1000 permutations, \emph{specaccum} function) were estimated for
each habitat for \emph{h} , 3\% and 15\%-level clusters, and total
extrapolated richness (Chao equation, \emph{specpool} function) by
habitat was estimated. Total community dissimilarity across the
communities of each habitat was estimated at all clustering levels, and
pairwise community matrices were generated using total \selectlanguage{greek}β \selectlanguage{english}diversity
(Sorensen index, \selectlanguage{greek}β\selectlanguage{english}sor) and its additive turnover (Simpson index, \selectlanguage{greek}β\selectlanguage{english}sim)
and nestedness (\selectlanguage{greek}β\selectlanguage{english}sne) components (Baselga \& Orme, 2012). Community
composition matrices were used for non-parametric multidimensional
scaling (NMDS) for \emph{h} , 3\% and 15\%-level clusters, and plots
were created with the \emph{ordispider} option to visualise the
compositional ordination of communities according to their respective
habitat. Permutational ANOVAs were conducted over the community
dissimilarity matrices using 999 permutations and the habitat as the
grouping factor.
Variation in community composition with spatial distance was assessed
following the 'multi-hierarchical macroecology' approach of
Baselga\emph{et al.} (2013), where distance decay of similarity is
contrasted across hierarchical levels. For each habitat, the
relationship between community similarity and spatial distance between
sampled sites (1 -- pairwise \selectlanguage{greek}β \selectlanguage{english}diversity, see above) was assessed for
each clustering level. The spatial distance was calculated using the R
package \emph{gdistance}(van Etten, 2017), which uses Tobler's hiking
function to provide the shortest route between two points given the
slope of the terrain (\emph{m} ) (Tobler, 1993). Pairwise calculations
were made among sites within the same habitat. The lowest and highest
elevations of each habitat within our sampling sites were used to
constrain altitudinal movement, to avoid shortest paths transgressing a
different habitat. A negative exponential function was used to adjust a
generalised linear model (GLM) with Sorensen similarity as the response
variable, spatial distance as the predictor, log link and Gaussian
error, and maintaining untransformed spatial distances (G\selectlanguage{ngerman}ómez-Rodríguez
\& Baselga, 2018). Fractal patterning (power-law function) among
distance\selectlanguage{english}-decay curves was assessed by a log-log Pearson correlation
across clustering levels for (a) the number of lineages, (b) the initial
similarity, and (c) the mean similarity of the distance-decay curves.
High correlation values are indicative of self-similarity in lineage
branching (i.e., number of lineages) and spatial geometry of lineage
distributional ranges (i.e., initial and mean similarity; Baselga
\emph{et al.} , (2015)), which are predicted under a predominant neutral
process of community evolution. Analyses were also conducted to assess
the relationship between community similarity and environmental
distance, computed using Gower's distance over the elevation and 19
bioclimatic variables (from WORLDCLIM at 30 arc-seconds resolution),
characterising each sampling site (Table S3). When significant
relationships were found, variance partitioning was conducted to assess
the fractions of variance in community dissimilarity that are uniquely
and jointly explained by spatial and environmental distance.
Finally, we compared biodiversity measures for haplotypes and 3\% OTUs
for the four habitat types in Tenerife with those obtained by
Arribas\emph{et al.} (2020) in three forest and three grassland sampling
regions in a continental setting (n=12 for each habitat on each sampling
region). Kruskal-Wallis rank sum tests were used to compare \selectlanguage{greek}α \selectlanguage{english}diversity
by sample with insularity (Tenerife island n=52; continent n = 72) and
sampling region as grouping factors. Comparisons of \selectlanguage{greek}β \selectlanguage{english}diversity by
sampling region were restricted to a comparable spatial scale of 15 km,
conducting a Kruskal-Wallis rank sum test with insularity as a grouping
factor. Comparisons of \selectlanguage{greek}β \selectlanguage{english}diversity were repeated for intervals of
spatial distance between 0-5 km, 5-10 km, and 10-15 km. Finally, g
\selectlanguage{english}diversity (total species richness) was estimated for each habitat and
region using accumulated haplotypes and 3\% OTUs across 12 community
samples (using \emph{specaccum} function when the available number of
samples was higher). All analyses were performed using the
R-packages\emph{vegan} (Oksanen et al., 2016), \emph{cluster} (Maechler,
Rousseeuw, Struyf, Hubert, \& Hornik, 2021), \emph{PMCMR} (Pohlert,
2014), \emph{hier.part} (Mac Nally \& Walsh, 2004),
\emph{ecodist}(Goslee \& Urban, 2007), and \emph{betapart} (Baselga \&
Orme, 2012).
\emph{Lineages characterisation and meta-phylogeographical patterns}
Three per cent and a 15\% similarity clusters were used, whereby 3\%
clusters are considered a proxy to species, and from here on referred to
as ''OTUs''; while 15\% clusters are lineages of one or more species and
are hereon referred to as ''15\% lineages''. We evaluated the genetic
diversity, distribution, and degree of habitat specificity for each OTU
and 15\% lineage. We then tested the relative roles of the habitat and
the geographical distance in the diversification of soil fauna within
the island. The number of haplotypes was recorded as a measure of the
genetic richness of each OTU, and OTUs were classified as ''single
haplotype'' or ''multiple haplotypes''. At the level of 15\% lineages
and under the assumption that each arises from a single colonisation of
Tenerife, the number of OTUs within each 15\% lineage was used to
classify each lineage as ''non-diversified'' or ''diversified''
according to whether they included one or multiple OTUs within the
island. BLAST search (\emph{blastn -outfmt 5 -evalue 0.001} ) against a
reference library including all sequences on BOLD (database downloaded
at 3-07-2020), together with COI sequences from southern Iberia
(Arribas\emph{et al.,} 2020), and COI Collembola sequences from
Cicconardi et al. (2017) from outside the Canary Islands, were used to
classify OTUs as 'non-endemic' if similarity with non-Canarian sequences
was [?]97\%; and 'likely introduced' if the similarity was [?]99\%.
To explore OTU and 15\% lineage distributions, the number of sampling
sites with a presence (number of occurrences), the maximum geographical
distance of occurrences, and the different habitats with occurrences
were recorded for each OTU and 15\% lineage, the latter summarised using
Venn diagrams. Habitat specificity was estimated for each entity using
the proportion of occurrences in a particular habitat, considering those
with 80-100\% of occurrences in one habitat as entities with high
habitat specificity. Estimations of habitat specificity were performed
for those entities sampled in \emph{n} or more sites, with \emph{n} = 3,
4, 5, and 6. Finally, we explore the structure of genetic diversity for
each OTU and 15\% lineage with a product of its number of sites by its
number of haplotypes [?] 15. Firstly, we tested the relationship between
the genetic distance (F84 model) and geographic distance (Euclidean
distance between sampling sites). The relationship between both
distances was estimated by randomising spatial distances 1000 times and
computing the proportion of times in which the model deviance was
smaller than the randomised model deviance, adjusting a linear model
using the \emph{glm} function (link = ''identity'') as in
G\selectlanguage{ngerman}ómez-Rodríguez \& Baselga (2018). Geographic distances were calculated
using the R package \emph{gdistance} as before, with calculations
performed for each pair of sites with the lowest and highest limit of
permitted movements restricted to the highest (plus 100 meters) and
lowest (minus 400 meters) values of the two sites. We applied these
restrictions to avoid shortest paths transgressing unfavourable habitats
over the top of the island, while also allowing paths to cross the
valley separating the central region of Tenerife from the Anaga
peninsula, and facilitating connectivity over cliffs separating coastal
sites. In addition we also tested the correlation between genetic
distance (F84 model) among haplotypes and their distribution in the four
habitats, using permutational ANOVAs with 999 permutations and the
habitat as a grouping factor. To graphically summarise patterns of
haplotype relatedness and habitat association, we estimated and plotted
haplotype networks for all 15\% lineages including four or more
haplotypes using the function\emph{mjn} of the R package \emph{pegas}
(Paradis, 2010). For 15\% lineages with more than 40 haplotypes (four
cases), the \emph{mjn}function could not be applied, and networks were
alternatively estimated with the \emph{haploNet} function\emph{,} which
uses an infinite site model and uncorrected distances\emph{.}
\textbf{Results}
\emph{Metabarcode data}
Overall, 12,621,754 raw reads were obtained, distributed across 104
libraries, of which 1,405,224 passed initial cleaning and denoising
steps and were classified as Acari, Collembola, and Coleoptera,
resulting in 19,304 ASVs. Of these, 1,813 ASVs (1,278,294 reads) passed
metaMATE filtering, applying parameters to maximize the number of
surviving va-ASVs while limiting na-ASVs to comprise \selectlanguage{english}[?]5\% of the final
dataset. Parameters used and estimated contributions of a-ASVs and
na-ASVs to the filtered dataset are provided in Table 1. Final filtering
to remove records with less than five reads in a library resulted in the
retention of 1791 ASVs (i.e. 98.7\%) in the final community dataset.
Summary data per library is provided in supplementary table S4, and the
final set of ASVs and the community table is provided as a supplementary
file.
\emph{Community richness and structure at multiple thresholds of genetic
similarity}
Superficial layers tend to have higher richness than their corresponding
deep soil layers across all four habitats, with significant richness
differences between soil layers found for thermophilous woodland and
pine forest (Fig. S1). After combining superficial and deep soil layers
for all 52 sites, mean site richness (\selectlanguage{greek}α \selectlanguage{english}diversity) within habitats
ranged 55 - 73.5 for haplotypes, 38.5 - 49 for 3\% clusters and 34.5 -
43 for 15\% clusters (Fig. 2A). Differences in richness by sample among
habitats were small and maintained across different clustering
thresholds, and pointed to dry scrubland community samples as poorer
(lower richness by site) compared to the other habitats (Fig. 2A).
Mean endemicity by site (proportion of lineages that occur exclusively
in that site) ranged from 24.0\% to 48.8\% at the haplotype level, from
13.5\% to 22.7\% for 3\% clusters, and from 6.8\% to 15.4\% for 15\%
clusters (Fig. 2B). Comparisons among habitats revealed that endemicity
was significantly higher for dry scrubland communities than for laurel
forest communities (Fig. 2B). Compositional dissimilarity among
communities (\selectlanguage{greek}β \selectlanguage{english}diversity, \selectlanguage{greek}β\selectlanguage{english}sor) was high and was dominated by lineage
turnover (\selectlanguage{greek}β\selectlanguage{english}sim), rather than nestedness (\selectlanguage{greek}β\selectlanguage{english}sne), with \selectlanguage{greek}β\selectlanguage{english}sor values ranging
0.87-0.96 across all clustering levels and habitats. Dry scrubland
communities showed the highest levels of compositional dissimilarity
across the different clustering thresholds (Fig. 2C).
Total observed richness at the island scale (g \selectlanguage{english}diversity) by habitat
ranged from 534 - 588 haplotypes, 278 - 316 lineages at 3\% and 194 -
255 lineages at 15\% (Fig. 2C), while extrapolated values (Chao index)
nearly doubled observed values (Fig. 2D). Differences in g \selectlanguage{english}diversity
among habitats were not consistent across different clustering
thresholds, with thermophilous woodland showing the lowest number of
haplotypes but the highest number of lineages at the 15\% clustering
threshold (Fig. 2C). Accumulation curves reveal no plateau in the
accumulation of entities across samples for any habitat or genetic
threshold, with the laurel forest showing the lowest rates of
accumulation (Fig. 2D).
Comparisons with biodiversity measures obtained by Arribas \emph{et
al.}(2020) in forest and grassland sites in a continental setting
revealed that richness by sample (\selectlanguage{greek}α \selectlanguage{english}diversity) was lower in the samples
of Tenerife compared with continental soils (Kruskal p \textless{}
0.001; Fig. S2). Comparisons of \selectlanguage{greek}β \selectlanguage{english}diversity values restricted to a
comparable spatial scale of 15 km resulted in significantly lower \selectlanguage{greek}β
\selectlanguage{english}diversity values in Tenerife for haplotypes (p \textless{} 0.001) but
not for 3\% OTUs (Fig. S2). Finally, g \selectlanguage{english}diversity by sampling region, as
estimated by the total number of haplotypes and OTUs recorded, was
similar for the different habitats of Tenerife (534 -- 588 haplotypes
and from 278 - 316 3\% OTUs) and the six continental settings in Arribas
et al. (2020) (558 - 623 haplotypes, and 276 -- 319 OTUs) (Fig. S2).
NMDS for the compositional dissimilarity of the communities of Tenerife
soils showed habitat as a major driver of the ordination of samples, and
accordingly, for all clustering levels, a significant proportion of
variance (0.18 \textless{} r2 \textless{} 0.28; p \textless{} 0.001) was
explained by the habitat factor (Fig. 3A). In addition, dry scrubland
communities showed the highest dispersal, while the laurel forest
communities were the least scattered (Fig. 3A).
Analyses of community similarity (1-pairwise \selectlanguage{greek}β \selectlanguage{english}diversity) with spatial
distance within each habitat revealed significant distance decay for all
clustering levels in all habitats, except for dry scrubland (Fig. 3B).
For laurel forest, pine forest, and thermophilous woodland, slopes of
the exponential decay curves were very similar at all threshold levels,
and assemblage similarity increased with each level (Fig. 3B). Genetic
similarity showed a high and significant log-log correlation with the
number of lineages (0.97 \textless{} r2 \textless{} 0.99, p \textless{}
0.001), initial similarity (0.92 \textless{} r2 \textless{} 0.99; p
\textless{} 0.001), and mean similarity of communities (0.97 \textless{}
r2 \textless{} 0.99; p \textless{} 0.001) (Table S5), as expected if
community variation across hierarchical levels of similarity is
described by a fractal geometry (Baselga et al., 2013, 2015).
A decrease in community similarity with environmental distance (Fig. S3)
was only significant for the laurel forest and some clustering levels in
the pine forest (Table S6). However, variance partitioning showed that
variance uniquely explained by environmental distance (i.e.
independently of the spatial distance) was lower (3.2\% - 9.0\% of
explained variation at all levels) than the uniquely explained variance
by the spatial distance (6.9\% - 45.0\% of explained variation).
\emph{Lineages characterisation and meta-phylogeographical patterns}
Across all 52 samples across Tenerife island, a total of 813 OTUs (3\%
clustering) and 533 15\% lineages (15\% clustering) were found, with a
mean of 2.2 haplotypes by OTU and a mean of 1.5 OTUs by 15\% lineage.
Table 2 shows the number of OTUs and 15\% lineages obtained and
extrapolated values (Chao index) for Acari, Collembola, and Coleoptera
across the 52 sites. Among OTUs, 488 (60\%) included a single haplotype
(single-haplotype OTUs), and 325 (40\%) were classified
as\emph{multi-haplotype OTUs} (Fig. 4). The most diverse OTU included 40
haplotypes and corresponded to a species of Acari from the order
Sarcoptiformes, not represented in public sequence repositories. Among
the 533 15\% lineages, 413 (77\%) included a single OTU
(\emph{non-diversified lineages} ), and 122 (23\%) included 2 or more
OTUs and were classified as \emph{diversified lineages} . (Fig. 4). The
most diverse 15\% lineage included 21 OTUs (77 haplotypes),
corresponding to the weevil genus \emph{Laparocerus} Schoenherr, 1834,
the most diverse beetle genus in Tenerife (Machado, Rodr\selectlanguage{ngerman}íguez-Expósito,
López, \& Hernández, 2017). Among the 813 OTUs, 135 (16.6\%) were
classified as \emph{non-endemic OTUs} because they have a similarity
\selectlanguage{english}[?]97\% with sequences of external (non-Canarian) databases. Of these, 115
OTUs (14.1\%) showed a similarity \selectlanguage{english}[?]99\% and so were additionally
categorised as \emph{likely introduced OTUs} (Table S7).
Each OTUs was found on average on 2.9 sampling sites and each 15\%
lineage on 3.9 sites. Four hundred and five OTUs (49.8\%) were detected
in a single site, and the remaining 408 (50.2\%) in two or more sites
(Fig. 4). Two-hundred 15\% lineages (37.5\%) were detected in a single
site, and the remaining 333 (63.5\%) were found in two or more sites.
The most widespread 15\% lineage, including a single OTU, was found in
37 sites and corresponded to a likely introduced species identified
as\emph{Ceratophysella gibbosa} (Bagnall, 1940), having similarity
\textgreater{}99\% with specimens from France and Australia (Table S7).
Regarding the distributions of OTUs and 15\% lineages across the
habitats, habitat specificity was estimated for those entities sampled
in \emph{n} or more sites, with \emph{n} = 3, 4, 5, and 6. The
percentage of OTUs considered \emph{OTUs with high habitat specificity}
, with at least 80\% of occurrences within the same habitat, ranged from
31\% to 29\% (Fig. 5A). Similarly, \emph{lineages with high habitat
specificity} ranged from 30\% to 26\% (Fig. 5B). Patterns of shared OTUs
and 15\% lineages among habitats revealed that spatially (and
climatically) adjacent habitats presented higher numbers of shared OTUs
and lineages (e.g., laurel forest and thermophilous woodland; 102 shared
15\% lineages), compared to spatially disconnected habitats (e.g. laurel
forest and dry scrubland; 50 shared 15\% lineages, of which 45 are also
shared with the thermophilous woodland typically located in between)
(Fig. 5). Individual Venn diagrams for Acari, Collembola, and Coleoptera
were highly consistent with this general pattern (Fig. S4).
Regarding the structure of the genetic diversity within OTUs and 15\%
lineages, the analyses were restricted to those entities showing a
product of the number of sites by the number of haplotypes \selectlanguage{english}[?] 15; n = 107
OTUs and 128 15\% lineages. The proportion of these entities with a
significant geographical structure of genetic diversity constituted
29.0\% of the OTUs and 30.5\% of the 15\% lineages (Fig. 6). The
proportion of entities with a significant structure of genetic diversity
associated with the factor habitat was lower and represented 8.4\% of
OTUs and 16.4\% of 15\% lineages (Fig. 6). The overlap between the
entities structured by spatial distance and habitat revealed that 13 of
the 21 entities structured by habitat were also structured by spatial
distance (Fig. 6).
\textbf{Discussion}
Using the recently developed metaMATE pipeline (Andujar et al., 2021),
we have generated a stringently filtered dataset of amplicon sequence
variants (ASVs) for mesofaunal soil communities sampled across an
oceanic island. By achieving a level of spurious sequences estimated to
be no more than 5\% of ASVs in the final dataset, we have been able to
undertake both phylogeographic and community ecological analyses at
different hierarchical levels of relatedness. These data reveal both
ecological patterns and evolutionary processes, providing novel insights
into community assembly within soil mesofauna at an unprecedented
taxonomic scale. In doing so, we demonstrate wocDNA metabarcoding to be
a powerful tool for understanding ecological and evolutionary processes
within dark taxa -- highly diversified lineages for which described
species are estimated to be only a limited proportion of true species
richness (Hartop, Srivathsan, Ronquist, \& Meier, 2021).
\emph{The (unknown) diversity of soil mesofauna within insular soils}
The diversity of soil mesofauna within oceanic islands has been poorly
explored. Literature on the topic is limited (i.e. Koh \emph{et al.} ,
2002; Maraun \emph{et al.} , 2007; Fattorini, 2009; Cicconardi \emph{et
al.} , 2017), and even basic species inventory data are in general
scarce for this ecologically important biodiversity fraction. Within the
Canary Islands, the Biodiversity Databank of the Canary Islands
(\emph{https://www.biodiversidadcanarias.es/biota/}; from hereon
referred to as BIOTA) is a constantly updated public database containing
all species records for the archipelago published in the scientific
literature. BIOTA currently reports 287 species of Acari, 88 of
Collembola, and 1360 species of Coleoptera from the island of Tenerife.
Our results demonstrate that current knowledge of species diversity
within the island is greatly underestimated. By sampling only 52 soil
communities (approximately a 2.6 x 10\textsuperscript{-8} of total
island surface area) across the four dominant habitats of the island of
Tenerife, we have recovered nearly 1800 mtDNA haplotypes from Acari,
Collembola, and Coleoptera that clustered into 813 putative species
(OTUs at 3\%), 434 Acari, 129 Collembola and 250 Coleoptera (Table 2).
Even with a substantially more conservative dissimilarity threshold of
15\%, total lineage number remains above 500. For Acari and Collembola,
sampled OTU numbers exceed the number of species recorded until the date
for the island (Table 2). In the case of Coleoptera, it should be noted
that many of the 1360 recorded species in BIOTA for Tenerife are not
associated with soil, while our sampling is strictly focussed on soil
lineages, so a direct comparison is difficult. Overall, our results
reveal that the soils of Tenerife are much richer in mesofauna than
previously reported, and highlight the generally appreciated problems of
the Linnaean and Wallacean shortfalls (Cardoso, Erwin, Borges, \& New,
2011; Hortal et al., 2015) for soil arthropod biodiversity.
By comparing obtained ASVs against public molecular repositories, we
found that 135 (16.7\%) of the 813 OTUs matched (97\% similarity)
non-Canarian records, and can thus be considered as non-endemic species,
being either native or introduced (Table S7). However, attributing all
remaining OTUs to endemic species is not possible, because of the
incomplete nature of public molecular repositories. Comparison to public
molecular repositories identifies 34 Acari (8\%), 39 Coleoptera (16\%),
and 49 Collembola (38\%) OTUs with high sequence similarity (\selectlanguage{english}[?]99\%) to
individuals from other regions. It is plausible that most, if not all,
are recent human-mediated introductions, rather than worldwide
distributed species requiring unrealistic passive dispersal kernels to
maintain species cohesion. These findings are in agreement with
Cicconardi et al. (2017), who concluded from genome skimming data that
88\% of the 25 Collembola species they sampled from laurel forests in
Tenerife result from human-mediated introductions. Distinguishing
between native and introduced origins for soil-adapted species is
challenging, and focused studies are needed to elucidate the extent of
species introductions within oceanic islands (Andersen et al., 2019).
Alpha, beta, and gamma diversity estimations at the OTU and haplotype
levels point to lower diversity values in island soils compared to
continental soils (Fig. S2). In contrast, high endemicity by sample and
significant community differences among habitats are similar to patterns
found in continental soils (see the section below). Using the same
field, laboratory, and bioinformatic protocols, Arribas \emph{et
al.}(2020) sampled 12 sites within each of three forested and three
grassland areas in Iberia. While sample sizes are comparable between
both studies, spatial scale differs, with maximum distances between
samples of 70 km within Tenerife habitats compared to only 15 km in
Arribas \emph{et al.} (2020). Within habitats, both \selectlanguage{greek}α \selectlanguage{english}and \selectlanguage{greek}β \selectlanguage{english}diversity
(restricted to a comparable spatial scale of 15 km) were significantly
lower in Tenerife, suggesting that insular soil mesofaunal diversity may
be lower, compared to continental areas of a similar size, consistent
with previous suggestions for aboveground plant and animal communities
(Kier et al., 2009; Whittaker \& Fern\selectlanguage{ngerman}ández-Palacios, 2007). While
interesting, the generality of this pattern awaits further
investigation. With appropriate measures to harmonise methodology and
optimise data comparability, the generality of patterns observed here
can feasibly be assessed across independent studies (Arribas, Andújar,
Bidartondo, et al., 2021).
\emph{Dispersal limitation as a driver of the assemblage and
diversification of insular soil mesofauna}
Dispersal is a key process shaping island biotas, being fundamental for
colonization and consequential within islands for the geographic
structuring of genetic variation within species, speciation, and
intra-island diversification (Gillespie et al., 2012; Salces-Castellano
et al., 2020; Warren et al., 2014). Integrating across the distances and
frequencies over which active and passive dispersal processes contribute
to species cohesion and speciation (Fig. 1) provides a predictive
framework for evolutionary trajectories at the level of individual
lineages (Gillespie et al., 2012). Competing models can be proposed for
the likely shape of the dispersal kernel for the typically tiny and
flightless component of mesofaunal soil species, with differing
implications for their spatial patterns of diversity (Fig. 1) (Andújar
et al., 2017). The first is a model of limited active but high passive
dispersal potential, mediated by the small size of soil mesoarthropods,
according to the ''everything is everywhere hypothesis'' (Fenchel \&
Finlay, 2004; Finlay, 2002) which predicts large spatial distances for
species cohesion. The second model is one of limited active and passive
dispersal potential, and thus predicts a limited spatial scale for
speciation (Andújar et al., 2017; Arribas, Andújar, Salces-Castellano,
et al., 2021).
Analyses of mesofauna from continental soils have led to contrasting
inferences for how dispersal shapes their community assembly and
diversification. Strong dispersal constraints have rarely been
recognised for soil mesofauna, and long-distance dispersal has been
considered to characterise soil mesofauna, largely mediated by passive
dispersal by air, water or in marine plankton (Decaëns, 2010; Thakur et
al., 2019; Wardle, 2002). In contrast, molecular studies of soil
mesofaunal lineages and communities frequently reveal dispersal
limitation, associated with both diversification and community turnover
across limited spatial scales (Andújar et al., 2017; Arribas et al.,
2021; Francesco Cicconardi, Fanciulli, \& Emerson, 2013; Collins et al.,
2019). The BIOTA inventory for the island of Tenerife reveals that 236
of 297 recorded species of Acari (79\%) , 62 of 88 Collembola (70\%),
and 699 of 1360 Coleoptera (51\%), are considered to be non-endemic,
having populations outside of the Canary Islands. These data are more
consistent with a model of high dispersal potential for soil mesofauna.
However, our metabarcode data provide greater support for a model where
dispersal is limited, where island populations are evolutionarily
independent entities, within which futher diversification can occur.
Following island colonisation and establishment, dispersal limitation
may favour subsequent intra-island genetic differentiation, the extent
of which will be mediated by species traits (e.g. niche,
species-specific dispersal ability), and the selective landscape (e.g.
spatial variation in biotic and abiotic conditions). Under this model,
spatially structured lineages and communities are expected to emerge,
and there are clear signatures for this within our data. Within each of
the studied habitats, for haplotype, species, and supraespecific levels
of variation, community similarity is a function of geographic distance
(Fig. 2C and 3). This self-similarity of distance decay at haplotype and
species level (Fig. 3B) is consistent with a role for dispersal
limitation driving community assembly (Baselga et al., 2015;
Gómez-Rodríguez \& Baselga, 2018). The influence of dispersal
constraints within the soil matrix appears to act at short spatial
distances, and the evident high turnover with physical distance suggests
that our sampled communities within each habitat are not from a single
panmictic metacommunity. At the lineage level, our results reveal
multiple signals of dispersal limitation constraining diversification.
Many of the soil mesofaunal OTUs recovered from our wocDNA metabarcode
data are not recorded (at least molecularly) outside the island (Fig. 4;
Table S7), have restricted distributions within the island, and present
spatially structured genetic variation (Fig. 6). Additionally, among the
533 15\% lineages recovered, 122 comprises two or more OTUs. If we
assume each 15\% lineage represents a single colonization event into
Tenerife, 49.2\% of all OTUs may be derived from intra-island divergence
events. Thirty-nine OTUs show a significant correlation between genetic
and spatial distances, 34 of these comprising two or more OTUs, further
supporting \emph{in situ} spatial structuring and diversification within
lineages (Fig. 4 and 6).
\emph{Habitat and the diversity and structure of insular soil mesofauna}
Across all communities, the greatest differentiation was among
communities from the different habitats (Fig. 3A), and we find up to
30\% of OTUs and lineages that are consistent with high habitat
specialisation (Fig. 5). Habitat specificity in soil mesofauna has been
previously reported, with strong evidence for specialisation between
open versus forested vegetation types (Arribas \emph{et al.} , 2020,
Caruso et al. 2012) and different forest types (Noguerales et al.,
2021). Our results extend the generality of these patterns to the soils
of an oceanic island.
Islands have been suggested to favour generalist species, either by
colonization and persistence filters that select for species with wide
niche breadth (ecological tolerance) (Gaston, 2003; Reaka, 1980) or
through lower levels of competition favouring ecological release
following colonisation (Olesen et al., 2002). However, our results are
not consistent with these proposals, revealing that for much of the soil
mesofauna, habitat features could be driving a scenario of species
sorting (Leibold et al., 2004), with the existence of largely separate
(still overlapping) metacommunities inhabiting the different habitats
within the island. Two contrasting but not mutually exclusive models can
be evoked to explain these patterns of habitat specificity. The first
involves niche conservatism, with colonising species establishing into
habitats to which they are preadapted and with intraisland
diversification primarily constrained within the same habitats (Lim et
al., 2021; Salces-Castellano et al., 2020). The second involves niche
lability, and it has been described as an essential process generating
diversity within oceanic island biotas through selection gradients
across different habitats (Gillespie et al., 2001). Our results reveal
that among the 533 lineages that are assumed to be independent
colonisations to Tenerife, 312 are restricted to a single habitat.
Furthermore, among the 128 15\% lineages where genetic differentiation
associated with habitat type was tested for, 21 presented a significant
association (Fig. 6). Thus, our data provides only limited evidence for
habitat shifts promoting diversification, suggesting an important role
for climatic niche conservatism driving ecological assembly of soil
mesofauna within the island.
Despite contrasting biotic and abiotic features among the sampled
habitats (del Arco Aguilar, González-González, Garzón-Machado, \&
Pizarro-Hernández, 2010), \selectlanguage{english}g and mean \selectlanguage{greek}α \selectlanguage{english}diversities were similar within
each, albeit with some differences between dry scrublands and the
remaining three habitats (Fig. 2). Dry scrubland soils have
significantly lower species richness by sample, whereas lineage
accumulation across multiple sites resulted in similar values of g
\selectlanguage{english}diversity. This pattern is mediated by significantly higher local
endemicity within dry scrubland soils, and thus higher turnover not
spatially structured (Fig. 3). Habitat specific differences related
current and past habitat patchiness and connectivity could be driving
such differences. Under the habitat stability hypothesis (Ribera \&
Vogler, 2000; Southwood, 1977), lineages with high dispersal potential
are expected to be primarily selected within more ephemeral habitats.
Within this framework, it can be hypothesised that a lower habitat
stability for scrublands, due to higher exposure to sea-level changes in
geological times, could be contributing to their observed lower local
richness and more limited spatial structuring of their soil mesofaunal
communities. However, the number of spatially structured OTUs and 15\%
lineages was very similar among habitats, and habitat specificity at
different hierarchical levels of analysis was comparable among them
(data not shown). Other factors, such as fine-scale habitat
heterogeneity, may have eroded the signature of geography into the
assembly of mesofaunal communities within the dry scrublands. Further
studies are needed to explore the contrasting ecological and
evolutionary processes that drive the community assembly within
different habitat types, such as those described here.
\par\null
\textbf{Conclusions}
Our results reveal an important role for niche conservatism as a driver
of insular soil mesofaunal community assembly, with limited evidence for
habitat shifts promoting diversification. These results also support a
fundamental role of habitat features in the assembly of soil mesofauna,
in agreement with previous studies (Arribas et al., 2021; Noguerales et
al., 2021), with much habitat specialism being explained as the result
of independent colonisation and establishment of preadapted species.
Hierarchical patterns of distance decay at the community level and
metaphylogeographical analyses reveal geographic structuring over
limited spatial scales from the level of haplotypes through to species
and lineages, as expected for taxa under strong dispersal limitations.
We also reveal broadly consistent patterns of local-scale species
richness across different insular habitats and find that local insular
richness is lower than in broadly comparable continental settings. These
results demonstrate the potential for wocDNA metabarcoding to advance
our understanding of biodiversity, particularly for the so called dark
taxa -- important fractions of biodiversity that have traditionally been
difficult to work with.
\par\null\par\null
\textbf{Acknowledgements}
This work was supported by project CGL2015-74178-JIN (AEI, Spain/FEDER,
EU) awarded to CA, and CGL2017-85718-P and PID2020-116788GB-I00 (AEI,
Spain/FEDER, EU) awarded to BCE. CA was additionally supported by
Fundaci\selectlanguage{ngerman}ón Caja Canarias/Obra Social ``La Caixa'' (2017RCE03). BCE,
together with PA was additionally supported by the H2020 iBioGen
project, funded by the European Research Council, Award Number: 810729.
We extend our gratitude to the regional governments of Andalucía,
Canarias, and and the local council (Cabildo) of Tenerife for
facilitating collecting of samples, and to Jesús Arribas for assistance
with field sampling and drawings of habitats.
\textbf{Author contributions}
CA, PA and BCE conceived the study, that was led by CA; CA and PA
designed the methodology; CA, PA, HL, AP-D, PO, APV, and BCE provided
the data; CA, PA, and YA analysed the data. CA, PA, and BCE interpreted
results and wrote the manuscript. All authors contributed critically to
the drafts and gave final approval for publication.
\textbf{Orcid}
Carmelo Andújar\emph{https://orcid.org/0000-0001-9759-7402}
Paula Arribas\emph{https://orcid.org/0000-0002-0358-8271}
Alfried P. Vogler\emph{https://orcid.org/0000-0002-2462-3718}
Brent C. Emerson\emph{https://orcid.org/0000-0003-4067-9858}
Antonio J. Pérez-Delgado\emph{https://orcid.org/0000-0002-3797-4058}
Heriberto López\emph{https://orcid.org/0000-0001-6988-5204}
Yurena Arjona\emph{https://orcid.org/0000-0002-1851-1664}
\par\null
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\textbf{Data Accessibility and Benefit-Sharing}
The data that support the findings of this study are openly available in
Dryad at\emph{https://doi.org/10.5061/dryad.mw6m905z3}
\textbf{Tables}
\textbf{Table 1.} Filtering parameters selected with MetaMATE and
summary results on the number of va-ASVs, vna-ASVs, a-ASVs, and na-ASVs
in the dataset before and after filtering. Filtering parameters were
selected for a final contribution of na-ASVs below 5\%. Values for
a-ASVs and na-ASVs are estimated within MetaMATE. Removed va-ASVs were
added to the final dataset; surviving vna-ASVs were excluded from the
final dataset.
\par\null\selectlanguage{english}
\begin{longtable}[]{@{}lllllllllll@{}}
\toprule
\begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
Before MetaMATE\strut
\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
Before MetaMATE\strut
\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
Before MetaMATE\strut
\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
Before MetaMATE\strut
\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
After MetaMATE\strut
\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
After MetaMATE\strut
\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
After MetaMATE\strut
\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
After MetaMATE\strut
\end{minipage} & \begin{minipage}[b]{0.07\columnwidth}\raggedright\strut
After MetaMATE\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
Filtering criteria (always by library)\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
va-ASV\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
vna-ASV\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
a-ASV\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
na-ASV\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
va-ASV\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
vna-ASV\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
a-ASV\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
na-ASV\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
Contibution of na-ASV\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
Acari\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
Minimum N reads = 40 and Minimum percentage abundace = 0.7\%\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
32\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
514\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
1594.0\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
9508.0\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
16\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
2\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
797.0\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
37.0\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
4.40\%\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
Collembola\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
Minimum N reads = 50 andMinimum percentage abundace by 20\% lineage =
4\%\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
67\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
235\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
512.5\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
2887.5\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
35\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
1\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
267.7\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
12.3\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
4.40\%\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
Coleoptera\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
Minimum N reads = 8 and Minimum percentage abundace by 20\% lineage =
3\%\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
77\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
763\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
734.9\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
4067.1\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
65\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
5\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
620.3\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
26.7\strut
\end{minipage} & \begin{minipage}[t]{0.09\columnwidth}\raggedright\strut
4.10\%\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}
\textbf{Table 2.} Number of species of Acari, Collembola, and Coleoptera
recorded on the Biodiversity Databank of the Canary Islands
(\emph{https://www.biodiversidadcanarias.es/biota/}; BIOTA) and number
of OTUs (clusters 3\%) and lineages (cluster 15\%) observed and
extrapolated (Chao index) across the 52 sampling sites.\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
\begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
BIOTA Canarias\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
BIOTA Tenerife\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
OTUs (3\%) Observed/Chao\strut
\end{minipage} & \begin{minipage}[b]{0.19\columnwidth}\raggedright\strut
Lineages (15\%) Observed/Chao\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Acari\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
469\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
287\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
434/733\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
276/382\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Collembola\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
138\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
88\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
129/172\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
105/129\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
Coleoptera\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
2234\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
1360\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
250/503\strut
\end{minipage} & \begin{minipage}[t]{0.17\columnwidth}\raggedright\strut
152/225\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}
\textbf{Figure legends}
\textbf{Figure 1.} Hypothetical dispersal kernels for soil mesofaunal
lineages with different passive dispersal potential (\textbf{A} ) and
map of Tenerife with the distribution of sampled sites (left) and zonal
distribution of habitats on the island (right) (\textbf{B} ). Within
(\textbf{A} ), the high passive dispersal of species in (left) allows
species cohesion over larger geographic distances than in (right), with
a lower passive dispersal. Modified from And\selectlanguage{ngerman}újar \emph{et al.} (2017).
\textbf{Figure 2.} Richness of soil mesofaunal lineages by sample (alpha
diversity, \textbf{A} ), mean endemicity by sample (\textbf{B} ), total
accumulated richness (local-scale richness or gamma diversity,\textbf{C}
), mean \selectlanguage{greek}β \selectlanguage{english}diversity among samples (\textbf{C} ), and richness
accumulation curves (\textbf{D} ) for haplotypes (left), 3\% OTUs
(middle), and 15\% lineages (right) by habitat (laurel forest, pine
forest, dry scrublands, and thermophilous woodlands). The significance
of Kruskal-Wallis rank sum test (post-hoc comparisons using Bonferroni
correction) is indicated for panels (\textbf{A} ) and (\textbf{B} ).
\textbf{Figure 3.} Non-parametric multidimensional scaling (NMDS)
ordinations (\textbf{A} ) and distance decay of genetic similarity
(\textbf{B} ) for soil mesofaunal samples. NMDSs represent the variation
in community composition (Simpson index, \selectlanguage{greek}β\selectlanguage{english}sim) for haplotype, 3\%
similarity OTUs, and 15\% similarity lineages. Explained variation (r2)
and significance (p) of habitat as a grouping factor from the
permutational ANOVAs over the community dissimilarity matrixes are
shown. Distance decay is plotted at multiple levels of genetic
similarity (from haplotypes, black, to 15\% genetic similarity, pale
grey) within the four habitats (laurel forest, La; pine forest, Pi; dry
scrubland, Ds; and thermophilous woodland, Tw).
\textbf{Figure 4.} Histograms showing the distribution of OTUs
(\textbf{A} ) and 15\% lineages (\textbf{B} ) according to the number of
sites (left) and the number of haplotypes (middle). Graphs on the right
indicate the number of OTUs and 15\% lineages found in either one or
multiple sites, and the number of OTUs and 15\% lineages with one or
multiple haplotypes. Also indicated for (\textbf{A} ) is the number of
OTUs with a similarity match [?] 97\% and [?] 99 with reference sequences
from outside the Canary Islands, and for (B) the number of lineages with
one or multiple OTUs.
\textbf{Figure 5.} Venn diagrams showing distribution of OTUs
(\textbf{A} ) and 15\% lineages (\textbf{B} ) among habitats (laurel
forest, La; pine forest, Pi; dry scrubland, Ds; and thermophilous
woodland, Tw). Venn diagrams illustrate patterns of exclusive and shared
OTUs and 15\% lineages. In parenthesis the following are respectively
indicated: the total number of OTUs and 15\% lineages (``L''), and the
number of those collected in a single site (``S''). Barplots on the
right represent the proportion and number of OTUs and 15\% lineages
sampled in more than two, three, four, and five sites that are
considered to have high habitat specificity ([?] 80\% of sampled sites
from a single habitat).
\textbf{Figure 6.} Graph showing the number of 15\% lineages with
genetic diversity significantly structured with increasing spatial
distance (orange) and with genetic diversity significantly structured by
habitat (purple) (\textbf{A} ), and examples of lineages with and
without significant habitat and spatial structure (\textbf{B} ,
\textbf{C} , and\textbf{D} ). Estimations in A considered only lineages
for which the product of the number of sites by the number of haplotypes
is [?] 15. Graphs in B, C, and D represent the correlation between
corrected geographical distance (x axis) and genetic similarity (y axis)
(left) and haplotype networks (right). Circle size represents the number
of sites where haplotypes are found, and colour represents the habitat
(laurel forest, yellow; pine forest, red; dry scrubland, blue; and
thermophilous woodland, green).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/figure-1/figure-1}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/figure-2/figure-2}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/figure-3/figure-3}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/figure-4/figure-4}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/figure-5/figure-5}
\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/figure-6/figure-6}
\end{center}
\end{figure}
\selectlanguage{english}
\FloatBarrier
\end{document}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%
%%% "Mathematische Physik II"
%%%
%%% im Sommersemester 2003
%%%
%%% Bei Rainer Wuest, Ulrike Scheerer (Assi)
%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt]{article}
\usepackage{german,a4wide,amssymb,amsmath, enumerate}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\pagestyle{empty} % Keine Seitenzahlen oder
% aehnliches
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% zw. zwei Absaetzen
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% vier Zeilen
\addtolength{\textheight}{+8\baselineskip} % Textblock vier Zeilen
% laenger
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%% n\"utzliche Definitionen von Aufgaben
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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% Aufgaben mit Punktzahl als Parameter.
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\begin{document}
{\large\bf 2. \"Ubungsblatt zur Mathematischen Physik II} \hfill
W\"ust/Scheerer \par\medskip
Abgabe am Montag, den 5.5.03 (vor der \"Ubung)\hfill SS 2003
\par\medskip
\hrule
\begin{aufgabe}{3}
Sei $A$ ein selbstadjungierter Operator und sei
$f\in\cal{S}(\mathbb{R})$. Zeige, dass dann gilt:
\[ f(A) = \frac{1}{\sqrt{2\pi}} \int_{-\infty}^\infty
\hat{f}(t)e^{itA}dt. \]
\end{aufgabe}
\begin{ueaufgabe}
Sei $\cal H$ ein Hilbertraum und $A\in B(\cal H)$ ein beschr\"ankter
Operator.
\begin{enumerate}[a)]
\item Zeige, dass das Spektrum $\sigma(A)\subset\mathbb{C}$ in der
abgeschlossenen Kreisscheibe vom Radius $\| A\|$ um 0 enthalten ist.
\item Sei $K$ eine offene Kreisscheibe um 0, die $\sigma(A)$ enth\"alt
und sei $\Gamma=\partial K$ die geschlossene Randkurve von
$K$. Seien $f,g$ Funktionen, die analytisch in einer offenen
Umgebung von $\bar{K}$ sind. Damit sei der folgende Operator definiert:
\[ f(A) = \frac{1}{2\pi i} \oint\limits_\Gamma f(z)(z-A)^{-1}dz
.\]
\begin{enumerate}[(i)]
\item Zeige, dass die Definition von $f(A)$ unabh\"angig von der Wahl
der Randkurve $\Gamma$ (mit den obigen Eigenschaften) ist.
\item Zeige, dass $A= \frac{1}{2\pi i} \oint\limits_\Gamma
z(z-A)^{-1}dz$ gilt.
\item Zeige, dass $f(A)g(A) = (fg)(A)$ gilt.
\item Sei nun $A$ zus\"atzlich selbstadjungiert. Zeige, dass die obige
Definition von $f(A)$ mit dem Ausdruck $\int_{-\infty}^\infty
f(\lambda) dE_\lambda$ \"ubereinstimmt. Dabei bezeichnet
$\{E_\lambda\}_{\lambda\in\mathbb{R}}$ die zu $A$ geh\"orige Spektralschar.
\end{enumerate}
\end{enumerate}
\end{ueaufgabe}
\begin{aufgabe}{2}
Sei $G\subset \mathbb{C}$ ein Gebiet und $\Gamma=\partial G$ die
Randkurve von $G$. Sei $T$ ein abgeschlossener Operator, dessen
Spektrum (zum Teil) in $G$ enthalten ist: $\sigma(T)\cap
G\not=\emptyset$ und dist$(\sigma(T),\Gamma)>0$. Sei dann folgender
Operator $P$ gegeben durch
\[P = -\frac{1}{2\pi i} \oint\limits_\Gamma (T-z)^{-1}dz .\]
\begin{enumerate}[a)]
\item Zeige, dass $P^2=P$ gilt.
\item (\"Ubung) Sei nun $T$ ein selbstadjungierter
Operator. Zeige, dass dann $P=P^*$ gilt. \\
Zeige au\ss erdem, dass f\"ur
$\sigma(T)\cap G = \{E\}$ mit $E\in\mathbb{R}$ gilt $TP=EP$.
\end{enumerate}
\end{aufgabe}
\begin{aufgabe}{5}
F\"ur $t\in\mathbb{R}$ sei der lineare Operator $U(t)$ auf
$L^2(\mathbb{R})$ definiert durch $(U(t)f)(x) = f(x-t)$, ($f\in
L^2(\mathbb{R})$, $x\in\mathbb{R}$).
\begin{enumerate}[a)]
\item Zeige, dass $U(\cdot)$ eine unit\"are Gruppe definiert.
\item Sei nun eine unit\"are Gruppe $U(\cdot)$ gegeben, die schwach
stetig ist,
d.h. f\"ur alle $t_0\in\mathbb{R}$ und alle $f,g\in L^2(\mathbb{R})$
gelte $\lim\limits_{t\rightarrow t_0} (U(t)f,g) = (U(t_0)f,g)$.\\
Zeige, dass $U(\cdot)$ dann auch stark stetig ist,
d.h. f\"ur alle $t_0\in\mathbb{R}$ und $f\in L^2(\mathbb{R})$ gilt
$\lim\limits_{t\rightarrow t_0} \|U(t)f -U(t_0)f\| = 0$.
\end{enumerate}
\end{aufgabe}
\end{document}
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\title{Washington, DC: Report from the Anti-Fascist Mobilization of July 6}
\date{July 17, 2019}
\author{CrimethInc.}
\subtitle{With Reflections on the Past and Future of Anti-Fascist Tactics}
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On July 6, fascists attempted to hold a “Demand Free Speech”\footnote{Rather than the kind of state-enforced “freedom of speech” in which armored riot police bludgeon the general population while fascists call for even more violence, anarchists seek real freedom, in which communities can self-organize to defend against the violence of police and fascists alike.} rally in Washington, DC. Anarchists and DC Black Lives Matter mobilized in response. Although the massive police presence hampered what anti-fascists could do, the fascist rally was not a success, confirming that anti-fascists have largely succeeded in thwarting the street-level fascist movement that many feared would emerge in the Trump era. The question, now, is how we can employ the tactics we have popularized in the anti-fascist movement—black blocs, de-platforming, and investigative research—in other movements and contexts.
Here, we offer a short report from DC on how previous mobilizations in DC have informed the strategies they employ today and an account of the day’s events.
\hairline
A group of neo-fascists promised to bring out 1000 people to a rally in Washington, DC on July 6, including 100 or more battle-ready Proud Boys. The rally had big names associated with it and local, institutional, and regional support; the organizers included Enrique Tarrio, chairman of the Proud Boys. Because it was sandwiched between Donald Trump’s 4\textsuperscript{th} of July rally and “Christians United For Israel,” a major gathering of the Christian right, DC anti-fascists feared that this event could draw a much larger turnout than recent far-right events had in the area. If a new right-wing coalition emerged that could put numbers back into the streets, that would mean another round of draining confrontations between fascists and anti-fascists.
In the end, the “Demand Free Speech Rally” was disorganized and undermined by infighting. Their event was a bust: it brought out dozens, not a thousand. The Proud Boys, which seemed like the last group standing among the fascist organizations of two years ago, couldn’t muster the 100 attendees they promised. Identity Evropa also made a last-minute push and failed to bring out the numbers they anticipated.
While repugnant characters like Adrienna DiCioccio and Jack Posebiac have huge Twitter followings, that doesn’t translate into an on-the-ground street movement. They are social media grifters who consider a trending hashtag more important than the kind of organizing that can mobilize people. They can use their platforms to create misinformation campaigns that cause police to crack down on anti-fascists, but they lack the relationships, skills, and political savvy to produce mass turnout for their own events.
Of course, that doesn’t mean they can’t learn those skills. We were not born with those skills either. We have to remain vigilant, mobilizing resistance whenever there is a chance that fascists will succeed in their publicity stunts, so they are never positively reinforced for their efforts to recruit.
\hairline
Over the past two and a half years, organizers in DC have learned harsh lessons about the risks of collective action. DC anarchists were shocked when DC police chose to mass-arrest and then mass-prosecute over 200 people for the courageous actions of January 20, 2017. The preceding 10 years of comparatively restrained police tactics had made DC anarchists complacent. We had conceded ground to liberals when we should have been developing strategies that could prepare us for the moment when that scenario changed.
The now infamous Anti-Capitalist and Anti-fascist Bloc that responded to the inauguration of Donald Trump was isolated from the majority of people on the street in DC that day, leaving it vulnerable to repression. In setting out to deepen our relations with other social movements, local anarchists aim to ensure that we cannot be isolated from our communities.
One role the black bloc can play is to defend those who are vulnerable to police attacks. On July 6, the bloc mobilized when we heard a large number of fascists were close to the park; our goal was to preserve space for people from targeted communities.
Just a year ago, anarchists were asked to stay out of the sites of permitted demonstrations for fear of mass arrests and police violence. Our recent efforts to connect with others have enabled us to make space for anarchist actions in a way that was not possible between 2016 and 2018. Building and deepening relationships takes time, patience, and understanding. We hope that these efforts will equip us to be better prepared for demonstrations in DC in the future, the next time anarchists and anti-fascists mobilize.
When we act, ideally we should be swimming in a sea of people, not alone against the stream of police repression. Different tactics make us stronger and sharing space makes those tactical experiments safer. It’s harder to make a bugaboo out of the black bloc when other people in the streets know who we are and what we are there to do.
\section{Don’t Mute DC}
Earlier this year, white gentrifiers demanded that Shaw’s Metro PCS store shut off the music it has been playing for over 20 years. In the 1990s, Metro PCS was rocking Go-Go outside and selling beepers, back when the U Street corridor was known as “Black Broadway.” Formerly known as Chocolate City, DC has been seeing Go-Go events pushed out of the city into Maryland.
Go-Go is a historically Black music that developed in Washington, DC—a homegrown variant of funk, old-school hip-hop, and R\&B music, founded by DC legend Chuck Brown. The disappearance of Go-Go from nearly every neighborhood illustrates the displacement of Black people from DC.
A massive community-led response to the pressure on Metro PCS brought thousands out to U Street in the face of policing and gentrification. Later in spring 2019, Metro PCS returned Go-Go music to Chuck Brown Way. While police have been shutting down these shows, communities have been using Go-Go dance parties as a form of resistance to gentrification in DC.
When anarchists, Black Lives Matter activists, and local anti-fascists became aware of the right-wing mobilization, we believed it would be important to hold space for Black, Brown, and Indigenous joy in the face of white supremacy. We set out to help bring together social movements in a way that would secure the safety of DC residents and give the protest a specifically DC atmosphere.
\section{Before the Demonstration}
While DC organizers believed that this could be a large mobilization, the material force the Proud Boys promised didn’t show. It was a far cry from the “thousand give or take a few hundred” they predicted to the park service.
Leading up to the mobilization, we intentionally did not discuss their politics. We just branded their event for them, clarifying for everyone that it was part of the same fascist current as previous such demonstrations in DC. A week before the demonstration, public pressure compelled the Spy Museum to publicly denounce the alt-right and cancel their VIP event.
In a last-minute attempt to mobilize supporters and escalate the likelihood of right-wing violence, far-right media began circulating rumors about “antifa acid attacks.” Numerous Proud Boys used this opportunity to announce that they planned to carry loaded guns. On the other hand, these rumors also provided cover for several of the speakers to cancel in the face of public pressure from DC antifascists. Jack Posebiac, Mike Cernovich, and Omar Navarrow all backed out before the rally.
Roger Stone was a no-show. Laura Loomer blamed antifa for the rally ending early. All in all, their rally was logistical failure, with numerous events ending early and lacking attendance.
\section{Tactically Unimpressive, But Politically Successful?}
\textbf{10:30 am, at the beginning of the rally}—Black Lives Matter organizers explicitly expressed support for the black bloc. “See the people with masks over their faces, and in all black? Those are our people. They are here to keep us safe because police don’t. We keep each other safe.” As the demonstration got underway, crowds gathered against the backdrop of DC’s homegrown funk while anti-fascist speakers from different communities alternated with bands brought in by anti-fascists of color and Black Lives Matter DC.
\textbf{Around 11:30 am,} as the rally was getting started, groups of right-wing journalists repeatedly attempted to film the demonstration without the consent of organizers or the community at large. Organizers aimed to create an atmosphere characterized by consent and solidarity in which the attendees would actively prevent fascists from attacking community members. At one point, a group of drag queens chased out a group of fascists.
This began a skirmish between fascists and anti-fascists. As this unfolded, some of them continued heckling and attempting to film. Numerous Proud Boys and their supporters repeatedly came over and were chased away by community members, not just participants in the bloc.
For those not in the streets in DC on July 6, the black bloc was hardly the wrecking ball we were on J20. There are no smashed windows to write about, no video clips of Richard Spencer being punched in the face. The most exciting moments of the day were funny rather than inspiring—for example, playing tug of war with police over a newspaper box.
\textbf{Around 1:30 pm,} the bloc took the streets to confront Identity Evropa, who were a block away from their permitted rally.
Other participants in the movement are no longer seeking to police our actions, but the police are still eager to suppress even the most mundane acts of rebellion.
\textbf{At 5:30 pm,} anarchists, Black Lives Matter, and other rally attendees met up again in front of Trump hotel. Despite hundreds of police, several dozen people were able to hold their ground in front of the VIP Bus, delaying its departure by over an hour and a half. The bus, which was supposed to leave by 6:30, left around 8 pm for an undisclosed location in Northern Virginia.
In short, while the day can hardly be said to be a tactical victory, we were able to diminish and contain the threat that the rally posed to our communities. It’s for the best that we didn’t have to physically fight a large group of fascists, in any case.
\section{Towards the Future}
Days after the \#DemandFreeSpeech rally, the “Mother of All Rallies” organizers cancelled their annual pro-Trump demonstration. They cited censorship on social media, but it’s significant that this occurred days after the flop of the “free speech” demonstration.
When we think about the future—about how to make sure we gain strength and our opposition lays dormant—we need to reflect on the limitations we have experienced in this era and how to make sure we don’t repeat the mistakes of the past.
In many different mobilizations, we have repeatedly confronted the same issues. When there is a small minority of people who feel comfortable standing up to the police, how do we relate to others who are not prepared to? How do we address the questions of timing and the safety of participants both inside and outside the bloc? When should we push the envelope, and when should we play a supporting role?
If we want to increase our ability to act and build relationships with other revolutionary movements, these are important questions. But we can think bigger about what we do next. How do we push for militant resistance in the face of border militarization, concentration camps, police killing, and state violence? How do we take the focus from individual fascists towards state-sanctioned racial violence.
As we look forward to replicating the success of mass anti-fascism, we can look to use the skills we’ve built in other arenas. The Republican Party’s vision, while not quite as overt as Richard Spencer’s program, still includes systemic ethnic cleansing and concentration camps for children. How can we map the relationships that create these policies? How can we build campaigns that make it unpopular to support racialized policing and mass incarceration? How do we use research, counter-intelligence, social mapping, and street militancy against ICE and prisons? How do we use these skills in mutual aid projects as well?
On one hand, we have to build the ability to sustain and intensify our offensive organizing. On the other hand, we have to support each another through crisis capitalism.
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Anti-Copyright
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CrimethInc.
Washington, DC: Report from the Anti-Fascist Mobilization of July 6
With Reflections on the Past and Future of Anti-Fascist Tactics
July 17, 2019
\bigskip
Retrieved on 17\textsuperscript{th} June 2021 from \href{https://crimethinc.com/2019/07/17/washington-dc-report-from-the-anti-fascist-mobilization-of-july-6-with-reflections-on-the-past-and-future-of-anti-fascist-tactics}{crimethinc.com}
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%% File: linSysGuide.tex
%%
%% Study Guide for Numerical Methods with MATLAB
%% Chapter 8, "Solving Systems of Equations"
%%
%% Gerald Recktenwald, [email protected]
%% August 2000
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\title{Study Guide for\\
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%\maketitle
% -------------------------
\section*{Bare Essentials}
At the end of this chapter you should be able to
\begin{enumerate}
\item Transform equations written in the natural variables of an
applied problem to the canonical $Ax=b$ of linear algebra.
\item Explain the condition of consistency in terms of linear combinations
of column vectors.
\item Explain the condition of singularity of an \matdim{n}{n} matrix
in terms of linear independence.
\item Express matrix rank as a measure of linear independence.
\item Relate rank of the coefficient matrix to the consistency of a \matdim{n}{n}
system of equations.
\item Write the formal solution to $Ax=b$.
\item Explain why it is not a good idea to use the formal solution as a
computational procedure for solving $Ax=b$.
\item Describe the most efficient procedures for solving $Lx=b$ or $Ux=b$
when $L$ is lower triangular and $U$ is upper triangular.
\item Name the solution algorithm most commonly used for solving $Ax=b$.
\item Write the equation that defines the residual vector.
\item Describe the significance of $\kappa(A)$ on the reliability of
the numerical solution to $Ax=b$.
\item Describe the significance of \norm{r} for a well-conditioned $A$.
\item Describe the significance of \norm{r} for a ill-conditioned $A$.
\item Describe the reason for pivoting. Is pivoting a remedy for
ill-conditioned systems?
\setcounter{enumSave}{\value{enumi}}
\end{enumerate}
\bigskip
\noindent To perform basic solutions of linear systems with \MATLAB\ you will need to
\begin{enumerate}
\setcounter{enumi}{\value{enumSave}}
\item Assign the elements of matrix \texttt{A}, and vector \texttt{b},
for a system of equations.
\item Write a compact (one line) statement that uses the recommended method for
solving $Ax=b$, given that $A$ and $b$ are already assigned to
\MATLAB\ variables.
\item Compute \norm{r} of a system given that $A$, $x$, and $b$
are already assigned to \MATLAB\ variables.
\end{enumerate}
% -------------------------
\section*{An Expanded Core of Knowledge}
After mastering the bare essentials you should move on to a deeper understanding
of the fundamentals. Doing so involves being able to
\begin{enumerate}
\item Describe the qualitative relationship between the magnitude of $\kappa(A)$
and the singularity of $A$.
\item Estimate the number of correct significant digits in the numerical
solution to $Ax=b$ given values of \epsm\ and $\kappa(A)$.
\item State conditions required for a successful LU factorization of $A$.
% $A$ is square, nonsingular
\item Write (describe) a procedure for solving $Ax=b$ given an LU factorization of $A$.
\item State conditions required for a successful Cholesky factorization of $A$.
% $A$ is symmetric and positive definite
\item Write (describe) a procedure for solving $Ax=b$ given a Cholesky factorization of $A$.
\setcounter{enumSave}{\value{enumi}}
\end{enumerate}
\bigskip
\noindent To perform more advanced solutions of linear systems with \MATLAB\ you will need to
\begin{enumerate}
\setcounter{enumi}{\value{enumSave}}
\item Write the preferred expression for solving $Lx=b$ or $Ux=b$
when $L$ is lower triangular and $U$ is upper triangular.
What algorithm does \MATLAB\ select to implement the solution
for these systems?
\item Use \MATLAB\ and the LU factorization of $A$ to solve several systems
of equations that have the same $A$ and a sequence of different $b$.
\item Use \MATLAB\ and a Cholesky factorization of $A$ to solve several systems
of equations that have the same $A$ and a sequence of different $b$.
\item Implement solutions of nonlinear systems of equations with
successive substitution.
\item Implement solutions of nonlinear systems of equations with
Newton's method.
\end{enumerate}
\clearpage
% -------------------------
\section*{Developing Mastery}
Working toward mastery of solving systems of equations you will need to
\begin{enumerate}
\item Given a variety of \matdim{m}{n} system of equations, where $m$ is not necessarily
equal to $n$, describe the method used by the \verb|\| operator to solve
$Ax=b$.
\item Given $L$, $U$, and permutation matrix $P$ from an LU factorization of $A$, apply
these to solve $Ax=b$. Specifically, use the $P$ appropriately.
\item Explain how \MATLAB\ uses the $L$ and $U$ factors returned from the \texttt{lu}
command to solve $Ax=b$ \emph{without} explicitly requiring $P$.
\item List the order of magnitude work estimates for Gaussian elimination with back
substitution, LU factorization, and Cholesky factorization.
\end{enumerate}
\end{document}
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\title{Aberration: The Automobile}
\date{}
\author{Lynne Clive (Marilynn Rashid)\forcelinebreak L’Encyclopedie des Nuisances}
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\begin{quote}
\textbf{Introduction}
\end{quote}
\begin{quote}
It is said that the automobile industry created and brought life to the cities, but once again official history dangerously misrepresents and distorts the facts. In reality, it is responsible for the destruction of viable human communities and emblematic of death culture all over the world. The auto industry’s monopolistic power kept Detroit and the rest of the world from creating alternative urban environments and consciously built car cities and a car world, chopped up and destroyed by incredible expressway systems—cities and a world for cars, not for people.
\end{quote}
\begin{quote}
The automobile is a murderous weapon, responsible for 40,300 to 50,000 deaths and one and-a-half to two million disabling injuries yearly in the U.S. alone. Between 1913 and 1985, 2,530,119 people were killed in the U.S. in traffic accidents. Countless others were killed and injured on the job, in the production of the automobile. These figures have been on the rise all over the “civilized” world for the past five years. They are called “accidents” and yet they are assumed, calculated, expected—acceptable losses that go with the territory. Sterile, cold statistics somehow blur the bitter reality of this horrifying yet preordained massive slaughter.
\end{quote}
\begin{quote}
The automobile is the embodiment of a culture of waste—the waste of human lives, of natural resources, the waste of people’s time and energies. A blatant example of material waste is the phenomenon of the abandoned car: 7 million cars are junked each year in the U.S. The automobile lays waste to the landscape with its unending network of roads and freeways. It lays waste to the environment, creating acid rains that critically threaten the ecosystem, polluting the city air and thus perilously increasing the incidence of respiratory diseases. In its production, each automobile creates 50 barrels of toxic waste.
\end{quote}
\begin{quote}
And we waste our days on this commodity that wastes us. Think of the time spent shopping for and purchasing a car, buying car insurance, getting licenses, driving permits; the time spent in maintenance and repair; the time spent waiting in your car in traffic jams, on the road, waiting at lights, in gas stations, in auto supply stores, in traffic court; the time spent looking for parking spaces and running nervously back and forth to parking meters; and worst of all, think of all the time spent at work making money to pay for it all. The car is clearly the focal commodity of capitalism. It purports to free us, but it binds us tightly in to a maniacal mechanical circle that parcels out our time in a frenetic stop and go.
\end{quote}
\begin{quote}
City dwellers grow up listening to the constant roar of freeway traffic, engine noises, squealing brakes, and horns outside their windows. Most seem unaware when the urban disquiet breaks the “acceptable” decibel levels. Most seem not to notice when yet another “necessary” invention, like the car alarm, fills the day and night with countless frantic interruptions. Perhaps they adapt somewhat to the noise. Perhaps they grow quietly insane.
\end{quote}
\begin{quote}
As the accompanying article from the French publication Encyclopedie des Nuisances recounts, the automobile, in its quick shift from luxury to necessity, has become an insidious obsession. Myopically perceiving the qualities of “speed” and “efficiency” as crucial and invaluable, people willingly accept the machine that leads them into a cycle of doom. They mindlessly allow it to control their lives and shape their environment.
\end{quote}
\begin{quote}
“Aberration” is a clever and poetic treatise on the automobile. It exposes the technological trick played on humankind at the expense of the earth’s natural beauty, at the expense of community, at the expense of our very lives. Although its focus is Europe, it adeptly expresses many of our own sentiments about this commodity that has so thoroughly and so devastatingly transformed our daily existence.
\end{quote}
\begin{quote}
—Lynne Clive
\end{quote}
\section{Aberration: The Automobile}
(This text was collectively translated in Montreal—with some stylistic changes by the FE staff.)
Aberration in people’s lives is not a passive phenomenon caused by a deceptive world that deludes their judgment. It is the consequence of non-reasoned acts whose sequence derives its logic from seemingly immediate constraints which are never questioned. Generalized aberration feeds on this lack of awareness. Since commodities and their circulation are the essence and basis of this social organization, we have chosen to consider the resulting aberration from a particular yet exemplary perspective—by looking at the circulation of a key commodity: the automobile.
Until the Second World War, the automobile was generally a luxury belonging to one class (FE note: this is more true of Europe than of the U.S.), which is apparent in the quality of production of that era and in the satisfactions one could experience from a landscape that had not yet been transformed for the automobile’s use. But the market has its own needs, and regardless of the sumptuousness of yesteryear, the privately-owned automobile was also destined for the poor, becoming what even an economist like E.J. Mishan (in \emph{The Costs of Economic Growth}) called the greatest catastrophe of this century.
An increasing proportion of activity was organized around and for it, and already the first indications of success were appearing: in the U.S., the victims of a three-day-long traffic jam were lifted provisions by helicopter, and, in 1953, filled with auto-non-mobiles for the first time, Rome glimpsed the end of its eternity when a several-hour-long horn concert demonstrated beyond a doubt the joy of her accession to the modern economic disorder, to the orgies of moving at a snail’s pace and to the revels of circulatory thrombosis.
In order to develop, the cities believed that they were honoring themselves by copying everything that was disagreeable from the suburbs. The expressway reached more people and regions, infesting the countryside and giving mass recreation an appearance of noble modernity. After this triumph, the addiction to the pneumatic tire had definitely shaped people’s habits. The central truth involved was no longer necessarily an inflated statement: “The automobile is a working tool, and not at all for driving around.” (Francois Michelin, Ingenieurs de \Slash{}Automobile, October 1979)
\section{An Implement for Self-Punishment}
The ordinary wage-earner must devote a quarter of his or her working time to support something that is definitely not a ballet dancer and that resembles an implement for self-punishment (see \emph{Le Temps Qu’on Nous Vole}, Jean Robert, 1980). And the infamy of this finality corresponds to what the economy calls well-being, which in this case is only the pseudo-facility of traversing the landscape of alienation. As always, this is only realized by damaging and altering the purpose of what had originally been produced. The luxury that the automobile was meant to incarnate, which, like any other luxury, implied privilege and ease, was never intended for modern wage-earners; it was through a fascinating aberration that luxury has lent its name to the inconvenience of modern objects. But for all their mobility, these common objects cannot yet circulate by themselves. To wear out and fall into decay, they need the very workforce they have to transport, which, in its turn, suffers the same fate.
We must therefore turn our attention to their owners and keepers—the drivers. Motorists, who work to go to work, are doubly ill-treated, and are directly controlled by the police down to their least significant actions. The network of roads is the drivers’ immense work camp, subject to the arbitrary annoyance of a corps which is specialized in repression. Feared as potential murderers, motorists personify to perfection the human model of a decadent society; they are obedient and aggressive; powerless and anxious to dominate; pathetic and narcissistic. They lack two virtues when operating their machines—sophistication and mastery. In effect, they exist only as representatives of the objects they put into motion. Isolated inside their machines, what is only a functional connection between things assumes for them the unreal form of relationships between people. However, they are confident in the future and believe in progress with the same conviction with which they drive—that is, until the statistically inevitable catastrophe.
Their religion does not vanish in a haze of theological quibbling; it is quite concrete and indisputable, with redemption through labor expressed in horsepower. But religion cannot exist without fanaticism and sweat, and boredom’s crystallization as a salary constitutes the only thing our motorized citizen is still able to fight and even kill for. In the zones where blasphemy, or at least impiety, reigns, the faithful even organize militias and, weapons in hand, sit up all night guarding their cars parked on gloomy lots. Because tomorrow they will have to fulfill the moral contract that ties them to their automobile-ball and chain and its developed terrain.
But, as with any other religion, what counts is the ritual and not the result. Otherwise, this one would have already broken down because of this simple observation: the average motorized city-dweller’s speed is approximately twice a pedestrian’s. However, if the social time required to produce the means of transport is added to the traveling time, the average global traveling speed ends up being inferior to that of people of the Paleolithic age. Such an objectively laughable result would legitimately disturb the user and planner if objectivity constituted a criterion of judgment in this society. No such luck! And what might only provoke a smile becomes a cruel joke when one realizes that reaching this point necessitated upsetting the rural and urban landscape from top to bottom.
\section{Everything is Organized in Military Fashion}
The initial development of the automobile attained its limits quite rapidly; its very success provoked a situation in which it was impossible for it to be used. The most visible cause of the destruction of cities has been their adaptation to automobile traffic. This adaptation summarizes quite well how the intensification of commodity traffic overcame the subsequent aberrations of its propagation, through aberrations which were even more vast, until new developments permitted new automobiles to be put into circulation, awaiting a new suffocation.
Henceforth, in the ideal city—ideal for the automobile, that is—one third of the surface is destined for the circulatory network, one third for parking and one third for residual activities. But this remarkable achievement reached perfection with the construction of a vast road network exclusively meant for the automobile—expressways. Here we find the subjective motive which justifies the objective aberration of the means of traveling—the journey.
It is well known that expressway construction and the motorization of the labor force was one of the components of the mobilization of the German proletariat under National Socialism. Both the Volkswagen and the Panzerwagon could circulate on the expressways, with the military excursion constituting the other original blemish that dominates the modern journey. Everything submits to the same demand for speed and efficiency and to the same reality of slowness and waste. One can be certain to lose time, at best, and at worst life itself. During the elaborate maneuvers of going on vacation, which for the great majority of motorists is the opportunity of a real trip, everything is organized in military fashion. On “D” Day, the general staff organizes radio guidance for the legions of vacationers. From the weather report to light aircraft reconnaissance flights, from reminders about necessary discipline to extrication itineraries in case the offense gets bogged down, everything has been foreseen for traversing hostile lands, from rescue squads to the installation of special tribunals.
Then the balance sheet is drawn up. Naturally, the losses are in proportion to the undertaking: during one year in a reasonably bellicose country like France, fatalities amount to the equivalent of a large infantry division, and the number of injured to several army corps. Such a criminal slaughter is perfectly accepted by the population as a natural disaster about which, by definition, nothing can be done. This incredible fatalism well demonstrates, once again, the general loss of common sense in our era.
\section{A Wandering Which is Purely Formal}
These grotesque or tragicomical aspects are the consequence of a journey that has become pure motion. What modern travelers find everywhere, because of their separation from the scenery they traverse only with their eyes, is their submission to a time that does not belong to them. Fragments of petrified reality which line the route—residues of a formerly-lived totality in which they once made sense—become literally inaccessible and are indicated solely to prevent the driver from dropping off to sleep. And since the remains of ancient history are not sufficiently displayed along the expressways, monuments that can’t be seen must be pointed out; it must be indicated that the motorists are in fact traversing the countryside of the region in which they find themselves.
Such an impoverished route, whose arbitrary pseudo-reality culminates in the noteworthy monument or the not-to-be-missed scenic view, still requires security-inducing rest stops where the neo-travelers will be able to satisfy their elementary needs with the usual products from the supermarket. The nomads’ wandering is purely formal, said Hegel; they carry their world with them. In the same way, the modern travelers move through their now motionless world and the familiar commodity precedes them everywhere; it is already there waiting for them.
At the next stage of this logic, one finds the ex nihilo fabrication of synthesized realities which are meant to be laid out along the expressway’s axis—typical villages, frontier towns, disneylands and the futuropolis. To get there—to these dumps of the spirit—the motorists should have acquired a new way of orienting themselves which no longer has anything to do with a concrete practical human sensitivity (even one as ludicrous as that which permits people to find their bearings because they turned right once and left twice). But there is no longer any other means of going from one point to another except by following the posted instructions—threatening subtitles of the misery of the landscape. Mistakes are not anticipated and they are irreparable; one is obliged to wait for the next exit leading to another labor or leisure unit. A sense of direction which might still bind us too much to primitiveness is substanceless and useless here. The different means that used to exist for “going straight ahead,” whether in the city or the country, choosing a series of streets or roads, are abolished by a redundancy of turn-offs which evoke psychotechnical tests for rats in a laboratory labyrinth.
\section{Autonomy Turned Into its Opposite}
What was promised by automobile motion—autonomy—turned into its opposite, not only in the pseudo-choice of a destination, but also with respect to mastering the machine. Once again, fairytale electronics promise to remedy everything, initially by giving a new look to technical and social archaisms. They make the machine speak, and already city nights are filled with screams of raped cars. But humanoidisation by computer systems tends to more directly entail technical aspects. “It is not so much technology itself that will upset the automobile world, but the substitution of human intelligence, which is too unreliable, with an artificial one.” (Michel Guegan, \emph{Les Dynasteurs, supplement to Les Echos}, October 9, 1985).
Since cars are work tools, their drivers have the same responsibilities towards the social organization that the operators of any other machine have, and whim becomes an offense and subject to prevention, as in any other labor activity. “Administering proper functioning, pampering the drivers, electronics will simplify their lives, but in a certain sense will place them in a state of supervised liberty. New traffic conditions in the year 2000 will ill-tolerate excesses of temperament, and electronics will help to establish a new social order on the roads” (ibidem).
We’ll forego comments about the police fantasies of these two bit standardizers. We will simply note that the sophistication and diversification of this archaic machine into hundreds of identical models evokes to a certain extent a kind of evolutionary dead-end similar to that of insects, rather than the arrival of a new age. The demiurge of this evolution in production remains the market, which today depends exclusively on the creation of pseudo-needs for which people agree to work. The automobile has fulfilled its role to perfection as motor of the economy for 50 years at the cost of harmful effects that we have only touched upon here, and of others that we have left out, for example, its massive contribution to generalized pollution or the insane waste of resources it demands. But it seems more important to us to conclude by denouncing what augurs rather than what has already occurred.
\section{People Continue on Their Way to Atomization}
As usual, our best arguments come from our enemies, in this case from those fanatical partisans of modern alienation who are satisfied with what exists and at the same time by what is being established. “The automobile has structured our physical space since the beginning of the century. Electronics and telecommunications are beginning to structure our intellectual space. Will they renew economic and human geography as well?” ask the editors of a special issue of the magazine \emph{Sciences et Techniques}, modestly entitled “Revolution of Intelligence.”
To prevent the apparent neutrality of the formulation and the final question mark from disturbing the reader’s positivist convictions, they specify further on in the text: “The steam engine has replaced the physical resources of people or animals. This time, it is a conceptual mastery which is promised, and no longer only the physical one offered by the automobile.” And a representative of MIT, Edward Fredkin, is able to conclude, with the brutal frankness which is specific to a continent that voluntarily ignores contradictions: “Artificial intelligence is evolution’s next step.”
Thus, fortunately, the abstract movement allowed by the automobile is completed by the motionless movement of telecommunications. People continue on their way to atomization, and separation increases to the point of being a programmed loss of independent thought. Through the gratification of needs that were created by its former development, the economy has discovered a new world with possibilities as limitless as the fundamental dissatisfaction it produces. Because what characterizes this new market is that the underlying modern commodity—the message (from software to banal TV entertainment)—has become purely non-material. All the madness of a mode of production which has become independent from people reaches its inevitable conclusion when none of its premises are radically questioned. And the disasters we have become used to are nothing in comparison to those we can expect.
L’Encyclopedie Des Nuisances
BP 188, 75665 Paris Cedex 14, France
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\strut
\vfill
\begin{center}
Lynne Clive (Marilynn Rashid)\forcelinebreak L’Encyclopedie des Nuisances
Aberration: The Automobile
\bigskip
\href{https://www.fifthestate.org/archive/325-spring-1987/aberration-the-automobile}{\texttt{https://www.fifthestate.org/archive/325-spring-1987/aberration-the-automobile}}
Fifth Estate \#325, Spring, 1987
\bigskip
\textbf{fifthestate.anarchistlibraries.net}
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% %%
%% Copyright (C) 2013 Texas Instruments Incorporated - http://www.ti.com %%
%% %%
%% Author: Felipe Balbi <[email protected]> %%
%% %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{The Sprint}
\label{chap:the-sprint}
\paragraph{}
The \textit{basic unit of development in Scrum}\cite{wikipediascrum} whose
duration is pre-defined in advance. All sprints have the same duration and,
usually, is defined to last between one week and one month.
\paragraph{}
In section \ref{sec:sprint-duration} we shall discuss about Sprint Duration and
what is Linux Core Product Development's accepted duration. In the following
section \ref{sec:scrum-process} we shall expose Scrum's development process and
discuss how to handle an iterative development model while development Linux
Kernel code. Lastly, on section \ref{sec:sprint-burndown-charts} we will expose
the idea of Sprint Burndown Charts; what they are and how to use them.
\paragraph{}
By the end of this chapter, we should have enough grounds to discuss about how
to successfully break backlog entries into bite-sized pieces in chapter
\ref{chap:breaking-tasks-up} and finally come up with our definition of done in
chapter \ref{chap:definition-of-done}.
\section{Sprint Duration}
\label{sec:sprint-duration}
\paragraph{}
Linux Core Product Development team will use sprint of 2 weeks with no
exceptions.
\section{Scrum Process}
\label{sec:scrum-process}
\paragraph{}
Figure \ref{fig:scrum-process} shows Scrum's process. Note that development is
carried away as a loop.
\begin{figure}[ht]
\centering
\includegraphics[width=0.8\textwidth]{images/scrum-process.png}
\caption{Scrum Process}
\label{fig:scrum-process}
\end{figure}
\section{Sprint Burndown Charts}
\label{sec:sprint-burndown-charts}
\paragraph{}
Figure \ref{fig:example-burndown-chart} shows an example Sprint Burndown Chart.
\begin{figure}[ht]
\centering
\pgfplotsset{width=12cm}
\pgfplotsset{grid style={dashed,lightgray}}
\begin{tikzpicture}
\begin{axis}[xlabel=Days,ylabel=Effort (hours),grid=both]
\addplot coordinates {
(1, 60)
(2, 51)
(3, 47)
(4, 40)
(5, 35)
(6, 22)
(7, 13)
(8, 8)
(9, 3)
(10, 0)
};
\end{axis}
\end{tikzpicture}
\caption{Example Sprint Burndown Chart}
\label{fig:example-burndown-chart}
\end{figure}
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%(If you can read TeX fluently, you may wish to fast-forward to a line
%beginning %file starts here or \frfilename
%if one of these is present.)
%TeXfiles with this heading are designed for processing with AmSTeX,
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% amsppt.sty and possibly amsppt.sti) during compilation.
%If these are unavailable, a readable
%version can (I hope) be produced by deleting the % before
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%Some special difficulties arise with the importation of .ps files
%in diagrams. Look for \picture below for notes on these.
%The following \count registers are used for special purposes:
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% \count2=\footnotenumber to count footnotes
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\DHFamstexloadedfalse
\newif\iffourteensavailable
\fourteensavailabletrue
\fourteensavailablefalse
\newif\ifathome
\athometrue %to cope with a quirk of my son's Ghostscript
\athomefalse
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%Style assignments
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\medskip}
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\hskip 3truein University of Essex\par
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\hskip 3truein England\par
%\hskip 3truein van Vleck Hall\par
%\hskip 3truein 480 Lincoln Drive\par
%\hskip 3truein Madison\par
%\hskip 3truein WI 53706-1388, U.S.A.\par
\hskip 3.2truein#1\par
\noindent e-mail: {\tt fremdh\@essex.ac.uk}
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{\parindent=0pt
#2
}
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\noindent Dear #3,
}
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\noindent e-mail: fremdh\@essex.ac.uk}
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%picture importation
%It is clear that there is no general standard. The syntax I
%use when calling for a .ps file is
% \def\Caption{<...>}
% \picture{<filename>}{<dim>}
%which is supposed to import <filename>.ps at height <dim> ,
%centered, and adding a one-line centered caption <...> .
%The basic macro listed in \def\picture is that which
%used to work at Essex. We now use the macro in
% \def\picturewithpsfig , which requires a previous input
% psfig.tex or psfig.sty ;
%if you interpolate \atUEssex early in your file it will
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% I find that psfig is erratic in its centering, especially
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% \centerpicturewithpsfig (which may be called from
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}}
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\sideshiftedpicture{#1}{#2}{\dimen3}{\dimen4}}
%alternative picture importation code
\def\atUEssex{
\input psfig.sty
\def\picture##1##2{\picturewithpsfig{##1}{##2}}
\def\sideshiftedpicture##1##2##3##4{
\sideshiftedpicturewithpsfig{##1}{##2}{##3}{##4}}
\def\centerpicture##1##2##3{
\centerpicturewithpsfig{##1}{##2}{##3}}
\def\startsideshiftedpicture{
\sideshiftedpicture{empty}{1pt}{1pt}{1pt}}
}
\def\athome{
\input psfig.sty
\def\picture##1##2{\picturewithpsfig{##1}{##2}}
\def\sideshiftedpicture##1##2##3##4{
\sideshiftedpicturewithpsfig{##1}{##2}{##3}{##4}}
\def\centerpicture##1##2##3{
\centerpicturewithpsfig{##1}{##2}{##3}}
\def\startsideshiftedpicture{
\sideshiftedpicture{empty}{1pt}{1pt}{1pt}}
\voffset=1cm}
\def\atUWMadison{
\fourteensavailablefalse
\input psfig
\def\picture##1##2{\picturewithpsfig{##1}{##2}}
}
\def\picturewithpsfig#1#2{\bigskip
\immediate\write0{importing diagram #1.ps at height #2 with psfig}
\vbox{\hsize\pagewidth
%\centerline{\discrversionA{\hskip1truein}{}\psfig{figure=#1.ps,height=#2
\centerline{\psfig{figure=#1.ps,height=#2
}}
\medskip
\centerline{\Caption}}
\medskip
\def\Caption{}}
\def\centerpicturewithpsfig#1#2#3{
\immediate\write0{importing diagram #1.ps at width #2
and height #3 with psfig}
\dimen 3 = #2
\dimen 4 = #3
\dimen 2 = \pagewidth
\advance\dimen2 by -\dimen3
\divide\dimen2 by 2
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\discrversionH{\shrinkonestep}{}
\discrversionF{\enlargeonestep}{}
\discrversionG{\enlargeonestep\enlargeonestep}{}
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\discrversionG{\vskip-\dimen6}{}
}
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\dimen6 = 7truein
\ifdim\dimen3<\dimen5
\ifdim\dimen4<\dimen6
\divide\dimen3 by 5
\multiply\dimen3 by 6
\divide\dimen4 by 5
\multiply\dimen4 by 6
\fi\fi}
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\multiply\dimen3 by 5
\divide\dimen4 by 6
\multiply\dimen4 by 5}
\def\sideshiftedpicturewithpsfig#1#2#3#4{
\immediate\write0{importing diagram #1.ps at height #4
and width #3 with psfig}
\dimen 4 = #4
\dimen 3 = #3
\discrversionB{\shrinkonestep}{}
\discrversionH{\shrinkonestep}{}
\discrversionF{\enlargeonestep}{}
\discrversionG{\enlargeonestep\enlargeonestep}{}
% \vbox{\psfig{figure=#1.ps,height=\dimen4,width=\dimen3}}
\psfig{figure=#1.ps,height=\dimen4,width=\dimen3}
\hskip #2 plus 0pt minus 0pt}
%alternative picture importation code; I've been told about this
%but have not used it successfully myself
%\def\Caption{}
%\input epsf
%\epsfverbosetrue
%\def\picture#1#2{\bigskip
% \bigskip
% \immediate\write0{importing diagram #1.ps at height #2 with epsf}
% \epsfysize=#2
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% }
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\font\fourteenrm=cmr14
\font\fourteenbf=cmbx14
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\else\font\fourteenrm=cmr10 scaled \magstep2
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\font\fourteenit=cmsl10 scaled \magstep2
\fi}
\def\smallerfonts{\eightpoint
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\textfont\eufmfam=\eighteufm
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\textfont\eurmfam=\eighteurm
\textfont\eusmfam=\eighteusm
}
\font\twelveex=cmex10 scaled \magstep1
%Copyright declarations
\newcount\oldcopyrightdate
\newcount\newcopyrightdate
\oldcopyrightdate=0
\def\copyrightdate#1{\newcopyrightdate=#1
\ifnum\newcopyrightdate=\oldcopyrightdate\else
\oldcopyrightdate=#1
\oldfootnote{}{\eightrm \copyright\ #1 D.~H.~Fremlin}
\immediate\write0{Copyright #1}
\fi}
%hyphenation
\hyphenation{equi-ver-id-ical}
%miscellaneous definitions
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\def\backstep#1{\hskip -0.#1em plus 0 pt minus 0 pt}
\def\Bang{\vthsp\hbox{\bf X\hskip0.07em\llap X\hskip0.07em\llap X\rm}}
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%\def\BbbN{\mathchoice{\hbox{$\Bbb N$\hskip0.02em}}
% {\hbox{$\Bbb N$\hskip0.02em}}
% {\hbox{$\scriptstyle\Bbb N$\hskip0.04em}}
% {\hbox{$\scriptscriptstyle\Bbb N$\hskip0.04em}}}
%\def\BbbQ{\mathchoice{\hbox{$\Bbb Q$\hskip0.02em}}
% {\hbox{$\Bbb Q$\hskip0.02em}}
% {\hbox{$\scriptstyle\Bbb Q$\hskip0.04em}}
% {\hbox{$\scriptscriptstyle\Bbb Q$\hskip0.04em}}}
%\def\BbbR{\mathchoice{\hbox{$\Bbb R$\hskip0.02em}}
% {\hbox{$\Bbb R$\hskip0.02em}}
% {\hbox{$\scriptstyle\Bbb R$\hskip0.04em}}
% {\hbox{$\scriptscriptstyle\Bbb R$\hskip0.04em}}}
% %do not use with $\Bbb R^X$
%\def\BbbZ{\mathchoice{\hbox{$\Bbb Z$\hskip0.02em}}
% {\hbox{$\Bbb Z$\hskip0.02em}}
% {\hbox{$\scriptstyle\Bbb Z$\hskip0.04em}}
% {\hbox{$\scriptscriptstyle\Bbb Z$\hskip0.04em}}}
\def\BbbN{\openBbb{N}}
\def\BbbQ{\openBbb{Q}}
\def\BbbR{\openBbb{R}}
%do not use with $\Bbb R^X$ do not use in $_{\Bbb R^r}$
\def\BbbZ{\openBbb{Z}}
\def\Bbbone{\hbox{1\hskip0.13em\llap{1}}}
\def\biggerint{\textfont3=\twelveex\intop\nolimits}
\def\biggeriint{\textfont3=\twelveex\iint}
% remarkable fact: this seems to infect all \int within a given
% $..$ ; but $\biggerint$$\int$ seems to leave the $\int$ normal
% really \textfont3=\twelveex is more reliable, since \biggerint
% as written gets turned off rather easily (e.g. by \cmmnt,
% or by an extra {...})
\def\bover#1#2{{{#1}\over{#2}}}
\def\Bover#1#2{\backstep4\hbox{
\biggerscriptfonts${\shortstrut#1}\over{\shortstrut#2}$}}
\def\bsp{/\penalty-100}
\def\Caption{}
\def\Center#1{
\Centerline{#1}
\noindent}
\def\Centerline#1{\vskip 5pt
{\def\int{\text{}$$\textfont3=\twelveex\intop\nolimits}
\def\underlineint{
\text{}$$\textfont3=\twelveex\underline{\intop\nolimits}}
\def\overlineint{
\text{}$$\textfont3=\twelveex\overline{\intop\nolimits}}
\def\iint{\text{}$$\textfont3=\twelveex
\intop\nolimits\hskip-0.5em\intop\nolimits}
\def\sum{\text{}$$\sumop}
\def\prod{\text{}$$\prodop}
\def\bigcap{\text{}$$\bigcapop}
\def\bigcup{\text{}$$\bigcupop}
\centerline{#1}}\vskip 5pt}
\mathchardef\bigcapop="1354
\mathchardef\bigcupop="1353
\mathchardef\prodop="1351
\mathchardef\sumop="1350
%in \Centerline{$\overline{}$, $\cmmnt{}$} must use \sumop etc
\def\cf{\mathop{\text{cf}}}
\def\circumflex#1{{\accent"5E #1}}
\def\closeplus{\hbox{+}}
\def\coint#1{\left[#1\right[}
\def\curl{\mathop{\text{curl}}}
\def\diam{\mathop{\text{diam}}}
\def\discrpage{\ifnum\stylenumber=1{\frnewpage} %redefined in mt.tex
\else\ifnum\stylenumber=3{\frnewpage}
\else\ifnum\stylenumber=5{\frnewpage}
\else\bigskip\fi\fi\fi}
\long\def\discrversionA#1#2{\ifnum\stylenumber=1#1\else#2\fi}
\long\def\discrversionB#1#2{\ifnum\stylenumber=2#1\else#2\fi}
\long\def\discrversionC#1#2{\ifnum\stylenumber=3#1\else#2\fi}
\long\def\discrversionD#1#2{\ifnum\stylenumber=4#1\else#2\fi}
\long\def\discrversionE#1#2{\ifnum\stylenumber=5#1\else#2\fi}
\long\def\discrversionF#1#2{\ifnum\stylenumber=6#1\else#2\fi}
\long\def\discrversionG#1#2{\ifnum\stylenumber=7#1\else#2\fi}
\long\def\discrversionH#1#2{\ifnum\stylenumber=8#1\else#2\fi}
% facility for special interpolations in particular output
% versions
\def\displaycause#1{\noalign{\noindent (#1)}}
\def\diverg{\mathop{\text{div}}}
\def\dom{\mathop{\text{dom}}}
\def\eae{=_{\text{a.e.}}}
\def\esssup{\mathop{\text{ess sup}}}
\mathchardef\Exists="3239
\def\family#1#2#3{\langle#3\rangle_{#1\in#2}}
\def\familyi#1#2{\family{i}{#1}{#2}}
\def\familyiI#1{\familyi{I}{#1}}
\def\frfilename#1{\ifnum\stylenumber=1{\noindent filename #1}
\else\ifnum\stylenumber=6{\noindent{\fiverm filename #1}}
\else\ifnum\stylenumber=7{\noindent{\fiverm filename #1}}
\else\fi\fi\fi}
\def\Footnote#1#2{\footnote"#1"{#2}}
\mathchardef\Forall="3238
\def\frakc{\frak c\phantom{.}}
\def\frnewpage{\vskip \pageheight plus 0pt minus\pageheight
\eject
\gdef\topparagraph{}
\gdef\bottomparagraph{}
}
\def\frsmallcirc{{\hskip 1pt plus 0pt minus 0pt}
{\raise 1pt\hbox{$\smallcirc$}}{\hskip 1pt plus 0pt minus 0pt}}
\def\geae{\ge_{\text{a.e.}}}
\def\grad{\mathop{\text{grad}}}
\def\grheada{{\bf ($\pmb{\alpha}$)}}
\def\grheadb{{\bf ($\pmb{\beta}$)}}
\def\grheadc{{\bf ($\pmb{\gamma}$)}}
\def\grheadd{{\bf ($\pmb{\delta}$)}}
\def\grheade{{\bf ($\pmb{\epsilon}$)}}
\def\grheadz{{\bf ($\pmb{\zeta}$)}}
\def\grv#1{{\accent"12 #1}}
\def\hfillall{\hskip\pagewidth plus0pt minus\pagewidth}
\def\Imag{\mathop{\Cal I\text{m}}}
\long\def\inset#1{{\narrower\narrower{\vskip 1pt plus 1pt minus 0pt}
#1
{\vskip 1pt plus 1pt minus 0pt}}}
\def\Int#1#2{\displaystyle{\int_{#1}^{#2}}}
\def\interior{\mathop{\text{int}}}
\def\leae{\le_{\text{a.e.}}}
\long\def\leaveitout#1{}
\def\LLcorner{\mathchoice{\,\hbox{\vrule height7pt width0.5pt depth0pt
\vrule height0.5pt width5pt depth0pt}\,}
{\,\hbox{\vrule height7pt width0.5pt depth0pt
\vrule height0.5pt width5pt depth0pt}\,}
{\,\hbox{\vrule height5pt width0.3pt depth0pt
\vrule height0.3pt width4pt depth0pt}\,}
{\,\hbox{\vrule height4pt width0.25pt depth0pt
\vrule height0.25pt width3pt depth0pt}\,}
}
\def\Ln{\mathop{\text{Ln}}}
\def\Loadfont#1{\font#1}
\def\loibr{\hbox{\hskip0.1em]\hskip0.05em}}
\def\Matrix#1{\pmatrix#1\endpmatrix}
\def\Mu{\text{M}}
\def\noamslogo{\def\amslogo{}}
\def\normalsubgroup{\vartriangleleft}
\def\Nu{\text{N}}
\def\ocint#1{\left]#1\right]}
\def\ofamily#1#2#3{\langle#3\rangle_{#1<#2}}
\let\oldcases=\cases
\let\oldfootnote=\footnote
\def\On{\mathop{\text{On}}}
\def\ooint#1{\left]#1\right[}
\def\openBbb#1{\Bbb#1\mskip1mu}
\def\pd#1#2{\bover{\partial#1}{\partial#2}}
\def\Pd#1#2{\Bover{\partial#1}{\partial#2}}
\def\pdd#1#2{{{\partial^2#1}\over{\partial#2^2}}}
\def\Pdd#1#2{\Bover{\partial^2#1}{\partial#2^2}}
\def\pde#1#2#3{{{\partial^2#1}\over{\partial#2\partial#3}}}
\def\Pde#1#2#3{\Bover{\partial^2#1}{\partial#2\partial#3}}
\def\Prf{\vthsp\hbox{\bf P\hskip0.07em\llap P\hskip0.07em\llap P\rm}}
\def\pushbottom#1{\hbox{\vrule height0pt depth#1 width0pt}}
\def\Qed{\vthsp\hbox{\bf Q\hskip0.07em\llap Q\hskip0.07em\llap Q\rm}}
\def\QeD{\Qed{\hskip 1.111pt plus 3.333pt minus 0pt}}
\def\Quer{\vthsp\hbox{\bf ?\hskip0.05em\llap ?\hskip0.05em\llap
?\thinspace\rm}}
\def\query{\discrversionA{\immediate\write0{query}
\global\advance\footnotenumber by 1
\oldfootnote{$^{\the\footnotenumber}$}{query}}{}}
\def\Real{\mathop{\Cal R\text{e}}}
\def\restr{\mathchoice
{\hbox{$\restriction$}\mskip 1mu}
{\hbox{$\restriction$}\mskip 1mu}
{\hbox{$\scriptstyle\restriction$}\mskip 1mu}
{\hbox{$\scriptscriptstyle\restriction$}\mskip 1mu}}
\def\restrp{\mathchoice
{\hbox{$\restriction$}\mskip 2mu}
{\hbox{$\restriction$}\mskip 2mu}
{\mskip-1mu\hbox{$\scriptstyle\restriction$}\mskip 2mu}
{\mskip-1mu\hbox{$\scriptscriptstyle\restriction$}\mskip 1mu}}
%use in \restrp V \restrp W \restrp Y
% \restrp\Cal P \restrp\Bbb R \restrp\frak
% \restrp\Omega \restrp\Tau \restrp\Upsilon
% \restrp[ \restrp\widehat{}
\def\Rho{\text{P}}
\def\roibr{\hbox{\hskip0.05em[\hskip0.1em}}
\def\RoverC{\hbox{\biggerscriptfonts${{\Bbb R}\atop{\Bbb C}}$}}
\def\sequence#1#2{\family{#1}{\Bbb N}{#2}}
\def\sequencen#1{\sequence{n}{#1}}
\def\setover#1#2{\overset{#1}\to{#2}}
\def\shortstrut{\hbox{\vrule height7.5pt depth2.5pt width0pt}}
\def\smallcirc{{\scriptstyle\circ}}
\def\ssbullet{{\scriptscriptstyle\bullet}}
\def\symmdiff{\triangle}
\def\thicken#1{#1\hskip0.03em\llap #1\hskip0.03em\llap
#1\hskip0.03em\llap #1}
\def\Tau{{\text{T}}}
\def\tbf#1{\text{\bf #1}}
\def\terror#1#2#3{#1: `#2' for `#3'}
\def\today{\number\day.\number\month.\number\year}
\def\UEMDRR{University of Essex Mathematics Department Research Report}
\def\UEMDWP{University of Essex Mathematics Department Working Paper}
\def\versiondate#1{
\immediate\write0{Version of #1}
\def\vsdate{#1}
\ifnum\stylenumber=1\hfill Version of #1
\medskip
\else\ifnum\stylenumber=2\hfill Version of #1
\medskip
\else\ifnum\stylenumber=6\hfill{\fiverm Version of #1}
\medskip
\else\ifnum\stylenumber=7\hfill{\fiverm Version of #1}
\medskip
\else\ifnum\stylenumber=8\hfill Version of #1
\medskip
\else\fi\fi\fi\fi\fi}
\def\versionhead{\ifnum\stylenumber=3 \hfill PRINTER'S VERSION
\bigskip
\else\ifnum\stylenumber=4 \hfill REFEREE'S VERSION
\bigskip
\else \fi\fi}
\def\versionref{\ifnum\stylenumber=1\workingref
\else\ifnum\stylenumber=2\workingref
\else\ifnum\stylenumber=8\workingref
\else\printerref\fi\fi\fi}
\def\vthsp{\hskip0.001em}
%Missing AmSTeX can be coped with by using the following lines:
\ifamstexsupported\else
\immediate\write0{AmSTeX-unsupported version of fremtex.tex }
\fourteensavailablefalse
\def\@{@}
\def\\{\cr}
\def\AmSTeX{AmS\TeX}
\def\Bbb#1{\underline{\hbox{\bf #1}}}
\def\biggerscriptfonts{}
\def\bumpeq{=_{\text{approx}}}
\def\Cal#1{{\cal#1}}
\def\curlyeqprec{\preceq}
\def\eightit{\it}
\def\eightrm{\rm}
\def\eightsmc{\smc}
\def\eurm#1{\underline{\cal#1}}
\def\Footnote#1#2{\footnote#1#2}
\def\frac#1#2{{#1}\over{#2}}
\def\frak#1{{\hbox{\bf #1\hskip0.1em\llap #1}}}
\def\frnewpage{\vfill\eject}
\def\iint{\int\!\!\!\int}
\def\iiint{\int\!\!\!\int\!\!\!\int}
\def\iiiint{\int\!\!\!\int\!\!\!\int\!\!\!\int}
\def\llcorner{\lfloor}
\def\loadeurm{}
\def\loadeusm{}
% \def\loadeusmx#1{\def\eusm#1{\underline{\cal#1}}}
\def\ltimes{|\hskip-0.5em\times}
\def\eusm#1{\underline{\cal#1}}
\def\ltimes{|\times}
\def\Loadfont#1{}
\def\Matrix#1{\pmatrix{#1}}
\def\noamslogo{}
\def\nrightarrow{\not\rightarrow}
\def\preccurlyeq{\preceq}
\def\query{\discrversionA{\immediate\write0{query}
\footnote{$^*$}{query}}{}}
\def\restriction{|}
\def\rtimes{\times|}
\def\setover#1#2{#2^{#1}}
\def\shortstrut{\strut}
\def\smallerfonts{\eightrm}
\def\smallfrown{\wedge}
\def\smc{}
\def\spcheck{\check{\phantom{x}}}
\def\sphat{\hat{\phantom{x}}}
\def\square{\mathchoice\sqr64\sqr64\sqr43\sqr33}
\def\sqr#1#2{{\vcenter{\vbox{\hrule height.#2pt
\hbox{\vrule width.#2pt height#1pt \kern#1pt
\vrule width.#2pt}
\hrule height.#2pt}}}}
\def\pmb#1{\underline{#1}}
\def\text#1{\hbox{#1}}
\def\therefore{\ssbullet\raise.9ex\hbox{$\ssbullet$}\ssbullet}
\def\twelvebf{\bf}
\def\vartriangleleft{\triangleleft}
\def\Vdash{|\vdash}
\def\vDash{|=}
\fi
%End of lines for missing AmSTeX
\ifamstexsupported\input amstex
% \documentstyle{amsppt}
\input amsppt.sty
\PSAMSFonts
\tenpoint
%\font\eightmsb=msym8
\font\eightmsb=msbm10
\font\eighteufm=eufm10 at 8 pt
\font\eighteurm=eurm8
\font\eighteusm=eusm8
\font\varseveneufm=eufm10 at 7 pt
\scriptfont\eufmfam=\varseveneufm
\def\=#1{{\accent"16 #1}}
\gdef\nopagenumbers{\NoPageNumbers}
\else\fi
\newcount\footmarkcount
\ifDHFamstexloaded
\loadmsBm
\def\eightBbb#1{{\fam\msBfam#1}}
\loadmsFm
\def\eightfrak#1{{\fam\msFfam#1}}
\else
\def\eightBbb#1{{\scriptstyle\Bbb#1}}
\def\eightfrak#1{{\scriptstyle\frak#1}}
\fi
\ifathome\athome\fi
%interword glue (\tenrm standard):
% (space) 3.333 + 1.667 - 1.111
% (,) 3.333 + 2.083 - 0.889
% (.) 4.444 + 5. - 0.370
%very odd fact: \discrversionA{2.}{} doesn't work,
% \discrversionA{ 2.}{} does
%The false line feed introduced by my Fortran programs, leading to
%a surplus ae at the top of the next page, seems to be associated
%with Fk(\032)3081 or something similar in the PostScript file.
\immediate\write0{Extract from MEASURE THEORY,
by D.H.Fremlin, University of Essex, Colchester.
This material is copyright. It is issued under the terms of the
Design Science License as published in
http://dsl.org/copyleft/dsl.txt. This is a development version and the
source files are not permanently archived.
For further information contact fremdh\@essex.ac.uk}
%hyphenations
\hyphenation{Archi-me-dean}
%new style
\def\smallprintersize{\magnification=\magstep0
\overfullrule0pt
\def\pagewidth{468pt}
\def\pageheight{9truein}
\hsize\pagewidth
\vsize\pageheight
\nologo
\def\copyrightdate##1{}
\def\folio{\ifnum\pageno<0{}\else\number\pageno\fi}
\def\mtfootline{}
\stylenumber=10
\pageno=-1000}
%With-proofs, results-only versions
\newif\ifwithproofs
\withproofstrue
\newif\ifresultsonly
\resultsonlyfalse
\def\noproofs{\withproofsfalse}
\def\resultsonly{\resultsonlytrue\withproofsfalse}
\long\def\proof#1{\ifwithproofs{\medskip
\noindent{\bf proof} #1}\else\fi}
\long\def\prooflet#1{\ifwithproofs{#1}\else\fi}
%\long\def\Prooflet#1{\prooflet{\Prf\ #1 \Qed\ }}
\long\def\cmmnt#1{\ifresultsonly\else{#1}\fi}
%the "\comment" macro in AMSTeX is a form of
%\leaveitout
\long\def\endnotes#1{\ifresultsonly\else{#1}\fi}
\long\def\exercises#1{\ifresultsonly\else{#1}\fi}
\long\def\references#1{\ifresultsonly\else{#1}\fi}
\long\def\dvro#1#2{\ifresultsonly{#1}\else{#2}\fi}
%Headline assignments
\def\topparagraph{}
\def\bottomparagraph{}
\def\newparagraph{}
\newif\ifpageturned
\newdimen\rhoffset
\newdimen\lhoffset
\lhoffset=0cm
\rhoffset=0cm
\newcount\volumeno
\volumeno=5
\let\plainpagebody=\pagebody
\def\pagebody{\ifodd\pageno\global\hoffset=\rhoffset
\else\global\hoffset=\lhoffset\fi
\plainpagebody
\turnpage}
\def\turnpage{
\iffirstpage\else{\baselineskip=24pt\line{\mtfootline}}\fi
% \immediate\write0{turning page \folio,
% top paragraph \topparagraph, bottom \bottomparagraph,
% new \newparagraph, pagetotal \the\pagetotal}
\global\pageturnedtrue
\global\firstpagefalse
\global\let\topparagraph=\bottomparagraph
\global\let\bottomparagraph=\newparagraph
}
\newdimen\pagetotalplus
\newdimen\pagetotalminus
%wheader parameters: #1 new paragraph name
% #2#3#4 \vskip parameters if page break not forced (no. of points)
% #5 spare height demanded if page break not to be forced
\long\def\wheader#1#2#3#4#5{\gdef\newparagraph{#1}
\pagetotalplus=\the\pagetotal
\advance\pagetotalplus by #5
\pagetotalminus=\the\pagetotal
\advance\pagetotalminus by -\pageshrink
%\showthe\pagetotal \showthe\pageshrink \showthe\pagegoal
\global\pageturnedfalse
\ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal
% \immediate\write0{forcing page break}
\eject
\else\vskip #2pt plus #3pt minus #4pt\fi
\else\vskip #2pt plus #3pt minus #4pt\fi
\ifpageturned\global\let\topparagraph=\newparagraph
% \immediate\write0{paragraph change on turning page}
\fi
% \immediate\write0{top paragraph \topparagraph, bottom
% \bottomparagraph, pagetotal \the\pagetotal}
\ifdim\pagetotal>\pagegoal\else\gdef\bottomparagraph{#1}\fi
% \immediate\write0{top paragraph \topparagraph, bottom
% \bottomparagraph, pagetotal \the\pagetotal}
}
\def\header#1{
\wheader{#1}{10}{4}{4}{24pt}}
\def\leader#1#2{\immediate\write0{#1 #2}
\wheader{#1}{14}{7}{5}{24pt}
{\bf #1 #2}}
\def\vleader#1#2#3{\immediate\write0{#2 #3}
\wheader{#2}{14}{7}{5}{#1}
{\bf #2 #3}}
\def\Notesheader#1{\immediate\write0{#1 Notes and comments}
\wheader{#1 Notes}{16}{8}{5}{48pt}
\noindent{\bf #1 Notes and comments}}
\def\vNotesheader#1#2{\immediate\write0{#2 Notes and comments}
\wheader{#2 Notes}{16}{8}{5}{#1}
\noindent{\bf #2 Notes and comments}}
\def\spheader#1#2#3#4#5{\header{#1#2#3#4#5}{\bf (#5)}}
\def\spstheader#1#2#3#4#5{\header{*#1#2#3#4#5}{\bf (#5)}}
\def\vspheader#1#2#3#4#5#6{
\wheader{#2#3#4#5#6}{10}{4}{4}{#1}{\bf (#6)}}
\def\sqheader#1#2#3#4#5{\header{#1#2#3#4#5}{$\pmb{>}${\bf (#5)}}}
\def\errheader#1#2#3{
\wheader{#3}{6}{4}{2}{24pt}
{\bf p #1 l #2}}
\def\indexmedskip{
\pagetotalplus=\the\pagetotal
\advance\pagetotalplus by 18 pt
\pagetotalminus=\the\pagetotal
\advance\pagetotalminus by -\pageshrink
\ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal
\else\medskip\fi\else\medskip\fi}
\def\indexheader#1{\immediate\write0{#1}\vindexheader{#1}{24}}
\def\vindexheader#1#2{
\gdef\newparagraph{#1}
\pagetotalplus=\the\pagetotal
\advance\pagetotalplus by #2 pt
\pagetotalminus=\the\pagetotal
\advance\pagetotalminus by -\pageshrink
%\showthe\pagetotal \showthe\pageshrink \showthe\pagegoal
\global\pageturnedfalse
% \ifdim\pagetotal>\pagegoal
% \immediate\write0{warning: pagetotal greater than pagegoal}
% \immediate\write0{... last paragraph reached is #1}
% \immediate\write0{... check for underfull vbox here}
% \else\fi
\ifdim\pagetotalminus<\pagegoal\ifdim\pagetotalplus>\pagegoal{\eject}
\fi\fi
\ifdim\pagetotal>\pagegoal\else\gdef\bottomparagraph{#1}\fi
\ifpageturned{\global\let\topparagraph=\newparagraph
% \immediate\write0{paragraph change on turning page}
}\else\fi
% \immediate\write0{top paragraph \topparagraph, bottom
% \bottomparagraph}
}
\def\leftheadline{\rlap{\folio}\hfill{\eightit\chaptername}\hfill
\llap{\eightbf\topparagraph}}
\def\rightheadline{\rlap{\eightbf\bottomparagraph}\hfill
{\eightit\headlinesectionname}\hfill\llap{\folio}}
%Footline assignments
\def\mtfootline{\ifodd\pageno\rightfootline\else\leftfootline\fi}
\def\rightfootline{\hfill\eightsmc D.H.Fremlin}
\def\leftfootline{\ifwithproofs{\eightsmc Measure Theory\hfill}
\else{\eightsmc Measure Theory \eightit (abridged version)\hfill}
\fi}
%Style assignments
\def\newchapter#1{\gdef\topparagraph{}
\gdef\bottomparagraph{Chap.\ #1 {\it intro.}}
\gdef\newparagraph{Chap.\ #1 {\it intro.}}
\footmarkcount=0
\gdef\sectionname{Introduction}
\gdef\headlinesectionname{Introduction}
\centerline{\bf Chapter #1}
\medskip
\centerline{\bf \chaptername}
\medskip
}
\def\newsection#1{\gdef\bottomparagraph{\S#1 {\it intro.}}
\noindent{\bf #1 \sectionname}
\smallskip
\let\headlinesectionname=\sectionname}
\def\newvolume#1{\gdef\topparagraph{}
\gdef\bottomparagraph{Vol.\ #1 {\it intro.}}
\Loadfont{\twelvebf=cmbx12}
\def\chaptername{Volume #1}
\def\sectionname{\volumename}
\def\headlinesectionname{\volumename}
\centerline{\twelvebf Volume #1: \volumename}
\bigskip
}
%font variations
\newif\ifvolwp
\volwpfalse
\def\tenrmstretch#1{\fontdimen3\tenrm=#1}
\font\elevenex=cmex10 at 11pt
%contents and introductions
\newif\iflargelogo
\def\vtmpb#1{}
\def\newvolume#1{}
\def\pagereference#1#2{\hfill\hbox{\ifresultsonly#2\qquad \hskip 1truein
\else#1\qquad \fi}}
\long\def\section#1#2#3#4#5#6{\ifresultsonly
\hskip1truein\qquad #1 #2\pagereference{#4}{#5}
\else\vskip 2pt plus 1pt minus 0pt
\vbox{\qquad #1 #2\pagereference{#4}{#5}\par
\abstr{#6}
}\vskip -1pt plus 1pt minus 0pt
\fi}
\long\def\chapintrosection#1#2#3{\ifresultsonly
\hskip1truein\qquad Introduction\pagereference{#2}{#3}
\else\vskip 1pt plus 1pt minus 0pt
\qquad Introduction\pagereference{#2}{#3}
\vskip 1pt plus 1pt minus 0pt
\fi}
\def\abstr#1{\ifresultsonly{}
\else{\advance\leftskip by\parindent
\advance\rightskip by\parindent
\advance\rightskip by\parindent
{\advance\leftskip by\parindent
\advance\rightskip by\parindent
{\advance\leftskip by\parindent
{\advance\leftskip by\parindent
{\vskip1pt plus 0pt minus0pt
\parindent=0pt
\smallerfonts #1
\hbox{\vrule height0pt depth2pt width0pt}
\vskip1pt plus 0pt minus0pt }}}}}\fi}
%language variations
\newif\ifUSEnglish
\USEnglishfalse
%\USEnglishtrue
%"Later editions only"
\newcount\latereditionpage
\newcount\latereditionmark
\latereditionpage=-1
\latereditionmark=0
\newcount\newpageno
%removing the \write0 instruction altogether disorganizes it!
\def\Latereditions{\newpageno=\pageno%
\ifdim\pagetotal>\pagegoal\advance\newpageno by 1\fi%
\divide\newpageno by 2%
\immediate\write0{%
% "Later editions only";
% pagetotal \the\pagetotal, pagegoal \the\pagegoal,
% pageno \number\pageno,\par newpageno \number\newpageno,
% latereditionpage \number\latereditionpage%
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\noindent{\eightit
\ifresultsonly{Extract from {\eightsmc Measure Theory,}
results-only version,
by {\eightsmc D.H.Fremlin,} University of Essex,
Colchester.}
\else\ifwithproofs{Extract from {\eightsmc Measure Theory,}
by {\eightsmc D.H.Fremlin,} University of Essex,
Colchester.}
\else{Extract from {\eightsmc Measure Theory,} abridged version,
by {\eightsmc D.H.Fremlin,} University of Essex,
Colchester.}
\fi\fi
This material is copyright. It is issued under the terms of the
Design Science License as published in
{\eighttt http://dsl.org/copyleft/dsl.txt}.
This is a development version and the
source files are not permanently archived, but current versions are
normally
accessible through
{\eighttt http://www.essex.ac.uk/maths/staff/fremdh/mt.htm}.
For further information contact {\eighttt fremdh\@essex.ac.uk}.}
}}
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\versiondate{26.11.08}
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\def\zz{}
\Centerline{\bf Ergodic averages, following Austin}
\Centerline{\smc D.H.Fremlin}
\Centerline{\it University of Essex, Colchester, England}
\medskip
I rewrite the main results of {\smc Austin p08a} and {\smc Austin p08b}, in
which a version of the multiple recurrence theorem is proved by a new
method based on ideas of T.Tao.
\bigskip
{\bf\zz1 Useful facts}
\medskip
{\bf\zz1A Lemma} Let $(\frak A,\bar\mu)$ be a measure
algebra. If $a\in\frak A$ and $u\in L^{\infty}(\frak A)$ is such that
$0\le u\le\chi 1$, there is an $\alpha\in\ooint{0,1}$ such that
$\bar\mu(a\Bsymmdiff\Bvalue{u>\alpha})\le\int|\chi a-u|$.
\medskip
\noindent{\bf proof} Set $\gamma=\int|\chi a-u|$. If $\gamma=\infty$ we
can stop. Otherwise,
we may suppose that $(\frak A,\bar\mu)$ is the measure
algebra of a measure space $(X,\Sigma,\mu)$. Express $a$ as
$E^{\ssbullet}$ and $u$ as $f^{\ssbullet}$ where $E\in\Sigma$ and
$f:X\to[0,1]$ is $\Sigma$-measurable. Then
$\int|\chi E-f|d\mu=\gamma$ is finite, so $H=\{x:\chi E(x)\ne f(x)\}$ is
expressible as a countable union of sets of finite measure. Set
$\Omega'_f=\{(x,\alpha ):x\in X$, $0\le \alpha<f(x)\}$ and
$W=(E\times[0,1])\symmdiff\Omega'_f$. Then $W\subseteq H\times\Bbb R$ is
measured by the product of the subspace measure
$\mu_H$ on $H$ and Lebesgue measure $\mu_L$ on $[0,1]$.
Because $\mu_H$ is $\sigma$-finite, we have
$$\eqalign{\gamma
&=\int_H|\chi E(x)-f(x)|\mu(dx)
=\int_H\mu_LW[\{x\}]\mu_H(dx)\cr
&=\int_0^1\mu_HW^{-1}[\{\alpha\}]\mu_L(d\alpha )
=\int_0^1\mu_H(E\symmdiff\{x:f(x)>\alpha\})\mu_L(d\alpha ),\cr}$$
\noindent and there must be an $\alpha\in\ooint{0,1}$ such that
\Centerline{$\gamma
\ge\mu_H(E\symmdiff\{x:f(x)>\alpha\})
=\mu(E\symmdiff\{x:f(x)>\alpha\})
=\bar\mu(a\Bsymmdiff\Bvalue{u>\alpha})$.}
\bigskip
{\bf\zz1B Lemma} Let $G$ be a topological group, $(\frak A,\bar\mu)$ a
measure algebra, and $\action$ a continuous action of $G$ on $\frak A$,
where $\frak A$ is given its measure-algebra topology
({\smc Fremlin 02}, \S323), such that
$a\mapsto g\action a$ is a measure-preserving Boolean automorphism for
every $g\in G$.
(a) We have an action of $G$ on $L^0=L^0(\frak A)$ defined by saying that
$\Bvalue{g\action u>\alpha}=g\action\Bvalue{u>\alpha}$ whenever $g\in G$,
$u\in L^0$ and $\alpha\in\Bbb R$; for
$g\in G$, $u\mapsto g\action u:L^0\to L^0$ is an $f$-algebra automorphism.
(b) For every $p\in[1,\infty]$, $L^p=L^p(\frak A,\bar\mu)$ and $\|\,\|_p$
are $G$-invariant. For $p\in\coint{1,\infty}$, the action is continuous.
(c) Let $B$ be the
unit ball of $L^{\infty}=L^{\infty}(\frak A)$, with the topology
$\frak T_s(L^{\infty},L^1)$ induced by the duality between $L^{\infty}$ and
$L^1=L^1(\frak A,\bar\mu)$. Then $B$ is $G$-invariant and the action of
$G$ on $B$ is continuous.
\medskip
\noindent{\bf proof (a)} For each $g\in G$, we have a measure-preserving
automorphism $\pi_g$ defined by saying that $\pi_g(a)=g\action a$ for
$a\in\frak A$, and a corresponding $f$-algebra isomorphism
$R_g:L^0\to L^0$, where $L^0=L^0(\frak A)$, given by saying that
$\Bvalue{R_gu>\alpha}=\pi_g\Bvalue{u>\alpha}$ for $u\in L^0$ and
$\alpha\in\Bbb R$.
If $g$, $h\in G$, then
\Centerline{$\pi_{gh}(a)=(gh)\action a=g\action(h\action a)
=\pi_g(\pi_h(a))$}
\noindent for every $a\in\frak A$, so $\pi_{gh}=\pi_g\pi_h$,
$R_{gh}=R_gR_h$ ({\smc Fremlin 02}, 364Re) and
$g\action(h\action u)=(gh)\action u$ for every $u\in L^0(\frak A)$. So
we have an action of $G$ on $L^0(\frak A)$.
\medskip
{\bf (b)} Every $R_g$ acts on every $L^p$ as a Banach lattice
automorphism ({\smc Fremlin 02}, 364R, 365O and 366H). If $p<\infty$,
this action is continuous for the norm topology on
$L^p$. \Prf\ Suppose that $g_0\in G$, $v_0\in L^p$ and $\epsilon>0$.
Then we can find a $v_1\in L^p$ such that $\|v_1-v_0\|_p\le\epsilon$ and
$v_1$ is expressible as $\sum_{i=0}^n\alpha_i\chi a_i$ where
$\bar\mu a_i<\infty$ for every $i\le n$.
Let $\eta>0$ be such that $(2\eta)^{1/p}\sum_{i=0}^n|\alpha_i|\le\epsilon$.
Because the action of $G$ on $\frak A$ is continuous, there is a
neighbourhood $V$ of $g_0$ such that
$\bar\mu(g\action a_i\Bcap g_0\action a_i)\ge\bar\mu(g_0\action a_i)-\eta$
whenever
$i\le n$ and $g\in V$. Since $\pi_g$ is measure-preserving for every $g$,
we see that $\bar\mu(g\action a_i\Bsymmdiff g_0\action a_i)\le 2\eta$
whenever $g\in V$ and $i\le n$, so that
$\|g\action v_1-g_0\action v_1\|_p\le\epsilon$ whenever $g\in V$.
Now if $g\in V$ and $v\in L^1$ is such that $\|v-v_0\|_p\le\epsilon$, we
shall have
$$\eqalign{\|g\action v-g_0\action v_0\|_p
&\le\|g\action v-g\action v_1\|_p
+\|g\action v_1-g_0\action v_1\|_p
+\|g_0\action v_1-g_0\action v_0\|_p\cr
&\le\|v-v_1\|_p
+\epsilon
+\|v_1-v_0\|_p
\le 4\epsilon.\cr}$$
\noindent As $g_0$, $v_0$ and $\epsilon$ are arbitrary, the action is
continuous.\ \Qed
\medskip
{\bf (c)} $R_g\restr L^{\infty}$ is a
norm-preserving automorphism of $L^{\infty}$, so we have an action
of $G$ on $B$. Now suppose that $u_0\in B$, $g_0\in G$, $v\in L^1$ and
$\epsilon>0$. Then there is a neighbourhood $V$ of $g_0$ such that
$\|g^{-1}\action v-g_0^{-1}\action v\|_1\le\epsilon$ whenever $g\in V$.
Suppose that $u\in B$ is such that
$|\int u\times(g_0^{-1}\action v)-\int u_0\times(g_0^{-1}\action v)|
\le\epsilon$. Then, for any $g\in V$,
$$\eqalignno{|\int(g\action u-g_0\action u_0)\times v|
&=|\int(g\action u)\times v-\int(g_0\action u_0)\times v|\cr
&=|\int g^{-1}\action((g\action u)\times v)
-\int g_0^{-1}\action((g_0\action u_0)\times v)|\cr
&=|\int u\times(g^{-1}\action v)
-\int u_0\times(g_0^{-1}\action v)|\cr
\displaycause{because $R_g$, $R_{g_0}$ are multiplicative}
&\le|\int u\times(g^{-1}\action v)-\int u\times(g_0^{-1}\action v)|\cr
&\mskip100mu
+|\int u\times(g_0^{-1}\action v)-\int u_0\times(g_0^{-1}\action v)|\cr
&\le\|g^{-1}\action v-g_0^{-1}\action v\|_1
+\epsilon
\le 2\epsilon.\cr}$$
\noindent As $u_0$, $g_0$, $v$ and $\epsilon$ are arbitrary, the action of
$G$ on $B$ is continuous.
\bigskip
{\bf\zz1C Remark} In this context, the following remark will be useful.
Suppose that $G$ is a topological group, $(\frak A,\bar\mu)$ a
probability algebra, and $\action$ an action of $G$ on $\frak A$ such that
$a\mapsto g\action a$ is a measure-preserving Boolean automorphism for
every $g\in G$. If $D\subseteq\frak A$ is such that the
subalgebra $\frak D$ of $\frak A$ generated by $D$ is dense
for the measure-algebra topology of $\frak A$,
and $g\mapsto g\action d:G\to\frak A$ is continuous for every $d\in D$,
then $\action$ is continuous. \Prf\ (i)
$\{d:d\in\frak A$, $g\mapsto g\action d$ is continuous$\}$ is
a subalgebra of $\frak A$ because the Boolean operations are uniformly
continuous ({\smc Fremlin 02}, 323B). So it includes $\frak D$.
(ii) Suppose that $g_0\in G$,
$a_0\in\frak A$ and $\epsilon>0$. Let $d\in\frak D$ be such that
$\bar\mu(d\Bsymmdiff a)\le\epsilon$, and $H\subseteq G$ a neighbourhood of
$g_0$ such that $\bar\mu(g\action d\Bsymmdiff g_0\action d)\le\epsilon$ for
every $g\in H$. Then if $g\in H$ and
$\bar\mu(a\Bsymmdiff a_0)\le\epsilon$,
$$\eqalign{\bar\mu(g\action a\Bsymmdiff g_0\action a_0)
&\le\bar\mu(g\action a\Bsymmdiff g\action d)
+\bar\mu(g\action d\Bsymmdiff g_0\action d)
+\bar\mu(g_0\action d\Bsymmdiff g_0\action a_0)\cr
&\le\bar\mu(a\Bsymmdiff d)
+\epsilon
+\bar\mu(d\Bsymmdiff a_0)
\le 4\epsilon.\cr}$$
\noindent As $g_0$, $a_0$ and $\epsilon$ are arbitrary, $\action$ is
continuous.\ \Qed
\bigskip
{\bf\zz1D Proposition}
Let $U$ and $V$ be Hausdorff locally convex linear
topological spaces, $A\subseteq U$ a convex set and
$\phi:A\to V$ a continuous
function such that $\phi[A]$ is bounded and
$\phi(\alpha x+(1-\alpha)y)=\alpha\phi(x)+(1-\alpha)\phi(y)$
for all $x$, $y\in A$ and
$\alpha\in[0,1]$. Let $\mu$ be a topological
probability measure on $A$ with a barycenter
$x^*$ in $A$. Then $\phi(x^*)$ is the barycenter of the image measure
$\mu\phi^{-1}$ on $V$.
\medskip
\noindent{\bf proof (a)} Suppose that $\familyiI{E_i}$ is a
finite partition of $A$ into
non-empty convex sets measured by $\mu$,
and set $\alpha_i=\mu E_i$ for each $i\in I$. Set
$C=\{\sum_{i\in I}\alpha_ix_i:x_i\in E_i$ for every $i\in I\}$. Then
$x^*\in\overline{C}$. \Prf\ Because each $E_i$ is convex, so is $C$.
If $g\in U^*$, then
$$\eqalign{g(x^*)
&=\int_Ag(x)\mu(dx)
=\sum_{i\in I}\int_{E_i}g(x)\mu(dx)\cr
&\le\sum_{i\in I}\alpha_i\sup_{x\in E_i}g(x)
=\sup\{\sum_{i\in I}\alpha_ig(x_i):x_i\in E_i\text{ for every }i\in I\}\cr
&=\sup\{g(\sum_{i\in I}\alpha_ix_i):x_i\in E_i\text{ for every }i\in I\}
=\sup_{z\in C}g(z).\cr}$$
\noindent By the Hahn-Banach theorem, $x^*\in\overline{C}$.\ \Qed
\medskip
{\bf (b)} Now suppose that $h\in V^*$ and $\epsilon>0$. Then $h[\phi[A]]$
is bounded; take $\alpha\in\Bbb R$ and $n\ge 1$ such that
$h[\phi[A]]\subseteq\coint{\alpha,\alpha+n\epsilon}$. For $i<n$ set
$F_i=\{y:y\in V$, $\alpha+i\epsilon\le h(y)<\alpha+(i+1)\epsilon\}$ and
$E_i=\phi^{-1}[F_i]$; set $I=\{i:i<n$, $E_i\ne\emptyset\}$. Then
$\familyiI{E_i}$ is a partition of $A$ into relatively Borel sets.
As in (a), set $\alpha_i=\mu E_i$ for $i\in I$ and
$C=\{\sum_{i\in I}\alpha_ix_i:x_i\in E_i$ for every $i\in I\}$.
Then $C\subseteq A$ and $x^*\in\overline{C}$; there must therefore be a
$z\in C$ such that $|h(\phi(z))-h(\phi(x^*))|\le\epsilon$. Express
$z$ as $\sum_{i\in I}\alpha_ix_i$ where $x_i\in E_i$ for each $i\in I$.
Then
$$\eqalignno{|h(\phi(x^*))-\int h\,d(\mu\phi^{-1})|
&\le\epsilon+|h(\phi(z))-\sum_{i\in I}\int_{F_i}h\,d(\mu\phi^{-1})|\cr
&=\epsilon+|h(\sum_{i\in I}\alpha_i\phi(x_i))
-\sum_{i\in I}\int_{F_i}h\,d(\mu\phi^{-1})|\cr
&\le\epsilon+\sum_{i\in I}|\alpha_ih(\phi(x_i))
-\int_{F_i}h\,d(\mu\phi^{-1})|\cr
&\le\epsilon+\sum_{i\in I}\alpha_i\sup_{y\in F_i}|h(\phi(x_i))-h(y)|\cr
\displaycause{because $\mu\phi^{-1}[F_i]=\alpha_i$ for each $i$}
&\le\epsilon+\sum_{i\in I}\alpha_i\epsilon\cr
\displaycause{by the choice of the $F_i$}
&=2\epsilon.\cr}$$
\noindent As $h$ and $\epsilon$ are arbitrary, $\phi(x^*)$ is the
barycenter of $\mu\phi^{-1}$.
\bigskip
{\bf\zz1E Lemma} Let $U$ be a uniformly convex Banach space,
$A\subseteq U$ a non-empty bounded set, and $C\subseteq U$ a non-empty
closed convex set. Set
\Centerline{$\delta_0=\inf\{\delta:$ there is some $w\in C$ such that
$A\subseteq B(w,\delta)\}$.}
\noindent Then there is a unique $w^*\in C$ such that
$A\subseteq B(w^*,\delta_0)$.
\medskip
\noindent{\bf proof (a)} For $\delta\ge\delta_0$, set
\Centerline{$C_{\delta}=C\cap\bigcap_{u\in A}B(u,\delta)\}$,}
\noindent so that $C_{\delta}$ is closed, and is non-empty if
$\delta>\delta_0$. Now
$\lim_{\delta\downarrow\delta_0}\diam C_{\delta}=0$. \Prf\ Of course
$\diam C_{\delta}\le 2\delta$, so if $\delta_0=0$ the result is trivial.
Otherwise, let $\epsilon>0$. Then there is an $\eta>0$ such that
$\|\Bover12(v_0+v_1)\|<\Bover{1-\eta}{1+\eta}$ whenever
$\|v_0\|$, $\|v_1\|\le 1$ and
$\|v_0-v_1\|\ge\epsilon\delta_0$. \Quer\ Suppose that
$\delta\le(1+\eta)\delta_0$ and $\diam C_{\delta}>\epsilon$.
Let $w_0$, $w_1\in C_{\delta}$ be such that $\|w_0-w_1\|\ge\epsilon$.
Then $\bover12(w_0+w_1)\in C$, so there is a $u\in A$ such that
$\|u-\bover12(w_0+w_1)\|\ge(1-\eta)\delta_0$, while
$\|u-w_0\|\le(1+\eta)\delta_0$ and $\|u-w_1\|\le(1+\eta)\delta_0$;
setting $v_j=\Bover1{(1+\eta)\delta_0}(u-w_j)$ for $j=0$ and $j=1$,
we see that this contradicts the choice of $\eta$.\ \Bang
So $\diam C_{\delta}\le\epsilon$ whenever $\delta\le(1+\eta)\delta_0$; as
$\epsilon$ is arbitrary, we have the result.\ \Qed
\medskip
{\bf (b)}
$\{C_{\delta}:\delta>\delta_0\}$ generates a Cauchy filter, which has a
limit $w^*\in\bigcap_{\delta>\delta_0}C_{\delta}$. Now
$w^*\in C_{\delta_0}$; since $C_{\delta_0}$ has zero diameter,
$w^*$ is its only member, that is, is the unique element of $U$ such that
$A\subseteq B(w^*,\delta_0)$.
\bigskip
{\bf\zz1F Proposition} (T.Austin, e-mail of 8.10.08)
Let $G$ be a group, $(\frak A,\bar\mu)$ a
probability algebra, and $\action$ an action of $G$ on $\frak A$
such that $a\mapsto g\action a$ is a
measure-preserving Boolean automorphism for every
$g\in G$. Let $\frak C$ be the fixed-point algebra
$\{c:c\in\frak A$, $g\action c=c$ for every $g\in G\}$. Then for every
$a\in\frak A$, there is a $c\in\frak C$ such that
$\bar\mu(a\Bsymmdiff c)\le\sup_{g\in G}\bar\mu(a\Bsymmdiff g\action a)$.
\medskip
\noindent{\bf proof (a)} Set
$\gamma=\sup_{g\in G}\bar\mu(a\Bsymmdiff g\action a)$. As in Lemma \zz1B,
we have an action of $G$ on $L^0(\frak A)$ defined by saying that
$\Bvalue{g\action u>\alpha}=g\action\Bvalue{u>\alpha}$ whenever $g\in G$,
$u\in B$ and $\alpha\in\Bbb R$. Set
\Centerline{$A
=\{\chi(g\action a):g\in G\}=\{g\action\chi a:g\in G\}$,
\quad$C=\{u:0\le u\le\chi 1$ in $L^0\}$.}
\noindent If $p\in\coint{1,\infty}$, $L^p=L^p(\frak A,\bar\mu)$ is
invariant under this action, and
$u\mapsto g\action u:L^p\to L^p$ is
a Banach lattice automorphism for every $g\in G$. If
$p\in\ooint{1,\infty}$, $L^p$ is uniformly convex
({\smc Fremlin 01}, 244P\footnote{Later editions only; see
{\tt http://www.essex.ac.uk/maths/staff/fremlin/mtcont.htm}.}, or
{\smc Clarkson 36}), so there is a unique $w_p\in C$ such that
\Centerline{$\sup_{u\in A}\|u-w_p\|_p
=\inf_{w\in C}\sup_{u\in A}\|u-w\|_p
\le\sup_{u\in A}\|u-\chi a\|_p=\gamma^{1/p}$}
\noindent (\zz1E). Because $A$ and $C$ and $\|\,\|_p$, are $G$-invariant,
so is $w_p$, and $w_p\in L^0(\frak C)$.
\medskip
{\bf (b)}
Recall now that there is a $w^*\in L^1_{\frak C}=L^1(\frak C,\bar\mu)$
such that
$\|\chi a-w^*\|_1=\inf\{\|\chi a-w\|:w\in L^1_{\frak C}$
(use Bukhvalov's theorem, {\smc Fremlin 02}, 367V/367Xx, or
Koml\'os' theorem, {\smc Fremlin 01},
276H). Replacing $w^*$ by $\med(0,w^*,\chi 1)$ if necessary, we may
suppose that $w^*\in C$. In this case,
\Centerline{$\|\chi a-w^*\|_1
\le\|\chi a-w_p\|_1
\le\|\chi a-w_p\|_p\le\gamma^{1/p}$}
\noindent for every $p>1$, and $\|\chi a-w^*\|_1\le\gamma$.
By Lemma \zz1A, there is an $\alpha\in\ooint{0,1}$ such that
$\bar\mu(a\Bsymmdiff\Bvalue{v>\alpha})\le\gamma$.
Set $c=\Bvalue{v>\alpha}$; then $c\in\frak C$ and
$\bar\mu(a\Bsymmdiff c)\le\gamma$, so we have the result.
\bigskip
{\bf\zz1G Proposition} ({\smc Austin p08a}, 2.1)
Let $(T,\le)$ be an upwards-directed partially
ordered set, $\family{t}{T}{(\frak A_t,\bar\mu_t)}$
a family of probability algebras and $G$ a group; suppose
that $\phi_{ji}:\frak A_t\to\frak A_j$ and
$\action^{(t)}:G\times\frak A_t\to\frak A_t$ are such that
\inset{(i) $\phi_{st}$ is a measure-preserving
Boolean homomorphism whenever $s\le t$ in $T$,
(ii) $\phi_{su}=\phi_{tu}\phi_{st}$ whenever $i\le j\le k$ in $T$,
(iii) $\action^{(t)}$ is an action of $G$ on $\frak A_t$ for each $t\in T$,
(iv) $g\action^{(t)}(\phi_{st}a)=\phi_{st}(g\action^{(s)}a)$
whenever $s\le t$ in $T$, $a\in\frak A_t$ and $g\in G$,
(v) $a\mapsto g\action^{(t)}a:\frak A_t\to\frak A_t$ is a
measure-preserving Boolean automorphism for each $t\in T$.}
(a) Writing $(\frak A,\bar\mu,\familyiI{\phi_t})$ for the
inductive limit of
$(\familyiI{(\frak A_t,\bar\mu_t)},\langle\phi_{st}\rangle_{s\le t})$ as in
{\smc Fremlin 02}, 328G\Latereditions, we have a
unique action $\action$ of $G$ on $\frak A$ such that
\inset{$a\mapsto g\action a:\frak A\to\frak A$ is a measure-preserving
Boolean automorphism for every $g\in G$,
$g\action(\phi_ta)=\phi_t(g\action^{(t)}a)$ whenever $t\in T$,
$a\in\frak A_t$ and $g\in G$.}
(b) For each $t\in T$, let
$\frak C_t=\{c:c\in\frak A_t$, $g\action^{(t)}c=c$ for every $g\in G\}$
be the fixed-point subalgebra of the action $\action^{(t)}$. Then the
fixed-point
subalgebra $\frak C$ of the action $\action$ is the closure of
$\bigcup_{t\in T}\phi_t[\frak C_t]$.
(c) If $G$ is a topological group and $\action^{(t)}$ is continuous for
every $t\in T$, then $\action$ is continuous.
\medskip
\noindent{\bf proof (a)} For $g\in G$ and $t\in T$, set
$\psi_{gt}(a)=\phi_t(g\action^{(t)}a)$ for every $a\in\frak A_t$. Then
$\psi_{gt}=\psi_{gt}\phi_{st}$ whenever $s\le t$, so by the defining
property of probability algebra inductive limit, there is a unique
measure-preserving Boolean homomorphism $\psi_g:\frak A\to\frak A$ such
that $\psi_g\phi_t=\psi_{gt}$ for every $t$. It is now elementary to
verify that $(g,a)\to\phi_g(a)$ is an action of $G$ on $\frak A$, as
required.
\medskip
{\bf (b)} If $i\in I$ and $a\in\phi_t[\frak C_t]$, set $c=\phi_t^{-1}a$;
then
\Centerline{$g\action a=\phi_t(g\action^{(t)}c)=a$}
\noindent so $a\in\frak C$. Now suppose that $c\in\frak C$ and
$\epsilon>0$. Then there are a $t\in T$ and an $a\in\frak A_t$ such that
$\bar\mu(c\Bsymmdiff\phi_ta)\le\epsilon$. If $g\in G$, then
$$\eqalign{\bar\mu_t(a\Bsymmdiff g\action^{(t)}a)
&=\bar\mu\phi_t(a\Bsymmdiff g\action^{(t)}a)
=\bar\mu(\phi_ta\Bsymmdiff g\action\phi_ta)\cr
&\le
\bar\mu(\phi_ta\Bsymmdiff c)+\bar\mu(g\action c\Bsymmdiff g\action\phi_ta)
=\bar\mu(\phi_ta\Bsymmdiff c)+\bar\mu(c\Bsymmdiff\phi_ta)
\le 2\epsilon.\cr}$$
\noindent By Lemma \zz1F, there is a $b\in\frak C_t$ such that
$\bar\mu_t(a\Bsymmdiff b)\le 2\epsilon$, so that
$\phi_tb\in\phi_t[\frak C_t]$ and
$\bar\mu(c\Bsymmdiff\phi_tb)\le 2\epsilon$.
As $c$ and $\epsilon$ are arbitrary,
$\frak C=\overline{\bigcup_{t\in T}\phi_t[\frak C_t]}$.
\medskip
{\bf (c)} Because $\{0,1\}\cup\bigcup_{t\in T}\phi_t[\frak A_t]$ is
dense in
$\frak A$ ({\smc Fremlin 02}, 328G), it will be enough to show that
$g\mapsto g\action(\phi_ta):G\to\frak A$ is continuous whenever $t\in T$
and $a\in\frak A_t$ (\zz1C). But this is just the function
$g\mapsto\phi_t(g\action^{(t)}a)$, which is continuous because
$\action^{(t)}$ and $\phi_t$ are continuous.
\bigskip
{\bf\zz1H Well-distributed limits} ({\smc Fremlin n08})
Let $G$ be an amenable
discrete group ({\smc Fremlin 03}, \S449) and $U$ a Banach space.
\medskip
{\bf (a)}
The {\bf left F{\o}lner filter} of $G$
is the filter $\CalFo$ on $[G]^{<\omega}\setminus\{\emptyset\}$
generated by sets of the form
\Centerline{$\{K:K\subseteq G$ is finite and not empty and
$\#(K\symmdiff hK)\le\epsilon\#(K)\}$}
\noindent where $h\in G$ and $\epsilon>0$. If $U$
is a Banach space and $f:G\to U$ is a bounded function, I write
\Centerline{$\WDL_{g\to G}f(g)
=\lim_{L\to\CalFo}\Bover1{\#(L)}\sum_{g\in L}f(g)$}
\noindent if the limit exists in $U$ for the norm of $U$.
Of course $\WDL_{g\to G}f(g)$, if defined, must belong to the closed convex
hull of the image $f[G]$, and we have
\Centerline{$\WDL_{g\to G}(f_1+f_2)(g)
=\WDL_{g\to G}f_1(g)+\WDL_{g\to G}f_2(g)$,}
\Centerline{$\WDL_{g\to G}(Tf)(g)
=T(\WDL_{g\to G}f(g))$}
\noindent whenever the right-hand sides are defined and
$T:U\to V$ is a bounded linear operator to another Banach space.
Also
\Centerline{$\|\WDL_{g\to G}f(g)\|\le\WDL_{g\to G}\|f(g)\|$}
\noindent whenever both sides are defined.
\medskip
{\bf (b)} If $f:G\to\Bbb R$ is any function I will write
\Centerline{$\overline{\WDL}_{g\to G}f(g)
=\limsup_{L\to\CalFo}\Bover1{\#(L)}\sum_{g\in L}f(g)$.}
\noindent Observe that of $U$ is a Banach space and $f:G\to U$
is a bounded function such that
$\overline{\WDL}_{g\to G}\|f(g)\|=0$ then $\WDL_{g\to G}f(g)=0$.
\medskip
{\bf (c)} For a bounded function $f:G\to\Bbb R$,
$$\eqalign{\overline{\WDL}(f)
&=\sup\{\int f\,d\mu:\mu\text{ is a translation-invariant
finitely additive functional}\cr
&\mskip100mu\text{ from }\Cal PG\text{ to }[0,1],
\text{ and }\mu G=1\}.\cr}$$
\noindent(Here the `integral' $\int f\,d\mu$ must be interpreted as in
{\smc Fremlin 02}, 363L.) \Prf\ For $f\in\BbbR^G$ and $g\in G$, define
$g\action_lf\in\BbbR^G$ by setting $(g\action_lf)(h)=f(g^{-1}h)$ for
every $h\in G$. Writing $P$ for the set of positive linear
functionals $p:\ell^{\infty}(G)\to\Bbb R$ such that
$p(\chi G)=1$ and $p(g\action_lf)=p(f)$
whenever $f\in\ell^{\infty}(G)$ and $g\in G$,
\Centerline{$\overline{\WDL}(f)=\sup_{p\in P}p(f)$}
\noindent for every $f\in\ell^{\infty}(G)$ ({\smc Fremlin n08}, \zz6Ia).
On the other hand, it is easy to check that we
have a one-to-one correspondence between positive linear functionals
on $\ell^{\infty}(X)$ and the
set of finitely additive measures
$\mu:\Cal PG\to\coint{0,\infty}$, given by setting
\Centerline{$\mu A=p(\chi A)$ for $A\subseteq X$,
\quad$p(f)=\int fd\mu$ for $f\in\ell^{\infty}(G)$}
\noindent(see the discussion in
{\smc Fremlin 02}, 363L); and $p\in P$ iff $\mu$ is translation-invariant
and $\mu G=1$. So we get
$$\eqalign{\overline{\WDL}(f)
&=\sup_{p\in P}p(f)\cr
&=\sup\{\int f\,d\mu:\mu\text{ is a translation-invariant
finitely additive functional}\cr
&\mskip100mu\text{ from }\Cal PG\text{ to }[0,1],
\text{ and }\mu G=1\}.\cr}$$
\medskip
{\bf (d)} If $G$ is infinite, and $f:G\to U$ is a bounded function such
that $\#(\{g:f(g)\ne 0\})<\#(G)$, then $\WDL_{g\to G}f(g)=0$.
\Prf\ Setting $A=\#(\{g:f(g)\ne 0\})$, we can choose inductively a sequence
$\sequencen{g_n}$ in $G$ such that $g_nA\cap\bigcup_{i<n}g_iA=\emptyset$
for every $n$. (When we come to choose $g_n$, only
$\bigcup_{i<n}g_iAA^{-1}$ is forbidden, and this has cardinal less than
$\#(G)$.) By (c), $\overline{\WDL}_{g\to G}\chi A(g)=0$, so
$\WDL_{g\to G}\|f(g)\|=0$ and $\WDL_{g\to G}f(g)=0$.\ \Qed
\bigskip
{\bf\zz1I Theorem} Let $G$ be an abelian group,
and $\action$ an action of $G$ on a Banach
space $U$ such that $u\mapsto g\action u$ is a linear operator of norm at
most $1$ for every $g\in G$. If $u\in U$ is such that
$\{g\action u:g\in G\}$ is relatively weakly compact, then
$w=\WDL_{g\to G}g\action u$ is defined in $U$ and $g\action w=w$ for
every $g\in G$.
\medskip
\noindent{\bf proof} {\smc Fremlin 08}, 6M. (To match between the
definition of $\WDL$ in \zz1H with that in {\smc Fremlin 08}, apply
{\smc Fremlin 08}, 6Ic to the discrete topology on $G$.)
\bigskip
{\bf\zz1J Notation (a)} We shall have a very large number of conditional
expectation operators in the work to follow. It will be convenient to
reserve a letter for these. If $(\frak A,\bar\mu)$ is a probability
algebra and $\frak B$ is a closed subalgebra of
$\frak A$,\footnote{\smallerfonts As noted
in {\smc Fremlin 02}, 323H,
a subalgebra of $\frak A$ is order-closed iff it is
topologically closed; so we can use the word `closed' without
qualification in this context.} I will write
$Q_{\frak B}$ for the associated conditional expectation operator from
$L^1(\frak A,\bar\mu)$ to
$L^1(\frak B,\bar\mu\restrp\frak B)\subseteq L^1(\frak A,\bar\mu)$
({\smc Fremlin 02}, 365R).
\medskip
{\bf (b)} It will also be convenient to have some notation for lattices
of closed subalgebras. If $(\frak A,\bar\mu)$ is a probability algebra
and $\family{t}{T}{\frak B_t}$ is a family of closed subalgebras of
$\frak A$, then I will write $\bigvee_{t\in T}\frak B_t$ for the closed
subalgebra of $\frak A$ generated by $\bigcup_{t\in T}\frak B_t$.
Similarly, if $\frak B$ and $\frak C$ are two closed subalgebras of
$\frak A$, $\frak B\vee\frak C$ will be the smallest closed subalgebra
including both $\frak B$ and $\frak C$.
\bigskip
{\bf\zz2 Measure-automorphism action systems}
\medskip
{\bf\zz2A Definitions (a)} An {\bf action system} is a triple
$(X,G,\familyiI{\action_i})$ where $X$ is a set, $G$ is a group and
$\action_i$ is an action of $G$ on $X$ for each $i\in I$.
\medskip
{\bf (b)} An action system $(X,G,\familyiI{\action_i})$ is {\bf commuting}
if $G$ is abelian and
$g\action_i(h\action_jx)=h\action_j(g\action_ix)$
whenever $g$, $h\in G$, $i$, $j\in I$ and $x\in X$.
\medskip
{\bf (c)} A {\bf measure-automorphism action system} is a
quadruple $(\frak A,\bar\mu,G,\familyiI{\action_i})$ such that
\inset{$(\frak A,\bar\mu)$ is a probability algebra,
$(\frak A,G,\familyiI{\action_i})$ is an action system,
$a\mapsto g\action_ia$ is a measure-preserving Boolean automorphism
for every $i\in I$ and $g\in G$.}
\bigskip
{\bf\zz2B Construction} Let $(\frak A,G,\familyiI{\action_i})$ be
an action system. Suppose that $\frak A$ is a Boolean algebra and that
$\mu:\frak A\to[0,1]$ an additive functional; suppose that
\inset{$a\mapsto g\action_ia$ is a Boolean automorphism whenever $g\in G$
and $i\in I$,
$\mu 1=1$,
$\mu(g\action_ia)=\mu a$ whenever $a\in\frak A$, $g\in G$ and $i\in I$.}
\noindent Set $\Cal I=\{a:a\in\frak A$, $\mu a=0\}$; then
$\Cal I\normalsubgroup\frak A$. Let $\frak C_0$ be the quotient
$\frak A/\Cal I$. Then we can define
$\action'_i:G\times\frak C_0\to\frak C_0$, for $i\in I$, by saying that
$g\action'_ia^{\ssbullet}=(g\action_ia)^{\ssbullet}$ whenever
$a\in\frak A$, $g\in G$ and $i\in I$. Each $\action'_i$ is an action of
$G$ on $\frak C_0$.
There is a strictly positive additive functional
$\bar\nu_0:\frak C_0\to[0,1]$ defined by saying that
$\bar\nu_0a^{\ssbullet}=\mu a$ for every $a\in\frak A$. Let $\frak C$ be
the completion of $\frak C_0$ under the metric
$(c,c')\mapsto\bar\nu_0(c\Bsymmdiff c')$, and $\bar\nu$ the continuous
extension of $\bar\nu_0$ to $\frak C$; then $(\frak C,\bar\nu)$ is a
probability algebra. Each $\action'_i$ has a unique
extension to a function $\tilde{\action}_i:G\times\frak C\to\frak C$
such that $c\mapsto g\tilde{\action}_ic$ is a measure-preserving Boolean
automorphism for every $g\in G$.
$(\frak C,\bar\nu,G,\familyiI{\tilde{\action}_i})$ is a
measure-preserving action
system. Setting $\phi a=^{\ssbullet}$ for $a\in\frak A$,
$\phi:\frak A\to\frak C$ is a Boolean homomorphism and
\Centerline{$g\tilde{\action}_i\phi(a)
=g\action'_i\phi(a)
=\phi(g\action_ia)$}
\noindent whenever $a\in\frak A$, $i\in I$ and $g\in G$.
If $(\frak A,G,\familyiI{\action_i})$ is commuting, so is
$(\frak C,G,\familyiI{\tilde{\action}_i})$.
\medskip
\noindent{\bf proof} The verifications are all elementary. We have to
confirm, for instance, that if $a$, $b\in\frak A$ and
$a^{\ssbullet}=b^{\ssbullet}$ in $\frak C_0$, then
$(g\action_ia)^{\ssbullet}=(g\action_ib)^{\ssbullet}$ whenever $g\in G$ and
$i\in I$. But for this all we need to know is that
\Centerline{$\mu((g\action_ia)\Bsymmdiff(g\action_ib))
=\mu(g\action_i(a\Bsymmdiff b))
=\mu(a\Bsymmdiff b)=0$.}
\noindent Because $a\mapsto g\action_ia:\frak A\to\frak A$ is always a
Boolean automorphism, so is $c\mapsto g\action'_ic:\frak C_0\to\frak C_0$.
We see at the same time that
\Centerline{$\bar\nu_0(g\action'_ia^{\ssbullet})
=\bar\nu_0(g\action_ia)^{\ssbullet}=\mu(g\action_ia)
=\mu a=\bar\nu_0a^{\ssbullet}$}
\noindent whenever $a\in\frak A$, $g\in G$ and $i\in I$. So all the maps
$c\mapsto g\action'_ic$ are isometries on $\frak C_0$, and extend uniquely
to isometries on the completion $\frak C$, which are again Boolean
automorphisms. (See {\smc Fremlin 02}, 392H\formerly{393B} for the
construction of $(\frak C,\bar\nu)$ from $(\frak C_0,\bar\nu_0)$.)
Now the confirmation that all the $\action'_i$ and
$\tilde{\action}_i$ are actions is just a matter of writing out the relevant
formulae with their interpretations, and the same is true of the
confirmation that if the
original system $(\frak A,G,\familyiI{\action_i})$ is commuting, so are
$(\frak C_0,G,\familyiI{\action'_i})$
and $(\frak C,G,\familyiI{\tilde{\action}_i})$.
\bigskip
{\bf\zz2C Definition} Let $(\frak A,\bar\mu,G,\familyiI{\action_i})$ be a
measure-preserving action system. A {\bf factor} of the system is a
closed subalgebra $\frak B$ of $\frak A$ which is $G$-invariant in the
sense that $g\action_ib\in\frak B$ whenever $b\in\frak B$, $g\in G$ and
$i\in I$.
\bigskip
{\bf\zz2D Lemma} Let $\Bbb A=(\frak A,\bar\mu,G,\familyiI{\action_i})$ be a
commuting measure-preserving action system.
(a) If $\frak B$ is a factor of $\Bbb A$,
then $(\frak B,\bar\mu\restrp\frak B,G,
\familyiI{\action_i\restr G\times\frak B})$ is a commuting
measure-preserving action system.
(b) If $\family{t}{T}{\frak B_t}$ is a non-empty family of factors of
$\Bbb A$, then
$\bigvee_{t\in T}\frak B_t$ and $\bigcap_{t\in T}\frak B_t$ are
factors of $\Bbb A$.
(c) If $J\subseteq I$, then $\frak B_J=\{a:a\in\frak A$,
$g\action_ib=g\action_jb$ for all $g\in G$ and $i$, $j\in J\}$ is a factor
of $\Bbb A$.
(d) Let $\frak B$ be a factor of $\Bbb A$. Then
\Centerline{$g\action_i(Q_{\frak B}u)=Q_{\frak B}(g\action_iu)$}
\noindent for all $g\in G$, $i\in I$ and $u\in L^1(\frak A,\bar\mu)$.
(e) Suppose that $J\subseteq I$ and that $\frak B$ is any factor of
$\Bbb A$. Then $Q_{\frak B}Q_{\frak B_J}=Q_{\frak B\cap\frak B_J}$.
\medskip
\noindent{\bf proof (a)-(b)} Elementary.
\medskip
{\bf (c)} Elementary, recalling that $\Bbb A$ is supposed to be commuting.
\medskip
{\bf (d)} Because $Q_{\frak B}u\in L^0(\frak B)$,
$g\action_i(Q_{\frak B}u)\in L^0(\frak B)$. \Prf\ For any
$\alpha\in\Bbb R$,
\Centerline{$\Bvalue{g\action_i(Q_{\frak B}u)>\alpha}
=g\action_i\Bvalue{Q_{\frak B}u>\alpha}\in\frak B$}
\noindent because $\Bvalue{Q_{\frak B}u>\alpha}\in\frak B$.\ \QeD\ Also,
for any $b\in\frak B$,
$$\eqalign{\int_bg\action_i(Q_{\frak B}u)\,d\bar\mu
&=\int_{g^{-1}\action_ib}Q_{\frak B}u\,d\bar\mu\cr
&=\int_{g^{-1}\action_ib}u\,d\bar\mu
=\int_bg\action_iu\,d\bar\mu;\cr}$$
\noindent as $b$ is arbitrary,
$g\action_i(Q_{\frak B}u)=Q_{\frak B}(g\action_iu)$.
\medskip
{\bf (e)} If $u\in L^1(\frak A,\bar\mu)$, then
$Q_{\frak B}Q_{\frak B_J}u\in L^0(\frak B_J)$. \Prf\ Set
$v=Q_{\frak B_J}u$. For any $g\in G$,
$\alpha\in\Bbb R$ and $i$, $j\in J$,
\Centerline{$\Bvalue{g\action_iv>\alpha}
=g\action_i\Bvalue{v>\alpha}
=g\action_j\Bvalue{v>\alpha}
=\Bvalue{g\action_jv>\alpha}$;}
\noindent so $g\action_iv=g\action_jv$.
It follows that, for any $\alpha\in\Bbb R$, $g\in G$ and $i$, $j\in J$,
$$\eqalignno{g\action_i\Bvalue{Q_{\frak B}v>\alpha}
&=\Bvalue{g\action_i(Q_{\frak B}v)>\alpha}
=\Bvalue{Q_{\frak B}(g\action_iv)>\alpha}\cr
\displaycause{by (d)}
&=\Bvalue{Q_{\frak B}(g\action_jv)>\alpha}
=g\action_j\Bvalue{Q_{\frak B}v>\alpha},\cr}$$
\noindent so that $\Bvalue{Q_{\frak B}v>\alpha}\in\frak B_J$; as $\alpha$
is arbitrary, $Q_{\frak B}v\in L^0(\frak B_J)$.\ \Qed
So in fact $Q_{\frak B}Q_{\frak B_J}u\in L^0(\frak B\cap\frak B_J)$. Now
if $b\in\frak B\cap\frak B_J$,
\Centerline{$\int_bQ_{\frak B}Q_{\frak B_J}u\,d\bar\mu
=\int_bQ_{\frak B_J}u\,d\bar\mu
=\int_bu\,d\bar\mu$,}
\noindent so $Q_{\frak B}Q_{\frak B_J}u=Q_{\frak B\cap\frak B_J}u$.
\bigskip
{\bf\zz2E Definition} Let $(\frak A,\bar\mu,G,\familyiI{\action_i})$ be a
measure-automorphism action system. An {\bf extension} of the
system $(\frak A,\bar\mu,G,\familyiI{\action_i})$ will be a
quintuple
$(\frak A',\bar\mu',G,\familyiI{\action'_i},\phi)$ such that
$(\frak A',\bar\mu',G,\familyiI{\action'_i})$ is a measure-automorphism
action system, $\phi:\frak A\to\frak A'$ is a measure-preserving
homomorphism and $g\action'_i(\phi a)=\phi(g\action_ia)$
whenever $a\in\frak A$, $g\in G$ and $i\in I$.
In this case, $\phi[\frak A]$ is a factor of
$(\frak A',\bar\mu',G,\familyiI{\action'_i})$.
\bigskip
{\bf\zz2F Inductive limits}
Elaborating on \zz1G, we have the following.
Let us say that an {\bf inductive system} of measure-automorphism action
systems is an object of the form
$(\family{t}{T}{(\frak A_t,\bar\mu_t,G,\familyiI{\action^{(t)}_i}},
\langle\phi_{st}\rangle_{s\le t\in T})$ where
\inset{$T$ is an upwards-directed set,
$I$ is a set, $G$ is a group,
$(\frak A_t,\bar\mu_t,G,\familyiI{\action^{(t)}_i})$ is a
measure-automorphism action system for each $t\in T$,
$(\frak A_t,\bar\mu_t,G,\familyiI{\action^{(t)}_i},\phi_{st})$ is an
extension of $(\frak A_s,\bar\mu_s,G,\familyiI{\action^{(s)}_i})$ whenever
$s\le t$ in $T$,
$\phi_{tu}\phi_{st}=\phi_{su}$ whenever $s\le t\le u$ in $T$.}
\noindent In this case, if $(\frak A,\bar\mu,\family{t}{T}{\phi_t})$ is
the inductive limit of
$(\family{t}{T}{(\frak A_t,\bar\mu_t)},\langle\phi_{st}\rangle_{s\le t})$,
we have a unique family $\familyiI{\action_i}$ of actions of $G$ on
$\frak A$ such that
$(\frak A,\bar\mu,G,\familyiI{\action_i},\phi_t)$ is an extension of
$(\frak A_t,\bar\mu_t,G,\familyiI{\action^{(t)}_i})$ for every $t\in T$
(\zz1Ga).
In this case I will call
$(\frak A,\bar\mu,G,\penalty-100\familyiI{\action_i},
\family{t}{T}{\phi_t})$ the
{\bf inductive limit} of
$(\family{t}{T}{(\frak A_t,\bar\mu_t,G,
\penalty-100\familyiI{\action^{(t)}_i})},
\langle\phi_{st}\rangle_{s\le t\in T})$.
\bigskip
{\bf\zz2G Proposition} Let
$(\family{t}{T}{(\frak A_t,\bar\mu_t,G,\familyiI{\action^{(t)}_i})},
\langle\phi_{st}\rangle_{s\le t\in T})$ be an inductive system of
measure-automorphism action systems, with inductive limit
$(\frak A,\bar\mu,G,\familyiI{\action_i},\family{t}{T}{\phi_t})$.
(a) Suppose that $J\subseteq I$. Set
\Centerline{$\frak B^{(t)}_J
=\{a:a\in\frak A_t$, $g\action^{(t)}_ia=g\action^{(t)}_ja$
whenever $g\in G$ and $i$, $j\in J\}$ for $t\in T$,}
\Centerline{$\frak B_J
=\{a:a\in\frak A$, $g\action_ia=g\action_ja$
whenever $g\in G$ and $i$, $j\in J\}$.}
\noindent Then
$\frak B_J=\overline{\bigcup_{t\in T}\phi_t[\frak B^{(t)}_J]}$.
(b) Suppose that $\Cal J\subseteq\Cal PI$. Then
\Centerline{$\bigvee_{J\in\Cal J}\frak B_J
=\overline{\bigcupop_{t\in T}
\phi_t[\bigvee_{J\in\Cal J}\frak B^{(t)}_J]}$.}
(c) If $(\frak A_t,\bar\mu_t,G,\familyiI{\action^{(t)}_i})$ is
commuting for every $t\in T$, then
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is commuting.
\medskip
\noindent{\bf proof (a)} It is easy to check that
$\phi_t[\frak B^{(t)}_J]=\frak B_J\cap\phi_t[\frak A_t]$ for every
$t\in T$. If $a\in\frak B_J$ and $\epsilon>0$,
there are a $t\in T$ and a $b\in\phi_t[\frak A_t]$ such that
$\bar\mu(a\Bsymmdiff b)\le\epsilon$. Let
$P:L^1(\frak A,\bar\mu)\to L^1(\frak A,\bar\mu)$ be the conditional
expectation defined by the factor $\phi_t[\frak A_t]$. Then
\Centerline{$\|P(\chi a)-\chi b\|_1=\|P(\chi a-\chi b)\|_1\le\epsilon$.}
\noindent By Lemma \zz1A, there is an $\alpha\in\ooint{0,1}$ such that
$\bar\mu(a\Bsymmdiff b')\le\epsilon$, where $b'=\Bvalue{P\chi a>\alpha}$.
Now recall from Lemma \zz2De that $P(\chi a)\in L^0(\frak B_J)$, so that
$b'$ belongs to $\frak B_J$ and therefore to $\phi_t[\frak B^{(t)}_J]$.
As $\epsilon$ is arbitrary,
$a\in\overline{\bigcup_{t\in T}\phi_t[\frak B^{(t)}_J]}$; as $a$ is
arbitrary, $\frak B_J=\overline{\bigcup_{t\in T}\phi_t[\frak B^{(t)}_J]}$.
\medskip
{\bf (b)} Of course
$\phi_s[\bigvee_{J\in\Cal J}\frak B^{(s)}_J]
\subseteq\phi_t[\bigvee_{J\in\Cal J}\frak B^{(t)}_J]$ whenever $s\le t$
in $T$, so
$\frak D=\bigcup_{t\in T}\phi_t[\bigvee_{J\in\Cal J}\frak B^{(t)}_J]$ is a
subalgebra of $\frak A$ and $\overline{\frak D}$ is a closed subalgebra
included in $\bigvee_{J\in\Cal J}\frak B_J$.
By (a), it includes $\frak B_J$ for each $J\in\Cal J$, so we have equality.
\medskip
{\bf (c)} If $g$, $h\in G$ and $i$, $j\in I$, then
$\{a:g\action_i(h\action_ja)=h\action_j(g\action_ia)\}$ is a closed
subalgebra of $\frak A$ including
\Centerline{$\bigcup_{t\in T}\phi_t[\{a:a\in\frak A_t,\,
g\action^{(t)}_i(h\action^{(t)}_ja)
=h\action^{(t)}_j(g\action^{(t)}_ia)\}
=\bigcup_{t\in T}\frak A_t$}
\noindent so is the whole of $\frak A$.
\bigskip
{\bf\zz2H Definitions (a)} A measure-automorphism action system
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is {\bf measure-averaging} if
\inset{$G$ is an abelian group,
$I$ is finite,
$\WDL_{g\to G}(\prod_{i\in I}g\action_iu_i)$
is defined, for the norm $\|\,\|_1$, for
every family $\family{i}{I}{u_i}$ in $L^{\infty}(\frak A)$.}
\noindent(If $I=\emptyset$, so that we need to interpret an empty product
$\prod_{i\in I}g^{-1}\action_iu_i$, I will take it to be the
multiplicative identity $\chi 1$ of $L^0(\frak A)$.)
\medskip
{\bf (b)} A measure-automorphism action system
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is
{\bf weakly measure-averaging} if
\inset{$G$ is an abelian group,
$I$ is finite,
$\WDL_{g\to G}\bar\mu(\inf_{i\in I}g\action_ia_i)$
is defined in $\Bbb R$ for
every family $\family{i}{I}{a_i}$ in $\frak A$.}
\bigskip
{\bf\zz2I Remark}
A measure-automorphism action system
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is measure-averaging whenever
$G$ is an abelian group and $\#(I)=1$, by Theorem \zz1I, since
$\|\,\|_{\infty}$-bounded sets are relatively weakly compact in
$L^1(\frak A,\bar\mu)$.
\bigskip
{\bf\zz2J Definition} ({\smc Austin p08a}, 4.1-4.2)
Let $(\frak A,\bar\mu,G,\familyiI{\action_i})$ be
a commuting measure-automorphism action system, with $I$ finite,
and $j\in I$. I will say
that $(\frak A,\bar\mu,G,\familyiI{\action_i})$ is {\bf$j$-pleasant} if,
taking $\frak B$ to be
the closed subalgebra of $\frak A$ generated by
\Centerline{$\{a:g\action_ja=a$ for every $g\in G\}
\cup\bigcup_{i\in I}\{a:g\action_ia=g\action_ja$
for every $g\in G\}$,}
\noindent then
\Centerline{$\WDL_{g\to G}\bigl(g\action_j(u_j-Q_{\frak B}u_j)
\times\prod_{i\in I\setminus\{j\}}g\action_iu_i\bigr)
=0$}
\noindent in $L^1(\frak A,\bar\mu)$
whenever $\langle u_i\rangle_{i\in I}$ is a family in
$L^{\infty}(\frak A)$.
\bigskip
{\bf\zz2K Lemma} In the context of Definition \zz2J,
\Centerline{$\|\Bover1{\#(L)}\sum_{g\in L}
\prod_{i\in I}g\action_iu_i\|_1
\le\|u_j\|_1\cdot\prod_{i\in I\setminus\{j\}}\|u_i\|_{\infty}$}
\noindent for every non-empty finite set $L\subseteq G$.
\medskip
\noindent{\bf proof} For each $g\in L$,
\Centerline{$\|\prod_{i\in I}g\action_iu_i\|_1
\le\|g\action_ju_j\|_1
\cdot\prod_{i\in I\setminus\{j\}}\|g\action_iu_i\|_{\infty}
\le\|u_j\|_1\cdot\prod_{i\in I\setminus\{j\}}\|u_i\|_{\infty}$.}
\bigskip
{\bf\zz2L Lemma} ({\smc Austin p08a}, 4.5) Suppose that $I$ is a
finite set, $j\in I$, and that
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$ is a
$j$-pleasant system such that
$(\frak A,\bar\mu,G,\family{i}{I\setminus\{j\}}{\action_i})$ is
measure-averaging.
Then $(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$ is
measure-averaging.
\medskip
\noindent{\bf proof} Take $\frak B$ as in \zz2J.
Take $u_i\in L^{\infty}(\frak A)$ for $i\in I$. Set $v=Q_{\frak B}u_j$. Set
\Centerline{$\frak B_j=\{a:g\action_ja=a$ for every $g\in G\}$,
\quad$\frak B_i=\{a:g\action_ia=g\action_ja$ for every $g\in G\}$}
\noindent for $i\in I\setminus\{j\}$, so that every $\frak B_i$ is a
closed subalgebra of
$\frak A$ and $\frak B=\bigvee_{i\in I}\frak B_i$
is the closed subalgebra of $\frak A$ generated by
$\bigcup_{i\in I}\frak B_i$. Taking $\frak D$ to be the subalgebra of
$\frak A$ generated by $\bigcup_{i\in I}\frak B_i$, $\frak B$ is the
closure of $\frak D$ for the measure-algebra topology. Let
$E\subseteq\frak A$ be the family of elements expressible as
$\inf_{i\in I}b_i$ where $b_i\in\frak B_i$ for every $i\in I$. Then every
element of $\frak D$ is expressible as the supremum of a finite disjoint
subset of $E$. Let $\epsilon>0$.
Then we have disjoint $e_0,\ldots,e_m\in E$ and
$\alpha_0,\ldots,\alpha_m\in\Bbb R$ such that $\|v-w\|_1\le\epsilon$, where
$w=\sum_{k=0}^m\alpha_k\chi e_k$.
For each $k\le m$, express $e_k$ as $\inf_{i\in I}b_{ki}$ where
$b_{ki}\in\frak B_i$ for each $i$. Then
$$\eqalignno{g\action_j\chi e_k
\times\prod_{i\in I\setminus\{j\}}g\action_iu_i
&=g\action_j(\prod_{i\in I}\chi b_{ki})
\times\prod_{i\in I\setminus\{j\}}g\action_iu_i
=\prod_{i\in I}g\action_k\chi b_{ki}
\times\prod_{i\in I\setminus\{j\}}g\action_iu_i\cr
&=\chi b_{kj}\times\prod_{i\in I\setminus\{j\}}g\action_i\chi b_{ki}
\times\prod_{i\in I\setminus\{j\}}g\action_iu_i\cr
&=\chi b_{kj}\times\prod_{i\in I\setminus\{j\}}
g\action_i(\chi b_{ki}\times u_i)\cr}$$
\noindent for each $g$, so
\Centerline{$\WDL_{g\to G}\bigl(g\action_j\chi e_k
\times\prod_{i\in I\setminus\{j\}}g\action_iu_i\bigr)
=\chi b_{kj}\times\WDL_{g\to G}\bigl(\prod_{i\in I\setminus\{j\}}
g\action_i(\chi b_{ki}\times u_i)\bigr)$}
\noindent is defined for $\|\,\|_1$ because
$(\frak A,\bar\mu,G,\family{i}{I\setminus\{j\}}{\action_i})$
is measure-averaging. Consequently
\Centerline{$\WDL_{g\to G}
\bigl(g\action_jw\times\prod_{i\in I\setminus\{j\}}g\action_iu_i\bigr)$}
\noindent is defined. As $\epsilon$ is arbitrary,
\Centerline{$\WDL_{g\to G}
\bigl(g\action_jv\times\prod_{i\in I\setminus\{j\}}g\action_iu_i\bigr)$}
\noindent is defined (use \zz2K). Because
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$ is a $j$-pleasant
system,
\Centerline{$\WDL_{g\to G}
\bigl(g\action_j(v-u_j)
\times\prod_{i\in I\setminus\{j\}}g\action_iu_i\bigr)=0$}
\noindent for $\|\,\|_1$. So
\Centerline{$\WDL_{g\to G}
\bigl(g\action_ju_j
\times\prod_{i\in I\setminus\{j\}}g\action_iu_i\bigr)$}
\noindent is defined. As $\langle u_i\rangle_{i\in I}$ is arbitrary,
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$ is
measure-averaging.
\bigskip
{\bf\zz2M Lemma} ({\smc Austin p08a}, \S3) Let $I$ be a finite set,
$j$ an element of $I$, and
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$ a commuting
measure-{\vthsp}automorphism action system such that
$(\frak A,\bar\mu,G,\family{i}{I\setminus\{j\}}{\action'_i})$ is
measure-averaging
whenever $\family{i}{I\setminus\{j\}}{\action'_i}$ is such that
$(\frak A,\bar\mu,G,\langle\action'_i\rangle_{i\in I\setminus\{j\}})$
is a commuting measure-automorphism action system. Then
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$ is weakly
measure-averaging.
\medskip
\noindent{\bf proof} Let $\langle a_i\rangle_{i\in I}$ be a family in
$\frak A$. For $i\in I\setminus\{j\}$, define
$\action'_i:G\times\frak A\to\frak A$ by setting
$g\action'_ia=g^{-1}\action_j(g\action_ia)$ for $g\in G$ and $a\in\frak A$.
If $g$, $h\in G$ and $a\in\frak A$, then
$$\eqalign{(gh)\action'_ia
&=(gh)^{-1}\action_j((gh)\action_ia)
=h^{-1}\action_jg^{-1}\action_jg\action_ih\action_ia\cr
&=g^{-1}\action_jg\action_ih^{-1}\action_jh\action_ia
=g\action'_ih\action'_ia,\cr}$$
\noindent so $\action'_i$ is an action. Similarly direct calculation
shows that
$(\frak A,\bar\mu,G,\langle\action'_i\rangle_{i\in I\setminus\{j\}})$
is a commuting measure-automorphism action system.
Accordingly
\Centerline{$\WDL_{g\to G}
\prod_{i\in I\setminus\{j\}}g\action'_i\chi a_i$}
\noindent is defined in $L^1(\frak A,\bar\mu)$, and
$$\eqalign{\WDL_{g\to G}
\bar\mu(\inf_{i\in I}g\action_ia_i)
&=\WDL_{g\to G}
\int\prod_{i\in I}g\action_i\chi a_i\,d\bar\mu\cr
&=\WDL_{g\to G}
\int g^{-1}\action_j(\prod_{i\in I}g\action_i\chi a_i)d\bar\mu\cr
&=\WDL_{g\to G}
\int\prod_{i\in I}g^{-1}\action_j(g\action_i\chi a_i)d\bar\mu\cr
&=\WDL_{g\to G}
\int_{a_j}\prod_{i\in I\setminus\{j\}}g\action'_i\chi a_i\,d\bar\mu\cr
&=\int_{a_j}\WDL_{g\to G}
\bigl(\prod_{i\in I\setminus\{j\}}g\action'_i\chi a_i\bigr)
\,d\bar\mu\cr}$$
\noindent is defined in $\Bbb R$.
\bigskip
\noindent{\bf\zz3 Furstenberg self-joinings}
\medskip
{\bf\zz3A Construction} ({\smc Austin p08a}, \S3) Let $G$ be an abelian
group and $(\frak A,\bar\mu,G,\familyiI{\action_i})$ a commuting
measure-{\vthsp}automorphism action system. Suppose that $J\subseteq I$
is a non-empty finite set such that
$(\frak A,\bar\mu,G,\family{i}{J}{\action_i})$
is weakly measure-averaging.
(a) Let $(\frak B,\family{i}{J}{\varepsilon_j})$
be the free power $\bigotimes_J\frak A$ of $J$
copies of $\frak A$ ({\smc Fremlin 03}, \S315).
Then we have an additive functional
$\nu:\frak B\to[0,1]$ defined by saying that
\Centerline{$\nu(\inf_{j\in J}\varepsilon_ja_j)
=\WDL_{g\to G}
\bar\mu(\inf_{j\in J}g\action_ja_j)$}
\noindent whenever $\family{j}{J}{a_j}$ is a family in $\frak A$,
writing $\CalFo$ for the F{\o}lner filter of $G$.
Note that $\nu\varepsilon_j(a)=\bar\mu a$ for every
$a\in\frak A$ and $j\in J$.
(b) Let $\frak C_0$ be the quotient Boolean algebra
$\frak B/\{b:\nu b=0\}$, $\bar\nu_0$ the strictly positive finitely
additive functional on $\frak C_0$ defined by saying that
$\bar\nu_0b^{\ssbullet}=\nu b$ for every $b\in\frak B$, and $\frak C$ the
metric completion of $\frak C_0$ under the associated metric;
let $\bar\nu$ be the continuous
extension of $\bar\nu_0$ to $\frak C$, so that $(\frak C,\bar\nu)$ is a
probability algebra.
For each $j\in J$, we have a measure-preserving Boolean homomorphism
$\pi_j:\frak A\to\frak C$ defined by saying that
$\pi_ja=(\varepsilon_ja)^{\ssbullet}$ for $a\in\frak A$.
(c)(i) $\bar\nu(\inf_{i\in J}a_i)
=\WDL_{g\to G}\bar\mu(\inf_{i\in J}g\action_ia_i)$ for any family
$\family{i}{J}{a_j}$ in $\frak A$.
\quad(ii) For $j\in J$ let $R_j:L^0(\frak A)\to L^0(\frak C)$ be the
multiplicative Riesz homomorphism corresponding to the Boolean homomorphism
$\pi_j:\frak A\to\frak C$. Then for any family $\family{j}{J}{u_j}$
in $L^{\infty}(\frak A)$,
\Centerline{$\int\prod_{j\in J}R_ju_j\,d\bar\nu
=\WDL_{g\to G}
\int\prod_{j\in J}g\action_ju_j\,d\bar\mu$.}
(d) We have a commuting measure-automorphism action system
$(\frak C,\bar\nu,G,
\langle\tilde{\action}_i\rangle_{i\in I\cup\{\infty\}})$ defined
by saying that
\Centerline{$g\tilde{\action}_i(\pi_ja)
=\pi_j(g\tilde{\action}_ia)$,}
\Centerline{$g\tilde{\action}_{\infty}(\pi_ja)
=\pi_j(g\action_ja)$}
\noindent whenever $i\in I$, $j\in J$ and $a\in\frak A$.\footnote{Here, and
later, I use the symbol $\infty$ unscrupulously to denote an object not
belonging to any relevant set previously mentioned.}
The corresponding actions on $L^0(\frak C)$ are defined by the formulae
\Centerline{$g\tilde{\action}_i(R_ku)
=R_k(g\action_iu)$,}
\Centerline{$g\tilde{\action}_{\infty}(R_ku)
=R_k(g\action_ku)$}
\noindent for $i\in I$, $k\in J$ and $u\in L^0(\frak A)$.
(e) Now fix on a member $j$ of $J$, and for $i\in I$ set
$$\eqalign{\hat{\action}_i&=\tilde{\action}_{\infty}\text{ if }i=j,\cr
&=\tilde{\action}_i\text{ otherwise}.\cr}$$
\noindent Then $(\frak C,\bar\nu,G,\familyiI{\hat{\action}_i},\pi_j)$ is
an extension
of $(\frak A,\bar\mu,G,\familyiI{\action_i})$.
\medskip
\noindent{\bf proof (a)} We know that the limit
\Centerline{$\WDL_{g\to G}
\bar\mu(\inf_{j\in J}g\action_ja_j)$}
\noindent is always defined, so we have a well-defined functional on
the set $\frak A^J$.
Since this is clearly additive in each variable separately, it
uniquely defines an additive functional on $\frak B$ ({\smc Fremlin 02},
326Q).
Taking $a_j=a$, $a_i=1$ for $i\in J\setminus\{j\}$ in the formula,
we get the correct value for $\nu\varepsilon_j(a)$.
\medskip
{\bf (b)} Elementary, in view of the results in {\smc Fremlin 02}.
\medskip
{\bf (c)(i)} This is just
the definition of $\nu$ translated into terms of $\bar\nu$.
\medskip
\quad{\bf (ii)} Both sides of the equation correspond to
$\|\,\|_{\infty}$-continuous multilinear functionals on
$L^{\infty}(\frak A)^J$, which agree on families of
the form $\family{i}{J}{u_i}=\family{i}{J}{\chi a_i}$.
\medskip
{\bf (d)(i)} The defining universal mapping property of
$\bigotimes_J\frak A$ tells us that we have functions
$\action^*_i$, $\action^*_{\infty}$
from $G\times\frak B$ to $\frak B$ defined by saying that
\Centerline{$g\action^*_i(\varepsilon_ja)=\varepsilon_j(g\action_ia)$,}
\Centerline{$g\action^*_{\infty}(\varepsilon_ja)
=\varepsilon_j(g\action_ja)$}
\noindent for $a\frak A$, $g\in G$, $i\in I$ and $j\in J$, and that
all the maps $b\mapsto g\action^*_ib$ (for $i\in I\cup\{\infty\})$
are Boolean homomorphisms. Direct
calculation shows that $\action^*_i$ is an action of $G$ on $\frak B$
for every $i\in I\cup\{\infty\}$.
\medskip
\quad{\bf (ii)} $\nu$ is $G$-invariant for all these actions.
\Prf\ If $i\in I$, $h\in G$ and $a_j\in\frak A$ for $j\in J$,
$$\eqalign{\nu(h\action^*_i(\inf_{j\in J}\varepsilon_ja_j))
&=\nu(\inf_{j\in J}\varepsilon_j(h\action_ia_j))\cr
&=\WDL_{g\to G}
\bar\mu(\inf_{j\in J}g\action_jh\action_ia_j)\cr
&=\WDL_{g\to G}
\bar\mu(\inf_{j\in J}h\action_ig\action_ja_j)\cr
&=\WDL_{g\to G}
\bar\mu(h\action_i(\inf_{j\in J}g\action_ja_j))\cr
&=\WDL_{g\to G}
\bar\mu(\inf_{j\in J}g\action_ja_j)
=\nu(\inf_{j\in J}\varepsilon_ja_j),\cr}$$
$$\eqalign{\nu(h\action^*_{\infty}(\inf_{j\in J}\varepsilon_ja_j))
&=\nu(\inf_{j\in J}\varepsilon_j(h\action_ja_j))\cr
&=\WDL_{g\to G}
\bar\mu(\inf_{j\in J}g\action_jh\action_ja_j)\cr
&=\WDL_{g\to G}
\bar\mu(\inf_{j\in J}(gh)\action_ja_j)\cr
&=\lim_{L\to\CalFo}\Bover1{\#(Lh)}\sum_{g\in Lh}
\bar\mu(\inf_{j\in J}g\action_ja_j)
=\nu(\inf_{j\in J}\varepsilon_ja_j)\cr}$$
\noindent because $\CalFo$ is invariant under translation.
Since an additive functional on $\frak B$ is determined by its values on
the basic elements $\inf_{j\in J}\varepsilon_ja_j$,
$\nu(h\action^*_ib)=\nu b$ for every $b\in\frak B$, $h\in G$ and
$i\in I\cup\{\infty\}$.\ \Qed
\medskip
\quad{\bf (iii)} Of course
$$\eqalign{g\action^*_i(h\action^*_k
(\inf_{j\in J}\varepsilon_ja_j))
&=g\action^*_i(\inf_{j\in J}\varepsilon_j(h\action_ka_j))
=\inf_{j\in J}\varepsilon_j(g\action_ih\action_ka_j))\cr
&=\inf_{j\in J}\varepsilon_j(h\action_kg\action_ia_j))
=h\action^*_k(g\action^*_i(\inf_{j\in J}\varepsilon_ja_j)),\cr}$$
$$\eqalign{g\action^*_i(h\action^*_{\infty}
(\inf_{j\in J}\varepsilon_ja_j))
&=g\action^*_i(\inf_{j\in J}\varepsilon_j(h\action_ja_j))
=\inf_{j\in J}\varepsilon_j(g\action_ih\action_ja_j))\cr
&=\inf_{j\in J}\varepsilon_j(h\action_jg\action_ia_j))
=h\action^*_{\infty}(g\action^*_i(\inf_{j\in J}\varepsilon_ja_j))\cr}$$
\noindent whenever $g$, $h\in G$, $i$, $k\in I$ and
$\family{j}{J}{a_j}\in\frak A^J$. So
the $\action^*_i$, for $i\in I\cup\{\infty\}$, are commuting actions.
\medskip
\quad{\bf (iv)} Applying the method of \zz2B to the system
$(\frak B,\nu,G,\langle\action^*_i\rangle_{i\in I\cup\{\infty\}})$,
we see that the declared formulae define
actions $\tilde{\action}_i$ of $G$ on $\frak C$ such that
$(\frak C,\bar\nu,G,\family{i}{I\cup\{\infty\}}{\tilde{\action}_i})$
is a commuting measure-automorphism action system.
\medskip
\quad{\bf (v)} The other formulae are now elementary.
\medskip
{\bf (e)} All we have to check is that, for $g\in G$ and $a\in\frak A$,
$$\eqalign{g\hat{\action}_i(\pi_ja)
&=g\tilde{\action}_{\infty}(\pi_ja)=\pi_j(g\action_ja)
\text{ if }i=j,\cr
&=g\tilde{\action}_i(\pi_ja)=\pi_j(g\action_ia)
\text{ otherwise}.\cr}$$
\bigskip
{\bf\zz3B Definition} In the context of \zz3A, I will call
$(\frak C,\bar\nu,G,\family{i}{I\cup\{\infty\}}{\tilde{\action}_i},
\family{j}{J}{\pi_j})$ the {\bf Furstenberg self-joining} of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ over $J$; in addition, I will
call $(\frak C,\bar\nu,G,\familyiI{\hat{\action}_i},\pi_j)$ the
{\bf$(J,j)$-Furstenberg extension} of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$.
(See {\smc Austin p08a} for some of the history of this construction.)
\bigskip
{\bf\zz3C Proposition} Let
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ be a commuting
measure-automorphism action system with an extension
$(\frak A',\bar\mu',G,\familyiI{\action'_i},\phi)$, and $J\subseteq I$ a
non-empty finite set.
Suppose that both $(\frak A,\bar\mu,G,\family{j}{J}{\action_j})$
and $(\frak A',\bar\mu',G,\family{j}{J}{\action'_j})$
are weakly measure-averaging, with Furstenberg self-joinings
$(\frak C,\bar\nu,G,\family{i}{I\cup\{\infty\}}{\tilde{\action}_i},
\family{j}{J}{\pi_j})$
and $(\frak C',\bar\nu',G,\family{i}{I\cup\{\infty\}}{\tilde{\action}'_i},
\family{j}{J}{\pi'_j})$ respectively.
Then there is a unique
measure-preserving Boolean homomorphism
$\psi:\frak C\to\frak C'$ such that $\psi\pi_j=\pi'_j\phi$ for every
$i\in I$, and
$(\frak C',\bar\nu',G,
\family{i}{I\cup\{\infty\}}{\tilde{\action}'_i},\psi)$
is an extension of
$(\frak C,\bar\nu,G,\family{i}{I\cup\{\infty\}}{\tilde{\action}_i})$.
\medskip
\noindent{\bf proof (a)} Taking $\frak B=\bigotimes_J\frak A$ and
$\frak B'=\bigotimes_J\frak A'$, we have a Boolean homomorphism
$\theta:\frak B\to\frak B'$ defined by saying that
$\theta\varepsilon_j=\varepsilon'_j\phi$ for every $j\in J$.
Now, writing $\CalFo$ for the F{\o}lner
filter of $G$,
$$\eqalign{\nu'\theta(\inf_{j\in J}\varepsilon_ja_j)
&=\nu'(\inf_{j\in J}\varepsilon'_j\phi a_j)
=\WDL_{g\to G}
\bar\mu'(\inf_{j\in J}g\action'_j\phi a_j)\cr
&=\WDL_{g\to G}
\bar\mu'(\inf_{j\in J}\phi(g\action_ja_j))
=\WDL_{g\to G}
\bar\mu'\phi(\inf_{j\in J}g\action_ja_j)\cr
&=\WDL_{g\to G}
\bar\mu(\inf_{j\in J}g\action_ja_j)
=\nu(\inf_{j\in J}\varepsilon_ja_j)\cr}$$
\noindent whenever $a_j\in\frak A$ for $j\in J$. So $\nu'\theta b=\nu b$
for every $b\in\frak B$. It follows that $\theta$ induces a Boolean
homomorphism $\psi_0:\frak C_0\to\frak C_0'$ such that
$\psi_0(b^{\ssbullet})=(\theta b)^{\ssbullet}$ for every $b\in\frak B$,
taking $\frak C_0$, $\frak C'_0$ to be the quotient algebras as in \zz3Ab;
and $\bar\nu'_0\psi_0c=\bar\nu_0c$ for every $c\in\frak C_0$. Accordingly
$\psi_0$ extends to a measure-preserving Boolean homomorphism
$\psi:\frak C\to\frak C'$. Tracing the definitions, we have
\Centerline{$\psi\pi_ja
=\psi_0\pi_ja
=\psi_0(\varepsilon_ja)^{\ssbullet}
=(\varepsilon'_j\phi a)^{\ssbullet}
=\pi'_j\phi a$}
\noindent for every $a\in\frak A$ and $j\in J$, and clearly this defines
$\psi$. Similarly, examining the actions of $G$ on these structures,
$$\eqalign{g\tilde{\action}'_i(\psi\pi_j a)
&=g\tilde{\action}'_i(\pi'_j\phi a)
=\pi'_j(g\action'_i(\phi a))\cr
&=\pi'_j\phi(g\action_i a)
=\psi\pi_j(g\action_i a)
=\psi(g\tilde{\action}_i(\pi_j a)),\cr}$$
$$\eqalign{g\tilde{\action}'_{\infty}(\psi\pi_j a)
&=g\tilde{\action}'_{\infty}(\pi'_j\phi a)
=\pi'_j(g\action'_j(\phi a))\cr
&=\pi'_j\phi(g\action_j a)
=\psi\pi_j(g\action_j a)
=\psi(g\tilde{\action}_{\infty}(\pi_j a))\cr}$$
\noindent whenever $a\in\frak A$, $g\in G$, $i\in I$ and
$j\in J$; consequently
\Centerline{$g\tilde{\action}'_i(\psi c)=\psi(g\tilde{\action}_ic)$}
\noindent whenever $c\in\frak C$, $g\in G$ and $i\in I\cup\{\infty\}$. So
$(\frak C',\bar\nu',G,\family{i}{I\cup\{\infty\}}{\tilde{\action}'_i},
\psi)$ is an extension of
$(\frak C,\bar\nu,G,\family{i}{I\cup\{\infty\}}{\tilde{\action}_i})$.
\bigskip
{\bf\zz3D Lemma} ({\smc Bergelson McCutcheon \& Zhang 97}, 4.2)
Let $G$ be an abelian group,
$\CalFo$ its F{\o}lner filter,
$U$ an inner product space and $g\mapsto u_g:G\to U$ a bounded
function such that
\Centerline{$\inf_{\emptyset\ne M\in[G]^{<\omega}}\Bover1{\#(M)^2}
\overline{\WDL}_{g\to G}
\sum_{h,h'\in M}\innerprod{u_{hg}}{u_{h'g}}\le 0$.}
\noindent Then $\WDL_{g\to G}u_g=0$.
\medskip
\noindent{\bf proof} Set $\gamma=\sup_{g\in G}\|u_g\|$.
Let $\epsilon>0$. Let $M\in[G]^{<\omega}\setminus\{\emptyset\}$
be such that
\Centerline{$\Bover1{\#(M)^2}\overline{\WDL}_{g\to G}
\sum_{h,h'\in M}\innerprod{u_{hg}}{u_{h'g}}\le\epsilon$.}
\noindent For non-empyt finite sets $L\subseteq G$ set
\Centerline{$v_L
=\sum_{g\in L}\Bover1{\#(M)}\sum_{h\in M}\Bover1{\#(L)}u_{hg}$.}
\noindent Then
$$\eqalign{\limsup_{L\to\CalFo}\|v_L-\Bover1{\#(L)}\sum_{g\in L}u_g\|
&\le\Bover1{\#(M)}\sum_{h\in M}\limsup_{L\to\CalFo}
\|\Bover1{\#(L)}(\sum_{g\in L}u_g-\sum_{g\in L}u_{hg})\|\cr
&\le\sup_{h\in M}\limsup_{L\to\CalFo}\Bover1{\#(L)}
\|\sum_{g\in L}u_g-\Bint_{hL}u_g\|\cr
&\le\sup_{h\in M}\limsup_{L\to\CalFo}\Bover1{\#(L)}
\gamma\#(L\symmdiff hL)
=0.\cr}$$
\noindent On the other hand, for every non-empty finite $L\subseteq G$,
$$\eqalign{\|v_L\|
&\le\Bover1{\#(L)}\sum_{g\in L}\|\Bover1{\#(M)}
\sum_{h\in M}u_{hg}\|\cr
&\le\Bover1{\#(L)}\cdot\sqrt{\#(L)}\cdot
\sqrt{\sum_{g\in L}\|\Bover1{\#(M)}\sum_{h\in M}u_{hg}\|^2}\cr}$$
\noindent (by the Cauchy-Schwartz inequality), so
$$\eqalign{\|v_L\|^2
&\le\Bover1{\#(L)}\sum_{g\in L}
\|\Bover1{\#(M)}\sum_{h\in M}u_{hg}\|^2\cr
&=\Bover1{\#(L)}\sum_{g\in L}\Bover1{\#(M)^2}
\sum_{h,h'\in M}\innerprod{u_{hg}}{u_{h'g}}\cr}$$
\noindent and
$$\eqalign{\limsup_{L\to\CalFo}\|v_L\|^2
&\le\Bover1{\#(M)^2}\limsup_{L\to\CalFo}
\Bover1{\#(L)}\sum_{g\in L}\sum_{h,h'\in M}\innerprod{u_{hg}}{u_{h'g}}\cr
&=\Bover1{\#(M)^2}\overline{\WDL}_{g\to G}
\sum_{h,h'\in M}\innerprod{u_{hg}}{u_{h'g}}
\le\epsilon.\cr}$$
\noindent Putting these together,
\Centerline{$\limsup_{L\to\CalFo}\|\Bover1{\#(L)}\sum_{g\in L}u_g\|
\le\sqrt{\epsilon}$;}
\noindent as $\epsilon$ is arbitrary, the limit is zero, and
\Centerline{$\WDL_{g\to G}u_g
=\lim_{L\to\CalFo}\Bover1{\#(L)}\sum_{g\in L}u_g=0$.}
\bigskip
{\bf\zz3E Lemma} ({\smc Austin p08a}, 4.7) Let
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$ be a commuting
measure-{\vthsp}automorphism action system, $J\subseteq I$ a finite
non-empty set such that
$(\frak A,\bar\mu,G,\family{i}{J}{\action_i})$ is weakly measure-averaging,
and $(\frak C,\bar\nu,G,
\family{i}{I\cup\{\infty\}}{\tilde{\action}_i},
\penalty-100\family{j}{J}{\pi_j})$ the
Furstenberg self-joining of
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$ over $J$.
Let $\frak D$ be the fixed-point algebra
$\{c:c\in\frak C$, $g\tilde{\action}_{\infty}c=c$ for every $g\in G\}$.
For $j\in J$ let
$R_j:L^0(\frak A)\to L^0(\frak C)$ be the multiplicative Riesz
homomorphism corresponding to $\pi_j:\frak A\to\frak C$.
If $\family{j}{J}{u_j}$ is a family in $L^{\infty}(\frak A)$
such that $Q_{\frak D}(\prod_{j\in J}R_ju_j)=0$, then
\Centerline{$\WDL_{g\to G}\prod_{j\in J}g\action_ju_j=0$}
\noindent in $L^1(\frak A,\bar\mu)$.
\medskip
\noindent{\bf proof}
Set $w=\prod_{j\in J}R_ju_j$; for $h\in G$, set
$w_h=h\tilde{\action}_{\infty}w$. Set
$\gamma=\prod_{j\in J}\|u_j\|_{\infty}$;
note that $\|w_h\|_{\infty}\le\gamma$ for every $h$.
\medskip
{\bf (a)} Note first that $Q_{\frak D}w_h=0$ for every $h\in G$.
\Prf\ If $d\in\frak D$,
\Centerline{$\int_d h\tilde{\action}_{\infty}w\,d\bar\nu
=\int_{h^{-1}\tilde{\action}_{\infty}d}w\,d\bar\nu
=0$}
\noindent because $h^{-1}\tilde{\action}_{\infty}d=d\in\frak D$.
As $d$ is arbitrary, $Q_{\frak D}w_h=0$.\ \Qed
\medskip
{\bf (b)} For any $h$, $h'\in G$,
$$\eqalign{\int w_h\times w_{h'}\,d\bar\nu
&=\int\prod_{j\in J}R_j(h\action_ju_j)
\times\prod_{j\in J}R_j(h'\action_ju_j)\,d\bar\nu\cr
&=\int\prod_{j\in J}R_j(h\action u_j\times h'\action_ju_j)\,
d\bar\nu\cr
&=\WDL_{g\to G}
\int\prod_{j\in J}g\action_j(h\action u_j\times h'\action_ju_j)
\,d\bar\mu\cr}$$
\noindent by \zz3Ac. Now
\Centerline{$w^*=\WDL_{h\to G}w_h$}
\noindent is defined for $\|\,\|_2$ and belongs to
$L^{\infty}(\frak D)$ (\zz1I).
\medskip
{\bf (c)} For $g\in G$ set $v_g=\prod_{j\in J}g\action_ju_j$,
We find that
\Centerline{$\inf_{\emptyset\ne M\in[G]^{<\omega}}
\Bover1{\#(M)^2}\overline{\WDL}_{g\to\infty}\sum_{h,h'\in M}
\int v_{hg}\times v_{h'g}\,d\bar\mu\le 0$.}
\noindent\Prf\ Let $\epsilon>0$. Then there is an non-empty finite
$M\subseteq G$ such that
$\|w^*-\Bover1{\#(M)}\sum_{h\in M}w_h\|_2\le\epsilon$. Now
$$\eqalignno{\Bover1{\#(M)^2}&\overline{\WDL}_{g\to G}
\sum_{h,h'\in M}\int v_{hg}\times v_{h'g}\,d\bar\mu\cr
&=\Bover1{\#(M)^2}\overline{\WDL}_{g\to G}
\sum_{h,h'\in M}
\int\prod_{j\in J}(hg)\action_ju_j\times\prod_{j\in J}(h'g)\action_ju_j)
\,d\bar\mu\cr
&=\Bover1{\#(M)^2}\overline{\WDL}_{g\to G}\sum_{h,h'\in M}
\int g\action_j\prod_{j\in J}(h\action_ju_j\times h'\action_ju_j)
\,d\bar\mu\cr
\displaycause{because the system is commuting}
&=\Bover1{\#(M)^2}\sum_{h,h'\in M}
\int w_h\times w_{h'}\,d\bar\nu\cr
\displaycause{by (b) above}
&=\Bover1{\#(M)}\sum_{h\in M}\int w_h
\times(\Bover1{\#(M)}\sum_{h\in M}w_{h'})
\,d\bar\nu\cr
&\le\Bover1{\#(M)}\sum_{h\in M}\int w_h\times w^*d\bar\nu
+\Bover1{\#(M)}\sum_{h\in M}\|w_h\|_2
\|w^*-\Bover1{\#(M)}\sum_{h\in M}w_{h'}\|_2\cr
&\le\Bover1{\#(M)}\sum_{h\in M}\gamma\epsilon\cr
\displaycause{because $w^*\in L^{\infty}(\frak D)$ and
$Q_{\frak D}w_h=0$, so $\int w_h\times w^*d\bar\nu=0$ for every $h$}
&\le\gamma\epsilon.\cr}$$
\noindent As $\epsilon$ is arbitrary, we have the result.\ \Qed
\medskip
{\bf (d)} By \zz3D,
$\lim_{L\to\CalFo}\|\Bover1{\mu L}\sum_{g\in L}v_g\|_2=0$. But
$\{v_g:g\in G\}$ is $\|\,\|_{\infty}$-bounded, so
$\{\Bover1{\mu L}\sum_{g\in L}v_g:L\in[G]^{<\omega}\setminus\{0\}\}$ also
is, and
$\lim_{L\to\CalFo}\|\Bover1{\mu L}\sum_{g\in L}v_g\|_1=0$, as required.
\bigskip
{\bf\zz3F Lemma} ({\smc Austin p08a}, 4.6) Let $G$ be an abelian
group, and suppose that $I$ is a non-empty finite set such that
every commuting measure-automorphism action system
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$ is
weakly measure-averaging. If $j\in I$,
every commuting measure-automorphism action system
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$ has a $j$-pleasant
extension.
\medskip
\noindent{\bf proof (a)} Set $\frak A_0=\frak A$,
$\bar\mu_0=\bar\mu$ and $\action_{0i}=\action_i$ for $i\in I$.
Given that
$(\frak A_m,\bar\mu_m,G,\langle\action^{(m)}_i\rangle_{i\in I})$
is a commuting measure-automorphism action system, then our hypothesis
tells us that it is weakly measure-averaging; let
$(\frak C_m,\bar\nu_m,G,
\langle\tilde{\action}^{(m)}_i\rangle_{i\in I\cup\{\infty\}},
\langle\pi^{(m)}_i\rangle_{i\in I})$ be its Furstenberg self-joining over
$I$, and
$(\frak A_{m+1},\bar\mu_{m+1},G,\familyiI{\action^{(m+1)}_i},
\phi_{m,m+1})$
the $(I,j)$-Furstenberg extension of
$(\frak A_m,\bar\mu_m,G,\langle\action^{(m)}_i\rangle_{i\in I})$.
Continue.
For $l\le m$, define $\phi_{lm}:\frak A_l\to\frak A_m$ by taking
$\phi_{ll}$ to be the identity on $\frak A_l$ and
$\phi_{l,m+1}=\phi_{m,m+1}\phi_{lm}$. Let
$(\frak A',\bar\mu',\sequence{m}{\phi_m})$ be the inductive limit of
$\sequence{m}{(\frak A_m,\bar\mu_m)},\langle\phi_{lm}\rangle_{l\le m})$.
For each $i\in I$ we have an action $\action'_i$ of $G$ on $\frak A'$
defined by saying that $g\action'_i(\phi_ma)=\phi_m(g\action^{(m)}_ia)$
for $g\in G$, $m\in\Bbb N$ and $a\in\frak A_m$ (\zz1Ga);
now $(\frak A',\bar\mu',G,\langle\action'_i\rangle_{i\in I})$ is a
measure-automorphism action system.
Because all the
systems $(\frak A_m,G,\langle\action^{(m)}_i\rangle_{i\in I})$ and
$(\frak C_m,G,\langle\tilde{\action}^{(m)}_i\rangle_{i\in I\cup\{\infty\}})$ are
commuting, so is
$(\frak A',G,\langle\action'_i\rangle_{i\in I})$.
Of course
$(\frak A',\bar\mu',G,\langle\action'_i\rangle_{i\in I},\phi_0)$ is an
extension of $(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$.
\medskip
{\bf (b)} Once again, the hypothesis of this lemma ensure
that $(\frak A',G,\langle\action'_i\rangle_{i\in I})$ is weakly
measure-averaging and has a Furstenberg self-joining
$(\frak C,\bar\nu,G,\langle\tilde{\action}_i\rangle_{i\in I\cup\{\infty\}},
\langle\pi_i\rangle_{i\in I})$ over $I$. Now we can identify
$(\frak C,\bar\nu,G,\langle\tilde{\action}_i\rangle_{i\in I\cup\{\infty\}})$ with the
inductive limit of
$(\sequence{m}{(\frak C_m,\bar\nu_m,G,\penalty-75
\langle\tilde{\action}^{(m)}_i\rangle_{i\in I\cup\{\infty\}})},\penalty-100
\langle\phi_{l+1,m+1}\rangle_{l\le m})$. \Prf\ By Proposition \zz3C, we
have measure-preserving Boolean homomorphisms
$\psi_{lm}:\frak C_l\to\frak C_m$ and $\psi_m:\frak C_m\to\frak C$, for
$l\le m$, such that
\Centerline{$\psi_{lm}\pi^{(l)}_i=\pi^{(m)}_i\phi_{lm}$,
\quad$\psi_m\pi^{(m)}_i=\pi_i\phi_m$}
\noindent for $l\le m$ and $i\in I$; and these homomorphisms are
consistent with the actions, that is,
\Centerline{$g\tilde{\action}^{(m)}_i(\psi_{lm}d)
=\psi_{lm}(g\tilde{\action}^{(l)}_id)$}
\noindent whenever $l\le m$, $i\in I\cup\{\infty\}$, $g\in G$ and
$d\in\frak C_l$. We need to check that
$\bigcup_{m\in\Bbb N}\psi_m[\frak C_m]$ is metrically dense in $\frak C$,
but this is easy; the closure of
$\bigcup_{m\in\Bbb N}\psi_m[\frak C_m]$ must include
\Centerline{$\bigcup_{m\in\Bbb N,i\in I}\psi_m[\pi^{(m)}_i[\frak A]]
=\bigcup_{i\in I}\pi_i[\bigcup_{m\in\Bbb N}\phi_m[\frak A]]$}
\noindent and therefore includes $\bigcup_{i\in I}\pi_i[\frak A']$ and the
subalgebra it generates, which is dense in $\frak C$ (see the construction
in \zz3Ab).\ \Qed
\medskip
{\bf (c)} The formulae of the rest of this proof will be easier to read if
I give names to the multiplicative Riesz homomorphisms corresponding to the
measure-preserving Boolean homomorphisms here:
\Centerline{$S_{lm}:L^0(\frak A_l)\to L^0(\frak A_m)$
from $\phi_{lm}:\frak A_l\to\frak A_m$,}
\Centerline{$S_m:L^0(\frak A_m)\to L^0(\frak A')$
from $\phi_m:\frak A_m\to\frak A'$,}
\Centerline{$R^{(m)}_i:L^0(\frak A_m)\to L^0(\frak C_m)=L^0(\frak A_{m+1})$
from $\pi^{(m)}_i:\frak A_m\to\frak C_m$,}
\Centerline{$R_i:L^0(\frak A')\to L^0(\frak C)$
from $\pi_i:\frak A'\to\frak C$,}
\Centerline{$T_{lm}:L^0(\frak C_l)\to L^0(\frak C_m)$
from $\psi_{lm}:\frak C_l\to\frak C_m$,}
\Centerline{$T_m:L^0(\frak C_m)\to L^0(\frak C)$
from $\psi_m:\frak C_m\to\frak C$}
\noindent for $l\le m$ and $i\in I$. The identities above become
\inset{$S_{m,m+1}=R^{(m)}_n$ because $\phi_{m,m+1}=\pi^{(m)}_n$,
$S_l=S_mS_{lm}$ because $\phi_l=\phi_m\phi_{lm}$,
$T_l=T_mT_{lm}$ because $\psi_l=\psi_m\psi_{lm}$,
$T_mR^{(m)}_i=R_iS_m$ because $\psi_m\pi^{(m)}_i=\pi_i\phi_m$.}
\noindent In addition, we shall have
\Centerline{$\int_dv\,d\bar\nu_m
=\int_{\psi_md}T_mv\,d\bar\mu'$ whenever $d\in\frak C_m$ and
$u\in L^1(\frak C_m,\bar\nu_m)$,}
\Centerline{$\int_au\,d\bar\mu_m
=\int_{\phi_md}S_mu\,d\bar\mu'$ whenever $a\in\frak A_m$ and
$u\in L^1(\frak A_m,\bar\mu_m)$.}
\medskip
{\bf (d)} For each $m\in\Bbb N$, let $\frak B_m$ be the closed subalgebra
of $\frak A_m$ generated by
$$\eqalign{\{a:a\in\frak A_m&,
\,g\action^{(m)}_ja=a\text{ for every }g\in G\}\cr
&\cup\bigcup_{i\in I\setminus\{j\}}\{a:a\in\frak A_m,\,
g\action^{(m)}_ia=g\action^{(m)}_ja\text{ for every }g\in G\},\cr}$$
\noindent and $P_m=Q_{\frak B_m}$. Similarly,
let $\frak B$ be the closed subalgebra of $\frak A'$ generated by
$$\eqalign{\{a:a\in\frak A'&,
\,g\action'_ja=a\text{ for every }g\in G\}\cr
&\cup\bigcup_{i\in I\setminus\{j\}}\{a:a\in\frak A',\,
g\action'_ia=g\action'_ja\text{ for every }g\in G\},\cr}$$
\noindent and $P=Q_{\frak B}$. If
$l\in\Bbb N$ and $u\in L^1(\frak A_l,\bar\mu_l)$, then
$PS_lu=\lim_{m\to\infty}S_mP_mS_{lm}u$ for $\|\,\|_1$. \Prf\
By 1Fb, $\{a:a\in\frak A'$, $g\action'_ja=a$ for every $g\in G\}$ is the
metric closure of
\Centerline{$\bigcup_{m\in\Bbb N}\{\phi_ma:
a\in\frak A_m$, $g\action^{(m)}_ja=a$ for every $g\in G\}$;}
\noindent applying the same result to the actions
$(g,a)\mapsto g^{-1}\action^{(m)}_j(g\action^{(m)}_ia)$, we see that
$\{a:a\in\frak A'$, $g\action'_ia=g\action'_ja$ for every $g\in G\}$ is the
metric closure of
\Centerline{$\bigcup_{m\in\Bbb N}\{\phi_ma:
a\in\frak A_m$, $g\action^{(m)}_ia=g\action^{(m)}_ja$
for every $g\in G\}$.}
\noindent So $\frak B$ is the closure of
$\bigcup_{m\in\Bbb N}\phi_m[\frak B_m]$. Of course
$\phi_{m,m+1}[\frak B_m]\subseteq\frak B_{m+1}$ for every $m$, so
$\sequence{m}{\phi_m[\frak B_m]}$ is non-decreasing. For each $m\ge l$,
$S_mP_mS_{lm}u$ is the conditional expectation of
$S_lu$ on $\phi_m[\frak B_m]$; the result follows at once, by the
martingale convergence theorem ({\smc Fremlin 02}, 367Qb).
\Qed
\medskip
{\bf (e)} Suppose that $m\in\Bbb N$, that
$u_i\in L^{\infty}(\frak A_m)$ for $i\in I$ and that
$d\in\frak C_m=\frak A_{m+1}$ is such that
$g\tilde{\action}^{(m)}_{\infty}d=d$ for every $g\in G$. Then
$$\eqalign{\int_{\psi_md}\prod_{i\in I}R_iS_mu_i\,d\bar\nu
&=\int_{\psi_md}\prod_{i\in I}T_mR^{(m)}_iu_i\,d\bar\nu
=\int_{\psi_md}T_m(\prod_{i\in I}R^{(m)}_iu_i)\,d\bar\nu\cr
&=\int_d\prod_{i\in I}R^{(m)}_iu_i\,d\bar\nu_m
=\int_d\prod_{i\in I}R^{(m)}_iu_i\,d\bar\mu_{m+1}\cr
&=\int_dS_{m,m+1}u_j\times\prod_{i\in I\setminus\{j\}}R^{(m)}_iu_i
\,d\bar\mu_{m+1}.\cr}$$
\noindent Now $\tilde{\action}^{(m)}_{\infty}=\action^{(m+1)}_j$, so
$d\in\frak B_{m+1}$. While if $i\in I\setminus\{j\}$, then
\Centerline{$g\action^{(m+1)}_i(\pi^{(m)}_ia)
=g\tilde{\action}^{(m)}_i(\pi^{(m)}_ia)
=g\tilde{\action}^{(m)}_{\infty}(\pi^{(m)}_ia)
=g\action^{(m+1)}_j(\pi^{(m)}_ia)$}
\noindent for every $a\in\frak A_m$ and $g\in G$, so that
$\pi^{(m)}_i[\frak A_m]\subseteq\frak B_{m+1}$ and
$P_{m+1}R^{(m)}_iu=R^{(m)}_iu$ for every
$u\in L^1(\frak A_m,\bar\mu_m)$. Accordingly
$$\eqalign{\int_{\psi_md}\prod_{i\in I}&R_iS_mu_i\,d\bar\nu
=\int_dS_{m,m+1}u_j\times\prod_{i\in I\setminus\{j\}}R^{(m)}_iu_i
\,d\bar\mu_{m+1}\cr
&=\int_dP_{m+1}(S_{m,m+1}u_j\times\prod_{i\in I\setminus\{j\}}R^{(m)}_iu_i)
\,d\bar\mu_{m+1}\cr
&=\int_dP_{m+1}S_{m,m+1}u_j\times\prod_{i\in I\setminus\{j\}}R^{(m)}_iu_i
\,d\bar\mu_{m+1}\cr
&=\int_{\phi_{m+1}d}S_{m+1}P_{m+1}S_{m,m+1}u_j
\times\prod_{i\in I\setminus\{j\}}S_{m+1}R^{(m)}_iu_i\,d\bar\mu'.\cr}$$
\medskip
{\bf (f)} Re-casting the formulae in (e) we get the following.
Suppose that $l\in\Bbb N$, that
$u_i\in L^{\infty}(\frak A_l)$ for $i\in I$ and that
$d\in\frak C_l=\frak A_{l+1}$ is such that
$g\action^{(l+1)}_jd=g\tilde{\action}^{(l)}_{\infty}d=d$ for every $g\in G$.
Then
$$\eqalignno{
\int_{\psi_ld}&\prod_{i\in I}R_iS_lu_i\,d\bar\nu
=\lim_{m\to\infty}\int_{\psi_m\psi_{lm}d}\prod_{i\in I}
R_iS_mS_{lm}u_i\,d\bar\nu\cr
&=\lim_{m\to\infty}\int_{\phi_{m+1}\psi_{lm}d}
S_{m+1}P_{m+1}S_{l,m+1}u_j
\times\prod_{i\in I\setminus\{j\}}
S_{m+1}R^{(m)}_iS_{lm}u_i\,d\bar\mu'\cr
\displaycause{by (e), because $g\tilde{\action}^{(m)}_{\infty}(\psi_{lm}d)
=\psi_{lm}(g\tilde{\action}^{(l)}_{\infty}d)=\psi_{lm}d$ for every $g\in G$}
&=\lim_{m\to\infty}
\int_{\phi_{m+1}\psi_{lm}d}S_mP_mS_{lm}u_j
\times\prod_{i\in I\setminus\{j\}}
S_{m+1}R^{(m)}_iS_{lm}u_i\,d\bar\mu'\cr
\displaycause{because
$\lim_{m\to\infty}S_mP_mS_{lm}u_j=PS_lu_j
=\lim_{m\to\infty}S_{m+1}P_{m+1}S_{l,m+1}u_j$ for the norm
$\|\,\|_1$, by (d)}
&=\lim_{m\to\infty}
\int_{\phi_{m+1}\psi_{lm}d}S_{m+1}P_{m+1}S_{m,m+1}
P_mS_{lm}u_j
\times\prod_{i\in I\setminus\{j\}}
S_{m+1}R^{(m)}_iS_{lm}u_i\,d\bar\mu'\cr
\displaycause{because
$\phi_{m,m+1}[\frak B_m]\subseteq\frak B_{m+1}$, so
$P_{m+1}S_{m,m+1}P_m=S_{m,m+1}P_m$}
&=\lim_{m\to\infty}\int_{\psi_ld}R_jS_mP_mS_{lm}u_j
\times\prod_{i\in I\setminus\{j\}}R_iS_lu_i\,d\bar\nu\cr
\displaycause{by (e) again, applied to $\psi_{lm}d$, $P_mS_{lm}u_j$ and
$\langle S_{lm}u_i\rangle_{i\in I\setminus\{j\}}$}
&=\int_{\psi_ld}R_jPS_lu_j
\times\prod_{i\in I\setminus\{j\}}R_iS_lu_i\,d\bar\nu.\cr}$$
\medskip
{\bf (g)} It follows that if $v_i\in L^{\infty}(\frak A')$ for $i\in I$
and $c\in\frak C$ is such that $g\tilde{\action}_{\infty}c=c$ for every
$g\in G$, then
\Centerline{$\int_c\prod_{i\in I}R_iv_i\,d\bar\nu
=\int_cR_jPv_j\times\prod_{i\in I\setminus\{j\}}R_iv_i\,d\bar\nu$.}
\noindent\Prf\ Set $\gamma =\max_{i\in I}\|v_i\|_{\infty}$. Let $\epsilon>0$.
Then $c$ belongs to the metric closure of $\{\psi_md:m\in\Bbb N$,
$d\in\frak C_m$, $g\tilde{\action}^{(m)}_{\infty}d=d$ for every $g\in G\}$
(\zz1Gb). Also every $v_i$ belongs to the $\|\,\|_1$-closure of
$\{S_mu:m\in\Bbb N$, $u\in L^{\infty}(\frak A_m)$, $\|u\|_{\infty}\le \gamma \}$.
So there are an $l\in\Bbb N$, a $d\in\frak C_l$ and
$u_i\in L^{\infty}(\frak A_l)$, for $i\in I$, such that
\Centerline{$g\tilde{\action}^{(l)}_{\infty}d=d$ for every $g\in G$,
\quad$\bar\nu(c\Bsymmdiff\psi_ld)\le\epsilon$,}
\Centerline{$\|u_i\|_{\infty}\le \gamma $,
\quad$\|v_i-S_lu_i\|_1\le\epsilon$ for every $i\in I$.}
\noindent It follows that
\Centerline{$\|\prod_{i\le j}R_iv_i-\prod_{i\le j}R_iS_lu_i\|_1
\le(j+1)\epsilon \gamma^j$}
\noindent for every $j\le n$ (induce on $j$, recalling that
$S_l$ and every $R_i$ are both $\|\,\|_1$-non-expanding and
$\|\,\|_{\infty}$-non-expanding), so that
\Centerline{$\|\prod_{i\in I}R_iv_i-\prod_{i\in I}R_iS_lu_i\|_1
\le(n+1)\epsilon \gamma^n$.}
\noindent Consequently
\Centerline{$|\int_c\prod_{i\in I}R_iv_i\,d\bar\nu
-\int_{\psi_ld}\prod_{i\in I}R_iS_lu_i\,d\bar\nu|
\le(n+1)\epsilon \gamma^n+\epsilon$.}
Similarly,
\Centerline{$|\int_cR_jv_j\times\prod_{i\in I\setminus\{j\}}R_iv_i\,d\bar\nu
-\int_{\psi_ld}R_jS_lu_j\times\prod_{i\in I\setminus\{j\}}R_iS_lu_i\,d\bar\nu|
\le(1+(n+1)\gamma^n)\epsilon$.}
\noindent Putting these together with (f), we get
\Centerline{$|\int_cR_jPv_j\times\prod_{i\in I\setminus\{j\}}R_iv_i\,d\bar\nu
-\int_cR_jv_j\times\prod_{i\in I\setminus\{j\}}R_iv_i\,d\bar\nu|
\le 2\epsilon(1+(n+1)\gamma^n)$.}
\noindent As $\epsilon$ is arbitrary,
\Centerline{$\int_c\prod_{i\in I}R_iv_i\,d\bar\nu
=\int_cR_jPv_j\times\prod_{i\in I\setminus\{j\}}R_iv_i\,d\bar\nu$. \Qed}
\medskip
{\bf (h)} We are nearly home. Take any
$v_i\in L^{\infty}(\frak A')$ for $i\in I$. Let $\frak D$
be the fixed-point algebra
$\{c:c\in\frak C$, $g\tilde{\action}_{\infty}c=c$ for every $g\in G\}$.
We know that
\Centerline{$\int_cR_j(v_j-Pv_j)
\times\prod_{i\in I\setminus\{j\}}R_iv_i\,d\bar\nu=0$}
\noindent for every $c\in\frak D$, that is, that
\Centerline{$Q_{\frak D}(R_j(v_j-Pv_j)
\times\prod_{i\in I\setminus\{j\}}R_iv_i)=0$.}
\noindent By Lemma \zz3E,
\Centerline{$\WDL_{g\to G}\bigl(g\action'_j(v_j-Pv_j)
\times\prod_{i\in I\setminus\{j\}}g\action'_iv_i\bigr)=0$}
\noindent But this means that
$(\frak A',\bar\mu',G,\langle\action'_i\rangle_{i\in I})$ is a $j$-pleasant
system. And we have known since (a) above that it is an extension of
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\in I})$.
\bigskip
{\bf\zz3G Theorem} ({\smc Austin p08a}, 1.1) Let $G$ be an abelian
group, $I$ a non-empty finite set and
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ a commuting
measure-automorphism action system. Then
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is measure-averaging.
\medskip
\noindent{\bf proof} We may suppose that $I=n+1$ for some $n\in\Bbb N$.
Induce on $n$. If $n=0$ the result is a special case of
Proposition \zz1I. For the inductive step to $n\ge 1$, the inductive
hypothesis tells us that the conditions of Lemma \zz2M are satisfied, so
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\le n})$ is weakly
measure-averaging whenever it is a commuting measure-automorphism action
system. Consequently, if
$(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\le n})$ is a commuting
measure-automorphism action system, it has an $n$-pleasant
extension $(\frak A',\bar\mu',G,\langle\action'_i\rangle_{i\le n})$,
by \zz3F. Take
$\phi:\frak A\to\frak A'$ witnessing the extension, and
$S:L^0(\frak A)\to L^0(\frak A')$ the associated multiplicative Riesz
homomorphism. By the inductive hypothesis and Lemma \zz2L,
$(\frak A',\bar\mu',G,\langle\action'_i\rangle_{i\le n})$
is measure-averaging. Let $\CalFo$ be the F{\o}lner filter of $G$.
If $u_0,\ldots,u_n$ belong to $L^{\infty}(\frak A)$,
\Centerline{$\WDL_{g\to G}\prod_{i\le n}g\action'_iSu_i$}
\noindent is defined in $L^1(\frak A',\bar\mu')$, so
$$\eqalign{\|\Bover1{\mu L}\sum_{g\in L}\prod_{i\le n}g\action_i&u_i\,d\bar\mu
-\Bover1{\mu M}\Bint_M\prod_{i\le n}g\action_iu_i\,d\bar\mu\|_1\cr
&=\|S(\Bover1{\mu L}\sum_{g\in L}\prod_{i\le n}g\action_iu_i\,d\bar\mu
-\Bover1{\mu M}\Bint_M\prod_{i\le n}g\action_iu_i)\,d\bar\mu\|_1\cr
&=\|\Bover1{\mu L}\sum_{g\in L}\prod_{i\le n}S(g\action_iu_i)\,d\bar\mu
-\Bover1{\mu M}\Bint_M\prod_{i\le n}S(g\action_iu_i)\,d\bar\mu\|_1\cr
&=\|\Bover1{\mu L}\sum_{g\in L}\prod_{i\le n}g\action'_iSu_i\,d\bar\mu
-\Bover1{\mu M}\Bint_M\prod_{i\le n}g\action'_iSu_i\,d\bar\mu\|_1
\to 0\cr}$$
\noindent as $L$, $M\to\CalFo$, and
\Centerline{$\WDL_{g\to G}
\prod_{i\le n}g\action_iu_i$}
\noindent is defined in $L^1(\frak A,\bar\mu)$. As $u_0,\ldots,u_n$ are
arbitrary, $(\frak A,\bar\mu,G,\langle\action_i\rangle_{i\le n})$ is
measure-averaging, and the induction proceeds.
\bigskip
{\bf\zz3H Corollary} Let $G$ be an abelian group and
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ a commuting
measure-automorphism action system. Then
$(\frak A,\bar\mu,G,\familyiI{\action_i})$
has a Furstenberg self-joining over $J$ for any finite set $J\subseteq I$.
\bigskip
\noindent{\bf\zz4 Agreeable and isotropized extensions}
\medskip
{\bf\zz4A Definition} ({\smc Austin p08b}, 4.1)
\medskip
{\bf (a)} Let $I$ be a set, $J$ a finite subset of $I$ and $j$ a member of
$J$ and $G$ an abelian group.
A commuting measure-automorphism action system
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is {\bf$(J,j)$-agreeable} if,
writing $\frak B$ for the closed subalgebra of $\frak A$
generated by
\Centerline{$\bigcup_{i\in J\setminus\{j\}}\{a:a\in\frak A$,
$g\action_ia=g\action_ja$ for every $g\in G\}$,}
\noindent we have
\Centerline{$\WDL_{g\to G}\int g\action_j(u_j-Q_{\frak B}u_j)
\times\prod_{i\in J\setminus\{j\}}g\action_iu_id\bar\mu
=0$}
\noindent whenever $\family{i}{J}{u_i}$ is a family in
$L^{\infty}(\frak A)$.
(Compare, but do not confuse, with \zz2J.)
\medskip
{\bf (b)} A commuting measure-automorphism action system
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is {\bf fully agreeable}
if it is $(J,j)$-agreeable whenever $j\in J\in[I]^{<\omega}$.
\bigskip
{\bf\zz4B Lemma} ({\smc Austin p08b}, \S4) Let $G$ be an abelian
group, $\kappa$ an ordinal and
$(\ofamily{\xi}{\kappa}
{(\frak A_{\xi},\bar\mu_{\xi},G,
\penalty-100\familyiI{\action^{(\xi)}_i})},
\penalty-100\langle\phi_{\eta\xi}\rangle_{\eta\le\xi<\kappa})$
an inductive system
of commuting measure-automorphism action systems with inductive limit
$(\frak A,\bar\mu,G,\penalty-100\familyiI{\action_i},
\ofamily{\xi}{\kappa}{\phi_{\xi}})$.
Suppose that $J\in[I]^{<\omega}$,
$j\in J$ and a cofinal set $M\subseteq\kappa$ are
such that, for $\xi\in M$,
$(\frak A_{\xi+1},\bar\mu_{\xi+1},G,\familyiI{\action^{(\xi+1)}_i},
\phi_{\xi,\xi+1})$
is the $(J,j)$-Furstenberg extension of
$(\frak A_\xi,\bar\mu_\xi,G,\familyiI{\action^{(\xi)}_i})$.
Then
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is $(J,j)$-agreeable.
\medskip
\noindent{\bf proof (a)} For $\xi<\kappa$ let $\frak B_{\xi}$ be the
closed subalgebra of $\frak A_{\xi}$ generated by
\Centerline{$\bigcup_{i\in J\setminus\{j\}}\{a:a\in\frak A_{\xi}$,
$g\action^{(\xi)}_ia=g\action^{(\xi)}_ja$ for every $g\in G\}$,}
\noindent and let $\frak B$ be the closed subalgebra of $\frak A$
generated by
\Centerline{$\bigcup_{i\in J\setminus\{j\}}\{a:a\in\frak A$,
$g\action_ia=g\action_ja$ for every $g\in G\}$;}
\noindent set $P_{\xi}=Q_{\frak B_{\xi}}$ and $P=Q_{\frak B}$.
Then $\phi_{\eta\xi}[\frak B_{\eta}]\subseteq\frak B_{\xi}$ whenever
$\eta\le\xi$, and
$\frak B$ is the closed subalgebra of $\frak A$ generated by
$\bigcup_{\xi<\kappa}\phi_{\xi}[\frak B_{\xi}]$ (\zz2Gb), so
$PS_{\eta}u=\lim_{\xi\to\kappa}S_{\xi}P_{\xi}S_{\eta\xi}u$ for $\|\,\|_1$
whenever $\eta<\kappa$ and
$u\in L^1(\frak A_{\eta},\bar\mu_{\eta})$, writing
$S_{\eta\xi}:L^0(\frak A_{\eta})\to L^0(\frak A_{\xi})$ and
$S_{\xi}:L^0(\frak A_{\xi})\to L^0(\frak A)$ for the multiplicative Riesz
homomorphisms corresponding to
$\phi_{\eta\xi}:\frak A_{\eta}\to\frak A_{\xi}$ and
$\phi_{\xi}:\frak A_{\xi}\to\frak A$, as in the proof of \zz3H.
\medskip
{\bf (b)} Suppose that $v_i\in L^{\infty}(\frak A)$ for each $i\in J$;
set $\gamma=\max_{i\in J}\|v_i\|_{\infty}$. Let $\epsilon>0$.
\medskip
\quad{\bf (i)} There are a $\xi\in M$ and
$u_i\in L^{\infty}(\frak A_{\xi})$, for $i\in J$, such that
\Centerline{$\|u_i\|_{\infty}\le\gamma$,
$\|v_i-S_{\xi}u_i\|_1\le\epsilon$ for every $i\in J$,}
\Centerline{$\|PS_{\xi}u_j-S_{\xi}P_{\xi}u_j\|_1\le\epsilon$,
\quad$\|PS_{\xi}u_j-S_{\xi+1}P_{\xi+1}S_{\xi,\xi+1}u_j\|_1\le\epsilon$.}
\noindent\Prf\ First, there are an $\eta<\kappa$ and
$u'_i\in L^{\infty}(\frak A_{\eta})$, for $i\in J$, such that
$\|v_i-S_{\eta}u_i\|_1\le\epsilon$ for every $i$;
replacing $u'_i$ by
$\med(-\gamma\chi 1,u'_i,\gamma\chi 1)$ if necessary, we can arrange that
$\|u'_i\|_{\infty}\le\gamma$ for every $i$. Next, by the martingale
convergence theorem, there is a $\zeta<\kappa$ such that $\eta\le\zeta$ and
$\|S_{\xi}P_{\xi}S_{\eta\xi}u'_i-PS_{\eta}u'_i\|_1\le\epsilon$ whenever
$i\in J$ and $\zeta\le\xi<\kappa$. Since $M$ is cofinal with $\kappa$,
there is a $\xi\in M$ such that $\xi\ge\zeta$; set $u_i=S_{\eta\xi}u'_i$
for each $i$.\ \Qed
\medskip
\quad{\bf (ii)} It follows that
$$\eqalign{\bigl|\int\prod_{i\in J}g\action_iv_i\,d\bar\mu
-\int\prod_{i\in J}g\action^{(\xi)}_iu_i\,d\bar\mu_{\xi}\bigr|
&=\bigl|\int\prod_{i\in J}g\action_iv_i
-\prod_{i\in J}g\action_iS_{\xi}u_i\,d\bar\mu\bigr|\cr
&\le\gamma^{\#(J)-1}\sum_{i\in J}\|g\action_iv_i-g\action_iS_{\xi}u_i\|_1\cr
&\le\gamma^{\#(J)-1}\#(J)\epsilon\cr}$$
\noindent for every $g\in G$, so that
\Centerline{$\bigl|\WDL_{g\to G}\int\prod_{i\in J}g\action_iv_i\,d\bar\mu
-\WDL_{g\to G}\int\prod_{i\in J}g\action^{(\xi)}_iu_i\,d\bar\mu_{\xi}\bigr|
\le\gamma^{\#(J)-1}\#(J)\epsilon$.}
\medskip
\quad{\bf (iii)} Writing $(\frak C,\bar\nu,G,
\family{i}{J\cup\{\infty\}}{\tilde{\action}_i},
\family{i}{J}{\pi_i})$ for the Furstenberg self-joining of
$(\frak A_{\xi},\bar\mu_{\xi},\penalty-100G,\familyiI{\action^{(\xi)}_i})$, and
$R_i:L^0(\frak A_{\xi})\to L^0(\frak C)$ for the Riesz homomorphism
corresponding to $\pi_i:\frak A_{\xi}\to\frak C$,
$$\eqalignno{\WDL_{g\to G}\int\prod_{i\in J}
g\action^{(\xi)}_iu_i\,d\bar\mu_{\xi}
&=\int\prod_{i\in J}R_iu_i\,d\bar\nu
=\int\prod_{i\in J}R_iu_i\,d\bar\mu_{\xi+1}\cr
&=\int P_{\xi+1}(R_ju_j\times\prod_{i\in I\setminus\{j\}}R_iu_i)
\,d\bar\mu_{\xi+1}\cr
&=\int P_{\xi+1}R_ju_j\times\prod_{i\in I\setminus\{j\}}R_iu_i
\,d\bar\mu_{\xi+1}\cr
\displaycause{because $g\action^{(\xi+1)}_iR_iu_i
=g\tilde{\action}_iR_iu_i
=R_i(g\action^{(\xi)}_iu_i)
=g\tilde{\action}_{\infty}(R_iu_i)
=g\action^{(\xi+1)}_jR_iu_i$ for every $g\in G$, $i\in J\setminus\{j\}$, so
$P_{\xi+1}R_iu_i=R_iu_i$ for every $i\in J\setminus\{j\}$}
&=\int P_{\xi+1}S_{\xi,\xi+1}u_j\times\prod_{i\in J\setminus\{j\}}R_iu_i\,d\bar\mu_{\xi+1}.
\cr}$$
\medskip
\quad{\bf (iv)}
\Centerline{$\|P_{\xi+1}S_{\xi,\xi+1}u_j-S_{\xi,\xi+1}P_{\xi}u_j\|_1
=\|S_{\xi+1}P_{\xi+1}S_{\xi,\xi+1}u_j-S_{\xi}P_{\xi}u_j\|_1
\le 2\epsilon$,}
\noindent so
$$\eqalign{
\bigl|\int P_{\xi+1}S_{\xi,\xi+1}&u_j
\times\prod_{i\in J\setminus\{j\}}R_iu_i\,d\bar\mu_{\xi+1}\cr
& -\int S_{\xi,\xi+1}P_{\xi}u_j
\times\prod_{i\in J\setminus\{j\}}R_iu_i\,d\bar\mu_{\xi+1}\bigr|
\le 2\epsilon\gamma^{\#(J)-1}.\cr}$$
\medskip
\quad{\bf (v)}
$$\eqalign{\int S_{\xi,\xi+1}P_{\xi}u_j
\times\prod_{i\in J\setminus\{j\}}R_iu_i
\,d\bar\mu_{\xi+1}
&=\int R_jP_{\xi}u_j\times\prod_{i\in J\setminus\{j\}}R_iu_i\,d\bar\nu\cr
&=\WDL_{g\to G}\int g\action^{(\xi)}_jP_{\xi}u_j
\times\prod_{i\in J\setminus\{j\}}g\action^{(\xi)}_iu_i\,d\bar\mu_{\xi}\cr
&=\WDL_{g\to G}\int g\action_jS_{\xi}P_{\xi}u_j
\times\prod_{i\in J\setminus\{j\}}g\action_iS_iu_i\,d\bar\mu.
\cr}$$
\medskip
\quad{\bf (vi)} Since
$$\eqalign{\|g\action_j(Pv_j)-g\action_jS_{\xi}P_{\xi}u_j\|_1
&=\|Pv_j-S_{\xi}P_{\xi}u_j\|_1\cr
&\le\|Pv_j-PS_{\xi}u_j\|_1+\|PS_{\xi}u_j-S_{\xi}P_{\xi}u_j\|_1\cr
&\le\|v_j-S_{\xi}u_j\|_1+\epsilon
\le 2\epsilon,\cr
\|g\action_iv_i-g\action_iS_{\xi}u_i\|_1
&=\|v_i-S_{\xi}u_i\|_1\le\epsilon\cr}$$
\noindent for every $g\in G$ and $i\in I\setminus\{j\}$,
$$\eqalign{\bigl|\WDL_{g\to G}\int g&\action_jS_{\xi}P_{\xi}u_j
\times\prod_{i\in J\setminus\{j\}}g\action_iS_iu_i\,d\bar\mu\cr
&-\WDL_{g\to G}\int g\action_jPv_j
\times\prod_{i\in J\setminus\{j\}}g\action_iv_i\,d\bar\mu\bigr|
\le\gamma^{\#(J)-1}(\#(J)+1)\epsilon.\cr}$$
\medskip
\quad{\bf (vii)} Assembling (ii)-(vi), we get
$$\eqalign{\bigl|\WDL_{g\to G}\int\prod_{i\in J}g\action_i&v_i
\,d\bar\mu
-\WDL_{g\to G}\int g\action_jPu_j
\times\prod_{i\in J\setminus\{j\}}g\action_iv_i\,d\bar\mu\bigr|\cr
&\le\gamma^{\#(J)-1}(\epsilon\#(J)+2\epsilon+(\#(J)+1)\epsilon).\cr}$$
\noindent As $\epsilon$ is arbitrary,
\Centerline{$\WDL_{g\to G}\int\prod_{i\in J}g\action_iv_i
\,d\bar\mu
=\WDL_{g\to G}\int g\action_jPu_j
\times\prod_{i\in J\setminus\{j\}}g\action_iv_i\,d\bar\mu$.}
\noindent As $\family{i}{J}{v_i}$ is arbitrary,
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is $(J,j)$-agreeable.
\bigskip
{\bf\zz4B Definition} ({\smc Austin p08b}, 5.1)
Let $G$ be a group, $I$ a set,
and $(\frak A,\bar\mu,G,\familyiI{\action_i})$ a
measure-automorphism action system. If $J\subseteq I$, write
\Centerline{$\frak B_J
=\{a:a\in\frak A$, $g\action_ia=g\action_ja$ for all $i$, $j\in J$
and $g\in G\}$.}
\noindent If $j\in J\subseteq I$, we say that
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is {\bf$(J,j)$-isotropized}
if
\Centerline{$\frak B_J\cap\bigvee_{i\in I\setminus J}\frak B_{\{i,j\}}
=\bigvee_{i\in I\setminus J}\frak B_{J\cup\{i\}}$.}
\noindent $(\frak A,\bar\mu,G,\familyiI{\action_i})$ is
{\bf fully isotropized} if it is $(J,j)$-isotropized whenever
$j\in J\subseteq I$.
\bigskip
{\bf\zz4D Construction (a)} Let
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ be a
measure-automorphism action system, and $j\in J\subseteq I$.
Set $\frak B_J=\{a:a\in\frak A$, $g\action_ia=g\action_ja$ whenever
$i$, $j\in J$ and $g\in G\}$.
The {\bf$(J,j)$-isotropizing extension} of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is
$(\frak A',\bar\mu',G,\familyiI{\action'_i},\psi_0)$, constructed as
follows. $(\frak A',\bar\mu',\psi_0,\psi_1)$ is the
relative free product of
$(\frak A,\bar\mu)$ with itself over $\frak B_J$ ({\smc Fremlin 03},
458N\formerly{458J}). For $i\in I$ and $g\in G$, we can define
$g\action'_ib$, for $b\in\frak A'$, by setting
$$\eqalign{g\action'_i(\psi_0a)
&=\psi_0(g\action_ia),\cr
g\action'_i(\psi_1a)
&=\psi_1(g\action_ia)\text{ if }i\in I\setminus J,\cr
&=\psi_1(g\action_ja)\text{ if }i\in J\cr}$$
\noindent whenever $g\in G$ and $a\in\frak A$,
and requiring that $b\mapsto g\action'_ib:\frak A'\to\frak A'$ is a
measure-preserving Boolean homomorphism for every $g\in G$. \Prf\
The point is that if $i\in J$ and $a\in\frak B_J$ then
$g\action_ja=g\action_ia\in\frak B_J$, so
$\psi_0(g\action_ia)=\psi_1(g\action_ja)$.
We can therefore apply the defining universal mapping theorem for the
relative free product ({\smc Fremlin 03}, 458O\formerly{458K}) to see that
there is indeed a (unique) measure-preserving Boolean homomorphism from
$\frak A'$ to itself satisfying the given formulae.\ \Qed
It is now elementary
to check that every $\action'_i$ is an action of $G$ on $\frak A'$, so that
$(\frak A',\bar\mu',G,\familyiI{\action'_i})$ is a measure-automorphism
action system.
And the formula for $g\action'_i(\psi_0a)$ is just what we need
to ensure that
$(\frak A',\bar\mu',G,\familyiI{\action'_i},\psi_0)$ is an extension of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$.
\medskip
{\bf (b)} If $(\frak A,\bar\mu,G,\familyiI{\action_i})$ is a commuting
system, then a similar calculation shows that
$(\frak A',\bar\mu',G,\familyiI{\action'_i})$ is also commuting.
\bigskip
{\bf\zz4E Lemma} Let $(\frak A,\bar\mu,G,\familyiI{\action_i})$
be a measure-automorphism action system, and $j\in J\subseteq I$.
Let $(\frak A',\bar\mu',G,\familyiI{\action'_i},\psi_0)$ be the
$(J,j)$-isotropizing extension of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$. For $K\subseteq I$, set
\Centerline{$\frak B_K=\{a:a\in\frak A$, $g\action_ia=g\action_ka$ for all
$i$, $k\in K$ and $g\in G\}$,}
\Centerline{$\frak B'_K=\{a:a\in\frak A'$, $g\action'_ia=g\action'_ka$
for all $i$, $k\in K$ and $g\in G\}$;}
\noindent set
\Centerline{$\frak D
=\frak B_J\cap\bigvee_{i\in I\setminus J}\frak B_{\{i,j\}}
\subseteq\frak A$,
\quad$\frak E
=\bigvee_{i\in I\setminus J}\frak B'_{J\cup\{i\}}
\subseteq\frak A'$.}
\noindent Then $\psi_0[\frak D]\subseteq\frak E$.
\medskip
\noindent{\bf proof} Take $d\in\frak D$ and $\epsilon>0$. Then there are
$n\in\Bbb N$, a finite set $K\subseteq I\setminus J$ and a family
$\langle c_{rk}\rangle_{r\le n,k\in K}$
such that $c_{rk}\in\frak B_{\{k,j\}}$ for $r\le n$ and
$k\in K$ and $\bar\mu(d\Bsymmdiff d')\le\epsilon$, where
$d'=\sup_{r\le n}\inf_{k\in K}c_{rk}$. Now if $r\le n$, $k\in K$,
$i\in J$ and $g\in G$,
\Centerline{$g\action'_k(\psi_1c_{rk})
=\psi_1(g\action_kc_{rk})
=\psi_1(g\action_jc_{rk})
=g\action'_i(\psi_1c_{rk})$,}
\noindent so $\psi_1c_{rk}\in\frak B'_{J\cup\{k\}}\subseteq\frak E$;
accordingly $\psi_1d'\in\frak E$. Also
$$\eqalignno{\bar\mu'(\psi_0d\Bsymmdiff\psi_1d')
&=\bar\mu'(\psi_1d\Bsymmdiff\psi_1d')\cr
\displaycause{because $d\in\frak B_J$}
&=\bar\mu(d\Bsymmdiff d')
\le\epsilon.\cr}$$
\noindent As $\epsilon$ is arbitrary and $\frak E$ is closed,
$\psi_0d\in\frak E$; as $d$ is arbitrary, we have the result.
\bigskip
{\bf\zz4F Lemma} ({\smc Austin p08b}, \S5) Let $G$ be an abelian group,
$\kappa$ an ordinal of uncountable cofinality, and
$(\ofamily{\xi}{\kappa}{(\frak A_{\xi},\bar\mu_{\xi},G,
\familyiI{\action^{(\xi)}_i})},
\langle\phi_{\eta\xi}\rangle_{\eta\le\xi<\kappa})$ an inductive system
of commuting measure-automorphism action systems with inductive limit
$(\frak A,\bar\mu,G,\familyiI{\action_i},
\ofamily{\xi}{\kappa}{\phi_{\xi}})$.
Suppose that $J\subseteq I$,
$j\in J$ and a cofinal set $M\subseteq\kappa$ are
such that, for $\xi\in M$,
$(\frak A_{\xi+1},\bar\mu_{\xi+1},G,\familyiI{\action^{(\xi+1)}_i},
\phi_{\xi,\xi+1})$
is the $(J,j)$-isotropizing extension of
$(\frak A_{\xi},\bar\mu_{\xi},G,\familyiI{\action^{(\xi)}_i})$.
Then
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is $(J,j)$-isotropized.
\medskip
\noindent{\bf proof (a)} For $K\subseteq I$ and $\xi<\kappa$ set
\Centerline{$\frak C^{(\xi)}_K=\{a:a\in\frak A_{\xi},\,
g\action^{(\xi)}_ia=g\action^{(\xi)}_ka$ for all $i$, $k\in K$ and
$g\in G\}$,}
\Centerline{$\frak C_K=\{a:a\in\frak A,\,
g\action_ia=g\action_ka$ for all $i$, $k\in K$ and
$g\in G\}$;}
\noindent set
\Centerline{$\frak D_{\xi}
=\frak C^{(\xi)}_J
\cap\bigvee_{i\in I\setminus J}\frak C^{(\xi)}_{\{i,j\}}$,
\quad$\frak E_{\xi}
=\bigvee_{i\in I\setminus J}\frak C^{(\xi)}_{J\cup\{i\}}
\subseteq\frak A_{\xi}$}
\noindent for $\xi<\kappa$ and
\Centerline{$\frak D
=\frak C_J\cap\bigvee_{i\in I\setminus J}\frak C_{\{i,j\}}$,
\quad$\frak E
=\bigvee_{i\in I\setminus J}\frak C_{J\cup\{i\}}
\subseteq\frak A$.}
\noindent Because $\cf\kappa>\omega$,
$\frak A=\bigcup_{\xi<\kappa}\phi_{\xi}[\frak A_{\xi}]$;
consequently $\frak C_K=\bigcup_{\xi<\kappa}\phi_{\xi}[\frak C^{(\xi)}_K]$
for every $K\subseteq I$, and
$\frak D=\bigcup_{\xi<\kappa}\phi_{\xi}[\frak D_{\xi}]$.
\medskip
{\bf (b)} Take any $d\in\frak D$. Then there is a $\xi\in M$ such that
$d\in\phi_{\xi}[\frak D_{\xi}]$; set $d'=\phi_{\xi}^{-1}(d)$.
By \zz4E, $\phi_{\xi,\xi+1}d'\in\frak E_{\xi+1}$. But this means that
\Centerline{$d=\phi_{\xi+1}\phi_{\xi,\xi+1}d'
\in\phi_{\xi+1}[\frak E_{\xi+1}]\subseteq\frak E$.}
\noindent As $d$ is arbitrary, $\frak D\subseteq\frak E$. It is
elementary to check from their definitions that $\frak D$ includes
$\frak E$, so they are equal, that is,
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is $(J,j)$-isotropized.
\bigskip
{\bf\zz4G Proposition} Let $G$ be an abelian
group and $(\frak A,\bar\mu,G,\familyiI{\action_i})$ a
commuting measure-automorphism action system. Then it has an extension
which is commuting, fully isotropized and fully agreeable.
\medskip
\noindent{\bf proof} Set $\kappa=\max(\omega_1,2^{\#(I)})$. Then we can
build inductively an inductive system
$(\ofamily{\xi}{\kappa}{(\frak A_{\xi},\bar\mu_{\xi},G,
\familyiI{\action^{(\xi)}_i)}},
\penalty-100\langle\phi_{\eta\xi}\rangle_{\eta\le\xi<\kappa})$
of commuting measure-automorphism action systems such that
$(\frak A_0,\bar\mu_0,G,\familyiI{\action^{(0)}_i})
=(\frak A,\bar\mu,G,\familyiI{\action_i})$ and
$$\eqalignno{\{&\xi:(\frak A_{\xi+1},\bar\mu_{\xi+1},G,
\familyiI{\action^{(\xi+1)}_i},\phi_{\xi,\xi+1})\cr
&\mskip100mu\text{is the }
(J,j)\text{-Furstenberg extension of }
(\frak A_{\xi},\bar\mu_{\xi},G,\familyiI{\action^{(\xi)}_i})\}\cr
\noalign{\noindent
is cofinal with $\kappa$ whenever $j\in J\in[I]^{<\omega}$, and}
\{&\xi:(\frak A_{\xi+1},\bar\mu_{\xi+1},G,
\familyiI{\action^{(\xi+1)}_i},\phi_{\xi,\xi+1})\cr
&\mskip100mu\text{is the }
(J,j)\text{-isotropizing extension of }
(\frak A_{\xi},\bar\mu_{\xi},G,\familyiI{\action^{(\xi)}_i})\}\cr}$$
\noindent is cofinal with $\kappa$ whenever $j\in J\subseteq I$.
Now if $(\frak A',\bar\mu',G,\familyiI{\action'_i},
\ofamily{\xi}{\kappa}{\phi_{\xi}})$ is the inductive limit
of this system, Lemmas \zz4B and \zz4F tell us that
$(\frak A',\bar\mu',G,\familyiI{\action'_i})$
is fully agreeable
and fully isotropized, and of course
$(\frak A',\bar\mu',G,\familyiI{\action'_i},\phi_0)$ is an extension of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$.
\bigskip
\noindent{\bf\zz5 More about Furstenberg self-joinings}
\medskip
{\bf\zz5A Alternative description of agreeable systems} Let $G$ be an
abelian group,
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ a commuting
measure-preserving action system, $J$ a finite subset of $I$, and $j$ a
member of $J$. Let
$(\frak C,\bar\nu,G,
\penalty-100\family{i}{I\cup\{\infty\}}{\tilde{\action}_i},
\family{i}{J}{\pi_i})$ be the Furstenberg self-joining of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ over $J$. Set
\Centerline{$\frak B=\bigvee_{i\in J\setminus\{j\}}
\{a:a\in\frak A$, $g\action_ia=g\action_ja$ for every $g\in G\}
\subseteq\frak A$.}
\noindent Then $(\frak A,\bar\mu,G,\familyiI{\action_i})$ is
$(J,j)$-agreeable iff $\pi_j[\frak A]$ and
$\bigvee_{i\in J\setminus\{j\}}\pi_i[\frak A]$ are relatively independent
over $\pi_j[\frak B]$.
\medskip
\noindent{\bf proof} For $j\in J$, let $R_j:L^0(\frak A)\to L^0(\frak C)$
be the Riesz homomorphism defined from $\pi_j:\frak A\to\frak C$. Set
$\frak D=\bigvee_{i\in I\setminus\{j\}}\pi_i[\frak A]\subseteq\frak C$.
We have
$$\eqalignno{(\frak A,\bar\mu,&G,\familyiI{\action_i})\text{ is }
(J,j)\text{-agreeable}\cr
&\iff\WDL_{g\to G}\int g\action_j(Q_{\frak B}u_j)
\times\prod_{i\in J\setminus\{j\}}g\action_iu_i\,d\bar\mu
=\WDL_{g\to G}\int\prod_{i\in J}g\action_iu_i\,d\bar\mu\cr
&\mskip100mu\text{ whenever }\family{i}{J}{u_i}\in L^{\infty}(\frak A)^J\cr
\displaycause{\zz4Aa}
&\iff\int R_jQ_{\frak B}u_j
\times\prod_{i\in J\setminus\{j\}}R_iu_i\,d\bar\nu
=\int\prod_{i\in J}R_iu_i\,d\bar\nu\cr
&\mskip100mu\text{ whenever }\family{i}{J}{u_i}\in L^{\infty}(\frak A)^J\cr
\displaycause{\zz3Ac}
&\iff\int R_jQ_{\frak B}\chi a_j
\times\prod_{i\in J\setminus\{j\}}R_i\chi a_i\,d\bar\nu
=\int\prod_{i\in J}R_i\chi a_id\bar\nu\cr
&\mskip100mu\text{ whenever }\family{i}{J}{a_i}\in \frak A^J\cr
&\iff\int_dR_jQ_{\frak B}\chi a_j\,d\bar\nu
=\bar\nu(d\Bcap\pi_ja_j)\cr
&\mskip100mu\text{ whenever }\family{i}{J}{a_i}\in \frak A^J\text{ and }
d=\inf_{i\in J\setminus\{j\}}\pi_ia_i\cr
&\iff\int_dR_jQ_{\frak B}\chi a_j\,d\bar\nu
=\bar\nu(d\Bcap\pi_ja_j)
\text{ whenever }a_j\in\frak A\text{ and }
d\in\frak D\cr
\displaycause{because $\{\inf_{i\in J\setminus\{j\}}\pi_ia_i:a_i\in\frak A$
for every $i\in J\setminus\{j\}\}$ is closed under finite infima and
generates $\frak D$}
&\iff\int_dQ_{\pi_j[\frak B]}R_j\chi a_j\,d\bar\nu
=\bar\nu(d\Bcap\pi_ja_j)
\text{ whenever }a_j\in\frak A\text{ and }
d\in\frak D\cr
\displaycause{because $R_jQ_{\frak B}=Q_{\pi_j[\frak B]}R_j$
({\smc Fremlin 02}, 365Xq\Latereditions)}
&\iff\int(Q_{\pi_j[\frak B]}\chi c)\times\chi d\,d\bar\nu
=\int\chi c\times\chi d\,d\bar\nu
\text{ whenever }c\in\pi_j[\frak A]\text{ and }
d\in\frak D\cr
&\iff\int(Q_{\pi_j[\frak B]}\chi c)\times(Q_{\pi_j[\frak B]}\chi d)
\,d\bar\nu
=\int\chi c\times\chi d\,d\bar\nu\cr
&\mskip100mu\text{ whenever }c\in\pi_j[\frak A]\text{ and }
d\in\frak D\cr
&\iff\pi_j[\frak A]\text{ and }\frak D\text{ are relatively independent
over }\pi_j[\frak B].\cr}$$
\bigskip
{\bf\zz5B Lemma} ({\smc Austin p08b}, 3.2) Let $G$ be an abelian group,
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ a commuting
measure-automorphism action system and $J$ a finite subset of $I$. Let
$(\frak C,\bar\nu,G,\family{i}{I\cup\{\infty\}}{\tilde{\action}_i},
\family{i}{J}{\pi_i})$ be the Furstenberg self-joining of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ over $J$.
(a) If $j$, $k\in J$ and $a\in\frak A$ is such that
$g\action_ja=g\action_ka$ for every $g\in G$, then $\pi_ja=\pi_ka$.
(b) If $K\subseteq J$
and $\frak B_K=\{a:a\in\frak A$, $g\action_ja=g\action_ka$ for all $g\in G$
and $j$, $k\in K\}$, then $\pi_j[\frak B_K]=\pi_k[\frak B_K]$ for all $j$,
$k\in K$.
\medskip
\noindent{\bf proof (a)} If $j=k$ this is trivial. Otherwise, by
\zz3A(c-i),
\Centerline{$\bar\nu(\pi_ja\Bcap\pi_ka)
=\WDL_{g\to G}\bar\mu(g\action_ja\Bcap g\action_ka)
=\WDL_{g\to G}\bar\mu(g\action_ja)
=\bar\nu\pi_ja$}
\noindent and $\pi_ja\Bsubseteq\pi_ka$; similarly,
$\pi_ka\Bsubseteq\pi_ja$ and the two are equal.
\medskip
{\bf (b)} follows at once.
\bigskip
{\bf\zz5C Definition} ({\smc Austin p08b}, 3.3)
In the context of part (b) of \zz5B,
I will call the common value $\pi_j[\frak B_K]$ the
{\bf divaricate copy} of $\frak B_K$ in $\frak C$. For definiteness, if
$K$ is empty, I will say that the divaricate copy of
$\frak B_{\emptyset}=\frak A$ is
$\frak C$.
%if K\subseteq L then \frak B_L\subseteq\frak B_K and
%\frak B_L^*\subseteq\frak B_K^*
%\frak B_{K\cup L}=\frak B_K\cap\frak B_L,
%\frak B_{K\cup L}^*=\frak B_K^*\cap\frak B_L^*
\bigskip
{\bf\zz5D Lemma} ({\smc Austin p08b}, 6.1) Let $G$ be an abelian
group, $I$ a finite set and
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ a
commuting measure-automorphism action system which is fully isotropized
and fully agreeable. Let
$(\frak C,\bar\nu,G,
\penalty-100\family{i}{I\cup\{\infty\}}{\tilde{\action}_i},
\familyiI{\pi_i})$ be the Furstenberg self-joining of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ over $I$.
For $J\subseteq I$ set
\Centerline{$\frak B_J=\{a:a\in\frak C$,
$g\action_ia=g\action_ja$ for all $i$, $j\in J$ and
$g\in G\}$,}
\noindent and let
$\frak B_J^*\subseteq\frak C$ be the divaricate copy of $\frak B$ (\zz5C).
Let $\Cal J\subset\Cal PI$ be such that $K\in\Cal J$ whenever $J\in\Cal J$
and $J\subseteq K\subseteq I$, and $L$ a maximal element of
$\Cal PI\setminus\Cal J$. Set
\Centerline{$\frak D=\bigvee_{J\in\Cal J}\frak B_J^*$,}
\Centerline{$\frak E=\bigvee_{L\subseteq J\in\Cal J}\frak B_J^*$.}
%\Tau^F_{\Cal I\cap<e>}
\noindent Then $\frak D$ and $\frak B_L^*$ are relatively independent over
$\frak E$.
\medskip
\noindent{\bf proof (a)} If $L$ is empty, then $\frak D=\frak E$ and
$\frak B_L^*=\frak C$, so the result is trivial.
If $\Cal J=\emptyset$ then $\frak D=\frak E=\{0,1\}$ and again the result
is trivial. Otherwise, fix $j\in L$. Set
$\frak B=\bigvee_{i\in I\setminus L}\frak B_{\{i,j\}}$ and
$\frak B'=\bigvee_{i\in I\setminus L}\frak B_{L\cup\{i\}}$. Because
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is $(L,j)$-isotropized,
\Centerline{$\frak B_L\cap\frak B=\frak B'$,}
\noindent so $Q_{\frak B'}=Q_{\frak B}Q_{\frak B_L}$ (\zz2De).
Because $L\notin\Cal J$,
$\bigvee_{L\subseteq J\in\Cal J}\frak B_J\subseteq\frak B'$; on the other
hand, by the maximality of $L$,
$\frak B'\subseteq\bigvee_{L\subseteq J\in\Cal J}\frak B_J$.
Now
\Centerline{$\frak E
=\bigvee_{L\subseteq J\in\Cal J}\frak B_J^*
=\bigvee_{L\subseteq J\in\Cal J}\pi_j[\frak B_J]
=\pi_j[\bigvee_{L\subseteq J\in\Cal J}\frak B_J]
=\pi_j[\frak B']$.}
Set $I'=(I\setminus L)\cup\{j\}$, and let
$(\frak C',\bar\nu',G,\family{i}{I'\cup\{\infty\}}{\tilde{\action}'_i},
\family{i}{I'}{\pi'_i})$ be the Furstenberg self-joining of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ over $I'$.
Let $J_0,\ldots,J_n$ enumerate the minimal elements of $\Cal J$. Since
$L\notin\Cal J$, we can find $i_m\in J_m\setminus L$ for each $m\le n$.
If $J\in\Cal J$, there is an $m\le n$ such that $J\supseteq J_m$ and
$\frak B_J\subseteq\frak B_{J_m}$. So
$\frak D=\bigvee_{m\le n}\frak B_{J_m}^*$. Suppose that
$a_m\in\frak B_{J_m}$ for $m\le n$, and that $b\in\frak B_L$.
Then
$$\eqalignno{\bar\nu(\pi_jb\Bcap\inf_{m\le n}\pi_{i_m}a_m)
&=\WDL_{g\to G}\bar\mu(g\action_jb\Bcap\inf_{m\le n}g\action_{i_m}a_m)\cr
\displaycause{note that it makes no difference if the $i_m$ are not all
distinct}
&=\bar\nu'(\pi'_jb\Bcap\inf_{m\le n}\pi'_{i_m}a_m)\cr
&=\int R'_jQ_{\frak B}\chi b
\times\chi(\inf_{m\le n}\pi'_{i_m}a_m)d\bar\nu'\cr
\displaycause{because $(\frak A,\bar\mu,G,\familyiI{\action_i})$ is
$(I',j)$-agreeable}
&=\int R_jQ_{\frak B}\chi b
\times\chi(\inf_{m\le n}\pi_{i_m}a_m)d\bar\nu\cr
&=\int R_jQ_{\frak B}Q_{\frak B_L}\chi b
\times\chi(\inf_{m\le n}\pi_{i_m}a_m)d\bar\nu\cr
&=\int R_jQ_{\frak B'}\chi b
\times\chi(\inf_{m\le n}\pi_{i_m}a_m)d\bar\nu\cr
&=\int Q_{\frak E}R_j\chi b
\times\chi(\inf_{m\le n}\pi_{i_m}a_m)d\bar\nu\cr
\displaycause{because
$\frak E=\pi_j[\frak B']$}
&=\int Q_{\frak E}\chi(\pi_jb)
\times\chi(\inf_{m\le n}\pi_{i_m}a_m)d\bar\nu.\cr}$$
\noindent Because $\pi_j[\frak B_L]=\frak B_L^*$ and
$\pi_{i_m}[\frak B_{J_m}]=\frak B_{J_m}^*$ for each $m$, we have
\Centerline{$\bar\nu(c\Bcap\inf_{m\le n}c_m)
=\int Q_{\frak E}(\chi c)\times\chi(\inf_{m\le n}c_m)d\bar\nu$}
\noindent whenever $c\in\frak B_L^*$ and $c_m\in\frak B_{J_m}^*$ for each
$m$. Because $\frak D=\bigvee_{m\le n}\frak B_{J_m}^*$,
\Centerline{$\bar\nu(c\Bcap d)
=\int Q_{\frak E}(\chi c)\times\chi d\,d\bar\nu
=\int Q_{\frak E}(\chi c)\times Q_{\frak E}(\chi d)\,d\bar\nu
$}
\noindent whenever $c\in\frak B_L^*$ and $d\in\frak D$.
But this is just what is required to ensure that $\frak B_L^*$ and
$\frak D$ are relatively independent over $\frak E$
({\smc Fremlin 03}, 458Lc\Latereditions).
\bigskip
{\bf\zz5E Lemma} ({\smc Austin p08b}, 6.2) Let $G$ be an abelian
group, $I$ a finite set and
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ a commuting measure-automorphism
action system which is fully isotropized and fully agreeable. Let
$(\frak C,\bar\nu,G,
\penalty-100\family{i}{I\cup\{\infty\}}{\tilde{\action}_i},
\familyiI{\pi_i})$ be the Furstenberg self-joining of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ over $I$.
For $J\subseteq I$ set
\Centerline{$\frak B_J=\{a:a\in\frak C$,
$g\action_ia=g\action_ja$ for all $i$, $j\in J$ and
$g\in G\}$,}
\noindent and let
$\frak B_J^*\subseteq\frak C$ be the divaricate copy of $\frak B_J$
(\zz5C). Let $\Cal J$,
$\Cal K\subseteq\Cal PI$ be sets such that $J'\in\Cal J$ whenever
$J\in\Cal J$ and $J\subseteq J'\subseteq I$ and
$K'\in\Cal K$ whenever
$K\in\Cal K$ and $K\subseteq K'\subseteq I$. Then
$\bigvee_{J\in\Cal J}\frak B_J^*$ and $\bigvee_{K\in\Cal K}\frak B_K^*$ are
relatively independent over $\bigvee_{L\in\Cal J\cap\Cal K}\frak B_L^*$.
\medskip
\noindent{\bf proof (a)} Induce on $\#(\Cal K\setminus\Cal J)$. If
$\Cal K\subseteq\Cal J$ the result is trivial. So the rest of the
argument will be the inductive step to $\#(\Cal K\setminus\Cal J)=n>0$.
\medskip
{\bf (b)} Take a maximal member $M$ of
$\Cal K\setminus\Cal J$, and set
$\Cal K'=\Cal K\setminus\{M\}$. If $M\subset J\subseteq I$ then
$J\in\Cal K$; thus $M$ is maximal in $\Cal PI\setminus\Cal K'$.
If $M\subseteq J\in\Cal J\cup\Cal K'$ then $J\in\Cal K$ because
$M\in\Cal K$, while $J\ne M$, so $J\in\Cal K'$. Thus $M$ is also
maximal in $\Cal PI\setminus(\Cal J\cup\Cal K')$. Set
\Centerline{$\frak D_1=\bigvee_{J\in\Cal J}\frak B_J^*$,
\quad$\frak D_2=\bigvee_{K\in\Cal K}\frak B_K^*$,
\quad$\frak E=\bigvee_{L\in\Cal J\cap\Cal K}\frak B_L^*
=\bigvee_{L\in\Cal J\cap\Cal K'}\frak B_L^*$,}
\Centerline{$\frak D'_2=\bigvee_{K\in\Cal K'}\frak B_K^*$,
\quad$\frak E'=\bigvee_{M\subseteq J\in\Cal J\cup\Cal K'}\frak B_J^*
=\bigvee_{M\subseteq J\in\Cal K'}\frak B_J^*$.}
\noindent By the inductive hypothesis,
$\frak D_1$ and $\frak D_2'$ are relatively independent over $\frak E$.
If $c\in\frak B^*_M$, then
$$\eqalignno{Q_{\frak D_1\vee\frak D'_2}(\chi c)
&=Q_{\frak E'}(\chi c)\cr
\displaycause{because $\frak D_1\vee\frak D_2'$ and $\frak B_M^*$
are relatively independent over $\frak E'$, by \zz5D}
&=Q_{\frak D_2'}(\chi c)\cr}$$
\noindent
because $\frak D_2'$ and $\frak B_M^*$ are relatively independent over
$\frak E'$, again by \zz5D. So if $c\in\frak B_M^*$ and
$d\in\frak D'_2$,
$$\eqalignno{Q_{\frak D_1}(\chi c\times\chi d)
&=Q_{\frak D_1}(Q_{\frak D_1\vee\frak D'_2}(\chi c\times\chi d))
=Q_{\frak D_1}(Q_{\frak D_1\vee\frak D'_2}(\chi c)\times\chi d)\cr
&=Q_{\frak D_1}(Q_{\frak D_2'}(\chi c)\times\chi d)
=Q_{\frak D_1}(Q_{\frak D_2'}(\chi c\times\chi d))
=Q_{\frak E}(\chi c\times\chi d)\cr}$$
\noindent because $\frak D_1$ and $\frak D'_2$ are relatively
independent over $\frak E$.
As $c$ and $d$ are arbitrary, $Q_{\frak D_1}$ and $Q_{\frak E}$ agree on
$\frak B^*_M\vee\frak D'_2=\frak D_2$. Rearranging the notation, we have
\Centerline{$\bar\nu(d_1\Bcap d_2)
=\int\chi d_1\times Q_{\frak D_1}(\chi d_2)\,d\bar\nu
=\int\chi d_1\times Q_{\frak E}(\chi d_2)\,d\bar\nu
=\int Q_{\frak E}(\chi d_1)\times Q_{\frak E}(\chi d_2)\,d\bar\nu$}
\noindent whenever $d_1\in\frak D_1$ and $d_2\in\frak D_2$, so
$\frak D_1$ and $\frak D_2$ are relatively independent over $\frak E$.
\bigskip
{\bf\zz5F Lemma} ({\smc Austin p08b}, 7.1) Let $G$ be an abelian group,
$I$ a finite set and
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ a commuting measure-automorphism
action system which is fully isotropized and fully agreeable. Let
$(\frak C,\bar\nu,G,
{\penalty-100}\family{i}{I\cup\{\infty\}}{\tilde{\action}_i},
\familyiI{\pi_i})$ be the Furstenberg self-joining of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ over $I$.
For $\Cal J\subseteq\Cal PI$ set
\Centerline{$\tilde{\frak B}_{\Cal J}=
\bigvee_{J\in\Cal J}\{a:a\in\frak A$,
$g\action_ia=g\action_ja$ for all $i$, $j\in J$ and $g\in G\}$}
\noindent(interpreting $\tilde{\frak B}_{\emptyset}$ as $\{0\}$, of
course). Let $\Bbb J\subseteq I\times\Cal P\Cal PI$ be such that
if $(i,\Cal J)\in\Bbb J$ then
\Centerline{$I\in\Cal J$,
\quad$i\in J$ for every $J\in\Cal J$,
\quad if $J\in\Cal J$ and $J\subseteq K\subseteq I$ then
$K\in\Cal J$.}
\noindent If
$\langle a_{i\Cal J}\rangle_{(i,\Cal J)\in\Bbb J}$ is a family in
$\frak A$
such that $a_{i\Cal J}\in\tilde{\frak B}_{\Cal J}$ for all
$(i,\Cal J)\in\Bbb J$, and
\Centerline{$\inf_{(i,\Cal J)\in\Bbb J}\pi_i(a_{i\Cal J})=0$,}
\noindent then
\Centerline{$\inf_{(i,J)\in\Bbb J}a_{i\Cal J}=0$.}
\medskip
\noindent{\bf proof (a)} Before starting on the main argument, it
will be helpful to explain the way in which Lemma \zz5F will be applied.
Import the notation of \zz5E, so that if $J\subseteq I$ then
\Centerline{$\frak B_J=\{a:g\action_ia=g\action_ja$ for all $i$, $j\in J$
and $g\in G\}$,
\quad$\frak B^*_J=\pi_i[\frak B_J]$ whenever $i\in J$,}
\noindent(with $\frak B^*_{\emptyset}=\frak C$); then
$\tilde{\frak B}_{\Cal J}=\bigvee_{J\in\Cal J}\frak B_J$ and
$\pi_i[\tilde{\frak B}_{\Cal J}]=\bigvee_{J\in\Cal J}\frak B_J^*$ whenever
$(i,\Cal J)\in\Bbb J$.
Take any $l_0\in\Bbb N$ and for $\Cal J\subseteq\Cal PI$ set
$\hat{\Cal J}=\{J:J\in\Cal J$, $\#(J)>l_0\}$. Suppose that for each
$J\subseteq I$ we are given a closed subalgebra $\frak G_J$ of
$\frak B_J$, and for $\Cal J\subseteq\Cal PI$ set
$\frak D_{\Cal J}
=\bigvee_{J\in\hat{\Cal J}}\frak B_J\vee\bigvee_{J\in\Cal J}\frak G_J$.
If $(l,\Cal L)\in\Bbb J$ then $\frak E_1=\pi_l[\tilde{\frak B}_{\Cal L}]$
and
$\frak E_2=\bigvee_{(i,\Cal J)\in\Bbb J,(i,\Cal J)\ne(l,\Cal L)}
\pi_i[\frak D_{\Cal J}]$ are relatively independent over
$\frak E=\pi_l[\frak D_{\Cal L}]$. \Prf\ Set
\Centerline{$\Cal K
=\bigcup_{(i,\Cal J)\in\Bbb J}\Cal J
\setminus(\Cal L\setminus\hat{\Cal L})$.}
\noindent Observe that if $K\in\Cal K$ and $K\subseteq K'\subseteq I$ then
$K'\in\Cal K$. By \zz5F, $\frak E_1=\bigvee_{J\in\Cal L}\frak B_J^*$ and
$\frak E'_2=\bigvee_{J\in\Cal K}\frak B_J^*$ are relatively independent
over
\Centerline{$\bigvee_{J\in\Cal K\cap\Cal L}\frak B_J^*
=\bigvee_{J\in\hat{\Cal L}}\frak B_J^*
\subseteq\frak E
=\bigvee_{J\in\hat{\Cal L}}\frak B_J^*
\vee\bigvee_{J\in\Cal L}\pi_l[\frak G_J]
\subseteq\frak E_1$.}
\noindent Consequently $\frak E_1$ and $\frak E'_2$ are relatively
independent over $\frak E$ ({\smc Fremlin 03}, 458Ld\Latereditions).
It follows that $\frak E_1$ and $\frak E'_2\vee\frak E$ are relatively
independent over $\frak E$ ({\smc Fremlin 03}, 458Ld again). But
\Centerline{$\frak E_2\subseteq\frak E_2'
\vee\bigvee_{J\in\Cal L}\pi_l[\frak G_J]
\subseteq\frak E'_2\vee\frak E$,}
\noindent so $\frak E_1$ and $\frak E_2$ are relatively independent over
$\frak E$.\ \Qed
\medskip
{\bf (b)} Now for the main line of the proof.
The case $\Bbb J=\emptyset$ is trivial; suppose that
$\Bbb J$ is non-empty. Induce on the triple $(\#(I)-l_0,l_1,l_2)$ where
\Centerline{$l_0=\min\{\#(J):
J\in\bigcupop_{(i,\Cal J)\in\Bbb J}\Cal J\}$,}
\Centerline{$l_1=\#(\{(i,\Cal J):(i,\Cal J)\in\Bbb J$,
$\min\{\#(J):J\in\Cal J\}=l_0$,
$\Cal J$ has no least element$\}$,}
\Centerline{$l_2=\#(\{(i,\Cal J):(i,\Cal J)\in\Bbb J$,
$\min\{\#(J):J\in\Cal J\}=l_0$,
$\Cal J$ has a least element$\}$.}
\noindent The case $l_1=l_2=0$ is vacuous.
Let $M$ be $\{(i,\Cal J):(i,\Cal J)\in\Bbb J$,
$\min\{\#(J):J\in\Cal J\}=l_0\}$.
\medskip
{\bf (c)} Suppose that there are an $\Cal L\subseteq\Cal PI$
and distinct $j$, $k\in I$ such that $(j,\Cal L)$ and $(k,\Cal L)$
both belong to $M$. In this case, every member of
$\Cal L$ must contain both $j$ and $k$, so
$\frak B_J\subseteq\frak B_{\{j,k\}}$ for every $J\in\Cal L$,
$\tilde{\frak B}_{\Cal L}\subseteq\frak B_{\{j,k\}}$,
$g\action_ja=g\action_ka$
whenever $a\in\tilde{\frak B}_{\Cal L}$ and $g\in G$, and
$\pi_j$ and $\pi_k$ agree on $\tilde{\frak B}_{\Cal L}$, by \zz5Ba.
Set $\Bbb J'=\Bbb J\setminus\{(k,\Cal L)\}$.
Then $\Bbb J'$ yields the triple
$(\#(I)-l_0,l'_1,l'_2)$ where $l'_1\le l_1$, $l'_2\le l_2$ and
$l'_1+l'_2<l_1+l_2$, so has been previously dealt with. Set
$$\eqalign{a'_{i\Cal J}
&=a_{j\Cal L}\Bcap a_{k\Cal L}\text{ if }i=j\text{ and }\Cal J=\Cal L,\cr
&=a_{i\Cal J}\text{ if }(i,\Cal J)\in\Bbb J'
\text{ and }(i,\Cal J)\ne(j,\Cal L).\cr}$$
\noindent Since $a_{k\Cal L}\in\tilde{\frak B}_{\Cal L}$,
$$\eqalign{\inf_{(i,\Cal J)\in\Bbb J'}\pi_ia'_{i\Cal J}
&=\inf_{(i,\Cal J)\in\Bbb J'}
\pi_i(a_{i\Cal J})\Bcap\pi_j(a_{k\Cal L})\cr
&=\inf_{(i,\Cal J)\in\Bbb J'}\pi_i(a_{i\Cal J})\Bcap\pi_k(a_{k\Cal L})
=\inf_{(i,\Cal J)\in\Bbb J}\pi_i(a_{i\Cal J})
=0.\cr}$$
\noindent By the inductive hypothesis,
\Centerline{$0
=\inf_{(i,\Cal J)\in\Bbb J'}a'_{i\Cal J}
=\inf_{(i,\Cal J)\in\Bbb J'}a_{i\Cal J}\Bcap a_{k\Cal L}
=\inf_{(i,\Cal J)\in\Bbb J}a_{i\Cal J}$}
\noindent and the induction proceeds.
We can therefore assume, for the rest of the argument, that
there are no such $\Cal L$, $j$ and $k$.
\medskip
{\bf (d) Inductive step to $(l_0,0,l_2)$ when $l_2>0$:}
In this case, for every $(i,\Cal J)\in M$, $\Cal J$ has a least member.
\medskip
\quad{\bf (i)}
Take any $(l,\Cal L)\in M$, and let $L$ be the least member of $\Cal L$.
Set $\hat{\Cal L}=\Cal L\setminus\{L\}$,
\Centerline{$\Bbb J'
=(\Bbb J\setminus\{(l,\Cal L)\})\cup\{(l,\hat{\Cal L})\}$.}
\noindent Then $\Bbb J'$ yields a triple
$(\#(I)-l_0',l'_1,l_2')$ where either $l_0'>l_0$
(because $(l,\Cal L)$ was the only member of $M$) or $l'_0=l_0$ and
$l'_1=0$ and $l'_2=l_2-1$; in either
case, it has already been dealt with.
Set $\frak D=\tilde{\frak B}_{\hat{\Cal L}}$,
$$\eqalignno{a'_{i\Cal J}
&=\upr(a_{l\Cal L},\frak D)\text{ if }i=l
\text{ and }\Cal J=\hat{\Cal L}\text{ and }(l,\Cal J)\notin\Bbb J,\cr
\displaycause{recall that
$\upr(a,\frak D)
=\inf\{d:a\Bsubseteq d\in\frak D\}$; see {\smc Fremlin},
313S\formerly{314V}}
&=a_{l\Cal J}\Bcap\upr(a_{l\Cal L},\frak D)
\text{ if }i=l\text{ and }\Cal J=\hat{\Cal L}
\text{ and }(l,\Cal J)\in\Bbb J,\cr
&=a_{l\Cal J}\text{ if }i=l\text{ and }
(l,\Cal J)\in\Bbb J'\text{ and }\Cal J\ne\hat{\Cal L},\cr
&=a_{i\Cal J}\text{ if }i\in I\setminus\{l\}
\text{ and }(i,\Cal J)\in\Bbb J.\cr}$$
\noindent
Then $a'_{i\Cal J}\in\tilde{\frak B}_{\Cal J}$ whenever
$(i,\Cal J)\in\Bbb J'$. \Prf\ If $i=l$ and $\Cal J=\hat{\Cal L}$, then
\Centerline{$\upr(a_{l\Cal L},\frak D)
\in\frak D
=\tilde{\frak B}_{\Cal J}$.}
\noindent If $\Cal J=\hat{\Cal L}$ and $(l,\Cal J)\notin\Bbb J$ then
$a'_{i\Cal J}=\upr(a_{l\Cal L},\frak D)$;
if $\Cal J=\hat{\Cal L}$ and $(l,\Cal J)\in\Bbb J$ then
$a'_{i\Cal J}
=a_{i\Cal J}\Bcap\upr(a_{l\Cal L},\frak D)$;
in either case it belongs to $\tilde{\frak B}_{\Cal J}$. In all other
cases, $a'_{i\Cal J}=a_{i\Cal J}\in\tilde{\frak B}_{\Cal J}$.\ \Qed
\medskip
\quad{\bf (ii)} Write $N$ for $\Bbb J\setminus\{(l,\Cal L)\}$.
In (a), set $\frak G_L=\{0\}$ and $\frak G_J=\frak B_J$ for
other $J\subseteq I$. Then
$\frak D_{\Cal J}=\tilde{\frak B}_{\Cal J}$ whenever
$(i,\Cal J)\in N$.
\Prf\ The point is
that $L\notin\Cal J$. For if $J\in\Cal J$ then either $\#(J)>l_0$
or $\#(J)=l_0$ is the least member of $\Cal J$; since
$\Cal J\ne\Cal L$, as settled in (b) above, and $\Cal J$ and $\Cal L$ both
have least members, their least members must be different, and $J\ne L$.
So
\Centerline{$\frak D_{\Cal J}
=\bigvee_{J\in\hat{\Cal J}}\frak B_J\vee\bigvee_{J\in\Cal J}\frak G_J
=\bigvee_{J\in\hat{\Cal J}}\frak B_J\vee\bigvee_{J\in\Cal J}\frak B_J
=\tilde{\frak B}_{\Cal J}$. \Qed}
\noindent On the other hand,
\Centerline{$\frak D_{\Cal L}
=\bigvee_{J\in\hat{\Cal L}}\frak B_J=\frak D$}
\noindent because $\Cal L=\hat{\Cal L}\cup\{L\}$ and $\frak G_L=\{0\}$.
Now observe that, in the notation of (a),
\Centerline{$\frak E_1
=\pi_l[\tilde{\frak B}_{\Cal L}]$}
\noindent contains $\pi_l(a_{l\Cal L})$,
\Centerline{$\frak E_2
=\bigvee_{(i,\Cal J)\in N}\pi_i[\frak D_{\Cal J}]
=\bigvee_{(i,\Cal J)\in N}\pi_i[\tilde{\frak B}_{\Cal J}]$}
\noindent contains $\inf_{(i,\Cal J)\in N}a_{i\Cal J}$, and
\Centerline{$\frak E
=\pi_l[\frak D_{\Cal L}]
=\pi_l[\frak D]$.}
\noindent Since
$\frak E_1$ and $\frak E_2$ are relatively independent over
$\frak E$, by (a),
and $\pi_l(a_{l\Cal L})\Bcap\inf_{(i,\Cal J)\in N}\pi_i(a_{i\Cal J})=0$,
we also have
$$\eqalignno{0
&=\upr(\pi_l(a_{l\Cal L}),\frak E)
\Bcap\inf_{(i,\Cal J)\in N}\pi_i(a_{i\Cal J})\cr
\displaycause{{\smc Fremlin 03}, 458Lf\Latereditions}
&=\pi_l(\upr(a_{l\Cal L}),\frak D)
\Bcap\inf_{(i,\Cal J)\in N}\pi_i(a_{i\Cal J})\cr
\displaycause{{\smc Fremlin 02}, 313Xs\Latereditions}
&=\pi_l(\upr(a_{l\Cal L}),\frak D)
\Bcap\inf_{(i,\Cal J)\in\Bbb J}\pi_i(a_{i\Cal J})
=\inf_{(i,\Cal J)\in\Bbb J'}
\pi_i(a'_{i\Cal J}).\cr}$$
\noindent By the inductive hypothesis,
$$0
=\inf_{(i,\Cal J)\in\Bbb J'}a'_{i\Cal J}
=\upr(a_{l\Cal L},\frak D)
\Bcap\inf_{(i,\Cal J)\in N}a_{i\Cal J}
\Bsupseteq\inf_{(i,\Cal J)\in\Bbb J}a_{i\Cal J}$$
\noindent and the induction proceeds in this case also.
\medskip
{\bf (e) Inductive step to $(l_0,l_1,l_2)$ when $l_1>0$:}
For $\Cal J\subseteq\Cal PI$, set
$\hat\Cal J=\{J:J\in\Cal J$, $\#(J)>l_0\}$.
Note that
$\tilde{\frak B}_{\Cal J}=\tilde{\frak B}_{\hat{\Cal J}}
\vee\bigvee_{J\in\Cal J,\#(J)=l_0}\frak B_J$ whenever
$(i,\Cal J)\in\Bbb J$.
\medskip
\quad{\bf case 1}
Suppose there is a pair $(l,\Cal L)\in M$ such that
$a_{l\Cal L}$ is of the form
$b\Bcap\inf_{L\in\Cal L,\#(L)=l_0}b_L$ where
$b\in\tilde{\frak B}_{\hat{\Cal L}}$ and $b_L\in\frak B_L$ for each
$L\in\Cal L\setminus\hat{\Cal L}$.
Set $\Cal K_L=\{J:L\subseteq J\subseteq I\}$ for $L\subseteq I$,
so that $\tilde{\frak B}_{\Cal K_L}=\frak B_L$ for each $L$,
and
\Centerline{$\Bbb J'
=(\Bbb J\setminus\{(l,\Cal L)\})
\cup\{(l,\Cal K_L):L\in\Cal L\setminus\hat{\Cal L}\}
\cup\{(l,\hat{\Cal L})\}$.}
\noindent Then $\Bbb J'$ yields a triple
$(\#(I)-l_0,l_1-1,l'_2)$, because every $\Cal K_L$ has a least element
of size $l_0$, while $\hat{\Cal L}$ contains no set of size $l_0$;
so $\Bbb J'$ has been previously dealt with. Set
$$\eqalign{a'_{i\Cal J}
&=b\text{ if }i=l,\,\Cal J=\hat{\Cal L}
\text{ and }(l,\Cal J)\notin\Bbb J,\cr
&=b\Bcap a_{l\Cal L}
\text{ if }i=l,\,\Cal J=\hat{\Cal L}
\text{ and }(l,\Cal J)\in\Bbb J,\cr
&=b_L\text{ if }i=l,\,L\in\Cal L\setminus\hat{\Cal L},\,
\Cal J=\Cal K_L\text{ and }(l,\Cal J)\notin\Bbb J,\cr
&=b_L\Bcap a_{l\Cal J}\text{ if }i=l,\,L\in\Cal L\setminus\hat{\Cal L},\,
\Cal J=\Cal K_L\text{ and }(l,\Cal J)\in\Bbb J,\cr
&=a_{l\Cal J}\text{ if }i=l,\,
(l,\Cal J)\in\Bbb J\text{ and }\Cal J\notin
\{\Cal L,\hat{\Cal L}\}
\cup\{\Cal K_L:L\in\Cal L\setminus\hat{\Cal L}\},\cr
&=a_{i\Cal J}\text{ if }i\in I\setminus\{l\}\text{ and }
(i,\Cal J)\in\Bbb J.\cr}$$
\noindent Then
$$\eqalign{\inf_{(i,\Cal J)\in\Bbb J'}\pi_i(a'_{i\Cal J})
&=\pi_{l}(b\Bcap\inf_{L\in\Cal L\setminus\hat{\Cal L}}b_L)
\Bcap\inf_{\Atop{(i,\Cal J)\in\Bbb J}{(i,\Cal J)\ne(l,\Cal L)}}
\pi_i(a_{i\Cal J})\cr
&=\inf_{(i,\Cal J)\in\Bbb J}\pi_i(a_{i\Cal J})
=0.\cr}$$
\noindent By the inductive hypothesis,
$$0
=\inf_{(i,\Cal J)\in\Bbb J'}a'_{i\Cal J}
=b\Bcap\inf_{L\in\Cal L\setminus\hat{\Cal L}}b_L
\Bcap\inf_{\Atop{(i,\Cal J)\in\Bbb J}{(i,\Cal J)\ne(l,\Cal L)}}
a_{i\Cal J}
=\inf_{(i,\Cal J)\in\Bbb J}a_{i\Cal J}$$
\noindent and again we can move forward.
\medskip
\quad{\bf case 2} Suppose there is a pair $(l,\Cal L)\in M$ such that
$a_{l\Cal L}$ belongs to the
{\it subalgebra} of $\frak A$ generated by
$\tilde{\frak B}_{\hat{\Cal L}}
\cup\bigcup\{\frak B_J:J\in\Cal L\}$.
Then it is a finite supremum of elements of the form considered in case 1
and, applying the argument above to each of these, we again find that
$\inf_{(i,\Cal J)\in\Bbb J}a_{i\Cal J}=0$.
\medskip
\quad{\bf case 3} Now for the case of general $a_{i\Cal J}$. Take any
$\epsilon\in\ocint{0,1}$. Set $\delta=\epsilon/2\#(\Bbb J)$.
For each $(i,\Cal J)\in\Bbb J$,
$$\eqalign{a_{i\Cal J}
&\in\tilde{\frak B}_{\Cal J}
=\tilde{\frak B}_{\hat{\Cal J}}
\vee\bigvee_{J\in\Cal J\setminus\hat{\Cal J}}\frak B_J\cr
&=\overline{\bigcup\{\tilde{\frak B}_{\hat{\Cal J}}
\vee\bigvee_{J\in\Cal J}\frak G_J:
\frak G_J\text{ is a finite subalgebra of }\frak B_J
\text{ for every }J\in\Cal J\}}.\cr}$$
\noindent We can therefore find families
$\langle\frak G_J\rangle_{J\subseteq I}$ and
$\langle b_{i\Cal J}\rangle_{(i,\Cal J)\in\Bbb J}$ such that $\frak G_J$ is
a finite subalgebra of $\frak B_J$ for every $J$,
$b_{i\Cal J}\in\tilde{\frak B}_{\hat{\Cal J}}
\vee\bigvee_{J\in\Cal J}\frak G_J$ for every
$(i,\Cal J)\in\Bbb J$, and
$\bar\mu(a_{i\Cal J}\Bsymmdiff b_{i\Cal J})\le\delta^2$ for every
$(i,\Cal J)\in\Cal J$. As in (a), set
$\frak D_{\Cal J}=\tilde{\frak B}_{\hat{\Cal J}}
\vee\bigvee_{J\in\Cal J}\frak G_J$
for $\Cal J\subseteq\Cal PI$. For $(i,\Cal J)\in\Bbb J$,
set $d_{i\Cal J}=\Bvalue{Q_{\frak D_{\Cal J}}(\chi a_{i\Cal J})>1-\delta}$.
Then
$$\eqalign{Q_{\frak D_{\Cal J}}\chi(d_{i\Cal J}\Bsetminus a_{i\Cal J})
&=Q_{\frak D_{\Cal J}}
\bigl(\chi(d_{i\Cal J})-\chi(d_{i\Cal J})\times\chi(a_{i\Cal J})\bigr)\cr
&=\chi d_{i\Cal J}
-\chi(d_{i\Cal J})\times Q_{\frak D_{\Cal J}}\chi(a_{i\Cal J})
\le\delta\chi d_{i\Cal J};\cr}$$
\noindent on the other hand,
$$\eqalign{\delta\bar\mu(a_{i\Cal J}\Bsetminus d_{i\Cal J})
&\le\int_{a\Bsetminus d}
\chi a_{i\Cal J}-Q_{\frak D_{\Cal J}}(\chi a_{i\Cal J})\,d\bar\mu
\le\|\chi a_{i\Cal J}-Q_{\frak D_{\Cal J}}(\chi a_{i\Cal J})\|_1\cr
&\le\|\chi a_{i\Cal J}-\chi b_{i\Cal J}\|_1
+\|\chi b_{i\Cal J}-Q_{\frak D_{\Cal J}}(\chi b_{i\Cal J})\|_1
+\|Q_{\frak D_{\Cal J}}(\chi b_{i\Cal J}-\chi a_{i\Cal J})\|_1\cr
&\le 2\|\chi a_{i\Cal J}-\chi b_{i\Cal J}\|_1
\le 2\delta^2,\cr}$$
\noindent so $\bar\mu(a_{i\Cal J}\Bsetminus d_{i\Cal J})\le 2\delta$.
Consider $c=\inf_{(i,\Cal J)\in\Bbb J}\pi_i(d_{i\Cal J})$. For
$(l,\Cal L)\in\Bbb J$, we know from (a) that
$\frak E_1=\pi_l[\tilde{\frak B}_{\Cal L}]$ and
$\frak E_2
=\bigvee_{(i,\Cal J)\in\Bbb J,(i,\Cal J)\ne(l,\Cal L)}
\pi_i[\frak D_{\Cal J}]$
are relatively independent over $\frak E=\pi_l[\frak D_{\Cal L}]$. Since
$\pi_l(d_{l\Cal L}\Bsetminus a_{l\Cal L})\in\frak E_1$ and
$e=\inf_{(i,\Cal J)\in\Bbb J,\,(i,\Cal J)\ne(l,\Cal L)}\pi_i(d_{i\Cal J})$
belongs to $\frak E_2$,
$$\eqalignno{\bar\nu(c\Bsetminus\pi_l(a_{l\Cal L}))
&=\int\chi\pi_l(d_{l\Cal L}\Bsetminus a_{l\Cal L})
\times\chi e\,d\bar\nu
=\int Q_{\frak E}(\chi\pi_l(d_{l\Cal L}\Bsetminus a_{l\Cal L}))
\times\chi e\,d\bar\nu\cr
&=\int R_lQ_{\frak D_{\Cal L}}\chi(d_{l\Cal L}\Bsetminus a_{l\Cal L})
\times\chi e\,d\bar\nu\cr
\displaycause{where $R_l:L^0(\frak A)\to L^0(\frak C)$ corresponds to
$\pi_l:\frak A\to\frak C$, as usual}
&\le\delta\int R_l\chi(d_{l\Cal L})
\times\chi e\,d\bar\nu
=\delta\int\chi(\inf_{(i,\Cal J)\in\Bbb J}
\pi_i(d_{i\Cal J}))\,d\bar\nu
=\delta\nu c.\cr}$$
\noindent Summing over $(l,\Cal L)\in\Bbb J$,
$$\eqalignno{\bar\nu c
&=\bar\nu(c\Bsetminus\inf_{(l,\Cal L)\in\Bbb J}\pi_la_{l\Cal L})\cr
\displaycause{because $\inf_{(l,\Cal L)\in\Bbb J}\pi_l(a_{l\Cal L})=0$}
&\le\sum_{(l,\Cal L)\in\Bbb J}\bar\nu(c\Bsetminus\pi_la_{l\Cal L})
\le\delta\#(\Bbb J)\bar\nu c\le\Bover12\bar\nu c,\cr}$$
\noindent and $\bar\nu c=0$, that is, $c=0$.
Now observe that, because every $\frak G_J$ is finite, the subalgebra
of $\frak A$ generated by
$\tilde{\frak B}_{\hat{\Cal J}}
\cup\bigcup_{J\in\Cal J}\frak G_J$ is closed,
and is equal to $\frak D_{\Cal J}$, for every $\Cal J\subseteq\Cal PI$.
Applying case 2 to the family
$\langle d_{i\Cal J}\rangle_{(i,\Cal J)\in\Bbb J}$ and any
$(l,\Cal L)\in M$, we see that $\inf_{(i,\Cal J)\in\Bbb J}d_{i\Cal J}=0$.
But this means that
\Centerline{$\bar\mu(\inf_{(i,\Cal J)\in\Bbb J}a_{i\Cal J})
\le\sum_{(i,\Cal J)\in\Bbb J}\bar\mu(a_{i\Cal J}\setminus d_{i\Cal J})
\le 2\delta\#(\Bbb J)\le\epsilon$.}
\noindent As $\epsilon$ is arbitrary,
$\inf_{(i,\Cal J)\in\Bbb J}a_{i\Cal J}=0$ and the induction proceeds in
this case also.
This completes the proof.
\bigskip
{\bf\zz5G Theorem} Let $G$ be an abelian group,
$I$ a finite set and $(\frak A,\bar\mu,G,\familyiI{\action_i})$ a
commuting measure-automorphism action system. Then
\Centerline{$\WDL_{g\to G}\bar\mu(\inf_{i\in I}g\action_ia)>0$}
\noindent for every non-zero $a\in\frak A$.
\medskip
\noindent{\bf proof (a)(i)} If $I=\emptyset$ we have to interpret
the infimum of the empty set in $\frak A$, but this is $1$, so we get
$\WDL_{g\to G}\bar\mu(\inf_{i\in I}g\action_ia)=1$ for every $a\in\frak A$.
\medskip
\quad{\bf (ii)} If $I=\{j\}$ is a singleton, then
\Centerline{$\WDL_{g\to G}\bar\mu(\inf_{i\in I}g\action_ia)
=\WDL_{g\to G}\bar\mu(g\action_ja)=\bar\mu a>0$}
\noindent for every non-zero $a$. So henceforth we can assume that
$\#(I)\ge 2$.
\medskip
\quad{\bf (iii)} It may make you more comfortable if I remind you that
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is measure-averaging, by Theorem
\zz3G, so
\Centerline{$\WDL_{g\to G}\chi(\inf_{i\in I}g\action_ia)
=\WDL_{g\to G}\prod_{i\in I}g\action_i\chi a$}
\noindent is defined in $L^1(\frak A,\bar\mu)$ for every $a\in\frak A$,
and $\WDL_{g\to G}\bar\mu(\inf_{i\in I}g\action_ia)$ is always defined.
\medskip
{\bf (b)} Suppose that
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is fully isotropized and fully
agreeable. Let $(\frak C,\bar\nu,G,
\penalty-100\family{i}{I\cup\{\infty\}}{\tilde{\action}_i},
\familyiI{\pi_i})$ be the Furstenberg self-joining of
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ over $I$.
Take $a\in\frak A$ such that
$\WDL_{g\to G}\bar\mu(\inf_{i\in I}g\action_ia)=0$. Because
$(\frak A,\bar\mu,G,\familyiI{\action_i})$ is $(I,j)$-agreeable for every
$j\in I$,
\Centerline{$0
=\WDL_{g\to G}\bar\mu(\inf_{i\in I}g\action_ia)
=\bar\nu(\inf_{i\in I}\pi_ia)
=\int\prod_{i\in I}R_iP_i\chi a\,d\bar\nu$}
\noindent where $R_i:L^0(\frak A)\to L^0(\frak C)$ is the Riesz
homomorphism corresponding to $\pi_i:\frak A\to\frak C$, and
$P_i$ is the conditional expectation operator corresponding to the
closed subalgebra
$\bigvee_{j\in I\setminus\{i\}}\{a:g\action_ja=g\action_ia$ for every
$g\in G\}\subseteq\frak A$. Set $a_i=\Bvalue{P_i\chi a>0}$ for each
$i$; then $\pi_ia_i=\Bvalue{R_iP_i\chi a}$ for each $i$, so
$\inf_{i\in I}\pi_ia_i=0$. Applying \zz5F with
$\Cal J_i=\{J:i\in J\subseteq I$, $\#(J)\ge 2\}$,
$\Bbb J=\{(i,\Cal J_i):i\in I\}$, we see that
\Centerline{$a_i
\in\bigvee_{j\in I\setminus\{i\}}\{a:g\action_ja=g\action_ia$ for
every $g\in G\}
=\tilde{\frak B}_{\Cal J_i}$}
\noindent for each $i$, so
$\inf_{i\in I}a_i=0$. But $a\Bsubseteq a_i$ for each $i$, so $a=0$.
\medskip
{\bf (c)} In general, $(\frak A,\bar\mu,G,\familyiI{\action_i})$ has
a fully isotropized and fully agreeable extension
$(\frak A',\bar\mu',G,\familyiI{\action'_i},\phi)$, by Proposition \zz4G.
If $a\in\frak A\setminus\{0\}$, then $\phi a\ne 0$ so
$$\eqalign{0
&<\WDL_{g\to G}\bar\mu'(\inf_{i\in I}g\action'_i\phi a)
=\WDL_{g\to G}\bar\mu'(\inf_{i\in I}\phi(g\action_ia))\cr
&=\WDL_{g\to G}\bar\mu'(\phi(\inf_{i\in I}g\action_ia))
=\WDL_{g\to G}\bar\mu(\inf_{i\in I}g\action_ia),\cr}$$
\noindent as required.
\medskip
\noindent{\bf Remark} The special case of this theorem in which
$G=\Bbb Z$ is the Multiple Recurrence Theorem
({\smc Furstenberg \& Katznelson 78}).
\bigskip
{\bf\zz5H Corollary} Let $G$ be an infinite abelian group,
$I$ a finite set and
$(X,G,\familyiI{\action_i})$ a commuting action system. Suppose that
there is a finitely additive functional $\mu:\Cal PX\to\coint{0,\infty}$
which is
$G$-invariant, that is, $\mu(g\hat{\action}_iA)=\mu A$ whenever
$A\subseteq X$, $i\in I$ and $g\in G$, writing $g\hat{\action}_iA$ for
$\{g\action_ix:x\in A\}$. If $A\subseteq X$ and $\mu A>0$,
there are a $g\in G$, not the identity,
and an $x\in X$ such that $g\action_ix\in A$ for every $i\in I$.
\medskip
\noindent{\bf proof} If $\mu X=0$ this is vacuous; otherwise, taking a
scalar multiple of $\mu$ if necessary, we can assume that $\mu X=1$.
Of course we can take it that $I$ is non-empty.
Applying \zz2B to the system
$(\Cal PX,G,\familyiI{\hat{\action}_i})$,
we get a commuting measure-preserving action system
$(\frak A,\bar\mu,G,\familyiI{\tilde{\action}_i})$ together with a
Boolean homomorphism $\phi:\Cal PX\to\frak A$ such that
$\bar\mu\phi(A)=\mu A$ for every $A\subseteq X$ and
$g\tilde{\action}_i\phi(A)=\phi(g\hat{\action}_iA)$ whenever
$A\subseteq X$, $i\in I$ and $g\in G$. If $\mu A>0$, then
$\bar\mu\phi(A)>0$ so
$$\eqalign{\WDL_{g\to G}\mu(\bigcap_{i\in I}g\hat{\action}_iA)
&=\WDL_{g\to G}\bar\mu(\phi(\bigcap_{i\in I}g\hat{\action}_iA))
=\WDL_{g\to G}\bar\mu(\inf_{i\in I}\phi(g\hat{\action}_iA))\cr
&=\WDL_{g\to G}\bar\mu(\inf_{i\in I}g\tilde{\action}_i\phi(A))
>0\cr}$$
\noindent by Theorem \zz5G. In particular, there is a $g\in G$, other
than the identity, such that
$\mu(\bigcap_{i\in I}g\hat{\action_i}A)>0$ (\zz1Hd);
in which case, there is surely
an $x\in\bigcap_{i\in I}g\hat{\action_i}A$. Now $g^{-1}\action_ix\in A$
for every $i\in I$.
\bigskip
{\bf\zz5J Corollary} Let $R$ be an infinite ring and $X$ an $R$-module.
Suppose that $I\subseteq X$ is a finite set
and that $A\subseteq X$ has $\overline{\WDL}_{x\to X}\chi A(x)>0$,
where
$\overline{\WDL}_{x\to X}$ is defined with respect to the additive group
$(X,+)$. Then there is a similar copy
$x+rI$ of $I$ included in $A$, where $x\in X$ and $r\in R\setminus\{0\}$.
\medskip
\noindent{\bf proof} By \zz1Hc, there is a translation-invariant
finitely additive functional
$\mu:\Cal PX\to[0,1]$ such that $\mu A>0$. For $i\in I$, $r\in R$ and
$x\in X$, set $r\action_ix=x+ri$. It is easy to check that
$(X,R,\familyiI{\action_i})$ is a commuting action system when $R$ is given
its additive group structure. Because
$\mu$ is translation-invariant, it is $R$-invariant. By \zz5I, there are
an $x\in X$ and an $r\in R\setminus\{0\}$ such that
$x+ri=r\action_ix\in A$ for every $i\in I$.
%\bigskip
%\noindent{\bf Problems}
\bigskip
\noindent{\bf References}
Austin T.\ [p08a] `On the norm convergence of nonconventional ergodic
averages', preprint, 2008 ({\tt arXiv:{\penalty-100}0805.0320}).
Austin T.\ [p08b] `Deducing the multidimensional Szemer\'edi theorem from
an infinitary removal lemma', preprint, 2008 ({\tt arXiv:0808.2267}).
Bergelson V., McCutcheon R.\ \& Zhang Q.\ [97]
`A Roth theorem for amenable groups',
Amer.\ J.\ Math.\ 119 (1997) 1173-1211.
Clarkson J.A.\ [36] `Uniformly convex spaces', Trans.\ Amer.\ Math.\ Soc.\
40 (1936) 396-414.
%Fremlin D.H.\ [00] {\it Measure Theory, Vol.\ 1: The Irreducible Minimum.}
%Torres Fremlin, 2000.
Fremlin D.H.\ [01] {\it Measure Theory, Vol.\ 2: Broad Foundations.}
Torres Fremlin, 2001.
Fremlin D.H.\ [02] {\it Measure Theory, Vol.\ 3: Measure Algebras.}
Torres Fremlin, 2002.
Fremlin D.H.\ [03] {\it Measure Theory, Vol.\ 4: Topological Measure
Spaces.} Torres Fremlin, 2003.
Fremlin D.H.\ [n08] `Well-distributed sequences and Banach density',
note of 22.10.08
({\tt http:{\bsp}/www.essex.ac.uk{\bsp}maths{\bsp}staff{\bsp}fremlin{\bsp}preprints.htm}).
%n02j23.tex
Furstenberg H.\ \& Katznelson Y.\ [78]
`An ergodic Szemer\'edi theorem for commuting transformations',
J.\ d'Analyse Math.\ 34 (1978) 275-291.
Furstenberg H.\ [81]
{\it Recurrence in Ergodic Theory and Combinatorial Number Theory.}
Princeton U.P., 1981.
% QA614
%Lindenstrauss J.\ \& Tzafriri L.\ [96] {\it Classical Banach Spaces I \&
%II.} Springer, 1996.
Szemer\'edi E.\ [75]
`On sets of integers containing no k elements in arithmetic progression',
Acta Arithmetica 27 (1975) 199-245.
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\title{As Gaddafi falls — Lessons from Libya}
\date{August 25, 2011}
\author{Andrew Flood}
\subtitle{Imperialism, anti-imperialism \& democratic revolution}
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{\usekomafont{title}{\huge As Gaddafi falls — Lessons from Libya\par}}%
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The sudden end of the Gaddafi regime some 6 months after the start of the Libyan revolt leaves some difficult questions unanswered for the left. Gaddafi’s determination to physically crush the revolt quickly transformed it into a civil war, a civil war that saw considerable imperialist intervention on the rebel side, intervention that was essential to their eventual victory. This and Gaddafi’s historic record led to some on the left taking his side in the civil war while other organisations tried to balance support for the ‘Arab spring’s’ arrival in Libya with opposition to imperialism.
This question of where the balance lies between international solidarity with pro-democracy movements and opposition to imperialism could well rapidly return to the top of the agenda in a very much bigger way as the regime in Syria continues its months long military suppression of the democracy movement there.
The spread of the Arab democratic revolution to Libya and the subsequent intervention by imperialist airpower against Gaddafi led to a major and heated debate on the revolutionary left on the question of imperialism. The very complexity of the situation in Libya means that as well as the specifics of this war and revolution it provides a useful starting point for a re-examination of what has become traditional anti-imperialism. Libya like Rwanda, Srebrenicia and more rhetorically Palestine has become one of those recent conflicts where many argue for rather than against intervention.
Part of this is down to a standard dogmatic polarization between pro-intervention liberals who think the bombs are being dropped to protect Libyans on the one hand and on the other the nationalists and hard core leninist’s who think Gadaffi’s past make him an enemy of imperialism today. Neither pole has much to say of relevance to those who found themselves facing Gaddafi’s tanks outside Benghazi at the start of the revolt with little more than AK47’s to stop them. But much more reasoned argument for and against intervention has been made by commentators with a strong record like Gilbert Achcar who argued for intervention and Noam Chomsky who argued against.
\section{\textbf{From facts to analysis to positions}}
I want to look at what anarchists can say about the specifics of the Libyan situation and what the Libyan situation tells us about the politics of anti-imperialism today in general. On the specifics of Libya this means starting with looking at what we know of Gaddafi’s actual relationship with the imperialist powers. We also need to ask who the rebels are, what their program is and in what way has their dependency on imperialist air power transformed them. Among other sources I’ve used are the cables from the US embassy in Tripoli which had been released by wikileaks. These are valuable in giving an idea of what US imperialism’s actual relationship was with the regime and what they really thought about the rebellion shorn of the layers of spin embedded in every official utterance from the White House. My other sources are anarchist writings (in one case from within Libya) and the better end of the mainstream media and business press.
Those who have openly proclaimed support for Gaddafi have done so in the language of anti-imperialism. But whatever about his claim to be anti-imperialist back in the 1980’s, today Gaddafi is the dictator who it was claimed had turned anti aircraft guns on democracy protesters, killing hundreds in the first days of the revolt against his rule. Footage was posted by Libyans in those early days, and the gruesome sight of bodies that had literally been ripped apart by the high calibre bullets appeared to leave no doubt of such use. As did the charred bodies of solders who had refused to follow such orders and as a result had been executed, hands cable tied behind their backs.
The BBC carried an interview with a orderly at a Libyan hospital mortuary who claimed 6–700 had been killed in Tripoli alone (Source: ) However a June 24\textsuperscript{th} article by Patrick Cockburn citied an Amnesty International report (which I have yet managed to find online) which instead put the death toll of the initial repression of the protests at around 200 and said anti-aircraft guns had not been used as spent cartridges recovered where protesters had been killed all came from AK-47’s and similar weapons.
It may well be that it will never become clear what exactly happened during the suppression of the first protests. But regardless of the weapons used or exact numbers killed Gaddafi’s forces did shoot down demonstrators. The fact that section’s of the left were willing to support Gaddafi despite this (and indeed when the higher death tolls went unchallenged) is not a new departure. In the name of anti-imperialism sections of the left have supported other and more brutal dictators in the past.
So at the start of 2011 was there any seriousness to this presentation of Gaddaf as a fighter against imperialism? I would tend to argue no, the so called anti-imperialism was a front for public consumption at home and abroad rather than a reflection of what Libya’s actual relation with the imperialist powers were pre-rebellion.
\section{\textbf{A deal with imperialism — Lockerbie forgiven}}
Pre-rebellion Gaddafi had managed to transform himself into the locally respectable protector of the oil corporations, even if for the imperialist powers he still had a shady past. Gaddafi was almost certainly behind the 21 December 1988 bombing of a Boeing 747–121 over the Scottish town of Lockerbie that killed all 243 passengers and 11 people on the ground. The bombing was widely seen as retaliation for the bombing of his compound in April 1986 during which an adopted baby daughter was killed. A couple of Gaddafi’s speeches during the early days of the insurrection were filmed in the ruins of the compound, no doubt intended to remind the international audience that he had stood up to and withstood imperialist assaults before. Near the start of the insurrection on 24 February 2011, in an interview with the Swedish newspaper Expressen, justice minister Mustafa Abdel-Jalil who had just resigned from the regime claimed that Gaddafi personally ordered the Lockerbie bombing.
Despite the fallout of the Lockerbie bombing the scale of the restoration of relations with Gaddafi before the rebellion was such that the British government had as far back as 2009 arranged the release of Abdel Baset who had been convicted for the bombing. Although the US government made a public show of kicking up a stink about the release the reality is that already in October 2008 President Bush had signed an \emph{“Executive Order restoring the Libyan government’s immunity from terror-related lawsuits and dismissing all of the pending compensation cases in the US.”} Wikileaks cables make clear that the US embassy in Tripoli was well aware, in advance, of the deal being hatched between Britain and Libya to release Abdel Baset, who one cable acknowledged was \emph{“effectively viewed as something of a folks hero in the eyes of the regime and many ordinary Libyans.”}
\section{\textbf{It’s all about Oil?}}
The eagerness to strike a deal with Gaddafi was because Libya has the largest oil reserves in all of Africa and is already the 12\textsuperscript{th} biggest oil exporter in the world. Oil and gas account for 25\% of the economy, 97\% of exports and 90\% of government revenue. As long as Gaddafi’s power was not seriously challenged there was a need to deal with him. Both Britain and the US were willing to overlook the killing of their own citizens so that their energy corporations could obtain a share of the profits ahead of those of Russia or China. Italian and French companies are the other major oil players in Libya. According to the (Libyan) National Oil Company website \emph{“More than 50 international oil companies are present in the market.”} In May 2007 Gaddafi visited then British Prime Minister Tony Blair, during which British Petroleum (BP) signed a \$900 million exploration and production agreement with the Libyan National Oil Company.
The size of the Libyan oil reserves have encouraged some rather crude anti-imperialist writings on the intervention that suggest it is simply all about access to oil. But as the facts above suggest the reality is a good deal more complex. Gaddafi had after all already given the major oil companies access to the Libyan oil fields so there was no need for a war to gain access.
\emph{In some respects business was easier for the oil companies in Libya under Gaddafi then is some of the other oil rich states. Yet another cable makes it clear that the US embassy regarded Gaddafi as considerably less corrupt then most of the other rules in the region, reporting; “Compared to egregious pillaging of State coffers elsewhere in Africa, or the lavish spending of Gulf Arabs, the Libyans don’t see much to complain about in their leader’s lifestyle, as long as he does a good job of making sure other people get a piece of the pie. And when Libyans do complain, they are removed from access to financial rewards.”}
Further at the start of the rebellion when the identity of the rebels was unclear and their program unknown there was a danger the oil corporations might have had access withdrawn or reduced under a new democratic government eager to see a greater share of profits being used for development. The war itself seriously disrupted the flow of oil and saw significant damage to the infrastructure required to export it. But if there was clearly no need for imperialist intervention to get access to the oil the flip side is the crude pro interventionist argument that tries to rule out any discussion of imperialist interest as being oil driven because of this existing access, the situation was a good deal more complex.
The reality is that Gaddafi’s historical record meant that Libya under his rule was always going to be seen as potentially unstable. He had raised concerns in the oil industry as recently as January 2009 when he told Georgetown University students that Libya \emph{“could nationalize their oil production in view of sharply plummeting petroleum prices.”} But as was revealed in a wikileaks cable the US embassy in Tripoli calcualted that while \emph{“Industry experts in Washington and Libya have not entirely dismissed the possibility that the GOL could nationalize its oil and gas sector.. they do not currently judge it to be a serious threat.”} The embassy thought that Gaddafi \emph{“may in fact be signaling more aggressive efforts by the GOl and NOC to secure greater shares of oil produced under existing contracts.”} The embassy concluded that \emph{“While it is never wise to rule out the possibility of seemingly irrational decisions by the GOL, we are not inclined to believe that nationalization is being seriously considered”}
The bulk of Libya’s proven oil and gas reserves lie in the Eastern half of the country. So if pre-rebellion the imperialist powers had to deal with Gaddafi because he had access to the oil once the rebellion was underway they increasingly had to deal with the rebels for the same reason. And as we shall see the US in particular had some serious pre-existing concerns with the opposition movement in Eastern Libya.
It would seem we can make two general comments in relation to the role of Oil in the intervention
\begin{enumerate}[1.]
\item\relax
There was no need for the intervention in order to gain access to the oil, over 50 international oil companies were already present. But intervention well have an impact on what conditions future access happens under, which ever way the situation unfolded.
\item\relax
The imperialist powers do not have identical interests. Had Gaddafi won it is quite possible the western imperialist powers would have been ‘punished’ by him favoring Russian and Chinese energy companies. The victorious rebels have suggested they will not look favourably on Chinese and Russian companies for the same reason.
\end{enumerate}
The early French support for the rebellion may in part be down to hoping this would see the interests of French companies being promoted over those of other NATO countries, in particular Italy. Indeed in March the Prime Minister of Turkey almost directly accused France of following such motivations when during the row about which NATO countries would have military decision making capabilities he declared \emph{“I wish that those who only see oil, gold mines and underground treasures when they look in [Libya’s] direction, would see the region through glasses of conscience from now on.”}
Likewise the late break Italy made with Gaddafi probably reflects the fact that the Italian oil corporation ENI is the current largest foreign oil company in Libya, and thus has the most to lose by any restructuring of access if the rebels won or Libya was partitioned. The wikileaks cables reveal that the other oil companies were annoyed with ENI for giving too much to the Gaddafi regime so the rebel victory is quite likely to result in losses for ENI because of the identification of ENI with the Gaddafi regime.
\section{\textbf{Anti-imperialist armed by Imperialism?}}
Oil is of course only part of the story of Gaddafi’s improving relationships with the imperialist powers prior to the rebellion. Sanctions against Libya were lifted in 2004 allowing arms sales to resume. Both Britain and France suppled the Gaddafi regime with weapons. An EU report released just before the rebellion revealed that Britain had sold \emph{”\$33 million worth of small arms, ammunition, ordinance, aviation components, armoured and protective equipment and military electronics.”} The EU as a whole issued licences for the sale of weapons and weapons systems valued at pounds \$462 million to Libya in 2009. In 2007 the French company Dassault Aviation was awarded a contract to put Libya’s 12 remaining F1 Mirages back into flying condition, these were subsequently used in the attacks against protesters and rebels by the regime although 2 of the 4 repaired were flown to Malta by their pilots when they were ordered to bomb protesters.
US Huey and Chinnok helicopters have also been delivered to Libya via Italy. In fact the first Tomahawk missile strikes seem to have included as targets the very expensive US military command and control facilitates sold to Gaddafi to equip his elite brigades with in the last couple of years before the rebellion. No doubt afterwards the US will be keen to sell the same systems to the new regime! Right at the end of 2008 the US Embassy in Tripoli reported on the keenness of the Gaddafi regime to buy additional US military equipment.
The cable also confirms US embassies opinion as to the shallow nature of Gaddafi’s anti-imperialist posture in revealing that \emph{“Muammar al-Qadhafi expressed reservations to Muatassim in mid-November about U.S.-Libya military-to-military cooperation that could lead to having large numbers of U.S. advisers and trainers present in Libya. He was keen that U.S. military personnel not be seen in uniform in Libya, a prospect with which he was particularly concerned given that the “evacuation” of U.S. and U.K. military bases (the Wheelus and el-Adem airbases, respectively) in 1970 was viewed as a key accomplishment of the revolution.”}
Another May 2009 cable describes a meeting between the US Africa Command General and the regime, in it is reported that \emph{“Al-Qadhafi expressed a desire for cooperation with U.S. Africa Command in the fields of counter-terrorism and counter-piracy.”} The US embassy also clearly saw Gaddafi’s links with Daniel Ortega and Chavez as symbolic bluster rather than a threat, writing after Ortega’s 2009 visit that while \emph{“Libya is keen to pursue symbolic alliances with anti-U.S. leaders to balance the perception that it has gone western by finalizing the U.S.-Libya claims compensation agreement”} there was no depth to this symbolism. Indeed in yet another cable in 2009 the embassy reported that Gaddafi had \emph{“been pressing for a broad agreement on security, including a commitment to come to Libya’s aid if it were attacked”} for several months and advised on how to avoid this issue as the US didn’t want that level of commitment.
Although its a minor part of overall support on 11 Sep 2009 ‘The Telegraph’ revealed that members of the SAS were training their Libyan counterparts and that this \emph{“will further raise suspicions about exactly what has been agreed behind the scenes between Tripoli and Britain.”} Given the role of the SAS in ‘shoot to kill’ operations against Irish republicans this should provide food for thought to those who were inclined to support Gaddafi against the rebels.
At the start of the rebellion Amnesty International revealed that the British company NMS International Group Ltd had supplied armored crowd control vehicles \emph{“that look identical to ones recently seen patrolling the streets of protest-hit streets in Libya.”} The imperialist powers had not only supplied him with military equipment prior to the rebellion they had also supplied him with the equipment used to attack protests with ‘less lethal’ force, as they had with the regimes on either side of Libya in Egypt and Tunisia.
So the arms and training the imperialist powers were supplying to Gaddafi were not simply the equipment to fight battles against his neighboors but also the equipment and training to crush domestic dissent. He certainly had a record in doing so, in 1996 more that 1200 prisoners were massacred as a riot at Abu Salim, which jailed many of Libya’s political prisoners was put down. The day after the riot the prisoners were forced into the courtyards and shot down over a two hour period. It was only after families in Benghazi had kept up protests for two years culminating in a march on the 13\textsuperscript{th} anniversary of 200 that the regime acknowledged to over 900 families that their relatives had indeed been killed and offered compensation.
With the west in 2010 viewing Gaddafi as a friend who could not only be supplied with weapons prior to the rebellion but also the tools to crush protest the claim that he was an enemy of imperialism has to be seriously challenged. Nor were all the imperialist powers all that eager to intervene on the side of the rebellion initially. The uprising began on February 15\textsuperscript{th} yet it was over a month later on the 17\textsuperscript{th} of March before the UN authorized member states to enforce a no-fly zone over Libya . And two more days before the first actual airstrikes took place. That timeline makes no sense if we are to see the rebels as a proxy army for imperialism taking directions from Washington, Paris or elsewhere.
\section{\textbf{Imperialist rivalry}}
As is the case elsewhere in North Africa the imperialists are not united on the way forward. Rivalry between French and US imperialism is common across the region and this is expressed in the differing support each offered for the Libyan insurrection. France recognised the Libyan rebel leadership, the National Transitional Council (NTC) as the legitimate government on March 10\textsuperscript{th} only five days after the NTC first made this claim itself. But it took the US over four months to follow the French lead, holding off until July 15\textsuperscript{th} by which time it was clear Gaddafi was very unlikely to win.
France opposed the idea that the intervention should be NATO led. It was French war planes that stopped the fall of Benghazi by bombing Gaddafi’s armour as it reached the outskirts of the city, without that prompt action the question of intervention might have died before it started.
Gaddafi played on such division and those with Russia and China. Only 48 hours before the UN vote he threatened that if attacked he would transfer Libya’s energy contracts to companies from Russia, India and China. Paolo Scaroni, chief executive of Italy’s Eni SpA, which has the largest Libyan investments of any Western oil company and the closest relationship to the regime told the Wall Street Journal that \emph{“Whoever is in power needs to pump oil to get revenues for his people,”} and while some companies are concerned that \emph{“due to the supportive stances of their home governments towards the opposition,”} they might lose access to Libyan oil but that \emph{“At least publicly, the companies have expressed confidence that they will be permitted to return to Libya.”}
\section{\textbf{China \& Russia}}
Nearly 36,000 Chinese citizens lived in Libya before the rebellion, they were extracted by China’s government which made use of the opportunity to expand its sphere of military operations, they dispatched four military transport planes and a guided-missile frigate, the Xuzhou, to Libya for the extraction. After Lockerbie China never imposed sanctions on the Gaddafi regime and dozens of Chinese companies operate in Libya with trade mainly centering on oil, but there are also wide range of other businesses giving a total \$6.6bn in bilateral trade. This includes Chinese rail companies which have signed railway contracts with Libya, including that for a rail line between Tripoli and Sirte for \$1.7bn.
Russian companies have contracts in Libya worth billions and the former Russian ambassador ti Libya described the Kremlin’s lack of opposition to the air strikes as \emph{“betrayal of Russia’s interests.”} All in all it would be a major mistake to imagine that all the imperialist powers look at Libya in the same way or have a common plan as to where they would like to see it develop. There are very significant tensions between them.
Rather than seeking the overthrowal of Gaddafi from the start it appears that at least some of the imperialist powers were concerned that the rebel forces would be less under their control than Gaddafi. Gaddafi had been co-operating at a level where they were happy to supply him with arms for almost seven years at the time of the insurrection. The rebel forces on the other hand were a largely unknown force, with the US in particular being very cautious about who they might actually be.
\section{\textbf{Fear of an Islamist planet}}
This fear is in part based on a study from the U.S. West Point Military Academy’s ‘Combating Terrorism’ Center on the resistance in Iraq which claims that \emph{“Libya contributed far more fighters per capita than any other nationality .. including Saudi Arabia”} and that \emph{“The vast majority of Libyan fighters .. resided in the country’s Northeast, particularly the coastal cities of Darnah .. and Benghazi.”} These cities were the center of the anti-Gaddafi rebellion. Andrew Exum, a US counterinsurgency specialist and former Army Ranger noted in a blog posting that \emph{“This might explain why those rebels from Libya’s eastern provinces are not too excited about U.S. military intervention. It might also give some pause to those in the United States so eager to arm Libya’s rebels.”}
Similar worries from before the rebellion are found in the US Embassy wikileaks cables, one from 2008 warning that \emph{“the inability of eastern Libyans to effectively challenge Qadhafi’s regime, together with a concerted ideological campaign by returned Libyan fighters from earlier conflicts, have played important roles in Derna’s development as a wellspring of Libyan foreign fighters in Iraq .. One Libyan interlocutor likened young men in Derna to Bruce Willis’ character in the action picture “Die Hard”, who stubbornly refused to die quietly. For them, resistance against coalition forces in Iraq is an important act of ‘jihad’ and a last act of defiance against the Qadhafi regime.” “Many easterners feared the U.S. would not allow Qadhafi’s regime to fall and therefore viewed direct confrontation with the GOL in the near-term as a fool’s errand\dots{}Fighting against U.S. and coalition forces in Iraq represented a way for frustrated young radicals to strike a blow against both Qadhafi and against his perceived American backers.”}
The rebels for their part were not exactly welcoming the imperialist powers with open arms. On the 6\textsuperscript{th} of March an 8 man British mission of 7 SAS and 1 MI6 were arrested by Army units in Benghazi that had joined the rebellion. The Guardian reported that \emph{“The mission backfired when rebel leaders in Benghazi objected to foreign interference from governments which had not yet formally recognised them as Libya’s legitimate rulers.”} Far from being a proxy force under the controls of one or the other imperialist powers even in the early days of the rising they looked far more like a grouping trying to influence imperialist policy rather than take direction from it.
\section{\textbf{What is the National Transition Council}}
The rebels are grouped together in the body known as the National Transition Council (NTC). The political and international affairs committee of the National Transition Council released a program for the revolt timed to coincide with the London summit at the end of March. It’s was first published in English but when it was pointed out this was odd, as few Libyans speak English, the NTC assured that the original was debated in Arabic. In any case its appearance was clearly intended to reassure those in the US who were nervous about what the real intent of the rebels might be.
The program declared the goal to be \emph{“building a free and democratic society and ensuring the supremacy of international humanitarian law and human rights declarations.”} And the economic section which includes \emph{“The development of genuine economic partnerships between a strong and productive public sector, a free private sector and a supportive and effective civil society, which overstands corruption and waste”} seems certain not to frighten the oil companies. But in case there was any doubt it also proclaimed that \emph{“The interests and rights of foreign nationals and companies will be protected.”}
\section{\textbf{The Islamist presence, a threat or an opportunity for the US?}}
An Economist article sees the Islamist element of the rebellion as being something of a golden opportunity for the west, describing how the Economist reporter found that on the ground in Darna \emph{“These jihadis enthusiastically back the NATO-led bombing campaign.} \emph{“A blessing,” says Sufian bin Qumu, an inmate for six years of a pen in Guantánamo Bay, who drove trucks for Osama bin Laden’s Sudanese haulage company before heading to the Afghan camps. “Excellent,” echoes Abdel Hakim al-Hisadi, a rebel commander who trained in Khost camp, Mr bin Laden’s base in Afghanistan. “It’s changed the way we look at the West. They saved our people and we have to say thanks.”}
The New York Times carried a report on a post rebellion visit to Darnah which found \emph{“Secular figures here were adamant in endorsing the Islamists’ right to form parties and, at the Sahaba Mosque, slogans were markedly bereft of religious sentiment. “Freedom, dignity and national unity,” read one. A leaflet circulated there pronounced demands almost identical to those uttered in Egypt: a transitional government, a constitution approved by referendum, parliamentary and presidential elections and a democratic state built on pluralism, the peaceful transfer of power, the rule of law and guarantees of human rights and the protection of freedoms.”}
A report in the Wall Street Journal at the start of April singled out a number of Islamists including once again \emph{“Abdel Hakim al-Hasady, an influential Islamic preacher and high-school teacher who spent five years at a training camp in eastern Afghanistan [who] oversees the recruitment, training and deployment of about 300 rebel fighters from Darna”.} This time he was quoted as saying \emph{“If we hated the Americans 100\%, today it is less than 50\%. They have started to redeem themselves for their past mistakes by helping us to preserve the blood of our children.”}
The New York Times also interviewed Shukri Abdel-Hamid, describing him as a cleric who had spent 10 years in prison under Gaddafi and who declared \emph{“We want a civil state, pluralism, with freedom enshrined by law,”} he said, before echoing a sentiment heard often in Egypt and Tunisia. “\emph{Extremism was a reaction to oppression and the violence of the state. Give us freedom and see what happens.”}
A cynic here could suggest Abdel-Hamid has a good sense of saying what might reassure western journalists but if Abdel-Hamid represents an Islamist element this appears to be a small minority, mostly confined to the East. More worrying many senior figures are defectors from the Gaddafi regime who only jumped ship once the rebellion was underway.
The first meeting of what became the NTC was chaired by former justice minister Mustafa Abdul Jalil. On 5\textsuperscript{th} March Mustafa was declared the chair of the council. But the bulk appear to be the school teachers, engineers, lawyers and doctors who tend to form the leadership of republican insurrections because of their education and extensive pre existing social contacts. The NTC is completely dominated by defectors from the old ruling class and the middle class, all appear to be male although the TNC has claimed there are a small number of women.
The Libyan working class may have fought for freedom in the streets but does not appear as yet to have a voice. A situation made worse by the fact that an estimated one third of workers in Libya were migrant workers and Gaddafi’s use of some migrant workers as mercenaries will make class unity that include migrants all the more difficult. This builds on the system under the old regime where Independent trade unions were banned, legal strikes were almost impossible to organise and union membership was limited to workers of Libyan nationality.
This aside it is clear the rebels are not a single organisation but instead a rather uneasy coalition.
\section{\textbf{Killing of Abdul Fatah Younis}}
A demonstration of just how uneasy that coalition is was given at the end of July when rebels from one faction killed general Abdul Fatah Younis, the head of the Free Libyan Army. Abdul Fatah Younis had previously been a Major General in Gaddafi’s army and the Minister of the Interior but had defected on 22 February after leading a Gaddafin relief column to Benghazi. The rebels who shot him were said to have shouted that he was responsible for the death of their father. This led on August 9\textsuperscript{th} to the chair of the NTC sacking the entire 14 man cabinet!
The TNC appears to have most support in the East around Benghazi, rebel forces in the south and east are openly critical of the TNC. Even as the rebels reached Tripoli the Independent reported that \emph{“rebel fighters in Misrata, who fought so long to defend their city, say privately that they have no intention of obeying orders from the TNC.”}
\section{\textbf{The nature of military support}}
In the early days of the rebellion rebel spokespeople emphasized that while they needed the no fly zone to stop Gadaffi attacking them with ground attack planes and helicopters they did not want to see any imperialist troops on the ground in Libya. They also welcomed the use of air strikes against concentrations of Gaddafi’s tanks and artillery. The rebel force itself was a poorly armed little more that an almost untrained hootch potch of volunteers using seized arms sprinkled with some army units that have defected. In particular given the huge distances and desert terrain of much of Libya such a force could not hope to advance against a modern army equipped with armor and artillery, one that could strike them down from a considerable distance.
Al Jazeera provided some excellent coverage from the front lines of the very mis matched battles that resulted when the rebels tried to take on Gaddafi’s forced without imperialist air support. But within a day of the air attacks starting the rebels were able to stop retreating and start the process of retaking the towns they had been driven out of — until Gaddafi’s forced adopted and switched from their heavy armour to civilian style vehicles that were hard for airpower to identify to target. The Guardian described how \emph{“it has become increasingly apparent that the real issue for the rebels is a lack of discipline, experience and tactics. Even where they have had the advantage, they have been outmanoeuvred in large part because there has been no plan for attack or defence. Instead, the young rebels, full of bravado, charge forward only to turn and flee when they come under fire, often conceding ground.”}
It is almost certainly no coincidence that the day the TNC released its US friendly program was the same day that US AC-130 gunships and A10 tankbusters were reported as being deployed for the first time (Tue 29 March). This was also significant because these airplanes are close air support weapons that require close co-operation with the forces on the growing, this suggested that US Special Forces had been deployed for such purposes.
Within a couple of days of this deployment being revealed the New York Times was able to report that American officials had admitted that \emph{“small groups of C.I.A. operatives have been working in Libya for several weeks as part of a shadow force of Westerners that the Obama administration hopes can help bleed Colonel Qaddafi’s military”} and that \emph{“former British officials said that dozens of British special forces and MI6 intelligence officers are working inside Libya. The British operatives have been directing airstrikes by British jets and gathering intelligence about the whereabouts of Libyan government tank columns, artillery pieces and missile installations.”} This activity according to the same report was also being used for \emph{“meeting with rebels to try to fill in gaps in understanding who their leaders are and the allegiances of the groups opposed to Colonel Qaddafi.”} All the same according to The Economist \emph{“Admiral James Stavridis, NATO’s supreme allied commander for Europe, recently told American senators that “flickers” of intelligence suggested the presence of al-Qaeda and Hizbullah, the Lebanese Shia guerrilla group, among Libya’s opposition. But he also said that its leadership appeared to be “responsible men and women.”}
The withdrawal of US military forces that began 3 April demonstrated just how unsure the US continued to be about the rebels and the long terms effects of a rebel victory. But Al Jazeera report at that time also revealed that \emph{“some accounts describe a growing US presence on the ground. In the east, which is largely free from the regime’s control, media reports have said that American and British clandestine intelligence officers are meeting and training rebel fighters.”} The air strikes offered the US little opportunity to study and influence the rebels, these training programs allowed both. Journalists reporting from the advance on Tripoli reported on not only the presence of British Special Forces but also other forces.
\forcelinebreak
\emph{“The “others” in question are the small groups of former special forces operatives, many with British accents, working for private security firms who have been seen regularly by reporters in the vanguard of the rebels’ haphazard journey from Benghazi towards Tripoli. These small detachments of Caucasian males, equipped with sunglasses, 4x4 vehicles and locally acquired weaponry, do not welcome prying eyes, not least because their presence threatened to give credence to the Gaddafi regime’s claims that the rebel assault was being directed by Western fifth-columnists.”}
\section{\textbf{Military realities}}
At an extreme this appearance has resulted in some on the left (including Irish republicans) publicly joining the calls from Nicaraguan President Daniel Ortega, Cuban political leader Fidel Castro and Venezuelan President Hugo Chávez in expressing support for Gaddafi. Fidel really went off the deep end writing that the intervention is worse that the fascist intervention in the Spanish Civil War! \emph{“Not even the fascist leaders of Germany and Italy were so blatantly shameless regarding the Spanish Civil War unleashed in 1936, an event that maybe a lot of people have been recalling over these past days. Almost 75 years to the day have passed since then, but nothing that has happened over the last 75 centuries, or even 75 millenniums of human life on our planet can compare.”} Of course for Fidel the prospect of a democratic revolt being used as the excuse for US military intervention is terrifying in itself, leaving aside his historical connections with Gaddafi.
\section{\textbf{Irish Republicanism \& Gaddafi}}
I’m writing from Ireland so as a starting point in understanding why some on the nationalist left have taken this position I’m also going to look at why Gaddafi had a layer of support amongst physical force nationalists in this country. This is not just a question for anarchists in Ireland as elsewhere in the colonial and post colonial world there would be similar attitudes.
When the Libyan insurrection against the Gaddafi regime started most of the Irish republican organisations were silent on it, as were their members on Facebook and similar social media. It was only when the imperialist forces, in particular Britain, started flying bombing missions against the Gaddafi regime that statements started to appear. These tended to focus on the hypocrisy of the imperialist powers and had little to say about the democratic rising against Gaddafi itself. But some of what has been written goes beyond this and takes the side of Gaddafi against the revolution. With the left internationally a number of organisations and high profile leaders like Chavez and Castro were seen to line up behind Gaddafi in his attempts to crush the rebellion.
The NATO intervention meant that Gaddafi was once again able to pose as an anti-imperialist. Once again because he had been a good friend to Irish republicanism in the 1980’s, sending at least four ship loads of modern weapons to the Provisional IRA. This included almost all the supplies of the military explosive Semtex that enabled a very much more effective bombing campaign in Britain. Some considered that campaign key to forcing the British state to engage with the Irish ‘Peace Process’. Even those nationalists that later broke from Sinn Fein often retained a loyality of sorts towards Gaddafi. To them the arms he supplied offered a counter strategy to the ‘Peace Process’ based on launching a ‘mini-Tet’ offensive which they hoped would have a similar political effect in Britain as the Tet offensive had in the US.
In an example with particular relevance to the WSM éirígí whom we work alongside in the 1\% Network and other struggles as a first reaction to the rebellion republished on 20 March a long rambling speech Fidel Castro which included phrases like \emph{“Even Gaddafi’s adversaries assure us that he stood out for his intelligence as a student;” and “the latent Libyan rebellion being promoted by Yankee intelligence.”} An actual statement from éirígí on March 21\textsuperscript{st} headed ‘Attack on Libya – Another War for Oil’ managed to avoid even mentioning the democratic revolution in Libya outside of the neutral \emph{“éirígí supports the right of the people of Libya to determine their own future without interference from outside powers.”}
Back in the 90’s Gaddafi was also funding groups on the British and Irish left. Gaddafi the anti-imperialist of the 1980’s whose house had been blown up one night by American jets with British aid and who had access to both substantial funds and weaponry was for obvious reasons an attractive potential ally. That would be some residual feelings of loyalty towards him is not surprising.
There would be a logic to this position if all we were witnessing in Libya was simply a civil war between two ruling factions. Why take sides in such a fight? But while the situation is complex it is clear what we are seeing is a republican democratic revolution that started with mass street protests and which only later saw the defection of significant sections of the ruling elite. The process was inspired by the democratic revolutions in Egypt and Tunisia but followed a different pattern because when faced with mass protest Gaddafi was willing to do what Mubarak and Ben Ali were or could not.
Unarmed protesters in the streets can and did fight riot police and lightly armed police and mobs with whatever weapons could be improvised in Egypt and Tunisia. Hundreds of them died doing so but they called the bluff of the regimes which fell. Gaddafi was not bluffing and you cannot defeat automatic weapons with numbers alone. Is it really the case that the left should respond to the use of massacre by backing those willing to go that far. In particular given Ireland’s recent history, Bloody Sunday in particular, does it make sense to simply maintain silence when faced with a dictator willing to use far worse measures against his own people?
\section{\textbf{The rebel need for military support}}
From a military perspective it is almost inevitable that the rebels would welcome air attacks aimed at preventing Gaddafi using the massive military advantage that air power and heavy weapons gives his forces. This is not the 1930’s, the last time when you might have believed that a rag tag army without air support could take on a modern one in a war and win. And even in the 30’s superior military equipments, training and supplies played a massive role in ensuring Franco’s victory over the republicans in Spain. From that perspective it would have been suicidal for the rebels (and the civilian cities they held) not to demand such strikes.
Al Jazeera reported that \emph{“Benghazi residents say they’re sure the coalition air strikes saved them from a massacre. “People were on edge all day (before the strikes), like not even able to smile, being absolutely sick to our stomachs,”} said Kadura, the American who returned to Benghazi, where his large extended family of aunts, uncles and cousins lives. \emph{“I don’t think we would have stood a chance. or weeks prior to the rapid diplomatic push for a resolution at the Security Council, Libyans had been pleading for international help. Their call for foreign air strikes was loud, unqualified and came from a broad swath of society. English teachers, bankers, and Islamists who had supported the insurgency in Iraq all said they would welcome US attacks on Gaddafi’s forces.”} This arises not from a lack of awareness of the role of imperialism but despite it, the the same article Kadura is quoted as saying \emph{“I’m not gonna hold the French flag and kiss it because I do know that these countries act in their own interest. But what is the alternative?”}
Even the most thoughtful of those taking the conventional Leninist line don’t seem to have any answer to what the alternative might have been. Tariq Ali was reduced to wishful thinking, writing \emph{“We will now never know how long Gaddafi’s crumbling and weakened army would have held together in the face of strong opposition. The reason he lost support within his armed forces was precisely because he ordered them to shoot their own people.”} Well yes, but that could only go so far, once he had organised the majority of the forces that had remained loyal they were rapidly rolling over rebel controlled cities and when the air strikes arrived heavy armor units had already reached the outskirts of Benghazi.
The nature of modern weaponry is such that even a tiny handful of loyal military personnel can slaughter thousands in minutes. Revolution in the age of the helicopter gunship is no longer a question of huge numbers being able to face down military units. The wishful thinking approach was taken up by the Socialist Workers Party, the SWP slogan that the \emph{‘Libyan people can do it on their own’} was incorrect to the point of being absurd. With Gaddafi going on TV to call on his followers to \emph{“cleanse Libya house by house” of “cockroaches\dots{} greasy rats and cats”} it would seem to be demanding rather a lot of the rebels that they refuse the military support that prevented us from finding out whether these threats were rhetorical or real.
All of which leaves the western left in a difficult position. We don’t want to see future imperialist intervention gain legitimacy because of the needs of the Libyan rebels for air support if they were to have any hope of winning. We know that support for the intervention in Libya will translate into greater public support for intervention in general.
\section{\textbf{It wasn’t a Humanitarian intervention}}
The problem is that for a large number of leftist commentators they can’t acknowledge the rebel need for military aid without it seems having to also convince themselves that this means the imperialist motivations were humanitarian. It’s hard to understand the need for them to make such arguments. As we have seen a significant number of rebels interviewed don’t share such illusions and instead emphasise the limits they want placed on imperialist intervention precisely because they understand the imperialist armies have a nasty habit of staying around long after any limited welcome has worn off.
Whatever about the military side of the conflict politically and economically the imperialist intervention is intended not to follow the rebel agenda but that of the imperialist powers. There is no such thing as a free lunch and even if the rebels were publicly keeping up an opposition to further imperialist control in Libya in the period before the imposition of the no-fly zone we quickly saw significant concessions being given.
The imperialist powers will seek to install as compliant a regime as possible, one bargining method they used for a long period was to refuse to supply the rebels with weapons. This put them constantly in the position of being able to decide who wins the struggle by turning on and off air strikes at will and thus putting enormous pressure on the rebel forces to create a ruling council \& program that will be acceptable to the west. Those liberals who feel the need to pretend the intervention is actually being made in the interests of the people of Libya (or idiotically that somehow there would be no ‘collateral damage’) are as guilty of making as absurd an argument as those on the left who imagined the defeat of Gaddafi’s forces as being possible without imperialist military intervention.
Whatever the language of the UN declaration it is impossible to believe that the main interest of the imperialist powers lies with Libyan civilians. Early in the intervention the casual way the imperialist military powers regard such civilians was revealed when during the recovery of two US personnel from a crashed aircraft. Aj Jazeera reported that as the Search and Rescue mission came to pick them up local civilians approached to aid the two downed airmen. \emph{“Marines on board the Ospreys sprayed them with gunfire, injuring eight people. Hospital sources told British reporters who arrived on the scene the next day that one man might need his leg amputated.The injured included a young boy who local hospital officials say may lose his leg”.}
\emph{As many pointed out there was no western intervention in other Arab states like Bahrain where large numbers of civilian protesters were also being shot down. This made clear there was no absolute principle of protecting civilians. Even Lord Craig the head of the British armed forces during 1991 Gulf war has admitted that because of the different policy towards Bahrain “Once again we will face the accusation that oil-starved colonialists are up to their knavish tricks.”}
When the Egyptian revolt broke out the US restricted itself to calls for calm for many days, right to the point where it became obvious that a continued refusal of Mubarak to step down was going to lead to major revolt and instability. Just as in South America at that point the US switched from supporting the dictatorship to steering the direction of the democratic struggle away from any fundamental threat to US interests. Thus the last favour the dictator does as he heads off stage is to be seen to be finally pushed on his way by the same White House that up to then backed him.
\section{\textbf{Getting beyond strawman arguments}}
It would be wrong to pretend that all those who argued for or against intervention are making shrill arguments. On the anti-intervention side Noam Chomsky, in a well considered interview coming down against supporting the intervention because of the record and intentions of the western powers says; \emph{“In the case of intervention by the triumvirate of imperial powers that are currently violating UN 1973 in Libya, the burden is particularly heavy, given their horrifying records. Nonetheless, it would be too strong to hold that it can never be satisfied in principle — unless, of course, we regard nation-states in their current form as essentially holy. Preventing a likely massacre in Benghazi is no small matter, whatever one thinks of the motives.”} He doesn’t think Libya meets that test but he thinks it can be met pointing \emph{“In the post-World War II period, there are two cases of resort to force which — though not qualifying as humanitarian intervention — might legitimately be supported: India’s invasion of East Pakistan in 1971, and Vietnam’s invasion of Cambodia in December 1978, in both cases, ending massive atrocities.”} It has to be noted here that Chomsky is looking at whether the imperialist intervention is humanitarian rather than whether the rebels should have looked for imperialist military support.
On the pro-side Gilbert Achcar’s ‘Libya: a legitimate and necessary debate from an anti-imperialist perspective’ was a well argued critical look at the case for supporting the air intervention to prevent a massacre even though we know that it not the only reason why the imperialists are intervening.
\forcelinebreak
\emph{“Every general rule admits of exceptions. This includes the general rule that UN-authorized military interventions by imperialist powers are purely reactionary ones, and can never achieve a humanitarian or positive purpose. Just for the sake of argument: if we could turn back the wheel of history and go back to the period immediately preceding the Rwandan genocide, would we oppose an UN-authorized Western-led military intervention deployed in order to prevent it? Of course, many would say that the intervention by imperialist\Slash{}foreign forces risks making a lot of victims. But can anyone in their right mind believe that Western powers would have massacred between half a million and a million human beings in 100 days?”}
Achar’s article is weakened though by his idea that public opinion played a significant role in pushing the imperialists into intervening. Amusingly he pushes this the hardest around one of the rhetorical devices that the anti-interventionists are often heard to use, the idea of the US imposing a no-fly zone on Israel. \emph{“One can safely bet that the present intervention in Libya will prove most embarrassing for imperialist powers in the future. As those members of the US establishment who opposed their country’s intervention rightly warned, the next time Israel’s air force bombs one of its neighbours, whether Gaza or Lebanon, people will demand a no-fly zone. I, for one, definitely will.”}
The very idea of the US doing such a thing because of public pressure seems so ludicrous that it exposes the weakness of the idea that ‘public opinion’ played a significant role in leading to the decision to intervene. But there are two other substantial pieces of evidence against this concept. Firstly that the massive displays of anti-war public opinion, particularly in Britain, had no visible impact on the decision to invade Iraq. And secondly that opinion polls actually reveal that there were initially only narrow majorities supporting the air strikes in Libya and that in both cases there were more opposed to further military involvement then for it. Quite certainly there was no overwhelming demand for intervention by their military that pressurised the imperialist powers into acting. By the start of April a Quinnipiac University survey \emph{“found that 47 per cent of registered voters now disapprove while 41 per cent support”} the US intervention.
\section{\textbf{The limits of Leninist anti-imperialism}}
The Libyan example in particualr has exposed the limit of conventional Leninist anti-imperialism, the convention that simply looks at such struggles in terms of what is bad for imperialism. This Leninist approach might have made some sense for the first 20 years of its existence (although it often relied on the theoretical slight of hand of insisting that Soviet Russia or China could not be considered imperialist). But any serious look as previous republican insurrections has to acknowledge that very often the revolutionaries used whatever support from imperialist powers that they could obtain.
Right back to the American Revolution it is probable that victory of that revolution was dependent on French intervention and in particular the French fleets imposition of a ‘no sail zone’ off the American coast that robbed the British forces of the easy mobility they had enjoyed against the American rebels. In the Irish context \textbf{every} republican insurrection looked to other imperialist powers as a counter weight to British imperialism. Before the 20\textsuperscript{th} century this was France, in the twentieth century it was Germany (referred to in the 1916 proclamation as ‘our gallant allies in Europe’) and in the 1940’s it was even Nazi Germany.
Is the decision of the Libyan rebels to demand imperialist air support really qualitatively worse than looking to Hitler for support, as the IRA did in the 1940’s? Apart perhaps from Fidel few could seriously answer yes to this question. On the other side of that equation can we really fault those Spanish anarchist exiles who joined the Free French army to fight fascism and to steal weapons to conduct an armed offensive after the war against Franco’s Spain. Their faith contains a strong warning of the problem of such alliances, after the war not only did the Allies leave Franco in power but they handed over lists of who had fought with them and was now suspected of seeking his overthrowal.
What became conventional leftist anti-imperialism really arose to serve the needs of Leninist Russia to shatter as far as possible the power of the other imperialist powers they faced. Within this the struggle for freedom of many republican insurrections were seen as secondary to the needs of the USSR. In republican Spain in the 1930’s this meant suppressing the revolutionary movement, in particular the anarchists, within the republican zone because this was a period where the Soviet Union sought an alliance with some imperial powers against others. At the end of World War Two it meant Moscow ordering the communist partisan units in countries like Greece and Italy that Stalin had signed over to the west at Yalta to suppress the demand for revolution. In Greece this involved the execution of communist cadres that refused these orders. In Yugoslavia the partisans were strong and independent enough to resist these orders and take power under Tito.
\section{\textbf{Some anarchist perspectives}}
Soon after the start of the air war there were a number of contributions to the debate on Libya from an anarchist perspective. I’m going to look at three made at the start of the intervention from material in English on Anarkismo.net.
The first is important as it is a translation from the blog of a Libyan anarchist Saoud Salem who on the eve of the intervention argued that \emph{“this intervention that will transform Libya into a real hell, even more than now. That intervention will also steal the revolution from the Libyans, a revolution that has cost them thousands of dead women and men so far. An intervention that will also divide the Libyan resistance..To be liberated from Qaddafi just to become slaves to those who armed him and empowered him during all those years of authoritarian violence and repression.”}
Saoud Salem’s blog gives their location as being Libya and while there is no indication that this is any more than a single critical voice it still provides a useful antidote to the pro-intervention debate that insists that all anti-Gaddafi Libyan’s were calling for intervention. As importantly it makes clear what the cost of intervention for the Libyan revolution is, a cost as we have seen that is already been paid.
Italy was closely connected with the Gaddafi regime and now provides the airport bases for the intervention. A statement from the FdCA in Italy concludes \emph{“What we, as revolutionary activists, are interested in is the potential for revolt and self-organization being expressed by populations whose demands are no longer set by clericalists or fundamentalists, but are instead concentrated on basic rights and the re-distribution of wealth\dots{} This is why we support the popular committees and our comrades who, at the cost of their own lives and liberty, fight in the streets and squares of Benghazi, Syria, Bahrain, Saudi Arabia, throughout the Middle East and North Africa. This is why we are firmly opposed to the war and the military intervention whose inevitable result is devastation and misery in Libya, and to the cruel repression being carried out in other countries in their attempt to normalize the situation there.“}
Like the better anti-intervention statements this statement tries to both support the democratic revolution and oppose the imperialist intervention but while this is a strong political position for the reasons outlined above it is a position that is ludicrous from a military point of view. How are we to solve this particular contradiction between the formally correct political position of opposing all intervention and the military reality of a democratic revolution that would have been snuffed out, with perhaps considerable loss of life without such intervention? In his reply to Gilbert Achcar Alex Collinicos of the British SWP essentially just shrugged his shoulders and said ‘shit happens.’ His reply to Achcar concludes \emph{“The sad fact is that massacres are a chronic feature of capitalism. The revolutionary left is, alas, too weak to stop them. Until we become stronger, we can at least offer political clarity about what’s at stake.”} This at least is considerably more honest then the Irish SWP’s assertion that the Libyian people can win on their own but while its an answer that may go down well in London it wouldn’t sound so convincing when Gaddaffi’s tanks were on the outskirts Benghazi.
A final point worth looking at is the interview on Anarkismo.net with Nejat Firat Zeyneloglu, a Kurdish libertarian based in Turkey. Nejat made the following observation on the polar debate the old style leninist and liberal left are engaged in. \emph{“Defending dictatorships or defending the imperialist intervention against dictators, are basically the same thing; it means to reject or to ignore the will of the masses of the people who fight for their liberty by themselves. I’d like to point out that there is on both arguments, distrust of the people, the masses, and their struggle. For the imperialist countries, naturally, the whole issue is to provide a so-called “stability”, because their interests depend on the “stability”. So, generally, as long as their benefits are protected, they do not care who the dominant power is; fascists, social democrats, conservatives, greens and so on.
\forcelinebreak
Remember that as far as a month ago, Sarkozy, Berlusconi, Erdogan and others were best pals with Gaddaffi. Because all of them have investments in Libya, and as you know, investment is more important than people’s life for capitalism.
\forcelinebreak
The imperialist countries are more worried about the Libyan people rather than Gaddaffi. Therefore, the aim of this war is to establish and to guarantee a new structure in Libya that is for the benefit of the imperialist countries. I think that we must support the struggle of the Libyan people that is based on their own will. We must support all kind of practice of direct democracy and self-management against any kind of oppression or authority. We have to acknowledge that the Libyan people have the right to self-determination, and we have to side with people, not Gaddaffi or the imperialists.”}
Within this is the seed of a different approach. Which is that while retaining the right to advise and criticise we should start off with a defense of the popular movement and an acceptance that the decision about how to balance political opposition to imperialism with the military need for imperialist intervention is theirs and theirs alone to make. In any case it is not as if the imperialist powers themselves are going to pay attention to what the miniscule groups of anarchists, leninists or other revolutionaries have to say anyway. It makes some sense for Castro or Chazez to come out with grand policy statements on what imperialists should not be allowed to do, Cuba or Venezula at least have a vote at the UN.
Leninist and nationalist parties imagine that one day they will have state power, that they will get to call the shots. So again from that perspective approaching these questions on the basis of imagine state policy has some credibility. For anarchists however we never expect to be in that position so why issue statements as if we were.
\section{\textbf{Cold hard realities}}
Historically democratic, anti-colonial and republican movements have always sought external support in order to achieve their goals. Revolutions require money and guns and for the proletarian element within such movements both are likely to be in short supply. This in itself means internally the proletarian element will often be forced into an alliance with the more national democratic end of the business class with one providing the resources and the other the number for a successful rebellion. The historical experience of such alliances is quite negative, whether it is with the domestic business class or a foreign power. Certainly this puts enormous limitations on what can be openly fought for, when the Spanish stalinists argued that the anarchist revolution was liable to reduce the liklihood of France or Britain lifting the arms embargo during the Civil War there was a logic to their argument. But it was one the anarchists rightly rejected for reasons that are outlined elsewhere.
To an extent I think both Chomsky and Achcar have the right argument, even if they disagree over how it applies in the case of Libya. There can not be some sort of absolute principle that insists not one crumb of aid can ever be sought from an imperialist power. Libya was not Iraq in 2003 which saw no popular insurrection or even Afghanistan in 2001 where the US supported ethnic warlords against a common enemy.
If the Leninists were in any way honest they would recognise that in practise if not in theory that was their historical approach. Not only did Lenin accept aid from imperialist Germany during the Russian revolution but it is only by the slight of hand of pretending that the Soviet Union (or China) was not imperialist that Leninists are unable to see the reliance of Cuba, Vietnam, Korea, Nicaragua etc on imperialist aid.
An anarchist approach to these questions needs to have a number of components
\forcelinebreak
1. An absolute political opposition to imperialism itself in either its military or economic forms and a rejection of the concept of humanitarian intervention from above.
\forcelinebreak
2. Defense of democratic republican movements in general
\forcelinebreak
3. Promotion and support for libertarian tendencies \& currents within such movements
\forcelinebreak
4. An acceptance that the question of how much military support it is permissable for those in struggle to accept from imperialist powers is not an absolute but rather dependent on the nature of those movements and what they are sacrificing for such support. And at the end of the day while we may advise and critique it is the movements themselves that will make these judgement calls
Such a position may be more complex, less based on ideological rigidity and more based on case by case judgement calls but it also reflects the actual rather than imagined history of the anarchist, the left and both anti-colonial and pro-democracy movements. As in other areas we cannot suspend activity until the perfect movement spontaneously emerges from the depths, rather we have to struggle with the movements that exist. And such a struggle is unlikely to have influence if it is purely an exercise in ideological restatement that if forced to ignore the realities on the ground.
\section{\textbf{The future for the Libyan people}}
At this point in time who can predict which way a post Gaddafi Libya will go. Clearly the rebels are not a single body and the TNC’s claim to represent all is already being challenged. Will the neo-liberal direction sketched out in the program released at the time of the London conference hold. Will the new found respect for the US cited by the Islamist’s crumble when US policy returns to business as usual. Will the TNC disintegrate in a fresh civil war? Will the masses succeed in breaking through the limits both these factions would impose and start to develop the organisation to create a Libya that is truly free?
What we can be certain of seeing is a process under which the new ruling class attempts to put the lid back on working class confidence that will have risen during the rebellion. They will attempt to limit discussion of what sort of ‘Free Libya’ was fought for and to limit the right of workers to organise in unions and other class bodies just as we have seen happen in Egypt. In the short term the challenge will be for the working class to fight for its interests in the New Libya rather than see their interests subsumed in the name of national unity whose only outcome will be the recreation of a Libya safe for the oil corporations.
The imperialists will clearly favour the TNC and the rapid imposition of a ‘business as usual’ stability like that they are imposing in Tunisia and Egypt. In that respect perhaps the greatest real hope is that victory in Libya links the revolts to either side in Tunisia and Egypt and adds new hope to those ongoing revolutions. If all three are to be succesful at some point they will have to deal with hostile imperalist intervention themselves, regardless of the alliances of necessity made in the course of this battle.
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The Anarchist Library
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Anti-Copyright
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Andrew Flood
As Gaddafi falls — Lessons from Libya
Imperialism, anti-imperialism \& democratic revolution
August 25, 2011
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Retrieved on 13\textsuperscript{th} August 2021 from \href{http://www.anarkismo.net/article/20392}{www.anarkismo.net}
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\textbf{theanarchistlibrary.org}
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\begin{document}
\begin{center}{{\tenbf EXTENDED TECHNICOLOR MODEL BUILDING}
\vglue 1.0cm
{\tenrm Thomas Appelquist and John Terning\footnote{Talk presented by
John Terning at the International Workshop for Electroweak Symmetry Breaking,
Nov. 12-15, 1991, Hiroshima.}\\}
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{\tenit Department of Physics,Yale University\\}
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{\tenit New Haven, CT 06511, U.S.A.\\}
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%{\tenit Group, Company, Address, City, State ZIP/Zone, Country\\}
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{\tenrm ABSTRACT}}
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\noindent
We discuss constraints on extended technicolor model building, and show how
to rule out classes of models. We also present some speculations on neutrino
masses and discuss a toy model of leptons.
\vglue 0.6cm}
{\elevenbf\noindent 1. Introduction}
\vglue 0.4cm
\baselineskip=14pt
\elevenrm
The recent resurgence of technicolor (TC) theories has produced a body of
work\cite{walking,strongETC,KMEN} that
demonstrates that there are no serious technical obstacles to describing
the observed particle mass spectrum (while avoiding flavor changing neutral
currents (FCNC's) and keeping $\rho$ close to 1) in terms of a TC model. That is,
given the ability to choose an arbitrary TC gauge group with an arbitrary number
of technifermions, and representing the extended technicolor (ETC) interactions
by four-Fermi interactions with arbitrary mass scales and arbitrary
couplings for each of the ordinary fermions one can produce the entire observed
range of fermion masses up to a few hundred GeV for the t quark
without any phenomenological disasters. Even the
${\tw S}$
parameter may be held at bay\cite{S}.
Though such an exercise is interesting as a sort of existence proof, one worries
that since there are more parameters than observables it is impossible to tell
whether this success is the result of having the qualitatively correct physics
or merely the power of parameter fitting.
To do better we have to be more
ambitious and construct models that explain the real world rather than just
describe it, i.e. models with fewer parameters than the Standard Model.
Such models should make testable predictions, and have the potential to be
ruled out by experiment. In principle an explicit ETC model would fit this bill,
since it would reduce all the parameters of the Higgs sector to just one
parameter for each ETC gauge group. In reality it is unlikely that given the
``right" ETC model we would be able to accurately analyze the non-perturbative
dynamics involved, so we might be forced to parameterize
at least some of our ignorance. Even so, we could still hope to have fewer
parameters than observables, and hence a testable model.
In this paper we will discuss some constraints on model building, show how to
rule out some classes of models, give some speculations on neutrino
masses and finish with a discussion of a toy model of leptons.
%\vglue 0.2cm
%\elevenit \noindent 1.2. Section Headings}
%\vglue 0.1cm
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{\elevenbf\noindent 2. Constraints on Model Building}
\vglue 0.4cm
There are several constraints that one might like to impose on a realistic
ETC model. First of all, we expect that there should be more than one ETC
scale. The absence of FCNC's (inferred from the $K-\overline{K}$
mass splitting) requires that the ETC scale associated with the the s quark
(that is the mass of the gauge boson that connects the s quark to
technifermions, and,
more importantly, the d) be at least 200 TeV. On the other hand, to obtain a
t quark mass of a few hundred GeV (without excessive fine tuning) we expect
an ETC scale of order 5 TeV or lower. Such arguments suggest that it is
natural to have
at least three different ETC scales, one for each family.
Another
constraint arises from trying to obtain a large t-b mass splitting, while
keeping $\rho$ within 0.5\% of 1. To do this without fine tuning it is
natural to require that the right-handed t and right-handed b be in different
representations of the ETC group ($SU(2)_L$ gauge invariance requires the
left-handed t and b to be in the same representation). Having t$_R$ and b$_R$
in different representations allows for the possibility that the t and b are
associated with different ETC scales (This scenario has been exploited by
King and Mannon and by Einhorn and Nash\cite{KMEN}).
A powerful constraint on ETC model building was originally
elucidated by Eichten and Lane\cite{EL},
who showed that the absence of a visible axion implies a limit on
the number of irreducible representations (irreps) of the ETC gauge group.
The number of exactly massless Nambu-Goldstone bosons is given by the number
of spontaneously broken global U(1)'s minus the number of global anomalies.
For
vector-like ETC theories of the form ETC${\twelverm\otimes SU(2)_L \otimes
U(1)}$ this implies that there
can be at most 3 conjugate pairs of irreps, which implies that quarks and
leptons
should not be in seperate irreps. Such considerations lead one to consider
grand unified technicolor theories (GUTT's) where all interactions
(including TC) are unified in a single gauge group. This is the simplest and,
in principle, most predictive type of ETC model.
We are allowed at most one irrep of the GUTT gauge group, again in order to
avoid axions\cite{EL}.
\vglue 0.5cm
{\elevenbf \noindent 3. Ruling Out Classes of Models}
\vglue 0.4cm
Combining the limit on the number of irreps discussed previously with the
requirement of three families, we can rule out classes of GUTT's. As we have
seen we are allowed only one irrep of the GUTT gauge group. In order to
protect ordinary fermions from gaining GUTT scale masses, this single irrep
must be complex. Having a complex irrep also allows us to imagine that the
GUTT gauge symmetry can be broken down with fermion bilinears, that is without
using any scalar fields. However our GUTT must be free of gauge anomalies, so
the search can be limited to $E_6$ and $O(4N+2)$ since these are the only
anomaly safe groups with complex representations. The {\bf 27} of $E_6$ and the
{\bf 64} of $O(14)$ are too small to contain three families, so the smallest
possible
GUTT is $O(18)$ with all fermions in the 256 dimensional spinor.
This type of
theory has been previously discussed\cite{Bag} as an ordinary GUT, and it may
be helpful to review that scenario. If we imagine $O(18)$ breaking to
$O(10)\otimes O(8)$ we find that the {\bf 256} decomposes as
$({\bf 16},{\bf 8}')+({\bf \overline{16}},{\bf 8}'')$. The {\bf 16}
of $O(10)$ contains one ordinary family (when $O(10)$ breaks to the Standard
Model in the usual fashion) with an extra right handed neutrino,
while the ${\tw {\bf \overline{16}}}$ is a family with left and right
interchanged.
We will refer to the ${\tw {\bf \overline{16}}}$ as a right-family and the
{\bf 16} as a left-family. At this
stage the 8 left-families and the 8 right-families are protected from
getting masses by gauge symmetries.
However, if the $O(8)$ gauge group breaks entirely, then nothing prevents the
left-families from getting masses with the right-families, and there will be no
light families left over to make up the observed three families.
We
can now see what must happen in a GUTT scenario. If the $O(8)$ breaks
down to a TC gauge group such that more of the right-families are technicolored
than the left-families, then there will be some excess left-families which
are protected from getting masses until $SU(2)_L$ is broken. Thus we can
proceed straightforwardly by simply listing all the possible TC groups,
embedding them (along with the Standard Model) in all possible ways
in $O(18)$ and then counting the
number of extra techni-singlet left-families\footnote{The tables of ref.
\citenum{group} are invaluable in this analysis.}.
(This family counting proceedure
was originally suggested by Georgi\cite{Georgi} for ordinary GUT's.)
In fact we can imagine breaking patterns that do not automatically generate a
family structure; in these cases we can simply count quarks and antiquarks.
It should be noted that this procedure does not rely on knowing the actual pattern of
symmetry breaking between the GUTT scale and the TC scale; different patterns
of breaking can result in the same unbroken TC group. The
result of this counting procedure is that there is an essentially
unique TC group that produces three
families from the $O(18)$ GUTT: $Sp(4)_{TC}$. (There exists an embedding of
$SU(2)$ into ${\tw Sp(4)}$ where ${\tw {\bf 4}\rightarrow {\bf 4}}$, and
${\tw {\bf 5}\rightarrow {\bf 5}}$, but
all of our further considerations will apply to this degenerate case as well.)
This pattern of breaking was originally discovered by Gell-Mann, Ramond, and
Slansky\cite{GMRS} and by Farhi and Susskind\cite{FS}. What we have shown is
that there are no other possibilities for $O(18)$.
In the $Sp(4)_{TC}$ scenario
the ${\bf 8}' \rightarrow {\bf 5}\oplus{\bf 1}\oplus{\bf 1}\oplus{\bf 1}$
and the
${\bf 8}'' \rightarrow {\bf 4}\oplus{\bf \overline{4}}$.
Since
all of the right-families are technicolored, we expect no super-heavy fermions,
that is, except for right-handed Majorana neutrinos, the largest fermion masses
are of the order of hundreds of GeV.
Above 1 TeV, there is the equivalent of 16 families contributing to beta
functions, so asymptotic freedom for QCD and $Sp(4)_{TC}$ is lost ($ \alpha_s$ will blow up
around 35 TeV). For this theory to make sense, it may be
necessary to rely on some non-perturbative dynamics, but computations then
become difficult. (For an attempt along these lines, assuming the the existence
of nontrivial ultraviolet fixed points, see ref.~\citenum{Raby}.)
One could also consider larger GUTT's like $O(22)$. It is easy to
find TC groups which, when embedded in ${\tw O(22)}$, give four families,
but a systematic search for three family models has not been made.
Since the smallest GUTT models seem beyond our computational abilities, it is
natural to consider the more general possibility of partial unification, that
is where color is unified with TC so that the gauge group is of the form
$SU(N)_{ETC}\otimes SU(2)_L \otimes U(1)$. We will consider a fermion content of the
form $({\bf R_1},{\bf 2}) \oplus ({\bf R_2},{\bf 1}) \oplus
({\bf R_3},{\bf 1})$. We will also assume that
$SU(3)_C$ is embedded in $SU(N)_{ETC}$ is the simplest possible way (i.e. the
{\bf N} decomposes to a {\bf 3} and ${\tw N-3}$ color singlets) and that
${\bf R_1}$, ${\bf R_2}$, and ${\bf R_3}$
are antisymmetric irreps. These assumptions ensure that the model will only
contain {\bf 3}'s and ${\bf \overline{3}}$'s of color\cite{Georgi}, that is,
we eschew quixes,
queights, etc. We also assume that ${\bf R_1}$, ${\bf R_2}$, and ${\bf R_3}$
are not all identical, while requiring that the $SU(N)_{ETC}$ gauge anomaly
vanishes.
Finally, by counting the number of quarks and antiquarks and
requiring three families we can
show that such models are ruled out for ${\tw N < 10}$.
Since such general group theoretical arguments do not get us very far towards
a realistic model, we will take a more phenomenological approach in the
remainder of the paper.
\vglue 0.5cm
{\elevenbf \noindent 4. Speculations on Neutrino Masses \hfil}
\vglue 0.4cm
The fact that only extremely light left-handed neutrinos are seen poses special
problems in
TC theories. Fortunately there is a simple explanation available in
the usual seesaw mechanism\cite{GMRS,models}. The idea is that right-handed
neutrinos
get large Majorana masses, so that the left-handed neutrinos end up with masses
given by a Dirac mass squared divided by the Majorana mass. Presumably, the
neutrino Dirac masses are of the same order as their charged leptonic partners'
masses. The remaining problem is to estimate the Majorana masses. One
possibility is that the Majorana masses are much larger than any ETC
scales, then
all the left-handed neutrinos are light, typically with a mass less than
$10^{-4}$ eV for $\nu_{e}$. This assumes that although there are right-handed
techni-neutrinos with ETC interactions, the right-handed neutrinos have no
ETC interactions or are exclusively in real representations of the ETC group.
A more interesting possibility\cite{models} is that the
Majorana masses are of the same order as some ETC scale (or scales). This
is natural if condensates of bilinears of right-handed neutrinos (Majorana
condensates) are involved
in the dynamical breaking of the ETC gauge symmetry. First we imagine a
hierarchy of ETC scales (1000 TeV, 200 TeV, 25 TeV) in order to naturally arrange for a
hierarchy of charged lepton masses ($m_e$, $m_\mu$, $m_\tau$). The 25 TeV
scale is a natural scale for producing the mass of the $\tau$, and may also
produce the mass of the b when color enhancement effects are included\cite{color}.
Now we can estimate the
masses of the assocated neutrinos in such a scheme:
\begin{equation}
m_{\nu_{e}} \approx {(0.5 {\rm MeV})^2 \over {1000 {\rm TeV}}} \approx
3 \times 10^{-4} {\rm eV}
\nonumber
\end{equation}
\begin{equation}
m_{\nu_{\mu}} \approx {(100 {\rm MeV})^2 \over {200 {\rm TeV}}} \approx
50 {\rm eV}
\nonumber
\end{equation}
\begin{equation}
m_{\nu_{\tau}} \approx {(1 {\rm GeV})^2 \over {25 {\rm TeV}}} \approx
40 {\rm keV} .
\end{equation}
From these estimates we see three interesting mass scales arising. The
$\nu_{e}$ mass is in the right range to impliment the MSW mechanism, provided
that there exists another light neutrino for $\nu_e$ to mix with.
A $\nu_{\mu}$ with a mass of the order of 10 eV is interesting as
a dark matter candidate. The $\nu_{\tau}$ mass is also in
an interesting mass range\cite{Simpson}. However if such a heavy Majorana $\nu_{\tau}$
has a substantial mixing with the $\nu_e$ then another heavy
neutrino\cite{Nelson} is required in order to stay below the experimental
limit on
neutrinoless double $\beta$ decay.
For example, if a 17 keV $\nu_{\tau}$ has
a 1\% mixing with the $\nu_e$, and we use the $\nu_\mu$ as the second heavy
neutrino, then the $\nu_\mu$ must have either a 17 keV mass, or a mass in
the range 150 - 250 keV for consistency with a variety of
experiments\cite{Nelson}.
\vglue 0.5cm
{\elevenbf \noindent 5. A Toy Model \hfil}
\vglue 0.4cm
Finally, we will present a toy model of leptons
which produces some of
the neutrino masses discussed above. As seen in section 2, quarks and
leptons should come from the same irrep in order to avoid massless
visible axions. Here we postpone quark lepton unification to some scale
higher than the ETC scale, so we will be able to discuss leptons seperately.
The ETC gauge group in our model
is $SU(5)$, which is assumed to commute with the standard model gauge
interactions. $SU(5)$ was chosen so that we can break it down to the smallest
TC group, $SU(2)_{TC}$, and produce three families.
The fermion content of the model is:
\begin{equation}{\nu_\sigma}_L ~~
\left( \begin{array}{lr}
N & E \\
N & E \\
\nu_\tau & \tau \\
\nu_\mu & \mu \\
\nu_e & e
\end{array} \right) _L ~~
\left( \begin{array}{lcccr}
0 & \nu_e & n & n & N \\
- \nu_e & 0 & n & n & N \\
- n & - n & 0 & \nu_\sigma & \nu_\tau \\
- n & - n & - \nu_\sigma & 0 & \nu_\mu \\
- N & - N & - \nu_\tau & - \nu_\mu & 0
\end{array} \right) _R ~~
\left( \begin{array}{l} E \\ E \\ \tau \\ \mu \\ e
\end{array} \right) _R ~~.
\end{equation}
Note that two extra sets of right-handed techni-neutrinos (the $n$'s) and
an extra singlet neutrino (${\nu_\sigma}_R$) are needed to fill out the
{\bf 10}. In general we can expect that the breaking of gauge symmetries at a
higher unification scale may leave some sterile particles. Here
we assume that the ${\nu_\sigma}_R$ has a corresponding (but sterile)
${\nu_\sigma}_L$.
The most attractive channel for condensation in this model is for
the {\bf 10} to condense with itself, breaking $SU(5)$ down to $SU(4)$. Each of the
{\bf 5}'s splits into a {\bf 4} and a singlet; these new singlets being the leptons of the
first generation. The
{\bf 10} splits into a {\bf 4} and a {\bf 6}:
\begin{equation}
\left( \begin{array}{l}
N \\ N\\ \nu_\tau \\ \nu_\mu
\end{array} \right) _R ~~
\left( \begin{array}{lccr}
0 & \nu_e & n & n \\
- \nu_e & 0 & n & n \\
- n & - n & 0 & \nu_\sigma \\
- n & - n & - \nu_\sigma & 0
\end{array} \right) _R ~~.
\end{equation}
The fermions in the {\bf 6} obtain Majorana mixing masses of order the breaking
scale which we take to be 1000 TeV. Ignoring the heavy {\bf 6}, the $SU(4)$
gauge interactions are vector-like. We next imagine that $SU(4)$ is broken
down to $SU(3)$ by some as yet unspecified sector of the model (eg. quarks
and techniquarks), which splits off the second generation at an assumed scale
of 200 TeV. Finally, we assume that the attractive channel (but not the most
attractive) for ${\nu_\tau}_R$ condensation (${\bf3}\otimes{\bf3}\rightarrow
{\bf {\overline 3}}$) gives a Majorana mass of order
25 TeV (the assumed breaking scale) to ${\nu_\tau}_R$. This condensation
breaks $SU(3)$ down to $SU(2)_{TC}$.
We can now discuss the neutrino mass spectrum of our model.
The ${\nu_\sigma}_L$ can only
receive a mass which is suppressed by scales higher than 1000 TeV so
it is approximately massless. The ${\nu_e}_L$ mass is of order
$3 \times 10^{-4}$ eV as discussed before, so the ${\nu_e}_L$ and
${\nu_\sigma}_L$ are both in the appropriate mass range to allow the MSW
mechanism to operate. Also
as before, the ${\nu_\tau}_L$ mass is of order
40 keV. The case of the $\nu_\mu$ is more interesting.
We have assumed that there is no Majorana mass produced directly, but one
may feed down from the ${\nu_\tau}_R$ through ETC interactions. The model
contains ETC gauge bosons with masses of 200 TeV that connect ${\nu_\mu}_R$
with the ${\nu_\tau}_R$. The dynamical Majorana mass connects the
${\nu_\tau}_R$ with the conjugate ${\nu_\tau}_R$, and the same gauge boson
takes the ${\nu_\tau}_R$ back to a ${\nu_\mu}_R$. $SU(3)$ symmetry prevents
us from connecting the gauge boson lines, but when $SU(3)$ breaks to
$SU(2)_{TC}$
there will in general be some mixing of the order of the breaking scale squared.
Thus we can estimate the ${\nu_\mu}_R$ Majorana mass by one
factor of the condensate (which we approximate by $4 \pi (25 $TeV$)^3$)
a factor of the gauge boson mixing and four inverse powers
of the gauge boson mass:
\begin{equation}
M \approx 4 \pi {{(25 {\rm TeV})}^5 \over {(200 {\rm TeV})^4}} \approx
80 {\rm GeV} ~~.
\end{equation}
So we find an approximate mass for the left-handed $\nu_\mu$ given by:
\begin{equation}
m_{\nu_\mu} \approx {{(0.1 {\rm GeV})^2} \over {80 {\rm GeV}}} \approx
130 {\rm keV} ~~.
\end{equation}
Which is close to the region required for the absence of neutrinoless double
beta decay (with an assumed 1\% $\nu_\tau-\nu_e$ mixing). This model does
not contain a dark matter candidate, but
the possibility of the $\nu_\mu$ being heavier than
the $\nu_\tau$, which at first sight seems implausible, actually arises
naturally in the model.
\vglue 0.5cm
{\elevenbf \noindent 6. Conclusions \hfil}
\vglue 0.4cm
We have discussed how some classes of simple ETC models can be ruled out by
simply counting quarks. We have made some general speculations on neutrino
masses and also presented a
toy model of leptons that gives interesting neutrino masses. We have not
achieved the goal we set in the introduction, but
perhaps these observations will help in the task of constructing
a realistic ETC model.
\vglue 0.5cm
{\elevenbf \noindent 7. Acknowledgements \hfil}
\vglue 0.4cm
J.T. would like to thank S. Chivukula, E. Eichten, A. Nelson, M. Soldate, and
M. White for helpful discussions.
\vglue 0.5cm
{\elevenbf\noindent 8. References \hfil}
\vglue 0.4cm
\begin{thebibliography}{9}
\bibitem{walking}
B. Holdom, {\elevenit Phys. Lett.} {\elevenbf B150} (1985) 301;
T. Appelquist, D. Karabali, and L.C.R. Wijewarhanda, {\elevenit Phys. Rev.
Lett.} {\elevenbf 57} (1986) 957;
K. Yamawaki, M. Bando, and K. Matumoto,{\elevenit Phys. Rev.Lett.}
{\elevenbf 56} (1986) 1335
T. Appelquist and L.C.R. Wijewarhanda, {\elevenit Phys. Rev} {\elevenbf D35}
(1987) 774;
T. Appelquist and L.C.R. Wijewarhanda, {\elevenit Phys. Rev} {\elevenbf D36}
(1987) 568.
\bibitem{strongETC}
T. Appelquist, M. Einhorn, T. Takeuchi, and L.C.R.
Wijewardhana, {\em Phys. Lett.} {\bf 220B}, 223 (1989);
V.A. Miransky and K. Yamawaki, {\elevenit Mod. Phys. Lett.} {\elevenbf A4}
(1989) 129;
K. Matumoto {\elevenit Prog. Theor. Phys. Lett.} {\elevenbf 81} (1989) 277;
V.A. Miransky, M. Tanabashi, and K. Yamawaki, {\elevenit Phys. Lett.}
{\elevenbf B221} (1989) 177;
V.A. Miransky, M. Tanabashi, and K. Yamawaki, {\elevenit Mod. Phys. Lett.}
{\elevenbf A4} (1989) 1043.
\bibitem{KMEN}S. King and S. Mannan, U. of Southampton preprint {\elevenbf
SHEP 90/91-22}; M. Einhorn and D. Nash, to be published in
{\elevenit Nucl. Phys.} {\elevenbf B}.
\bibitem{S}E. Gates and J. Terning {\elevenit Phys. Rev. Lett.} {\elevenbf 67}
(1991) 1840;
R. Sundrum and S. Hsu, Lawrence Berkeley Lab. preprint {\elevenbf
LBL-31066};
see also the talks by T. Appelquist and T. Takeuchi in these proceedings.
\bibitem{EL} E. Eichten and K. Lane, {\elevenit Phys. Lett.} {\elevenbf B90}
(1980) 125.
\bibitem{group}R. Slansky, {\elevenit Phys. Rep.} {\elevenbf 79} (1981) 1.
\bibitem{Georgi} H. Georgi, {\elevenit Nucl. Phys.} {\elevenbf B156} (1979)
126.
\bibitem{GMRS} M. Gell-Mann, P. Ramond, and R. Slansky, in {\elevenit
Supergravity}, ed. P. van Nieuwenhuizen and D. Freedman (North-Holland,
Amsterdam, 1979).
\bibitem{FS} E. Farhi and L. Susskind, {\elevenit Phys. Rev.} {\elevenbf D20}
(1979) 3404.
\bibitem{Bag} J. Bagger and S. Dimopoulos, {\elevenit Nucl. Phys.} {\elevenbf
B244} (1984) 247; and references therein.
\bibitem{Raby}G. Guidice and S. Raby, Ohio State University preprint
{\elevenbf OSU-DOE-ER-01545-550}.
\bibitem{models}
B. Holdom, {\elevenit Phys. Rev.} {\elevenbf D23} (1981) 1637;
{\elevenit Phys. Lett.} {\elevenbf B246} (1990) 169;
U. of Nagoya preprint {\elevenbf DPNU-91-27}
to be published in the 1991 Nagoya Spring School Lectures.
\bibitem{color}B. Holdom, {\elevenit Phys. Rev. Lett.} {\elevenbf 60} (1988)
1233;
T. Appelquist and O. Shapira, {\elevenit Phys. Lett.} {\elevenbf B249} (1990)
327.
\bibitem{Simpson}J.J. Simpson, {\elevenit Phys. Rev. Lett.} {\elevenbf 54}
(1985) 1891;
J.J. Simpson and A. Hime, {\elevenit Phys. Rev.} {\elevenbf D39} (1989) 1825;
A. Hime and J.J. Simpson, {\elevenit Phys. Rev.} {\elevenbf D39} (1989) 1837;
A. Hime and N.A. Jelley, {\elevenit Phys. Lett.} {\elevenbf B257} (1991) 441.
\bibitem{Nelson}
M. Dugan et. al. {\elevenit Phys. Rev. Lett.} {\elevenbf 54} (1985) 2302;
M. Dugan, A. Manohar, and A. Nelson {\elevenit Phys. Rev. Lett.}
{\elevenbf 55} (1985) 170;
A. Manohar and A. Nelson, {\elevenit Phys. Rev. Lett.}
{\elevenbf 66} (1991) 2847;
L.M. Krauss, {\elevenit Phys. Lett.} {\elevenbf B263} (1991) 441.
\end{thebibliography}
%\vglue 0.5cm
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\title{Edistys ja ydinvoima}
\date{1979}
\author{Fredy Perlman}
\subtitle{Mantereen ja sen kansojen tuho}
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\emph{Seuraava teksti ilmestyi ensi kerran Fifth Estate -lehden ydinvoiman vastaisessa erikoisnumerossa 8. huhtikuuta 1979. Se oli kirjoitettu aiemmin samana vuonna, heti Pennsylvanian osavaltiossa sattuneen Three Mile Islandin ydinvoimalaonnettomuuden jälkeen. Kun uutinen onnettomuudesta levisi, viralliset viestimet inttivät että ”ei ole tarvetta ylireagointiin, tilanne on vakaa, johtajilla on kaikki hallinnassaan,” mutta lopulta voimalan lähellä asuneet ihmiset oli evakuoitava. Tässä Fredy muistuttaa meitä miten alueen alkuperäisiä asukkaita kusetettiin ja heidät tuhottiin käyttäen hyväksi latteuksia, lupauksia ja poliisia jotka aina seuraavat Pääoman mukana.}
\texttt{=}==
Ihmisolentojen, maaperän ja muiden elollisten lajien harkitun myrkyttämisen selittäminen ”onnettomuutena” on karkeaa teeskentelyä. Vain silmänsä tahallaan ummistava voi väittää että tämä Tekniikan Edistyksen seuraus oli ”ennalta-arvaamaton”.
Tämän mantereen elävien asukkaiden myrkyttäminen ja siirtäminen ”korkeampien olentojen” tähden on saattanut alkaa itäisessä Pennsylvaniassa, mutta ei vasta aivan viime viikkoina.
Reilu pari sataa vuotta sitten alueella jota Three Mile Islandin säteily parhaillaan myrkyttää, Franklinin, Morrisin, Washingtonin ja Halen kaltaiset keinottelijat piileskelivät Vandalia-yhtiön ja Ohio-yhtiön kaltaisten julkisivujen takana. Näillä yhtiöillä oli yksi ainoa tarkoitus: myydä maata voitolla. Yhtiöiden takana olevilla yksilöillä oli yksi ainoa tavoite: poistaa kaikki esteet rajoittamattoman voitonteon tieltä, olivatpa nämä esteet sitten ihmisolentoja tai vuosituhantisia kulttuureja tai metsiä tai eläimiä tai vaikkapa jokia ja vuoria. Heidän tavoitteensa oli Sivistää tämä manner, panna alulle sellaisen toiminnan kierre jota täällä ei koskaan aiemmin oltu harjoitettu: Työtä, Säästämistä, Sijoittamista, Myymistä – Pääoman uusintamisen ja laajentamisen kierre.
Suurin este tälle toiminnalle olivat vuosituhansien ajan tällä mantereella eläneet ihmiset, jotka ilman Lakia tai Hallitusta tai Kirkkoa nauttivat auringosta, joista, metsistä, kasvi- ja eläinlajien runsaudesta ja toisistaan. Nämä ihmiset pitivät elämää itsetarkoituksena, eivät ”korkeampien” tarkoitusten palveluun valjastettavana välineenä. He eivät rynnänneet Sivilisaatioon kuin lapset keksipurkille, toisin kuin Franklinit ja Washingtonit odottivat. Päin vastoin. He halusivat varsin vähän siitä mitä Sivilisaatiolla oli tarjota. He halusivat joitain sen aseista, ja niitäkin vain jotta voisivat säilyttää vapautensa Sivilisaation syvempää tunkeutumista vastaan; he valitsivat mieluummin kuoleman kuin Työnteoksi, Säästämiseksi, Sijoittamiseksi ja Myymiseksi köyhdytetyn elämän. Viimeisessä epätoivoisessa yrityksessään ajaa Sivilisaatio ja sen Hyödyt mereen ja sen taakse, kapinassa joka nykyään muistetaan automerkin nimestä, heidän soturinsa karkottivat maan anastajat ja heidän sotilaansa Ontariosta, Michiganista, Ohiosta ja länsi-Pennsylvaniasta. Tämän peräänantamattoman vastarinnan johdosta sivistyneet nimesivät heidät Raakalaisiksi. Tämä nimi antoi Sivistäjille luvan tuhota empimättä ja ilman tunnontuskia: ”Lähettäkää niille isorokolla tartutettuja huopia,” määräsi eräs hävittämisestä vastanneista komentajista.
Amerikan itsenäisyyden kaksisatavuotisjuhlaa vietettiin vähän aikaa sitten. Tuolloin muisteltiin sitä päivää jolloin maan anastajat, keinottelijat ja heidän liittolaisensa päättivät kiihdyttää riippumattoman elämän hävittämistä Three Mile Islandista länteen olevalla alueella. Kuninkaan hallitus oli liian kaukana suojellakseen sijoituksia riittävästi, ja joka tapauksessa se oli feodaalinen eikä aina jakanut keinottelijoiden päämääriä; se meni jopa niin pitkälle että valvoi rajoja jotka oli perustettu sopimuksilla Raakalaisten kanssa. Tarvittiin tehokas koneisto joka olisi maan anastajien suorassa hallinnassa ja omistettu ainoastaan heidän yritystensä hyvinvoinnin turvaamiselle. Paxton Boysin tapaiset rajaseutujen epämuodolliset poliisijärjestöt olivat riittävän tehokkaita murhaamaan Conestogan kaltaisen eristyneen kylän heimoväkeä. Sellaiset rajaseudun joukkiot olivat kuitenkin pieniä ja väliaikaisia, ja ne olivat riippuvaisia jokaisen mukana olevan jatkuvasta suostumuksesta samalla tavalla kuin heimosoturitkin; siksi ne eivät olleet mitään oikeita poliisiorganisaatioita. Keinottelijat liittoutuivat idealistien ja haaveilijoiden kanssa, poliittisen julisteen takana johon oli painettu Vapaus, Itsenäisyys ja Onnellisuus, ja sieltä he ottivat hallitus-, sotilas- ja poliisivallan omiin käsiinsä.
Puolitoistasataa vuotta sitten Pääoman kehittymisen tehokoneisto kävi täysillä. Sotilas- ja poliisiorganisaatiot, jotka perustuivat tottelevaisuuteen ja alistumiseen eivätkä kenenkään jatkuvaan suostumukseen, olivat valmiit ryhtymään toimintaan niitä vastaan jotka olivat vastustaneet juuri sen kaltaista kurinalaisuutta ainakin kahdenkymmenentuhannen vuoden ajan. Kongressi Sääti erään suorasukaisimmista laeistaan: Intiaanien poistolain. Muutamassa vuodessa kaikki vastarinta, kaikki toiminta mikä ei ollut Pääoman toimintaa, oli poistettu alueelta joka ulottuu Three Mile Islandilta Mississippille ja siitä etelään, Michiganista Georgiaan. Hallitus, josta tuli nopeasti yksi maailman mahtavimmista, ei enää rajoittanut toimiaan isorokolla myrkyttämiseen tai yllättäviin kyläläisten joukkomurhiin; se pani Poiston toimeen tarkkaan harkitulla latteuksien, lupausten ja poliisin yhdistelmällä. Vielä jäljellä olleet vapaat heimot eivät voineet vastustaa tätä yhdistelmää omaksumatta sitä itse, mutta he eivät voineet omaksua sitä luopumatta vapaudestaan. He päättivät pysyä vapaina, ja niinpä viimeiset vapaat ihmisolennot Three Mile Islandin ja Missisippin välillä Poistettiin.
Kun uudisasukkaat muuttivat tyhjennetyille maille, jossa heidän hengittämänsä ilma antoi heille makua äskettäin poistetusta vapaudesta, he muuttivat laajat metsämaat suurennetuiksi kopioiksi siitä helvetistä jonka he olivat jättäneet taakseen. Poluista ja metsistä nauttiminen loppui: metsät poltettiin; poluista tuli esteratoja joiden poikki piti kulkea niin nopeasti kuin Pääoma vain suinkin mahdollisti. Riemu lakkasi olemasta elämän tarkoitus; elämästä itsestään tuli pelkkä väline; sen tarkoitus oli taloudellinen voitto. Satojen erilaisten kulttuurien kirjo pelkistettiin erikoisen rutiinin yhdenmukaisuuteen: tee työtä, säästä, sijoita, myy, joka päivä aamunkoitosta illan kajoon, ja laske rahat auringon laskettua. Joka ikinen aiempi toiminnan muoto ja lukuisia uusia muutettiin riemun lähteistä taloudellisen voiton lähteiksi. Maissi, pavut ja kesäkurpitsa, ”kolme siskoa” joita alueen aiemmat asukkaat kunnioittivat ja rakastivat, muuttuivat pelkäksi ruokamarkkinoiden kauppatavaraksi; niiden kylväjät ja niittäjät eivät enää kasvattaneet niitä nauttiakseen aterioista, pidoista ja juhlista, vaan myydäkseen niitä voitolla. Maanviljelyn uuras työ syrjäytti rattoisan puutarhuroinnin, kiskot syrjäyttivät polut, jättimäisten hiiltä polttavien, pyörillä kulkevien tulisten pätsien liikkuminen korvasi kävelemisen, kanoottien tilalle tulivat kelluvat kaupungit joita ei pysäyttänyt mikään este ja jotka täyttivät ilman palavalla hiilloksella ja mustalla savulla. ”Kolme siskoa” alennettiin muun perheensä tavoin pelkäksi kauppatavaraksi, samoin kuin puut joista tuli sahatavaraa, eläimet joista tuli lihaa, ja jopa mantereen uusien asukkaiden matkat, laulut, myytit ja tarinat.
Ja uusia asukkaitahan oli: ensin satoja, sitten tuhansia, lopulta miljoonia. Kun varsinaisten orjien maahantuonti lopulta päättyi, alettiin tuoda ylijäämä-pienviljelijöitä feodalismin jälkeisen Euroopan alas ajetuilta kartanoilta. Heidän esivanhempansa eivät olleet tunteneet vapautta niin moneen sukupolveen että koko sen muisto oli kadonnut. Tulokkaat, jotka olivat aiemmin olleet palvelijoita tai maatyöläisiä nyt kaupankäynnistä innostuneiden isäntiensä kartanoissa, olivat maahan saapuessaan oppineet jo valmiiksi haluamaan täsmälleen sitä mitä Pääomalla oli tarjota, ja Pääoman kurjistama elämä oli heille vapautta verrattuna heidän ainoaan kokemukseensa. Maasijoittajat myivät heille maatilkkuja, rautatiesijoittajat kuljettivat heidät tilkuilleen, maataloustavarasijoittajat varustivat heidät, pankkisijoittajat rahoittivat, samat tahot toimittivat heille kalusteet ja vaatteet, ja usein samat firmat olivat järjestäneet heille kaiken muunkin sellaisella voitolla jota mikään aiempi aikakausi ei olisi pitänyt ”oikeudenmukaisena”. Ja sitten he kirjoittivat kerskaillen sukulaisilleen vanhalla mantereella että heistä oli tullut oman itsensä herroja, että he olivat vapaita viljelijöitä, mutta vatsansa pohjassa ja väliin jääneessä sydämenlyönnissä he tunsivat totuuden: he olivat orjia herralle joka oli vieläkin taipumattomampi, epäinhimillisempi ja etäisempi kuin entiset isännät, herra jonka tappavan voiman saattoi radioaktiivisuuden tapaan tuntea mutta ei nähdä. Heistä oli tullut Pääoman palvelijoita. (Niillä jotka päätyivät ”koneenkäyttäjiksi” tai ”aputyöläisiksi” tarvekalut ja raiteet tuottaneisiin tehtaisiin ei puolestaan ollut paljoakaan kerskailtavaa; he olivat hengittäneet vapaammin lähtöpaikassaan.)
Kun Pontiacin nimestä tunnetusta kapinasta oli kulunut sata vuotta, vuosisata Pontiacin perillisten epätoivoista vastarintaa Pääoman laajempaa tunkeutumista vastaan, alkoivat jotkut maahantuoduista viljelijöistä vastustaa omaa pelkistämistään rautatien, tarvikkeiden ja finanssipääoman palvelijoiksi. Populistiset viljelijät paloivat halusta pidättää ja sulkea lukkojen taakse Rockefellerit, Morganit ja Gouldit jotka olivat suoraan vastuussa heidän kurjuudestaan, mutta heidän kapinansa oli vain kalpea aavistus ottowasien, chippewasien, delawarejen ja potawatomien aikaisemmista kapinoista. Viljelijät nousivat henkilöitä vastaan mutta jakoivat yhä kulttuurin joka oli vastuussa heidän kurjuudestaan. Tästä syystä he eivät onnistuneet yhdistämään voimiaan tasankojen väen aseellisen vastarinnan kanssa tai edes tunnistamaan sitä omakseen. Se oli viimeinen kamppailu joka vastusti mantereen muuttamista kokonaisuudessaan Pääoman saareksi – ja se lannistettiin muinaisilla assyrialaisilla (ja moderneilla neuvosto-sosialistisilla) menetelmillä: joukkosiirroilla, keskitysleireillä, aseettomien vankien joukkomurhilla sekä kovistelevien sotilaiden ja lähetystyöntekijöiden herkeämättömällä aivopesulla.
Vaikka monet kamppailevista viljelijöistä olivat militantteja ja rohkeita, he nostivat vain harvoin elämän ja siitä nauttimisen työn, säästämisen ja taloudellisen voiton edelle. Heidän liikkeensä suistui lopullisesti raiteilta kun radikaalit poliitikot ujuttautuivat siihen ja väänsivät uuden elämän kaipuun uuden Johtajan tarpeeksi. Raiteilta suistuneen populistisen liikkeen muodosta tuli myös seuraavan vuosisadan työväenliikkeen muoto. Populismille haudan kaivaneet poliitikot näyttivät tietä sille äärettömälle munkkimaisten lahkojen joukolle, jotka organisoituivat Jesuiittaveljeskunnan mallin mukaan mutta johtivat oppijärjestelmänsä ja uskontunnustuksensa yhdestä tai toisesta kommunistisesta, sosialistisesta tai anarkistisesta Kirjasta. Valmiina loikkaamaan välittömästi mihin tahansa tilanteeseen jossa ihmiset alkoivat kamppailla saavuttaakseen jälleen ihmisyytensä, he tekivät lopun yhdestä lupaavasta kapinasta toisensa jälkeen hautaamalla elämänsä puolesta kamppailevat ihmiset oppijärjestelmänsä, järjestönsä ja johtajuutensa alle. Näistä pelleistä, jotka halusivat vain saada omat naamansa ja puheensa sanomalehtien kansiin, tuli lopulta kapitalisteja jotka ottivat omakseen sen erityisen kauppatavaran markkinat jonka he olivat vallanneet: työvoiman.
Kun toimiva vastarinta oli pysyvästi poistettu ja korvattu muka-vastarinnalla, joka oli tosiasiassa vain keino muuttaa ihmiselämä lopullisesti pelkäksi Pääoman muuttujaksi, ei voittoja takovalla tehokoneistolla enää 1900-luvun alla ollut mitään ulkoisia esteitä. Sisäisiä esteitä sillä vielä oli: Pääoman eri osaset, Vanderbiltit, Gouldit ja Morganit käänsivät jatkuvasti aseensa toisiaan vastaan ja olivat vaarassa kaataa koko rakenteen sisältäpäin. Rockefeller ja Morgan olivat yhteensulautumisen edelläkävijöitä, eri osasten yhdistäjiä: rahakkaat sijoittajat levittivät varojaan toistensa yrityksiin; johtajat istuivat toinen toistensa johtokunnissa; ja koneiston jokaisen osan rajoittamaton käynti alkoi olla heidän yhteisen etunsa mukaista. Harvoja jäljelle jääneitä henkilökohtaisia ja perhe-imperiumeja lukuun ottamatta yrityksiä johtivat nyt pelkät palkkalaiset, jotka erosivat muista käsistä lähinnä vain palkkioidensa suuruuden puolesta. Johtajien tehtävä oli jyrätä yli kaikkien inhimillisten ja luonnollisten esteiden, vain yhdellä rajoituksella: Pääoman yhdessä muodostavien yhtiöiden tehokas toiminta.
Nelisenkymmentä vuotta sitten Pääoman palveluksessa olevat fysiikan ja kemian tutkijat huomasivat etteivät paljain silmin nähtävät aineet maan päällä ja alla ole ainoita voiton takomiseen soveltuvia aineita. Kävi ilmi että tiettyjen aineiden ”vapautetut” ytimet olivat Pääomalle erittäin käyttökelpoisia. Aineen tuhoamisesta atomitasolla, jota käytettiin ensin hirveimpänä ihmisten tähän mennessä luomana aseena, tuli uusin kauppatavara. Tähän mennessä koronkiskonta, rahtausmaksut ja viljelijöiden tarvikehankinnat, samoin kuin kauan sitten kadonneet puut ja metsän eläimet olivat jo lakanneet kiinnostamasta merkittävinä voiton tuottajina. Uraani- ja öljymonopoleihin kietoutuneista energiayhtiöistä tuli mahtavampia imperiumeja kuin yhdestäkään niiden sovittelijoina toimineista valtioista. Näiden impieriumien tietokoneissa punnittiin ”hyväksyttävää” määrää maaseudun ja kaupunkien asukkaita ”hyväksyttäviä” voittojen saavutuksia tai menetyksiä vastaan. Mahdollisia yleisiä reaktioita sellaisiin laskelmiin kontrolloitiin tarkkaan harkituilla latteuksien, lupausten ja poliisin yhdistelmillä.
\texttt{=}==
\begin{itemize}
\item\relax
Järjestelmä, joka keskittyy ”puolustautumaan” ulkopuolelta tulevaa ydinhyökkäystä vastaan, myrkyttää itse itäisen Pennsylvanian ihmisiä syöpää aiheuttavalla säteilyllä --
\item\relax
Mantereen asukkaiden syötäväksi tarkoitettu ruoka saastuu ja Pääomaa kiinnostavan kauppatavaran kasvattamiselle kuuliaisesti elämänsä omistaneiden viljelijöiden tulevaisuudennäkymät tuhotaan --
\item\relax
Manner, jolla joskus eläneiden ihmisten päämäärä elämässä oli nauttia ilmasta, auringosta, puista, eläimistä ja toisistaan, muutetaan ennennäkemättömän tappavilla myrkyillä ja räjähteillä kirjaimellisesti miinakentäksi --
\item\relax
Näkymä helvetillisten tulimyrskyjen peittämästä mantereesta, jossa heidän kovaäänisensä toistavat nauhoitettua viestiä hiiltyneelle maalle: ”Ei ole syytä ylireagointiin; tilanne on vakaa; johtajilla on kaikki hallinnassaan” --
\end{itemize}
Mikään tästä ei ole onnetonta sattumaa. Se on nykyinen edistyksen vaihe Teknologiassa, alias Pääomassa, jota Mary Wollstonecraft Shelley kutsui Frankensteiniksi. Tätä teknologiaa johtajapyrkyrit, jotka palavat halusta saada ”vallankumoukselliset” kätensä namikoille, kutsuvat ”neutraaliksi”. Kaksi sataa vuotta Pääoma kehittyi tuhoamalla luontoa, siirtämällä ja tuhoamalla ihmisiä. Pääoma on nyt aloittanut suoran hyökkäyksen omia palvelijoitaan vastaan; sen tietokoneet ovat alkaneet laskelmoida niiden uhraamista, joita on opetettu pitämään itseään sen hyötyjinä.
Jospa kuolleiden henget voisivat syntyä elävien keskuuteen. Ottawa- ja chippewa- ja potawatomi-soturit voisivat jatkaa taistelua siitä mihin he jäivät kaksi vuosisataa sitten, yhdistäen voimansa siouxien, dakotojen ja nez percejen, yanojen ja medocien ja niiden lukemattomien heimojen kanssa joiden kieliä ei enää puhuta. Sellainen voima voisi ajaa kokoon rikolliset joita ei muuten saatettaisi minkäänlaisen tuomioistuimen eteen. Pääoman lukuisat välikädet voisivat sitten jatkaa heidän työskentele-säästä-sijoita-myy -rutiiniaan, kiduttaen toisiaan latteuksilla, lupauksilla ja poliisilla vaarattomiksi tehtyjen ja verkosta irrotettujen voimalaitosten sisällä, plutonium-ovien takana.
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\begin{center}
Anarkistinen kirjasto
\smallskip
Anti-Copyright
\strut
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\strut
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\begin{center}
Fredy Perlman
Edistys ja ydinvoima
Mantereen ja sen kansojen tuho
1979
\bigskip
Haettu 16.8.2012 osoitteesta: http:\Slash{}\Slash{}takku.net\Slash{}article.php\Slash{}20080427165014899
Fredy Perlman: Progress and Nuclear Power.
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\textbf{fi.theanarchistlibrary.org}
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\documentclass[12pt]{article}
\newcommand{\ds}{\displaystyle}
\parindent=0pt
\begin{document}
{\bf Question}
Let ${\cal M}$ be any $\sigma$-algebra of sets. Show that
\begin{itemize}
\item[i)]
$E\epsilon {\cal M}, \,\, F\epsilon {\cal M} \Rightarrow E\cup
F\epsilon {\cal M}$
\item[ii)]
$E\epsilon {\cal M}, \,\, F\epsilon {\cal M} \Rightarrow E\cap
F\epsilon {\cal M}$
\item[iii)]
$E\epsilon {\cal M}, \,\, F\epsilon {\cal M} \Rightarrow
E-F\epsilon {\cal M}$
\item[iv)]
$\phi\epsilon {\cal M}$
\item[v)]
$\ds E_1,E_2,\cdots E_n\cdots \epsilon {\cal M} \Rightarrow
\bigcap_{n=1}^\infty E_n\epsilon {\cal M}$
\end{itemize}
\vspace{0.25in}
{\bf Answer}
${\cal M}$ is a $\sigma$-algebra of sets. i.e.
\begin{itemize}
\item[a)]
$E\epsilon {\cal M} \Rightarrow E^C\epsilon{\cal M}$
\item[b)]
$\ds E_1,E_2\cdots\epsilon {\cal M} \Rightarrow
\bigcup_{i=1}^\infty E_i\epsilon{\cal M}$
\end{itemize}
\begin{itemize}
\item[i)]
Suppose $E,F\epsilon {\cal M}$. Let $E_1=E, \,\,
E_2=E_3=\cdots=F$.
Then $\ds \bigcup_{i=1}^\infty E_i=E\cup F\epsilon {\cal M}$.
\item[ii)]
$E\cap F=(E^C\cup F^C)^C\epsilon {\cal M}$
\item[iii)]
$E,F\epsilon{\cal M} \Rightarrow E\cap F^C=E-F\epsilon{\cal M}$
\item[iv)]
$E\epsilon{\cal M} \Rightarrow E^C\epsilon{\cal M} \Rightarrow
E\cap E^C=\phi\epsilon{\cal M}$
\item[v)]
$\ds E_1,E_2\cdots E_n\cdots\epsilon{\cal M} \Rightarrow
E_1^C,E_2^C\cdots E_n^C\cdots\epsilon{\cal M}$
$\ds \Rightarrow
\left(\bigcup_{i=1}^\infty(E_i)^C\right)^C\epsilon{\cal M}
\Rightarrow \bigcap_{i=1}^\infty E_i\epsilon{\cal M}$
\end{itemize}
\end{document}
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%@metapost:aire604exo010.mp
%@Dif:3
Sur la figure ci-dessous, les droites $(BD)$ et $(AC)$ sont perpendiculaires.
\[\includegraphics{aire604exo010.1}\]
Calcule l'aire du quadrilatère $ABCD$ dans chacun des cas suivants :
\begin{myenumerate}
\item $AE=39$~m ; $BD=27$~m ; $EC=13$~m ; $ED=12$~m.
\item $AE=7$~mm ; $BD=2,4$~mm ; $EC=1,5$~mm ; $ED=2$~mm.
\end{myenumerate}
%@Commentaire: Travail sur le principe soustractif du calcul d'aire.
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Copyright 2017 Louis Paternault --- http://ababsurdo.fr
%
% Publié sous licence Creative Commons Attribution-ShareAlike 4.0 International (CC BY-SA 4.0)
% http://creativecommons.org/licenses/by-sa/4.0/deed.fr
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%$ lualatex $basename
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\usepackage{20192020-pablo-devoir}
\usepackage{20192020-pablo-paternault}
\usepackage[a5paper, margin=2cm]{geometry}
\setlength{\parindent}{0pt}
\title{\large Suites --- Corrigé}
\date{21/03/2017}
\classe{1\up{e}S}
\dsnum{DS \no7 --- B}
\pagestyle{empty}
\begin{document}
\maketitle
\begin{exercice}[Termes d'une suite --- 3 points]
\begin{em}
Pour chacune des suites $u$ suivantes :
\begin{enumerate*}[(a)]
\item calculer $u_2$ ;
\item calculer le troisième terme.
\end{enumerate*}
Si nécessaire, les résultats pourront être arrondis au centième.
\end{em}
\begin{enumerate}
\item \emph{La suite $u$ de premier terme $u_1=4$ et telle que, pour tout $n\in\mathbb{N}$, on a $u_{n+1}=3u_n-2$.}
\begin{enumerate}
\item $u_1=4$, donc $u_2=3u_1-2=3\times4-2=10$.
\item Le premier terme est $u_1$, donc le deuxième est $u_2$, et le troisième est $u_3=3u_2-2=3\times10-2=28$.
\end{enumerate}
\item \emph{La suite $u$ définie pour $n\geq2$ par $u_n=\frac{1}{3n}$.}
\begin{enumerate}
\item $u_2=\frac{1}{3\times2}=\frac{1}{6}$.
\item Le premier terme est $u_2$, donc le deuxième est $u_3$, et le troisième est $u_4=\frac{1}{3\times4}=\frac{1}{12}$.
\end{enumerate}
\end{enumerate}
\end{exercice}
\begin{exercice}[Variations --- 3 points]
\emph{Donner, en justifiant, les variations des suites suivantes.}
\begin{enumerate}
\item \begin{em}$u$ est la suite définie par :
\[\left\{\begin{array}{r@{=}ll}
u_1 & 7 \\
u_{n+1} & u_n -1& \text{ pour $n\in\mathbb{N}^*$}
\end{array}\right.\]
\end{em}
C'est une suite arithmétique de raison -1, négative : elle est donc décroissante.
\item \emph{$v$ est la suite géométrique de premier terme $v_1=5$ et de raison $1$.}
C'est une suite géométrique de premier terme strictement positif et de raison 1 : elle est constante.
\end{enumerate}
\end{exercice}
\begin{exercice}[Restitution organisée des connaissances --- 3 points]
\begin{em}
Démontrer que :
\[1+2+\cdots+n=\frac{n\left( n+1 \right)}{2}\]
\end{em}
Voir le cours.
\end{exercice}
\begin{exercice}[Suite géométrique --- 6 points]
\begin{em}
Une usine d'assemblage de voitures est rachetée en 2017 par un fond d'investissement qui exige que sa production augmente de 2~\% par an. La première année, l'usine fabrique \numprint{15000} voitures.
On appelle $v$ la suite, définie sur $\mathbb{N}^*$ par : $v_n$ est le nombre (en milliers) de voitures fabriquées la $n$\up{e} année après le rachat (donc $v_1$ est le nombre de voitures fabriquées la première année ; $v_2$ la deuxième année ; etc.).
On admet que $v$ est une suite géométrique de premier terme $v_1=15$.
\end{em}
\begin{enumerate}
\item \emph{Justifier que la raison de la suite $v$ est 1,02.}
Soit un nombre $n$ entier positif. Alors $v_{n+1}$ est égal au nombre de voitures produites l'année précédente (soit $v_n$), plus 2~\%. Donc :
\begin{align*}
v_{n+1} &= v_n + \frac{2}{100}v_n \\
&= v_n \left(1+\frac{2}{100}\right) \\
&= v_n\times 1,02
\end{align*}
Donc chaque terme est égal au précédent multiplié par 1,02 : la raison est 1,02.
\item \emph{Calculer le nombre de voitures (arrondi à l'unité) produites par l'usine la douzième année.}
On a : $v_{12}=v_1\times1,02^{12-1}=15\times1,02^{11}=18,651$. Donc la douzième année, \numprint{18651} voitures seront produites.
\item \emph{Calculer le nombre de voitures (arrondi à l'unité) produites par l'usine sur l'ensemble des douze premières années.}
Il s'agit de calculer la somme des douze premières valeurs de la suite $v$, c'est-à-dire : $v_1\times\frac{1-1,02^{12}}{1-1,02}=201,181$. Donc sur l'ensemble des douze premières années, \numprint{201181} voitures seront assemblées dans l'usine.
\end{enumerate}
\end{exercice}
\begin{exercice}[Suite arithmétique --- 5 points]
\begin{em}
Une association de défense de la nature remarque qu'une usine de voitures pollue les marécages alentours. Ces marécages hébergent une espèce menacée de grenouilles, et, vu l'augmentation de la production de l'usine, l'association estime que la population de ces grenouilles diminue de 850 individus chaque année.
La population de grenouilles est estimée par l'association à \numprint{45000} individus en 2017.
On appelle $u$ la suite définie sur $\mathbb{N}$ par : « $u_n$ est la population de grenouilles à l'année $2017+n$ » (ainsi, $u_0$ est la population en 2017, $u_1$ en 2018, et ainsi de suite).
\end{em}
\begin{enumerate}
\item \emph{Justifier que $u$ est une suite arithmétique dont on précisera le premier terme et la raison.}
Chaque année, le nombre de grenouille diminue du même nombre 850. Donc chaque terme de la suite est égal au précédent auquel on retranche 850 : c'est la définition d'une suite arithmétique de premier terme \numprint{45000} et de raison -$850$.
\item \emph{Calculer $u_{12}$ ; interpréter ce résultat en terme de grenouilles.}
On a : $u_{12}=u_0+12\times\left(-850\right)=45000-12\times850=34800$. En 2029, il n'y aura plus que \numprint{34800} grenouilles.
\item \emph{Si rien ne change, en quelle année la population de grenouilles aura-t-elle totalement disparue ?}
On cherche le dernier $n$ tel que $u_n$ soit positif. Donc :
\begin{align*}
u_n &\geq0 \\
u_0-850n&\geq0\\
45000-850n&\geq0\\
45000&\geq850n\\
\frac{45000}{850}&\geq n\\
52,9&\geq n \text{ arrondi au dixième}
\end{align*}
Donc le dernier $n$ tel que $u_n$ soit positif est 52. Cela signifie que 2069 (2017+52) sera la dernière année où il restera des grenouilles : elles auront disparu en 2070.
\end{enumerate}
\end{exercice}
\end{document}
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\title{We Give A Shit}
\date{2011}
\author{Tom Nomad}
\subtitle{An Analysis of the Pittsburgh G20}
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\section{Intro: So It Begins}
The primary critique of the summit-hopping era, (one that applies to me as well) is that we never expanded outside of the activist context, never moved beyond complaining loudly around summits, never moved from complaint to active engagement. But there was something in the summit era that did hold promise; in the concentration of numbers in space there was always this possibility of breaking out of the confinement of the downtown area, the confinement of the frontal conflict between police and anarchists, the confinement of pre-planned confrontation, and the limitations of the dates of the summit itself. There was this sense that activism could be transcended, that conflict could be amplified on the streets with the speed and magnitude, that conflict could multiply territorially and break the logistical capacity of the police to contain it. This is what many of us saw, if only briefly, during the Pittsburgh G20, the finale of the summit era, and it was this that both generated the current tactical impasse that we find ourselves in and that points to the way out. The multiplication of the terrains of conflict during the first day of action creates a problem; it became clear that this form of action was insufficient to break the forms of containment that typify the summit demonstration (even if we raised the stakes dramatically). This left many of us feeling as if the terrain of conflict in our own spaces, in our own towns, began to be everything, and that seems to have left us at a loss. But it was specifically this collapse of the attempts to contain the Pittsburgh demonstrations into the traditional forms that typified the summit demonstrations that points to a way out of a dead-end strategy based in complaint and activist tourism. To understand why this was the case we must do more than just look at the context of the actions, the recent tactical shifts that had occurred between 2007 and that point, or even the actions themselves. As with all actions we have to keep in mind that these occurred in a time and in a space, and it is those, combined with the actions taken within those dynamics, that shaped the trajectory of conflict during those two days in September 2009.
To get a handle on what happened there we have to begin with the political and historical terrain. The city of Pittsburgh, however, has a long history of struggle. It was the Pittsburgh Congress of 1883 that is widely credited with being a beginning of an organized anarchist movement in America. This was the site of the Homestead strike in 1892, a huge steel strike that involved shoot-outs between strikers and Pinkerton guards and was the place where Alexander Berkman attempted to assassinate Henry Clay Frick, who now has a park named after him. This is one of those events that is now immortalized on plaques in warehouse districts and “historic areas.” Pittsburgh is also where the United Steelworkers began (and are still based), the AFL (American Federation of Labor), the CIO (Congress of Industrial Organizations), and was also the site of their merger agreement.
This history of struggle has shaped the dynamics of the city and its structure of enforcement. Beginning during the Homestead strike, when the Pittsburgh police refused to break the strike, the bosses called in Pinkerton guards and deputized them, beginning a practice in the Rustbelt of deputization to deal with social ruptures, something that has become a day-to-day part of life there. Homestead was also the motivation for the Commonwealth of Pennsylvania to create the Pennsylvania State Police, a large element of the enforcement structure during the G-20.
Like all cities in the Rustbelt, Pittsburgh is a city that has been completely fucked over by capitalist globalization. Starting in the 1970s and 1980s, with the rise of neoliberalism, privatization, and globalization, production of steel—the basis of the city’s economy—began to shift overseas. Today, despite being the home of the United Steel Worker’s union and commonly referred to as “Steel City,” Pittsburgh is left with no functioning steel mills, aside from some part-time, scrap-melting mills. Massive unemployment and political marginalization was coupled with the market abandonment of these areas, leaving many with no hope in the ability of the market to provide for their daily needs. In the recent past Pittsburgh has seen a rising anarchist scene, with a series of long-running and well-known direct action groups and campaigns occurring through the late 1990s and into the 2000s, specifically the anti-war and counter-recruitment campaigns between 2002 and 2008. These often had actions turn into confrontations with the police.
Obama claimed to have chosen Pittsburgh due to its recent economic “revival” through something called the Pittsburgh Model. This model of development is based on using tax breaks and the restructuring and colonization of poor neighborhoods to provide “favorable market conditions,” (cheap or free land, cheap or free buildings, the lack of unions, tax breaks, etc.) to attract investment. In Pittsburgh this has primarily concentrated around “green building”, military engineering research, the biomedical field, and the building of large universities, as well as the demolition of a poor neighborhood to build a baseball stadium on the north side of the city. Development is also a major force in gentrifying parts of the city, particularly Oakland (the university district), parts of Garfield, and East Liberty, among others. This has created a situation in which a few research and university jobs are created while the majority of the city is left working low wage and\Slash{}or temporary jobs. In contrast, 40 years ago this population had access to high paying union jobs in the steel mills. The city looks like it is reviving on the physical level, but under the facade the Rustbelt reality is the rising of the poverty rate and the shrinking of the population (by almost half since 1950).
Pittsburgh has begun to undergo a series of profound changes, with the abandonment of large parts of the city used as an excuse to restructure its entire fabric. In the attempt to draw outside investment the city government has almost bankrupted itself pouring money into neighborhood redevelopment projects, based on so-called green condo developments, medical research facilities, university expansion, and massive expansion of the policing and surveillance apparatuses, framed in a context of community policing, also known as counter-insurgency. This has caused the fragmentation of many neighborhoods, massive population displacement, and the bulldozing of the city’s history in favor of housing for yuppies. It is this environment which has generated a profound sense of tension on the streets of certain areas of the city, and it is this environment that played a large role in shaping the preparation and trajectory of conflict during the summit itself.
\section{Police Preparations and General Operating Procedures}
Analyzing the tactics of police in Pittsburgh is difficult for a series of reasons. Firstly, there were so many actions going on in so many different places that the very possibility of being able to look at their tactics as a single strategic body is impossible. Secondly, many people have reported long gaps between police sightings, periods of time with little to no police coverage of their movements. This attests to our ability to challenge their control of the streets and create zones where police had little to no physical control, but also makes analysis difficult. However, from the Twitter feeds, from news reports, and from personal experience we can begin to cobble together some understanding of their thinking during the actions.
There are a few things to keep in mind here. Cops need to build cohesive forces, to be able to generalize their needs for a certain situation, and to build force to define a situation. This takes both time and control, the time to build a cohesive force and the ability to use that force to operate within a terrain, to contain actions with their planned strategy. If the situation cannot be made to conform, then their force ceases to be relevant and they have to improvise, or move to a posture of response. This is what it means for the police to enforce definition. The state sets the limits of allowable action and the police must develop a way of enforcing those limits in a situation that is always changing, even though their force is not. Their preparation time was limited in the case of the G20, since they had only four months (as opposed to the two years it took to prepare a comparable police force for the RNC). In contrast, we can quickly do outreach, plan in out affinity groups, and link up with other affinity groups, all in non-linear structures that can adapt to changing circumstances. More time to prepare can be a good thing, but it is not as important for us. We do not need to create and enforce definitions, we are able to be mobile.
Maybe to compensate for their lack of time to prepare, maybe as an intentional tactic, the cops early on defined their approach to this series of actions. Firstly they engaged in raids meant to accomplish the two goals of disrupting organizing and intimidation. In other words, they were meant to build the feeling that the cops were everywhere. They kept catching the Seeds of Peace bus, as well as other cars, on the street instead of at static spaces, trying to create the impression that they could find us whenever they wanted to. They coupled this with very public announcements whenever they seized equipment. This approach backfired however, and led not just to lawsuits but also to embarrassment. (Having very publically announced finding PVC pipe they claimed was for “sleeping dragons,” they discovered later that the pipes were being stored by a company for product testing.) This constant presense also heightened the eerie feeling within the city of the coming police state, to the dismay of many residents. This can partially explain the intense public support that many reported while marching through the streets. These disruptions, like the tactics used on Thursday afternoon, were as much based on intimidating anarchists and the general populace as they were on materially disrupting organizing work.
This phsychological tactic was increased by their tactic of posturing, especially through the media. In past mobilizations the press work by the cops beforehand was aimed at the general populace and meant to generate fear of anarchists coming to burn the city to the ground, and so on. The G20 pre-action press preparation was different; it was aimed at us. There were the obligatory warnings from the mayor against the people coming to “cause destruction,” but on top of that there was endless coverage of the police build-up, tours of the security perimeter, tours of their command center (something even the press was confused about), as well as constant police harrassment before the actions. But without being able to carry out any raids of material importance, this all came across as psturing and nothing more. They were forced to backpedal from a lot of these statements in the days before the action as lawsuits and complains started coming in from various groups, and business owners started boarding up stores. The police were trying to strike a balance between inflating fears of anarchist horde to justify the massive police buildup, and reassuring business owners that they were safe to remain open. They failed. Ordinary Pittsburgh residents were angry at the government for turning their city into a police state, leading many of them to side with the protesters, and most downtown businesses shut down for the duration of the G20.
The press coverage was combined with the use of weapons of intimidation and staged force during the actual actions. They had announced beforehand that they “would not be the spark.” It became obvious that they were anticipating the possibility of disruption and that they assumed it would happen downtown, or at least on the way to downtown. Twin Cities Indymedia, as well as a lot of people traveling to Arsenal Park on Thursday, reported seeing large columns of police behind the initial skirmish line at 34th and Liberty, between the residential parts of the East End of Pittsburgh and Downtown in an area known as the Strip District.These police were there in case the initial lines broke down. They used weapons that couldn’t be carefully or accurately targetted, so they could not pick and shoose targets until they began to bring out bean-bag rounds and rubber bullets on Thursday afternoon. The indiscriminate weapons were used to keep people away from areas. But dispersing a crowd into smaller groups makes the situation harder to define, so this tactic of dispersal combined with the air of total presense, was supposed to make us want to disperse ourselves. In other words, like all the pre-action preparation, these weapons are meant to have a psychological effect; they are used to demoralize crowds, to take the fight out of us by making us feel that resistance is futile. But these weapons backfired. Because a lot of people have seem them before, the weapons didn’t have the intimidating effect the police planned on. Even the helicopters and gas (which became constants at a certain point) did little to deter people who have been in situations like this, and at this point that is a lot of us. When gassed we noticed a lot of people calmly putting on goggles and helping others to do the same, then calmly and quickly moving into more open space. Police also relied heavily on the LRAD, which had such minimal effect that it became a joke on the Daily Show, not to mention in our internal circles. (Note: many of us have switched our ringtone to the sound of the LRAD.) Police approaches generally are based in staging force, using increasing physical force instead of psychological as the situation escapes their control, and this is what they ended up doing. They brought out armored personnel carries on Thursday, but didn’t really use them except to block roads, and changed to indiscriminate use of force on Friday night.
\section{Days Filled with Stones and Flowers}
\section{The People’s Uprising March}
To begin to attempt an analysis of the People’s Uprising march is difficult. There was such widely dispersed action after the first half hour or so that we need to look at the dynamics of the action instead of the actions themselves. In other words, the actions built a dynamic environment, and this is what to focus on. What we know now is that outside the large police presence at Arsenal Park there was a much larger and more concentrated presence of police between the initial point of contact (34th and Liberty St) and the perimeter downtown, staggered in increased concentrations the closer we got to the David L. Lawrence Convention Center.
The initial police contingent seemed willing to give the street to the march. This is not surprising within a new, modified police tactic of containment\Slash{}dispersal, or containment as dispersal, a tactic that we have encountered in Washington, DC. If the police think the march will be able to take the street, or is determined to do so, they will set up a zone of control, an area of the street that they will give to the march to avoid confrontation, while they try to contain everything outside this space. So they may give the street but surround the march on all sides, they may give a lane, etc. As this march moved out, some noticed this and redirected the march through the park to another exit point, which immediately frustrated the police attempt to contain the march. This was evident as we were passing small contingents of riot cops, spaced out on the corners of intersections, especially when we encountered two riot cops in a car as we turned onto Liberty (i.e. we were seeing the backside of their tactic). At this point the march split, some trying to head away from the massive police contingent in downtown and go to any number of recently gentrifying commercial districts. On reaching the corner of 34th and Liberty we saw a line of riot police, an armored personnel carrier (APC) equipped with an LRAD, and a series of other vehicles. Unknown to many at this point was that this was only an intial line, there was a much higher concentration of cops further on. The cops gave a dispersal warning and then sounded the LRAD for the first time.
The march diverted down an alley next to the Church Brew Works, where the first dumpsters were came out and barricades were built. This area of Pittsburgh, in a neighborhood called Lawrenceville, is characterized by narrow winding streets, often dead ending into one another, which only requires a single dumpster to completely block. As we rounded the corner again, to get back to Butler St at 37th (and thus began to move through the Strip District towards downtown), we were met with another line. That is when the cops first used high concentrations of gas. After they failed to contain the march at the park, they switched to a blocking tactic, one that is only meant to prevent access to certain areas. They used a show of force and shifting blockades to prevent access to downtown while also trying to convince us to disperse. This is speculation, but it seems as if they made some mistakes in their projections of our actions. Firstly, they seemed to assume our goal was to head into downtown, and they allocated force to prevent that movement. This became clear as the march formed into smaller groups; those who headed away from downtown saw almost no cops for a long period of time. Secondly, police made a big deal before the actions about training to defeat lockdowns, maybe expecting a repeat of the Republican National Convention activist tactics, which centered around blockading access to certain areas of downtown. For G20, few if any groups planned on locking down, opting to remain more flexible instead of using a tactic that immobilizes at the point of deployment.
As the march "dispersed" into smaller groups, the situation became really fluid and dynamic. The constant changes in the scenario kept cops from accounting for numerous groups in the streets. We can separate these trajectories of movement into two general movements. One group engaged with the cops in their own territory by trying to head downtown. This was a rather large section of the march; they got stopped in the Strip district. It seems like many cops were diverted to stop this group. Another set of groups started to head the opposite way towards the gentrified shopping areas of the East End through Lawrenceville and Bloomfield. These groups began to notice a series of things. The most astonishing was that people from the neighborhoods, and these were largely working class neighborhoods, began to come out onto the streets to engage with the events, both in cheering anarchists on and in certain instances helping to barricade off streets. These groups set up barricades to create space. A PNC Bank got its ATMs smashed, pulling more police into simple response actions, and away from the operation of a coherent strategy.
There are two fundamental aspects to these sets of movements. Firstly, in multiplying the terrain of conflict, in the organized and intentional dispersal across space, we were able to break the zone of containment that the police attempted to set up, and to eventually break outside of their ability to contain the terrain of conflict at all. This forced police to respond to a series of points of conflict, often too slowly to actually catch anyone or to even engage, which constancy of movement stretched their capacity to maintain logistical coherence of strategic initiative. Secondly, the use of barricades and property destruction occurred in a way that had not really been seen in American summit demonstrations. Barricades proliferated on side streets as groups began to move off main roads and into the twisted tangles that characterizes this part of Pittsburgh. Barricades limited police movement to major roads. When combined with the loss of tactical initiative, which forced them to respond to points of engagement in small groups, usually on city buses, police lost the ability to project through space. As the terrain became more resistant, as the movement of motorized units was constrained, and as the terrain of conflict widened, the police were forced to move through whatever space they could, as fast as possible in as many groups as possible, to as many points as possible, and lost their ability to occupy, to move, or to maintain logistical coherence.
A call went out over Twitter to meet in Friendship Park, on the border of the Bloomfield, Friendship, and Garfield neighborhoods. A trickle of people ballooned to hundreds. The park became a space to rest, get treated for injuries, and plan next moves. Cops began showing up in droves, hoping to surround the park, but again the crowd was too large for them to box it. That march began, and headed down Liberty Ave, away from downtown, in the direction of the Oakland neighborhood, or the university district. At the intersection of Liberty and Baum Ave the march turned right and began to speed up, with many groups breaking off. The police began to fire rubber bullets into the crowd, causing some affinity groups to spread out, resulting in a trail of broken windows all up Baum, including hits on Boston Market and various other chain restaurants. During these confrontations police attempted to target certain individuals (including the now famous footage of police in camo fatigues jumping out of a car, grabbing someone and driving off with them); these stopped after groups began to double back and pelt the police with chunks of concrete. As people filtered into Oakland the police presense increased dramatically, beginning the trajectory of conflict that would result in large scale rioting a couple of hours later.
This concentration of police was bolstered by contingents of cops tasked with protecting a State Dinner at Philips Conservatory (a building in Schenley Park), which borders the University of Pittsburgh in the heart of Oakland. Students began to be harassed by police who, in response to events earlier in the day, were attempting to clear the campus of any students not in their dorm rooms. This caused a conflict between students at Pitt and the police on their campus; a conflict that would set the stage for what was about to occur.
\section{Bash Back!}
As the cops were tear gassing the first groups of protesting students in Schenley Park, the Bash Back! march began to gather at the corner of Desoto and Fifth Ave, three blocks south-west of the Plaza. The march rolled out around 10, only a half hour after the disturbances began on Pitt's campus. The march began down Meyran Ave to Forbes and along the way picked up six dumpsters. The first police vehicle arrived and was stopped by four of the dumpsters being turned over in the intersection. Corporate shops were attacked, with windows busted out of Subway, McDonald's, and American Apparel, among others. While the cops were still stopped at the first barricade, two more dumpsters were overturned, one on fire, at the intersection of Forbes and Desoto, which created even more space away from the cops. Students and bystanders crowded the sidewalks as the police substation got its windows busted out. The march then saw its first police line, a line of vehicles, about a block ahead. Instead of engaging, the crowd began to move through university property across the street from Schenley Plaza. The crowd took a right and headed up past a university vivisection lab, which got its windows smashed out, then a left, a right, ending up on a street with three banks and a Quiznos, all of which got windows broken.
At this point, you could begin to see the police cordon setting up at the intersections: a couple of cops per intersection, a car, and usually some form of wooden barricade with reinforcements of riot police down the streets. We did not know at the time that there was a much larger disturbance back at Pitt. In the wake of the crowd, students had swarmed the streets, chanting "cops off campus," "Go Pitt, Fuck the Police" and "we love Pittsburgh, fuck the G20." Contrary to media reports, students were not just swept up in the events but were actively participating.
The cops were split once again, trying to deal with dynamic situations moving in two different directions. Vehicles had caught up with the crowd again and were attempting to run it off the streets. Many small groups started to disperse down alleys and work their way back toward Schenley Plaza where cops had begun gassing students again. Around midnight, around when they called "All units to Oakland" over the police scanner, they decided to cordon off the area. They set up skirmish lines on Forbes and Fifth and pushed students away from the commercial district and back onto campus. They began by pushing people down the sidewalk but that quickly escalated into firing tear gas down the street and even gassing students trying to enter their dorms. This escalated the situation and brought more students out into the street. It took till 2:30 for them to finally quell the unrest in Oakland. What few of us knew whas that when the march began, they were trying to get Obama out of Schenley Park, the entrance to which is Schenley Plaza, ground zero for the rioting, and at this point many units ran out of their gas requisition, freezing them in place for a period of time. This even further escalated the situation until they began to completely clear streets, driving vehicles down residential streets in Oakland, repeating the dispersal warning from loudspeakers.
The point when they ran out of gas is an important moment, the point where their security plan broke down completely. In a single day we had exceeded their projection of the worst possible scenario for the entire weekend. When creating a summit security plan, police will requisition supplies based on what they consider to be the worst possible scenario for the entire time of potential conflict (in this case, a weekend). The fact that they ran out of gas makes it clear that in a single day we exceeded the worst possible scenario projection for the entire weekend. This wasn't because of the volume of property destroyed or the magnitude of any individual action; it wwas a result of the speed of movement through terrain, the ability to limit police projection, and the multiplication of terrains of conflict that ruptured the coherence of police logistics and eviscerated any concept of tactical initiative on their part. As is often discussed in relation to asymmetric conflict, when conflict spreads throughout a terrain, gaps in police coverage open up, and these gaps are where conflict can proliferate; but in the creation of these gaps conflict becomes a potential in all space and police movement through space becomes uncertain and difficult. It was in specifically breaking the containment of the summit demonstration, breaking the planned demonstration zones, the containment strategy of police strategy, and the containment of political identity, that these actions pushed police logistics to the breaking point. The only tragedy of that day was that we did not push this further, through the night and into the following days, and in failing to pursue, to continue to amplify conflict. We allowed the police time to regroup, resupply, and call in reinforcements.
\section{The Permit March}
The next morning the permitted march began to gather. At the gathering point itself there were relatively few police, but just blocks away were hundreds of riot cops, spaced out in groups of 30-50, surrounding vehicles so that they could be mobile, and accompanied by K-9 units. These mobile units were to deal with anyone who diverged from the agreed-upon plan for the day. As the march moved downtown we noticed more and more cops, in higher concentrations, 'til we got downtown and then they lined the streets, standing in front of barriers that held back crowds of people who had gathered along the march route. When the march stopped in front of the City County Building, the cops began to show a little of what they had in store for later that night. The crowd stood in a downtown street while 50-100 riot cops began to move off a side street, one (backed up by one of the LRADs) even moving into the crowd. The bloc assembled and moved towards the cops to form a buffer between the cops and the rest of the crowd.
As the march moved the police presense thinned out. They moved squads of riot cops into the positions that we had occupied minutes before and drove Hummers with fences attached to their fronts to block off the bridges to everything but foot traffic. As the gathering in the park wore on, and as the time for the permit to expire approached, we noticed lines of riot police beginning to surround the park and a large contingent getting off a school bus and gearing up in the southwest corner of the park. These shows of force were further foreshadowing of the actions later that night.
\section{Go Pitt, Fuck The Police}
That night a large group of Pitt students, along with assorted anarchists and activists, gathered in Schenley Plaza to demonstrate against the police brutality from the previous night. Hours before the gathering, we could see large groups of riot cops gearing up in the Oakland neighborhood and hiding down side streets, particularly around Forbes between Meyran and Desoto. As people began to gather, the park became immediately surrounded. After 45 minutes the dispersal warnng was sounded and the LRAD blared, but there was nowher for anyone to go. The cops began to move in but not as a unit. They sent small tactical teams into the crowd to secure an area, while cops behind them gassed that area, and pepper sprayed or attacked anyone in range. Those they caught were cuffed and arrested. Larger lines would move in behind them to secure the area and process the arrestees. Groups managed to break through and head both out of the area and further into Pitt's campus. Those groups that ended up on campus were chased down by riot cops and beaten if caught. The cops beat and gassed people indiscriminately, including at least one instance of launching tear gas canisters into open dorm windows.
The gathering in itself was relatively innocuous, being largely people playing drums and giving speeches, but that is not the point. The police response was meant to send a message not only against causing distubances that night, but the make anyone present think twice about stepping out of line again. The response was meant to psychologically damage and generate fear, not just to stabilize a situation. And this is a good lesson to learn. If we are going to be successful we have to be ready for and expect this type of response in subsequent gatherings. While difficult to deal with, it is inevitable. The police are trying to stabilize a situation, and for them that means preserving control. That means constructing us as subjects to be organized, to be positioned to preserve the flows of the city, and if we can't be organized, to be forced back into stability. The police actions on Friday night accomplished their goal. There were few popular actions Friday night and the energy of the actions dissipated quickly, but we doubt the resonance of those actions will fade as easily.
\section{The End\dots{} or The Dawn of New Beginnings}
There is little doubt that these were some of the most successful actions that we have undertaken in recent memory. Not because anarchists barricaded streets and created space, or because we fought back against the cops and actually held our ground. Not because we forced the cops into a stalemate by the middle of Thursday or the scale of the property destruction. Rather it is that we were able to glimpse a form of action. Unlike past summit demonstrations, isolated in downtown areas like the summits themselves, these actions were both visible and invisible simultaneously. They engaged on a plane of daily life that our actions rarely touch (outside of our own lives). The actions were dispersed and mobile, escaping the ability of the state to impose order on them.
During the Greek uprising a government minister complained most about the inability to have an object of group to negotiate with, no demands to mediate. Those actions existed on a different plane than the state. Against the state's imposition of sameness, people in the streets created divergence and multiplicity. The streets became indefinable as actions proliferated, changing the environment with the participants themselves. It became a terrain impossible to define, impossible to limit as the very structures of control had broke down. The inability of the state to mediate these actions was precisely due to the existence of the actions of a plane that could not be mediated. It was not for anything specific but for the possibility of possibility, the very energy that destroys limits. This is a strategy of disappearance unable to be defined, unable to be categorized, and therefore unable to be policed. It was a fight over the possibility of control.
Not that the G20 was anywhere close to the intensity of Greece, but that type of situation can only exist to the degree that it is invisible to the state, that there are too many dynamics, too many actions to stabilize. But this disappearance from the plane of the state, from the state's gaze, is also an appearance on the level of daily life, a level where life and action link up in ways that can only create dynamic situations. Resistance struck a chord, it resonated, and that resonance built itself into an energy that shook the city. It escaped the bounds of the removed specialists of political action and broke out, it became social war, or at least a glimpse of what that resonance may feel like. It opened a window into something else. What that is, is up to us to decide.
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Library.Anarhija.Net
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\begin{center}
Tom Nomad
We Give A Shit
An Analysis of the Pittsburgh G20
2011
\bigskip
The following article was written by Tom Nomad and originally published as a zine entitled “We Give A Shit” in 2011 and later revised as a chapter of \emph{The Master’s Tools: Warfare and Insurgent Possibility} (Little Black Cart, 2013)
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\textbf{lib.anarhija.net}
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\title{\textbf{Effectiveness of acupressure on pain management among mediosternotomy patients}}
\author{Pritha~L\textsuperscript{*}\textsuperscript{1},
Valliammal~S\textsuperscript{2},
Vijayaraghavan~R\textsuperscript{3}~\\[5pt]{\textsuperscript{1}Research Scholar\unskip, Saveetha Institute of Medical and Technical Sciences, 162, Poonamalle High Road, Velappanchavadi, Chennai -600 077 , Tamil Nadu, India}~\\{\textsuperscript{2}NIMHANS\unskip, College of Nursing, Hosur Road, 560029, Bangalore, India}~\\{\textsuperscript{3}Saveetha Institute of Medical and Technical Sciences, 162, Poonamalle High Road, Velappanchavadi, Chennai-600 077, Tamil Nadu, India}}
\begin{abstract}
Acupressure is an alternative therapy that uses fingers and hands to stimulate acupoints and maintains the balance of energy. It was well documented that acupressure is effective in relieving different types of pains in patients with different conditions. The present study was undertaken to evaluate the effect of acupressure in pain and improving the bio-physiological parameters among mediosternotomy patients. The present study was conducted at Sri Sathya Sai Institute of Higher Medical Sciences, Whitefield, Bangalore. A total of twenty male and female patients\textbf{\space }those who undergo open-heart surgery via median sternotomy were part of the study after obtaining the written informed consent. After recording the demographic data, the participants were randomly grouped into control and intervention groups using random numbers generated by computer with 10 participants in each group. The intervention will be provided at PC6 acupressure point, situated on the inner side of the forearm, three fingers below the wrist joint, three times a day for four days. Numerical Rating Scale of pain was used to assess the pain of the Participants. There was a significant decrease in the pain score of the participants\textbf{, }followed by the acupressure. The study provides further evidence for the effectiveness of the acupressure in pain management and also recommends detailed research in this area.
\end{abstract}\def\keywordstitle{Keywords}
\begin{keywords}Acupressure,\newline Pain management,\newline Mediosternotomy patients
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\section{Introduction}
Acupressure is an alternative therapy that uses fingers and hands to stimulate acupoints and maintains the balance of energy. It was well documented that acupressure is effective in relieving different types of pains in patients with different conditions. Earlier studies have recommended applying the acupressure as an alternative therapy in the management of the pain of the patients \unskip~\citep{558485:12830479}. Musculoskeletal pain was reported to decrease, followed by the acupressure. Further, there were no side effects followed by the acupressure. The major advantage of using the acupressure is that it reduces the usage of the analgesics and anti-inflammatory agents \unskip~\citep{558485:12830480}. Worldwide nearly 2 million people undergo sternotomy for surgery of heart every year. Followed by the surgery, the majority of the patients develop pain which must be managed. Analgesics are the commonly used treatment for the management of the pain. However, it is associated with severe side effects. Hence, there is a strong need for alternative therapy, which is effective and associated with no side effects. Acupressure PC6 acupoint stimulation has been practiced in various surgeries, but there is no study done by stimulating PC6 acupoint through touch on open-heart surgery patients. In this regard, the investigator is also interested in finding out the effect of using acupressure on pain and bio-physiological parameters for mediosternotomy patients. Hence, the present study was undertaken to evaluate the effect of acupressure in pain and improving the bio-physiological parameters among mediosternotomy patients.
\section{Materials and Methods}
\textbf{Study design: }Experimental study with pre and post with control design.
\textbf{Study setting: }The present study was conducted at Sri Sathya Sai Institute of Higher Medical Sciences, Whitefield, Bangalore.
\textbf{Study population: }A total of twenty male and female patients\textbf{\space }those who undergo open-heart surgery via median sternotomy were part of the study after obtaining the written informed consent. After recording the demographic data, the participants were randomly grouped into control and intervention groups using random numbers generated by computer with 10 participants in each group.
Group Control (= 10): No intervention applied
Group Experimental (= 10): Acupressure was applied
The participants were recruited using the following criteria.
\textbf{Inclusion criteria: }
\begin{enumerate}
\item \relax Male and female participants between the age group of 19 and 60 years.
\item \relax Patients those who report pain during deep inspiration with an intensity of at least 3 on a 0{\textendash}10 rating scale under standard analgesia.
\end{enumerate}
\begin{itemize}
\item \relax Patients those who are extubated.
\item \relax Patients on the first operative day and conscious.
\item \relax Patients who can communicate through English, Tamil, Hindi and Kannada.
\item \relax Patients those who are willing to participate in a study.
\end{itemize}
\textbf{Exclusion criteria:}
\begin{itemize}
\item \relax Patients who are having post-operative complications.
\item \relax Patients with the emergency operation, chronic pain and hemodynamically unstable.
\item \relax Patients who underwent graft donor site from hands for CABG.
\end{itemize}
\textbf{Acupressure: }The intervention will be provided at PC6 acupressure point, situated on the inner side of the forearm, three fingers below the wrist joint, three times a day.
\textbf{Assessment of pain: }Numerical Rating Scale of pain was used to assess the pain of the Participants \unskip~\citep{558485:12830481}.
\textbf{Ethical consideration:}
The study was approved by the Institutional Ethics Committee. A written, informed consent was obtained from all the participants. The study was performed in accordance with the "Ethical Guidelines for Biomedical Research on Human Participants, 2006" by the Indian Council of Medical Research and the Declaration of Helsinki, 2008.
\textbf{Data analysis:} Data was analyzed using SPSS 20.0. Data were expressed as frequency and percentage. Pain scores were expressed as mean and SD. A probability value of less than 0.05 was considered significant.
\section{Results and Discussion}
Distribution of participants on the basis of gender was presented in Table~\ref{tw-bee217dc47e4}. Distribution of participants on the basis of age is presented in Table~\ref{tw-f5a5494f7c37}. Distribution of participants on the basis of education was presented in Table~\ref{tw-1c6b4bc2ee8f} . Distribution of participants on the basis of occupation was presented in Table~\ref{tw-4e034b8a6b50}.\textbf{\space }Distribution of participants on the basis of family income was presented in Table~\ref{tw-6ef1eb271b55}. Distribution of participants on the basis of family religion was presented in Table~\ref{tw-1a960058aaa2}. Prior information on acupressure was presented in Table~\ref{tw-f9aeb79192c3}. Co-morbid illness in the participants was presented in Table~\ref{tw-213d354fd973}. Duration of illness in the participants was presented in Table~\ref{tw-64a1be2c96e2}. Surgical diagnosis in the participants was presented in Table~\ref{tw-82606a7cb681}. Name of the surgery undergone by the participants was presented in Table~\ref{tw-ed5e2b2f472e}.\textbf{\space }Location of water seal drainages and the number of drains undergone by the participants were presented in Table~\ref{tw-aeb3cf45f88a}. Location of pain in the participants was presented in Table~\ref{tw-437ff4b401ee}. Frequency of experiencing pain in the participants was presented in Table~\ref{tw-4737ec2b2030}. Activities associated with pain sensation duration in the participants were presented in Table~\ref{tw-35b392593b69}. Medication given to the participants was presented in Table~\ref{tw-0c194f3d33cb}. Comparison of pre and post values of pain score among control and experimental groups was presented in Table~\ref{tw-b8f3cce34e73}. Comparison of pre -values of pain score among control and experimental groups were presented in Table~\ref{tw-ca503781cadb}. Comparison of post- values of pain score among control and experimental groups, were presented in Table~\ref{tw-833124d8b576}.
\begin{table}[!htbp]
\caption{\boldmath {Distribution of participants on the basis of gender (Data was presented as frequency and percentage). (Exp-experimental group, Con- control group)} }
\label{tw-bee217dc47e4}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.1795\linewidth-2\tabcolsep}p{\dimexpr.31619999999999997\linewidth-2\tabcolsep}p{\dimexpr.5043\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Gender & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
Male &
6 (60) &
9 (90)\\
Female &
4 (40) &
1 (10)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Distribution of participants on the basis of age (Data was presented as frequency and percentage).(Exp-experimental group, Con- control group)} }
\label{tw-f5a5494f7c37}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2348\linewidth-2\tabcolsep}p{\dimexpr.266\linewidth-2\tabcolsep}p{\dimexpr.4992\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Age (years) & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
19-28 &
1 (10) &
1 (10)\\
29-38 &
3 (30) &
4 (40)\\
39-48 &
2 (20) &
1 (10)\\
49-58 &
2 (20) &
1 (10)\\
59-68 &
2 (20) &
3 (30)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Distribution of participants on the basis of education (Data was presented as frequency and percentage).(Exp-experimental group, Con- control group)} }
\label{tw-1c6b4bc2ee8f}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.33\linewidth-2\tabcolsep}p{\dimexpr.2712\linewidth-2\tabcolsep}p{\dimexpr.3988\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Education & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
No formal education &
0 (0) &
1 (10)\\
Primary &
6 (60) &
6 (60)\\
Secondary &
1 (10) &
3 (30)\\
Higher secondary &
2 (20) &
0 (0)\\
Graduate &
1 (10) &
0 (0)\\
Post Graduate &
0 (0) &
0 (0)\\
Other specify &
0 (0) &
0 (0)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Distribution of participants on the basis of occupation (Data was presented as frequency and percentage).(Exp-experimental group, Con- control group)} }
\label{tw-4e034b8a6b50}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2366\linewidth-2\tabcolsep}p{\dimexpr.31440000000000005\linewidth-2\tabcolsep}p{\dimexpr.449\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Occupation & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
Daily wages &
5 (50) &
3 (30)\\
Business &
0 (0) &
3 (30)\\
Private &
2 (20) &
3 (30)\\
Government &
0 (0) &
0 (0)\\
Retired &
0 (0) &
0 (0)\\
Homemaker &
3 (30) &
1 (10)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Distribution of participants on the basis of family income (Data was presented as frequency and percentage). (Exp-experimental group, Con- control group)} }
\label{tw-6ef1eb271b55}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.3058\linewidth-2\tabcolsep}p{\dimexpr.2971\linewidth-2\tabcolsep}p{\dimexpr.3971\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Family income \mbox{}\protect\newline (rupees per month) & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
{\textless}10000 &
6 (60) &
1 (10)\\
10001-20000 &
3 (30) &
5 (5)\\
20001-30000 &
1 (10) &
4 (4)\\
30001-40000 &
0 (0) &
0 (0)\\
{\textgreater}40000 &
0 (0) &
0 (0)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Distribution of participants on the basis of family religion (Data was presented as frequency and percentage). (Exp-experimental group, Con- control group)} }
\label{tw-1a960058aaa2}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2348\linewidth-2\tabcolsep}p{\dimexpr.3403999999999999\linewidth-2\tabcolsep}p{\dimexpr.42480000000000004\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Religion & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
Hindu &
8 (80) &
5 (50)\\
Muslim &
1 (10) &
4 (40)\\
Christian &
1 (10) &
1 (10)\\
Others &
0 (0) &
0 (0)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Prior information on acupressure (Data was presented as frequency and percentage). (Exp-experimental group, Con-control group)} }
\label{tw-f9aeb79192c3}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.33\linewidth-2\tabcolsep}p{\dimexpr.2954\linewidth-2\tabcolsep}p{\dimexpr.3746\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Have you received any information on acupressure & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
Yes &
0 (0) &
0 (0)\\
No &
10 (100) &
10 (100)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Co-morbid illness inthe participants (Data was presented as frequency and percentage).(Exp-experimental group, Con- control group)} }
\label{tw-213d354fd973}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.33\linewidth-2\tabcolsep}p{\dimexpr.259\linewidth-2\tabcolsep}p{\dimexpr.411\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Co-morbid illness & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
Angina or Exertion &
3(30) &
7 (70)\\
Hypertension &
5 (50) &
1 (10)\\
Dyspnea or exertion &
2 (20) &
2 (20)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Duration of illness in the participants (Data was presented as frequency and percentage). (Exp-experimental group, Con- control group)} }
\label{tw-64a1be2c96e2}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.21409999999999997\linewidth-2\tabcolsep}p{\dimexpr.25390000000000004\linewidth-2\tabcolsep}p{\dimexpr.532\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Duration of illness & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
\textless\ 1 year &
3(30) &
1(10)\\
1-5 years &
6(60) &
6(60)\\
6-10 years &
1(10) &
3(30)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Surgical diagnosis in the participants (Data was presented as frequency and percentage).(Exp-experimental group, Con- control group)} }
\label{tw-82606a7cb681}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.33\linewidth-2\tabcolsep}p{\dimexpr.2469\linewidth-2\tabcolsep}p{\dimexpr.4231\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Surgical diagnosis & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
Coronary artery disease &
8(80) &
8(80)\\
Mitral valve stenosis &
1(10) &
1(10)\\
Aortic valve stenosis &
1(10) &
1(10)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Name of the surgery undergone by the participants (Data was presented as frequency and percentage).(Exp-experimental group, Con- control group)} }
\label{tw-ed5e2b2f472e}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.33\linewidth-2\tabcolsep}p{\dimexpr.2573\linewidth-2\tabcolsep}p{\dimexpr.41269999999999996\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Name of the surgery & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
On pump coronary artery bypass graft &
7(70) &
7(70)\\
Mitral valve replacement &
1(10) &
1(10)\\
Aortic valve replacement &
1(10) &
1(10)\\
Off-pump coronary artery bypass graft &
1(10) &
1(10)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Location of water seal drain ages and the number of drains undergone by the participants (Data was presented as frequency and percentage). (Exp-experimental group, Con- control group)} }
\label{tw-aeb3cf45f88a}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.285\linewidth-2\tabcolsep}p{\dimexpr.2954\linewidth-2\tabcolsep}p{\dimexpr.4196\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Location of water seal drainage \& number & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
Right \& Left chest &
10(100) &
10(100)\\
2 Drains &
10(100) &
10(100)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Location of pain in the participants (Data was presented as frequency and percentage).(Exp-experimental group, Con- control group)} }
\label{tw-437ff4b401ee}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2452\linewidth-2\tabcolsep}p{\dimexpr.2609\linewidth-2\tabcolsep}p{\dimexpr.4939\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Location of pain & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
Sternum \& Leg &
8(80) &
8(80)\\
Only Sternum &
2(20) &
2(20)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Frequency of experiencing pain in the participants (Data was presented as frequency and percentage). (Exp-experimental group, Con- control group)} }
\label{tw-4737ec2b2030}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2314\linewidth-2\tabcolsep}p{\dimexpr.2625\linewidth-2\tabcolsep}p{\dimexpr.5061\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Frequency of Experiencing pain & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
Never &
0(0) &
0(0)\\
Sometime &
0(0) &
0(0)\\
Mostly &
0(0) &
0(0)\\
Always &
10(100) &
10(100)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Activities associated with pain sensation duration in the participants (Data was presented as frequency and percentage). (Exp-experimental group, Con- control group)} }
\label{tw-35b392593b69}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2816\linewidth-2\tabcolsep}p{\dimexpr.22440000000000005\linewidth-2\tabcolsep}p{\dimexpr.494\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Activities associated with pain sensation duration & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
Cough &
10(100) &
10(100)\\
Physiotherapy &
10(100) &
10(100)\\
Wound dressing &
2(20) &
4(40)\\
Movement &
7(70) &
7(70)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table}[!htbp]
\caption{\boldmath {Medication given to the participants (Data was presented as frequency and percentage).(Exp-experimental group, Con- control group)} }
\label{tw-0c194f3d33cb}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2764\linewidth-2\tabcolsep}p{\dimexpr.2625\linewidth-2\tabcolsep}p{\dimexpr.4611\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Medication & Exp-Frequency (percentage) & Con- Frequency (Percentage)\\
\tblmidrule
T. Paracetamol &
10(100) &
10(100)\\
Dose (1 Gram) &
10(100) &
10(100)\\
Route (Oral) &
10(100) &
10(100)\\
Frequency (BD) &
10(100) &
10(100)\\
\tblbottomrule
\end{tabulary}\par
\end{table}
\begin{table*}[!htbp]
\caption{\boldmath {Comparison of pre and post values of pain score among control and experimental groups (Data was presented as frequency and percentage). (Exp-experimental group, Con- control group)} }
\label{tw-b8f3cce34e73}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2706\linewidth-2\tabcolsep}p{\dimexpr.0476\linewidth-2\tabcolsep}p{\dimexpr.24309999999999995\linewidth-2\tabcolsep}p{\dimexpr.2221\linewidth-2\tabcolsep}p{\dimexpr.21659999999999997\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{Day-1}\\
\rowcolor{kwdboxcolor}\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{Experiment group}\\
\tblmidrule
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{Test time} &
Pre &
Post &
p-value\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{5-6:00pm} &
9.10\ensuremath{\pm}0.32 &
6.70\ensuremath{\pm}0.48 &
{\textless}0.0001***\\
\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Control group}}\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{5-6:00pm} &
9.10\ensuremath{\pm}1.32 &
9.10\ensuremath{\pm}1.32 &
1\\
\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{Day-2}\\
\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Experiment group}}\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{Test time} &
Pre &
Post &
p-value\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{5-6:00am} &
8.40\ensuremath{\pm}0.52 &
6.20\ensuremath{\pm}0.42 &
{\textless}0.0001***\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{12-1:00pm} &
7.70\ensuremath{\pm}0.67 &
6\ensuremath{\pm}0.00 &
{\textless}0.0001***\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{5-6:00pm} &
7.20\ensuremath{\pm}0.42 &
5.50\ensuremath{\pm}0.53 &
{\textless}0.0001***\\
\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Control group}}\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{5-6:00am} &
8.20\ensuremath{\pm}0.42 &
8.20\ensuremath{\pm}0.42 &
1\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{12-1:00pm} &
7.40\ensuremath{\pm}0.70 &
7.40\ensuremath{\pm}0.70 &
1\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{5-6:00pm} &
6.60\ensuremath{\pm}0.70 &
6.60\ensuremath{\pm}0.70 &
1\\
\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{Day-3}\\
\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Experiment group}}\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{Test time} &
Pre &
Post &
p-value\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{5-6:00am} &
7\ensuremath{\pm}0 &
5.30\ensuremath{\pm}0.48 &
{\textless}0.0001***\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{12-1:00pm} &
6.10\ensuremath{\pm}0.32 &
3.90\ensuremath{\pm}0.57 &
{\textless}0.0001***\\
\multicolumn{2}{p{\dimexpr(.3182\linewidth-2\tabcolsep)}}{5-6:00pm} &
5.10\ensuremath{\pm}0.32 &
3\ensuremath{\pm}0.00 &
{\textless}0.0001***\\
\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Control group}}\\
5-6:00am &
\multicolumn{2}{p{\dimexpr(.29069999999999993\linewidth-2\tabcolsep)}}{6.10\ensuremath{\pm}0.74} &
6.10\ensuremath{\pm}0.74 &
1\\
12-1:00pm &
\multicolumn{2}{p{\dimexpr(.29069999999999993\linewidth-2\tabcolsep)}}{5.80\ensuremath{\pm}0.63} &
5.80\ensuremath{\pm}0.63 &
1\\
5-6:00pm &
\multicolumn{2}{p{\dimexpr(.29069999999999993\linewidth-2\tabcolsep)}}{5.10\ensuremath{\pm}0.74} &
5.10\ensuremath{\pm}0.74 &
1\\
\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{Day-4}\\
\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Experiment group}}\\
Test time &
\multicolumn{2}{p{\dimexpr(.29069999999999993\linewidth-2\tabcolsep)}}{Pre} &
Post &
p-value\\
5-6:00am &
\multicolumn{2}{p{\dimexpr(.29069999999999993\linewidth-2\tabcolsep)}}{3.90\ensuremath{\pm}0.57} &
2.60\ensuremath{\pm}0.70 &
0.0002\\
12-1:00pm &
\multicolumn{2}{p{\dimexpr(.29069999999999993\linewidth-2\tabcolsep)}}{3.70\ensuremath{\pm}0.48} &
2.10\ensuremath{\pm}0.32 &
{\textless}0.0001***\\
5-6:00pm &
\multicolumn{2}{p{\dimexpr(.29069999999999993\linewidth-2\tabcolsep)}}{2.90\ensuremath{\pm}0.32} &
1.40\ensuremath{\pm}0.52 &
{\textless}0.0001***\\
\multicolumn{5}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Control group}}\\
5-6:00am &
\multicolumn{2}{p{\dimexpr(.29069999999999993\linewidth-2\tabcolsep)}}{4.70\ensuremath{\pm}0.67} &
4.70\ensuremath{\pm}0.67 &
1\\
12-1:00pm &
\multicolumn{2}{p{\dimexpr(.29069999999999993\linewidth-2\tabcolsep)}}{4\ensuremath{\pm}0.67} &
4\ensuremath{\pm}0.67 &
1\\
5-6:00pm &
\multicolumn{2}{p{\dimexpr(.29069999999999993\linewidth-2\tabcolsep)}}{3.80\ensuremath{\pm}0.42} &
3.90\ensuremath{\pm}0.32 &
0.5567 \\
\tblbottomrule
\end{tabulary}\par
\end{table*}
\begin{table*}[!htbp]
\caption{\boldmath {Comparison of pre values of pain score among control and experimental groups (Data was presented asfrequency and percentage). (Exp-experimental group, Con- control group)} }
\label{tw-ca503781cadb}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2362\linewidth-2\tabcolsep}p{\dimexpr.2482\linewidth-2\tabcolsep}p{\dimexpr.2656\linewidth-2\tabcolsep}p{\dimexpr.25\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}\multicolumn{4}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{Day-1}\\
\tblmidrule
Test time &
Con Pre &
Exp Pre &
p-value\\
5-6:00pm &
9.10\ensuremath{\pm}1.32 &
9.10\ensuremath{\pm}0.32 &
1\\
\multicolumn{4}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Day-2}}\\
Test time &
Con Pre &
Exp Pre &
p-value\\
5-6:00am &
8.20\ensuremath{\pm}0.42 &
8.40\ensuremath{\pm}0.52 &
0.3566\\
12-1:00pm &
7.40\ensuremath{\pm}0.70 &
7.70\ensuremath{\pm}0.67 &
0.3405\\
5-6:00pm &
6.60\ensuremath{\pm}0.70 &
7.20\ensuremath{\pm}0.42 &
0.0320*\\
\multicolumn{4}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Day-3}}\\
Test time &
Con Pre &
Exp Pre &
p-value\\
5-6:00am &
6.10\ensuremath{\pm}0.74 &
7\ensuremath{\pm}0 &
0.0012**\\
12-1:00pm &
5.80\ensuremath{\pm}0.63 &
6.10\ensuremath{\pm}0.32 &
0.0937\\
5-6:00pm &
5.10\ensuremath{\pm}0.74 &
5.10\ensuremath{\pm}0.32 &
1\\
\multicolumn{4}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Day-4}}\\
Test time &
Con Pre &
Exp Pre &
p-value\\
5-6:00am &
4.70\ensuremath{\pm}0.67 &
3.90\ensuremath{\pm}0.57 &
0.0101*\\
12-1:00pm &
4\ensuremath{\pm}0.67 &
3.70\ensuremath{\pm}0.48 &
0.2648\\
5-6:00pm &
3.80\ensuremath{\pm}0.42 &
2.90\ensuremath{\pm}0.32 &
{\textless}0.0001***\\
\tblbottomrule
\end{tabulary}\par
\end{table*}
\begin{table*}[!htbp]
\caption{\boldmath {Comparison of post values of pain score among control and experimental groups (Data was presented as frequency and percentage). (Exp-experimental group, Con- control group)} }
\label{tw-833124d8b576}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2344\linewidth-2\tabcolsep}p{\dimexpr.25\linewidth-2\tabcolsep}p{\dimexpr.2656\linewidth-2\tabcolsep}p{\dimexpr.25\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}\multicolumn{4}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{Day-1}\\
\tblmidrule
Test time &
Con Post &
Exp Post &
p-value\\
5-6:00pm &
9.10\ensuremath{\pm}1.32 &
6.70\ensuremath{\pm}0.48 &
{\textless}0.0001***\\
\multicolumn{4}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Day-2}}\\
Test time &
Con Post &
Exp Post &
p-value\\
5-6:00am &
8.20\ensuremath{\pm}0.42 &
6.20\ensuremath{\pm}0.42 &
{\textless}0.0001***\\
12-1:00pm &
7.40\ensuremath{\pm}0.70 &
6\ensuremath{\pm}0.00 &
{\textless}0.0001***\\
5-6:00pm &
6.60\ensuremath{\pm}0.70 &
5.50\ensuremath{\pm}0.53 &
0.0009\\
\multicolumn{4}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Day-3}}\\
Test time &
Con Post &
Exp Post &
p-value\\
5-6:00am &
6.10\ensuremath{\pm}0.74 &
5.30\ensuremath{\pm}0.48 &
0.0102\\
12-1:00pm &
5.80\ensuremath{\pm}0.63 &
3.90\ensuremath{\pm}0.57 &
{\textless}0.0001***\\
5-6:00pm &
5.10\ensuremath{\pm}0.74 &
3\ensuremath{\pm}0.00 &
{\textless}0.0001***\\
\multicolumn{4}{p{\dimexpr(1\linewidth-2\tabcolsep)}}{\textbf{Day-4}}\\
Test time &
Con Post &
Exp Post &
p-value\\
5-6:00am &
4.70\ensuremath{\pm}0.67 &
2.60\ensuremath{\pm}0.70 &
{\textless}0.0001***\\
12-1:00pm &
4\ensuremath{\pm}0.67 &
2.10\ensuremath{\pm}0.32 &
{\textless}0.0001***\\
5-6:00pm &
3.90\ensuremath{\pm}0.32 &
1.40\ensuremath{\pm}0.52 &
{\textless}0.0001***\\
\tblbottomrule
\end{tabulary}\par
\end{table*}
The present study was undertaken to observe the\textbf{\space }effectiveness of acupressure on pain management among mediosternotomy patients. There was a significant decrease in the pain scores followed by the acupressure. Patients are undergoing the heart surgery experiences immense pain due to the opening of the sternum, damage of the tissues and inflammation \unskip~\citep{558485:12830483,558485:12830482}. Management of postoperative pain of these patients is essential as ill management not only delays the discharge of the patient but also causes disorders of respiratory, digestive, cardiovascular and other systems \unskip~\citep{558485:12830484,558485:12830485}. Both pharmacological and non-pharmacological therapies are currently used for the management of the post-operative pain of the patients.
Opioids are the most commonly used drugs in the management of the pain of the patients \unskip~\citep{558485:12830486}. Though opioids relieve pain effectively, they are associated with side effects and also need a long-term stay of the patient in the hospital. Hence, the alternative therapies like acupuncture, music therapy, exercises etc. are widely used as they are associated with minimal or no side effects. One such method is acupressure. However, there are limited studies in this aspect. Hence, the present study was undertaken to provide scientific evidence to implement the acupressure also as a complementary therapy in the management of post-operative pain of the patients. Acupressure was reported as a safe method in relieving the pain of the patients \unskip~\citep{558485:12830487,558485:12830488,558485:12830489,558485:12830490}. Acupressure is performed by applying the physical pressure on specific points of the body surface by means of energy circulation. Though it appears similar to the acupuncture, in this method, the pressure is applied only using the fingers, hands, palms, wrists and knees in order to provide an internal flow of energy. Thus, the method is non-invasive, effective and safer method compared to acupuncture.
\textbf{\space }The present study results provide scientific evidence for the effectiveness of acupressure in the management of the post-operative pain of the patients. The study results are in accordance with the earlier studies which explained that the acupressure is an effective method in relieving different kinds of pain \unskip~\citep{558485:12830491,558485:12830492}.
\section{Conclusion}
There was a significant decrease in the pain score of the participants\textbf{, } followed by acupressure. The study provides further evidence for the effectiveness of the acupressure in pain management and also recommends detailed research in this area.
\bibliographystyle{pharmascope_apa-custom}
\bibliography{\jobname}
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\section*{2003}
Oriol Rodriguez-Leon, Josefina Mauri, Eduard Fernandez-Nofrerias, C.Garcia, R.Villuendas, Vicente del Valle, et al. (2003).\textbf{" Reconstruction of a spatio-temporal model of the intima layer from intravascular ultrasound sequences" } . \textit{European Heart Journal}, .
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\clan {Primož Potočnik}
%--------------------------------------------------------
% A. objavljene znanstvene monografije
%--------------------------------------------------------
%\begin{skupina}{A}
%\disertacija
% {NASLOV}
% {UNIVERZA}
% {FAKULTETA}
% {ODDELEK}
% {KRAJ} {DRZAVA} {LETO}
%\magisterij
% {NASLOV}
% {UNIVERZA}
% {FAKULTETA}
% {ODDELEK}
% {KRAJ} {DRZAVA} {LETO}
%\monografija
% {AVTORJI}
% {NASLOV}
% {ZALOZBA}
% {KRAJ} {DRZAVA} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% B. raziskovalni clanki sprejeti v objavo v znanstvenih
% revijah in v zbornikih konferenc
%--------------------------------------------------------
%\begin{skupina}{B}
%\sprejetoRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA}
%\sprejetoZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
\begin{skupina}{B}
\sprejetoRevija
{B.~Kuzman, A.~Malni\v{c},\crta}
{On vertex- and edge-transitive graphs over doubled cycles}
{J. Combin.\ Theory Ser.\ B}
\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% C. raziskovalni clanki objavljeni v znanstvenih revijah
% in v zbornikih konferenc
%--------------------------------------------------------
%\begin{skupina}{C}
%\objavljenoRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA} {LETNIK} {LETO} {STEVILKA} {STRANI}
%\objavljenoZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
% {ZBORNIK} {STRANI}
%\end{skupina}
\begin{skupina}{C}
\objavljenoRevija
% 1.01: {\bf 1}. POTO\v{C}NIK, Primo\v{z}, \v{S}PARL, Primo\v{z}. On the radius and the attachment number of tetravalent half-arc-transitive graphs. {\it Discrete Math.}, 2017, vol. 340, iss. 12, str. 2967-2971. $[$COBISS.SI-ID 18142297$]$\\
{\crta, P.~\v{S}parl}
{On the radius and the attachment number of tetravalent half-arc-transitive graphs}
{Discrete Math.} {340} {2017} {} {2967--2971}
\objavljenoRevija
% 1.01: %list {\bf 4}. POTO\v{C}NIK, Primo\v{z}, SPIGA, Pablo, VERRET, Gabriel. Asymptotic enumeration of vertex-transitive graphs of fixed valency. {\it J. Combin.\ Theory Ser.\ B}, 2017, vol. 122, str. 221-240. $[$COBISS.SI-ID 17743705$]$\\
{\crta, P.~Spiga, G.~Verret}
{Asymptotic enumeration of vertex-transitive graphs of fixed valency}
{J. Combin.\ Theory Ser.\ B} {122} {2017} {} {2{2}1--240}
\objavljenoRevija
% 1.01: %list {\bf 2}. POTO\v{C}NIK, Primo\v{z}, WILSON, Stephen. The separated box product of two digraphs. {\it European J. Combin.}, 2017, vol. 62, str. 35-49. $[$COBISS.SI-ID 17840729$]$\\
{\crta, S.~Wilson}
{The separated box product of two digraphs}
{European J. Combin.} {62} {2017} {} {35--49}
\objavljenoRevija
% 1.01: {\bf 3}. POTO\v{C}NIK, Primo\v{z}, PO\v{Z}AR, Rok. Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order 8. {\it J. Combin. \Theory Ser.\ A}, 2017, vol. 145, str. 172-183. $[$COBISS.SI-ID 1538771652$]$\\
{\crta, R.~Požar}
{Smallest tetravalent half-arc-transitive graphs with the vertex-stabiliser isomorphic to the dihedral group of order 8}
{J. Combin.\ Theory Ser.\ A} {145} {2017} {} {172--183}
\end{skupina}
%--------------------------------------------------------
% D. urednistvo v znanstvenih revijah in zbornikih
% znanstvenih konferenc
%--------------------------------------------------------
%\begin{skupina}{D}
%\urednikRevija
% {OPIS}
% {REVIJA}
%\urednikZbornik
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
\begin{skupina}{D}
\urednikRevija
{Urednik}
{Ars Mathematica Contemporanea}
\urednikRevija
{Urednik}
{The Art of Discrete and Applied Mathematics}
\urednikRevija
{Gostujoči urednik}
{Discrete Applied Mathematics}
\hspace{-8.5mm} D2. Član uredniškega odbora spletne enciklopedije {\it Encyclopedia of graphs}, \url{http://atlas.gregas.eu}
\end{skupina}
%--------------------------------------------------------
% E. organizacija mednarodnih in domacih znanstvenih
% srecanj
%--------------------------------------------------------
%\begin{skupina}{E}
%\organizacija
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% F. vabljena predavanja na tujih ustanovah in
% mednarodnih konferencah
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%\begin{skupina}{F}
%\predavanjeUstanova
% {NASLOV}
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%\predavanjeKonferenca
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% G. aktivne udelezbe na mednarodnih in domacih
% konferencah
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%\begin{skupina}{G}
%\konferenca
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%\end{skupina}
%\begin{skupina}{G}
%\konferenca
% 1.12: {\bf 5}. MALNI\v{C}, Aleksander, KUZMAN, Bo\v{s}tjan, POTO\v{C}NIK, Primo\v{z}. Vertex and edge transitive graphs over doubled cycles. V: 2nd Malta Conference in Graph Theory and Combinatorics 2017 (MCGTC-2017), 26-30 June 2017, Qawra, Malta. {\it $[$Abstracts$]$}}. Msida: University of Malta, Department of Mathematics, Faculty of Science, 2017, str. 20-21. $[$COBISS.SI-ID 18067801$]$\\
% {Vertex and edge transitive graphs over doubled cycles}
% {2\und Malta Conference in Graph Theory and Combinatorics}
% {Qawra} {Malta} {junij} {2017}
%\end{skupina}
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% H. strokovni clanki
%--------------------------------------------------------
%\begin{skupina}{H}
%\clanekRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA} {LETNIK} {LETO} {STEVILKA} {STRANI}
%\clanekZbornik
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% {KONFERENCA}
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% {ZBORNIK} {STRANI}
%\end{skupina}
% Ni podatkov za to sekcijo
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% I. razno
%--------------------------------------------------------
%\begin{skupina}{I}
%\razno
% {OPIS}
%\end{skupina}
%\begin{skupina}{I}
%POZOR: Bibliografija2017.tex > 2017\rac\clani\potocnik.tex 8849/218: Stevilo neopredeljenih zadetkov: 3
%\razno
% Ured: {\bf 6}. {\it Ars Math.\ Contemp.}. Poto\v{c}nik, Primo\v{z} (\v{c}lan uredni\v{s}kega odbora 2012-). $[$Tiskana izd.$]$. Ljubljana: Dru\v{s}tvo matematikov, fizikov in astronomov, 2008-. ISSN 1855-3966. $[$COBISS.SI-ID 239049984$]$\\
%\razno
% Ured: %list {\bf 7}. {\it Discrete Appl.\ Math.}. Poto\v{c}nik, Primo\v{z} (gostujo\v{c}i urednik 2017). $[$Print ed.$]$. Amsterdam: Elsevier, 1979-. ISSN 0166-218X. $[$COBISS.SI-ID 25342464$]$\\
%\razno
% Ured: %list {\bf 8}. {\it Encyclopedia of graphs}. Poto\v{c}nik, Primo\v{z} (\v{c}lan uredni\v{s}kega odbora 2012-). Ljubljana: Abelium, raziskave in razvoj, 2012-. ISSN 2335-2825. $[$COBISS.SI-ID 264926976$]$\\
%\end{skupina}
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% gostovanja
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%\begin{seznam}
%\gostovanje {IME} {TRAJANJE} {USTANOVA} {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{seznam}
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%\documentclass[12pt]{book}
\input{../preamble-regulation.tex}
%------------------------------------------------------------
\begin{document}
\parindent 16pt
%--------------------------------------
\setcounter{chaptercounter}{7}
\refstepcounter{chaptercounter} \label{chap07}
\setcounter{page}{1}
\pagestyle{fancy} \lhead[\ref{chap07}--\thepage]{} \chead{Monopoly} \rhead[]
{\ref{chap07}--\thepage} \cfoot{}
%--------------------------------------
\vspace*{\fill}
\begin{center}
\begin{Huge}
{\bf \ref{chap07}: Monopoly} \label{s05.0} \\
\end{Huge}
\vspace*{\fill}
\begin{quotation}
\begin{Large}
{\it ``He that withholdeth corn, the people shall curse him: but blessing shall be upon the head of
him that selleth it." }
\end{Large}
\end{quotation}
\vspace*{\fill}
\includegraphics[width=\linewidth]{fig06-bond-US-Steel.jpg}
\end{center}
\vspace*{\fill}
{\small \hfill {\it April 4, 2018 }}\\
{\small \hfill \url{http://rasmusen.org/g406/reg-rasmusen.htm}, [email protected]}
%------------------------------------------------------
\newpage
\noindent
{\bf \ref{chap07}.1. Choosing Price and Output to Maximize Profits}\label{s05.1}
Monopoly is a classic source of market failure. Being the only firm in the industry means that the firm has market power, defined as the ability to increase the price without losing all its customers. All firms have this to some extent, but if market power is significant, the firm may decide to restrict output in order to drive up the price. In diagrams, we depict market power as a downward-sloping demand curve facing the individual firm, as opposed to a downward-sloping demand curve for the market as a whole.
Since the demand curve facing it slopes downward, a firm with market power can increase its producer surplus by reducing output and raising the price. This power is not value-creating in itself. It just shifts surplus from consumer to producer. Moreover, since the producer purposely foregoes some sales that would generate surplus, to push up the price, consumers lose more than producers gain. Still another problem is that producers may devote effort to acquiring market power, creating additional waste.\footnote{An important exception is effort in the form of innovation. If the inventor of a new product is given a monopoly on it, that actually raises surplus if he wouldn't invent the product otherwise, so it's worth providing him with profit as an incentive. If a firm in a competitive market puts effort into discovering a new market as yet unserved, so it has a monopoly, that too is innovation that deserves reward. }
A firm without market power simply takes the market price as given and to maximize profit compares the cost of output to that price. It increases output $Q$ until the marginal cost equals the price: $MC(Q) =P$. A firm with market power needs to think about {\bf marginal revenue}\index{marginal revenue}: the change in revenue from an increase in quantity. Marginal revenue equals the market price if the firm has no market power: sell one more unit, and revenue increases by the price. Marginal revenue is less than the price for a firm with market power: produce one more unit, and the price drops a little.
%\vspace*{24pt}
\noindent
%\begin{minipage}[c]{\linewidth}
\begin{figure}[htb]
\begin{center}
{\sc Figure \ref{chap07}.\ref{fig05-1.pdf} \\
Marginal Revenue }
% \vspace*{-12pt}
\refstepcounter{figurecounter} \label{fig05-1.pdf}
\includegraphics[width=.8\linewidth]{fig05-1-monopoly.jpg}
\end{center}
\end{figure}
%\vspace*{24pt}
The firm's tradeoff is between (a) selling more quantity and (b) keeping the price high, as Figure \ref{chap07}.\ref{fig05-1.pdf} shows. If $Q=4$ and $P=8$ and the firm increases its output to $Q= 5$, revenue won't rise by 8, because the price will fall. The price falls from 8 to 7, so revenue only rises from 32 to 35, an increase of 3. The old 4 units are now sold at 7 instead of 8, so they are collecting 28 for the firm instead of 32, a loss of 4. The new unit is collecting the new price of 7, so it is adding 7. There is a net gain of 3 in revenue. That gain is the marginal revenue: the change in revenue when quantity rises.
The firm's marginal cost is 4 at $Q=4$, however, so increasing output to 5 reduces producer surplus.
To maximize surplus, the seller should set $Q$ so that marginal revenue equals marginal cost. Since the
marginal revenue curve lies below the demand curve, this means that the marginal revenue curve will hit marginal cost before the demand curve, so the MR=MC rule produces less output than the P=MC rule.
The simplest case is when the seller has zero marginal cost. Then its task is to set $MR=0$, which
means to maximize revenue because it picks quantity so that the slope of the revenue function (which is marginal revenue) is zero. An example is a stadium selling tickets to a football game. If the
stadium is not filled up, the marginal cost of admitting an extra person is approximately zero. To
maximize profit, however, some seats should be left unfilled. Increasing ticket sales would require
a drop in price that would actually reduce revenue. The stadium example illustrates why market power
leads to market failure: it would cost nothing to let more people enjoy the football game, but the
seller keeps them out anyway.
We often use linear (straight line diagonal) demand curves in diagrams. When demand is linear, marginal
revenue is linear too, with twice as steep a slope (as in Figure \ref{chap07}.1).\footnote{Calculus
shows this. If demand is $P= a- b P$, then revenue is $R= PQ =(a- b Q)Q = aQ- bQ^2$,
and marginal revenue is $\frac{dR}{dQ} = a - 2bQ$, just like demand but with a 2 so it descends
twice as fast.} In drawing, remember that if the demand curve hits the quantity axis at $Q=10$,
then the marginal revenue curve hits at $Q=5$. Also, though the demand curve becomes flat at $P=0$, the marginal revenue curve keeps going down below the $P=0$ axis. That is because marginal revenue can be negative, which just means that increasing output reduces revenue because the quantity rises by less than the price falls.
Another way to think of pricing with market power is using elasticities. The{\bf price elasticity of
demand} (which is easier to remember if you think of it as ``price sensitivity of demand'') is
\begin{equation}
Elasticity = \frac{\text{ \% change in quantity demanded}}{ \text{\% change in price}}
\end{equation}
Iff the quantity demanded falls by 20\% after the price rises by 10\%, the elasticity of demand
is -2. Economists often just say that the elasticity is ``2'',
because the quantity demanded falls with price so we can assume the elasticity is negative. A firm with
market power is trading off price against quantity, so the elasticity is relevant to its decision.
If the elasticity is -1, marginal revenue is zero: if the firm increases output by 1\%, the price
falls by 1\% and revenue is unchanged. We say that demand is {\bf inelastic} \index{inelastic}if
the price elasticity of demand is less than 1, in which case the quantity demanded falls less than
the price rises. A monopoly never wants to choose a price and quantity where demand is inelastic,
because reducing sales at that point by 1\% would drive up the price more than 1\% and thus raise
revenue--- and better yet, total production cost would because of the 1\% reduction. Demand is {\bf perfectly inelastic}
\index{perfectly inelastic} if quantity demanded does not change at all when the price increases. If demand is perfectly inelastic, the demand curve is vertical and the elasticity is 0.
A monopoly would want to raise its price if demand
were perfectly inelastic. Demand is {\bf perfectly elastic} if the quantity changes infinitely (to
zero) as the price rises. Then, the demand curve facing the firm has an elasticity of $-$infinity
and is perfectly flat, just as under perfect competition.
A general rule relating the monopoly markup of price over marginal cost to the elasticity of demand
is the {\bf Lerner Rule}\index{Lerner Rule}.\footnote{The original paper is Abba P.
Lerner, \index{Lerner} ``The Concept of Monopoly and the Measurement of Monopoly Power,''
{\it Review
of Economic Studies}, 1: 157--175 (1934). A good verbal discussion of its modern use is Kenneth G.
Elzinga,\index{Elzinga} \& David E. Mills,\index{Mills}
\href{http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1884993}{``The Lerner Index of
Monopoly Power: Origins and Uses,'' }{\it American Economic Review: Papers and Proceedings},
101(3): 558--564 (2011). } It says that at the profit-maximizing price:
\footnote{The rule $MR=MC$ can be written as $\frac{dP}{dQ} Q + P =MC$, so $-\frac{dP}{dQ}Q = P -MC$
and $-\frac{dP}{dQ}\frac{Q}{P} = \frac{P -MC}{P}$, but $Elasticity = \frac{dQ}{dP}\frac{P}{Q}$ and
so we get the Lerner Rule. }
\begin{equation} \label{lerner1}
\frac{ P - MC}{P} = -\left( \frac{ 1}{ Elasticity} \right)
\end{equation}
Remember: at the optimal price, demand must have gotten elastic, more negative than $-1$.
If the elasticity of demand is $-2$, the Lerner Rule tells us that the price is such that $\frac{ P - MC}{P} = -\left( \frac{ 1}{ Elasticity} \right) = - \frac{1}{-2}$, so $P-MC = \frac{P}{2}$ and $P = 2MC$. If the elasticity of demand is $-1.5$, the price
is three times marginal cost, by the same working out of the equation.
The Lerner Rule can be used in two ways. The firm can use it to help choose its price. The
analyst can use it to estimate the elasticity of demand facing the firm. The expression $\frac{P-MC}
{P} $ on the left-hand-side of equation \eqref{lerner1} is known as the {\bf Lerner Index} of
market power. Its biggest possible value is 1 (if MC=0, so $\frac{P-MC}{P}=1 $) and its smallest
possible value is 0 (if $P=MC$). An analyst who saw that a firm's price was 20 and its marginal
cost was 15 could use equation \eqref{lerner1} to conclude that the Lerner Index was $\frac{P-MC}
{P}= \frac{20-15}{20} = .25$ and the elasticity of demand was $-4$. Thus,
to find out the elasticity of demand facing the firm, the measure of market power we are after, we
just need to know the price and the marginal cost.
\bigskip
\noindent
{\bf \ref{chap07}.2: Oligopoly and the Cournot Model}\label{s05.2}
Price fixing is a form of {\bf collusion}\index{collusion}--- firms agreeing on prices, outputs, or
sales territories--- and is {\it per se} illegal. Even if firms do not collude, however, prices and
profits can still be above marginal cost if entry into the industry is difficult and the
number of firms is small. If the cola industry has just two firms, then even if they never discuss
prices with each other, they know that it is to their mutual benefit to keep prices high and that a
price cut of one of them will be followed by a price cut of the other. Without communication between
them, though, it is hard to agree on a price in the first place and even harder to monitor cheating
than under a cartel. Thus, we expect prices in a two-firm industry--- a {\bf duopoly}
\index{duopoly}--- or a several-firm industry--- an {\bf oligopoly}\index{oligopoly}--- to be lower
than in a monopoly or cartel, but higher than in a perfectly competitive industry. Note, however, that this
depends on the {\bf incumbent firms}\index{incumbent firms}--- the ones already in the industry---
being safe from {\bf entrants}\index{entrants}--- new firms who start selling in the industry
attracted by the high profits. If it is easy to start up a new business in the industry ({\bf free
entry})\index{free entry} then even a monopoly could not get away with earning high profits very long.
The {\bf Cournot model}\index{Cournot model} is the basic model economists use to explain why
prices fall with the number of firms when there is not free entry. In the model, each firm chooses
its output given its expectations about the other firms' outputs. The market price is
determined by how much consumers will pay for the total output of all the firms.
Let's look at a specific example of a Cournot model.
Let there be $n$ firms, each firm producing its output $q_i$ $ i = 1, \ldots, n$ with no
fixed cost and with a constant marginal cost of 20. Let market demand be
\begin{equation} \label{eqx}
Q = 140-P,
\end{equation}
where $Q$ is the total output: $Q = q_1+q_2+ \ldots q_n$. Firm $i$'s profit function is then
\begin{equation} \label{eqx}
\pi_i = Pq_i - 20q_i .
\end{equation}
\bigskip
%-------------------------------------------------------
\newpage
\noindent
{\bf One Firm: Monopoly}
First, suppose $n=1$, so Firm 1 is a monopoly. Then, $P = 140-q_1$, so Firm 1's profit, shown
in Figure \ref{chap07}.\ref{fig06-6-profit.jpg}, is
\begin{equation} \label{eqx}
\pi_1 = (140-q_1)q_1 - 20q_1 = (140q_1 -q_1^2) - 20q_1
\end{equation}
\begin{center}
\vspace*{24pt}
\begin{minipage}[c]{.8\linewidth}
\begin{center}
{\sc Figure \ref{chap07}.\ref{fig06-6-profit.jpg} \\
Monopoly Profits with One Seller } \\
\refstepcounter{figurecounter} \label{fig06-6-profit.jpg}
\includegraphics[width=\linewidth]{fig06-6-profit.jpg}
\end{center}
\end{minipage}
%\vspace*{24pt}
\end{center}
To maximize the firm's profit, use a little calculus. Take profit's derivative with respect to
$q_1$ and set the derivative equal to zero, the equation called the {\bf first-order condition}
\index{first-order condition} for maximization. That yields
\begin{equation} \label{eqx}
\frac{d \pi_1}{d q_1} = (140 -2q_1 ) - 20 =0
\end{equation}
which solving out for $q_1$ yields $q_1 = 60$. Then, using the demand function, $P=80$, so $\pi_1 =
(80)(60) - 20(60) = 3,600.$
This is the same result as we would get by setting marginal revenue equal to marginal cost,
because the last equation can be rewritten as
\begin{equation} \label{eqx}
\frac{d \pi_1}{d q_1} = (Marginal \; revenue) - Marginal \; cost =0
\end{equation}
Recall that with linear demand,the marginal revenue curve slopes down twice as fast as the demand
curve, so with $P=140-Q$, we get $MR = 140-2Q$. So calculus is a way to generate that rule.
\bigskip
%-------------------------------------------------------
\noindent
{\bf Two Firms: Duopoly--- The Asymmetric-Costs Case}
So much for monopoly. Let's do the same thing for duopoly. We will start with an asymmetric case, in which Firm 1 has a cost advantage over Firm 2: Firm 1's marginal cost is $c_1=20$ and Firm 2's is $c_2=40$.
As before, $P = 140-(q_1+q_2)$, so Firm
1's profit is:
\begin{equation} \label{eqx}
\pi_1 (q_1)= (140-q_1-q_2)q_1 - 20q_1 = (140q_1 -q_1^2 - q_2q_1) - 20q_1.
\end{equation}
Take profit's derivative with respect to $q_1$ and set it to zero:
\begin{equation} \label{eqx}
\frac{d \pi_1}{d q_1} = (140 -2q_1 -q_2) - 20 =0.
\end{equation}
This gives us the {\bf reaction function } \index{reaction function} or {\bf reaction curve}
\index{reaction curve} for $q_1$ as a function
of $q_2$. The more output Firm 1 expects from Firm 2, the smaller will Firm 1 choose its own output
to be:
\begin{equation} \label{eqx}
q_1 (q_2) = 60 -\frac{q_2}{2}.
\end{equation}
At the extremes, if Firm 1 thinks Firm 2 will set $q_2=0$, then Firm 1 chooses (``reacts with'')
its monopoly output of 60; but if Firm 1 thinks Firm 2 will set $q_2=120$ and drive the price down
to marginal cost, then Firm 1 chooses to produce zero. Firm 1's reaction curve goes
from 60 on the y-axis to 120 on the x-axis.
\bigskip
Now let's do the same for Firm 2. Firm
2's profit is:
\begin{equation} \label{eqx}
\pi_2(q_2) = (140-q_1-q_2)q_2 - 40q_2 = (140q_2 -q_2^2 - q_2q_1) - 40q_2.
\end{equation}
Take profit's derivative with respect to $q_2$ and set it to zero:
\begin{equation} \label{eqx}
\frac{d \pi_2}{d q_2} = (140 - 2q_2 - q_1 ) - 40 =0.
\end{equation}
This gives us the {\bf reaction function } for $q_2$ as a function
of $q_1$. The more output Firm 2 expects from Firm 1, the smaller will Firm 2 choose its own output
to be:
\begin{equation} \label{eqx}
q_2 (q_1) = 50 -\frac{q_1}{2}.
\end{equation}
At the extremes, if Firm 2 thinks Firm 1 will set $q_1=0$, then Firm 2 chooses (``reacts using'')
its monopoly output of 50 ($not$ Firm 1's monopoly output, which is 60---why not?). If Firm 2 thinks Firm 1 will set $q_1=100$ and drive the price down
to Firm 2's marginal cost of 40, then Firm 2 chooses to produce zero. Firm 2's reaction curve goes
from 100 on the y-axis to 50 on the x-axis.
What is the value of $q_2$ that Firm 2 will actually pick?
The two reaction equations give us two equations for two unknowns, $q_1$ and $q_2$. We can substitute to get:
\begin{equation} \label{eqx}
q_2 (q_1) = 50 - \frac{60-\frac{q_2}{2} } {2} = 50 - 30+ \frac{q_2}{4} = 20+\frac{q_2}{4}.
\end{equation}
Solving this last equation yields $ q_2^*= \frac{80}{3} = 26 \frac{2}{3}$. We can substitute $q_2^*$ into $q_1(q_2) $ to get $q_1^*= 60 - \frac{\frac{80}{3} }{2} =\frac{140}{3}= 46 \frac{2}{3}$.
Using the demand function, $P = 140- \frac{140}{3} - \frac{80}{3} =67 \frac{1}{3} $.
%\vspace*{24pt}
\begin{center}
\begin{minipage}[c]{.8\linewidth}
\begin{center}
{\sc Figure \ref{chap07}.\ref{fig05-2a-asym-cournot.jpg} \\
Asymmetric Cournot Duopoly Reaction Curves } \\
\medskip
\refstepcounter{figurecounter} \label{fig05-2a-asym-cournot.jpg}
\includegraphics[width=\linewidth]{fig05-2a-asym-cournot.jpg}
\end{center}
\end{minipage}
\end{center}
\vspace*{24pt}
\bigskip
%-------------------------------------------------------
\noindent
{\bf Two Firms: Duopoly--- The Symmetric-Costs Case}
Let's do the same thing for a duopoly in which both firms have the same cost curves. This is simpler than the asymmetric-cost Cournot duopoly, but its ease is deceptive, because it doesn't show so clearly how each firm is acting independently to maximize its own profits. As before, $P = 140-(q_1+q_2)$, so Firm
1's profit is
\begin{equation} \label{eqx}
\pi_1 = (140-q_1-q_2)q_1 - 20q_1 = (140q_1 -q_1^2 - q_2q_1) - 20q_1
\end{equation}
Again take profit's derivative with respect to $q_1$ and set it to zero.
\begin{equation} \label{eqx}
\frac{d \pi_1}{d q_1} = (140 -2q_1 -q_2) - 20 =0
\end{equation}
This gives us a reaction function (as shown in Figure \ref{chap07}.\ref{fig06-2.jpg}) for $q_1$ as a function
of $q_2$. The more output Firm 1 expects from Firm 2, the smaller will Firm 1 choose its own output
to be.
\begin{equation} \label{eqx}
q_1 (q_2) = 60 -\frac{q_2}{2}.
\end{equation}
At the extremes, if Firm 1 thinks Firm 2 will set $q_2=0$, then Firm 1 chooses (``reacts with'')
the monopoly output of 60; but if Firm 1 thinks Firm 2 will set $q_2=120$ and drive the price down
to marginal cost, then Firm 1 chooses to produce zero. That is why Firm 1's reaction curve goes
from 60 on the y-axis to 120 on the x-axis.
%\vspace*{24pt}
\begin{center}
\begin{minipage}[c]{.8\linewidth}
\begin{center}
{\sc Figure \ref{chap07}.\ref{fig06-2.jpg} \\
Cournot Duopoly Reaction Curves } \\
\medskip
\refstepcounter{figurecounter} \label{fig06-2.jpg}
\includegraphics[width=\linewidth]{fig06-2.jpg}
\end{center}
\end{minipage}
\end{center}
\vspace*{24pt}
What is the value of $q_2$ that Firm 2 will actually pick? If we were to set up the same problem for Firm 2's choice of $q_2$ to maximize
profit and solve the two first order conditions together, it turns out that we would find that $q_1=
q_2$. Figure \ref{chap07}.\ref{fig06-2.jpg} shows Firm 2's reaction curve, and how it crosses Firm
1's reaction curve at $q_1=q_2=40$. Rather than finding the equation for Firm 2's reaction
curve, though, we can set $q_2=q_1$ in Firm 1's reaction curve, so
\begin{equation} \label{eqx}
q_1 = 60 -\frac{q_1}{2},
\end{equation}
which solves out to $q_1 = 40$. As a result, $q_2=q_1=40$ and $Q = q_1+q_2 = 80.$ Using the demand
function, $P = 60$. Profits are $pi_1 = \pi_2 = (60)(40) - 20 (40) = 1,600$ each. Duopoly industry
profit is 3,200, which is below the monopoly industry profit of 3,600, but above the competitive
profit of 0 that would result if $P =MC =20$.
Figure \ref{chap07}.\ref{fig06-2.jpg} shows another line going from 30 on the y-axis to 30 on the
x-axis, labelled ``Cartel Output''. This is the combinations of $q_1$ and $q_2$ that add up to 60,
the monopoly output. Any combination on that line maximizes industry profit. The natural cartel
outcome if collusion were legal would be $q_1=q_2=30$, but if Firm 1 is better at bargaining and
threatens to break up a cartel, it could get $q_1=32, q_2=28$. The price would be $P=80$, so Firm
2's profit would be (80-20) (28) = 1,680, better than Firm 2 could get from Cournot duopoly if it
refused Firm 1's offer. That kind of threatening and bargaining, though, is one reason cartels
often do not form or form but break up as the result of bickering.
\bigskip
%-------------------------------------------------------
\noindent
{\bf Three or More Firms: Oligopoly to Competition}
The beauty of the Cournot model is that it can be extended to any number of firms. Now let there
be $n$ firms, so
$P = 140-(q_1+q_2 + \ldots + q_n)$, and Firm 1's profit is
\begin{equation} \label{eqx}
\pi_1 = \left(140- (q_1+q_2+\ldots q_n) \right)q_1 - 20q_1 = \left(140q_1 -q_1^2 - q_1 (
q_2+\ldots q_n)\right) - 20q_1
\end{equation}
Again take profit's derivative with respect to $q_1$ and set it to zero.
\begin{equation} \label{eqx}
\frac{d \pi_1}{d q_1} = \left(140 -2q_1 - ( q_2+\ldots q_n)\right) - 20 =0
\end{equation}
Firm 1's reaction curve is
\begin{equation} \label{eqx}
q_1 = 60 - \frac{( q_2+\ldots q_n)}{2}.
\end{equation}
If there are 3 firms the reaction curves
are actually reaction planes in 3-dimensional space. The reaction curves are $q_1 =
60 - \frac{q_2 + q_3}{2}$, $q_2 = 60 - \frac{q_1 + q_3}{2}$, and $q_3 = 60 -
\frac{q_2 + q_3}{2}$.
Since the firms all are identical and solve the same kind of profit maximization problem, it turns
out that $q_1 = q_2 = \ldots = q_n$, so $( q_2+\ldots q_n) = (n-1) q_1$ and we can write
\begin{equation} \label{eqx}
q_1 = 60 -\frac{(n-1) q_1}{2} = \frac{120}{n+1}
\end{equation}
Table \ref{chap07}.\ref{tab05-04} shows what happens as the number of firms increases from 1 to 99.
\vspace*{24pt}
\noindent
\begin{minipage}[c]{\linewidth}
\begin{center}
\refstepcounter{tablecounter} \label{tab05-04}
{\sc Table \ref{chap07}.\ref{tab05-04} \\
Cournot Industry Output Rises with the Number of Firms}\\
\bigskip
\begin{tabular}{l|cr|cr r}
\hline
\hline
& & &&& \\
Number of & Output &Profit & Total & Price & Industry \\
firms ($n$) &per firm & per firm & output &&profit \\
& & &&& \\
\hline
& & &&& \\
\: 1 & 60& 3,600 &60&80 &3,600 \\
\: 2 & 40 & 1,600 & 80& 60& 3,200 \\
\: 3 & 30 & 900 &90 & 50 &2,700 \\
\: 4 & 24 & 576 & 96 &44 &2,304 \\
\: 5 &20 & 200 & 100 &40 &2,000 \\
\: 9 & 12 & 144 &108 & 32& 1,296 \\
99 &$\approx 1$ & $\approx 1$ & $\approx 119$ & $\approx 21$ & $\approx 143$ \\
& & &&& \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\vspace*{12pt}
Table \ref{chap07}.\ref{tab05-04} shows that if the number of firms becomes large, the Cournot
model predicts that the price will approach marginal cost and profits will approach zero. It is
always true that as the number of firms increases, output per firm falls, industry output rises, and
the price falls.
\bigskip
%-------------------------------------------------------
\noindent
{\bf \ref{chap07}.3: Price Fixing} \label{s05.5}
Price fixing is the most frequent
antitrust violation, and perhaps the most important. The two most common kinds of cases involve
industries for homogeneous products that have few sellers and bidders in auctions, whether auctions
to sell objects where the high bid wins or auctions for procurement contracts where the low bid
wins. (A {\bf homogeneous}\index{homogeneous} product such as salt is the same whichever firm
produces it; a {\bf heterogeneous }\index{heterogeneous} of {\bf differentiated}
\index{differentiated}product such as perfume differs across firms.) Probably most cases go
undetected because they are on a local scale and too small for federal authorities to intervene
(though there do exist state antitrust laws as well). Even when there is little fear of punishment,
however, cartels have a problem: making sure members obey the cartel rules.\footnote{By the way, the
now-common practice of referring to drug gangs as drug cartels misuses the word completely. A cartel
is a group of sellers that agree to all sell at the same price. If a group of gangs agree to all
sell cocaine at a certain price, that is a cartel. If they compete ferociously using price cuts and
automatic weapons, that is the opposite of a cartel.}
\vspace*{16pt}
\noindent
\begin{minipage}[c]{\linewidth}
\begin{center}
\begin{footnotesize}
\refstepcounter{tablecounter} \label{table5-2}
{\sc Table \ref{chap07}.\ref{table5-2}\\
The Prisoner's Dilemma}\\
\bigskip
\begin{tabular}{lll ccc}
\hline
\hline
\multicolumn{6}{l}{\it } \\
& & &\multicolumn{3}{c}{\bf Column}\\
& & & {\it Not Confess} & & {\it Confess} \\
\multicolumn{6}{l}{\it } \\
& & {\it Not Confess} & -1,-1 & & -10, 0 \\ & {\bf Row} && & & \\
& & {\it Confess} & 0,-10 & & {\bf -8,-8} \\
\multicolumn{6}{l}{\it } \\
\multicolumn{6}{l}{\it Payoffs to: (Row,Column) } \\
\multicolumn{6}{l}{\it } \\
\hline
\hline
\end{tabular}
\end{footnotesize}
\end{center}
\end{minipage}
\vspace*{16pt}
The cartel enforcement problem is a good example of the Prisoner's Dilemma\index{Prisoner's
Dilemma}. In the original Prisoner's Dilemma, illustrated in Table \ref{chap07}.\ref{table5-2}, two
criminals, Mr. Row and Mr. Column, have been captured and accused of carrying out a crime together.
They are kept in separate cells and each offered a choice: Confess or Not Confess. If neither
confess, there is still enough evidence to sentence each to 1 year in prison. If both confess, they
will each get 8 years in prison. But if one confesses and the other does not, the confessor gets off
free and the nonconfessor is sentenced to 10 years. Clearly the best thing for the prisoners jointly
is to Not Confess. For each one acting individually, however, Confess is a dominant strategy. Think
of Prisoner Row's choice. If Prisoner Column confesses,Row gets 8 years in prison by confessing
too, and 10 years if he chooses Not Confess, so he should confess. If Prisoner Column doesn't
confess, Row gets 0 years in prison by confessing, and 1 years if he chooses Not Confess, so he
should confess. Either way, Prisoner Row does better by confessing. And so they both confess and
get a total of 16 years in prison.
\vspace*{24pt}
\noindent
\begin{minipage}[c]{\linewidth}
\begin{center}
\refstepcounter{tablecounter} \label{table5-3}
{\sc Table \ref{chap07}.\ref{table5-3}\\
The Price-Fixer's Dilemma}\\
\bigskip
\begin{tabular}{lllccc}
\hline
\hline
\multicolumn{6}{l}{\it } \\
& & &\multicolumn{3}{c}{\bf Column}\\
& & & {\it High Price} & & {\it Low Price} \\
\multicolumn{6}{l}{\it } \\
& & {\it High Price} & 400, 400 & & 200, 500 \\
& {\bf Row} && & & \\
& & {\it Low Price} & 500, 200 & & {\bf 300, 300} \\
\multicolumn{6}{l}{\it } \\
\multicolumn{6}{l}{\it Payoffs to: (Row, Column) }\\
\multicolumn{6}{l}{\it } \\
\hline
\hline
\end{tabular}
\end{center}
\end{minipage}
\vspace*{24pt}
Now consider price fixers Row and Column in Table \ref{chap07}.\ref{table5-3}. They have a constant
marginal cost of \$4 per unit for a differentiated product. If they each charge the cartel price of
\$8 that they have agreed upon each will sell 100 units (for a profit of \$800 - \$400 each), but
if they both sell at the low, discounted price of \$5 each will sell 300 units (for a profit of
\$1,500 - \$1,200 each). If one sells at the low price and one sells at the high price, then the
low-priced seller sells 500 and the high-priced seller sells 50. The product is differentiated,
which is why even a high-priced seller still gets some sales.
If both price fixers charge high prices, they each will have profits of 400. If both choose low
prices, they will each get 300. But if one prices high and the other prices low, the low-price firm
has profits of 500 and the high-price firm has profits of 200. Clearly the best thing for the firms
jointly is to price high. For each one acting individually, however, Low Price is a dominant
strategy. Think of Row's choice. If Column prices high, Row's profits are 400 if he too prices
high, but 500 if he prices low. If Column prices low, Row's profits are 200 if he prices high,
but 300 if he prices low. Either way, Row does better by pricing low. And so both firms price
low, and industry profits are 600 instead of the 800 they could be if both stuck to the cartel
price.
%\begin{figure}[htb]
%\end{figure}
If Row and Column could make a legal agreement to keep prices high, their task would be much
easier, but the courts would not enforce such a contract even before the Sherman
Act.\footnote{Courts in the common law countries of England the United States would not enforce
cartel contracts, as being ``against public policy'' like contracts to commit crimes. Germany, on
the other hand, did enforce cartel contracts, which were common. See Steven B. Webb \href{http://www.jstor.org/stable/2120181 }{``Tariffs, Cartels, Technology, and Growth in the German Steel Industry, 1879 to 1914,''} {\it Journal of Economic History}
40: 309-330 (1980). }
Because of the antitrust laws, they cannot even hire an accounting firm to audit each other and
find out if someone has been cheating on the cartel. To even detect cheating, price fixers have to
rely on gossip from customers--- who would like to stir up competition--- and fluctuations in their
own demand. If Row's sales go down, however, he cannot tell whether it is because Column has been
making secret price cuts to steal customers or because of random fluctuations in market demand. For
this reason, cartels tend to break down eventually.
I
suggested earlier that industries with homogeneous products would have the most successful cartels. The reason will now be clearer: if the
product sold by each firm is identical, it is easier to police a cartel and detect when one member
is trying to undercut the others. Prices are most easily compared when the product is homogeneous:
Row's salt should sell a pound of salt at the same price as Column, but if Honda and Toyota tried to
fix prices, should a Honda Accord sell for the same price as a Toyota Camry, particularly if
different sellers include different options such as leather seating? In addition, if products are
heterogeneous, a seller can gain an advantage by increasing the quality of his product or giving the
consumer a better version at the same price.
If there are few sellers, it is easier to come to an agreement. When there are many firms, one
firm can stay out of the cartel and still profit from the high prices generated by the cartel's
reduced output, so there is a free rider problem. If the good is sold by auction, then the cartel
members can see who won and perhaps even see the price, depending on the auction rules. This aids
in detecting who cheated on the cartel agreement. If Row wins the auction for a government
contract this week, and Column was supposed to win, Row has some explaining to do. Thus, the
structure of the market affects whether successful cartels form.
%-------------------------------------------------------
\bigskip
\noindent
{\bf \ref{chap07}.4: Regulating Monopoly: The History of Antitrust Law} \label{s06.1}
What to do about market power has long been controversial. Perhaps the earliest discussion is in Aristotle's {\it Politics},\href{http://www.perseus.tufts.edu/hopper/text?doc=Perseus:text:1999.01.0058:book=1:section= 1259a&highlight=thales}{1.1259a}, where he writes about Thales, the first philosopher, and a ``man of Sicily".
\begin{bigquote}
Thales, so the story goes, because of his poverty was taunted with the uselessness of philosophy; but from his knowledge of astronomy he had observed while it was still winter that there was going to be a large crop of olives, so he raised a small sum of money and paid round deposits for the whole of the olive-presses in Miletus and Chios, which he hired at a low rent as nobody was running him up; and when the season arrived, there was a sudden demand for a number of presses at the same time, and by letting them out on what terms he liked he realized a large sum of money, so proving that it is easy for philosophers to be rich if they choose, but this is not what they care about. Thales then is reported to have thus displayed his wisdom, but as a matter of fact this device of taking an opportunity to secure a monopoly is a universal principle of business; hence even some states have recourse to this plan as a method of raising revenue when short of funds: they introduce a monopoly of marketable goods.
There was a man in Sicily who used a sum of money deposited with him to buy up all the iron from the iron mines, and afterwards when the dealers came from the trading- centers he was the only seller, though he did not greatly raise the price, but all the same he made a profit of a hundred talents on his capital of fifty. When Dionysius came to know of it he ordered the man to take his money with him but clear out of Syracuse on the spot,6 since he was inventing means of profit detrimental to the tyrant's own affairs. Yet really this device is the same as the discovery of Thales, for both men alike contrived to secure themselves a monopoly.
\end{bigquote}
Modern regulation of monopolies started in
1603, when Queen Elizabeth I of England granted her groom a legal monopoly on selling playing cards.
A court voided the grant, saying that monopolies were counter to the common law. ({\it Darcy v.
Allein}, 77 Eng. Rep. 1260 (K.B. 1603)) Another source of market power was {\bf price-fixing
agreements}\index{price-fixing agreements} or {\bf cartels}:\index{cartels} agreements between
sellers to jointly keep their prices high. Forming a cartel was not a criminal offense, but the
courts would not enforce an agreement ``in restraint of trade'' unless it could be shown that the
agreement benefitted the public. An example of such a beneficial agreement would be if someone sold
a law firm under the condition that he not start a new law firm in the same city and steal back his
old clients.
\begin{wrapfigure}{L}{2.4in}
\noindent
\begin{boxedminipage}[c]{\linewidth}
\renewcommand{\baselinestretch}{.9}
\begin{footnotesize}
\refstepcounter{sidebarcounter} \label{sidebar6-4}
\noindent
{\sc Box \ref{chap07}.\ref{sidebar6-4}\\
An Indiana Cement Conspiracy }\\
\hspace*{16pt}
Gus B.
(Butch) Nuckols of Builder's Concrete and Supply Co. agreed with another company, IMI, to fix
cement prices from 2000 till 2004. Since cement is so heavy, competition tends to be local,
high profits will not attract new entrants for some time. The FBI found that meetings were held
at the Nuckols horse barn in
Fishers, Indiana to discuss price, discounts, and conditions of sale.
\hspace*{16pt}IMI made \$225 million from the conspiracy. In a
plea bargain, Nuckols turned on his co-conspirators in exchange for a lighter sentence: a
\$50,000 fine personally, a 14-month prison sentence, and a fine of \$4 million for his company.
\hspace*{16pt}Four executives of IMI pled guilty too. They received fines of one to two
hundred thousand dollars, and went to jail for 5 months and the IMI itself was fined \$29
million.
\end{footnotesize}
\end{boxedminipage}
\end{wrapfigure}
Excluding cartels from the benefits of contract rights does have an effect, because members of a
cartel are always tempted to cheat by reducing prices to increase the quantity they sell, at the
expense of other members of the cartel.
In 1890, the Sherman Act\index{Sherman Act} was passed in the United States, and it remains one
of the two most important antitrust laws. Section 1 of the Act, in its 2010 form (the fine limit
has been increased from the 1890 version), says
\begin{bigquote}
{\bf Every contract}, combination in the form of trust or otherwise, or conspiracy, {\bf in
restraint of trade or commerce among the several States, or with foreign nations, is declared to be
illegal.} Every person who shall make any contract or engage in any combination or conspiracy hereby
declared to be illegal shall be deemed guilty of a felony, and, on conviction thereof, shall be
punished by fine not exceeding \$100,000,000 if a corporation, or, if any other person, \$1,000,000,
or by imprisonment not exceeding 10 years, or by both said punishments, in the discretion of the
court. (U.S.C. Title 15, Chapter 1, \S 1)
\end{bigquote}
Section 1 is aimed directly at price-fixing agreements. At least two parties must be involved,
and they must make some definite agreement to restraint trade by such actions as directly
restricting output or raising prices. It says:
\begin{bigquote}
{\bf Every person who shall monopolize, or attempt to monopolize,} or combine or conspire with any
other person or persons, to monopolize {\bf any part of the trade or commerce among the several
States,} or with foreign nations, {\bf shall be deemed guilty of a felony}, and, on conviction
thereof, shall be punished by fine not exceeding \$100,000,000 if a corporation, or, if any other
person, \$1,000,000, or by imprisonment not exceeding 10 years, or by both said punishments, in the
discretion of the court. (U.S.C. Title 15, Chapter 1, \S 2, as of 2010)
\end{bigquote}
Section 2 is harder to interpret than Section 1. It is directed not just against agreements, but
also at single parties who try to monopolize a market. What ``monopolize'' means is left unclear,
and so has had to be interpreted by executive branch policy and by the courts. Early interpretation
ruled that if firms merged to form a monopoly, that was not ``monopolizing'' and so was legal.
Courts treat Section 1 and Section 2 violations differently. Section 1 violations are {\bf per se}
\index{per se} illegal, meaning that they are illegal even if they do not cause any harm. If two gas
stations agree on a minimum price, they violate Section 1 even if they can show that there are many
other competing gas stations and they did not hurt any consumers. Section 2 violations, on the
other hand, are subject to {\bf the rule of reason}\index{the rule of reason}: if the defendant can
show that his actions were not really intended to monopolize or did not have a bad effect, he can
escape punishment.
Since ``monopolizing'' is a vague term, the
Clayton Act\index{Clayton Act} was passed in 1914 to try to pin down monopolizing practices more
clearly. One section, which incorporates amendments in the 1930's called the {\bf Robinson-Patman
Act}\index{Robinson-Patman Act}\footnote{For more on this, see Donald S. Clark, ``The Robinson-
Patman Act:
General Principles, Commission Proceedings,
and Selected Issues,''\url{http://www.ftc.gov/speeches/other/patman.shtm}, June 7, 1995.} attacks
{\bf price discrimination}: \index{price discrimination}
\begin{bigquote}
{\bf It shall be unlawful for any person} engaged in commerce, in the course of such commerce,
either directly or indirectly, {\bf to discriminate in price between different purchasers of
commodities of like grade and quality}, where either or any of the purchases involved in such
discrimination are in commerce, where such commodities are sold for use, consumption, or resale
within the United States or any Territory thereof or the District of Columbia or any insular
possession or other place under the jurisdiction of the United States, and {\bf where the effect of
such discrimination may be substantially to lessen competition or tend to create a monopoly} in any
line of commerce, {\bf or to injure, destroy, or prevent competition with any person who either
grants or knowingly receives the benefit of such discrimination, or with customers of either of
them} \ldots (U.S.C. Title 15, Chapter 1, \S 13)
\end{bigquote}
Section 13 does not apply to every case where a seller charges different customers different
prices, only to cases where the effect is to substantially lessen competition. {\bf Predatory
pricing} \index{Predatory pricing} is the best example of this: a monopoly charges below-cost prices
in some markets to drive out competing firms and high prices in other markets where competitors
do not operate.
Another section of the Clayton Act tackles special contractual provisions that could help a firm
monopolize a market. One of these is {\bf exclusive dealing contracts} \index{exclusive dealing
contracts}or {\bf total-requirements contracts}\index{total-requirements contracts} under which the
buyer must buy the good exclusively from the one seller, even if his competitor offers a lower
price:
\begin{bigquote}
It shall be unlawful for any person engaged in commerce, in the course of such commerce, to lease or
make a sale or contract for sale of goods, ... on the condition, agreement, or understanding that
the lessee or purchaser thereof shall not use or deal in the goods, wares, merchandise, machinery,
supplies, or other commodities of a competitor or competitors of the lessor or seller, where the
effect \ldots may be to substantially lessen competition or tend to create a monopoly in any line of
commerce. (U.S.C. Title 15, Chapter 1, \S 14)
\end{bigquote}
The most important part of the Clayton Act deals with mergers.
\begin{bigquote}
No person engaged in commerce or in any activity affecting commerce shall acquire, directly or
indirectly, the whole or any part of the stock or other share capital and no person subject to the
jurisdiction of the Federal Trade Commission shall acquire the whole or any part of the assets of
another person engaged also in commerce or in any activity affecting commerce, where in any line of
commerce or in any activity affecting commerce in any section of the country, the effect of such
acquisition may be substantially to lessen competition, or to tend to create a monopoly. (15 U.S.C.
\S 18)
\end{bigquote}
As with other Clayton Act provisions, this is subject to the rule of reason. Mergers are legal
unless they substantially lessen competition. Merger regulation is the most active area of
antitrust enforcement in the United States today. Businesses over a certain size which wish to merge
are required to notify the Justice Department and the Federal Trade Commission, who scrutinize the
merger for anticompetitive effects. Either agency may then challenge the legality of the merger in
court. Ordinarily, if the agencies decide the merger would substantially reduce competition, the
companies either drop the merger or change its terms in consultation with the agencies. If some
cities are only served by two chain store chains, for example, a merger of those two chains might
be required to sell some of their stores in the ``captive'' cities to a third chain to preserve
competition. Occasionally firms that wish to merge think that the FTC and Justice are so
unreasonable that they say they will merge anyway. In that case, the agency goes to a federal
court and tries to persuade a judge to issue an injunction blocking the merger on the grounds
that it would substantially lessen competition.
A major purpose of the Clayton Act was actually to reduce competition in one area of the economy:
labor markets. The Sherman Act and the common law had been used to punish labor unions, which are
cartels in which workers agree to sell their labor jointly at a price above the market level. These
are heavily regulated by labor laws, but are exempt from antitrust laws and much of labor law is
devoted to encouraging workers to organize, to increase the price at which they sell their labor,
and to be able to use exclusive-dealing contracts to shut out workers not in the union. The Clayton
Act says:
\begin{bigquote}
The labor of a human being is not a commodity or article of commerce. {\bf Nothing contained in the
antitrust laws shall be construed to forbid the existence and operation of labor, agricultural, or
horticultural organizations, instituted for the purposes of mutual help, and not having capital
stock or conducted for profit,} or to forbid or restrain individual members of such organizations
from lawfully carrying out the legitimate objects thereof; {\bf nor shall such organizations, or the
members thereof, be held or construed to be illegal combinations or conspiracies in restraint of
trade,} under the antitrust laws. (15 U.S.C. \S 17)
\end{bigquote}
As one would expect, since labor cartels are legal, they are common. They are heavily regulated,
however, by the National Labor Relations Board, which uses a multitude of rules to determine how a
labor union may be organized, what happens if competing unions both wish to one company's workers,
whether the union can require the firm to fire competing workers who refuse to the join the union,
and so forth.
An example that shows the market power of a labor cartel can exercise is
Local 1 of the International Alliance of Theatrical Stage Employees in New York City. Carnegie Hall
has five stagehands, who hang the lights, move furniture around, and so forth. The highest-paid is
the props manager, who earns
\$423,000 cash per year plus \$107,000 in benefits and deferred compensation. The lowest-paid, an
electrician, earns \$403,000. Admittedly, the workweek can last up to 80 hours, during busy times
during the heavy season, but the contrast to the incomes of aspiring musicians is amazing. Of
course, Carnegie Hall pays especially well.
Average stagehand compensation at Avery Fisher Hall and the Alice Tully Hall in Lincoln Center
is only \$290,000 per year. But a successful monopoly can do well even in a market with demand as
seemingly fragile as classical music backstage
labor.\footnote{\href{http://www.northjersey.com/news/opinions/68164552.html?page=all} {``For
Backstage Labor, Rich Rewards,''} James Ahearn, {\it Northjersey.com} (November 1, 2009).}
One of the most important things to note about antitrust law is that it does not make monopoly
illegal. Rather, it makes {\bf monopolizing}\index{monopolizing} illegal. If a company grows to
dominate its industry because it has low costs or good products, that is perfectly legal. Moreover,
the firm can restrict its output by following the MR(Q) = MC(Q) rule and that is legal too. There is
still market failure, but the potential for government failure in breaking up large, successful
firms is too great to make a law against mere size a good idea. Letting the government control the
firm's prices and practices is also too fraught with risk. Both sources of government failure are
acute: the government would not know how to run the business, and the government might have the
objective of favoring politically powerful groups. Thus, the law focuses on practices which add to a
firm's market power for reasons that do not help consumers, practices such as practically all price-
fixing agreements and some mergers. The law goes well beyond price-fixing and mergers to suspicious
practices such as predatory pricing, tie-in requirements, and exclusive dealing, but those
practices, while still monitored and sometimes punished, are not the focus of antitrust activity.
\bigskip
%-------------------------------------------------------
\noindent
{\bf Antitrust Law in Europe}
Article 101 (formerly article 81) of the Treaty on the Functioning of the European Union
prohibits cartels and other ``concerted practices,'' in a way comparable to the Sherman Act's
Section 1.\footnote{Treaty on the Functioning of the European Union (consolidated text), Official
Journal of the European Union September 5, 2008,\url{http://eur-
lex.europa.eu/LexUriServ/LexUriServ.do?uri=OJ:C:2008:115:0047:0199:EN:PDF.}}
Article 102 is like the Sherman Act's Section 2. It says, ``Any abuse by one or more undertakings of
a dominant position within the common market... shall be prohibited as incompatible with the common
market insofar as it may affect trade between Member States.''\footnote{For more, see
Baumgardner\index{Baumgardner}, Larry (2005) ``Antitrust Law in the European Union
The law is changing---but to what effect?'' {\it Graziadio Business Report}, 11, 3. \url{
http://gbr.pepperdine.edu/2010/08/antitrust-law-in-the-european-union/}.} The laws are enforced
by the European Commission and by the antitrust authorities of each of the 20-plus nations of the
European Union.
EU antitrust policy is administered by the {\bf Competition Commissioner}\index{Competition
Commissioner}, one of 27 commissioners of the {\bf European Commission}\index{European Commission}.
He is appointed by the {\bf Council of the European Union}\index{Council of the European Union}
(informally, the {\bf Council of Ministers})\index{Council of Ministers} which is composed of one
representative of each EU country.\footnote{EU governance is confusing. There also exists a Council
of Europe and a European Council, which are different from the Council of the European Union.} In
addition, member countries continue to have their own antitrust agencies, e.g. the United Kingdom's
\url{http://www.competition-commission.org.uk/}{Competition Commission}.
The European Union's main legislative branch is the {\bf European Parliament}\index{European
Parliament}, which is elected directly by voters according to a country's population. {\bf EU
directives}\index{EU directives} are rules which member countries are supposed to implement by
passing their own national laws and regulations in accord with the rule. {\bf EU regulations}
\index{EU regulations} are rules which take effect immediately and bind every member country (note
that ``regulation'' has a particular legal meaning here, contrasting with directives). The
European Parliament does not make antitrust law, the rules for which are issued by the European
Commission and the Council of Ministers.\footnote{European Parliament, ``General Competition Policy
and Concerted
Practices,''\url{http://www.europarl.europa.eu/parliament/expert/displayFtu.do?language=en&id=
74&ftuId=FTU_3.3.1.html} (September 2006). }
Regulations say that the Competition Commissioner can block any merger which ``significantly impede
effective competition,'' a standard similar to the Clayton Act's and equally open to widely
varying interpretations.\footnote{Council Regulation 139/2004 on The Control of Concentrations
Between Undertakings (the EC Merger Regulation), art. 1, 2004 O.J. (L 24) 1.} The Commission has
issued Merger Guidelines based on how many firms are in a market to let businesses know which kinds
of mergers it is likely to challenge.\footnote{Guidelines on the Assessment of Horizontal Mergers
under the Council Regulation on the Control of Concentrations between Undertakings, 2004 O.J. (C 31)
5.}
In 2001 the EU blocked a merger of General Electric and Honeywell, even though American authorities
had already approved it. GE's CEO said, ``The European regulators' demands exceeded anything I or
our European advisers imagined and differed sharply from antitrust counterparts in the U.S. and
Canada.''\footnote{Andrew Ross Sorkin, \href{http://www.nytimes.com/2004/05/30/business/dealbook-a-trustbuster-s-song-is-ending-but-a-coda-is-possible.html}{``A Trustbuster's Song Is Ending, but a Coda Is Possible,''}
{\it New York Times}, May 30, 2004, C3.} The personal views of whoever is Competition
Commissioner are extremely important. Another notable case involved Microsoft, which in 2004 was
fined \$600 million for anti-competitive tactics, a much more severe penalty than in its settlement
with U.S. antitrust authorities.
\bigskip
%-------------------------------------------------------
\noindent
{\bf Enforcement}
The first way the antitrust laws are enforced is the same way as most federal laws are enforced:
by the {\bf Department of Justice}.\index{Department of Justice} One division of the Justice
Department is the {\bf Antitrust Division}, \index{Antitrust Division}which has a staff of
economists and lawyers ready to find and prosecute violations of the Sherman and Clayton Acts. The
Justice Department can bring both criminal and civil actions.
\begin{wrapfigure}{L}{.5\linewidth}
\noindent
\begin{boxedminipage}[c]{\linewidth}
\renewcommand{\baselinestretch}{.9}
\begin{footnotesize}
\refstepcounter{sidebarcounter} \label{sidebar05-2}
\noindent
{\sc Box \ref{chap07}.\ref{sidebar05-2} \\
A Rogue's Gallery of Price Fixers, and Regulators}\\
\includegraphics[height=.18\linewidth]{fig05BjornSjaastadOdfjell.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05brooks_2.jpg}\hfill
\includegraphics[height=.18\linewidth]{fig05davidge_2.jpeg}\hfill
\includegraphics[height=.18\linewidth]{fig05debney.jpg}\hfill
\includegraphics[height=.18\linewidth]{fig05kim.jpg}\\
\includegraphics[height=.18\linewidth]{fig05koehler.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05whitacre.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05tennant.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05taubman.jpg} \hfill
%\includegraphics[height=.18\linewidth]{fig05sommer.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05rpratt.jpg}\\
\includegraphics[height=.18\linewidth]{fig05baye-michael.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05carlto.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05farrell.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05katz_michael.jpg} \hfill
\includegraphics[height=.18\linewidth]{fig05Salinger-Michael.jpg}
\end{footnotesize}
\end{boxedminipage}
\end{wrapfigure}
Violations of the Sherman Act can be prosecuted as {\bf criminal violations}\index{criminal
violations}. This means that the accused must be found guilty ``beyond a reasonable doubt,'' and
may be punished with prison time of up to 20 years (since Sarbanes-Oxley raised the limit in 2002)
as well as fines. The {\bf reasonable doubt standard}\index{reasonable doubt standard} is a high
hurdle. The government may prefer to bring the case as a {\bf civil action}\index{civil action}.\footnote{Examples of complaints by the Justice Department and the FTC are:
\href{http://www.jdsupra.com/legalnews/complaint-in-us-v-apple-and-hachette-e-26600/}{ ``Complaint: United States v. Apple Inc., Hachette Book Group, et al.''
United States District Court for the Southern District of New York'' (April 12, 2011) } and
\href{http://www.ftc.gov/os/caselist/0610197/complaint.pdf}{``TC Group-Riverstone-Carlyle-Kinder Morgan Complaint''} (2007).}
There, the firm or person must be found to have done something illegal {\bf by the preponderance
of the evidence}\index{by the preponderance of the evidence}; that is, more likely than not. A
civil action can result in fines, and also in orders from the court called {\bf injunctions}
\index{injunctions}that can call for the firm to have special government oversight, to cease some
business practice, or even to split up into smaller firms. Or, the government and the firm may
settle a suit by a {\bf consent decree}\index{consent decree}, an agreement enforced by the court
under which the firm agrees to behave in a certain way.
The other agency that enforces antitrust law is the {\bf Federal Trade Commission}\index{Federal
Trade Commission} or FTC, which was created in 1914 at the same time as the Clayton Act was passed.
The FTC only brings civil cases, not criminal cases. If it brings a complaint, the complaint is
judged by an FTC hearing overseen by an {\bf administrative law judge} \index{administrative law
judge}from within the FTC who recommends a course of action to the Commission's five members at the
end of the hearing. The Commission then votes on a decision similar to a court injunction.
Besides the two agencies, private parties can enforce the antitrust laws through civil actions. The
private party, typically a customer or a competitor of the company charged with wrongdoing, must
show that it has been harmed, and by how much. Ordinarily if someone is found liable in a civil suit
he must pay the party who brought the suit (the plaintiff) the damage the court decides he caused.
(Punitive damages are possible, but unusual.) The antitrust laws, however, say that private
parties can collect {\bf treble damages}:\index{treble damages} that is, if the plaintiff can prove
he lost \$1 million dollars due to the defendant's monopolizing practices, he can collect \$3
million from the defendant.\footnote{On the other hand, treble damages aren't always high. A jury
agreed that the National Football League (NFL) did ``willfully acquire or maintain monopoly
power,'' but estimated the resulting damage to the rival United States Football League at only a
nominal \$1 (it would have been in financial trouble anyway). When trebled, that came to a measly
\$3 award. \href{http://sportsillustrated.cnn.com/vault/article/magazine/MAG1065099/index.htm}
{``The Award Was Only Token:
The USFL Proved Part of Its Case against the NFL only to See the Jury Throw the Winners for a
Loss,''}, Craig Neff, {\it Sports Illustrated} (August 11, 1986).} Defendants are rarely sentenced
to prison time in antitrust cases and criminal and civil fines are low relative to the potential
gains from violations, so the treble damages from private suits are perhaps the biggest punishment
violators face. Often, private parties can wait until the government has proved a case and then
pile on with private suits, much of the work having been done for them.
Who can sue for damages is an interesting legal question. The U.S. Supreme Court ruled in {\it
Hanover Shoe, Inc. v. United Shoe Machinery Corp.} (392 U.S. 481 [1968]) that the direct purchaser
of the overpriced product can sue for the price difference even if it resold the product and passed
the cost on to its own customers. Thus, if a cartel of plastic companies sold plastic to bottle
companies which then raised their prices to soft drink companies, the bottle companies could sue for
the entire plastic overcharge. What about the indirect purchasers, the soft drink companies? The
U.S. Supreme Court ruled in {\it Illinois Brick Co. v. Illinois} (431 U.S. 720 [1977]) that only
the direct purchasers could sue, since otherwise there would be duplicative suits and it would be
hard to track down all the indirect purchasers. Many states, however, have passed ``Illinois Brick
repealer'' statutes that allow the indirect purchasers to sue in state court.
\footnote{See Dennis W. Carlton, , \index{Carlton}\href{http://www.jstor.org/stable/30033739}
{``Does Antitrust Need To Be Modernized?''} {\it The Journal of Economic Perspectives}, 21(3):
155--176 (Summer 2007); Edward D. Cavanagh,\index{Cavanagh}\href{http://heinonline.org/HOL/Page?handle=hein.journals/lyclr17&div=8&g_sent=1&collection=journals}{ ``Illinois Brick: A Look Back
and a Look Ahead,''} {\it Loyola Consumer Law Review}, 17: 1--51 (2004).}
Why have this threefold enforcement? We must go back to government failure and the principal-
agent problem. Each of these three enforcers--- the Justice Department, the FTC, and private
parties--- has different incentives. The Justice Department is headed by the Attorney-
General\index{Attorney-General}, who is appointed by the President and who can be fired by him.
Thus, the Justice Department is subject to political influence. This is not an entirely bad thing,
because the President has, after all, been elected to execute the laws in a certain way, so if his
degree of enforcement is more severe or more lax, that is the result of voter choice. There is
potential, however, for favoritism in prosecutions.
Thus, when Congress established the Federal Trade Commission in 1914, they set it up as an {\bf
independent agency}\index{independent agency}. What this means is that although its five members are
appointed by the President, they cannot be fired by him, and their seven-year terms are staggered so
that when the Presidency changes parties, the President inherits the old Commission members
appointed by his predecessor. He must wait for their terms to end before he can appoint his own
people. Moreover, no more than three of the five members may be of the same political party.
If the Justice Department decides not to investigate a tying arrangement or challenge a merger,
the FTC can do so instead, and vice versa. Moreover, if both of them are too reluctant to enforce
the laws, private parties can bring treble-damage suits. Thus, the overlapping authority can
overcome the problem of one agency, an agent for the voters as principal, is slack in its effort.
In practice there do not seem to have been important policy differences between the FTC and the
Justice Department. They cooperate well, and they split investigation of proposed mergers so each
can specialize in certain industries rather than duplicating effort.\footnote{For more on the history and politics and anti-trust regulation, see
\href{http://www.nber.org/chapters/c12565.pdf}{``Antitrust and Regulation,''}
Dennis W. Carlton \& Randal C. Picker, a chapter in {\it Economic Regulation and Its Reform: What
Have We Learned?} National Bureau of Economic Research (2011).}
\bigskip
\noindent
{\bf \ref{chap07}.5: Mergers} \label{s06.2}
Certain business action might be either
surplus-increasing or surplus-reducing, and so are subject to the ``rule of reason'' in antitrust
law.
Section 1 of the Sherman Act makes illegal ``every contract, combination in the
form of trust or otherwise, or conspiracy, in restraint of trade,'' while Section 2 makes it illegal
to
``monopolize, or attempt to monopolize, or combine or conspire with any other person or persons, to
monopolize.''
Not only are there lots
of monopolizing practices, but for each one we could talk about several surplus-increasing
motivations and
several surplus-reducing motivations and how common each of these is. This makes
monopolizing a fun and important area for academic researchers, but it runs into diminishing
returns. Prosecutors are not eager to bring complicated cases, and judges do not look forward to
having to decide them, since the cases turn on difficult arguments being explained by expert
witnesses using mathematics and game theory. This is not laziness, but rationality; prosecutors and
courts have limited resources and if they can deal with five price-fixing cases in the time
required to analyze one predatory pricing case, they should do it--- particularly since they might
end up punishing someone for predatory pricing when he actually was engaged in surplus-increasing
severe price competition. In practice, by far the most important monopolizing practice regulated
is merger, since unlimited merger would clearly hurt competition severely and the least justifiable
mergers are easy to detect. The illegality of other practices is important, though, because they
too are sometimes punished and the mere threat of government action curtails the most abusive
behavior.
The most important suspicious practice is the merger of two or more firms. After it became
clear that the courts would use the Sherman Act to punish cartels firms turned to merger, and {\bf
the Great Merger Wave} \index{the Great Merger Wave} took place around 1900, mergers that involved a
substantial fraction of U.S. manufacturing.\footnote{See George Bittlingmayer,\index{Bittlingmayer} \href{http://heinonline.org/HOL/Page?handle=hein.journals/jlecono28&div=9&g_sent=1&collection=journals}{
``Did Antitrust Policy Cause the Great Merger Wave?'' } {\it The Journal of Law and Economics},
28: 77--118 (April 1985).} The hope was that the courts would interpret the Sherman Act to
mean that although agreements between firms to raise prices were illegal, it would still be legal to
merge the firms--- in which case, as has been explained, it is not illegal for a firm to use its
market power to raise its own prices unilaterally. In response, the courts began using the Sherman
Act against mergers in 1904 in the {\it Northern Securities}\index{Northern Securities} case and in
1914 the Clayton Act clarified that mergers could be illegal monopolization, saying, you will
recall, that:
\begin{bigquote}
\hspace*{16pt} No person \ldots shall acquire, directly or indirectly, the whole or any part of the
stock or other share capital and no person\ldots shall acquire the whole or any part of the assets
of another person \ldots where\ldots in any activity affecting commerce in any section of the
country, the effect of such acquisition may be substantially to lessen competition, or to tend to
create a monopoly.(15 U.S.C. \S 18)
\end{bigquote}
Mergers {\it can be}, not {\it are}, illegal monopolization. It is hard enough to think of surplus-
maximizing reasons for competitors to agree with each other to raise prices that the law makes such
agreements per se illegal. It is easy, though, to think of surplus-maximizing reasons for two
competitors to agree to merge. The new merged firm might have lower costs, either because of
economies of scale or because one of the firms has managers who are better at managing. On the other
hand, with fewer firms in the industry, competition falls, so the new firm might have cost curves
just the same or worse but be better able to raise prices, a bad motivation.
\bigskip
\noindent
{\bf Concentration Levels}
Since the number of firms matters to the amount of competition, it is useful to quantify the {\bf concentration}
\index{concentration} of a market: how few firms sell in it.
A simple measure is the {\bf four-firm concentration ratio}\index{four-firm concentration ratio}:
the percentage of market sales by the top four firms. Thus, if the firms in an industry have market
shares of 30\%, 25\%, 20\%, 15\%, 5\%, 4\%, and 1\%, the four-firm concentration ratio is
90\%. That is rather crude, since it doesn't care about how evenly the top four firms split the
market. The same level would be reached by an industry in which one firm had an 87\% market shares
and 13 other firms had shares of 1\% each.
\begin{footnotesize}
\vspace*{24pt}
\noindent
\begin{minipage}[c]{.8\linewidth}
\refstepcounter{tablecounter} \label{tab05-05}
\begin{center}
{\sc Table \ref{chap07}.\ref{tab05-05} \\
Concentration Ratios }\\
\bigskip
\begin{tabular}{lr | r| rrrr | r }
\hline
\hline
& & & & & & & \\
Industry & NAICS & Estab's & \multicolumn{4}{|c|}{No. of Firms } & Herf. \\
& Code & & 4 & 8 &20 & 50 & Index\\
& & & & & & & \\
\hline
& & & & & & & \\
Food Manufacturing& 311 & 21,355 & 14 &22&37& 50 &102\\
Sugar and Confectionery & 3113 & 1,631 &37 &49&66& 83 & 667 \\
Product Manufacturing & & & & & & & \\
Sugar Manufacturing& 31131 & 37 &59 &78&96& 100 &904\\
Beet Sugar Manufacturing & 311313 & 12 & 81 &98&100& 100 &x\\
& & & & & & & \\
\hline
& & & & & & & \\
Dog and Cat Food & 31111& 199 & 71 & 83&92& 98& 2,325 \\
Manufacturing & & & & & & & \\
Retail Bakeries &311812 & 6,101& 3 &6 &9&18 & 7 \\
Accommodation and & 72 & 634,361 & 5 & 10&17 & 23 &x \\
Food Services & & & & & & & \\
Hotels (except casino hotels) & 72111 & 48,108 &22 &28 & 35& 42 &x\\
and motels & & & & & & & \\
National Commercial & 5221101 & 46,809 & 55 &69 &83 & 90&x \\
Banks & & & & & & & \\
& & & & & & & \\
\hline
\hline
\end{tabular}\\
\end{center}
{\it Notes:} \url{http://factfinder.census.gov/}. One company can have many establishments, e.g. Hilton hotels. For some industries
the Herfindahl Index is not published.
\end{minipage}
\end{footnotesize}
\vspace*{24pt}
A more sophisticated approach is the {\bf Herfindahl Index}\index{Herfindahl
Index}:\footnote{The government and others call this the ``{\bf Herfindahl-Hirschman Index}
\index{Herfindahl-Hirschman Index}'' or ``HHI'' in documents, but that's too long a name, and in
common practice government people and economists just say ``Herfindahl Index''--- Too bad for
Professor Hirschman, a more important scholar than Herfindahl, but better for everyone else.}
The Herfindahl is calculated by summing the squares of the individual firms' market shares, and
thus gives proportionately greater weight to the larger market shares. Table
\ref{chap07}.\ref{tab05-09} shows how that works out. Table \ref{chap07}.\ref{tab05-05} also has
examples. When using the Herfindahl, the agencies consider both the post-merger level of the
Herfindahl and the increase in the Herfindahl resulting from the merger. The increase in the
Herfindahl is equal to twice the product of the market shares of the merging firms. \footnote{The
Herfindahl Index does have a theoretical justification. If the firms are Cournot competitors with
different marginal costs, they will have different market shares. The Herfindahl Index equals the
weighted average of their market shares multiplied by the industry elasticity of demand (and
multiplied by -10,000). }
\begin{center}
\begin{minipage}[c]{ \linewidth}
\begin{center}
\begin{footnotesize}
\refstepcounter{tablecounter} \label{tab05-09}
{\sc Table \ref{chap07}.\ref{tab05-09} \\
Herfindahl Examples} \\
\bigskip
\begin{tabular}{llr}
\hline
\hline
& & \\
Industry & Computation & Herfindahl Index\\
& & \\
\hline
& & \\
Entirely monopolized &(100*100) & 10,000 \\
& & \\
One firm has 70\% of sales and the other has 30\% & (70*70+30*30) & 5,800 \\
& & \\
Two firms split the market evenly &(50*50 + 50*50) &5,000\\
& & \\
One firm has 50\% and 50 others have 1\% each & (50*50+ 50*1*1) &2,550 \\
& & \\
Five firms split the market evenly & (5*(20*20)) &2,000\\
& & \\
Ten firms split the market evenly &(10*(10*10)) & 1,000 \\
& & \\
100 firms split the market evenly & (100*(1*1)) & 100 \\
& & \\
\hline
\hline
\end{tabular}
\end{footnotesize}
\end{center}
\end{minipage}
\end{center}
Thus, antitrust law allows mergers, but is suspicious of them. Federal law requires any two large
firms that wish to merge to submit information to the FTC and the Justice Department on the likely
effects on price. If the FTC or Justice object to the merger, they can try to block it.
Since the statute is not clear on what it means for a merger to substantially reduce competition,
the agencies have to decide what mergers to block and the courts have to decide whether the agencies
are interpreting the law correctly. Companies would like to know which mergers are going to be
allowed, since setting up a merger only to have it blocked by the FTC results in a lot of wasted
effort. Thus, the FTC and Justice have issued regulations declaring what kinds of mergers are
likely to be blocked. A regulation called ``Horizontal Merger Guidelines'' was issued in 2010 for
{\bf horizontal mergers}\index{horizontal mergers}, which are mergers between firms selling
competing products.\footnote{U.S. Department of Justice and the Federal Trade Commission,
\url{http://www.justice.gov/atr/public/guidelines/hmg-2010.html} {``Horizontal Merger
Guidelines,''} (August 9, 2010).} An example would be a merger between Ford Motor Company and
Toyota, which both sell cars. Another memo, ``Non-Horizontal Merger Guidelines,'' not revised
since 1984, covers vertical and conglomerate mergers. \footnote{U.S. Department of Justice,
\url{http://www.justice.gov/atr/public/guidelines/2614.htm}{``Non-Horizontal Merger Guidelines,''}
(June 14, 1984). } {\bf Vertical mergers} \index{Vertical mergers} merge two companies when one
company (the {\bf upstream firm}\index{upstream firm}) sells to another company (the {\bf downstream
firm}\index{downstream firm} which in turn sells to consumers. An example would be a merger of
U.S. Steel and with the Ford Motor Company. {\bf Conglomerate mergers} \index{Conglomerate
mergers} merge companies that sell unrelated products, such as a merger of the Ford Motor Company
with Pepsi. Horizontal mergers generally reduce competition, and the main question is whether
they substantially reduce it. Conglomerate mergers do not reduce competition, since the two firms
are in different markets and market concentration is unchanged by the merger. Vertical mergers are
the most complicated to analyze, since they can reduce competition but in more subtle
ways.\footnote{ One's first thought is that once the firms merge vertically, the upstream firm will
stop selling to competitors of the downstream firm, so downstream competition will be reduced. The
flaw in that reasoning is that if the upstream firm found it profitable to sell to all the
downstream firms before the merger, it will still be profitable afterwards, so trying to stifle
competition there would actually hurt the merged firm's profits. There are counter-arguments, but
they would take too long to explain.}
The {\it 2010 Horizontal Guidelines}\index{ 2010 Horizontal Guidelines} try to inform businesses
what mergers will be allowed without tying the hands of the antitrust agencies. The {\it
Guidelines} talk about objective criteria for market concentration, for example, but they are also
careful to say that if the Justice Department finds a memo saying ``Of course, the main reason why
we're merging is so we can increase prices and gouge consumers,'' the merger can be declared
illegal. The {\it Guidelines} are written for companies trying to comply with the law in good faith.
An unprincipled firm cannot say, ``But the guidelines didn't say I couldn't do X!'' and be excused.
The {\it Guidelines} establish that what the agencies really care about is whether the merger will
raise prices, which is what economists focus on.
Earlier antitrust policy focussed on a more legalistic definition of the market, trying to use a
more objective and predictable measure of how concentrated the market was. An example of such an
approach (not a real example--- just a simple one for discussion) would be to say that two firms
can merge if and only if they wouldn't have over 30\% of the market after the merger. This
rule would make for neither good economics nor good law.
The economic problem is that such a simple rule ignores whether the merger really affects market
power or not. In one market, two firms with market shares of 16\% would be forbidden to merge even
though neither could raise price above cost because entry by new competitors is easy. In another
market, two firms with market shares of 14\% would be allowed to merger even though there
exists only one other firm in the market (with the remaining 72\% of sales) so a triopoly would be
converted to a duopoly.
The legal problem is that the rule is not actually so simple. It depends completely on how one
defines ``market''. Lawyers can spend endless hours arguing about that. In the Cellophane Case,
was Dupont selling in the market for ``cellophane'' (of which it had a 75\% market share) or
``flexible packaging material'' (of which cellophane was only 20\%). {\it U.S. v. E. I. du Pont},
351 U.S. 377 (1956). In the ReaLemon Case, was Borden selling in the market for lemon juice
in plastic lemons or did the market include natural lemons too? (are ReaLemons like real lemons?)
{\it Borden. v. Federal Trade Commission},
{674 F.2d 498} (6th
Cir. 1982). In the 2000's, were Whole Foods and Wild Oats competing with ``premium, natural, and
organic supermarkets'' or with
``grocery stores and supermarkets''? {\it FTC v. Whole Foods Markets, Inc.},
{
533 F.3d 869} (D.C. Cir. 2008).
The {\it Guidelines} use a different approach to decide what group of products makes up ``a
market'' when trying to figure out how many firms currently are in the market of the proposed
merger. The
{\bf Hypothetical Monopolist Test}\index{Hypothetical Monopolist Test} asks whether if a single
firm was the only seller of a group of products, it ``likely would impose at least a small but
significant and non-transitory increase in price ({\bf ``SSNIP''})\index{SSNIP} on at least one
product in the market, including at least one product sold by one of the merging firms.'' A ten-
percent increase counts as significant. So, perhaps would be a 4\% increase. The Guidelines are
vague on that point.
The {\it Guidelines} make use of the Herfindahl Index as a way to screen mergers. They say:
\begin{bigquote}
\hspace*{16pt}The Agencies generally classify markets into three types:
\begin{itemize}
\item Unconcentrated Markets: HHI below 1,500
\item Moderately Concentrated Markets: HHI between 1,500 and 2,500
\item Highly Concentrated Markets: HHI above 2,500
\end{itemize}
\hspace*{16pt}The Agencies employ the following general standards for the relevant markets they have
defined:
\begin{itemize}
\item Small Change in Concentration: Mergers involving an increase in the HHI of less than 100
points are unlikely to have adverse competitive effects and ordinarily require no further analysis.
\item Unconcentrated Markets: Mergers resulting in unconcentrated markets are unlikely to have
adverse competitive effects and ordinarily require no further analysis.
\item Moderately Concentrated Markets: Mergers resulting in moderately concentrated markets that
involve an increase in the HHI of more than 100 points potentially raise significant competitive
concerns and often warrant scrutiny.
\item Highly Concentrated Markets: Mergers resulting in highly concentrated markets that involve
an increase in the HHI of between 100 points and 200 points potentially raise significant
competitive concerns and often warrant scrutiny. Mergers resulting in highly concentrated markets
that involve an increase in the HHI of more than 200 points will be presumed to be likely to enhance
market power. The presumption may be rebutted by persuasive evidence showing that the merger is
unlikely to enhance market power.
\end{itemize}
\end{bigquote}
%\end{minipage}
Consider an example.
One firm has 50\% and 50 others have 1\% each.
The Herfindahl is 2,550 (50*50+ 50*1*1) so the industry is {\it concentrated}.
Can 12 of the small firms merge?
They would contribute 144 to the Herfindahl instead of 12,
an increase of 132.
Such a merger {\it often warrants scrutiny} because it is in a {\it concentrated industry} even tho
it is small.
Concentration measures are not really what we're worried about, but they are numbers, and
relatively objective numbers, and that tends to focus the attention of policymakers, lawyers,
judges, and people generally. The subjective element gets shifted, but it doesn't disappear. In fact
it is crucial. Where it goes is into the definition of ``market'' that you use to measure the market
shares that go into the Herfindahl Index. If Microsoft's market is ``software'', its market
share is moderate. If the market is ``operating systems'' its market share is huge. If the
market is ``perfectly Windows-compatible operating systems'', it is an absolute monopoly.
Despite all of the details on measuring concentration,
the {\it Guidelines} say not to rely too much on the Herfindahl Index:
\begin{bigquote}
The purpose of these thresholds is not to provide a rigid screen to separate competitively benign
mergers from anticompetitive ones, although high levels of concentration do raise concerns. Rather,
they provide one way to identify some mergers unlikely to raise competitive concerns and some others
for which it is particularly important to examine whether other competitive factors confirm,
reinforce, or counteract the potentially harmful effects of increased concentration.
\end{bigquote}
%-------------------------------------------------------
\bigskip
\noindent
{\bf The Staples-Office Depot Merger}
When Staples and Office Depot proposed to merge in 1997, they presented evidence to the FTC and
the Justice Department that their costs would fall enough to increase total surplus. The two firms
were by far the biggest office supply firms, though, so it was clear their market power would
increase. The companies argued that the market should include anybody selling office supplies,
including big-box companies like Wal-Mart and K-Mart, but this case was not decided by looking at
concentration ratios.\footnote{See Harrington et al,; and Federal Trade Commission,
\url{http://www.ftc.gov/opa/1997/04/stapdep.shtm}{``FTC Rejects Proposed Settlement in
Staples/Office Depot Merger:
Says Deal Would Still Violate Antitrust Laws and Lead to Higher Prices for Office Supplies,''}
(April 4, 1997); John Broder, \href{http://www.nytimes.com/1997/07/01/business/office-depot-and-staples-merger-halted.html}{ ``Office Depot and Staples Merger Halted,''} {\it The New York Times}
(July 1, 1997). }
The companies said their joint costs would fall. The Justice Department let the FTC decide what to
do, and the FTC was skeptical about how big the cost savings would be. The FTC said that average
cost would fall by 1.4\% and the merger would raise prices by 7.3\%. Staples
said that if there weren't any cost savings at all, prices would rise by 2.4\%, but there
would be cost savings, so the ultimate effect would be a 2.2\% fall in prices.
What about total surplus, though? Obviously, if Staples were right it would rise because both
consumer and producer surplus would rise. But what if the FTC was right?
Staples and the FTC went to court to decide whether the merger was legal. The judge ruled that it
was not. He ruled that total surplus was not relevant, only consumer surplus, and that the merger
would raise prices and so would violate the Clayton Act. Indeed, Professor Carlton reports that only
New Zealand and Canada have antitrust laws which hold total surplus to be the goal, not consumer
surplus.\footnote{See p.\ 157 of Carlton,\index{Carlton} Dennis W.,
\href{http://www.jstor.org/stable/30033739}{``Does Antitrust Need to Be Modernized?''} {\it The
Journal of Economic Perspectives}, 21: 155--176 (Summer 2007). }
Thus, companies that want to merge have to show not only that their costs would fall but that
costs would fall enough that even consumers would benefit. Most mergers can pass this test, but not
all, as the Staples case shows.
%-------------------------------------------------------
\bigskip
%-------------------------------------------------------
\noindent
{\bf \ref{chap07}.6: Conclusion} \label{s06.5}
The theme in this chapter has been that there are business practices such as merging with another
firm that sometimes have good, surplus-increasing motivations and sometimes have bad,
surplus-reducing (but producer-surplus-{\it increasing}) motivations. The enforcement agencies
and the judges have to decide which is the real motive in a given situation. Knowing some economics
is clearly helpful to figuring that out. A major purpose of antitrust law is to deter egregiously
bad behavior, but to the extent that the law is successful, we would not actually observe that bad
behavior, and in fact we do not. Cartels can be kept secret and still work, but mergers and
predatory pricing cannot, so the cases most commonly brought involve secret price cutting.
Antitrust law addresses the market failure caused by market power. It does not address high
monopoly prices directly. Having a monopoly is legal, and so is charging high prices using market
power. Because of government failure, it would be imprudent for the antitrust authorities to try to
regulate prices or to break up companies simply because they are large and successful. Rather,
antitrust law addresses the creation of market power by means which do not create surplus, only
redistribute it. In this chapter, we have looked at the behavior of firms with market power, whether
in isolation as a monopoly or in strategic interactions with each other as a duopoly or oligopoly.
We've also looked at how antitrust laws are enforced and at its first priority: preventing price
fixing. In the next chapter we will look at other monopoly-creating behaviors such as mergers and
predatory pricing.
%-----------------------------
%\vfill
%\cleardoublepage
\begin{center}
{\sc Review Questions }
\end{center}
\begin{enumerate}
\item How does a firm with market power choose price and output to maximize
profits?
\item How does the Prisoner's Dilemma describe the incentive problem of cartels?
\item What is the Cournot Model and how does one solve for its equilibrium price?
\item
Under what conditions are mergers illegal in the United States?
\item How would you decide whether a merger would increase market power too
much?
\item How has the law treated cartels in the 19th century and in the present?
\item How do the Sherman Act and the Clayton Act differ?
\item How is antitrust law enforced in Europe and in the United States?
\end{enumerate}
%\newpage
% \vspace{1in}
%\cleardoublepage
\begin{center}
{\sc Readings }
\end{center}
\begin{enumerate}
\item
\href{https://www.law360.com/articles/875833/print?section=competition} {``\$54B Deal Will Spawn Daunting Anthem-Cigna, Expert Says,''} {\it Law360}.
\item
\href{http://www.wsj.com/articles/eu-to-fine-truck-makers-over-price-fixing-and-other-collusion-1468861361} { `` EU To Fine Truck Makers over Price-Fixing and Other Collusion:
Total Fines of Around \$3.3 Billion after a Probe by Antitrust Authorities,'' } {\it The Wall Street Journal}.
\item
\href{http://www.bbc.com/news/business-35028380} { ``Canada's Maple Syrup 'Rebels' ,''} {\it The BBC}.
\item
\href{http://www.economist.com/news/finance-and-economics/21568364-how-antitrust-economists-are-getting-better-spotting-cartels-scam-busters}{``The Scam Busters:
How Antitrust Economists Are Getting Better at Spotting Cartels,''} {\it The Economist}.
\item
\href{https://www.washingtonpost.com/opinions/george-will-a-strike-against-rent-seeking/2014/12/31/ba5a1686-9109-11e4-ba53-a477d66580ed_story.html?utm_term=.dd6d67cc1820}{``A Strike against Rent-Seeking," } George Will, {\it The Washington Post}.
\end{enumerate}
\end{document}
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\begin{document}
I)\ Policz drugie pochodne\ (w zadaniu by\l o pierwsze, ale drugie te\.{z} mo%
\.{z}na...):
Nie znam do nich odpowiedzi
1) $x^{2}(\ln x-\frac{1}{2})$
2) $x-\sin x\cos x$
3) $\arccos \sqrt{1-x^{2}}$
4) $x\sqrt{a^{2}-x^{2}}+a^{2}\arcsin \frac{x}{a}$
5)\ $(x^{2}+1)^{x+1/x}$
\bigskip 6) $\frac{1-\cos x}{\sin x}$
\qquad Odp: $f^{\prime \prime }=\frac{(1-\cos x)^{2}}{\sin ^{3}x}$
\bigskip
II)\ Znajdz n-ta pochodna funkcji:(to tez bylo u Kaski, a w kazdym razie nie
zglupiejesz)
1) $f=\sqrt{x}$
\qquad Odp: $f^{(n)}=(-1)^{n-1}\frac{1\cdot 3\cdot ...\cdot (2n-3)}{2^{n}}$
2) $f=e^{ax+b}$
\qquad Odp:\ $f^{(n)}=a^{n}e^{ax+b}$
3) $f=x\ln x$
\qquad Odp:\ $f^{(n)}=(-1)^{n}\frac{(n-2)!}{x^{n-1}}$
\bigskip
III)\ Znajdz eksterma i punkty przegi\k{e}cia:
1) $x\sqrt{4-x^{2}}$
2) $x(a-x)^{2}$ dla $a>0$
3) $\frac{5-3x}{x^{2}-1}$
4) $x-\frac{2}{x}-3\ln x$
5) $3x^{4}-8x^{3}-18x^{2}+1$
6)\ $(x-2)^{2}\cdot e^{x+x^{2}/2}$
7) $\sin x+\sin 2x$
\bigskip Odpowiedzi:
\begin{tabular}{llll}
Zadanie & min dla x= & max & przegiecie \\
1 & $-\sqrt{2}$ & $\sqrt{2}$ & $0$ \\
2 & $a$ & $\frac{1}{3}a$ & $\frac{2}{3}a$ \\
3 & $3$ & $\frac{1}{3}$ & info nizej \\
4 & $2$ & $1$ & $\frac{4}{3}$ \\
5 & $-1$ & $0$ & $\frac{1}{2}(2\pm \sqrt{13})$ \\
6 & $0$ i $2$ & $1$ & infonizej2 \\
7 & $\alpha $ i $2\pi -\beta $ & $\beta $ i $2\pi -\alpha $ & $\cos x=-\frac{%
1}{8}$, $\sin x=0$%
\end{tabular}
\bigskip $\alpha =\arccos \frac{\sqrt{33}-1}{8}$
$\beta =\arccos \frac{\sqrt{33}+1}{8}$
\bigskip infonizej:\ punkt przegiecia w tym punkcie przedzialu $(3,\inf )$,
w kt\'{o}rym $3x^{3}-15x^{2}+9x-5=0$
infonizej2: 2 punkty przegiecie w tych punktach przedia\l \'{o}w $(0,1)$ i $%
(1,2)$, gdzie $x^{4}-2x^{3}+2x^{2}-4x+2=0$
\end{document}
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%% Uebersetzung: Copyright (C) 2005 Claudia Krysztofiak,
%% Rebecca Stiels und Frank Mittelbach
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\cite{LGC97} \\
\cite{LGC97} \hfill zweiter \\
\cite*{LGC97}\hfill erzwungen \\
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\documentclass[a4paper]{article}
\def\npart {IA}
\def\nterm {Lent}
\def\nyear {2015}
\def\nlecturer {W.\ T.\ Gowers}
\def\ncourse {Analysis I}
\input{header}
\begin{document}
\maketitle
{\small
\noindent\textbf{Limits and convergence}\\
Sequences and series in $\R$ and $\C$. Sums, products and quotients. Absolute convergence; absolute convergence implies convergence. The Bolzano-Weierstrass theorem and applications (the General Principle of Convergence). Comparison and ratio tests, alternating series test.\hspace*{\fill} [6]
\vspace{10pt}
\noindent\textbf{Continuity}\\
Continuity of real- and complex-valued functions defined on subsets of $\R$ and $\C$. The intermediate value theorem. A continuous function on a closed bounded interval is bounded and attains its bounds.\hspace*{\fill} [3]
\vspace{10pt}
\noindent\textbf{Differentiability}\\
Differentiability of functions from $\R$ to $\R$. Derivative of sums and products. The chain rule. Derivative of the inverse function. Rolle's theorem; the mean value theorem. One-dimensional version of the inverse function theorem. Taylor's theorem from $\R$ to $\R$; Lagrange's form of the remainder. Complex differentiation.\hspace*{\fill} [5]
\vspace{10pt}
\noindent\textbf{Power series}\\
Complex power series and radius of convergence. Exponential, trigonometric and hyperbolic functions, and relations between them. *Direct proof of the differentiability of a power series within its circle of convergence*.\hspace*{\fill}[4]
\vspace{10pt}
\noindent\textbf{Integration}\\
Definition and basic properties of the Riemann integral. A non-integrable function. Integrability of monotonic functions. Integrability of piecewise-continuous functions. The fundamental theorem of calculus. Differentiation of indefinite integrals. Integration by parts. The integral form of the remainder in Taylor's theorem. Improper integrals.\hspace*{\fill} [6]}
\tableofcontents
\setcounter{section}{-1}
\section{Introduction}
In IA Differential Equations, we studied calculus in a non-rigorous setting. While we did define differentiation (properly) as a limit, we did not define what a limit was. We didn't even have a proper definition of integral, and just mumbled something about it being an infinite sum.
In Analysis, one of our main goals is to put calculus on a rigorous foundation. We will provide unambiguous definitions of what it means to take a limit, a derivative and an integral. Based on these definitions, we will prove the results we've had such as the product rule, quotient rule and chain rule. We will also rigorously prove the fundamental theorem of calculus, which states that the integral is the inverse operation to the derivative.
However, this is not all Analysis is about. We will study all sorts of limiting (``infinite'') processes. We can see integration as an infinite sum, and differentiation as dividing two infinitesimal quantities. In Analysis, we will also study infinite series such as $1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \cdots$, as well as limits of infinite sequences.
Another important concept in Analysis is \emph{continuous functions}. In some sense, continuous functions are functions that preserve limit processes. While their role in this course is just being ``nice'' functions to work with, they will be given great importance when we study metric and topological spaces.
This course is continued in IB Analysis II, where we study uniform convergence (a stronger condition for convergence), calculus of multiple variables and metric spaces.
\section{The real number system}
We are all familiar with real numbers --- they are ``decimals'' consisting of infinitely many digits. When we really want to study real numbers, while this definition is technically valid, it is not a convenient definition to work with. Instead, we take an \emph{axiomatic} approach. We define the real numbers to be a set that satisfies certain properties, and, if we really want to, show that the decimals satisfy these properties. In particular, we define real numbers to be an ordered field with the least upper bound property.
We'll now define what it means to be an ``ordered field with the least upper bound property''.
\begin{defi}[Field]
A \emph{field} is a set $\F$ with two binary operations $+$ and $\times$ that satisfies all the familiar properties satisfied by addition and multiplication in $\Q$, namely
\begin{enumerate}
\item $(\forall a, b, c)\,a + (b + c) = (a + b) + c$ and $a\times (b\times c) = (a\times b)\times c$ \hfill (associativity)
\item $(\forall a, b)\,a + b = b + a$ and $a\times b= b\times a$ \hfill (commutativity)
\item $\exists 0, 1$ such that $(\forall a)a + 0 = a$ and $a\times 1 = a$ \hfill (identity)
\item $\forall a\,\exists (-a)$ such that $a + (-a) = 0$. If $a\not= 0$, then $\exists a^{-1}$ such that $a\times a^{-1} = 1$. \hfill (inverses)
\item $(\forall a, b, c)\, a\times (b + c) = (a\times b) + (a\times c)$ \hfill (distributivity)
\end{enumerate}
\end{defi}
\begin{eg}
$\Q, \R, \C$, integers mod $p$, $\{a + b\sqrt{2}: a, b\in \Z\}$ are all fields.
\end{eg}
\begin{defi}[Totally ordered set]
An \emph{(totally) ordered set} is a set $X$ with a relation $<$ that satisfies
\begin{enumerate}
\item If $x, y, z\in X$, $x < y$ and $y < z$, then $x < z$ \hfill (transitivity)
\item If $x, y\in X$, exactly one of $x < y, x = y, y < x$ holds \hfill (trichotomy)
\end{enumerate}
\end{defi}
We call it a \emph{totally} ordered set, as opposed to simply an ordered set, when we want to emphasize the order is total, i.e.\ every pair of elements are related to each other, as opposed to a \emph{partial order}, where ``exactly one'' in trichotomy is replaced with ``at most one''.
Now to define an ordered field, we don't simply ask for a field with an order. The order has to interact nicely with the field operations.
\begin{defi}[Ordered field]
An \emph{ordered field} is a field $\F$ with a relation $<$ that makes $\F$ into an ordered set such that
\begin{enumerate}
\item if $x, y, z \in \F$ and $x < y$, then $x + z < y + z$
\item if $x, y, z \in \F$, $x < y$ and $z > 0$, then $xz < yz$
\end{enumerate}
\end{defi}
\begin{lemma}
Let $\F$ be an ordered field and $x\in \F$. Then $x^2 \geq 0$.
\end{lemma}
\begin{proof}
By trichotomy, either $x < 0$, $x = 0$ or $x > 0$. If $x = 0$, then $x^2 = 0$. So $x^2 \geq 0$. If $x > 0$, then $x^2 > 0\times x = 0$. If $x < 0$, then $x - x < 0 - x$. So $0 < -x$. But then $x^2 = (-x)^2 > 0$.
\end{proof}
\begin{defi}[Least upper bound]
Let $X$ be an ordered set and let $A\subseteq X$. An \emph{upper bound} for $A$ is an element $x\in X$ such that $(\forall a\in A)\,a \leq x$. If $A$ has an upper bound, then we say that $A$ is \emph{bounded above}.
An upper bound $x$ for $A$ is a \emph{least upper bound} or \emph{supremum} if nothing smaller that $x$ is an upper bound. That is, we need
\begin{enumerate}
\item $(\forall a\in A)\,a \leq x$
\item $(\forall y < x)(\exists a\in A)\,a \geq y$
\end{enumerate}
We usually write $\sup A$ for the supremum of $A$ when it exists. If $\sup A\in A$, then we call it $\max A$, the maximum of $A$.
\end{defi}
\begin{eg}
Let $X = \Q$. Then the supremum of $(0, 1)$ is $1$. The set $\{x: x^2 < 2\}$ is bounded above by $2$, but has no supremum (even though $\sqrt{2}$ seems like a supremum, we are in $\Q$ and $\sqrt{2}$ is non-existent!).
$\max [0, 1] = 1$ but $(0, 1)$ has no maximum because the supremum is not in $(0, 1)$.
\end{eg}
We can think of the supremum as a point we can get arbitrarily close to in the set but cannot pass through.
\begin{defi}[Least upper bound property]
An ordered set $X$ has the \emph{least upper bound property} if every non-empty subset of $X$ that is bounded above has a supremum.
\end{defi}
Obvious modifications give rise to definitions of lower bound, greatest lower bound (or infimum) etc. It is simple to check that an ordered field with the least upper bound property has the greatest lower bound property.
\begin{defi}[Real numbers]
The \emph{real numbers} is an ordered field with the least upper bound property.
\end{defi}
Of course, it is \emph{very} important to show that such a thing exists, or else we will be studying nothing. It is also nice to show that such a field is unique (up to isomorphism). However, we will not prove these in the course.
In a field, we can define the ``natural numbers'' to be $2 = 1 + 1$, $3 = 1 + 2$ etc. Then an important property of the real numbers is
\begin{lemma}[Archimedean property v1)]
Let $\F$ be an ordered field with the least upper bound property. Then the set $\{1, 2, 3, \cdots\}$ is not bounded above.
\end{lemma}
\begin{proof}
If it is bounded above, then it has a supremum $x$. But then $x - 1$ is not an upper bound. So we can find $n\in \{1, 2, 3, \cdots\}$ such that $n> x - 1$. But then $n + 1 > x$, but $x$ is supposed to be an upper bound.
\end{proof}
Is the least upper bound property required to prove the Archimedean property? It seems like \emph{any} ordered field should satisfy this even if they do not have the least upper bound property. However, it turns out there are ordered fields in which the integers are bounded above.
Consider the field of rational functions, i.e.\ functions in the form $\frac{P(x)}{Q(x)}$ with $P(x), Q(x)$ being polynomials, under the usual addition and multiplication. We order two functions $\frac{P(x)}{Q(x)}, \frac{R(s)}{S(x)}$ as follows: these two functions intersect only finitely many times because $P(x)S(x) = R(x)Q(x)$ has only finitely many roots. After the last intersection, the function whose value is greater counts as the greater function. It can be checked that these form an ordered field.
In this field, the integers are the constant functions $1, 2, 3, \cdots$, but it is bounded above since the function $x$ is greater than all of them.
\section{Convergence of sequences}
Having defined real numbers, the first thing we will study is sequences. We will want to study what it means for a sequence to \emph{converge}. Intuitively, we would like to say that $1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4},\cdots$ converges to $0$, while $1, 2, 3, 4, \cdots$ diverges. However, the actual formal definition of convergence is rather hard to get right, and historically there have been failed attempts that produced spurious results.
\subsection{Definitions}
\begin{defi}[Sequence]
A \emph{sequence} is, formally, a function $a: \N \to \R$ (or $\C$). Usually (i.e.\ always), we write $a_n$ instead of $a(n)$. Instead of $a$, we usually write it as $(a_n)$, $(a_n)_1^\infty$ or $(a_n)_{n = 1}^\infty$ to indicate it is a sequence.
\end{defi}
\begin{defi}[Convergence of sequence]
Let $(a_n)$ be a sequence and $\ell\in \R$. Then $a_n$ \emph{converges to} $\ell$, \emph{tends to} $\ell$, or $a_n \to \ell$ , if for all $\varepsilon > 0$, there is some $N \in \N$ such that whenever $n > N$, we have $|a_n - \ell| < \varepsilon$. In symbols, this says
\[
(\forall \varepsilon > 0)(\exists N)(\forall n\geq N)\,|a_n - \ell| < \varepsilon.
\]
We say $\ell$ is the \emph{limit} of $(a_n)$.
\end{defi}
One can think of $(\exists N)(\forall n\geq N)$ as saying ``eventually always'', or as ``from some point on''. So the definition means, if $a_n\to \ell$, then given any $\varepsilon$, there eventually, everything in the sequence is within $\varepsilon$ of $\ell$.
We'll now provide an alternative form of the Archimedean property. This is the form that is actually useful.
\begin{lemma}[Archimedean property v2]
$1/n \to 0$.
\end{lemma}
\begin{proof}
\textcolor{red}{Let $\varepsilon > 0$}. We want to find an $N$ such that $|1/N - 0| = 1/N < \varepsilon$. So \textcolor{red}{pick $N$} such that $N > 1/\varepsilon$. There exists such an $N$ by the Archimedean property v1. Then \textcolor{red}{for all $n \geq N$}, we have $0 < 1/n \leq 1/N < \varepsilon$. So \textcolor{red}{$|1/n - 0| < \varepsilon$}.
\end{proof}
Note that the red parts correspond to the \emph{definition} of convergence of a sequence. This is generally how we prove convergence from first principles.
\begin{defi}[Bounded sequence]
A sequence $(a_n)$ is \emph{bounded} if
\[
(\exists C)(\forall n)\,|a_n| \leq C.
\]
A sequence is \emph{eventually bounded} if
\[
(\exists C)(\exists N)(\forall n\geq N)\, |a_n| \leq C.
\]
\end{defi}
The definition of an \emph{eventually bounded} sequence seems a bit daft. Clearly every eventually bounded sequence is bounded! Indeed it is:
\begin{lemma}
Every eventually bounded sequence is bounded.
\end{lemma}
\begin{proof}
Let $C$ and $N$ be such that $(\forall n\geq N)\,|a_n| \leq C$. Then $\forall n \in \N$, $|a_n| \leq \max\{|a_1|, \cdots, |a_{N - 1}|, C\}$.
\end{proof}
The proof is rather trivial. However, most of the time it is simpler to prove that a sequence is eventually bounded, and this lemma saves us from writing that long line every time.
\subsection{Sums, products and quotients}
Here we prove the things that we think are obviously true, e.g.\ sums and products of convergent sequences are convergent.
\begin{lemma}[Sums of sequences]
If $a_n \to a$ and $b_n \to b$, then $a_n + b_n \to a + b$.
\end{lemma}
\begin{proof}
Let $\varepsilon > 0$. We want to find a clever $N$ such that for all $n \geq N$, $|a_n + b_n - (a+b)| < \varepsilon$. Intuitively, we know that $a_n$ is very close to $a$ and $b_n$ is very close to $b$. So their sum must be very close to $a + b$.
Formally, since $a_n\to a$ and $b_n \to b$, we can find $N_1, N_2$ such that $\forall n \geq N_1$, $|a_n - a| < \varepsilon/2$ and $\forall n \geq N_2$, $|b_n - b| < \varepsilon/2$.
Now let $N = \max\{N_1, N_2\}$. Then by the triangle inequality, when $n \geq N$,
\[
|(a_n + b_n) - (a + b)| \leq |a_n - a| + |b_n - b| < \varepsilon. \qedhere
\]
\end{proof}
We want to prove that the product of convergent sequences is convergent. However, we will not do it in one go. Instead, we separate it into many smaller parts.
\begin{lemma}[Scalar multiplication of sequences]
Let $a_n \to a$ and $\lambda \in \R$. Then $\lambda a_n \to \lambda a$.
\end{lemma}
\begin{proof}
If $\lambda = 0$, then the result is trivial.
Otherwise, let $\varepsilon > 0$. Then $\exists N$ such that $\forall n \geq N$, $|a_n - a| < \varepsilon/|\lambda|$. So $|\lambda a_n - \lambda a| < \varepsilon$.
\end{proof}
\begin{lemma}
Let $(a_n)$ be bounded and $b_n \to 0$. Then $a_nb_n \to 0$.
\end{lemma}
\begin{proof}
Let $C\not=0$ be such that $(\forall n)\, |a_n|\leq C$. Let $\varepsilon > 0$. Then $\exists N$ such that $(\forall n\geq N)\, |b_n| < \varepsilon/C$. Then $|a_nb_n| < \varepsilon$.
\end{proof}
\begin{lemma}
Every convergent sequence is bounded.
\end{lemma}
\begin{proof}
Let $a_n \to l$. Then there is an $N$ such that $\forall n \geq N$, $|a_n - l| \leq 1$. So $|a_n| \leq |l| + 1$. So $a_n$ is eventually bounded, and therefore bounded.
\end{proof}
\begin{lemma}[Product of sequences]
Let $a_n\to a$ and $b_n\to b$. Then $a_nb_n\to ab$.
\end{lemma}
\begin{proof}
Let $a_n = a + \varepsilon_n$. Then $a_nb_n = (a + \varepsilon_n)b_n = ab_n + \varepsilon_n b_n$.
Since $b_n \to b$, $ab_n \to ab$. Since $\varepsilon_n \to 0$ and $b_n$ is bounded, $\varepsilon_nb_n \to 0$. So $a_nb_n \to ab$.
\end{proof}
\begin{proof}
(alternative) Observe that $a_nb_n - ab = (a_n - a) b_n + (b_n - b)a$. We know that $a_n - a \to 0$ and $b_n - b\to 0$. Since $(b_n)$ is bounded, so $(a_n - a)b_n + (b_n - b)a \to 0$. So $a_nb_n \to ab$.
\end{proof}
Note that in this proof, we no longer write ``Let $\varepsilon > 0$''. In the beginning, we have no lemmas proven. So we must prove everything from first principles and use the definition. However, after we have proven the lemmas, we can simply use them instead of using first principles. This is similar to in calculus, where we use first principles to prove the product rule and chain rule, then no longer use first principles afterwards.
\begin{lemma}[Quotient of sequences]
Let $(a_n)$ be a sequence such that $(\forall n)\, a_n \not= 0$. Suppose that $a_n \to a$ and $a\not = 0$. Then $1/a_n \to 1/a$.
\end{lemma}
\begin{proof}
We have
\[
\frac{1}{a_n} - \frac{1}{a} = \frac{a - a_n}{aa_n}.
\]
We want to show that this $\to 0$. Since $a - a_n \to 0$, we have to show that $1/(aa_n)$ is bounded.
Since $a_n \to a$, $\exists N$ such that $\forall n\geq N$, $|a_n - a| \leq a/2$. Then $\forall n\geq N$, $|a_n| \geq |a|/2$. Then $|1/(a_na)| \leq 2/|a|^2$. So $1/(a_na)$ is bounded. So $(a - a_n)/(aa_n)\to 0$ and the result follows.
\end{proof}
\begin{cor}
If $a_n \to a, b_n \to b$, $b_n, b\not= 0$, then $a_n/b_n = a/b$.
\end{cor}
\begin{proof}
We know that $1/b_n \to 1/b$. So the result follows by the product rule.
\end{proof}
\begin{lemma}[Sandwich rule]
Let $(a_n)$ and $(b_n)$ be sequences that both converge to a limit $x$. Suppose that $a_n \leq c_n \leq b_n$ for every $n$. Then $c_n \to x$.
\end{lemma}
\begin{proof}
Let $\varepsilon > 0$. We can find $N$ such that $\forall n \geq N$, $|a_n - x| < \varepsilon$ and $|b_n - x| < \varepsilon$.
Then $\forall n\geq N$, we have $x - \varepsilon < a_n \leq c_n \leq b_n < x + \varepsilon$. So $|c_n - x| < \varepsilon$.
\end{proof}
\begin{eg}
$1/2^n \to 0$. For every $n$, $n < 2^n$. So $0 < 1/2^n < 1/n$. The result follows from the sandwich rule.
\end{eg}
\begin{eg}
We want to show that
\[
\frac{n^2 + 3}{(n + 5)(2n - 1)} \to \frac{1}{2}.
\]
We can obtain this by
\[
\frac{n^2 + 3}{(n + 5)(2n - 1)} = \frac{1 + 3/n^2}{(1 + 5/n)(2 - 1/n)} \to \frac{1}{2},
\]
by sum rule, sandwich rule, Archimedean property, product rule and quotient rule.
\end{eg}
\begin{eg}
Let $k\in \N$ and let $\delta > 0$. Then
\[
\frac{n^k}{(1 + \delta)^n}\to 0.
\]
This can be summarized as ``exponential growth beats polynomial growth eventually''.
By the binomial theorem,
\[
(1 + \delta)^n \geq \binom{n}{k + 1}\delta^{k + 1}.
\]
Also, for $n\geq 2k$,
\[
\binom{n}{k + 1} = \frac{n(n - 1)\cdots(n - k)}{(k + 1)!} \geq \frac{(n/2)^{k + 1}}{(k + 1)!}.
\]
So for sufficiently large $n$,
\[
\frac{n^k}{(1 + \delta)^n} \leq \frac{n^k 2^{k + 1} (k+1)!}{n^{k + 1}\delta^{k + 1}} = \frac{2^{k + 1} (k + 1)!}{\delta^{k + 1}} \cdot \frac{1}{n} \to 0.
\]
\end{eg}
\subsection{Monotone-sequences property}
Recall that we characterized the least upper bound property. It turns out that there is an alternative characterization of real number using sequences, known as the \emph{monotone-sequences property}. In this section, we will show that the two characterizations are equivalent, and use the monotone-sequences property to deduce some useful results.
\begin{defi}[Monotone sequence]
A sequence $(a_n)$ is \emph{increasing} if $a_n\leq a_{n + 1}$ for all $n$.
It is \emph{strictly increasing} if $a_n < a_{n + 1}$ for all $n$. \emph{(Strictly) decreasing} sequences are defined analogously.
A sequence is \emph{(strictly) monotone} if it is (strictly) increasing or (strictly) decreasing.
\end{defi}
\begin{defi}[Monotone-sequences property]
An ordered field has the \emph{monotone sequences property} if every increasing sequence that is bounded above converges.
\end{defi}
We want to show that the monotone sequences property is equivalent to the least upper bound property.
\begin{lemma}
Least upper bound property $\Rightarrow$ monotone-sequences property.
\end{lemma}
\begin{proof}
Let $(a_n)$ be an increasing sequence and let $C$ an upper bound for $(a_n)$. Then $C$ is an upper bound for the set $\{a_n: n \in \N\}$. By the least upper bound property, it has a supremum $s$. We want to show that this is the limit of $(a_n)$.
Let $\varepsilon > 0$. Since $s = \sup \{a_n: n\in \N\}$, there exists an $N$ such that $a_N > s - \varepsilon$. Then since $(a_n)$ is increasing, $\forall n \geq N$, we have $s - \varepsilon < a_N \leq a_n \leq s$. So $|a_n - s| < \varepsilon$.
\end{proof}
We first prove a handy lemma.
\begin{lemma}
Let $(a_n)$ be a sequence and suppose that $a_n \to a$. If $(\forall n)\, a_n \leq x$, then $a\leq x$.
\end{lemma}
\begin{proof}
If $a > x$, then set $\varepsilon = a - x$. Then we can find $N$ such that $a_N > x$. Contradiction.
\end{proof}
Before showing the other way implication, we will need the following:
\begin{lemma}
Monotone-sequences property $\Rightarrow$ Archimedean property.
\end{lemma}
\begin{proof}
We prove version 2, i.e.\ that $1/n \to 0$.
Since $1/n > 0$ and is decreasing, by MSP, it converges. Let $\delta$ be the limit. By the previous lemma, we must have $\delta \geq 0$.
If $\delta > 0$, then we can find $N$ such that $1/N < 2\delta$. But then for all $n \geq 4N$, we have $1/n \leq 1/(4N) < \delta/2$. Contradiction. Therefore $\delta = 0$.
\end{proof}
\begin{lemma}
Monotone-sequences property $\Rightarrow$ least upper bound property.
\end{lemma}
\begin{proof}
Let $A$ be a non-empty set that's bounded above. Pick $u_0, v_0$ such that $u_0$ is not an upper bound for $A$ and $v_0$ is an upper bound. Now do a repeated bisection: having chosen $u_n$ and $v_n$ such that $u_n$ is not an upper bound and $v_n$ is, if $(u_n + v_n)/2$ is an upper bound, then let $u_{n + 1} = u_n$, $v_{n + 1} = (u_n + v_n)/2$. Otherwise, let $u_{n + 1} = (u_n + v_n)/2$, $v_{n + 1} = v_n$.
Then $u_0 \leq u_1 \leq u_2 \leq \cdots$ and $v_0\geq v_1 \geq v_2 \geq \cdots$. We also have
\[
v_n - u_n = \frac{v_0 - u_0}{2^n} \to 0.
\]
By the monotone sequences property, $u_n\to s$ (since $(u_n)$ is bounded above by $v_0$). Since $v_n - u_n \to 0$, $v_n \to s$. We now show that $s = \sup A$.
If $s$ is not an upper bound, then there exists $a\in A$ such that $a > s$. Since $v_n \to s$, then there exists $m$ such that $v_m < a$, contradicting the fact that $v_m$ is an upper bound.
To show it is the \emph{least} upper bound, let $t < s$. Then since $u_n \to s$, we can find $m$ such that $u_m > t$. So $t$ is not an upper bound. Therefore $s$ is the least upper bound.
\end{proof}
Why do we need to prove the Archimedean property first? In the proof above, we secretly used the it. When showing that $v_n - u_n \to 0$, we required the fact that $\frac{1}{2^n} \to 0$. To prove this, we sandwiched it with $\frac{1}{n}$. But to show $\frac{1}{n}\to 0$, we need the Archimedean property.
\begin{lemma}
A sequence can have at most 1 limit.
\end{lemma}
\begin{proof}
Let $(a_n)$ be a sequence, and suppose $a_n \to x$ and $a_n\to y$. Let $\varepsilon > 0$ and pick $N$ such that $\forall n \geq N$, $|a_n - x| < \varepsilon/2$ and $|a_n - y| < \varepsilon/2$. Then $|x - y| \leq |x - a_N| + |a_N - y| < \varepsilon/2 + \varepsilon/2 = \varepsilon$. Since $\varepsilon$ was arbitrary, $x$ must equal $y$.
\end{proof}
\begin{lemma}[Nested intervals property]
Let $\F$ be an ordered field with the monotone sequences property. Let $I_1\supseteq I_2 \supseteq \cdots$ be closed bounded non-empty intervals. Then $\bigcap_{n = 1}^\infty I_n \not= \emptyset$.
\end{lemma}
\begin{proof}
Let $T_n = [a_n, b_n]$ for each $n$. Then $a_1 \leq a_2 \leq\cdots$ and $b_1 \geq b_2\geq \cdots$. For each $n$, $a_n \leq b_n \leq b_1$. So the sequence $a_n$ is bounded above. So by the monotone sequences property, it has a limit $a$. For each $n$, we must have $a_n\leq a$. Otherwise, say $a_n > a$. Then for all $m \geq n$, we have $a_m \geq a_n > a$. This implies that $a > a$, which is nonsense.
Also, for each fixed $n$, we have that $\forall m\geq n$, $a_m \leq b_m \leq b_n$. So $a \leq b_n$. Thus, for all $n$, $a_n \leq a \leq b_n$. So $a\in I_n$. So $a\in \bigcap_{n = 1}^\infty I_n$.
\end{proof}
We can use this to prove that the reals are uncountable:
\begin{prop}
$\R$ is uncountable.
\end{prop}
\begin{proof}
Suppose the contrary. Let $x_1, x_2, \cdots$ be a list of all real numbers. Find an interval that does not contain $x_1$. Within that interval, find an interval that does not contain $x_2$. Continue \emph{ad infinitum}. Then the intersection of all these intervals is non-empty, but the elements in the intersection are not in the list. Contradiction.
\end{proof}
A powerful consequence of this is the \emph{Bolzano-Weierstrass theorem}. This is formulated in terms of subsequences:
\begin{defi}[Subsequence]
Let $(a_n)$ be a sequence. A \emph{subsequence} of $(a_n)$ is a sequence of the form $a_{n_1}, a_{n_2}, \cdots$, where $n_1 < n_2 < \cdots$.
\end{defi}
\begin{eg}
$1, 1/4, 1/9, 1/16, \cdots$ is a subsequence of $1, 1/2, 1/3, \cdots$.
\end{eg}
\begin{thm}[Bolzano-Weierstrass theorem]
Let $\F$ be an ordered field with the monotone sequences property (i.e.\ $\F = \mathbb{R}$).
Then every bounded sequence has a convergent subsequence.
\end{thm}
\begin{proof}
Let $u_0$ and $v_0$ be a lower and upper bound, respectively, for a sequence $(a_n)$. By repeated bisection, we can find a sequence of intervals $[u_0, v_0] \supseteq [u_1, v_1]\supseteq [u_2,v_2] \supseteq\cdots$ such that $v_n - u_n = (v_0 - u_0)/2^n$, and such that each $[u_n, v_n]$ contains infinitely many terms of $(a_n)$.
By the nested intervals property, $\bigcap_{n = 1}^\infty [u_n, v_n] \not= \emptyset$. Let $x$ belong to the intersection. Now pick a subsequence $a_{n_1}, a_{n_2}, \cdots$ such that $a_{n_k} \in [u_k, v_k]$. We can do this because $[u_k, v_k]$ contains infinitely many $a_n$, and we have only picked finitely many of them. We will show that $a_{n_k} \to x$.
Let $\varepsilon > 0$. By the Archimedean property, we can find $K$ such that $v_K - u_K = (v_0 - u_0)/2^K \leq \varepsilon$. This implies that $[u_K, v_K] \subseteq (x - \varepsilon, x + \varepsilon)$, since $x\in [u_K, v_K]$.
Then $\forall k \geq K$, $a_{n_k}\in [u_k, v_k] \subseteq [u_K, v_K] \subseteq (x - \varepsilon, x + \varepsilon)$. So $|a_{n_k} - x| < \varepsilon$.
\end{proof}
\subsection{Cauchy sequences}
The third characterization of real numbers is in terms of Cauchy sequences. Cauchy convergence is an alternative way of defining convergent sequences without needing to mention the actual limit of the sequence. This allows us to say $\{3, 3.1, 3.14, 3.141, 3.1415, \cdots\}$ is \emph{Cauchy convergent} in $\Q$ even though the limit $\pi$ is not in $\Q$.
\begin{defi}[Cauchy sequence]
A sequence $(a_n)$ is \emph{Cauchy} if for all $\varepsilon$, there is some $N \in \N$ such that whenever $p, q \geq N$, we have $|a_p - a_q| < \varepsilon$. In symbols, we have
\[
(\forall \varepsilon > 0)(\exists N)(\forall p, q\geq N)\, |a_p - a_q| < \varepsilon.
\]
\end{defi}
Roughly, a sequence is Cauchy if all terms are eventually close to each other (as opposed to close to a limit).
\begin{lemma}
Every convergent sequence is Cauchy.
\end{lemma}
\begin{proof}
Let $a_n \to a$. Let $\varepsilon > 0$. Then $\exists N$ such that $\forall n \geq N$, $|a_n - a| < \varepsilon/2$. Then $\forall p, q\geq N$, $|a_p - a_q| \leq |a_p - a| + |a - a_q| < \varepsilon/2 + \varepsilon/2 = \varepsilon$.
\end{proof}
\begin{lemma}
Let $(a_n)$ be a Cauchy sequence with a subsequence $(a_{n_k})$ that converges to $a$. Then $a_n\to a$.
\end{lemma}
\begin{proof}
Let $\varepsilon > 0$. Pick $N$ such that $\forall p, q\geq N$, $|a_p - a_q| < \varepsilon/2$. Then pick $K$ such that $n_K \geq N$ and $|a_{n_K} - a| < \varepsilon/2$.
Then $\forall n \geq N$, we have
\[
|a_n - a| \leq |a_n - a_{n_K}| + |a_{n_K} - a| < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon.\qedhere
\]
\end{proof}
An important result we have is that in $\R$, Cauchy convergence and regular convergence are equivalent.
\begin{thm}[The general principle of convergence]
Let $\F$ be an ordered field with the monotone-sequence property. Then every Cauchy sequence of $\F$ converges.
\end{thm}
\begin{proof}
Let $(a_n)$ be a Cauchy sequence. Then it is eventually bounded, since $\exists N$, $\forall n \geq N$, $|a_n - a_N| \leq 1$ by the Cauchy condition. So it is bounded. Hence by Bolzano-Weierstrass, it has a convergent subsequence. Then $(a_n)$ converges to the same limit.
\end{proof}
\begin{defi}[Complete ordered field]
An ordered field in which every Cauchy sequence converges is called \emph{complete}.
\end{defi}
Hence we say that $\R$ is a complete ordered field.
However, not every complete ordered field is (isomorphic to) $\R$. For example, we can take the rational functions as before, then take the Cauchy completion of it (i.e.\ add all the limits we need). Then it is already too large to be the reals (it still doesn't have the Archimedean property) but is a complete ordered field.
To show that completeness implies the monotone-sequences property, we need an additional condition: the Archimedean property.
\begin{lemma}
Let $\F$ be an ordered field with the Archimedean property such that every Cauchy sequence converges. The $\F$ satisfies the monotone-sequences property.
\end{lemma}
\begin{proof}
Instead of showing that every bounded monotone sequence converges, and is hence Cauchy, We will show the equivalent statement that every increasing non-Cauchy sequence is not bounded above.
Let $(a_n)$ be an increasing sequence. If $(a_n)$ is not Cauchy, then
\[
(\exists \varepsilon > 0)(\forall N)(\exists p, q > N)\,|a_p - a_q| \geq \varepsilon.
\]
wlog let $p > q$. Then
\[
a_p \geq a_q + \varepsilon \geq a_N + \varepsilon.
\]
So for any $N$, we can find a $p > N$ such that
\[
a_p \geq a_N + \varepsilon.
\]
Then we can construct a subsequence $a_{n_1}, a_{n_2}, \cdots$ such that
\[
a_{n_{k + 1}} \geq a_{n_k} + \varepsilon.
\]
Therefore
\[
a_{n_k} \geq a_{n_1} + (k - 1)\varepsilon.
\]
So by the Archimedean property, $(a_{n_k})$, and hence $(a_n)$, is unbounded.
\end{proof}
Note that the definition of a convergent sequence is
\[
(\exists l)(\forall \varepsilon > 0)(\exists N)(\forall n\geq N)\, |a_n - l| < \varepsilon,
\]
while that of Cauchy convergence is
\[
(\forall \varepsilon > 0)(\exists N)(\forall p, q\geq N)\, |a_p - a_q| < \varepsilon.
\]
In the first definition, $l$ quantifies over all real numbers, which is uncountable. However, in the second definition, we only have to quantify over natural numbers, which is countable (by the Archimedean property, we only have to consider the cases $\varepsilon = 1/n$).
Since they are equivalent in $\R$, the second definition is sometimes preferred when we care about logical simplicity.
\subsection{Limit supremum and infimum}
Here we will define the limit supremum and infimum. While these are technically not part of the course, eventually some lecturers will magically assume students know this definition. So we might as well learn it here.
\begin{defi}[Limit supremum/infimum]
Let $(a_n)$ be a bounded sequence. We define the \emph{limit supremum} as
\[
\limsup_{n\to \infty} a_n = \lim_{n\to \infty}\left(\sup_{m \geq n} a_m\right).
\]
To see that this exists, set $b_n = \sup_{m\geq n}a_m$. Then $(b_n)$ is decreasing since we are taking the supremum of fewer and fewer things, and is bounded below by any lower bound for $(a_n)$ since $b_n \geq a_n$. So it converges.
Similarly, we define the \emph{limit infimum} as
\[
\liminf_{n\to \infty}a_n = \lim_{n\to\infty}\left(\inf_{m\geq n} a_m\right).
\]
\end{defi}
\begin{eg}
Take the sequence
\[
2, -1, \frac{3}{2}, -\frac{1}{2}, \frac{4}{3}, -\frac{1}{3}, \cdots
\]
Then the limit supremum is $1$ and the limit infimum is $0$.
\end{eg}
\begin{lemma}
Let $(a_n)$ be a sequence. The following two statements are equivalent:
\begin{itemize}
\item $a_n\to a$
\item $\limsup a_n = \liminf a_n = a$.
\end{itemize}
\end{lemma}
\begin{proof}
If $a_n \to a$, then let $\varepsilon > 0$. Then we can find an $n$ such that
\[
a - \varepsilon \leq a_m \leq a + \varepsilon\text{ for all } m \geq n
\]
It follows that
\[
a - \varepsilon \leq \inf_{m \geq n}a_m \leq \sup_{m\geq n} a_m \leq a + \varepsilon.
\]
Since $\varepsilon$ was arbitrary, it follows that
\[
\liminf a_n = \limsup a_n = a.
\]
Conversely, if $\liminf a_n = \limsup a_n = a$, then let $\varepsilon > 0$. Then we can find $n$ such that
\[
\inf_{m\geq n} a_m > a - \varepsilon\text{ and }\sup _{m \geq n} a_m < a + \varepsilon.
\]
It follows that $\forall m\geq n$, we have $|a_m - a| < \varepsilon$.
\end{proof}
\section{Convergence of infinite sums}
In this chapter, we investigate which infinite \emph{sums}, as opposed to sequences, converge. We would like to say $1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots = 2$, while $1 + 1 + 1 + 1 + \cdots$ does not converge. The majority of the chapter is coming up with different tests to figure out if infinite sums converge.
\subsection{Infinite sums}
\begin{defi}[Convergence of infinite sums and partial sums]
Let $(a_n)$ be a real sequence. For each $N$, define
\[
S_N = \sum_{n = 1}^N a_n.
\]
If the sequence $(S_N)$ converges to some limit $s$, then we say that
\[
\sum_{n = 1}^\infty a_n = s,
\]
and we say that the series $\displaystyle\sum_{n = 1}^\infty a_n$ \emph{converges}.
We call $S_N$ the $N$th \emph{partial sum}.
\end{defi}
There is an immediate necessary condition for a series to converge.
\begin{lemma}
If $\displaystyle\sum_{n = 1}^\infty a_n$ converges. Then $a_n \to 0$.
\end{lemma}
\begin{proof}
Let $\displaystyle\sum_{n = 1}^{\infty} a_n = s$. Then $S_n \to s$ and $S_{n - 1} \to s$. Then $a_n = S_n - S_{n - 1} \to 0$.
\end{proof}
However, the converse is false!
\begin{eg}[Harmonic series]
If $a_n = 1/n$, then $a_n \to 0$ but $\sum a_n = \infty$.
We can prove this as follows:
\[
S_{2^n} - S_{2^{n -1 }} = \frac{1}{2^{n - 1} + 1} + \cdots + \frac{1}{2^n} \geq \frac{2^{n - 1}}{2^n} = \frac{1}{2}.
\]
Therefore $S_{2^n} \geq S_1 + n/2$. So the partial sums are unbounded.
\end{eg}
\begin{eg}[Geometric series]
Let $|\rho| < 1$. Then
\[
\sum_{n = 0}^\infty \rho^n = \frac{1}{1 - \rho}.
\]
We can prove this by considering the partial sums:
\[
\sum_{n = 0}^N \rho^n = \frac{1 - \rho^{N + 1}}{1 - \rho}.
\]
Since $\rho^{N + 1} \to 0$, this tends to $1/(1 - \rho)$.
\end{eg}
\begin{eg}
$\displaystyle \sum_{n = 2}^\infty \frac{1}{n(n - 1)}$ converges. This is since
\[
\frac{1}{n(n -1 )} = \frac{1}{n - 1} - \frac{1}{n}.
\]
So
\[
\sum_{n = 2}^{N}\frac{1}{n(n - 1)} = 1 - \frac{1}{N} \to 1.
\]
\end{eg}
\begin{lemma}
Suppose that $a_n \geq 0$ for every $n$ and the partial sums $S_n$ are bounded above. Then $\sum_{n=1}^\infty a_n$ converges.
\end{lemma}
\begin{proof}
The sequence $(S_n)$ is increasing and bounded above. So the result follows form the monotone sequences property.
\end{proof}
The simplest convergence test we have is the \emph{comparison test}. Roughly speaking, it says that if $0 \leq a_n \leq b_n$ for all $n$ and $\sum b_n$ converges, then $\sum a_n$ converges. However, we will prove a much more general form here for convenience.
\begin{lemma}[Comparison test]
Let $(a_n)$ and $(b_n)$ be non-negative sequences, and suppose that $\exists C, N$ such that $\forall n\geq N$, $a_n \leq Cb_n$. Then if $\sum b_n$ converges, then so does $\sum a_n$.
\end{lemma}
\begin{proof}
Let $M > N$. Also for each $R$, let $S_R = \sum_{n = 1}^R a_n$ and $T_R = \sum_{n = 1}^R b_n$. We want $S_R$ to be bounded above.
\[
S_M - S_N = \sum_{n = N + 1}^M a_n \leq C\sum _{n = N + 1}^M b_n \leq C\sum_{n = N + 1}^\infty b_n.
\]
So $\forall M\geq N$, $S_M \leq S_n + C\sum_{n = N + 1}^\infty b_n$. Since the $S_M$ are increasing and bounded, it must converge.
\end{proof}
\begin{eg}\leavevmode
\begin{enumerate}
\item $\sum \frac{1}{n2^n}$ converges, since $\sum \frac{1}{2^n}$ converges.
\item $\sum \frac{n}{2^n}$ converges.
If $n \geq 4$, then $n \leq 2^{n/2}$. That's because $4 = 2^{4/2}$ and for $n\geq 4$, $(n + 1)/n < \sqrt{2}$, so when we increase $n$, we multiply the right side by a greater number by the left. Hence by the comparison test, it is sufficient to show that $\sum 2^{n/2}/2^n = \sum 2^{-n/2}$ converges, which it does (geometric series).
\item $\sum \frac{1}{\sqrt{n}}$ diverges, since $\frac{1}{\sqrt{n}} \geq \frac{1}{n}$. So if it converged, then so would $\sum \frac{1}{n}$, but $\sum \frac{1}{n}$ diverges.
\item $\sum \frac{1}{n^2}$ converges, since for $n \geq 2$, $\frac{1}{n^2} \leq \frac{1}{n(n - 1)}$, and we have proven that the latter converges.
\item Consider $\displaystyle \sum_{n = 1}^\infty \frac{n + 5}{n^3 - 7n^2/2}$. We show this converges by noting
\[
n^3 - \frac{7n^2}{2} = n^2\left(n - \frac{7}{2}\right).
\]
So if $n \geq 8$, then
\[
n^3 - \frac{7n^2}{2} \geq \frac{n^3}{2}.
\]
Also, $n + 5 \leq 2n$. So
\[
\frac{n + 5}{n^3 - 7n^2/2} \leq 4/n^2.
\]
So it converges by the comparison test.
\item If $\alpha > 1$, then $\sum 1/n^\alpha$ converges.
Let $S_N = \sum_{n = 1}^N 1/n^\alpha$. Then
\begin{align*}
S_{2^n} - S_{2^{n - 1}} &= \frac{1}{(2^{n - 1} + 1)^\alpha} + \cdots + \frac{1}{(2^{n})^\alpha}\\
&\leq \frac{2^{n - 1}}{(2^{n - 1})^\alpha}\\
&= (2^{n - 1})^{1 - \alpha}\\
&= (2^{1 - \alpha})^{n - 1}.
\end{align*}
But $2^{1 - \alpha} < 1$. So
\[
S_{2^n} = (S_{2^n} - S_{2^{n - 1}}) + (S_{2^{n - 1}} - S_{2^{n -2 }}) + \cdots (S_2 - S_1) + S_1
\]
and is bounded above by comparison with the geometric series $1 + 2^{1 - \alpha} + (2^{1 - \alpha})^2 + \cdots$
\end{enumerate}
\end{eg}
\subsection{Absolute convergence}
Here we'll consider two stronger conditions for convergence --- absolute convergence and unconditional convergence. We'll prove that these two conditions are in fact equivalent.
\begin{defi}[Absolute convergence]
A series $\sum a_n$ \emph{converges absolutely} if the series $\sum |a_n|$ converges.
\end{defi}
\begin{eg}
The series $\sum \frac{(-1)^{n + 1}}{n} = 1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \cdots$ converges, but not absolutely.
To see the convergence, note that
\[
a_{2n - 1} + a_{2n} = \frac{1}{2n - 1} - \frac{1}{2n} = \frac{1}{2n(2n- 1)}.
\]
It is easy to compare with $1/n^2$ to get that the partial sums $S_{2n}$ converges. But $S_{2n + 1} - S_{2n} = 1/(2n + 1) \to 0$, so the $S_{2n + 1}$ converges to the same limit.
It does not converge absolutely, because the sum of the absolute values is the harmonic series.
\end{eg}
\begin{lemma}
Let $\sum a_n$ converge absolutely. Then $\sum a_n$ converges.
\end{lemma}
\begin{proof}
We know that $\sum |a_n|$ converges. Let $S_N = \sum_{n = 1}^N a_n$ and $T_N = \sum_{n = 1}^N |a_n|$.
We know two ways to show random sequences converge, without knowing what they converge to, namely monotone-sequences and Cauchy sequences. Since $S_N$ is not monotone, we shall try Cauchy sequences.
If $p > q$, then
\[
|S_p - S_q| = \left|\sum_{n = q + 1}^p a_n\right| \leq \sum_{n = q + 1}^p |a_n| = T_p - T_q.
\]
But the sequence $T_p$ converges. So $\forall \varepsilon > 0$, we can find $N$ such that for all $p > q \geq N$, we have $T_p - T_q < \varepsilon$, which implies $|S_p - S_q| < \varepsilon$.
\end{proof}
\begin{defi}[Unconditional convergence]
A series $\sum a_n$ \emph{converges unconditionally} if the series $\sum_{n = 1}^{\infty} a_{\pi(n)}$ converges for every bijection $\pi: \N \to \N$, i.e.\ no matter how we re-order the elements of $a_n$, the sum still converges.
\end{defi}
\begin{thm}
If $\sum a_n$ converges absolutely, then it converges unconditionally.
\end{thm}
\begin{proof}
Let $S_n = \sum_{n = 1}^N a_{\pi (n)}$. Then if $p > q$,
\[
|S_p - S_q| = \left|\sum_{n = q + 1}^p a_{\pi(n)}\right| \leq \sum_{n = q + 1}^\infty|a_{\pi (n)}|.
\]
Let $\varepsilon > 0$. Since $\sum |a_n|$ converges, pick $M$ such that $\sum_{n = M + 1}^\infty|a_n| < \varepsilon$.
Pick $N$ large enough that $\{1, \cdots, M\}\subseteq \{\pi (1), \cdots, \pi(N)\}$.
Then if $n > N$, we have $\pi(n) > M$. Therefore if $p > q \geq N$, then
\[
|S_p - S_q| \leq \sum_{n = q + 1}^p |a_{\pi(n)}| \leq \sum_{n = M + 1}^\infty |a_n| < \varepsilon.
\]
Therefore the sequence of partial sums is Cauchy.
\end{proof}
The converse is also true.
\begin{thm}
If $\sum a_n$ converges unconditionally, then it converges absolutely.
\end{thm}
\begin{proof}
We will prove the contrapositive: if it doesn't converge absolutely, it doesn't converge unconditionally.
Suppose that $\sum |a_n| = \infty$. Let $(b_n)$ be the subsequence of non-negative terms of $a_n$, and $(c_n)$ be the subsequence of negative terms. Then $\sum b_n$ and $\sum c_n$ cannot both converge, or else $\sum |a_n|$ converges.
wlog, $\sum b_n = \infty$. Now construct a sequence $0 = n_0 < n_1 < n_2 < \cdots$ such that $\forall k$,
\[
b_{n_{k - 1} + 1} + b_{n_{k - 1} + 2} + \cdots + b_{n_k} + c_k \geq 1,
\]
This is possible because the $b_n$ are unbounded and we can get it as large as we want.
Let $\pi$ be the rearrangement
\[
b_1, b_2, \cdots b_{n_1}, c_1, b_{n_1 + 1}, \cdots b_{n_2}, c_2, b_{n_2 + 1}, \cdots b_{n_3}, c_3,\cdots
\]
So the sum up to $c_k$ is at least $k$. So the partial sums tend to infinity.
\end{proof}
We can prove an even stronger result:
\begin{lemma}
Let $\sum a_n$ be a series that converges absolutely. Then for any bijection $\pi: \N\to \N$,
\[
\sum_{n = 1}^\infty a_n = \sum_{n = 1}^\infty a_{\pi(n)}.
\]
\end{lemma}
\begin{proof}
Let $\varepsilon > 0$. We know that both $\sum |a_n|$ and $\sum|a_{\pi(n)}|$ converge. So let $M$ be such that $\sum_{n > M}|a_n| < \frac{\varepsilon}{2}$ and $\sum_{n > M}|a_{\pi(n)}| < \frac{\varepsilon}{2}$.
Now $N$ be large enough such that
\[
\{1, \cdots, M\}\subseteq \{\pi(1), \cdots, \pi(N)\},
\]
and
\[
\{\pi(1), \cdots, \pi(M)\}\subseteq \{1, \cdots, N\}.
\]
Then for every $K\geq N$,
\[
\left|\sum_{n = 1}^K a_n - \sum_{n = 1}^K a_{\pi(n)}\right| \leq \sum_{n = M + 1}^K |a_n| + \sum_{n = M + 1}^K |a_{\pi(n)}| < \frac{\varepsilon}{2} + \frac{\varepsilon}{2} = \varepsilon.
\]
We have the first inequality since given our choice of $M$ and $N$, the first $M$ terms of the $\sum a_n$ and $\sum a_{\pi(n)}$ sums are cancelled by some term in the huge sum.
So $\forall K \geq N$, the partial sums up to $K$ differ by at most $\varepsilon$. So $|\sum a_n - \sum a_{\pi(n)}| \leq \varepsilon$.
Since this is true for all $\varepsilon$, we must have $\sum a_n = \sum a_{\pi(n)}$.
\end{proof}
\subsection{Convergence tests}
We'll now come up with a \emph{lot} of convergence tests.
\begin{lemma}[Alternating sequence test]
Let $(a_n)$ be a decreasing sequence of non-negative reals, and suppose that $a_n \to 0$. Then $\displaystyle \sum_{n = 1}^\infty (-1)^{n + 1}a_n$ converges, i.e.\ $a_1 - a_2 + a_3 - a_4 + \cdots$ converges.
\end{lemma}
\begin{proof}
Let $\displaystyle S_N = \sum_{n = 1}^N(-1)^{n + 1}a_n$. Then
\[
S_{2n} = (a_1 - a_2) + (a_3 - a_4) + \cdots + (a_{2n - 1} - a_{2n}) \geq 0,
\]
and $(S_{2n})$ is an increasing sequence.
Also,
\[
S_{2n + 1} = a_1 - (a_2 - a_3) - (a_4 - a_5) - \cdots - (a_{2n} - a_{2n + 1}),
\]
and $(S_{2n + 1})$ is a decreasing sequence. Also $S_{2n + 1} - S_{2n} = a_{2n + 1} \geq 0$.
Hence we obtain the bounds $0 \leq S_{2n} \leq S_{2n + 1} \leq a_1$. It follows from the monotone sequences property that $(S_{2n})$ and $(S_{2n + 1})$ converge.
Since $S_{2n + 1} - S_{2n} = a_{2n + 1} \to 0$, they converge to the same limit.
\end{proof}
\begin{eg}
\[
1 - \frac{1}{\sqrt[3]{2}} + \frac{1}{\sqrt[3]{3}} - \frac{1}{\sqrt[3]{4}} + \cdots \text{converges}.
\]
\end{eg}
\begin{lemma}[Ratio test]
We have three versions:
\begin{enumerate}
\item If $\exists c < 1$ such that
\[
\frac{|a_{n + 1}|}{|a_n|} \leq c,
\]
for all $n$, then $\sum a_n$ converges.
\item If $\exists c < 1$ and $\exists N$ such that
\[
(\forall n \geq N)\, \frac{|a_{n + 1}|}{|a_n|} \leq c,
\]
then $\sum a_n$ converges. Note that just because the ratio is always less than $1$, it doesn't necessarily converge. It has to be always less than a fixed number $c$. Otherwise the test will say that $\sum 1/n$ converges.
\item If $\exists \rho \in (-1, 1)$ such that
\[
\frac{a_{n + 1}}{a_n} \to \rho,
\]
then $\sum a_n$ converges. Note that we have the \emph{open} interval $(-1, 1)$. If $\frac{|a_{n + 1}|}{|a_n|} \to 1$, then the test is inconclusive!
\end{enumerate}
\end{lemma}
\begin{proof}\leavevmode
\begin{enumerate}
\item $|a_n| \leq c^{n - 1}|a_1|$. Since $\sum c^n$ converges, so does $\sum |a_n|$ by comparison test. So $\sum a_n$ converges absolutely, so it converges.
\item For all $k\geq 0$, we have $|a_{N + k}|\leq c^k|a_N|$. So the series $\sum |a_{N + k}|$ converges, and therefore so does $\sum |a_k|$.
\item If $\frac{a_{n + 1}}{a_n} \to \rho$, then $\frac{|a_{n + 1}|}{|a_n|} \to |\rho|$. So (setting $\varepsilon = (1 - |\rho|)/2$) there exists $N$ such that $\forall n \geq N$, $\frac{|a_{n + 1}|}{|a_n|} \leq \frac{1 + |\rho|}{2} < 1$. So the result follows from (ii).\qedhere
\end{enumerate}
\end{proof}
\begin{eg}
If $|b| < 1$, then $\sum nb^n$ converges, since
\[
\frac{a_{n + 1}}{a_n} = \frac{(n + 1) b^{n + 1}}{nb^n} = \left(1 + \frac{1}{n}\right) b\to b < 1.
\]
So it converges.
We can also evaluate this directly by considering $\displaystyle\sum_{i = 1}^\infty \sum_{n = i}^\infty b^n$.
\end{eg}
The following two tests were taught at the end of the course, but are included here for the sake of completeness.
\begin{thm}[Condensation test]
Let $(a_n)$ be a decreasing non-negative sequence. Then $\sum_{n = 1}^\infty a_n < \infty$ if and only if
\[
\sum_{k = 1}^\infty 2^k a_{2^k} < \infty.
\]
\end{thm}
\begin{proof}
This is basically the proof that $\sum \frac{1}{n}$ diverges and $\sum \frac{1}{n^{\alpha}}$ converges for $\alpha < 1$ but written in a more general way.
We have
\begin{align*}
&a_1 + a_2 + (a_3 + a_4) + (a_5 + \cdots + a_8) + (a_9 + \cdots + a_{16}) + \cdots\\
\geq & a_1 + a_2 + 2a_4 + 4a_8 + 8 a_{16} + \cdots
\end{align*}
So if $\sum 2^k a_{2^k}$ diverges, $\sum a_n$ diverges.
To prove the other way round, simply group as
\begin{align*}
& a_1 + (a_2 + a_3) + (a_4 + \cdots + a_7) + \cdots\\
\leq & a_1 + 2a_2 + 4a_4 + \cdots .\qedhere
\end{align*}
\end{proof}
\begin{eg}
If $a_n = \frac{1}{n}$, then $2^k a_{2^k} = 1$. So $\sum_{k = 1}^\infty 2^k a_{2^k} = \infty$. So $\sum_{n = 1}^\infty \frac{1}{n} = \infty$.
\end{eg}
After we formally define integrals, we will prove the integral test:
\begin{thm}[Integral test]
Let $f: [1, \infty] \to \R$ be a decreasing non-negative function. Then $\sum_{n = 1}^\infty f(n)$ converges iff $\int_1^\infty f(x)\;\d x < \infty$.
\end{thm}
\subsection{Complex versions}
Most definitions in the course so far carry over unchanged to the complex numbers. e.g.\ $z_n \to z$ iff $(\forall \varepsilon > 0)(\exists N)(\forall n\geq N)\, |z_n - z| < \varepsilon$.
Two exceptions are least upper bound and monotone sequences, because the complex numbers do not have an ordering! (It cannot be made into an ordered field because the square of every number in an ordered field has to be positive) Fortunately, Cauchy sequences still work.
We can prove the complex versions of most theorems so far by looking at the real and imaginary parts.
\begin{eg}
Let $(z_n)$ be a Cauchy sequence in $\C$. Let $z_n = x_n+ iy_n$. Then $(x_n)$ and $(y_n)$ are Cauchy. So they converge, from which it follows that $z_n = x_n + iy_n$ converges.
\end{eg}
Also, the Bolzano-Weierstrass theorem still holds: If $(z_n)$ is bounded, let $z_n = x_n + y_n$, then $(x_n)$ and $(y_n)$ are bounded. Then find a subsequence $(x_{n_k})$ that converges. Then find a subsequence of $(y_{n_k})$ that converges.
Then nested-intervals property has a ``nested-box'' property as a complex analogue.
Finally, the proof that absolutely convergent sequences converge still works. It follows that the ratio test still works.
\begin{eg}
If $|z| < 1$, then $\sum nz^n$ converges. Proof is the same as above.
\end{eg}
However, we do have an extra test for complex sums.
\begin{lemma}[Abel's test]
Let $a_1 \geq a_2 \geq \cdots \geq 0$, and suppose that $a_n \to 0$. Let $z\in \C$ such that $|z| = 1$ and $z \not= 1$. Then $\sum a_n z^n$ converges.
\end{lemma}
\begin{proof}
We prove that it is Cauchy. We have
\begin{align*}
\sum_{n = M}^N a_n z^n &= \sum_{n = M}^N a_n\frac{z^{n + 1} - z^n}{z - 1}\\
&= \frac{1}{z - 1}\sum_{n = M}^N a_n (z^{n + 1} - z^n)\\
&= \frac{1}{z - 1}\left(\sum_{n = M}^N a_n z^{n + 1} - \sum_{n = M}^N a_n z^n\right)\\
&= \frac{1}{z - 1}\left(\sum_{n = M}^N a_n z^{n + 1} - \sum_{n = M - 1}^{N - 1} a_{n + 1} z^{n + 1}\right)\\
&= \frac{1}{z - 1}\left(a_N z^{N + 1} - a_M z^M + \sum_{n = M}^{N - 1} (a_n - a_{n + 1})z^{n + 1}\right)\\
\intertext{We now take the absolute value of everything to obtain}
\left|\sum_{n = M}^N a_nz^n\right| &\leq \frac{1}{|z - 1|} \left(a_N + a_M + \sum_{n = M}^{N - 1}(a_n - a_{n + 1})\right)\\
&= \frac{1}{|z - 1|}\left(a_N + a_M + (a_M - a_{M + 1}) + \cdots + (a_{N - 1} - a_{N})\right)\\
&= \frac{2a_M}{|z - 1|} \to 0.
\end{align*}
So it is Cauchy. So it converges
\end{proof}
Note that here we transformed the sum $\sum a_n(z^{n + 1} - z^n)$ into $a_N z^{N + 1} - a_M z^M + \sum (a_n - a_{n + 1})z^{n + 1}$. What we have effectively done is a discrete analogue of integrating by parts.
\begin{eg}
The series $\sum z^n/n$ converges if $|z| < 1$ or if $|z| = 1$ and $z \not= 1$, and it diverges if $z = 1$ or $|z| > 1$.
The cases $|z| < 1$ and $|z| > 1$ are trivial from the ratio test, and Abel's test is required for the $|z| = 1$ cases.
\end{eg}
\section{Continuous functions}
\subsection{Continuous functions}
\begin{defi}[Continuous function]
Let $A\subseteq \R$, $a\in A$, and $f: A\to \R$. Then $f$ is \emph{continuous at} $a$ if for any $\varepsilon > 0 $, there is some $\delta > 0$ such that if $y \in A$ is such that $|y - a| < \delta$, then $|f(y) - f(a)| < \varepsilon$. In symbols, we have
\[
(\forall \varepsilon > 0)(\exists \delta > 0)(\forall y\in A)\, |y - a| < \delta \Rightarrow |f(y) - f(a)| < \varepsilon.
\]
$f$ is \emph{continuous} if it is continuous at every $a\in A$. In symbols, we have
\[
(\forall a\in A)(\forall \varepsilon > 0)(\exists \delta > 0)(\forall y\in A)\, |y - a| < \delta \Rightarrow |f(y) - f(a)| < \varepsilon.
\]
\end{defi}
Intuitively, $f$ is continuous at $a$ if we can obtain $f(a)$ as accurately as we wish by using more accurate values of $a$ (the definition says that if we want to approximate $f(a)$ by $f(y)$ to within accuracy $\varepsilon$, we just have to get our $y$ to within $\delta$ of $a$ for some $\delta$).
For example, suppose we have the function
\[
f(x) = \begin{cases} 0 & x \leq \pi\\ 1& x > \pi\end{cases}.
\]
Suppose that we don't know what the function actually is, but we have a computer program that computes this function. We want to know what $f(\pi)$ is. Since we cannot input $\pi$ (it has infinitely many digits), we can try $3$, and it gives $0$. Then we try $3.14$, and it gives $0$ again. If we try $3.1416$, it gives $1$ (since $\pi = 3.1415926\cdots < 3.1416$). We keep giving more and more digits of $\pi$, but the result keeps oscillating between $0$ and $1$. We have no hope of what $f(\pi)$ might be, even approximately. So this $f$ is discontinuous at $\pi$.
However, if we have the function $g(x) = x^2$, then we \emph{can} find the (approximate) value of $g(\pi)$. We can first try $g(3)$ and obtain $9$. Then we can try $g(3.14) = 9.8596$, $g(3.1416) = 9.86965056$ etc. We can keep trying and obtain more and more accurate values of $g(\pi)$. So $g$ is continuous at $\pi$.
\begin{eg}\leavevmode
\begin{itemize}
\item Constant functions are continuous.
\item The function $f(x) = x$ is continuous (take $\delta = \varepsilon$).
\end{itemize}
\end{eg}
The definition of continuity of a function looks rather like the definition of convergence. In fact, they are related by the following lemma:
\begin{lemma}
The following two statements are equivalent for a function $f: A\to \R$.
\begin{itemize}
\item $f$ is continuous
\item If $(a_n)$ is a sequence in $A$ with $a_n \to a$, then $f(a_n) \to f(a)$.
\end{itemize}
\end{lemma}
\begin{proof}
(i)$\Rightarrow$(ii) Let $\varepsilon > 0$. Since $f$ is continuous at $a$,
\[
(\exists \delta > 0)(\forall y\in A)\, |y-a|< \delta \Rightarrow |f(y) - f(a)| < \varepsilon.
\]
We want $N$ such that $\forall n \geq N$, $|f(a_n) - f(a)| < \varepsilon$. By continuity, it is enough to find $N$ such that $\forall n\geq N$, $|a_n - a| < \delta$. Since $a_n \to a$, such an $N$ exists.
(ii)$\Rightarrow$(i) We prove the contrapositive: Suppose $f$ is not continuous at $a$. Then
\[
(\exists \varepsilon > 0)(\forall \delta > 0)(\exists y\in A)\, |y - a| < \delta \text{ and }|f(y) - f(a)| \geq \varepsilon.
\]
For each $n$, we can therefore pick $a_n \in A$ such that $|a_n - a| < \frac{1}{n}$ and $|f(a_n) - f(a)| \geq \varepsilon$. But then $a_n \to a$ (by Archimedean property), but $f(a_n) \not\to f(a)$.
\end{proof}
\begin{eg}\leavevmode
\begin{enumerate}
\item Let $f(x) = \begin{cases} -1 & x < 0 \\ 1 & x\geq 0\end{cases}$. Then $f$ is not continuous because $-\frac{1}{n} \to 0$ but $f(-\frac{1}{n}) \to -1 \not= f(0)$.
\item Let $f: \Q \to \R$ with
\[
f(x) =
\begin{cases}
1 & x^2 > 2\\
0 & x^2 < 2
\end{cases}
\]
Then $f$ is continuous. For every $a\in \Q$, we can find an interval about $a$ on which $f$ is constant. So $f$ is continuous at $a$.
\item Let
\[
f(x) =
\begin{cases}
\sin \frac{1}{x} & x \not= 0\\
0 & x = 0
\end{cases}
\]
Then $f(a)$ is discontinuous. For example, let $a_n = 1/[(2n + 0.5)\pi]$. Then $a_n\to 0$ and $f(a_n) \to 1 \not= f(0)$.
\end{enumerate}
\end{eg}
We can use this sequence definition as the definition for continuous functions. This has the advantage of being cleaner to write and easier to work with. In particular, we can reuse a lot of our sequence theorems to prove the analogous results for continuous functions.
\begin{lemma}
Let $A\subseteq \R$ and $f, g: A\to \R$ be continuous functions. Then
\begin{enumerate}
\item $f + g$ is continuous
\item $fg$ is continuous
\item if $g$ never vanishes, then $f/g$ is continuous.
\end{enumerate}
\end{lemma}
\begin{proof}\leavevmode
\begin{enumerate}
\item Let $a\in A$ and let $(a_n)$ be a sequence in $A$ with $a_n \to a$. Then
\[
(f + g)(a_n) = f(a_n) + g(a_n).
\]
But $f(a_n) \to f(a)$ and $g(a_n) \to g(a)$. So
\[
f(a_n) + g(a_n) \to f(a) + g(a) = (f + g)(a).
\]
\end{enumerate}
(ii) and (iii) are proved in exactly the same way.
\end{proof}
With this lemma, from the fact that constant functions and $f(x) = x$ are continuous, we know that all polynomials are continuous. Similarly, rational functions $P(x)/Q(x)$ are continuous except when $Q(x) = 0$.
\begin{lemma}
Let $A, B\subseteq \R$ and $f: A\to B$, $g: B\to \R$. Then if $f$ and $g$ are continuous, $g\circ f: A\to \R$ is continuous.
\end{lemma}
\begin{proof}
We offer two proofs:
\begin{enumerate}
\item Let $(a_n)$ be a sequence in $A$ with $a_n \to a\in A$. Then $f(a_n) \to f(a)$ since $f$ is continuous. Then $g(f(a_n)) \to g(f(a))$ since $g$ is continuous. So $g\circ f$ is continuous.
\item Let $a\in A$ and $\varepsilon > 0$. Since $g$ is continuous at $f(a)$, there exists $\eta > 0$ such that $\forall z\in B$, $|z - f(a)| < \eta \Rightarrow |g(z) - g(f(a))| < \varepsilon$.
Since $f$ is continuous at $a$, $\exists \delta > 0$ such that $\forall y\in A$, $|y - a| < \delta \Rightarrow |f(y) - f(a)| < \eta$. Therefore $|y - a| < \delta \Rightarrow |g(f(y)) - g(f(a))| < \varepsilon$.\qedhere
\end{enumerate}
\end{proof}
There are two important theorems regarding continuous functions --- the maximum value theorem and the intermediate value theorem.
\begin{thm}[Maximum value theorem]
Let $[a, b]$ be a closed interval in $\R$ and let $f: [a, b] \to \R$ be continuous. Then $f$ is bounded and attains its bounds, i.e.\ $f(x) = \sup f$ for some $x$, and $f(y) = \inf f$ for some $y$.
\end{thm}
\begin{proof}
If $f$ is not bounded above, then for each $n$, we can find $x_n\in [a, b]$ such that $f(x_n) \geq n$ for all $n$.
By Bolzano-Weierstrass, since $x_n \in [a, b]$ and is bounded, the sequence $(x_n)$ has a convergent subsequence $(x_{n_k})$. Let $x$ be its limit. Then since $f$ is continuous, $f(x_{n_k}) \to f(x)$. But $f(x_{n_k}) \geq n_k \to \infty$. So this is a contradiction.
Now let $C = \sup\{f(x): x\in [a, b]\}$. Then for every $n$, we can find $x_n$ such that $f(x_n) \geq C - \frac{1}{n}$. So by Bolzano-Weierstrass, $(x_n)$ has a convergent subsequence $(x_{n_k})$. Since $C - \frac{1}{n_{k}}\leq f(x_{n_k}) \leq C$, $f(x_{n_k})\to C$. Therefore if $x = \lim x_{n_k}$, then $f(x) = C$.
A similar argument applies if $f$ is unbounded below.
\end{proof}
\begin{thm}[Intermediate value theorem]
Let $a < b\in \R$ and let $f: [a, b] \to \R$ be continuous. Suppose that $f(a) < 0 < f(b)$. Then there exists an $x\in (a, b)$ such that $f(x) = 0$.
\end{thm}
\begin{proof}
We have several proofs:
\begin{enumerate}
\item Let $A = \{x: f(x) < 0\}$ and let $s = \sup A$. We shall show that $f(s) = 0$ (this is similar to the proof that $\sqrt{2}$ exists in Numbers and Sets). If $f(s) < 0$, then setting $\varepsilon = |f(s)|$ in the definition of continuity, we can find $\delta > 0$ such that $\forall y$, $|y - s| < \delta \Rightarrow f(y) < 0$. Then $s + \delta/2 \in A$, so $s$ is not an upper bound. Contradiction.
If $f(s) > 0$, by the same argument, we can find $\delta > 0$ such that $\forall y$, $|y - s| < \delta \Rightarrow f(y) > 0$. So $s - \delta/2$ is a smaller upper bound.
\item Let $a_0 = a$, $b_0 = b$. By repeated bisection, construct nested intervals $[a_n, b_n]$ such that $b_n - a_n = \frac{b_0 - a_0}{2^n}$ and $f(a_n) < 0 \leq f(b_n)$. Then by the nested intervals property, we can find $x\in \cap_{n = 0}^\infty [a_n, b_n]$. Since $b_n - a_n \to 0$, $a_n, b_n \to x$.
Since $f(a_n) < 0$ for every $n$, $f(x) \leq 0$. Similarly, since $f(b_n) \geq 0$ for every $n$, $f(x) \geq 0$. So $f(x) = 0$.\qedhere
\end{enumerate}
\end{proof}
It is easy to generalize this to get that, if $f(a) < c < f(b)$, then $\exists x\in (a, b)$ such that $f(x) = c$, by applying the result to $f(x) - c$. Also, we can assume instead that $f(b) < c < f(a)$ and obtain the same result by looking at $-f(x)$.
\begin{cor}
Let $f: [a, b]\to [c, d]$ be a continuous strictly increasing function with $f(a) = c$, $f(b) = d$. Then $f$ is invertible and its inverse is continuous.
\end{cor}
\begin{proof}
Since $f$ is strictly increasing, it is an injection (suppose $x \not= y$. wlog, $x < y$. Then $f(x) < f(y)$ and so $f(x) \not= f(y)$). Now let $y\in (c, d)$. By the intermediate value theorem, there exists $x\in (a, b)$ such that $f(x) = y$. So $f$ is a surjection. So it is a bijection and hence invertible.
Let $g$ be the inverse. Let $y\in [c, d]$ and let $\varepsilon > 0$. Let $x = g(y)$. So $f(x) = y$. Let $u = f(x - \varepsilon)$ and $v = f(x + \varepsilon)$ (if $y = c$ or $d$, make the obvious adjustments). Then $u < y < v$. So we can find $\delta > 0$ such that $(y - \delta , y + \delta) \subseteq (u, v)$. Then $|z - y| < \delta \Rightarrow g(z) \in (x - \varepsilon, x + \varepsilon) \Rightarrow |g(z) - g(y)| < \varepsilon$.
\end{proof}
With this corollary, we can create more continuous functions, e.g.\ $\sqrt{x}$.
\subsection{Continuous induction*}
Continuous induction is a generalization of induction on natural numbers. It provides an alternative mechanism to prove certain results we have shown.
\begin{prop}[Continuous induction v1]
Let $a < b$ and let $A\subseteq [a, b]$ have the following properties:
\begin{enumerate}
\item $a\in A$
\item If $x\in A$ and $x\not= b$, then $\exists y\in A$ with $y > x$.
\item If $\forall \varepsilon > 0$, $\exists y\in A: y\in (x - \varepsilon, x]$, then $x\in A$.
\end{enumerate}
Then $b\in A$.
\end{prop}
\begin{proof}
Since $a\in A$, $A\not= \emptyset$. $A$ is also bounded above by $b$. So let $s = \sup A$. Then $\forall \varepsilon > 0$, $\exists y\in A$ such that $y > s - \varepsilon$. Therefore, by (iii), $s\in A$.
If $s\not= b$, then by (ii), we can find $y\in A$ such that $y > s$.
\end{proof}
It can also be formulated as follows:
\begin{prop}[Continuous induction v2]
Let $A\subseteq [a, b]$ and suppose that
\begin{enumerate}
\item $a\in A$
\item If $[a, x]\subseteq A$ and $x\not = b$, then there exists $y > x$ such that $[a, y]\subseteq A$.
\item If $[a, x)\subseteq A$, then $[a, x]\subseteq A$.
\end{enumerate}
Then $A = [a, b]$
\end{prop}
\begin{proof}
We prove that version 1 $\Rightarrow$ version 2.
Suppose $A$ satisfies the conditions of v2. Let $A' = \{x\in [a, b]: [a, x]\subseteq A\}$.
Then $a\in A'$. If $x\in A'$ with $x \not= b$, then $[a, x]\subseteq A$. So $\exists y > x$ such that $[a, y] \subseteq A$. So $\exists y > x$ such that $y\in A'$.
If $\forall \varepsilon > 0, \exists y\in (x - \varepsilon, x]$ such that $[a, y]\subseteq A$, then $[a, x)\subseteq A$. So by (iii), $[a, x]\subseteq A$, so $x\in A'$. So $A'$ satisfies properties (i) to (iii) of version 1. Therefore $b\in A'$. So $[a, b]\subseteq A$. So $A = [a, b]$.
\end{proof}
We reprove intermediate value theorem here:
\begin{thm}[Intermediate value theorem]
Let $a < b\in \R$ and let $f: [a, b] \to \R$ be continuous. Suppose that $f(a) < 0 < f(b)$. Then there exists an $x\in (a, b)$ such that $f(x) = 0$.
\end{thm}
\begin{proof}
Assume that $f$ is continuous. Suppose $f(a) < 0 < f(b)$. Assume that $(\forall x)\, f(x) \not =0$, and derive a contradiction.
Let $A = \{x: f(x) < 0\}$ Then $a\in A$. If $x\in A$, then $f(x) < 0$, and by continuity, we can find $\delta > 0$ such that $|y - x| < \delta\Rightarrow f(y) < 0$. So if $x\not= b$, then we can find $y\in A$ such that $y > x$.
We prove the contrapositive of the last condition, i.e.
\[
x\not\in A\Rightarrow (\exists \delta > 0)(\forall y\in A)\, y\not\in(x - \delta, x].
\]
If $x\not\in A$, then $f(x) > 0$ (we assume that $f$ is never zero. If not, we're done). Then by continuity, $\exists \delta > 0$ such that $|y - x| < \delta \Rightarrow f(y) > 0$. So $y\not\in A$.
Hence by continuous induction, $b\in A$. Contradiction.
\end{proof}
Now we prove that continuous functions in closed intervals are bounded.
\begin{thm}
Let $[a, b]$ be a closed interval in $\R$ and let $f: [a, b] \to \R$ be continuous. Then $f$ is bounded.
\end{thm}
\begin{proof}
Let $f: [a, b]$ be continuous. Let $A = \{x: f\text{ is bounded on }[a, x]\}$. Then $a\in A$. If $x\in A, x\not= b$, then $\exists \delta > 0$ such that $|y - x| < \delta \Rightarrow |f(y) - f(x)| < 1$. So $\exists y > x$ (e.g.\ take $\min\{x + \delta/2, b\}$) such that $f$ is bounded on $[a, y]$, which implies that $y\in A$.
Now suppose that $\forall \varepsilon > 0$, $\exists y\in (x, - \varepsilon, x]$ such that $y\in A$. Again, we can find $\delta > 0$ such that $f$ is bounded on $(x - \delta, x + \delta)$, and in particular on $(x - \delta, x]$. Pick $y$ such that $f$ is bounded on $[a, y]$ and $y > x - \delta$. Then $f$ is bounded on $[a, x]$. So $x\in A$.
So we are done by continuous induction.
\end{proof}
Finally, we can prove a theorem that we have not yet proven.
\begin{defi}[Cover of a set]
Let $A\subseteq \R$. A \emph{cover} of $A$ by open intervals is a set $\{I_\gamma: \gamma\in \Gamma\}$ where each $I_\gamma$ is an open interval and $A \subseteq \bigcup_{\gamma\in \Gamma}$ $I_\gamma$.
A \emph{finite subcover} is a finite subset $\{I_{\gamma_1}, \cdots, I_{\gamma_n}\}$ of the cover that is still a cover.
\end{defi}
Not every cover has a finite subcover. For example, the cover $\{(\frac{1}{n}, 1): n\in \N\}$ of $(0, 1)$ has no finite subcover.
\begin{thm}[Heine-Borel*]
Every cover of a closed, bounded interval $[a, b]$ by open intervals has a finite subcover. We say closed intervals are \emph{compact} (cf.\ Metric and Topological Spaces).
\end{thm}
\begin{proof}
Let $\{I_\gamma: \gamma\in \Gamma\}$ be a cover of $[a, b]$ by open intervals. Let $A = \{x: [a, x]$ can be covered by finitely many of the $I_\gamma\}$.
Then $a\in A$ since $a$ must belong to some $I_\gamma$.
If $x\in A$, then pick $\gamma$ such that $x\in I_\gamma$. Then if $x\not = b$, since $I_\gamma$ is an open interval, it contains $[x, y]$ for some $y > x$. Then $[a, y]$ can be covered by finitely many $I_\gamma$, by taking a finite cover for $[a, x]$ and the $I_\gamma$ that contains $x$.
Now suppose that $\forall \varepsilon > 0, \exists y\in A$ such that $y\in (x - \varepsilon, x]$.
Let $I_\gamma$ be an open interval containing $x$. Then it contains $(x - \varepsilon, x]$ for some $\varepsilon > 0$. Pick $y\in A$ such that $y\in (x - \varepsilon, x]$. Now combine $I_\gamma$ with a finite subcover of $[a, y]$ to get a finite subcover of $[a, x]$. So $x\in A$.
Then done by continuous induction.
\end{proof}
We can use Heine-Borel to prove that continuous functions on $[a, b]$ are bounded.
\begin{thm}
Let $[a, b]$ be a closed interval in $\R$ and let $f: [a, b] \to \R$ be continuous. Then $f$ is bounded and attains it bounds, i.e.\ $f(x) = \sup f$ for some $x$, and $f(y) = \inf f$ for some $y$.
\end{thm}
\begin{proof}
Let $f: [a, b]\to \R$ be continuous. Then by continuity,
\[
(\forall x\in [a, b])(\exists \delta_x > 0)(\forall y)\, |y - x| < \delta_x\Rightarrow |f(y) - f(x)| < 1.
\]
Let $\gamma = [a, b]$ and for each $x\in \gamma$, let $I_x = (x - \delta_x, x + \delta_x)$. So by Heine-Borel, we can find $x_1, \cdots, x_n$ such that $[a, b]\subseteq \bigcup_1^n (x_i - \delta_{x_i}, x_i + \delta_{x_i})$.
But $f$ is bounded in each interval $(x_i - \delta_{x_i}, x_i + \delta_{x_i})$ by $|f(x_i)| + 1$. So it is bounded on $[a, b]$ by $\max|f(x_i)| + 1$.
\end{proof}
\section{Differentiability}
In the remainder of the course, we will properly develop calculus, and put differentiation and integration on a rigorous foundation. Every notion will be given a proper definition which we will use to prove results like the product and quotient rule.
\subsection{Limits}
First of all, we need the notion of limits. Recall that we've previously had limits for \emph{sequences}. Now, we will define limits for functions.
\begin{defi}[Limit of functions]
Let $A\subseteq \R$ and let $f: A\to \R$. We say
\[
\lim_{x\to a}f(x) = \ell,
\]
or ``$f(x) \to \ell$ as $x \to a$'', if
\[
(\forall \varepsilon > 0)(\exists \delta > 0)(\forall x\in A)\, 0 < |x - a| < \delta \Rightarrow |f(x) - \ell| < \varepsilon.
\]
We couldn't care less what happens when $x = a$, hence the strict inequality $0 < |x - a|$. In fact, $f$ doesn't even have to be defined at $x = a$.
\end{defi}
\begin{eg}
Let
\[
f(x) =
\begin{cases}
x & x \not = 2\\
3 & x = 2
\end{cases}
\]
Then $\lim\limits_{x\to 2} = 2$, even though $f(2) = 3$.
\end{eg}
\begin{eg}
Let $f(x) = \frac{\sin x}{x}$. Then $f(0)$ is not defined but $\lim\limits_{x\to 0}f(x) = 1$.
We will see a proof later after we define what $\sin$ means.
\end{eg}
We notice that the definition of the limit is suspiciously similar to that of continuity. In fact, if we define
\[
g(x) =
\begin{cases}
f(x) & x \not =a\\
\ell & x = a
\end{cases}
\]
Then $f(x) \to \ell$ as $x \to a$ iff $g$ is continuous at $a$.
Alternatively, $f$ is continuous at $a$ if $f(x) \to f(a)$ as $x \to a$. It follows also that $f(x) \to \ell$ as $x\to a$ iff $f(x_n) \to \ell$ for every sequence $(x_n)$ in $A$ with $x_n\to a$.
The previous limit theorems of sequences apply here as well
\begin{prop}
If $f(x)\to \ell$ and $g(x)\to m$ as $x\to a$, then $f(x) + g(x) \to \ell+ m$, $f(x)g(x) \to \ell m$, and $\frac{f(x)}{g(x)}\to \frac{\ell}{m}$ if $g$ and $m$ don't vanish.
\end{prop}
\subsection{Differentiation}
Similar to what we did in IA Differential Equations, we define the derivative as a limit.
\begin{defi}[Differentiable function]
$f$ is \emph{differentiable} at $a$ with derivative $\lambda$ if
\[
\lim_{x\to a}\frac{f(x) - f(a)}{x - a} = \lambda.
\]
Equivalently, if
\[
\lim_{h\to 0}\frac{f(a + h) - f(a)}{h} = \lambda.
\]
We write $\lambda = f'(a)$.
\end{defi}
Here we see why, in the definition of the limit, we say that we don't care what happens when $x = a$. In our definition here, our function is 0/0 when $x = a$, and we can't make any sense out of what happens when $x = a$.
Alternatively, we write the definition of differentiation as
\[
\frac{f(x + h) - f(x)}{h} = f'(x) + \varepsilon(h),
\]
where $\varepsilon(h) \to 0$ as $h \to 0$. Rearranging, we can deduce that
\[
f(x + h) = f(x) + hf'(x) + h\varepsilon(h),
\]
Note that by the definition of the limit, we don't have to care what value $\varepsilon$ takes when $h = 0$. It can be $0$, $\pi$ or $10^{10^{10}}$. However, we usually take $\varepsilon(0) = 0$ so that $\varepsilon$ is continuous.
Using the small-$o$ notation, we usually write $o(h)$ for a function that satisfies $\frac{o(h)}{h}\to 0$ as $h\to 0$. Hence we have
\begin{prop}
\[
f(x + h) = f(x) + hf'(x) + o(h).
\]
\end{prop}
We can interpret this as an approximation of $f(x + h)$:
\[
f(x + h) = \underbrace{f(x) + hf'(x)}_{\text{linear approximation}} + \underbrace{o(h)}_{\text{error term}}.
\]
And differentiability shows that this is a very good approximation with small $o(h)$ error.
Conversely, we have
\begin{prop}
If $f(x + h) = f(x) + hf'(x) + o(h)$, then $f$ is differentiable at $x$ with derivative $f'(x)$.
\end{prop}
\begin{proof}
\[
\frac{f(x + h) - f(x)}{h} = f'(x) + \frac{o(h)}{h} \to f'(x).\qedhere
\]
\end{proof}
We can take derivatives multiple times, and get multiple derivatives.
\begin{defi}[Multiple derivatives]
This is defined recursively: $f$ is $(n + 1)$-times differentiable if it is $n$-times differentiable and its $n$th derivative $f^{(n)}$ is differentiable. We write $f^{(n + 1)}$ for the derivative of $f^{(n)}$, i.e.\ the $(n + 1)$th derivative of $f$.
Informally, we will say $f$ is $n$-times differentiable if we can differentiate it $n$ times, and the $n$th derivative is $f^{(n)}$.
\end{defi}
We can prove the usual rules of differentiation using the small $o$-notation. It can also be proven by considering limits directly, but the notation will become a bit more daunting.
\begin{lemma}[Sum and product rule]
Let $f, g$ be differentiable at $x$. Then $f + g$ and $fg$ are differentiable at $x$, with
\begin{align*}
(f + g)'(x) &= f'(x) + g'(x)\\
(fg)'(x) &= f'(x)g(x) + f(x)g'(x)
\end{align*}
\end{lemma}
\begin{proof}
\begin{align*}
(f + g)(x + h) ={}& f(x + h) + g(x + h)\\
={}& f(x) +hf'(x) + o(h) + g(x) + hg'(x) + o(h)\\
={}& (f + g)(x) + h(f'(x) + g'(x)) + o(h)\\
fg(x + h) ={}& f(x + h)g(x + h)\\
={}& [f(x) + hf'(x) + o(h)][g(x) + hg'(x) + o(h)]\\
={}& f(x)g(x) + h[f'(x)g(x) + f(x)g'(x)]\\
&+ \underbrace{o(h)[g(x) + f(x) + hf'(x) + hg'(x) + o(h)] + h^2f'(x)g'(x)}_{\text{error term}}\\
\intertext{By limit theorems, the error term is $o(h)$. So we can write this as}
={}& fg(x) + h(f'(x)g(x) + f(x)g'(x)) + o(h).\qedhere
\end{align*}
\end{proof}
\begin{lemma}[Chain rule]
If $f$ is differentiable at $x$ and $g$ is differentiable at $f(x)$, then $g\circ f$ is differentiable at $x$ with derivative $g'(f(x))f'(x)$.
\end{lemma}
\begin{proof}
If one is sufficiently familiar with the small-$o$ notation, then we can proceed as
\[
g(f(x + h)) = g(f(x) + h f'(x) + o(h)) = g(f(x)) + h f'(x) g'(f(x)) + o(h).
\]
If not, we can be a bit more explicit about the computations, and use $h\varepsilon(h)$ instead of $o(h)$:
\begin{align*}
(g\circ f)(x + h) ={}& g(f(x + h))\\
={}& g[f(x) + \underbrace{hf'(x) + h\varepsilon_1(h)}_{\text{the ``}h\text{'' term}}]\\
={}& g(f(x)) + \big(fg'(x) + h\varepsilon_1(h)\big)g'(f(x))\\
&+ \big(hf'(x) + h\varepsilon_1(h)\big)\varepsilon_2(hf'(x) + h\varepsilon_1(h))\\
={}& g\circ f(x) + hg'(f(x))f'(x)\\
&+ \underbrace{h\Big[\varepsilon_1(h)g'(f(x)) + \big(f'(x) + \varepsilon_1(h)\big)\varepsilon_2\big(hf'(x) + h\varepsilon_1(h)\big)\Big]}_{\text{error term}}.
\end{align*}
We want to show that the error term is $o(h)$, i.e.\ it divided by $h$ tends to $0$ as $h\to 0$.
But $\varepsilon_1(h)g'(f(x))\to 0$, $f'(x) + \varepsilon_1(h)$ is bounded, and $\varepsilon_2(hf'(x) + h\varepsilon_1(h))\to 0$ because $hf'(x) + h\varepsilon_1(h) \to 0$ and $\varepsilon_2(0) = 0$. So our error term is $o(h)$.
\end{proof}
We usually don't write out the error terms so explicitly, and just use heuristics like $f(x + o(h)) = f(x) + o(h)$; $o(h) + o(h) = o(h)$; and $g(x) \cdot o(h) = o(h)$ for any (bounded) function $g$.
\begin{eg}\leavevmode
\begin{enumerate}
\item Constant functions are differentiable with derivative $0$.
\item $f(x) = \lambda x$ is differentiable with derivative $\lambda$.
\item Using the product rule, we can show that $x^n$ is differentiable with derivative $nx^{n - 1}$ by induction.
\item Hence all polynomials are differentiable.
\end{enumerate}
\end{eg}
\begin{eg}
Let $f(x) = 1/x$. If $x\not = 0$, then
\[
\frac{f(x + h) - f(x)}{h} = \frac{\frac{1}{x + h} - \frac{1}{x}}{h} = \frac{\left(\frac{-h}{x(x + h)}\right)}{h} = \frac{-1}{x(x + h)} \to \frac{-1}{x^2}
\]
by limit theorems.
\end{eg}
\begin{lemma}[Quotient rule]
If $f$ and $g$ are differentiable at $x$, and $g(x) \not = 0$, then $f/g$ is differentiable at $x$ with derivative
\[
\left(\frac{f}{g}\right)'(x) = \frac{f'(x)g(x) - g'(x)f(x)}{g(x)^2}.
\]
\end{lemma}
\begin{proof}
First note that $1/g(x) = h(g(x))$ where $h(y) = 1/y$. So $1/g(x)$ is differentiable at $x$ with derivative $\displaystyle \frac{-1}{g(x)^2}g'(x)$ by the chain rule.
By the product rule, $f/g$ is differentiable at $x$ with derivative
\[
\frac{f'(x)}{g(x)} - f(x)\frac{g'(x)}{g(x)^2} = \frac{f'(x)g(x) - f(x)g'(x)}{g(x)^2}.\qedhere
\]
\end{proof}
\begin{lemma}
If $f$ is differentiable at $x$, then it is continuous at $x$.
\end{lemma}
\begin{proof}
As $y\to x$, $\displaystyle \frac{f(y) - f(x)}{y - x} \to f'(x)$. Since, $y - x \to 0$, $f(y) - f(x) \to 0$ by product theorem of limits. So $f(y) \to f(x)$. So $f$ is continuous at $x$.
\end{proof}
\begin{thm}
Let $f:[a, b]\to [c, d]$ be differentiable on $(a, b)$, continuous on $[a, b]$, and strictly increasing. Suppose that $f'(x)$ never vanishes. Suppose further that $f(a) = c$ and $f(b) = d$. Then $f$ has an inverse $g$ and for each $y\in (c, d)$, $g$ is differentiable at $y$ with derivative $1/f'(g(y))$.
In human language, this states that if $f$ is invertible, then the derivative of $f^{-1}$ is $1/f'$.
\end{thm}
Note that the conditions will (almost) always require $f$ to be differentiable on \emph{open} interval $(a, b)$, continuous on \emph{closed} interval $[a, b]$. This is because it doesn't make sense to talk about differentiability at $a$ or $b$ since the definition of $f'(a)$ requires $f$ to be defined on both sides of $a$.
\begin{proof}
$g$ exists by an earlier theorem about inverses of continuous functions.
Let $y, y + k\in (c, d)$. Let $x = g(y)$, $x + h = g(y + k)$.
Since $g(y + k) = x + h$, we have $y + k = f(x + h)$. So $k = f(x + h) - y = f(x + h) - f(x)$. So
\[
\frac{g(y + k) - g(y)}{k} = \frac{(x + h) - x}{f(x + h) - f(x)} = \left(\frac{f(x + h) - f(x)}{h}\right)^{-1}.
\]
As $k \to 0$, since $g$ is continuous, $g(y + k) \to g(y)$. So $h \to 0$. So
\[
\frac{g(y + k) - g(y)}{k} \to [f'(x)]^{-1} = [f'(g(y)]^{-1}.\qedhere
\]
\end{proof}
\begin{eg}
Let $f(x) = x^{1/2}$ for $x > 0$. Then $f$ is the inverse of $g(x) = x^2$. So
\[
f'(x) = \frac{1}{g'(f(x))} = \frac{1}{2x^{1/2}} = \frac{1}{2}x^{-1/2}.
\]
Similarly, we can show that the derivative of $x^{1/q}$ is $\frac{1}{q}x^{1/q - 1}$.
Then let's take $x^{p/q} = (x^{1/q})^p$. By the chain rule, its derivative is
\[
p(x^{1/q})^{p - 1}\cdot \frac{1}{q}x^{1/q - 1} = \frac{p}{q}x^{\frac{p - 1}{q} + \frac{1}{q} - 1} = \frac{p}{q}x^{\frac{p}{q} - 1}.
\]
\end{eg}
\subsection{Differentiation theorems}
Everything we've had so far is something we already know. It's just that now we can prove them rigorously. In this section, we will come up with genuinely new theorems, including but not limited to \emph{Taylor's theorem}, which gives us Taylor's series.
\begin{thm}[Rolle's theorem]
Let $f$ be continuous on a closed interval $[a, b]$ (with $a < b$) and differentiable on $(a, b)$. Suppose that $f(a) = f(b)$. Then there exists $x\in (a, b)$ such that $f'(x) = 0 $.
\end{thm}
It is intuitively obvious: if you move up and down, and finally return to the same point, then you must have changed direction some time. Then $f'(x) = 0$ at that time.
\begin{proof}
If $f$ is constant, then we're done.
Otherwise, there exists $u$ such that $f(u) \not= f(a)$. wlog, $f(u) > f(a)$. Since $f$ is continuous, it has a maximum, and since $f(u) > f(a) = f(b)$, the maximum is not attained at $a$ or $b$.
Suppose maximum is attained at $x\in (a, b)$. Then for any $h \not = 0$, we have
\[
\frac{f(x + h) - f(x)}{h}
\begin{cases}
\leq 0 & h > 0\\
\geq 0 & h < 0
\end{cases}
\]
since $f(x + h) - f(x) \leq 0$ by maximality of $f(x)$. By considering both sides as we take the limit $h\to 0$, we know that $f'(x) \leq 0$ and $f'(x) \geq 0$. So $f'(x) = 0$.
\end{proof}
\begin{cor}[Mean value theorem]
Let $f$ be continuous on $[a, b]$ ($a < b$), and differentiable on $(a, b)$. Then there exists $x\in (a, b)$ such that
\[
f'(x) = \frac{f(b) - f(a)}{b - a}.
\]
Note that $\frac{f(b) - f(a)}{b - a}$ is the slope of the line joining $f(a)$ and $f(b)$.
\begin{center}
\begin{tikzpicture}
\draw (0, 0) -- (3, 2);
\draw [dashed] (1, 1.26666) -- (2.7, 2.4);
\draw (0, 0) .. controls (3, 0) and (0, 2) .. (3, 2) node [above, pos = 0.8] {$f(x)$};
\node [circ] {};
\node [left] {$f(a)$};
\node at (3, 2) [circ] {};
\node at (3, 2) [right] {$f(b)$};
\end{tikzpicture}
\end{center}
\end{cor}
The mean value theorem is sometimes described as ``rotate your head and apply Rolle's''. However, if we actually rotate it, we might end up with a non-function. What we actually want is a shear.
\begin{proof}
Let
\[
g(x) = f(x) - \frac{f(b) - f(a)}{b - a}x.
\]
Then
\[
g(b) - g(a) = f(b) - f(a) - \frac{f(b) - f(a)}{b - a}(b - a) = 0.
\]
So by Rolle's theorem, we can find $x\in (a, b)$ such that $g'(x) = 0$. So
\[
f'(x) = \frac{f(b) - f(a)}{b - a},
\]
as required.
\end{proof}
We've always assumed that if a function has a positive derivative everywhere, then the function is increasing. However, it turns out that this is really hard to prove directly. It does, however, follow quite immediately from the mean value theorem.
\begin{eg}
Suppose $f'(x) > 0$ for every $x\in (a, b)$. Then for $u, v$ in $[a, b]$, we can find $w\in (u, v)$ such that
\[
\frac{f(v) - f(u)}{v - u} = f'(w) > 0.
\]
It follows that $f(v) > f(u)$. So $f$ is strictly increasing.
Similarly, if $f'(x) \geq 2$ for every $x$ and $f(0) = 0$, then $f(1) \geq 2$, or else we can find $x\in (0, 1)$ such that
\[
2\leq f'(x) = \frac{f(1) - f(0)}{1 - 0} = f(1).
\]
\end{eg}
\begin{thm}[Local version of inverse function theorem]
Let $f$ be a function with continuous derivative on $(a, b)$.
Let $x\in (a, b)$ and suppose that $f'(x) \not= 0$. Then there is an open interval $(u, v)$ containing $x$ on which $f$ is invertible (as a function from $(u, v)$ to $f((u, v))$). Moreover, if $g$ is the inverse, then $g'(f(z)) = \frac{1}{f'(z)}$ for every $z\in (u, v)$.
This says that if $f$ has a non-zero derivative, then it has an inverse locally and the derivative of the inverse is $1/f'$.
\end{thm}
Note that this not only requires $f$ to be differentiable, but the derivative itself also has to be continuous.
\begin{proof}
wlog, $f'(x) > 0$. By the continuity, of $f'$, we can find $\delta > 0$ such that $f'(z) > 0$ for every $z\in (x - \delta, x + \delta)$. By the mean value theorem, $f$ is strictly increasing on $(x - \delta, x + \delta)$, hence injective. Also, $f$ is continuous on $(x - \delta, x + \delta)$ by differentiability.
Then done by the inverse function theorem.
\end{proof}
Finally, we are going to prove Taylor's theorem. To do so, we will first need some lemmas.
\begin{thm}[Higher-order Rolle's theorem]
Let $f$ be continuous on $[a, b]$ ($a < b$) and $n$-times differentiable on an open interval containing $[a, b]$. Suppose that
\[
f(a) = f'(a) = f^{(2)}(a) = \cdots = f^{(n - 1)}(a) = f(b) = 0.
\]
Then $\exists x\in (a, b)$ such that $f^{(n)}(x) = 0$.
\end{thm}
\begin{proof}
Induct on $n$. The $n = 0$ base case is just Rolle's theorem.
Suppose we have $k < n$ and $x_k\in (a, b)$ such that $f^{(k)}(x_k) = 0$. Since $f^{(k)}(a) = 0$, we can find $x_{k + 1}\in (a, x_k)$ such that $f^{(k + 1)}(x_{k + 1}) = 0$ by Rolle's theorem.
So the result follows by induction.
\end{proof}
\begin{cor}
Suppose that $f$ and $g$ are both differentiable on an open interval containing $[a, b]$ and that $f^{(k)}(a) = g^{(k)}(a)$ for $k = 0, 1, \cdots, n - 1$, and also $f(b) = g(b)$. Then there exists $x\in (a, b)$ such that $f^{(n)}(x) = g^{(n)}(x)$.
\end{cor}
\begin{proof}
Apply generalised Rolle's to $f - g$.
\end{proof}
Now we shall show that for any $f$, we can find a polynomial $p$ of degree at most $n$ that satisfies the conditions for $g$, i.e.\ a $p$ such that $p^{(k)}(a) = f^{(k)}(a)$ for $k = 0, 1, \cdots, n - 1$ and $p(b) = f(b)$.
A useful ingredient is the observation that if
\[
Q_k(x) = \frac{(x - a)^k}{k!},
\]
then
\[
Q_k^{(j)}(a) =
\begin{cases}
1 & j = k\\
0 & j \not= k
\end{cases}
\]
Therefore, if
\[
Q(x) = \sum_{k = 0}^{n - 1}f^{(k)}(a) Q_k(x),
\]
then
\[
Q^{(j)}(a) = f^{(j)}(a)
\]
for $j = 0, 1, \cdots, n - 1$. To get $p(b) = f(b)$, we use our $n$th degree polynomial term:
\[
p(x) = Q(x) + \frac{(x - a)^n}{(b - a)^n}\big(f(b) - Q(b)\big).
\]
Then our final term does not mess up our first $n - 1$ derivatives, and gives $p(b)= f(b)$.
By the previous corollary, we can find $x\in (a, b)$ such that
\[
f^{(n)}(x) = p^{(n)}(x).
\]
That is,
\[
f^{(n)}(x) = \frac{n!}{(b - a)^n}\big(f(b) - Q(b)\big).
\]
Therefore
\[
f(b) = Q(b) + \frac{(b - a)^n}{n!}f^{(n)}(x).
\]
Alternatively,
\[
f(b) = f(a) + (b - a)f'(a) + \cdots + \frac{(b - a)^{n - 1}}{(n- 1)!}f^{(n - 1)}(a) + \frac{(b - a)^n}{n!}f^{(n)}(x).
\]
Setting $b = a + h$, we can rewrite this as
\begin{thm}[Taylor's theorem with the Lagrange form of remainder]
\[
f(a + h) = \underbrace{f(a) + hf'(a) + \cdots + \frac{h^{n - 1}}{(n - 1)!}f^{(n - 1)}(a)}_{(n - 1)\text{-degree approximation to }f\text{ near }a} + \underbrace{\frac{h^n}{n!}f^{(n)}(x)}_{\text{error term}}.
\]
for some $x\in (a, a + h)$.
\end{thm}
Strictly speaking, we only proved it for the case when $h > 0$, but we can easily show it holds for $h < 0$ too by considering $g(x) = f(-x)$.
Note that the remainder term is \emph{not} necessarily small, but this often gives us the best $(n - 1)$-degree approximation to $f$ near $a$. For example, if $f^{(n)}$ is bounded by $C$ near $a$, then
\[
\left|\frac{h^n}{n!}f^{(n)}(x)\right| \leq \frac{C}{n!}|h|^n = o(h^{n - 1}).
\]
\begin{eg}
Let $f: \R \to \R$ be a differentiable function such that $f(0) = 1$ and $f'(x) = f(x)$ for every $x$ (intuitively, we know it is $e^x$ , but that thing doesn't exist!). Then for every $x$, we have
\[
f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n = 0}^\infty \frac{x^n}{n!}.
\]
While it seems like we can prove this works by differentiating it and see that $f'(x) = f(x)$, the sum rule only applies for finite sums. We don't know we can differentiate a sum term by term. So we have to use Taylor's theorem.
Since $f'(x) =f(x)$, it follows that all derivatives exist. By Taylor's theorem,
\[
f(x) = f(0) + f'(0) x + \frac{f^{(2)}(0)}{2!}x^2 + \cdots + \frac{f^{(n - 1)}(0)}{(n - 1)!}x^{n - 1} + \frac{f^{(n)}(u)}{n!}x^n.
\]
for some $u$ between $0$ and $x$. This equals to
\[
f(x) = \sum_{k = 0}^{n - 1}\frac{x^k}{k!} + \frac{f^{(n)}(u)}{n!}x^n.
\]
We must show that the remainder term $\frac{f^{(n)}(u)}{n!}x^n \to 0$ as $n \to \infty$. Note here that $x$ is fixed, but $u$ can depend on $n$.
But we know that $f^{(n)}(u) = f(u)$, but since $f$ is differentiable, it is continuous, and is bounded on $[0, x]$. Suppose $|f(u)| \leq C$ on $[0, x]$. Then
\[
\left|\frac{f^{(n)(u)}}{n!}x^n\right| \leq \frac{C}{n!}|x|^n \to 0
\]
from limit theorems. So it follows that
\[
f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots = \sum_{n = 0}^\infty \frac{x^n}{n!}.
\]
\end{eg}
\subsection{Complex differentiation}
\begin{defi}[Complex differentiability]
Let $f: \C \to \C$. Then $f$ is differentiable at $z$ with derivative $f'(z)$ if
\[
\lim_{h \to 0}\frac{f(z + h) - f(z)}{h}\text{ exists and equals }f'(z).
\]
Equivalently,
\[
f(z + h) = f(z) + hf'(z) + o(h).
\]
\end{defi}
This is exactly the same definition with real differentiation, but has very different properties!
All the usual rules --- chain rule, product rule etc. also apply (with the same proofs). Also the derivatives of polynomials are what you expect. However, there are some more interesting cases.
\begin{eg}
$f(z) = \bar z$ is \emph{not} differentiable.
\[
\frac{\overline{z + h} - \overline{z\vphantom{h}}}{h} = \frac{\bar h}{h} =
\begin{cases}
1 & h\text{ is real}\\
-1 & h\text{ is purely imaginary}
\end{cases}
\]
\end{eg}
If this seems weird, this is because we often think of $\C$ as $\R^2$, but they are not the same. For example, reflection is a linear map in $\R^2$, but not in $\C$. A linear map in $\C$ is something in the form $x \mapsto bx$, which can only be a dilation or rotation, not reflections or other weird things.
\begin{eg}
$f(z) = |z|$ is also not differentiable. If it were, then $|z|^2$ would be as well (by the product rule). So would $\frac{|z|^2}{z} = \bar z$ when $z \not= 0$ by the quotient rule. At $z = 0$, it is certainly not differentiable, since it is not even differentiable on $\R$.
\end{eg}
\section{Complex power series}
Before we move on to integration, we first have a look at complex power series. This will allow us to define the familiar exponential and trigonometric functions.
\begin{defi}[Complex power series]
A \emph{complex power series} is a series of the form
\[
\sum_{n = 0}^{\infty}a_n z^n.
\]
when $z\in \C$ and $a_n\in \C$ for all $n$. When it converges, it is a function of $z$.
\end{defi}
When considering complex power series, a very important concept is the \emph{radius of convergence}. To make sense of this concept, we first need the following lemma:
\begin{lemma}
Suppose that $\sum a_nz^n$ converges and $|w| < |z|$, then $\sum a_n w^n$ converges (absolutely).
\end{lemma}
\begin{proof}
We know that
\[
|a_n w^n| = |a_nz^n|\cdot \left|\frac{w}{z}\right|^n.
\]
Since $\sum a_nz^n$ converges, the terms $a_nz^n$ are bounded. So pick $C$ such that
\[
|a_nz^n| \leq C
\]
for every $n$. Then
\[
0 \leq \sum_{n = 0}^\infty |a_nw^n| \leq \sum_{n = 0}^\infty C\left|\frac{w}{z}\right|^n,
\]
which converges (geometric series). So by the comparison test, $\sum a_nw^n$ converges absolutely.
\end{proof}
It follows that if $\sum a_nz^n$ does not converge and $|w| > |z|$, then $\sum a_nw^n$ does not converge.
Now let $R = \sup\{|z|: \sum a_nz^n$ converges $\}$ ($R$ may be infinite). If $|z| < R$, then we can find $z_0$ with $|z_0|\in (|z|, R]$ such that $\sum_n^\infty a_nz_0^n$ converges. So by lemma above, $\sum a_n z^n$ converges. If $|z| > R$, then $\sum a_nz^n$ diverges by definition of $R$.
\begin{defi}[Radius of convergence]
The \emph{radius of convergence} of a power series $\sum a_nz^n$ is
\[
R = \sup\left\{|z|: \sum a_nz^n\text{ converges }\right\}.
\]
$\{z: |z| < R\}$ is called the \emph{circle of convergence}.\footnote{Note to pedants: yes it is a disc, not a circle}.
If $|z| < R$, then $\sum a_nz^n$ converges. If $|z| > R$, then $\sum a_nz^n$ diverges. When $|z| = R$, the series can converge at some points and not the others.
\end{defi}
\begin{eg}
$\displaystyle\sum_{n = 0}^\infty z^n$ has radius of convergence of $1$. When $|z| = 1$, it diverges (since the terms do not tend to $0$).
\end{eg}
\begin{eg}
$\displaystyle\sum_{n = 0}^\infty \frac{z^n}{n}$ has radius of convergence $1$, since the ratio of $(n + 1)$th term to $n$th is
\[
\frac{z^{n + 1}/(n + 1)}{z^n /n} = z\cdot\frac{n}{n + 1} \to z.
\]
So if $|z| < 1$, then the series converges by the ratio test. If $|z| > 1$, then eventually the terms are increasing in modulus.
If $z = 1$, then it diverges (harmonic series). If $|z| = 1$ and $z \not= 1$, it converges by Abel's test.
\end{eg}
\begin{eg}
The series $\displaystyle \sum_{n = 1}^\infty \frac{z^{n}}{n^2}$ converges for $|z| \leq 1$ and diverges for $|z| > 1$.
\end{eg}
As evidenced by the above examples, the ratio test can be used to find the radius of convergence. We also have an alternative test based on the $n$th root.
\begin{lemma}
The radius of convergence of a power series $\sum a_nz^n$ is
\[
R = \frac{1}{\limsup \sqrt[n]{|a_n|}}.
\]
Often $\sqrt[n]{|a_n|}$ converges, so we only have to find the limit.
\end{lemma}
\begin{proof}
Suppose $|z| < 1/\limsup \sqrt[n]{|a_n|}$. Then $|z| \limsup \sqrt[n]{|a_n|} < 1$. Therefore there exists $N$ and $\varepsilon > 0$ such that
\[
\sup_{n \geq N}|z|\sqrt[n]{|a_n|} \leq 1 - \varepsilon
\]
by the definition of $\limsup$. Therefore
\[
|a_n z^n| \leq (1 - \varepsilon)^n
\]
for every $n \geq N$, which implies (by comparison with geometric series) that $\sum a_n z^n$ converges absolutely.
On the other hand, if $|z|\limsup\sqrt[n]{|a_n|} > 1$, it follows that $|z|\sqrt[n]{|a_n|} \geq 1$ for infinitely many $n$. Therefore $|a_nz^n| \geq 1$ for infinitely many $n$. So $\sum a_nz^n$ does not converge.
\end{proof}
\begin{eg}
The radius of convergence of $\displaystyle \frac{z^n}{2^n}$ is $2$ because $\sqrt[n]{|a_n|} = \frac{1}{2}$ for every $n$. So $\limsup \sqrt[n]{|a_n|} = \frac{1}{2}$. So $1/\limsup \sqrt[n]{|a_n|} = 2$.
\end{eg}
But often it is easier to find the radius convergence from elementary methods such as the ratio test, e.g.\ for $\sum n^2 z^n$.
\subsection{Exponential and trigonometric functions}
\begin{defi}[Exponential function]
The \emph{exponential function} is
\[
e^z = \sum_{n = 0}^\infty \frac{z^n}{n!}.
\]
By the ratio test, this converges on all of $\C$.
\end{defi}
A fundamental property of this function is that
\[
e^{z + w} = e^ze^w.
\]
Once we have this property, we can say that
\begin{prop}
The derivative of $e^z$ is $e^z$.
\end{prop}
\begin{proof}
\begin{align*}
\frac{e^{z + h} - e^z}{h} &= e^z \left(\frac{e^h - 1}{h}\right)\\
&= e^z\left(1 + \frac{h}{2!} + \frac{h^2}{3!} + \cdots\right)
\end{align*}
But
\[
\left|\frac{h}{2!} + \frac{h^2}{3!} + \cdots \right| \leq \frac{|h|}{2} + \frac{|h|^2}{4} + \frac{|h|^3}{8} + \cdots = \frac{|h|/2}{1 - |h|/2} \to 0.
\]
So
\[
\frac{e^{z + h} - e^z}{h} \to e^z.\qedhere
\]
\end{proof}
But we must still prove that $e^{z + w} = e^ze^w$.
Consider two sequences $(a_n), (b_n)$. Their \emph{convolution} is the sequence $(c_n)$ defined by
\[
c_n = a_0b_n + a_1b_{n - 1} + a_2b_{n - 2} + \cdots + a_nb_0.
\]
The relevance of this is that if you take
\[
\left(\sum_{n = 0}^N a_nz^n\right)\left(\sum_{n = 0}^N b_nz^n\right)\text{ and }\sum_{n = 0}^N c_n z^n,
\]
and equate coefficients of $z^n$, you get
\[
c_n = a_0b_n + a_1b_{n - 1} + a_2b_{n - 2} + \cdots + a_nb_0.
\]
\begin{thm}
Let $\sum_{n = 0}^\infty a_n$ and $\sum_{n = 0}^\infty b_n$ be two absolutely convergent series, and let $(c_n)$ be the convolution of the sequences $(a_n)$ and $(b_n)$. Then $\sum_{n = 0}^\infty c_n$ converges (absolutely), and
\[
\sum_{n = 0}^{\infty} c_n = \left(\sum_{n = 0}^\infty a_n\right)\left(\sum_{n = 0}^\infty b_n \right).
\]
\end{thm}
\begin{proof}
We first show that a rearrangement of $\sum c_n$ converges absolutely. Hence it converges unconditionally, and we can rearrange it back to $\sum c_n$.
Consider the series
\[
(a_0b_0) + (a_0 b_1 + a_1b_1 + a_1b_0) + (a_0 b_2 + a_1 b_2 + a_2b_2 + a_2b_1 + a_2b_0) + \cdots\tag{$*$}
\]
Let
\[
S_N = \sum_{n = 0}^{N}a_n, \quad T_N = \sum_{n = 0}^N b_n,\quad U_N = \sum_{n = 0}^N | a_n|,\quad V_N = \sum_{n = 0}^N|b_n|.
\]
Also let $S_N \to S, T_N \to T, U_N \to U, V_N \to V$ (these exist since $\sum a_n$ and $\sum b_n$ converge absolutely).
If we take the modulus of the terms of $(*)$, and consider the first $(N + 1)^2$ terms (i.e.\ the first $N + 1$ brackets), the sum is $U_NV_N$. Hence the series converges absolutely to $UV$. Hence $(*)$ converges.
The partial sum up to $(N + 1)^2$ of the series $(*)$ itself is $S_NT_N$, which converges to $ST$. So the whole series converges to $ST$.
Since it converges absolutely, it converges unconditionally. Now consider a rearrangement:
\[
a_0 b_0 + (a_0b_1 + a_1b_0) + (a_0b_2 + a_1b_1 + a_2b_0) + \cdots
\]
Then this converges to $ST$ as well. But the partial sum of the first $1 + 2 + \cdots + N$ terms is $c_0 + c_1 + \cdots + c_N$. So
\[
\sum_{n = 0}^N c_n \to ST = \left(\sum_{n = 0}^\infty a_n\right)\left(\sum_{n = 0}^\infty b_n \right). \qedhere
\]
\end{proof}
\begin{cor}
\[
e^z e^w = e^{z + w}.
\]
\end{cor}
\begin{proof}
By theorem above (and definition of $e^z$),
\begin{align*}
e^z e^w &= \sum_{n = 0}^\infty \left(1\cdot \frac{w^n}{n!} + \frac{z}{1!}\frac{w^{n - 1}}{(n- 1)!} + \frac{z^2}{2!}\frac{w^{n - 2}}{(n- 2)!} + \cdots + \frac{z^{n}}{n!}\cdot 1\right)\\
e^z e^w &= \sum_{n = 0}^\infty \frac{1}{n!}\left(w^n + \binom{n}{1} zw^{n - 1} + \binom{n}{2}z^2 w^{n - 2} + \cdots + \binom{n}{n}z^n\right)\\
&= \sum_{n = 0}^\infty (z + w)^n \text{ by the binomial theorem}\\
&= e^{z + w}. \qedhere
\end{align*}
\end{proof}
Note that if $(c_n)$ is the convolution of $(a_n)$ and $(b_n)$, then the convolution of $(a_nz^n)$ and $(b_nz^n)$ is $(c_nz^n)$. Therefore if both $\sum a_n z^n$ and $\sum b_nz^n$ converge absolutely, then their product is $\sum c_n z^n$.
Note that we have now completed the proof that the derivative of $e^z$ is $e^z$.
Now we define $\sin z$ and $\cos z$:
\begin{defi}[Sine and cosine]
\begin{align*}
\sin z &= \frac{e^{iz} - e^{-iz}}{2i} = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} + \cdots\\
\cos z &= \frac{e^{iz} + e^{-iz}}{2} = 1 - \frac{z^2}{2!} + \frac{z^4}{4!} - \frac{z^6}{6!} + \cdots.
\end{align*}
\end{defi}
We now prove certain basic properties of $\sin$ and $\cos$, using known properties of $e^z$.
\begin{prop}
\begin{align*}
\frac{\d}{\d z}\sin z &= \frac{ie^{iz} + ie^{-iz}}{2i} = \cos z\\
\frac{\d}{\d z}\cos z &= \frac{ie^{iz} - ie^{-iz}}{2} = -\sin z\\
\sin^2 z + \cos ^2 z &= \frac{e^{2iz} + 2 + e^{-2iz}}{4} + \frac{e^{2iz} - 2 + e^{-2iz}}{-4} = 1.
\end{align*}
\end{prop}
It follows that if $x$ is real, then $|\cos x|$ and $|\sin x|$ are at most $1$.
\begin{prop}
\begin{align*}
\cos(z + w) &= \cos z\cos w - \sin z\sin w\\
\sin(z + w) &= \sin z \cos w + \cos z \sin w
\end{align*}
\end{prop}
\begin{proof}
\begin{align*}
\cos z\cos w - \sin z \sin w &= \frac{(e^{iz} + e^{-iz})(e^{iw} + e^{-iw})}{4} + \frac{(e^{iz} - e^{-iz})(e^{iw} - e^{-iw})}{4}\\
&= \frac{e^{i(z + w)} + e^{-i(z + w)}}{2}\\
&= \cos (z + w). \qedhere
\end{align*}
Differentiating both sides wrt $z$ gives
\[
-\sin z \cos w - \cos z \sin w = -\sin (z + w).
\]
So
\[
\sin(z + w) = \sin z\cos w + \cos z \sin w.\qedhere
\]
\end{proof}
When $x$ is real, we know that $\cos x \leq 1$. Also $\sin 0 = 0$, and $\frac{\d }{\d x}\sin x = \cos x \leq 1$. So for $x \geq 0$, $\sin x \leq x$, ``by the mean value theorem''. Also, $\cos 0 = 1$, and $\frac{\d}{\d x}\cos x = -\sin x$, which, for $x \geq 0$, is greater than $-x$. From this, it follows that when $x \geq 0$, $\cos x \geq 1 - \frac{x^2}{2}$ (the $1 - \frac{x^2}{2}$ comes from ``integrating'' $-x$, (or finding a thing whose derivative is $-x$)).
Continuing in this way, we get that for $x \geq 0$, if you take truncate the power series for $\sin x$ or $\cos x$, it will be $\geq \sin x, \cos x$ if you stop at a positive term, and $\leq$ if you stop at a negative term. For example,
\[
\sin x \geq x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} - \frac{x^{11}}{11!}.
\]
In particular,
\[
\cos 2 \leq 1 - \frac{2^2}{2!} + \frac{2^4}{4!} = 1 - 2 + \frac{2}{3} < 0.
\]
Since $\cos 0 = 1$, it follows by the intermediate value theorem that there exists some $x\in (0, 2)$ such that $\cos x = 0$. Since $\cos x \geq 1 - \frac{x^2}{2}$, we can further deduce that $x > 1$.
\begin{defi}[Pi]
Define the smallest $x$ such that $\cos x = 0$ to be $\frac{\pi}{2}$.
\end{defi}
Since $\sin^2 z + \cos ^2 z = 1$, it follows that $\sin \frac{\pi}{2} = \pm 1$. Since $\cos x > 0$ on $[0, \frac{\pi}{2}]$, $\sin \frac{\pi}{2} \geq 0$ by the mean value theorem. So $\sin \frac{\pi}{2} = 1$.
Thus
\begin{prop}
\begin{align*}
\cos \left(z + \frac{\pi}{2}\right) &= -\sin z\\
\sin \left(z + \frac{\pi}{2}\right) &= \cos z\\
\cos (z + \pi) &= -\cos z\\
\sin (z + \pi) &= -\sin z\\
\cos (z + 2\pi) &= \cos z\\
\sin (z + 2\pi) &= \sin z
\end{align*}
\end{prop}
\begin{proof}
\begin{align*}
\cos\left(z + \frac{\pi}{2}\right) &= \cos z\cos \frac{\pi}{2} - \sin z\sin \frac{\pi}{2}\\
&= -\sin z\sin \frac{\pi}{2}\\
&= -\sin z
\end{align*}
and similarly for others.
\end{proof}
\subsection{Differentiating power series}
We shall show that inside the circle of convergence, the derivative of $\sum_{n = 0}^\infty a_z^n$ is given by the obvious formula $\sum_{n = 1}^\infty na_n z^{n - 1}$.
We first prove some (seemingly arbitrary and random) lemmas to build up the proof of the above statement. They are done so that the final proof will not be full of tedious algebra.
\begin{lemma}
Let $a$ and $b$ be complex numbers. Then
\[
b^n - a^n - n(b - a)a^{n - 1} = (b - a)^2(b^{n - 2} + 2ab^{n - 3} + 3a^2 b^{n - 4} + \cdots + (n - 1)a^{n - 2}).
\]
\end{lemma}
\begin{proof}
If $b = a$, we are done. Otherwise,
\[
\frac{b^n - a^n}{b - a} = b^{n - 1} + ab^{n - 2} + a^2b^{n - 3} + \cdots + a^{n - 1}.
\]
Differentiate both sides with respect to $a$. Then
\[
\frac{-na^{n - 1}(b - a) + b^n - a^n}{(b - a)^2} = b^{n - 2} + 2ab^{n - 3} + \cdots + (n - 1)a^{n - 2}.
\]
Rearranging gives the result.
Alternatively, we can do
\[
b^n - a^n = (b - a)(b^{n - 1} + ab^{n - 2} + \cdots + a^{n - 1}).
\]
Subtract $n(b - a)a^{n - 1}$ to obtain
\[
(b - a)[b^{n - 1} - a^{n - 1} + a(b^{n - 2} - a^{n - 2}) + a^2(b^{n - 3} - a^{n -3 }) + \cdots]
\]
and simplify.
\end{proof}
This implies that
\[
(z + h)^n - z^n - nhz^{n - 1} = h^2((z + h)^{n - 2} + 2z(z + h)^{n - 3} + \cdots + (n - 1)z^{n - 2}),
\]
which is actually the form we need.
\begin{lemma}
Let $a_n z^n$ have radius of convergence $R$, and let $|z| < R$. Then $\sum na_n z^{n - 1}$ converges (absolutely).
\end{lemma}
\begin{proof}
Pick $r$ such that $|z| < r < R$. Then $\sum |a_n| r^n$ converges, so the terms $|a_n|r^n$ are bounded above by, say, $C$. Now
\[
\sum n|a_n z^{n - 1}| = \sum n|a_n|r^{n - 1}\left(\frac{|z|}{r}\right)^{n - 1} \leq \frac{C}{r}\sum n\left(\frac{|z|}{r}\right)^{n - 1}
\]
The series $\sum n\left(\frac{|z|}{r}\right)^{n - 1}$ converges, by the ratio test. So $\sum n|a_n z^{n - 1}|$ converges, by the comparison test.
\end{proof}
\begin{cor}
Under the same conditions,
\[
\sum_{n = 2}^\infty \binom{n}{2}a_n z^{n -2 }
\]
converges absolutely.
\end{cor}
\begin{proof}
Apply Lemma above again and divide by 2.
\end{proof}
\begin{thm}
Let $\sum a_n z^n$ be a power series with radius of convergence $R$. For $|z| < R$, let
\[
f(z) = \sum_{n = 0}^\infty a_n z^n\text{ and }g(z) = \sum_{n = 1}^\infty na_nz^{n - 1}.
\]
Then $f$ is differentiable with derivative $g$.
\end{thm}
\begin{proof}
We want $f(z + h) - f(z) - hg(z)$ to be $o(h)$. We have
\[
f(z + h) - f(z) - hg(z) = \sum_{n = 2}^\infty a_n ((z + h)^n - z^n - hnz^n).
\]
We started summing from $n = 2$ since the $n = 0$ and $n = 1$ terms are 0. Using our first lemma, we are left with
\[
h^2\sum_{n = 2}^\infty a_n \big((z + h)^{n - 2} + 2z(z + h)^{n - 3} + \cdots + (n - 1)z^{n - 2}\big)
\]
We want the huge infinite series to be bounded, and then the whole thing is a bounded thing times $h^2$, which is definitely $o(h)$.
Pick $r$ such that $|z| < r < R$. If $h$ is small enough that $|z + h| \leq r$, then the last infinite series is bounded above (in modulus) by
\[
\sum_{n = 2}^\infty|a_n|(r^{n - 2} + 2r^{n - 2} + \cdots + (n - 1)r^{n - 2}) = \sum_{n = 2}^\infty |a_n|\binom{n}{2}r^{n -2 },
\]
which is bounded. So done.
\end{proof}
In IB Analysis II, we will prove the same result using the idea of uniform convergence, which gives a much nicer proof.
\begin{eg}
The derivative of
\[
e^z = 1 + z + \frac{z^2}{2!} + \frac{z^3}{3!} + \cdots
\]
is
\[
1 + z + \frac{z^2}{2!} + \cdots = e^z.
\]
So we have another proof that of this fact.
Similarly, the derivatives of $\sin z$ and $\cos z$ work out as $\cos z$ and $-\sin z$.
\end{eg}
\subsection{Hyperbolic trigonometric functions}
\begin{defi}[Hyperbolic sine and cosine]
We define
\begin{align*}
\cosh z &= \frac{e^z + e^{-z}}{2} = 1 + \frac{z^2}{2!} + \frac{z^4}{4!} + \frac{z^6}{6!} + \cdots\\
\sinh z &= \frac{e^z - e^{-z}}{2} = z + \frac{z^3}{3!} + \frac{z^5}{5!} + \frac{z^7}{7!} + \cdots
\end{align*}
\end{defi}
Either from the definition or from differentiating the power series, we get that
\begin{prop}
\begin{align*}
\frac{\d}{\d z}\cosh z &= \sinh z\\
\frac{\d }{\d z}\sinh z &= \cosh z
\end{align*}
\end{prop}
Also, by definition, we have
\begin{prop}
\begin{align*}
\cosh iz &= \cos z\\
\sinh iz &= i\sin z
\end{align*}
\end{prop}
Also,
\begin{prop}
\[
\cosh^2 z - \sinh^2 z = 1,
\]
\end{prop}
\section{The Riemann Integral}
Finally, we can get to integrals. There are many ways to define an integral, which can have some subtle differences. The definition we will use here is the \emph{Riemann integral}, which is the simplest definition, but is also the weakest one, in the sense that many functions are \emph{not} Riemann integrable but integrable under other definitions.
Still, the definition of the Riemann integral is not too straightforward, and requires a lot of preliminary definitions.
\subsection{Riemann Integral}
\begin{defi}[Dissections]
Let $[a, b]$ be a closed interval. A \emph{dissection} of $[a, b]$ is a sequence $a = x_0 < x_1 < x_2 < \cdots < x_n = b$.
\end{defi}
\begin{defi}[Upper and lower sums]
Given a dissection $\mathcal{D}$, the \emph{upper sum} and \emph{lower sum} are defined by the formulae
\begin{align*}
U_\mathcal{D} (f) &= \sum_{i = 1}^{n}(x_i - x_{i - 1}) \sup_{x\in [x_{i - 1}, x_{i}]}f (x)\\
L_\mathcal{D} (f) &= \sum_{i = 1}^{n}(x_i - x_{i - 1}) \inf_{x\in [x_{i - 1}, x_{i}]}f (x)
\end{align*}
Sometimes we use the shorthand
\[
M_i = \sup_{x\in [x_{i - 1}, x_i]} f(x), \quad m_i = \inf_{x\in [x_{i - 1} - x_i]} f(x).
\]
\end{defi}
The upper sum is the total area of the red rectangles, while the lower sum is the total area of the black rectangles:
\begin{center}
\begin{tikzpicture}
\draw [->] (-1, 0) -- (5, 0) node [right] {$x$};
\draw [->] (0, -0.5) -- (0, 5) node [above] {$y$};
\draw [domain=-1:5] plot (\x, {(\x + 1)*(\x + 1)/10 + 1});
\draw (0.5, 0) node [below] {$a$} -- (0.5, 1.225) -- (1, 1.225);
\draw (1, 0) node [below] {$x_1$} -- (1, 1.4) -- (1.5, 1.4);
\draw (1.5, 0) node [below] {$x_2$} -- (1.5, 1.625) -- (2, 1.625) -- (2, 0) node [below] {$x_3$};
\node at (2.4, 0.8) {$\cdots$};
\draw (2.75, 0) node [below] {$x_i$} -- (2.75, 2.40625) -- (3.25, 2.40625) -- (3.25, 0) node [anchor = north west] {$\!\!\!\!\!x_{i + 1}\cdots$};
\node at (3.65, 1.2) {$\cdots$};
\draw (4, 0) -- (4, 3.5) -- (4.5, 3.5) -- (4.5, 0) node [below] {$b$};
\draw [red] (0.5, 1.225) -- (0.5, 1.4) -- (1, 1.4) -- (1, 1.625) -- (1.5, 1.625) -- (1.5, 1.9) -- (2, 1.9) -- (2, 1.625);
\draw [red] (2.75, 2.40625) -- (2.75, 2.80625) -- (3.25, 2.80625) -- (3.25, 2.40625);
\draw [red] (4, 3.5) -- (4, 4.025) -- (4.5, 4.025) -- (4.5, 3.5);
\end{tikzpicture}
\end{center}
\begin{defi}[Refining dissections]
If $\mathcal{D}_1$ and $\mathcal{D}_2$ are dissections of $[a, b]$, we say that $\mathcal{D}_2$ refines $\mathcal{D}_1$ if every point of $\mathcal{D}_1$ is a point of $\mathcal{D}_2$.
\end{defi}
\begin{lemma}
If $\mathcal{D}_2$ refines $\mathcal{D}_1$, then
\[
U_{\mathcal{D}_2} f \leq U_{\mathcal{D}_1}f\text{ and }L_{\mathcal{D}_2 f}\geq L_{\mathcal{D}_1}f.
\]
\end{lemma}
Using the picture above, this is because if we cut up the dissections into smaller pieces, the red rectangles can only get chopped into shorter pieces and the black rectangles can only get chopped into taller pieces.
\begin{center}
\begin{tikzpicture}
\draw [->] (-0.5, 0) -- (3, 0) node [right] {$x$};
\draw [->] (0, -0.5) -- (0, 3) node [above] {$y$};
\draw [domain=-0.5:3] plot (\x, {(\x + 1)*(\x + 1)/10 + 1});
\draw (0.5, 0) node [below] {$x_0$} -- (0.5, 1.225) -- (2, 1.225) -- (2, 0) node [below] {$x_1$};
\draw [red] (0.5, 1.225) -- (0.5, 1.9) -- (2, 1.9) -- (2, 1.225);
\draw [->] (3.5, 1.5) -- (4.5, 1.5);
\begin{scope}[shift={(5.5, 0)}]
\draw [->] (-0.5, 0) -- (3, 0) node [right] {$x$};
\draw [->] (0, -0.5) -- (0, 3) node [above] {$y$};
\draw [domain=-0.5:3] plot (\x, {(\x + 1)*(\x + 1)/10 + 1});
\draw (0.5, 0) node [below] {$x_0$} -- (0.5, 1.225) -- (1, 1.225);
\draw (1, 0) node [below] {$x_1$} -- (1, 1.4) -- (1.5, 1.4);
\draw (1.5, 0) node [below] {$x_2$} -- (1.5, 1.625) -- (2, 1.625) -- (2, 0) node [below] {$x_3$};
\draw [red] (0.5, 1.225) -- (0.5, 1.4) -- (1, 1.4) -- (1, 1.625) -- (1.5, 1.625) -- (1.5, 1.9) -- (2, 1.9) -- (2, 1.625);
\end{scope}
\end{tikzpicture}
\end{center}
\begin{proof}
Let $\mathcal{D}$ be $x_0 < x_1 < \cdots < x_n$. Let $\mathcal{D}_2$ be obtained from $\mathcal{D}_1$ by the addition of one point $z$. If $z\in (x_{i - 1}, x_i)$, then
\begin{align*}
U_{\mathcal{D}_2}f - U_{\mathcal{D}_1}f &= \left[(z - x_{i - 1}) \sup_{x\in [x_{i - 1}, z]} f(x)\right]\\
&+ \left[(x_i - z)\sup_ {x\in[z, x_i]}f(x)\right] - (x_i - x_{i - 1})M_i.
\end{align*}
But $\sup_{x\in [x_{i - 1}, z]} f(x)$ and $\sup_{x\in [z, x_i]} f(x)$ are both at most $M_i$. So this is at most $M_i( z - x_{i - 1} + x_i - z - (x_i - x_{i - 1})) =0 $. So
\[
U_{\mathcal{D}_2} f\leq U_{\mathcal{D}_1}f.
\]
By induction, the result is true whenever $\mathcal{D}_2$ refines $\mathcal{D}_1$.
A very similar argument shows that $L_{\mathcal{D}_2f} \geq L_{\mathcal{D}_1}f$.
\end{proof}
\begin{defi}[Least common refinement]
If $\mathcal{D}_1$ and $\mathcal{D}_2$ be dissections of $[a, b]$. Then the least common refinement of $\mathcal{D}_1$ and $\mathcal{D}_2$ is the dissection made out of the points of $\mathcal{D}_1$ and $\mathcal{D}_2$.
\end{defi}
\begin{cor}
Let $\mathcal{D}_1$ and $\mathcal{D}_2$ be two dissections of $[a, b]$. Then
\[
U_{\mathcal{D}_1}f \geq L_{\mathcal{D}_2}f.
\]
\end{cor}
\begin{proof}
Let $\mathcal{D}$ be the least common refinement (or indeed any common refinement). Then by lemma above (and by definition),
\[
U_{\mathcal{D}_1}f \geq U_{\mathcal{D}}f \geq L_{\mathcal{D}}f \geq L_{\mathcal{D}_2}f.\qedhere
\]
\end{proof}
Finally, we can define the integral.
\begin{defi}[Upper, lower, and Riemann integral]
The \emph{upper integral} is
\[
\overline{\int_a^b} f(x)\;\d x = \inf_{\mathcal{D}}U_{\mathcal{D}}f.
\]
The \emph{lower integral} is
\[
\underline{\int_a^b} f(x)\;\d x = \sup_{\mathcal{D}}L_{\mathcal{D}}f.
\]
If these are equal, then we call their common value the \emph{Riemann integral} of $f$, and is denoted $\int_a^b f(x)\;\d x$.
If this exists, we say $f$ is \emph{Riemann integrable}.
\end{defi}
We will later prove the \emph{fundamental theorem of calculus}, which says that integration is the reverse of differentiation. But why don't we simply define integration as anti-differentiation, and prove that it is the area of the curve? There are things that we cannot find (a closed form of) the anti-derivative of, like $e^{-x^2}$. In these cases, we wouldn't want to say the integral doesn't exist --- it surely does according to this definition!
There is an immediate necessary condition for Riemann integrability --- boundedness. If $f$ is unbounded above in $[a, b]$, then for any dissection $\mathcal{D}$, there must be some $i$ such that $f$ is unbounded on $[x_{i - 1}, x_i]$. So $M_i = \infty$. So $U_\mathcal{D} f = \infty$. Similarly, if $f$ is unbounded below, then $L_{\mathcal{D}} f = -\infty$. So unbounded functions are not Riemann integrable.
\begin{eg}
Let $f(x) = x$ on $[a, b]$. Intuitively, we know that the integral is $(b^2 - a^2)/2$, and we will show this using the definition above. Let $\mathcal{D} = x_0 < x_1 < \cdots < x_n$ be a dissection. Then
\begin{align*}
U_{\mathcal{D}}f &= \sum_{i = 1}^n (x_i - x_{i - 1})x_i\\
\intertext{We \emph{know} that the integral is $\frac{b^2 - a^2}{2}$. So we put each term of the sum into the form $\frac{x_i^2 - x_{i - 1}^2}{2}$ plus some error terms.}
&= \sum_{i = 1}^n \left(\frac{x_i^2}{2} - \frac{x_{i - 1}^2}{2} + \frac{x_i^2}{2} - x_{i - 1}x_i + \frac{x_{i - 1}^2}{2}\right)\\
&= \frac{1}{2}\sum_{i = 1}^n (x_i^2 - x_{i - 1}^2 + (x_i - x_{i - 1})^2)\\
&= \frac{1}{2}(b^2 - a^2) + \frac{1}{2}\sum_{i = 1}^n(x_i - x_{i - 1})^2.
\end{align*}
\begin{defi}[Mesh]
The \emph{mesh} of a dissection $\mathcal{D}$ is $\max_i (x_{i+1} - x_i)$.
\end{defi}
Then if the mesh is $ < \delta$, then
\[
\frac{1}{2}\sum_{i = 1}^n (x_i - x_{i - 1})^2 \leq \frac{\delta}{2}\sum_{i = 1}^n (x_i - x_{i - 1}) = \frac{\delta}{2}(b - a).
\]
So by making $\delta$ small enough, we can show that for any $\varepsilon > 0$,
\[
\overline{\int_a^b} x\;\d x < \frac{1}{2}(b^2 - a^2) + \varepsilon.
\]
Similarly,
\[
\underline{\int_a^b} x\;\d x > \frac{1}{2}(b^2 - a^2) - \varepsilon.
\]
So
\[
\int_a^b x\;\d x = \frac{1}{2}(b^2 - a^2).
\]
\end{eg}
\begin{eg}
Define $f: [0, 1] \to \R$ by
\[
f(x) =
\begin{cases}
1 & x \in \Q\\
0 & x \not\in \Q
\end{cases}.
\]
Let $x_0 < x_1 < \cdots < x_n$ be a dissection. Then for every $i$, we have $m_i = 0$ (since there is an irrational in every interval), and $M_i = 1$ (since there is a rational in every interval). So
\[
U_{\mathcal{D}}f = \sum_{i = 1}^nM_i(x_i - x_{i - 1}) = \sum_{i = 1}^n (x_i - x_{i - 1}) = 1.
\]
Similarly, $L_\mathcal{D} f = 0$. Since $\mathcal{D}$ was arbitrary, we have
\[
\overline{\int_0^1}f(x)\;\d x = 1, \quad \underline{\int_0^1}f(x)\;\d x = 0.
\]
So $f$ is \emph{not} Riemann integrable.
Of course, this function is not interesting at all. The whole point of its existence is to show undergraduates that there are some functions that are not integrable!
\end{eg}
Note that it is important to say that the function is not \emph{Riemann} integrable. There are other notions for integration in which this function is integrable. For example, this function is \emph{Lebesgue-integrable}.
Using the definition to show integrability is often cumbersome. Most of the time, we use the \emph{Riemann's integrability criterion}, which follows rather immediately from the definition, but is much nicer to work with.
\begin{prop}[Riemann's integrability criterion]
This is sometimes known as Cauchy's integrability criterion.
Let $f: [a, b] \to \R$. Then $f$ is Riemann integrable if and only if for every $\varepsilon > 0$, there exists a dissection $\mathcal{D}$ such that
\[
U_\mathcal{D} - L_\mathcal{D} < \varepsilon.
\]
\end{prop}
\begin{proof}
$(\Rightarrow)$ Suppose that $f$ is integrable. Then (by definition of Riemann integrability), there exist $\mathcal{D}_1$ and $\mathcal{D}_2$ such that
\[
U_{\mathcal{D}_1} < \int_a^b f(x)\;\d x + \frac{\varepsilon}{2},
\]
and
\[
L_{\mathcal{D}_2} > \int_a^b f(x)\;\d x - \frac{\varepsilon}{2}.
\]
Let $\mathcal{D}$ be a common refinement of $\mathcal{D}_1$ and $\mathcal{D}_2$. Then
\[
U_\mathcal{D} f - L_\mathcal{D} f \leq U_{\mathcal{D}_1} f- L_{\mathcal{D}_2} f < \varepsilon.
\]
$(\Leftarrow)$ Conversely, if there exists $\mathcal{D}$ such that
\[
U_\mathcal{D} f - L_\mathcal{D}f < \varepsilon,
\]
then
\[
\inf U_\mathcal{D} f - \sup L_\mathcal{D} f < \varepsilon,
\]
which is, by definition, that
\[
\overline{\int_a^b} f(x)\;\d x - \underline{\int_a^b} f(x)\;\d x < \varepsilon.
\]
Since $\varepsilon > 0$ is arbitrary, this gives us that
\[
\overline{\int_a^b} f(x)\;\d x = \underline{\int_a^b} f(x)\;\d x.
\]
So $f$ is integrable.
\end{proof}
The next big result we want to prove is that integration is linear, ie
\[
\int_a^b (\lambda f(x) + \mu g(x))\;\d x = \lambda\int_a^b f(x)\;\d x + \mu\int_a^bg(x)\;\d x.
\]
We do this step by step:
\begin{prop}
Let $f: [a, b] \to \R$ be integrable, and $\lambda \geq 0$. Then $\lambda f$ is integrable, and
\[
\int_a^b \lambda f(x)\;\d x = \lambda\int_a^b f(x)\;\d x.
\]
\end{prop}
\begin{proof}
Let $\mathcal{D}$ be a dissection of $[a, b]$. Since
\[
\sup_{x\in [x_{i - 1}, x_i]}\lambda f(x) = \lambda\sup_{x\in [x_{i - 1}, x_i]}f(x),
\]
and similarly for inf, we have
\begin{align*}
U_{\mathcal{D}}(\lambda f) &= \lambda U_{\mathcal{D}} f\\
L_{\mathcal{D}}(\lambda f) &= \lambda L_\mathcal{D} f.
\end{align*}
So if we choose $\mathcal{D}$ such that $U_{\mathcal{D}}f - L_\mathcal{D} f < \varepsilon/\lambda$, then $U_\mathcal{D}(\lambda f) - L_\mathcal{D}(\lambda f) < \varepsilon$. So the result follows from Riemann's integrability criterion.
\end{proof}
\begin{prop}
Let $f: [a, b] \to \R$ be integrable. Then $-f$ is integrable, and
\[
\int_a^b -f(x)\;\d x = -\int_a^bf(x)\;\d x.
\]
\end{prop}
\begin{proof}
Let $\mathcal{D}$ be a dissection. Then
\begin{align*}
\sup_{x\in [x_{i - 1}, x_i]}-f(x) &= -\inf_{x\in [x_{i - 1}, x_i]} f(x)\\
\inf_{x\in [x_{i - 1}, x_i]}-f(x) &= -\sup_{x\in [x_{i - 1}, x_i]} f(x).
\end{align*}
Therefore
\[
U_\mathcal{D}(-f) = \sum_{i = 1}^n (x_i - x_{i - 1})(-m_i) = -L_\mathcal{D}(f).
\]
Similarly,
\[
L_\mathcal{D}(-f) = -U_\mathcal{D}f.
\]
So
\[
U_\mathcal{D}(-f) - L_\mathcal{D}(-f) = U_\mathcal{D}f - L_\mathcal{D}f.
\]
Hence if $f$ is integrable, then $-f$ is integrable by the Riemann integrability criterion.
\end{proof}
\begin{prop}
Let $f, g: [a, b] \to \R$ be integrable. Then $f + g$ is integrable, and
\[
\int_a^b(f(x) + g(x))\;\d x = \int_a^b f(x)\;\d x + \int_a^b g(x)\;\d x.
\]
\end{prop}
\begin{proof}
Let $\mathcal{D}$ be a dissection. Then
\begin{align*}
U_\mathcal{D}(f + g) &= \sum_{i = 1}^n (x_i - x_{i - 1})\sup_{x\in [x_{i - 1}, x_i]}(f(x) + g(x))\\
&\leq \sum_{i = 1}^n (x_i - x_{i - 1}) \left(\sup_{u\in [x_{i - 1}, x_i]}f(u) + \sup_{v\in [x_{i - 1}, x_i]}g(v)\right)\\
&= U_\mathcal{D}f + U_\mathcal{D} g
\end{align*}
Therefore,
\[
\overline{\int_a^b}(f(x) + g(x))\;\d x \leq \overline{\int_a^b} f(x)\;\d x + \overline{\int_a^b} g(x)\;\d x = \int_a^b f(x)\;\d x + \int_a^bg(x)\;\d x.
\]
Similarly,
\[
\underline{\int_a^b}(f(x) + g(x))\;\d x \geq \int_a^b f(x)\;\d x + \int_a^b g(x)\;\d x.
\]
So the upper and lower integrals are equal, and the result follows.
\end{proof}
So we now have that
\[
\int_a^b (\lambda f(x) + \mu g(x))\;\d x = \lambda\int_a^b f(x)\;\d x + \mu\int_a^bg(x)\;\d x.
\]
We will prove more ``obvious'' results.
\begin{prop}
Let $f, g: [a, b] \to \R$ be integrable, and suppose that $f(x) \leq g(x)$ for every $x$. Then
\[
\int_a^b f(x)\;\d x \leq \int_a^b g(x)\;\d x.
\]
\end{prop}
\begin{proof}
Follows immediately from the definition.
\end{proof}
\begin{prop}
Let $f: [a, b] \to \R$ be integrable. Then $|f|$ is integrable.
\end{prop}
\begin{proof}
Note that we can write
\[
\sup_{x\in [x_{i - 1}, x_i]}f(x) - \inf_{x\in [x_{i - 1}, x_i]}f(x) = \sup_{u, v\in [x_{i - 1}, x_i]}|f(u) - f(v)|.
\]
Similarly,
\[
\sup_{x\in [x_{i - 1}, x_i]}|f(x)| - \inf_{x\in [x_{i - 1}, x_i]}|f(x)| = \sup_{u, v\in [x_{i - 1}, x_i]}||f(u)| - |f(v)||.
\]
For any pair of real numbers, $x, y$, we have that $||x| - |y|| \leq |x - y|$ by the triangle inequality. Then for any interval $u, v\in [x_{i - 1}, x_i]$, we have
\[
||f(u)| - |f(v)|| \leq |f(u) - f(v)|.
\]
Hence we have
\[
\sup_{x\in [x_{i - 1}, x_i]}|f(x)| - \inf_{x\in [x_{i - 1}, x_i]}|f(x)| \leq \sup_{x\in [x_{i - 1}, x_i]}f(x) - \inf_{x\in [x_{i - 1}, x_i]} f(x).
\]
So for any dissection $\mathcal{D}$, we have
\[
U_\mathcal{D} (|f|) - L_\mathcal{D}(|f|) \leq U_\mathcal{D}(f) - L_\mathcal{D}(f).
\]
So the result follows from Riemann's integrability criterion.
\end{proof}
Combining these two propositions, we get that if
\[
|f(x) - g(x)| \leq C,
\]
for every $x\in[a, b]$, then
\[
\left|\int_a^bf(x)\;\d x - \int_a^b g(x)\;\d x\right| \leq C(b - a).
\]
\begin{prop}[Additivity property]
Let $f: [a, c] \to \R$ be integrable, and let $b\in (a, c)$. Then the restrictions of $f$ to $[a, b]$ and $[b, c]$ are Riemann integrable, and
\[
\int_a^b f(x)\;\d x + \int_b^c f(x)\;\d x = \int_a^c f(x) \;\d x
\]
Similarly, if $f$ is integrable on $[a, b]$ and $[b, c]$, then it is integrable on $[a, c]$ and the above equation also holds.
\end{prop}
\begin{proof}
Let $\varepsilon> 0$, and let $a = x_0 < x_1 < \cdots < x_n = c$ be a dissection of $\mathcal{D}$ of $[a, c]$ such that
\[
U_\mathcal{D}(f) \leq \int_a^c f(x)\;\d x + \varepsilon,
\]
and
\[
L_\mathcal{D}(f) \geq \int_a^c f(x)\;\d x - \varepsilon.
\]
Let $\mathcal{D}'$ be the dissection made of $\mathcal{D}$ plus the point $b$. Let $\mathcal{D}_1$ be the dissection of $[a, b]$ made of points of $\mathcal{D}'$ from $a$ to $b$, and $D_2$ be the dissection of $[b, c]$ made of points of $\mathcal{D}'$ from $b$ to $c$. Then
\[
U_{\mathcal{D}_1}(f) + U_{\mathcal{D}_2}(f) = U_{\mathcal{D}'}(f) \leq U_{\mathcal{D}}(f),
\]
and
\[
L_{\mathcal{D}_1}(f) + L_{\mathcal{D}_2}(f) = L_{\mathcal{D}'}(f) \geq L_\mathcal{D} (f).
\]
Since $U_\mathcal{D}(f) - L_\mathcal{D}(f) < 2\varepsilon$, and both $U_{\mathcal{D}_2}(f) - L_{\mathcal{D}_2} (f)$ and $U_{\mathcal{D}_1}(f) - L_{\mathcal{D}_1} (f)$ are non-negative, we have $U_{\mathcal{D}_1} (f) - L_{\mathcal{D}_1} (f)$ and $U_{\mathcal{D}_2}(f) - L_{\mathcal{D}_2}(f)$ are less than $2\varepsilon$. Since $\varepsilon$ is arbitrary, it follows that the restrictions of $f$ to $[a, b]$ and $[b, c]$ are both Riemann integrable. Furthermore,
\begin{multline*}
\int_a^b f(x)\;\d x + \int_b^c f(x)\;\d x \leq U_{\mathcal{D}_1}(f) + U_{\mathcal{D}_2}(f) = U_{\mathcal{D}'}(f) \leq U_{\mathcal{D}}(f)\\
\leq \int_a^c f(x)\;\d x + \varepsilon.
\end{multline*}
Similarly,
\begin{multline*}
\int_a^bf(x)\;\d x + \int_b^cf(x)\;\d x \geq L_{\mathcal{D}_1}(f) + L_{\mathcal{D}_2}(f) = L_{\mathcal{D}'}(f) \geq L_{\mathcal{D}}(f)\\
\geq \int_a^c f(x)\;\d x - \varepsilon.
\end{multline*}
Since $\varepsilon$ is arbitrary, it follows that
\[
\int_a^b f(x)\;\d x + \int_b^c f(x)\;\d x = \int_a^c f(x)\;\d x.
\]
The other direction is left as an (easy) exercise.
\end{proof}
\begin{prop}
Let $f, g: [a, b] \to \R$ be integrable. Then $fg$ is integrable.
\end{prop}
\begin{proof}
Let $C$ be such that $|f(x)|, |g(x)| \leq C$ for every $x\in [a, b]$. Write $L_i$ and $\ell_i$ for the $\sup$ and $\inf$ of $g$ in $[x_{i - 1}, x_i]$. Now let $\mathcal{D}$ be a dissection, and for each $i$, let $u_i$ and $v_i$ be two points in $[x_{i - 1}, x_i]$.
We will pretend that $u_i$ and $v_i$ are the maximum and minimum when we write the proof, but we cannot assert that they are, since $fg$ need not have maxima and minima. We will then note that since our results hold for arbitrary $u_i$ and $v_i$, it must hold when $fg$ is at its supremum and infimum.
We find what we pretend is the difference between the upper and lower sum:
\begin{align*}
&\quad \left|\sum_{i = 1}^n \big(x_i - x_{i - 1})(f(v_i)g(v_i) - f(u_i)g(u_i)\big)\right| \\
&= \left|\sum_{i = 1}^{n}(x_i - x_{i - 1})\big(f(v_i)(g(v_i) - g(u_i)) + (f(v_i - f(u_i))g(u_i))\big)\right|\\
&\leq \sum_{i = 1}^n \big(C(L_i - \ell_i) + (M_i - m_i)C\big)\\
&=C(U_\mathcal{D}g - L_\mathcal{D}g + U_\mathcal{D}f - L_\mathcal{D}f).
\end{align*}
Since $u_i$ and $v_i$ are arbitrary, it follows that
\[
U_\mathcal{D}(fg) - L_\mathcal{D}(fg) \leq C(U_\mathcal{D}f - L_\mathcal{D}f + U_\mathcal{D}g - L_\mathcal{D}g).
\]
Since $C$ is fixed, and we can get $U_\mathcal{D} f - L_\mathcal{D}f$ and $U_\mathcal{D}g - L_\mathcal{D}g$ arbitrary small (since $f$ and $g$ are integrable), we can get $U_\mathcal{D}(fg) - L_\mathcal{D}(fg)$ arbitrarily small. So the result follows.
\end{proof}
\begin{thm}
Every continuous function $f$ on a closed bounded interval $[a, b]$ is Riemann integrable.
\end{thm}
\begin{proof}
wlog assume $[a, b] = [0, 1]$.
Suppose the contrary. Let $f$ be non-integrable. This means that there exists some $\varepsilon$ such that for every dissection $\mathcal{D}$, $U_{\mathcal{D}} - L_{\mathcal{D}} > \varepsilon$. In particular, for every $n$, let $\mathcal{D}_n$ be the dissection $0, \frac{1}{n}, \frac{2}{n}, \cdots, \frac{n}{n}$.
Since $U_{\mathcal{D}_n} - L_{\mathcal{D}_n} > \varepsilon$, there exists some interval $\left[\frac{k}{n}, \frac{k + 1}{n}\right]$ in which $\sup f - \inf f > \varepsilon$. Suppose the supremum and infimum are attained at $x_n$ and $y_n$ respectively. Then we have $|x_n - y_n| < \frac{1}{n}$ and $f(x_n) - f(y_n) > \varepsilon$.
By Bolzano Weierstrass, $(x_n)$ has a convergent subsequence, say $(x_{n_i})$. Say $x_{n_i}\to x$. Since $|x_n - y_n| < \frac{1}{n}\to 0$, we must have $y_{n_i}\to x$. By continuity, we must have $f(x_{n_i}) \to f(x)$ and $f(y_{n_i}) \to f(x)$, but $f(x_{n_i})$ and $f(y_{n_i})$ are always apart by $\varepsilon$. Contradiction.
\end{proof}
With this result, we know that a lot of things are integrable, e.g.\ $e^{-x^2}$.
To prove this, we secretly used the property of \emph{uniform continuity}:
\begin{defi}[Uniform continuity*]
Let $A\subseteq \R$ and let $f: A\to \R$. Then $f$ is \emph{uniformly continuous} if
\[
(\forall \varepsilon)(\exists \delta > 0)(\forall x)(\forall y)\;|x - y| < \delta \Rightarrow |f(x) - f(y)| \leq \varepsilon.
\]
\end{defi}
This is different from regular continuity. Regular continuity says that at any point $x$, we can find a $\delta$ that works for this point. Uniform continuity says that we can find a $\delta$ that works for \emph{any} $x$.
It is easy to show that a uniformly continuous function is integrable, since by uniformly continuity, as long as the mesh of a dissection is sufficiently small, the difference between the upper sum and the lower sum can be arbitrarily small by uniform continuity. Thus to prove the above theorem, we just have to show that continuous functions on a closed bounded interval are uniformly continuous.
\begin{thm}[non-examinable]
Let $a < b$ and let $f: [a, b] \to \R$ be continuous. Then $f$ is uniformly continuous.
\end{thm}
\begin{proof}
Suppose that $f$ is not uniformly continuous. Then
\[
(\exists \varepsilon)(\forall \delta > 0)(\exists x)(\exists y)\;|x - y| < \delta \text{ and } |f(x) - f(y)| \geq \varepsilon.
\]
Therefore, we can find sequences $(x_n), (y_n)$ such that for every $n$, we have
\[
|x_n - y_n| \leq \frac{1}{n}\text{ and }|f(x_n) - f(y_n)| \geq \varepsilon.
\]
Then by Bolzano-Weierstrass theorem, we can find a subsequence $(x_{n_k})$ converging to some $x$. Since $|x_{n_k} - y_{n_k}| \leq \frac{1}{n_k}$, $y_{n_k}\to x$ as well. But $|f(x_{n_k}) - f(y_{n_k})| \geq \varepsilon$ for every $k$. So $f(x_{n_k})$ and $f(y_{n_k})$ cannot both converge to the same limit. So $f$ is not continuous at $x$.
\end{proof}
This proof is very similar to the proof that continuous functions are integrable. In fact, the proof that continuous functions are integrable is just a fuse of this proof and the (simple) proof that uniformly continuously functions are integrable.
\begin{thm}
Let $f: [a, b] \to \R$ be monotone. Then $f$ is Riemann integrable.
\end{thm}
Note that monotone functions need not be ``nice''. It can even have infinitely many discontinuities. For example, if $f: [0, 1] \to \R$ maps $x$ to the $1/(\text{first non-zero digit in the binary expansion of }x)$, with $f(0) = 0$.
\begin{proof}
let $\varepsilon > 0$. Let $\mathcal{D}$ be a dissection of mesh less than $\frac{\varepsilon}{f(b) - f(a)}$. Then
\begin{align*}
U_\mathcal{D} f - L_\mathcal{D}f &= \sum_{i = 1}^n (x_i - x_{i - 1})(f(x_i) - f(x_{i - 1}))\\
&\leq \frac{\varepsilon}{f(b) - f(a)} \sum_{i = 1}^n (f(x_i) - f(x_{i - 1}))\\
&= \varepsilon.\qedhere
\end{align*}
\end{proof}
Pictorially, we see that the difference between the upper and lower sums is total the area of the red rectangles.
\begin{center}
\begin{tikzpicture}
\draw [->] (-1, 0) -- (5, 0) node [right] {$x$};
\draw [->] (0, -0.5) -- (0, 5) node [above] {$y$};
\draw [domain=-1:5] plot (\x, {(\x + 1)*(\x + 1)/10 + 1});
\draw (0.5, 0) rectangle (1, 1.225);
\draw (1, 0) rectangle (1.5, 1.4);
\draw (1.5, 0) rectangle (2, 1.625);
\draw (2, 0) rectangle (2.5, 1.9);
\draw (2.5, 0) rectangle (3, 2.225);
\draw (3, 0) rectangle (3.5, 2.6);
\draw (3.5, 0) rectangle (4, 3.026);
\draw (4, 0) rectangle (4.5, 3.5);
\draw [red] (0.5, 1.225) rectangle (1, 1.4);
\draw [red] (1, 1.4) rectangle (1.5, 1.625);
\draw [red] (1.5, 1.625) rectangle (2, 1.9);
\draw [red] (2, 1.9) rectangle (2.5, 2.225);
\draw [red] (2.5, 2.225) rectangle (3, 2.6);
\draw [red] (3, 2.6) rectangle (3.5, 3.025);
\draw [red] (3.5, 3.025) rectangle (4, 3.5);
\draw [red] (4, 3.5) rectangle (4.5, 4.025);
\end{tikzpicture}
\end{center}
To calculate the total area, we can stack the red areas together to get something of width $\frac{\varepsilon}{f(b) - f(a)}$ and height $f(b) - f(a)$. So the total area is just $\varepsilon$.
\begin{lemma}
Let $a < b$ and let $f$ be a bounded function from $[a, b] \to \R$ that is continuous on $(a, b)$. Then $f$ is integrable.
\end{lemma}
An example where this would apply is $\int_0^1 \sin \frac{1}{x}$. It gets nasty near $x = 0$, but its ``nastiness'' is confined to $x = 0$ only. So as long as its nastiness is sufficiently contained, it would still be integrable.
The idea of the proof is to integrate from a point $x_1$ very near $a$ up to a point $x_{n - 1}$ very close to $b$. Since $f$ is bounded, the regions $[a, x_1]$ and $[x_{n - 1}, b]$ are small enough to not cause trouble.
\begin{proof}
Let $\varepsilon > 0$. Suppose that $|f(x)| \leq C$ for every $x\in [a, b]$. Let $x_0 = a$ and pick $x_1$ such that $x_1 - x_0 < \frac{\varepsilon}{8C}$. Also choose $z$ between $x_1$ and $b$ such that $b - z < \frac{\varepsilon}{8C}$.
Then $f$ is continuous $[x_1, z]$. Therefore it is integrable on $[x_1, z]$. So we can find a dissection $\mathcal{D}'$ with points $x_1 < x_2 < \cdots < x_{n - 1} = z$ such that
\[
U_{\mathcal{D}'}f - L_{\mathcal{D}'}f < \frac{\varepsilon}{2}.
\]
Let $\mathcal{D}$ be the dissection $a = x_0 < x_1 < \cdots < x_n = b$. Then
\[
U_\mathcal{D} f - L_\mathcal{D} f < \frac{\varepsilon}{8C}\cdot 2C + \frac{\varepsilon}{2} + \frac{\varepsilon}{8C}\cdot 2C = \varepsilon.
\]
So done by Riemann integrability criterion.
\end{proof}
\begin{eg}\leavevmode
\begin{itemize}
\item $f(x) =
\begin{cases}
\sin \frac{1}{x}& x \not = 0\\
0 & x = 0
\end{cases}$ defined on $[-1, 1]$ is integrable.
\item $g(x) =
\begin{cases}
x & x \leq 1\\
x^2 + 1 & x > 1
\end{cases}$ defined on $[0, 1]$ is integrable.
\end{itemize}
\end{eg}
\begin{cor}
Every piecewise continuous and bounded function on $[a, b]$ is integrable.
\end{cor}
\begin{proof}
Partition $[a, b]$ into intervals $I_1, \cdots, I_k$, on each of which $f$ is (bounded and) continuous. Hence for every $I_j$ with end points $x_{j - 1}$, $x_j$, $f$ is integrable on $[x_{j - 1}, x_j]$ (which may not equal $I_j$, e.g.\ $I_j$ could be $[x_{j - 1}, x_j)$). But then by the additivity property of integration, we get that $f$ is integrable on $[a, b]$
\end{proof}
We defined Riemann integration in a very general way --- we allowed \emph{arbitrary} dissections, and took the extrema over all possible dissection. Is it possible to just consider some particular nice dissections instead? Perhaps unsurprisingly, yes! It's just that we opt to define it the general way so that we can easily talk about things like least common refinements.
\begin{lemma}
Let $f: [a, b] \to \R$ be Riemann integrable, and for each $n$, let $\mathcal{D}_n$ be the dissection $a = x_0 < x_1 < \cdots < x_n = b$, where $x_i = a + \frac{i(b - a)}{n}$ for each $i$. Then
\[
U_{\mathcal{D}_n}f \to \int_a^b f(x)\;\d x
\]
and
\[
L_{\mathcal{D}_n}f \to \int_a^b f(x)\;\d x.
\]
\end{lemma}
\begin{proof}
Let $\varepsilon > 0$. We need to find an $N$. The only thing we know is that $f$ is Riemann integrable, so we use it:
Since $f$ is integrable, there is a dissection $\mathcal{D}$, say $u_0 < u_1 < \cdots < u_m$, such that
\[
U_\mathcal{D} f - \int_a^b f(x)\;\d x < \frac{\varepsilon}{2}.
\]
We also know that $f$ is bounded. Let $C$ be such that $|f(x)| \leq C$.
For any $n$, let $\mathcal{D}'$ be the least common refinement of $\mathcal{D}_n$ and $\mathcal{D}$. Then
\[
U_{\mathcal{D}'}f \leq U_\mathcal{D} f.
\]
Also, the sums $U_{\mathcal{D}_n}f$ and $U_\mathcal{D'}f$ are the same, except that at most $m$ of the subintervals $[x_{i - 1}, x_i]$ are subdivided in $\mathcal{D}'$.
For each interval that gets chopped up, the upper sum decreases by at most $\frac{b - a}{n}\cdot 2C$. Therefore
\[
U_{\mathcal{D}_n}f - U_{\mathcal{D}'}f \leq \frac{b - a}{n}2C\cdot m.
\]
Pick $n$ such that $2Cm(b - a)/n < \frac{\varepsilon}{2}$. Then
\[
U_{\mathcal{D}_n} f - U_\mathcal{D}f < \frac{\varepsilon}{2}.
\]
So
\[
U_{\mathcal{D}_n}f - \int_a^b f(x)\;\d x < \varepsilon.
\]
This is true whenever $n > \frac{4C(b - a)m}{\varepsilon}$. Since we also have $U_{\mathcal{D}_n} f \geq \int_a^b f(x)\;\d x$, therefore
\[
U_{\mathcal{D}_n}f \to \int_a^b f(x)\;\d x.
\]
The proof for lower sums is similar.
\end{proof}
For convenience, we define the following:
\begin{notation}
If $b > a$, we define
\[
\int_b^a f(x)\;\d x = -\int_a^b f(x)\;\d x.
\]
\end{notation}
We now prove that the fundamental theorem of calculus, which says that integration is the reverse of differentiation.
\begin{thm}[Fundamental theorem of calculus, part 1]
Let $f: [a, b]\to \R$ be continuous, and for $x\in [a, b]$, define
\[
F(x) = \int_a^x f(t)\;\d t.
\]
Then $F$ is differentiable and $F'(x) = f(x)$ for every $x$.
\end{thm}
\begin{proof}
\[
\frac{F(x + h) - F(x)}{h} = \frac{1}{h}\int_x^{x + h}f(t)\;\d t
\]
Let $\varepsilon > 0$. Since $f$ is continuous, at $x$, then there exists $\delta$ such that $|y - x| < \delta$ implies $|f(y) - f(x)| < \varepsilon$.
If $|h| < \delta$, then
\begin{align*}
\left|\frac{1}{h}\int_x^{x + h}f(t) \;\d t - f(x)\right| &= \left|\frac{1}{h}\int_x^{x + h}(f(t) - f(x))\;\d t\right|\\
&\leq \frac{1}{|h|}\left|\int_x^{x + h}|f(t) - f(x)|\;\d t\right|\\
&\leq \frac{\varepsilon|h|}{|h|}\\
&= \varepsilon.\qedhere
\end{align*}
\end{proof}
\begin{cor}
If $f$ is continuously differentiable on $[a, b]$, then
\[
\int_a^b f'(t)\;\d t = f(b) - f(a).
\]
\end{cor}
\begin{proof}
Let
\[
g(x) = \int_a^x f'(t)\;\d t.
\]
Then
\[
g'(x) = f'(x) = \frac{\d }{\d x}(f(x) - f(a)).
\]
Since $g'(x) - f'(x) = 0$, $g(x) - f(x)$ must be a constant function by the mean value theorem. We also know that
\[
g(a) = 0 = f(a) - f(a)
\]
So we must have $g(x) = f(x) - f(a)$ for every $x$, and in particular, for $x = b$.
\end{proof}
\begin{thm}[Fundamental theorem of calculus, part 2]
Let $f: [a, b] \to \R$ be a differentiable function, and suppose that $f'$ is integrable. Then
\[
\int_a^b f'(t)\;\d t = f(b) - f(a).
\]
\end{thm}
Note that this is a stronger result than the corollary above, since it does not require that $f'$ is continuous.
\begin{proof}
Let $\mathcal{D}$ be a dissection $x_0 < x_1 < \cdots < x_n$. We want to make use of this dissection. So write
\[
f(b) - f(a) = \sum_{i = 1}^n (f(x_i) - f(x_{i - 1})).
\]
For each $i$, there exists $u_i\in (x_{i - 1}, x_i)$ such that $f(x_i) - f(x_{i - 1j}) = (x_i - x_{i - 1})f'(u_i)$ by the mean value theorem. So
\[
f(b) - f(a) = \sum_{i = 1}^n (x_i - x_{i - 1})f'(u_i).
\]
We know that $f'(u_i)$ is somewhere between $\sup\limits_{x\in[x_i, x_{i - 1}]}f'(x)$ and $\inf\limits_{x\in[x_i, x_{i - 1}]}f'(x)$ by definition. Therefore
\[
L_\mathcal{D} f' \leq f(b) - f(a) \leq U_\mathcal{D} f'.
\]
Since $f'$ is integrable and $\mathcal{D}$ was arbitrary, $L_\mathcal{D}f'$ and $U_\mathcal{D}f'$ can both get arbitrarily close to $\int_a^b f'(t)\;\d t$. So
\[
f(b) - f(a) = \int_a^b f'(t)\;\d t.\qedhere
\]
\end{proof}
Note that the condition that $f'$ is integrable is essential. It is possible to find a differentiable function whose derivative is not integrable! You will be asked to find it in the example sheet.
Using the fundamental theorem of calculus, we can easily prove integration by parts:
\begin{thm}[Integration by parts]
Let $f, g:[a, b]\to \R$ be integrable such that everything below exists. Then
\[
\int_a^b f(x)g'(x)\;\d x = f(b)g(b) - f(a)g(a) - \int_a^b f'(x)g(x)\;\d x.
\]
\end{thm}
\begin{proof}
By the fundamental theorem of calculus,
\[
\int_a^b (f(x)g'(x) + f'(x)g(x))\;\d x = \int_a^b(fg)'(x)\;\d x = f(b)g(b) - f(a)g(a).
\]
The result follows after rearrangement.
\end{proof}
Recall that when we first had Taylor's theorem, we said it had the Lagrange form of the remainder. There are many other forms of the remainder term. Here we will look at the integral form:
\begin{thm}[Taylor's theorem with the integral form of the remainder]
Let $f$ be $n + 1$ times differentiable on $[a, b]$ with with $f^{(n + 1)}$ continuous. Then
\begin{align*}
f(b) &= f(a) + (b - a)f'(a) + \frac{(b - a)^2}{2!}f^{(2)}(a) + \cdots \\
&+ \frac{(b - a)^n}{n!}f^{(n)}(a) + \int_a^b \frac{(b - t)^n}{n!}f^{(n + 1)}(t)\;\d t.
\end{align*}
\end{thm}
\begin{proof}
Induction on $n$.
When $n = 0$, the theorem says
\[
f(b) - f(a) = \int_a^b f'(t)\;\d t.
\]
which is true by the fundamental theorem of calculus.
Now observe that
\begin{align*}
\int_a^b \frac{(b - t)^n}{n!}f^{(n + 1)}(t)\;\d t ={}& \left[\frac{-(b - t)^{n + 1}}{(n + 1)!}f^{(n + 1)}(t)\right]_a^b\\
&+ \int_a^b \frac{(b - t)^{n + 1}}{(n + 1)!}f^{(n + 1)}(t)\;\d t \\
={}& \frac{(b - a)^{n + 1}}{(n + 1)!} f^{(n + 1)}(a) + \int_a^b \frac{(b - t)^{n + 1}}{(n + 1)!}f^{(n + 2)}(t)\;\d t.
\end{align*}
So the result follows by induction.
\end{proof}
Note that the form of the integral remainder is rather weird and unexpected. How could we have come up with it? We might start with the fundamental theorem of algebra and integrate by parts. The first attempt would be to integrate $1$ to $t$ and differentiate $f'(t)$ to $f^{(2)}(t)$. So we have
\begin{align*}
f(b) &= f(a) + \int_a^b f'(t)\;\d t\\
&= f(a) + [tf'(t)]_a^b - \int_a^b tf^{(2)}(t)\;\d t\\
&= f(a) + bf'(b) - af'(a) - \int_a^b tf^{(2)}(t)\;\d t\\
\intertext{We want something in the form $(b - a)f'(a)$, so we take that out and see what we are left with.}
&= f(a) + (b - a)f'(a) + b(f'(b) - f'(a)) - \int_a^b tf^{(2)}(t)\;\d t\\
\intertext{Then we note that $f'(b) - f'(a) = \int_a^b f^{(2)}(t)\;\d t$. So we have}
&= f(a) + (b - a)f'(a) + \int_a^b (b - t)f^{(2)}(t)\;\d t.
\end{align*}
Then we can see that the right thing to integrate is $(b - t)$ and continue to obtain the result.
\begin{thm}[Integration by substitution]
Let $f: [a, b] \to \R$ be continuous. Let $g: [u, v] \to \R$ be continuously differentiable, and suppose that $g(u) = a, g(v) = b$, and $f$ is defined everywhere on $g([u, v])$ (and still continuous). Then
\[
\int_a^b f(x)\;\d x = \int_u^v f(g(t))g'(t)\;\d t.
\]
\end{thm}
\begin{proof}
By the fundamental theorem of calculus, $f$ has an anti-derivative $F$ defined on $g([u, v])$. Then
\begin{align*}
\int_u^v f(g(t))g'(t) \;\d t &= \int_u^v F'(g(t))g'(t)\;\d t \\
&= \int_u^v (F\circ g)'(t)\;\d t \\
&= F\circ g(v) - F\circ g(u)\\
&= F(b) - F(a)\\
&= \int_a^b f(x)\;\d x.\qedhere
\end{align*}
\end{proof}
We can think of ``integration by parts'' as what you get by integrating the product rule, and ``integration by substitution'' as what you get by integrating the chain rule.
\subsection{Improper integrals}
It is sometimes sensible to talk about integrals of unbounded functions or integrating to infinity. But we have to be careful and write things down nicely.
\begin{defi}[Improper integral]
Suppose that we have a function $f: [a, b] \to \R$ such that, for every $\varepsilon > 0$, $f$ is integrable on $[a + \varepsilon, b]$ and $\lim\limits_{\varepsilon \to 0}\int_{a + \varepsilon}^b f(x)\;\d x$ exists. Then we define the improper integral
\[
\int_a^bf(x)\;\d x \text{ to be } \lim_{\varepsilon \to 0}\int_{a + \varepsilon}^b f(x)\;\d x.
\]
even if the Riemann integral does not exist.
We can do similarly for $[a, b - \varepsilon]$, or integral to infinity:
\[
\int_a^\infty f(x)\;\d x = \lim_{b \to \infty} \int_a^b f(x)\;\d x.
\]
when it exists.
\end{defi}
\begin{eg}
\[
\int_\varepsilon^1 x^{-1/2}\;\d x = \left[2x^{-1/2}\right]^1_\varepsilon = 2 - 2\varepsilon^{1/2} \to 2.
\]
So
\[
\int_0^1 x^{-1/2}\;\d x = 2,
\]
even though $x^{-1/2}$ is unbounded on $[0, 1]$.
Note that officially we are required to make $f(x) = x^{-1/2}$ a function with domain $[0, 1]$. So we can assign $f(0) = \pi$, or any number, since it doesn't matter.
\end{eg}
\begin{eg}
\[
\int_1^x \frac{1}{t^2}\;\d t = \left[-\frac{1}{t}\right]_1^x = 1 - \frac{1}{x} \to 1\text{ as }x\to \infty
\]
by the fundamental theorem of calculus. So
\[
\int_1^{\infty}\frac{1}{x^2}\;\d x = 1.
\]
\end{eg}
Finally, we can prove the integral test, whose proof we omitted when we first began.
\begin{thm}[Integral test]
Let $f: [1, \infty] \to \R$ be a decreasing non-negative function. Then $\sum_{n = 1}^\infty f(n)$ converges iff $\int_1^\infty f(x)\;\d x < \infty$.
\end{thm}
\begin{proof}
We have
\[
\int_n^{n + 1}f(x)\;\d x \leq f(n) \leq \int_{n -1}^n f(x)\;\d x,
\]
since $f$ is decreasing (the right hand inequality is valid only for $n\geq 2$). It follows that
\[
\int_1^{N + 1}f(x)\;\d x \leq \sum_{n = 1}^N f(n) \leq \int_1^N f(x)\;\d x + f(1)
\]
So if the integral exists, then $\sum f(n)$ is increasing and bounded above by $\int_1^\infty f(x)\;\d x$, so converges.
If the integral does not exist, then $\int_1^N f(x)\;\d x$ is unbounded. Then $\sum_{n = 1}^N f(n)$ is unbounded, hence does not converge.
\end{proof}
\begin{eg}
Since $\int_1^x \frac{1}{t^2}\;\d t < \infty$, it follows that $\sum_{n = 1}^\infty \frac{1}{n^2}$ converges.
\end{eg}
\end{document}
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The key $k_2$ would be stored at $v$'s parent in this case. This is done in order to maintain the ordering of the keys within the tree. By storing $k_2$ at $v$'s parent, all of the children would still be ordered such that the children to the left of the key are less than the key and those to the right of the key are greater than the key.
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\begin{document}
\def\RunningAuthor{Ahmed Abdul jabbar jaloob Aljanaby et al.,}
\firstPage{1847}
\articleType{Original Article}
\receivedDate{15.03.2019}
\acceptedDate{24.06.2019}
\revisedDate{20.06.2019}
\journalVolume{10}
\journalIssue{3}
\journalDoi{ijrps.v10i3.1382}
\copyrightYear{2019}
\def\authorCount{4}
\def\affCount{2}
\def\journalTitle{International Journal of Research in Pharmaceutical Sciences}
\title{Synthesis and characterization of new imidazole azo ligand with some of transition metal ions, and their biological effect on two pathogenic bacteria of burn patients.}
\author{Israa N. Witwit\textsuperscript{1},
Husham M. Mubark\textsuperscript{1},
Hutham Mahmood Yousif Al-Labban\textsuperscript{1},
Ahmed abduljabbar jaloob aljanaby\textsuperscript{*}\textsuperscript{2}~\\[5pt]{\textsuperscript{1}Department of Chemistry\unskip, University of Kufa, Factualy of Science, Najaf, Iraq}~\\{\textsuperscript{2}Department of Biology\unskip, University of Kufa, Faculty of Science, Iraq}}
\begin{abstract}
New imidazole azo ligand (DPIDA) was prepared by coupling reaction between 4,5-di phenyl imidazole and N1,N1-dimethylbenzene1,4-diamine di hydrochloride and studied the complexation of this ligand with Mn(II) , Co(II) , Ni(II) , Cu(II) , Zn(II) , Cd(II) , and Hg(II) ions , The free ligand and it's complexes characterized by Mass, \ensuremath{^{1}}HNMR, IR, UV-Vis. , and molar conductivity that indicated the octahedral geometry of them with a bidentate ligand which coordinated from (N3) atom of imidazole ring and one nitrogen atom of azo group.Biological activity of ligand (DPIDA) and it's complexes tested against two multi-drug resistant aerobic pathogenic bacteria isolated from patients with a burn. Three concentrations were selected (50, 100, 150) mg/ml for each crud synthesized derivative compounds. The derivative compound (4,5-diphenyl imidazole) with concentration 150mg/ml had an excellent antibacterial effect against \textit{Staphylococcus aureus } and\textit{ Pseudomonas aeruginosa} with inhibition zone 21.83 \ensuremath{\pm} 0.1764 mm and 24.30 \ensuremath{\pm} 0.4163 mm respectively.
\end{abstract}\def\keywordstitle{Keywords}
\begin{keywords}imidazole azo,\newline metal complexes,\newline burn patients,\newline biological activity,\newline Staphylococcus aureus,\newline Pseudomonas aeruginosa
\end{keywords}
\twocolumn[ \maketitle {\printKwdAbsBox}]
\makeatletter\textsuperscript{*}Corresponding Author\par Name:\ Ahmed abduljabbar jaloob aljanaby~\\ Phone:\ ~\\ Email:\ [email protected]
\par\vspace*{-11pt}\hrulefill\par{\fontsize{12}{14}\selectfont ISSN: 0975-7538}\par%
\textsc{DOI:}\ \href{https://doi.org/10.26452/\@journalDoi}{\textcolor{blue}{\underline{\smash{https://doi.org/10.26452/\@journalDoi}}}}\par%
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\section{Introduction}
Heterocyclic azo compounds have a lot of scientific attention especially in the last five decades because of their applications in different of applied and academic fields such as Analytical reagents\unskip~\citep{549750:12574443}, antibacterial\unskip~\citep{549750:12574448}, antifungal \unskip~\citep{549750:12574447} anticancer \unskip~\citep{549750:12574445} and Optical electrical switching of liquid crystals\unskip~\citep{549750:12574444}. Imidazole azo ligands considered as an essential type of heterocyclic azo ligands chiefly in coordination chemistry due to their ability to form stable complexes with metal ions in various of oxidation states \unskip~\citep{549750:12574434}, formation of stable five-member ring with each ion through (N3) atom of imidazole ring and one of nitrogen atoms of azo group\unskip~\citep{549750:12574440}. As well as the contribution of imidazole molecule in preparation of many ligands which have \ensuremath{\pi } {\textendash} conjugated system that increases their stability and follows the colour change of them before and after the coordination with metal ions \unskip~\citep{549750:12574441}. The goal of this research is representing by preparation and characterization of new ligand as a derivative of 4,5-diphenylimidazole, studying it's coordination behavior with Mn(II) , Co(II) , Ni(II) , Cu(II) , Zn(II) , Cd(II) , and Hg(II), and experienced their biological activity against two types of multi-drug resistant aerobic pathogenic bacteria have isolated from burn patients.
\section{Materials and Methods}
\textbf{Chemicals and Instruments }
All chemicals and solvents were equipped with high purity from Sigma Aldrich, BDH and Merck companies. Mass Spectrum was measured using AB SCIEX 3200 QTRAP Mass analyzer , FT-IR carried out by Shimadzu FTIR8400 using KBr disks from (400-4000)cm\ensuremath{^{-,}} Electronic spectrum measured by Shimadzu UV-1650 UV-Vis Spectrophotometer , The element analysis performed on Costech ECS Elemental 4010, magnetic measurements of prepared complexes recorded by Balance Magnetic Susceptibility Model {\textendash}M.S.B Auto , Molar conductivity menstruated via 720(WTW), and \ensuremath{^{1}}HNMR carried out by Bruker Avance-111 300 MHz NMR Spectrometer.
\textbf{Preparation of (DPIDA) ligand} \textbf{\space }
Two and thirty-one gram of N1, N1-dimethylbenzene1,4-diamine dihydrochloride was dissolved in twenty-five ml of distilled water than one ml of hydrochloric acid added gradually to this solution which cooled in ice bath 0-5 \ensuremath{^{o}}C, the formation of diazonium salt occurred by addition the solution of sodium nitrate which prepared by dissolved 0.70 gm of it in 10 ml of distilled water drop by drop with stirring, This solution leaved in the ice bath for 30 minute then coupled with alcoholic solution of 4,5- diphenyl imidazole which prepared by dissolving 2.21 gm of imidazole derivative and 0.44 gm of sodium hydroxide in 25 ml of ethanol, orange precipitate was appeared after the completing of addition, filtered and dried then recrestalysid from ethanol yield percentage 71 \% as shown in scheme 1.
\bgroup
\fixFloatSize{images/ed4545e6-677a-4b86-bdb3-9a7c420ff00b-upicture1.png}
\begin{scheme*}[!htbp]
\centering \makeatletter\IfFileExists{images/ed4545e6-677a-4b86-bdb3-9a7c420ff00b-upicture1.png}{\includegraphics{images/ed4545e6-677a-4b86-bdb3-9a7c420ff00b-upicture1.png}}{}
\makeatother
\caption{\boldmath {Preparation of (DPIDA) ligand}}
\label{f-0351e154a898}
\end{scheme*}
\egroup
\textbf{Preparation of complexes (general method)}
All complexes were prepared with mole ratio 1:2 (metal: ligand) by mixing 1 mmole of metal chlorides in twenty-five ml of distilled water with 2 m moles of (DPIDA) in twenty-five ml of ethanol with stirring until the precipitations of the complexes were appeared, filtered and dried the yield percentage of them were shown in Table~\ref{tw-8bdbc95d26cb}.
\begin{table*}[!htbp]
\caption{\boldmath {Some of the physicochemical properties of (DPIDA) ligand and it's complexes} }
\label{tw-8bdbc95d26cb}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.23519999999999992\linewidth-2\tabcolsep}p{\dimexpr.07400000000000001\linewidth-2\tabcolsep}p{\dimexpr.0681\linewidth-2\tabcolsep}p{\dimexpr.10969999999999997\linewidth-2\tabcolsep}p{\dimexpr.10440000000000005\linewidth-2\tabcolsep}p{\dimexpr.126\linewidth-2\tabcolsep}p{\dimexpr.10089999999999998\linewidth-2\tabcolsep}p{\dimexpr.18169999999999998\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}{\multirow{2}{*}{Compound (Empirical Formula)}} & {\multirow{2}{*}{Mwt}} & {\multirow{2}{*}{Yield (\%)}} & \multicolumn{4}{p{\dimexpr(.441\linewidth-2\tabcolsep)}}{\cAlignHack Elemental Analysis Calcl. (Found)} & {\multirow{2}{*}{m.p (oC)}}\\
\rowcolor{kwdboxcolor} & & & C\% & H\% & N\% & M\% & \\
\tblmidrule
(DPIDA) \mbox{}\protect\newline (C\ensuremath{_{23}}H\ensuremath{_{21}}N\ensuremath{_{5}}) &
367.46 &
71 &
75.18 \mbox{}\protect\newline (75.20) &
5.76 \mbox{}\protect\newline (5.73) &
19.06 \mbox{}\protect\newline (19.08) &
------ &
232-234\\
{}[Mn(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] \mbox{}\protect\newline (C\ensuremath{_{46}}H\ensuremath{_{42}}Cl\ensuremath{_{2}}N\ensuremath{_{10}}Mn) &
860.75 &
68 &
64.19 \mbox{}\protect\newline (64.22) &
4.92 \mbox{}\protect\newline (4.94) &
16.27 \mbox{}\protect\newline (16.25) &
6.38 \mbox{}\protect\newline (6.40) &
310-312\\
{}[Co(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] \mbox{}\protect\newline (C\ensuremath{_{46}}H\ensuremath{_{42}}Cl\ensuremath{_{2}}N\ensuremath{_{10}}Co) &
864.75 &
75 &
63.89 \mbox{}\protect\newline (63.00) &
4.90 \mbox{}\protect\newline (4.90) &
16.20 \mbox{}\protect\newline (16.23) &
6.82 \mbox{}\protect\newline (6.79) &
325-328\\
{}[Ni(DPIDA)\ensuremath{_{2}}Cl2] \mbox{}\protect\newline (C\ensuremath{_{46}}H\ensuremath{_{42}}Cl\ensuremath{_{2}}N\ensuremath{_{10}}Ni) &
864.51 &
78 &
63.91 \mbox{}\protect\newline (63.95) &
4.90 \mbox{}\protect\newline (4.92) &
16.20 \mbox{}\protect\newline (16.20) &
6.79 \mbox{}\protect\newline (6.82) &
332-334\\
{}[Cu(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] \mbox{}\protect\newline (C\ensuremath{_{46}}H\ensuremath{_{42}}Cl\ensuremath{_{2}}N\ensuremath{_{10}}Cu) &
869.36 &
72 &
63.55 \mbox{}\protect\newline (63.52) &
4.87 \mbox{}\protect\newline (4.90) &
16.11 \mbox{}\protect\newline (16.12) &
7.31 \mbox{}\protect\newline (7.34) &
346-348\\
{}[Zn(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] \mbox{}\protect\newline (C\ensuremath{_{46}}H\ensuremath{_{42}}Cl\ensuremath{_{2}}N\ensuremath{_{10}}Zn) &
871.19 &
74 &
63.42 \mbox{}\protect\newline (63.42) &
4.86 \mbox{}\protect\newline (4.88) &
16.08 \mbox{}\protect\newline (16.05) &
7.50 \mbox{}\protect\newline (7.52) &
353-355\\
{}[Cd(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] \mbox{}\protect\newline (C\ensuremath{_{46}}H\ensuremath{_{42}}Cl\ensuremath{_{2}}N\ensuremath{_{10}}Cd) &
918.22 &
71 &
60.17 \mbox{}\protect\newline (60.18) &
4.61 \mbox{}\protect\newline (4.60) &
15.25 \mbox{}\protect\newline (15.26) &
12.24 \mbox{}\protect\newline (12.30) &
364-367\\
{}[Hg(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] \mbox{}\protect\newline (C\ensuremath{_{46}}H\ensuremath{_{42}}Cl\ensuremath{_{2}}N\ensuremath{_{10}}Hg) &
1006.40 &
84 &
54.90 \mbox{}\protect\newline (54.93) &
4.21 \mbox{}\protect\newline (4.19) &
13.92 \mbox{}\protect\newline (14.00) &
19.93 \mbox{}\protect\newline (19.88) &
375-377\\
\tblbottomrule
\end{tabulary}\par
\end{table*}
\textbf{Biological Activity}
Biological activity testing was done to detect the antibacterial activity of four synthesized derivatives compounds against two multi-drug resistant aerobic pathogenic bacteria isolated from patients with burn infection; Staphylococcus aureus (\textit{S.aureus}) is a gram-positive bacteria and pseudomonas aeruginosa (\textit{P.aeruginosa}) as a gram harmful bacteria. The two pathogenic bacteria were provided with kindly from the university of Kufa, Faculty of science, department of microbiology, Iraq. Antibacterial activity test was done according to the agar well diffusion method\unskip~\citep{549750:12574436,549750:12574437}. Three concentrations were selected (50, 100, 150) mg/ml for each crud synthesized derivative compounds. Four wells were made by crock-poorer (Oxoid, UK) in Muller-Hinton agar surface (Oxoid, UK) and swabbed with two pathogenic bacteria with turbidity according to 0.5 McFarland tube. Fifty \textmu l of each dilution was transferred to each well and left at (20)\ensuremath{^{o}}C for 3 hours and incubated at (37)\ensuremath{^{o}}C for 24 hours. Four replicates were done for each test. The inhibition zone around each well was measured in millimetres\unskip~\citep{549750:12574433} ;\unskip~\citep{549750:12574438}. \textbf{\space }
\textbf{Statically analysis }
Graph pad prism V.6 windows soft were was used in statically analysis to compare between diameters of inhibition zone (mm) according to T-test. P-value \textless\ 0.05 was considered indicative of statistically significant \unskip~\citep{549750:12574433}. \textbf{\space }
\section{Results and Discussion}
\ensuremath{^{1}}HNMR spectra of free ligand (DPIDA) in (d\ensuremath{^{6}} DMSO) inhibit a singlet signal in (3.03) ppm due to the protons of (N-CH\ensuremath{_{3}}) groups, whilst the siglet signal of (N-H) for imidazole ring \unskip~\citep{549750:12575700} appeared in (12.63) ppm, this spectrum confirms the number of protons in the molecular structure, as shown in Figure~\ref{f-a887ccebac23}.
\bgroup
\fixFloatSize{images/eb236737-2283-4afc-bc31-6f46f70f41b1-upicture1.png}
\begin{figure*}[!htbp]
\centering \makeatletter\IfFileExists{images/eb236737-2283-4afc-bc31-6f46f70f41b1-upicture1.png}{\includegraphics{images/eb236737-2283-4afc-bc31-6f46f70f41b1-upicture1.png}}{}
\makeatother
\caption{\boldmath {\ensuremath{^{1}}HNMR spectrum of (DPIDA) ligand in d\ensuremath{^{6}}DMSOsolvent}}
\label{f-a887ccebac23}
\end{figure*}
\egroup
Mass spectrum of (DPIDA) ligand showed molecular ion peak(M+1) at m/e (368) , the initial fragmentation started by losing (-N\ensuremath{_{2}}) molecule at m/e (340) , while the base peak appeared at ( e/z= 221) corresponding to 4,5- diphenylimidazole fragment ( C\ensuremath{_{15}}H\ensuremath{_{12}}N\ensuremath{_{2}})\unskip~\citep{549750:12575663}. The spectrum of Mn(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}} complex exhibited molecular peak at (e/z = 860) that affirmed the molecular weight of this complex, the fragmentation also started by losing the nitrogen's of the two coordinated azo ligands at (e/z=804) and continued to the last step which showed the fragment of 4,5- diphenylimidazole as base peak, theFigures~\ref{f-acfa06e0b567} and~\ref{f-63bee69cc0f1}andSchemes~\ref{f-a52e1b7f8b94} and~\ref{f-bd447cfb55a4} illustrated the fragmentation of ligand, and it's complicated.
\bgroup
\fixFloatSize{images/35002862-5306-4ad2-a0ee-906a01cfd7a6-upicture2.png}
\begin{figure*}[!htbp]
\centering \makeatletter\IfFileExists{images/35002862-5306-4ad2-a0ee-906a01cfd7a6-upicture2.png}{\includegraphics{images/35002862-5306-4ad2-a0ee-906a01cfd7a6-upicture2.png}}{}
\makeatother
\caption{\boldmath {Mass spectrum of (DPIDA) ligan}}
\label{f-acfa06e0b567}
\end{figure*}
\egroup
\bgroup
\fixFloatSize{images/b437d4e7-2e8a-4475-bf2a-6a422c4c8d95-upicture2.png}
\begin{scheme*}[!htbp]
\centering \makeatletter\IfFileExists{images/b437d4e7-2e8a-4475-bf2a-6a422c4c8d95-upicture2.png}{\includegraphics{images/b437d4e7-2e8a-4475-bf2a-6a422c4c8d95-upicture2.png}}{}
\makeatother
\caption{\boldmath {Mass Fragmentation of (DPIDA) ligand.}}
\label{f-a52e1b7f8b94}
\end{scheme*}
\egroup
\bgroup
\fixFloatSize{images/06221420-6629-4d54-80ca-139602bc008c-upicture3.png}
\begin{figure*}[!htbp]
\centering \makeatletter\IfFileExists{images/06221420-6629-4d54-80ca-139602bc008c-upicture3.png}{\includegraphics{images/06221420-6629-4d54-80ca-139602bc008c-upicture3.png}}{}
\makeatother
\caption{\boldmath {Mass spectrum of Mn (II) complex}}
\label{f-63bee69cc0f1}
\end{figure*}
\egroup
\bgroup
\fixFloatSize{images/a8cb0b44-95d4-4f59-b276-9609af662c91-upicture4.png}
\begin{scheme*}[!htbp]
\centering \makeatletter\IfFileExists{images/a8cb0b44-95d4-4f59-b276-9609af662c91-upicture4.png}{\includegraphics{images/a8cb0b44-95d4-4f59-b276-9609af662c91-upicture4.png}}{}
\makeatother
\caption{\boldmath {Mass Fragmentation of Mn (II) Complex}}
\label{f-bd447cfb55a4}
\end{scheme*}
\egroup
\begin{table*}[!htbp]
\caption{\boldmath {Molar Conductivity, Magnetic Susbtibility and Electronic Transitions of ligands and their complexes.} }
\label{tw-3ba28fa62103}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.18349999999999998\linewidth-2\tabcolsep}p{\dimexpr.0846\linewidth-2\tabcolsep}p{\dimexpr.10150000000000002\linewidth-2\tabcolsep}p{\dimexpr.11009999999999998\linewidth-2\tabcolsep}p{\dimexpr.1264\linewidth-2\tabcolsep}p{\dimexpr.13690000000000005\linewidth-2\tabcolsep}p{\dimexpr.257\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}Compound & \multicolumn{2}{p{\dimexpr(.18610000000000003\linewidth-2\tabcolsep)}}{\cAlignHack Molar Conductivity \mbox{}\protect\newline S.Cm\ensuremath{^{2}}.mole-} & \textmu .eff. \mbox{}\protect\newline (B.M.) & \ensuremath{\lambda } max \mbox{}\protect\newline (nm) \mbox{}\protect\newline & Transitions & Geometry\\
\rowcolor{kwdboxcolor} & DMF & DMSO & & & & \\
\tblmidrule
(DPIDA) &
----- &
------- &
----- &
284, 257, 238 \mbox{}\protect\newline 466 &
\ensuremath{\pi }-\ensuremath{\pi }* \mbox{}\protect\newline C.T &
-----\\
{}[Mn(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
23.6 &
20.6 &
5.74 &
281,242, 304 \mbox{}\protect\newline 472 \mbox{}\protect\newline &
\ensuremath{\pi }-\ensuremath{\pi }* \mbox{}\protect\newline MLCT \mbox{}\protect\newline &
Octahedral\\
{}[Co(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
21.8 &
18.3 &
4.70 &
280, 250, 230 \mbox{}\protect\newline 536 &
\ensuremath{\pi }-\ensuremath{\pi }* \mbox{}\protect\newline MLCT \mbox{}\protect\newline &
Octahedral\\
{}[Ni(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
21.7 &
18.2 &
2.84 &
281, 254 , 230 \mbox{}\protect\newline 550 &
\ensuremath{\pi }-\ensuremath{\pi }* \mbox{}\protect\newline MLCT \mbox{}\protect\newline &
Octahedral\\
{}[Cu(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
20.5 &
17.1 &
1.73 &
278, 254, 232 \mbox{}\protect\newline 536 &
\ensuremath{\pi }-\ensuremath{\pi }* \mbox{}\protect\newline MLCT \mbox{}\protect\newline &
Distorted Octahedral\\
{}[Zn(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
18.3 &
14.6 &
Dia &
274,250,232 \mbox{}\protect\newline 485 &
\ensuremath{\pi }-\ensuremath{\pi }* \mbox{}\protect\newline MLCT \mbox{}\protect\newline &
Octahedral\\
{}[Cd(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
17.8 &
13.2 &
Dia &
244, 276 , 308 \mbox{}\protect\newline 468 &
\ensuremath{\pi }-\ensuremath{\pi }* \mbox{}\protect\newline MLCT \mbox{}\protect\newline &
Octahedral\\
{}[Hg(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
15.4 &
11.7 &
Dia &
286 , 254, 230 \mbox{}\protect\newline 504 &
\ensuremath{\pi }-\ensuremath{\pi }* \mbox{}\protect\newline MLCT \mbox{}\protect\newline &
Octahedral\\
\tblbottomrule
\end{tabulary}\par
\end{table*}
Uv-Vis spectrum of free ligand (DPIDA) show up three bands at 284, 257, 238 nm which attributed to (\ensuremath{\pi } {\textendash} \ensuremath{\pi }*) transitions of aromatic rings which shifted to higher wavelengths with little changes of values in complexes spectrums , while the bands at 466 nm of (n-\ensuremath{\pi }*) transitions that exhibited redshift in the ranges of complexes as a result of charge transfer transitions after coordination as shown in Table~\ref{tw-3ba28fa62103} .
\begin{table*}[!htbp]
\caption{\boldmath {IR frequencies of ligand and it'scomplexed.} }
\label{tw-42ecc01fe904}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{LLLLLL}
\tbltoprule \rowcolor{kwdboxcolor} Compound & v(N-H) \mbox{}\protect\newline imidazole & v (C=N) imidazole & v(N=N) & v(C-N) imidazole & v(M-N)\\
\tblmidrule
(DPIDA) &
3400 w &
1588 m &
1498 m &
1315 m &
-\\
{}[Mn(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
3403 w &
1572 m &
1489m &
1325 m &
543 w\\
{}[Co(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
3400 w &
1570 m &
1486 m &
1328 m &
540 w\\
{}[Ni(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
3405 w &
1566 m &
1484 m &
1320 m &
435 w\\
{}[Cu(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
3400 w &
1575 m &
1489 m &
1323 m &
511 w\\
{}[Zn(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
3402 w &
1568 m &
1488 m &
1321 m &
525 w\\
{}[Cd(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
3402 w &
1570 m &
1482 m &
1324 m &
532 w\\
{}[Hg(DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] &
3400 w &
1565 m &
1485 m &
1324 m &
523 w\\
\tblbottomrule
\end{tabulary}\par
\end{table*}
\bgroup
\fixFloatSize{images/2527c34c-f13c-4797-9eea-ef424f3dc3b2-upicture4.png}
\begin{figure*}[!htbp]
\centering \makeatletter\IfFileExists{images/2527c34c-f13c-4797-9eea-ef424f3dc3b2-upicture4.png}{\includegraphics{images/2527c34c-f13c-4797-9eea-ef424f3dc3b2-upicture4.png}}{}
\makeatother
\caption{\boldmath {IR spectrum of (DPIDA) ligand}}
\label{f-c55ccbd4dc77}
\end{figure*}
\egroup
\bgroup
\fixFloatSize{images/1150d836-fa21-4eb4-a165-d1d5641ce7bb-upicture5.png}
\begin{figure*}[!htbp]
\centering \makeatletter\IfFileExists{images/1150d836-fa21-4eb4-a165-d1d5641ce7bb-upicture5.png}{\includegraphics{images/1150d836-fa21-4eb4-a165-d1d5641ce7bb-upicture5.png}}{}
\makeatother
\caption{\boldmath {IR spectrum of Co (II) complex}}
\label{f-c21d2241f6f9}
\end{figure*}
\egroup
\bgroup
\fixFloatSize{images/b2dab3d3-6fa6-4884-b772-61c5ce0f2f02-upicture6.png}
\begin{figure*}[!htbp]
\centering \makeatletter\IfFileExists{images/b2dab3d3-6fa6-4884-b772-61c5ce0f2f02-upicture6.png}{\includegraphics{images/b2dab3d3-6fa6-4884-b772-61c5ce0f2f02-upicture6.png}}{}
\makeatother
\caption{\boldmath {IR spectrum of Cu (II) complex}}
\label{f-c9e753a08fa6}
\end{figure*}
\egroup
\bgroup
\fixFloatSize{images/fc07fcdf-59a4-4f2d-874e-57d7d6f4e95b-upicture7-uemf.png}
\begin{figure*}[t!]
\centering \makeatletter\IfFileExists{images/fc07fcdf-59a4-4f2d-874e-57d7d6f4e95b-upicture7-uemf.png}{\includegraphics{images/fc07fcdf-59a4-4f2d-874e-57d7d6f4e95b-upicture7-uemf.png}}{}
\makeatother
\caption{\boldmath {Suggested structure of [M(DPIDA)\ensuremath{_{2\ }} Cl\ensuremath{_{2}}]}}
\label{f-32589475abd2}
\end{figure*}
\egroup
\begin{table*}[!htbp]
\caption{\boldmath {Antibacterial activity of four derivative compounds against two types of aerobic pathogenic bacteria isolated from patients with burns infections.} }
\label{tw-c1adec9740eb}
\centering
\begin{threeparttable}
\def\arraystretch{1.1}
\ignorespaces
\centering
\begin{tabulary}{\linewidth}{p{\dimexpr.2734\linewidth-2\tabcolsep}p{\dimexpr.15159999999999998\linewidth-2\tabcolsep}p{\dimexpr.21240000000000002\linewidth-2\tabcolsep}p{\dimexpr.16259999999999998\linewidth-2\tabcolsep}p{\dimexpr.20\linewidth-2\tabcolsep}}
\tbltoprule \rowcolor{kwdboxcolor}{\multirow{3}{*}{Derivative compounds}} & \multicolumn{4}{p{\dimexpr(.7266\linewidth-2\tabcolsep)}}{\cAlignHack Multi-drug resistance aerobic pathogenic bacteria}\\
\rowcolor{kwdboxcolor} & \multicolumn{2}{p{\dimexpr(.364\linewidth-2\tabcolsep)}}{\cAlignHack S.aureus } & \multicolumn{2}{p{\dimexpr(.3626\linewidth-2\tabcolsep)}}{\cAlignHack P.aeruginosa }\\
\rowcolor{kwdboxcolor} & Concentration & ME\ensuremath{\pm} SE, R=4 & Concentration & ME\ensuremath{\pm} SE, R=4\\
\tblmidrule
\multicolumn{1}{p{\dimexpr(.2734\linewidth-2\tabcolsep)}}{\multirow{3}{\linewidth}{(DPIDA)}} &
50 mg/ml &
5.4667 \ensuremath{\pm} 0.42557 &
50 mg/ml &
4.8667 \ensuremath{\pm} 0.43333\\
&
100 mg/ml &
7.7667 \ensuremath{\pm} 0.29627 &
100 mg/ml &
7.6333 \ensuremath{\pm} 0.088192\\
&
150 mg/ml &
8.7333 \ensuremath{\pm} 0.12019 &
150 mg/ml &
8.9000 \ensuremath{\pm} 0.11547\\
\multicolumn{1}{p{\dimexpr(.2734\linewidth-2\tabcolsep)}}{\multirow{3}{\linewidth}{Hg complex}} &
50 mg/ml &
9.4333 \ensuremath{\pm} 0.23333 &
50 mg/ml &
9.7333 \ensuremath{\pm} 0.088192\\
&
100 mg/ml &
9.6667 \ensuremath{\pm} 0.12018 &
100 mg/ml &
10.033 \ensuremath{\pm} 0.12019\\
&
150 mg/ml &
10.367 \ensuremath{\pm} 0.12019 &
150 mg/ml &
10.833 \ensuremath{\pm} 0.14530\\
\multicolumn{1}{p{\dimexpr(.2734\linewidth-2\tabcolsep)}}{\multirow{3}{\linewidth}{Zn Complex}} &
50 mg/ml &
11.500 \ensuremath{\pm} 0.20817 &
50 mg/ml &
11.960 \ensuremath{\pm} 0.070238\\
&
100 mg/ml &
12.037 \ensuremath{\pm} 0.051747 &
100 mg/ml &
12.617 \ensuremath{\pm} 0.29946 \\
&
150 mg/ml &
12.033 \ensuremath{\pm} 0.10899 &
150 mg/ml &
12.593 \ensuremath{\pm} 0.19548 \\
\multicolumn{1}{p{\dimexpr(.2734\linewidth-2\tabcolsep)}}{\multirow{3}{\linewidth}{Cu Complex}} &
50 mg/ml &
12.17 \ensuremath{\pm} 0.1901 &
50 mg/ml &
12.95 \ensuremath{\pm} 0.02333 N\\
&
100 mg/ml &
12.42 \ensuremath{\pm} 0.1654 &
100 mg/ml &
12.43 \ensuremath{\pm} 0.2010 \\
&
150 mg/ml &
12.72 \ensuremath{\pm} 0.1352 &
150 mg/ml &
12.72 \ensuremath{\pm} 0.1251 \\
\multicolumn{1}{p{\dimexpr(.2734\linewidth-2\tabcolsep)}}{\multirow{3}{\linewidth}{Co Complex}} &
50 mg/ml &
11.767 \ensuremath{\pm} 0.24037 &
50 mg/ml &
12.567 \ensuremath{\pm} 0.20276\\
&
100 mg/ml &
12.000 \ensuremath{\pm} 0.26458 &
100 mg/ml &
12.833 \ensuremath{\pm} 0.033333\\
&
150 mg/ml &
14.900 \ensuremath{\pm} 0.40415 &
150 mg/ml &
15.733 \ensuremath{\pm} 0.088192\\
\multicolumn{1}{p{\dimexpr(.2734\linewidth-2\tabcolsep)}}{\multirow{3}{\linewidth}{Mn Complex}} &
50 mg/ml &
14.830.09536=3 &
50 mg/ml &
14.72 \ensuremath{\pm} 0.1844 \\
&
100 mg/ml &
15.28 \ensuremath{\pm} 0.1802 &
100 mg/ml &
16.07 \ensuremath{\pm} 0.07142 \\
&
150 mg/ml &
16.11 \ensuremath{\pm} 0.1068 &
150 mg/ml &
16.74 \ensuremath{\pm} 0.1949 \\
\multicolumn{1}{p{\dimexpr(.2734\linewidth-2\tabcolsep)}}{\multirow{3}{\linewidth}{Cd Complex}} &
50 mg/ml &
18.61 \ensuremath{\pm} 0.2275 &
50 mg/ml &
18.52 \ensuremath{\pm} 0.2217\\
&
100 mg/ml &
18.81 \ensuremath{\pm} 0.1757 &
100 mg/ml &
18.70 \ensuremath{\pm} 0.1695 \\
&
150 mg/ml &
18.86 \ensuremath{\pm} 0.2781 &
150 mg/ml &
19.51 \ensuremath{\pm} 0.2623 \\
\multicolumn{1}{p{\dimexpr(.2734\linewidth-2\tabcolsep)}}{\multirow{3}{\linewidth}{4,5-diphenyl imidazol}} &
50 mg/ml &
17.833 \ensuremath{\pm} 0.17638 &
50 mg/ml &
18.700 \ensuremath{\pm} 0.11547\\
&
100 mg/ml &
18.767 \ensuremath{\pm} 0.17638 &
100 mg/ml &
19.533 \ensuremath{\pm} 0.17638\\
&
150 mg/ml &
21.83 \ensuremath{\pm} 0.1764 &
150 mg/ml &
24.30 \ensuremath{\pm} 0.4163\\
\tblbottomrule
\end{tabulary}\par
\begin{tablenotes}\footnotesize
\item{C: Concentrationsof derivative compounds, R: Numbers of replicates, M: Mean of the diameter ofinhibition zone (mm), SE: Standard error of the mean.}
\end{tablenotes}
\end{threeparttable}
\end{table*}
\bgroup
\fixFloatSize{images/d000b8b9-b2ce-4121-83f1-22534391d436-upicture8.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/d000b8b9-b2ce-4121-83f1-22534391d436-upicture8.png}{\includegraphics{images/d000b8b9-b2ce-4121-83f1-22534391d436-upicture8.png}}{}
\makeatother
\caption{\boldmath {: Antibacterial activity test of the 4,5-diphenyl imidazol derivative compound with concentration 150gm/ml against multi-drug resistance S.aureus isolated from patients with burns infections.}}
\label{f-6e5343e8133b}
\end{figure}
\egroup
\bgroup
\fixFloatSize{images/d63517bd-c55a-416a-b2cd-1d1f1535668a-uimage.png}
\begin{figure}[!htbp]
\centering \makeatletter\IfFileExists{images/d63517bd-c55a-416a-b2cd-1d1f1535668a-uimage.png}{\includegraphics{images/d63517bd-c55a-416a-b2cd-1d1f1535668a-uimage.png}}{}
\makeatother
\caption{\boldmath {Antibacterial activity test of the 4,5-diphenyl imidazol derivative compound with concentration 150 gm/ml against multi-drug resistance P.aeruginosa isolated from patients with burns infections.}}
\label{f-09a3fb71f3d8}
\end{figure}
\egroup
IR spectra of ligand (DPIDA) showed v(N-H) peak of imidazole ring at (3400) cm\ensuremath{^{-}}\unskip~\citep{549750:12574432}two peeks at (3034) and (3061) cm- due to the vibrations of aromatic v(C-H) , and one peak at (2991) cm- of aliphatic v(C-H) that showed no significant changes in the spectra of the complexes frequencies. The peaks at (1588) cm- and (1315) cm- exhibited a stretching of v(C=N) and v(C-N) \unskip~\citep{549750:12574435} respectively of imidazole ring that proceed a changing in position and intensity in complexes which is an indicate the participation of (N3) atom in coordination, also the values of v(N=N) \unskip~\citep{549750:12575246}; \unskip~\citep{549750:12574446} peak at (1498) cm- shifted to lower values in the complexes that's considered as an evidence on coordination proceed through one nitrogen atom of azo group, new values of v(M-N) frequencies appeared between (543 -511) cm\ensuremath{^{-}} in the complexes that consider as additional evidence on coordination process as shown inTable~\ref{tw-42ecc01fe904} andFigures~\ref{f-c55ccbd4dc77}, \ref{f-c21d2241f6f9} and~\ref{f-c9e753a08fa6}.
Conductivity measurements at 25\ensuremath{^{o}}C in both of DMF and DMSO solvents for (10\ensuremath{^{-3}}) M encouraged non- ionic character of all complexes , The values of molar conductivity ranged between 23.6-15.4 S.cm\ensuremath{^{2}}.mole in DMF , while their values within 20.6-11.7 S.cm\ensuremath{^{2}}.mole, as well no white precipitate of AgCl observed when a drops of 0.1 N from AgNO\ensuremath{_{3}} solution to metal complexes solutions added which also confirms the absence of counter ion outside the coordination sphere \unskip~\citep{549750:12575594}\unskip~\citep{549750:12574442} as apparent in Table~\ref{tw-3ba28fa62103}. The suggested structure of the complexes showed the octrahyral geometry that two (DPIDA) ligands coordinated with central metal ion as bidentate through nitrogen atom number 3 of imidazole ring and one nitrogen atom of the azo group as explained inFigure~\ref{f-32589475abd2}.
The results of biological activity demonstrated that most the ligand and complexes have good antibacterial activity against two pathogenic bacteria with inhibition zones in three concentrationsTable~\ref{tw-c1adec9740eb}andFigure~\ref{f-6e5343e8133b} and Figure~\ref{f-09a3fb71f3d8} While, the derivative compound 4,5-diphenyl imidazol with concentration 150mg/ml had excellent antibacterial effect against S.aureus and P.aeruginosa with inhibition zone 21.83 \ensuremath{\pm} 0.1764 mm and 24.30 \ensuremath{\pm} 0.4163 mm respectively.
\section{Conclusion}
Seven complexes of Mn(II) , Co(II) , Ni(II) , Cu(II) , Zn(II) , Cd(II) , and Hg(II) ions with new imidazole azo ligand (DPIDA) were prepared as an octahedral geometry of them with bidentate ligand with general formal [M (DPIDA)\ensuremath{_{2}}Cl\ensuremath{_{2}}] the prepared compounds showed excellent biological activity against two types of bacteria \textit{S.aureus and P.aeruginosa}.
\bibliographystyle{pharmascope_apa-custom}
\bibliography{\jobname}
\end{document}
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\clan {Iztok Banič}
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% A. objavljene znanstvene monografije
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% 1.01: {\bf 1}. BANI\v{C}, Iztok, \v{C}REPNJAK, Matev\v{z}, MERHAR, Matej, MILUTINOVI\'{C}, Uro\v{s}, SOVI\v{C}, Tina. The closed subset theorem for inverse limits with upper semicontinuous bonding functions. {\it Bull.\ Malays.\ Math.\ Sci.\ Soc.}, 2017, str. 1-12, $[$COBISS.SI-ID 23281928$]$\\
{\crta, M.~Črepnjak, M.~Merhar, U.~Milutinović, T.~Sovič}
{The closed subset theorem for inverse limits with upper semicontinuous bonding functions}
{Bull.\ Malays.\ Math.\ Sci.\ Soc.}
\sprejetoRevija
{\crta, M.~Črepnjak}
{Markov pairs, quasi Markov functions and inverse limits}
{Houston J. Math.}
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\objavljenoRevija
% 1.01: %list {\bf 4}. BANI\v{C}, Iztok. Integrations on rings. {\it Open Math.}, 2017, vol. 15, iss. 1, str. 365-373. $[$COBISS.SI-ID 23042568$]$\\
{}
{Integrations on rings}
{Open Math.} {15} {2017} {} {365--373}
\objavljenoRevija
% 1.01: %list {\bf 3}. BANI\v{C}, Iztok, \v{C}REPNJAK, Matev\v{z}, MERHAR, Matej, MILUTINOVI\'{C}, Uro\v{s}, SOVI\v{C}, Tina. An Anderson-Choquet-type theorem and a characterization of weakly chainable continua. {\it Mediterr\. J. Math.}, 2017, vol. 14, iss. 2, str. 1-14. $[$COBISS.SI-ID 22997512$]$\\
{\crta, M.~\v{C}repnjak, M.~Merhar, U.~Milutinovi\'{c}, T.~Sovi\v{c}}
{An Anderson-Choquet-type theorem and a characterization of weakly chainable continua}
{Mediterr.\ J. Math.} {14} {2017} {} {1--14}
\objavljenoRevija
% 1.01: {\bf 2}. BANI\v{C}, Iztok, TARANENKO, Andrej. Measuring closeness of graphs - the Hausdorff distance. {\it Bull.\ Malays.\ Math.\ Sci.\ Soc.}, 2017, vol. 40, iss. 1, str. 75-95. $[$COBISS.SI-ID 21722376$]$\\
{\crta, A.~Taranenko}
{Measuring closeness of graphs - the Hausdorff distance}
{Bull.\ Malays.\ Math.\ Sci.\ Soc.} {40} {2017} {} {75--95}
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% Ment: {\bf 5}. \v{S}TESL, Da\v{s}a{\it . Uporaba in posplo\v{s}itve Anderson-Choquetovega izreka : magistrsko delo : na \v{s}tudijskem programu 2. stopnje Matematika}. Maribor: $[$D. \v{S}tesl$]$, 2016. IX, 44 f., ilustr. $[$COBISS.SI-ID 23053832$]$\\
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\documentclass[10pt]{article}
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\begin{document}
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\rhead{\textbf{A. P{}. M. E. P{}.}}
\lhead{\small Baccalauréat ES}
\lfoot{\small{Liban}}
\rfoot{\small{29 mai 2012}}
\renewcommand \footrulewidth{.2pt}
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\begin{center}
{ \Large \textbf{\decofourleft~Corrigé du baccalauréat ES Liban ~\decofourright\\29 mai 2012}}
\end{center}
\bigskip
\textbf{Exercice 1 \hfill 4 points}
\textbf{Commun à tous les candidats}
\medskip
%\emph{Cet exercice est un questionnaire à choix multiples. Pour chacune des questions posées, une seule des trois réponses est exacte. Pour chaque question, indiquer par a), b), c) l'unique bonne réponse. Aucune justification n'est demandée.\\
%Une réponse exacte rapporte $1$ point. Une réponse fausse ou l'absence de réponse n'enlève ni ne rapporte aucun point.}
%
%\medskip
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% \item[$\bullet~~$] $f$ s'annule en $0,5$.
% \item[$\bullet~~$] La courbe représentative de $f$ admet une tangente horizontale au point d'abscisse $-2$ et une tangente horizontale au point d'abscisse $2$.
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\begin{center}
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%
%On notera $f'$ la la fonction dérivée de $f$.
\begin{enumerate}
\item %Sur $[-5~;~5]$, l'équation $f'(x) = 0$ admet exactement :
%\begin{tabularx}{\linewidth}{*{3}{@{}X}}
%\textbf{a)}\quad 0 solution & \textbf{b)}\quad 1 solution & \textbf{c)}\quad 2 solutions \\
%\end{tabularx}
Il y a deux tangentes horizontales, donc deux nombres dérivés nuls, l'équation a deux solutions.
\medskip
\item %Sur $[-5~;~5]$, l'inéquation $f^{\prime}(x)\geqslant 0$ admet pour ensemble de solutions :
%\begin{tabularx}{\linewidth}{*{3}{@{}X}}
%\textbf{a)}\quad $[-2~;~2]$ & \textbf{b)}\quad $[0~;~5]$ & \textbf{c)}\quad $[0,5~;~5]$ \\
%\end{tabularx}
Le nombre dérivé est positif signifie que la fonction est croissante ; on voit qu'elle est croissante sur $[- 2~;~2]$.
\medskip
\item %La fonction $g$ telle que $g(x) = \ln\left(f(x)\right)$ est définie sur :
%\begin{tabularx}{\linewidth}{*{3}{@{}X}}
%\textbf{a)}\quad $[-2~;~2]$ & \textbf{b)}\quad $]0~;~1]$ & \textbf{c)}\quad $]0,5~;~5]$ \\
%\end{tabularx}
La fonction $g$ est définie si $f(x) > 0$, donc quand la courbe est au dessus de l'axe des abscisses soit sur l'intervalle $]0,5~;~5]$.
\medskip
\item %On note $S=\displaystyle\int_1^{3}{f(x) \: \mathrm {d} x}$ alors :
%\begin{tabularx}{\linewidth}{*{3}{@{}X}}
%\textbf{a)}\quad $0 < S < 1$ & \textbf{b)}\quad $1 < S < 2$ & \textbf{c)}\quad $2 < S < 3$ \\
%\end{tabularx}
Sur l'intervalle [1~;~3], la fonction $f$ est positive, donc l'intégrale est l'aire en unités d'aire de la surface limitée par la coure, l'axe des abscisses et les verticales $x = 1$ et $x = 3$. L'unité d'aire contient 4 carreaux, et l'on compte pour la surface à peu près 7 carreaux soit environ 1,75 unité d'aire : la bonne réponse est la deuxième. (voir la figure)
\end{enumerate}
\bigskip
\textbf{Exercice 2 \hfill 6 points}
\textbf{Commun à tous les candidats}
\bigskip
\textbf{1\up{re} partie : Étude d'une fonction}
\medskip
On considère la fonction $f$ définie sur $[0~;~+\infty[$ par $f(x) = x \mathrm {e}^x - \mathrm {e}^x - 8$.
\begin{enumerate}
\item %En écrivant que $f(x) = \mathrm {e}^x (x - 1) - 8$, déterminer la limite de $f$ en $+\infty$.
On a $\displaystyle\lim_{x \to + \infty} \text{e}^x = \displaystyle\lim_{x \to + \infty} (x - 1) = + \infty$, d'où par produit $\displaystyle\lim_{x \to + \infty} \text{e}^x (x - 1) = + \infty$ et enfin $\displaystyle\lim_{x \to + \infty} f(x) = + \infty$.
\item %Montrer que $f'(x) = x \mathrm {e}^x $ où $f'$ désigne la fonction dérivée de $f$ sur $[0;+\infty[$.
Sur $[0~;~+\infty[$ la fonction $f$ est dérivable et sur intervalle :
$f'(x) = \text{e}^x + x\text{e}^x - \text{e}^x = x\text{e}^x$.
\item %Dresser le tableau de variations complet de de $f$ sur $[0~;~+\infty[$.
D'après le résultat précédent $f'(x)$ est le produit de deux nombres positifs ou nul, donc $f'(x) \geqslant 0$ : la fonction $f$ est donc croissante sur $[0~;~+\infty[$, de $f(0) = - 9$ à $+ \infty$.
\item
\begin{enumerate}
\item %Montrer que l'équation $f(x) = 0$ admet sur $[0~;~+\infty[$ une unique solution $a$.
Sur $[0~;~+\infty[$ la fonction $f$ est dérivable, donc continue et croissante de $- 9$ à plus l'infini : elle s'annule donc une seule fois sur cet intervalle. L'équation $f(x) = 0$ a une solution unique.
\item %Montrer que $2,040 < a < 2,041$.
La calculatrice donne $f(2)\approx -0,6$ et $f(3) \approx 32$, donc $2 < a < 3$ ;
$f(2,0)\approx -0,6$ et $f(2,1) \approx 0,98$, donc $2,0 < a 2,1$ ;
$f(2,04) \approx - 0,002$ et $f(2,05) \approx 0,16$, donc $2,04 < a < 2,05$ ;
$f(2,040) \approx - 0,002$ et $f(2,041) \approx 0,014$, donc finalement
\[ 2,040 < a < 2,041.\]
\item %En utilisant les questions précédentes, déduire le signe de $f(x)$ en fonction des valeurs de $x$ sur $[0~;~+\infty[$.
Puisque $f(a) = 0$ et que la fonction est croissante on a donc :
\begin{itemize}
\item $f(x) < 0$ sur $[0~;~a[$ et
\item $f(x) > 0$ sur $]a~;~+ \infty[$.
\end{itemize}
\end{enumerate}
\item
\begin{enumerate}
\item %Montrer que la fonction $g$ définie sur $[0~;~+\infty[$ par $g(x)= x \mathrm {e}^x - 2\mathrm {e}^x - 8x$ est une primitive de $f$ sur $[0~;~+\infty[$.
La fonction $g$ est dérivable sur $[0~;~+\infty[$ et sur cet intervalle :
$g'(x) = \text{e}^x + x\text{e}^x - 2\text{e}^x - 8 = x\text{e}^x - \text{e}^x - 8 = f(x)$.
Conclusion $g$ est une primitive de $f$.
\item %Calculer la valeur exacte de $\displaystyle\int_3^{5}{f(x) \: \mathrm {d}x}$.
D'après le résultat précédent : $\displaystyle\int_3^{5}{f(x) \: \mathrm {d}x} = g(5) - g(3) = 5\text{e}^5 - 2\text{e}^5 - 8 \times 5 - \left(3\text{e}^3 - 2\text{e}^3 - 8 \times 3\right) = 3\text{e}^5 - \text{e}^3 - 16.$
%Comme $3 > a$, la fonction $f$ est positive sur l'intervalle [3~;~5], donc l'intégrale est égale à la mesure de la surface (en unités d'aire) comprise entre la courbe représentative, l'axe des abscisses et les verticales $x = 3$ et $ x = 5$.
\end{enumerate}
\end{enumerate}
\medskip
\textbf{2\up{e} partie : Application à une situation économique}
\medskip
%Une entreprise fabrique $x$ milliers d'objets avec $x$ appartenant à $[0;5]$.
%
%La fonction $f$ de la 1\up{re} partie modélise les bénéfices ou les pertes de l'entreprise en centaine d'euros.
%
%Pour une quantité $x$ donnée, si $f(x)$ est positif, l'entreprise réalise un bénéfice, et si $f(x)$ est négatif, l'entreprise subit une perte.
%
%\medskip
%
%En utilisant les résultats de la 1\up{re} partie, répondre aux questions suivantes en justifiant :
\begin{enumerate}
\item %À partir de combien d'objets produits, l'entreprise commence-t-elle à réaliser des bénéfices ?
On a vu que $f(a) = 0$ et que $f(2,041) > 0$ ; donc l'entreprise commence à réaliser des bénéfices à partir de \np{2041} objets produits.
\item %L'entreprise pense produire régulièrement entre 3 et 5 milliers d'objets.
%Déterminer la valeur moyenne du bénéfice sur $[3,5]$ (On donnera le résultat arrondi à l'euro près).
On sait que la valeur moyenne de la fonction $f$ sur l'intervalle [3~;~5] est égale à :
\[m = \dfrac{1}{5 - 3}\displaystyle\int_3^5 f(x)\:\text{d}x = \dfrac{3\text{e}^5 - \text{e}^3 - 16}{2} \approx 204,58\]
À l'euro près la valeur moyenne du bénéfice sera de \np{20458}~\euro.
\end{enumerate}
\bigskip
\textbf{Exercice 3 \hfill 5 points}
\textbf{Commun à tous les candidats}
\bigskip
%Dans un salon de coiffure pour femmes, le coloriste propose aux clientes qui viennent pour une coupe deux prestations supplémentaires :
%
%\begin{itemize}
% \item une coloration naturelle à base de plantes qu'il appelle \og couleur-soin \fg{},
% \item des mèches blondes pour donner du relief à la chevelure, qu'il appelle \og effet coup de soleil \fg{}.
%\end{itemize}
%
%\medskip
%
%Ce coloriste a fait le bilan suivant sur ces prestations :
%
%\begin{itemize}
% \item 40\,\% des clientes demandent une \og couleur-soin \fg{}.
% \item parmi celles qui n'en veulent pas, 30\,\% des clientes demandent un \og effet coup de soleil \fg{}.
% \item de plus, 24\,\% des clientes demandent les deux à la fois.
%\end{itemize}
%
%\medskip
%
%On considère une de ces clientes.
%
%On notera C l'évènement \og \emph{la cliente souhaite une "couleur-soin"}\fg{}.
%
%On notera M l'évènement \og \emph{la cliente souhaite un "effet coup de soleil"}\fg{}.
%
%\medskip
\begin{enumerate}
\item %Calculer la probabilité de M sachant C notée $P_C\left(M\right)$.
On sait que $P_C\left(M\right) = \dfrac{P(M \cap C)}{P(C)} = \dfrac{0,24}{0,4} = 0,6$.
\item %Construire un arbre pondéré qui illustre la situation.
~
\begin{center}
\pstree[treemode=R,nodesep=2.5pt]{\TR{}}
{\pstree{\TR{$C$}\taput{0,4}}
{
\TR{$M$}\taput{0,6}
\TR{$\overline{M}$}\tbput{0,4}
}
\pstree{\TR{$\overline{C}$}\tbput{0,6}}
{
\TR{$M$}\taput{0,3}
\TR{$\overline{M}$}\tbput{0,7}
}
}
\end{center}
\medskip
\item %Calculer la probabilité que la cliente ne souhaite ni une \og couleur-soin \fg{}, ni un \og effet coup de soleil \fg{}.
On a $P\left(\overline{C} \cap \overline{M}\right) = P_{\overline{C}}\left(\overline{M}\right) \times P\left(\overline{C} \right) = 0,7 \times 0,6 = 0,42$.
\item %Montrer que la probabilité de l'évènement M est égale à 0,42.
Les évènements $C$ et $M$ déterminent une partition de l'ensemble des résultats, donc d'après la formule des probabilités totales :
$P(M) = P(M \cap C) + P\left(M \cap \overline{C}\right) = 0,4 \times 0,6 + 0,6 \times 0,3 = 0,24 + 0,18 = 0,42$.
\item %Les évènements C et M sont-ils indépendants ?
On a $P(M) = 0,42$ et $P_C(M) = 0,6$, donc les évènements $M$ et $C$ ne sont pas indépendants.
\item %Une \og couleur-soin \fg{} coûte 35 euros et un \og effet coup de soleil \fg{} coûte 40 euros.
\begin{enumerate}
\item %Recopier puis compléter sans justifier le tableau suivant donnant la loi de probabilité du gain en euros du coloriste par client :
On a $\left(C \cap \overline{M}\right) = 0,4 \times 0,4 = 0,16$ : cet évènement correspond à un coût de 35 euros.
\medskip
\begin{tabularx}{\linewidth}{|m{1cm}|*{5}{>{\centering \arraybackslash}X|}} \hline
$x_{i}$ & 75 &40 &35 & 0 \\ \hline
$p_{i}$ & 0,24 &0,18 &0,16 & 0,42\\ \hline
\end{tabularx}
\medskip
\item %Donner l'espérance E de cette loi.
On a $\text{E} = 75 \times 0,24 + 40 \times 0,18 + 35 \times 0,16 = 30,80$~(\euro)
La dépense moyenne par client sera de $30,80$~(\euro)
\item %\emph{Pour cette question, toute trace de recherche même incomplète, ou d'initiative même non fructueuse, sera prise en compte dans l'évaluation}.
%Combien le coloriste doit-il facturer la réalisation d'un \og effet coup de soleil \fg{} pour que l'espérance de gain par client augmente de 15\% ?
On reprend le tableau précédent en appelant $x$ la somme demandée pour la réalisation d'un \og effet coup de soleil \fg{}
\medskip
\begin{tabularx}{\linewidth}{|m{1cm}|*{5}{>{\centering \arraybackslash}X|}} \hline
$x_{i}$ & $35 + x $ &$x$ &35 & 0 \\ \hline
$p_{i}$ & 0,24 &0,18 &0,16 & 0,42\\ \hline
\end{tabularx}
\medskip
On veut une espérance de $30,8 \times 1,15 = 35,42$.
Il faut donc que $35,42 = 0,24(35 + x) + 0,18x + 35 \times 0,16+ 0$ soit
$35,42 = 8,4 + 0,24x + 0,18x + 5,6$ ou encore $21,42 = 0,42 x$ soit $x = 51$~(\euro)
Il faut donc augmenter la prestation \og effet coup de soleil \fg{} de $51 - 35 = 16$~(\euro). \end{enumerate}
\end{enumerate}
\bigskip
\textbf{Exercice 4 \hfill 5 points}
\textbf{Candidats n'ayant pas suivi l'enseignement de spécialité}
%\bigskip
%
%En pédiatrie (médecine des enfants), des études statistiques sur des enfants de moins de 36 mois ont permis de tracer les deux courbes fournies en annexe. Pour un âge $x$ donné en mois, la courbe inférieure $C_1$ donne le périmètre crânien minimal en centimètres, et la courbe supérieure $C_2$ donne le périmètre crânien maximal en centimètres.
%
%Ces deux courbes sont souvent utilisées pour observer le développement des enfants.
%
%\medskip
\textbf{A) Lectures graphiques}
\medskip
%À l'aide du graphique fourni en annexe, répondre aux deux questions suivantes en laissant les traits de construction apparents :
\begin{enumerate}
\item %Un enfant a un périmètre crânien égal à 41 cm.
%Déterminer l'âge minimum et l'âge maximum que peut avoir cet enfant.
On trace la droite horizontale d'équation $y = 41$ qui coupe les deux courbes en deux points d'abscisses respectives approximatives 2 et 7 : l'enfant a entre 2 et 7 mois.
\item %Un enfant a un âge compris entre 15 et 21 mois.
Pour 15 mois on lit entre 43 et 49,5 cm.
pour 21 mois on lit entre 45 et 51~cm.
Donc un enfant entre 15 et 21 mois aura un périmètre crânien compris entre 43 et 51~cm.
%Déterminer le périmètre crânien minimum et le périmètre crânien maximum que peut avoir cet enfant.
\end{enumerate}
\medskip
\textbf{B) Étude d'un modèle}
\medskip
%Dans cette partie, les résultats seront arrondis à $10^{-2}$ près.
%
%Le pédiatre ne disposant pas de données d'individus de plus de 36 mois sur son lieu d'étude, il considère les valeurs moyennes des deux courbes précédentes.
%
%Il obtient les mesures suivantes :
%\medskip
%
%\begin{tabularx}{\linewidth}{|l|*{4}{>{\centering \arraybackslash}X|}}\hline
%âge $x$ (en mois) & 0 & 12 & 24 & 36 \\ \hline
%Périmètre crânien $y$ (en cm) & 36 & 46 & 48 & 50 \\ \hline
%\end{tabularx}
%
%\medskip
\begin{enumerate}
\item %On considère $z=\ln \left(54 - y \right)$.
%Recopier puis compléter le tableau suivant :
\medskip
\begin{tabularx}{\linewidth}{|l|*{4}{>{\centering \arraybackslash}X|}}\hline
âge $x$ en mois & 0 &12 &24 &36 \\ \hline
$z$ &$\ln 18$ &$\ln 8$ &$\ln 6$ &$\ln 4$ \\ \hline
\end{tabularx}
\medskip
\item
\begin{enumerate}
\item %À l'aide de la calculatrice, déterminer une équation de la droite de régression de $z$ en fonction de $x$, obtenue par la méthode des moindres carrés.
La calculatrice livre $z = - 0,04x + 2,76$.
\item %En déduire que $y=54 - \mathrm{e}^{ax+b}$ avec $a\approx -0,04$ et $b\approx 2,76$.
On a pour $y < 5$, \: $z=\ln \left(54 - y \right) \iff \text{e}^z = 54 - y \iff y = 54 - \text{e}^z$ soit finalement :
\[y = 54 - \text{e}^{- 0,04x + 2,76}.\]
\end{enumerate}
\end{enumerate}
\medskip
\textbf{C) Utilisation du modèle précédent}
\medskip
Dans cette partie, on utilisera le modèle établi dans la question \textbf{2. b)} de la partie \textbf{B}.
\begin{enumerate}
\item %Un enfant a un périmètre crânien de 53 cm.
%Déterminer par le calcul une approximation de l'âge en mois de cet enfant.
On a quel que soit le nombre $x$, \: $54 - \text{e}^{- 0,04x + 2,76} = 53 \iff \text{e}^{- 0,04x + 2,76} = 1$ et par croissance de la fonction logarithme népérien :
$- 0,04x + 2,76 = 0 \iff 0,04x = 2,76 \iff x = \dfrac{2,76}{0,04} = 69$~(mois).
\item %Les scientifiques estiment que la structure osseuse crânienne se rigidifie dès l'âge de 15 ans, le périmètre crânien cesse alors de croître.
%Déterminer par le calcul une approximation du périmètre crânien correspondant. Arrondir au centimètre près.
15 mois correspondent à $15 \times 12 = 180$~(mois).
Le périmètre crânien moyen est alors :
$54 - \text{e}^{- 0,04 \times 180 + 2,76} \approx 54 - 0,0118$ soit à peu près 53,99~(cm).
À quinze ans le périmètre crânien est à peu près de 54 cm.
\end{enumerate}
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\end{center}
\bigskip
\textbf{Exercice 4 \hfill 5 points}
\textbf{Candidats ayant suivi l'enseignement de spécialité}
%\medskip
%
%Dans cet exercice, les résultats seront arrondis à $10^{-2}$ près.
%
%\medskip
%
%Cet exercice consiste à étudier la propagation d'une information d'une personne à l'autre, thème souvent abordé en sciences sociales. Cette information se transmet avec un risque d'erreur, c'est-à-dire avec une probabilité de propagation de l'information contraire.
%
%\medskip
%
%Dans cet exercice, on considère l'information suivante, notée E : \og Paul a réussi son examen \fg{}.
\medskip
\textbf{Partie A : Propagation symétrique (de type \og neutre \fg{})}
\medskip
%Dans cette partie, on suppose que, pour une information reçue (E ou $\overline{\mathrm{E}}$), la probabilité de communiquer cette information à l'identique vaut 0,9 et la probabilité de relayer l'information contraire vaut 0,1.
%
%\medskip
%
%On note $p_n$ la probabilité de recevoir l'information E au bout de $n$ étapes ($n$ étant le nombre de personnes ayant transmis l'information) et on note $q_n$ la probabilité de recevoir l'information $\overline{\mathrm{E}}$ au bout de $n$ étapes.
%
%\medskip
%
%On suppose que Paul a réussi son examen, on pose $p_0=1$ et $q_0=0$.
%
%\medskip
\begin{enumerate}
\item %Recopier puis compléter le graphe probabiliste relatif à la propagation de l'information suivant :
~
\begin{center}
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\newrgbcolor{bleu}{0.1 0.05 .5}
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\rput(2.5,0){0,1} \rput(2.5,2){0,1}
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\end{center}
\item %Préciser la matrice de transition M telle que $ \begin{pmatrix} p_{n+1} & q_{n+1} \\ \end{pmatrix} = \begin{pmatrix} p_{n} & q_{n} \\ \end{pmatrix} \mathrm{M}$.
La matrice de transition est $M = \begin{pmatrix}0,9&0,1\\0,1&0,9\end{pmatrix}$.
\item %À l'aide de la calculatrice, trouver le plus petit entier naturel $n$ tel que $p_n < 0,8$.
On a successivement :
$P_1 = P0 \times M = \begin{pmatrix}1&0\end{pmatrix} \times \begin{pmatrix}0,9&0,1\\0,1&0,9\end{pmatrix} = \begin{pmatrix}0,9&0,1\end{pmatrix}$ ;
$P_2 = P0 \times M^2 = \begin{pmatrix}1&0\end{pmatrix} \times \begin{pmatrix}0,9&0,1\\0,1&0,9\end{pmatrix}^2 = \begin{pmatrix}0,82&0,18\end{pmatrix}$ ;
$P_3 = P0 \times M^3 = \begin{pmatrix}1&0\end{pmatrix} \times \begin{pmatrix}0,9&0,1\\0,1&0,9\end{pmatrix}^3 = \begin{pmatrix}0,756&0,244\end{pmatrix}$.
Donc le plus petit entier naturel $n$ tel que $p_n < 0,8$ est $n = 3$.
\item %Déterminer par le calcul, l'état stable.
Les termes de la matrice étant non nuls il y a un état stable $P = \begin{pmatrix}p&q\end{pmatrix}$ tel que $p + q = 1$.
De $P = P \times M$ on déduit que $\begin{pmatrix}p&q\end{pmatrix} = \begin{pmatrix}p&q\end{pmatrix} \times \begin{pmatrix}0,9&0,1\\0,1&0,9\end{pmatrix}$.
On en déduit que :
$\left\{\begin{array}{l c l}
p&=&0,9p + 0,1q\\
q&=&0,1p + 0,9q\\
p + q &=& 1
\end{array}\right. \iff \left\{\begin{array}{l c l}
0,1p&=& 0,1q\\
0,1q&=&0,1p \\
p + q &=& 1
\end{array}\right.\iff \left\{\begin{array}{l c l}
p&=&q\\
q&=&p \\
p + q &=& 1
\end{array}\right. \Rightarrow p = q = 0,5$.
Conclusion $P = \begin{pmatrix}0,5&0,5\end{pmatrix}$.
\end{enumerate}
\medskip
\textbf{Partie B : Propagation asymétrique (de type \og rumeur \fg)}
\medskip
%Dans cette partie, on suppose toujours que la probabilité de transmission correcte de l'information E est égale à 0,9. Toutefois, il circule la fausse rumeur $\overline{\mathrm{E}}$. Dans ces conditions, on suppose que si l'information reçue est $\overline{\mathrm{E}}$, la probabilité de transmettre cette information $\overline{\mathrm{E}}$ est égale à 1.
%
%\medskip
%
%On suppose de nouveau que $p_0=1$ et $q_0=0$.
%
%\medskip
\begin{enumerate}
\item %Représenter cette situation par un graphe probabiliste.
~
\begin{center}
\psset{unit=1cm}
\begin{pspicture}(5,2)
\newrgbcolor{bleu}{0.1 0.05 .5}
\psset{radius=.3cm,linewidth=0.75pt, linecolor=bleu}
\cnodeput(0.5,1){E}{\small{\textsf{\bleu{E}}}}\cnodeput(4.5,1){F}{\small{\textsf{\bleu{$\overline{\mathrm{E}}$}}}}
\rput{90}{\nccircle{->}{E}{0.5}}\rput{-90}{\nccircle{->}{F}{0.5}}\ncarc[arcangle=40]{->}{E}{F}
%\ncarc[arcangle=40]{->}{F}{E}
\rput(-0.8,1){0,9}\rput(5.8,1){1}
\rput(2.5,2){0,1}
\end{pspicture}
\end{center}
\item %Préciser la matrice de transition N telle que $ \begin{pmatrix} p_{n+1} & q_{n+1} \\ \end{pmatrix} = \begin{pmatrix} p_{n} & q_{n} \\ \end{pmatrix} \mathrm{N}$.
On a $N = \begin{pmatrix}0,9&0,1\\0&1\end{pmatrix}$.
\item %Montrer que $p_{n+1} = 0,9p_{n}$. Quelle est la nature de la suite $\left(p_n\right)$ ?
$ \begin{pmatrix} p_{n+1} & q_{n+1} \\ \end{pmatrix} = \begin{pmatrix} p_{n} & q_{n} \\ \end{pmatrix} N = \begin{pmatrix} p_{n} & q_{n} \end{pmatrix} \times \begin{pmatrix}0,9&0,1\\0&1\end{pmatrix} = \begin{pmatrix} 0,9p_{n} & 0,1p_n + q_{n} \\ \end{pmatrix}$.
Donc $p_{n+1} = 0,9p_{n}$ : la suite $\left(p_n\right)$ est géométrique de raison 0,9.
\item %Exprimer $p_{n}$ en fonction de $n$.
Comme $p_0 = 1$, on sait que $p_n = p_0 \times 0,9^n = 0,9^n$.
\item %Trouver par le calcul, le plus petit entier naturel $n$ tel que $p_n < 0,5$.
$p_n < 0,5 \iff 0,9^n < 0,5 \iff n \ln 0,9 < \ln 0,5 \iff n > \dfrac{\ln 0,5}{\ln 0,9}$ (car $\ln 0,9 < $).
Comme $\dfrac{\ln 0,5}{\ln 0,9} \approx 6,6$, le plus petit entier naturel $n$ tel que $p_n < 0,5$ est $n = 7$.
\item %Déterminer la limite de $\left(p_{n}\right)$ lorsque $n$ tend vers $+\infty$ puis interpréter le résultat obtenu.
On sait que comme $0 < 0,9 < 1$,\: $\displaystyle\lim_{n \to + \infty} 0,9^n = 0$.
Conclusion : sur le long terme seule la fausse rumeur sera transmise.
\end{enumerate}
\end{document}
|
http://axiom-developer.org/axiom-website/CATS/richder12d.input.pamphlet
|
axiom-developer.org
|
CC-MAIN-2022-49
|
unk
|
application/x-tex
|
crawl-data/CC-MAIN-2022-49/segments/1669446711108.34/warc/CC-MAIN-20221206124909-20221206154909-00135.warc.gz
| 5,755,291 | 109,095 |
\documentclass{article}
\usepackage{axiom}
\setlength{\textwidth}{400pt}
\begin{document}
\title{\$SPAD/src/input richder12d.input}
\author{Albert Rich and Timothy Daly}
\maketitle
\begin{abstract}
\verb|(d+e x)^m (f+g x)^n (a+b x+c x^2)^p|
All derivatives match
\end{abstract}
\eject
\tableofcontents
\eject
\begin{chunk}{*}
)set break resume
)sys rm -f richder12d.output
)spool richder12d.output
)set message test on
)set message auto off
)clear all
--S 1 of 764
t0:=(b+2*c*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)
--R
--R
--R +--------------+
--R +-------+ | 2
--R (1) (2c x + b)\|e x + d \|c x + b x + a
--R Type: Expression(Integer)
--E 1
--S 2 of 764
--r0:=4/7*(a+b*x+c*x^2)^(3/2)*sqrt(d+e*x)-2/105*(8*c^2*d^2+b^2*e^2-_
-- c*e*(11*b*d-10*a*e)-3*c*e*(2*c*d-b*e)*x)*sqrt(d+e*x)*sqrt(a+b*x+_
-- c*x^2)/(c*e^2)-1/105*elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(c*(5*c*e*(b*d-2*a*e)^2-_
-- (2*c*d-b*e)*(4*b*c*d^2-b^2*d*e-2*a*c*d*e-a*b*e^2))+(2*c*d-b*e)*_
-- (4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-_
-- 4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-_
-- 4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^(5/2)*e^3*sqrt(a+b*x+_
-- c*x^2))-1/105*(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*_
-- elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^(5/2)*e^4*sqrt(a+b*x+c*x^2))
--E 2
--S 3 of 764
--d0:=D(m0,x)
--E 3
)clear all
--S 4 of 764
t0:=(b+2*c*x)*sqrt(a+b*x+c*x^2)/sqrt(d+e*x)
--R
--R
--R +--------------+
--R | 2
--R (2c x + b)\|c x + b x + a
--R (1) ---------------------------
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 4
--S 5 of 764
--r0:=-2/15*(8*c*d-7*b*e-6*c*e*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/e^2-_
-- 1/15*elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-_
-- e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c))))*(2*c*(7*b^2*d*e+4*a*c*d*e-8*b*(c*d^2+a*e^2))+_
-- (16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/_
-- (2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^(3/2)*e^3*sqrt(2)*sqrt(a+b*x+_
-- c*x^2))-1/15*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*_
-- elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-_
-- sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+_
-- sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))/(c^(3/2)*e^4*sqrt(2)*sqrt(a+b*x+c*x^2))
--E 5
--S 6 of 764
--d0:=D(m0,x)
--E 6
)clear all
--S 7 of 764
t0:=(b+2*c*x)*sqrt(a+b*x+c*x^2)/(d+e*x)^(3/2)
--R
--R
--R +--------------+
--R | 2
--R (2c x + b)\|c x + b x + a
--R (1) ---------------------------
--R +-------+
--R (e x + d)\|e x + d
--R Type: Expression(Integer)
--E 7
--S 8 of 764
--r0:=2/3*(8*c*d-3*b*e+2*c*e*x)*sqrt(a+b*x+c*x^2)/(e^2*sqrt(d+e*x))-_
-- 1/3*elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))*sqrt(2)*(b*e*(b-4*sqrt(b^2-4*a*c))+c*(-4*a*e+_
-- 8*d*sqrt(b^2-4*a*c)))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/_
-- (e^3*sqrt(c)*sqrt(a+b*x+c*x^2))+4/3*(2*c*d-b*e)*elliptic_e(asin(_
-- sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*_
-- sqrt(2)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-_
-- sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+_
-- sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))/(e^4*sqrt(c)*sqrt(a+b*x+c*x^2))
--E 8
--S 9 of 764
--d0:=D(m0,x)
--E 9
)clear all
--S 10 of 764
t0:=(b+2*c*x)*sqrt(a+b*x+c*x^2)/(d+e*x)^(5/2)
--R
--R
--R +--------------+
--R | 2
--R (2c x + b)\|c x + b x + a
--R (1) ------------------------------
--R 2 2 2 +-------+
--R (e x + 2d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 10
--S 11 of 764
--r0:=2/3*(8*c*d-b*e+6*c*e*x)*sqrt(a+b*x+c*x^2)/(e^2*(d+e*x)^(3/2))-_
-- 2/3*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*sqrt(a+b*x+c*x^2)/_
-- (e^2*(c*d^2-b*d*e+a*e^2)*sqrt(d+e*x))-1/3*elliptic_f(asin(sqrt(2)*_
-- sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(2*c*(7*b^2*d*e+_
-- 4*a*c*d*e-8*b*(c*d^2+a*e^2))+(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-_
-- 3*a*e))*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/_
-- (2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(2*c*d-b*e+e*sqrt(b^2-4*a*c))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/_
-- (e^3*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))-_
-- 1/3*(16*c^2*d^2+b^2*e^2-4*c*e*(4*b*d-3*a*e))*elliptic_e(asin(sqrt(2)*_
-- sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-_
-- 4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-_
-- 4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-_
-- 4*a*c))))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(e^4*(c*d^2-b*d*e+a*e^2)*_
-- sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))
--E 11
--S 12 of 764
--d0:=D(m0,x)
--E 12
)clear all
--S 13 of 764
t0:=(b+2*c*x)*sqrt(a+b*x+c*x^2)/(d+e*x)^(7/2)
--R
--R
--R +--------------+
--R | 2
--R (2c x + b)\|c x + b x + a
--R (1) ----------------------------------------
--R 3 3 2 2 2 3 +-------+
--R (e x + 3d e x + 3d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 13
--S 14 of 764
--r0:=2/15*(16*c^2*d^2-b^2*e^2-4*c*e*(4*b*d-5*a*e))*sqrt(a+b*x+c*x^2)/_
-- (e^2*(c*d^2-b*d*e+a*e^2)*(d+e*x)^(3/2))-2/5*(8*c*d+b*e+10*c*e*x)*_
-- sqrt(a+b*x+c*x^2)/(e^2*(d+e*x)^(5/2))+4/15*(2*c*d-b*e)*(4*c^2*d^2-_
-- b^2*e^2-4*c*e*(b*d-2*a*e))*sqrt(a+b*x+c*x^2)/(e^2*(c*d^2-b*d*e+a*_
-- e^2)^2*sqrt(d+e*x))+1/15*elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+_
-- e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*_
-- a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(-c*(b^3*d*e^2-4*a*_
-- c*e*(c*d^2+5*a*e^2)+4*b*c*d*(2*c*d^2+5*a*e^2)-b^2*(11*c*d^2*e-a*e^3))+_
-- (2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-2*a*e))*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(2*c*d-b*e+e*sqrt(b^2-4*a*c))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e^3*(c*d^2-b*d*e+a*e^2)^2*sqrt(c)*_
-- sqrt(a+b*x+c*x^2))+1/15*(2*c*d-b*e)*(4*c^2*d^2-b^2*e^2-4*c*e*(b*d-_
-- 2*a*e))*elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*_
-- (b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(e^4*(c*d^2-b*d*e+a*e^2)^2*_
-- sqrt(c)*sqrt(a+b*x+c*x^2))
--E 14
--S 15 of 764
--d0:=D(m0,x)
--E 15
)clear all
--S 16 of 764
t0:=(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/sqrt(d+e*x)
--R
--R
--R +--------------+
--R 2 3 2 2 | 2
--R (2c x + 3b c x + (2a c + b )x + a b)\|c x + b x + a
--R (1) -------------------------------------------------------
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 16
--S 17 of 764
--r0:=-2/63*(16*c*d-15*b*e-14*c*e*x)*(a+b*x+c*x^2)^(3/2)*sqrt(d+e*x)/e^2-_
-- 2/315*(128*c^3*d^3-b^3*e^3+3*b*c*e^2*(37*b*d-36*a*e)-12*c^2*d*_
-- e*(20*b*d-11*a*e)-3*c*e*(32*c^2*d^2+b^2*e^2-4*c*e*(8*b*d-7*a*e))*x)*_
-- sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c*e^4)-1/315*elliptic_f(asin(sqrt(2)*_
-- sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(-c*((4*b*c*_
-- d^2-b^2*d*e-2*a*c*d*e-a*b*e^2)*(32*c^2*d^2+b^2*e^2-4*c*e*(8*b*d-_
-- 7*a*e))+5*c*e*(b*d-2*a*e)*(15*b^2*d*e+4*a*c*d*e-16*b*(c*d^2+a*e^2)))+_
-- (128*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(64*b*d-57*a*e)-b^2*c*e^3*(7*b*d-_
-- 15*a*e)+3*c^2*e^2*(45*b^2*d^2-76*a*b*d*e+28*a^2*e^2))*(b-sqrt(b^2-_
-- 4*a*c)))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-_
-- 4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-_
-- 4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^(5/2)*e^5*sqrt(a+b*x+_
-- c*x^2))-1/315*(128*c^4*d^4-b^4*e^4-4*c^3*d^2*e*(64*b*d-57*a*e)-_
-- b^2*c*e^3*(7*b*d-15*a*e)+3*c^2*e^2*(45*b^2*d^2-76*a*b*d*e+28*a^2*_
-- e^2))*elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*_
-- (b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*_
-- (b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*_
-- (b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c)))/(c^(5/2)*e^6*sqrt(a+b*x+c*x^2))
--E 17
--S 18 of 764
--d0:=D(m0,x)
--E 18
)clear all
--S 19 of 764
t0:=(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/(d+e*x)^(3/2)
--R
--R
--R +--------------+
--R 2 3 2 2 | 2
--R (2c x + 3b c x + (2a c + b )x + a b)\|c x + b x + a
--R (1) -------------------------------------------------------
--R +-------+
--R (e x + d)\|e x + d
--R Type: Expression(Integer)
--E 19
--S 20 of 764
--r0:=2/7*(16*c*d-7*b*e+2*c*e*x)*(a+b*x+c*x^2)^(3/2)/(e^2*sqrt(d+e*x))+_
-- 2/35*(128*c^2*d^2+51*b^2*e^2-4*c*e*(44*b*d-5*a*e)-48*c*e*(2*c*d-_
-- b*e)*x)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/e^4+1/35*elliptic_f(asin(_
-- sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),_
-- (2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- (-2*c*(51*b^3*d*e^2-8*a*c*e*(8*c*d^2+5*a*e^2)+4*b*c*d*(32*c*d^2+_
-- 45*a*e^2)-2*b^2*(88*c*d^2*e+27*a*e^3))+(2*c*d-b*e)*(128*c^2*d^2+_
-- 3*b^2*e^2-4*c*e*(32*b*d-29*a*e))*(b-sqrt(b^2-4*a*c)))*sqrt(2*c*d-_
-- e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-_
-- b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c))))/(c^(3/2)*e^5*sqrt(2)*sqrt(a+b*x+c*x^2))+_
-- 1/35*(2*c*d-b*e)*(128*c^2*d^2+3*b^2*e^2-4*c*e*(32*b*d-29*a*e))*_
-- elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-_
-- sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+_
-- sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))/(c^(3/2)*e^6*sqrt(2)*sqrt(a+b*x+c*x^2))
--E 20
--S 21 of 764
--d0:=D(m0,x)
--E 21
)clear all
--S 22 of 764
t0:=(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/(d+e*x)^(5/2)
--R
--R
--R +--------------+
--R 2 3 2 2 | 2
--R (2c x + 3b c x + (2a c + b )x + a b)\|c x + b x + a
--R (1) -------------------------------------------------------
--R 2 2 2 +-------+
--R (e x + 2d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 22
--S 23 of 764
--r0:=2/15*(16*c*d-5*b*e+6*c*e*x)*(a+b*x+c*x^2)^(3/2)/(e^2*(d+e*x)^(3/2))-_
-- 2/15*(128*c^2*d^2+15*b^2*e^2-4*c*e*(28*b*d-9*a*e)+16*c*e*(2*c*d-_
-- b*e)*x)*sqrt(a+b*x+c*x^2)/(e^4*sqrt(d+e*x))-1/15*elliptic_f(_
-- asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),_
-- (2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2)*(b^2*e^2*(8*b-23*sqrt(b^2-4*a*c))-64*c^2*d*(-a*e+2*d*_
-- sqrt(b^2-4*a*c))-4*c*e*(4*b^2*d+8*a*b*e-32*b*d*sqrt(b^2-4*a*c)+_
-- 9*a*e*sqrt(b^2-4*a*c)))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/_
-- (e^5*sqrt(c)*sqrt(a+b*x+c*x^2))-1/15*(128*c^2*d^2+23*b^2*e^2-_
-- 4*c*e*(32*b*d-9*a*e))*elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(e^6*sqrt(c)*sqrt(a+b*x+c*x^2))
--E 23
--S 24 of 764
--d0:=D(m0,x)
--E 24
)clear all
--S 25 of 764
t0:=(b+2*c*x)*(a+b*x+c*x^2)^(3/2)/(d+e*x)^(7/2)
--R
--R
--R +--------------+
--R 2 3 2 2 | 2
--R (2c x + 3b c x + (2a c + b )x + a b)\|c x + b x + a
--R (1) -------------------------------------------------------
--R 3 3 2 2 2 3 +-------+
--R (e x + 3d e x + 3d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 25
--S 26 of 764
--r0:=2/15*(16*c*d-3*b*e+10*c*e*x)*(a+b*x+c*x^2)^(3/2)/(e^2*(d+e*x)^(5/2))-_
-- 2/15*(128*c^2*d^2+3*b^2*e^2-20*c*e*(4*b*d-a*e)+48*c*e*(2*c*d-b*e)*x)*_
-- sqrt(a+b*x+c*x^2)/(e^4*(d+e*x)^(3/2))+2/15*(2*c*d-b*e)*(128*c^2*d^2+_
-- 3*b^2*e^2-4*c*e*(32*b*d-29*a*e))*sqrt(a+b*x+c*x^2)/(e^4*(c*d^2-_
-- b*d*e+a*e^2)*sqrt(d+e*x))+1/15*elliptic_f(asin(sqrt(2)*sqrt(c)*_
-- sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(-2*c*(51*b^3*d*e^2-_
-- 8*a*c*e*(8*c*d^2+5*a*e^2)+4*b*c*d*(32*c*d^2+45*a*e^2)-2*b^2*(88*c*_
-- d^2*e+27*a*e^3))+(2*c*d-b*e)*(128*c^2*d^2+3*b^2*e^2-4*c*e*(32*b*d-_
-- 29*a*e))*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/_
-- (2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(2*c*d-b*e+e*sqrt(b^2-4*a*c))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/_
-- (e^5*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))+_
-- 1/15*(2*c*d-b*e)*(128*c^2*d^2+3*b^2*e^2-4*c*e*(32*b*d-29*a*e))*_
-- elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-_
-- sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+_
-- sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))/(e^6*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(c)*_
-- sqrt(a+b*x+c*x^2))
--E 26
--S 27 of 764
--d0:=D(m0,x)
--E 27
)clear all
--S 28 of 764
t0:=(b+2*c*x)*(d+e*x)^(5/2)/sqrt(a+b*x+c*x^2)
--R
--R
--R 2 3 2 2 2 2 +-------+
--R (2c e x + (b e + 4c d e)x + (2b d e + 2c d )x + b d )\|e x + d
--R (1) ------------------------------------------------------------------
--R +--------------+
--R | 2
--R \|c x + b x + a
--R Type: Expression(Integer)
--E 28
--S 29 of 764
--r0:=2/7*(2*c*d-b*e)*(d+e*x)^(3/2)*sqrt(a+b*x+c*x^2)/c+4/7*(d+e*x)^(5/2)*_
-- sqrt(a+b*x+c*x^2)+4/21*(3*c^2*d^2+2*b^2*e^2-c*e*(3*b*d+5*a*e))*_
-- sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/c^2-1/21*elliptic_f(asin(sqrt(2)*_
-- sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-_
-- e*(b-sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(2*c*(4*b^3*_
-- d*e^2+2*a*c*e*(27*c*d^2-5*a*e^2)-b*c*d*(3*c*d^2+25*a*e^2)-b^2*_
-- (9*c*d^2*e-4*a*e^3))+(2*c*d-b*e)*(3*c^2*d^2+8*b^2*e^2-c*e*(3*b*d+_
-- 29*a*e))*(b-sqrt(b^2-4*a*c)))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/_
-- (c^(7/2)*e*sqrt(2)*sqrt(a+b*x+c*x^2))-1/21*(2*c*d-b*e)*(3*c^2*d^2+_
-- 8*b^2*e^2-c*e*(3*b*d+29*a*e))*elliptic_e(asin(sqrt(2)*sqrt(c)*_
-- sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+_
-- e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^(7/2)*_
-- e^2*sqrt(2)*sqrt(a+b*x+c*x^2))
--E 29
--S 30 of 764
--d0:=D(m0,x)
--E 30
)clear all
--S 31 of 764
t0:=(b+2*c*x)*(d+e*x)^(3/2)/sqrt(a+b*x+c*x^2)
--R
--R
--R 2 +-------+
--R (2c e x + (b e + 2c d)x + b d)\|e x + d
--R (1) -----------------------------------------
--R +--------------+
--R | 2
--R \|c x + b x + a
--R Type: Expression(Integer)
--E 31
--S 32 of 764
--r0:=4/5*(d+e*x)^(3/2)*sqrt(a+b*x+c*x^2)+2/5*(2*c*d-b*e)*sqrt(d+e*x)*_
-- sqrt(a+b*x+c*x^2)/c+1/5*elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(c*(b*c*d^2+b^2*d*e-8*a*c*_
-- d*e+a*b*e^2)-(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/_
-- (2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^(5/2)*e*sqrt(a+b*x+c*x^2))-_
-- 1/5*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*elliptic_e(asin(sqrt(2)*_
-- sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*_
-- (b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-_
-- e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-_
-- b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^(5/2)*_
-- e^2*sqrt(a+b*x+c*x^2))
--E 32
--S 33 of 764
--d0:=D(m0,x)
--E 33
)clear all
--S 34 of 764
t0:=(b+2*c*x)*sqrt(d+e*x)/sqrt(a+b*x+c*x^2)
--R
--R
--R +-------+
--R (2c x + b)\|e x + d
--R (1) --------------------
--R +--------------+
--R | 2
--R \|c x + b x + a
--R Type: Expression(Integer)
--E 34
--S 35 of 764
--r0:=4/3*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)+1/3*elliptic_f(asin(sqrt(2)*_
-- sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-_
-- e*(b-sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(b*e*(b-_
-- sqrt(b^2-4*a*c))+2*c*(-2*a*e+d*sqrt(b^2-4*a*c)))*sqrt(2*c*d-e*_
-- (b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-_
-- b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c))))/(c^(3/2)*e*sqrt(2)*sqrt(a+b*x+c*x^2))-_
-- 1/3*(2*c*d-b*e)*elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b-sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^(3/2)*e^2*sqrt(2)*sqrt(a+b*x+c*x^2))
--E 35
--S 36 of 764
--d0:=D(m0,x)
--E 36
)clear all
--S 37 of 764
t0:=(b+2*c*x)/(sqrt(d+e*x)*sqrt(a+b*x+c*x^2))
--R
--R
--R 2c x + b
--R (1) ---------------------------
--R +--------------+
--R +-------+ | 2
--R \|e x + d \|c x + b x + a
--R Type: Expression(Integer)
--E 37
--S 38 of 764
--r0:=elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(b^2-4*a*c)*sqrt(-e*(b+2*c*x-_
-- sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(2*c*d-b*e+_
-- e*sqrt(b^2-4*a*c))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))/(e*sqrt(c)*sqrt(a+b*x+c*x^2))-_
-- elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(e^2*sqrt(c)*sqrt(a+b*x+c*x^2))
--E 38
--S 39 of 764
--d0:=D(m0,x)
--E 39
)clear all
--S 40 of 764
t0:=(b+2*c*x)/((d+e*x)^(3/2)*sqrt(a+b*x+c*x^2))
--R
--R
--R 2c x + b
--R (1) ------------------------------------
--R +--------------+
--R +-------+ | 2
--R (e x + d)\|e x + d \|c x + b x + a
--R Type: Expression(Integer)
--E 40
--S 41 of 764
--r0:=2*(2*c*d-b*e)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*sqrt(d+e*x))-_
-- elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))*(b*e*(b-sqrt(b^2-4*a*c))+c*(-4*a*e+2*d*sqrt(b^2-_
-- 4*a*c)))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-_
-- 4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-_
-- 4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(c*d^2-b*d*e+a*e^2)*_
-- sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))+(2*c*d-b*e)*_
-- elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-_
-- sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+_
-- sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))/(e^2*(c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(c)*_
-- sqrt(a+b*x+c*x^2))
--E 41
--S 42 of 764
--d0:=D(m0,x)
--E 42
)clear all
--S 43 of 764
t0:=(b+2*c*x)/((d+e*x)^(5/2)*sqrt(a+b*x+c*x^2))
--R
--R
--R 2c x + b
--R (1) -----------------------------------------------
--R +--------------+
--R 2 2 2 +-------+ | 2
--R (e x + 2d e x + d )\|e x + d \|c x + b x + a
--R Type: Expression(Integer)
--E 43
--S 44 of 764
--r0:=2/3*(2*c*d-b*e)*sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)*(d+e*x)^(3/2))+_
-- 4/3*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*sqrt(a+b*x+c*x^2)/((c*d^2-_
-- b*d*e+a*e^2)^2*sqrt(d+e*x))-1/3*elliptic_f(asin(sqrt(2)*sqrt(c)*_
-- sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(c*(b*c*d^2+_
-- b^2*d*e-8*a*c*d*e+a*b*e^2)-(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*(b-_
-- sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*_
-- sqrt(b^2-4*a*c)))*sqrt(2*c*d-b*e+e*sqrt(b^2-4*a*c))*sqrt(-e*(b+_
-- 2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(e*(c*d^2-_
-- b*d*e+a*e^2)^2*sqrt(c)*sqrt(a+b*x+c*x^2))+1/3*(c^2*d^2+b^2*e^2-_
-- c*e*(b*d+3*a*e))*elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(e^2*(c*d^2-b*d*e+a*e^2)^2*_
-- sqrt(c)*sqrt(a+b*x+c*x^2))
--E 44
--S 45 of 764
--d0:=D(m0,x)
--E 45
)clear all
--S 46 of 764
t0:=(b+2*c*x)*(d+e*x)^(7/2)/(a+b*x+c*x^2)^(3/2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3
--R 2c e x + (b e + 6c d e )x + (3b d e + 6c d e)x + (3b d e + 2c d )x
--R +
--R 3
--R b d
--R *
--R +-------+
--R \|e x + d
--R /
--R +--------------+
--R 2 | 2
--R (c x + b x + a)\|c x + b x + a
--R Type: Expression(Integer)
--E 46
--S 47 of 764
--r0:=-2*(d+e*x)^(7/2)/sqrt(a+b*x+c*x^2)+14/5*e^2*(d+e*x)^(3/2)*_
-- sqrt(a+b*x+c*x^2)/c+56/15*e^2*(2*c*d-b*e)*sqrt(d+e*x)*sqrt(a+_
-- b*x+c*x^2)/c^2-7/15*elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))*(-2*c*(15*c^2*d^3+4*b*e^2*(b*d+a*e)-_
-- c*d*e*(11*b*d+17*a*e))+e*(23*c^2*d^2+8*b^2*e^2-c*e*(23*b*d+9*a*e))*_
-- (b-sqrt(b^2-4*a*c)))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/_
-- (c^(7/2)*sqrt(2)*sqrt(a+b*x+c*x^2))-7/15*(23*c^2*d^2+8*b^2*e^2-_
-- c*e*(23*b*d+9*a*e))*elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b-sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^(7/2)*sqrt(2)*sqrt(a+b*x+c*x^2))
--E 47
--S 48 of 764
--d0:=D(m0,x)
--E 48
)clear all
--S 49 of 764
t0:=(b+2*c*x)*(d+e*x)^(5/2)/(a+b*x+c*x^2)^(3/2)
--R
--R
--R 2 3 2 2 2 2 +-------+
--R (2c e x + (b e + 4c d e)x + (2b d e + 2c d )x + b d )\|e x + d
--R (1) ------------------------------------------------------------------
--R +--------------+
--R 2 | 2
--R (c x + b x + a)\|c x + b x + a
--R Type: Expression(Integer)
--E 49
--S 50 of 764
--r0:=-2*(d+e*x)^(5/2)/sqrt(a+b*x+c*x^2)+10/3*e^2*sqrt(d+e*x)*_
-- sqrt(a+b*x+c*x^2)/c-5/3*elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(-c*(3*c*d^2-e*(b*d+a*e))+_
-- e*(2*c*d-b*e)*(b-sqrt(b^2-4*a*c)))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/_
-- (c^(5/2)*sqrt(a+b*x+c*x^2))-5/3*(2*c*d-b*e)*elliptic_e(asin(sqrt(2)*_
-- sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-e*_
-- (b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+_
-- e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^(5/2)*_
-- sqrt(a+b*x+c*x^2))
--E 50
--S 51 of 764
--d0:=D(m0,x)
--E 51
)clear all
--S 52 of 764
t0:=(b+2*c*x)*(d+e*x)^(3/2)/(a+b*x+c*x^2)^(3/2)
--R
--R
--R 2 +-------+
--R (2c e x + (b e + 2c d)x + b d)\|e x + d
--R (1) -----------------------------------------
--R +--------------+
--R 2 | 2
--R (c x + b x + a)\|c x + b x + a
--R Type: Expression(Integer)
--E 52
--S 53 of 764
--r0:=-2*(d+e*x)^(3/2)/sqrt(a+b*x+c*x^2)+3*elliptic_f(asin(sqrt(2)*sqrt(c)*_
-- sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))^(3/2)*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-_
-- b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c))))/(c^(3/2)*sqrt(2)*sqrt(a+b*x+c*x^2))-3*_
-- elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-_
-- sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+_
-- sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))/(c^(3/2)*sqrt(2)*sqrt(a+b*x+c*x^2))
--E 53
--S 54 of 764
--d0:=D(m0,x)
--E 54
)clear all
--S 55 of 764
t0:=(b+2*c*x)*sqrt(d+e*x)/(a+b*x+c*x^2)^(3/2)
--R
--R
--R +-------+
--R (2c x + b)\|e x + d
--R (1) ---------------------------------
--R +--------------+
--R 2 | 2
--R (c x + b x + a)\|c x + b x + a
--R Type: Expression(Integer)
--E 55
--S 56 of 764
--r0:=-2*sqrt(d+e*x)/sqrt(a+b*x+c*x^2)+elliptic_f(asin(sqrt(2)*sqrt(c)*_
-- sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(2*c*d-_
-- e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+_
-- e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))/(sqrt(c)*sqrt(a+b*x+c*x^2))
--E 56
--S 57 of 764
--d0:=D(m0,x)
--E 57
)clear all
--S 58 of 764
t0:=(b+2*c*x)/((a+b*x+c*x^2)^(3/2)*sqrt(d+e*x))
--R
--R
--R 2c x + b
--R (1) -------------------------------------------
--R +--------------+
--R 2 +-------+ | 2
--R (c x + b x + a)\|e x + d \|c x + b x + a
--R Type: Expression(Integer)
--E 58
--S 59 of 764
--r0:=-2*((b^2-4*a*c)*(c*d-b*e)-c*(b^2-4*a*c)*e*x)*sqrt(d+e*x)/((b^2-4*a*c)*_
-- (c*d^2-b*d*e+a*e^2)*sqrt(a+b*x+c*x^2))-elliptic_f(asin(sqrt(2)*_
-- sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))^(3/2)*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+_
-- e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))/((c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(c)*sqrt(a+b*x+_
-- c*x^2))+elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-_
-- sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+_
-- sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))/((c*d^2-b*d*e+a*e^2)*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))
--E 59
--S 60 of 764
--d0:=D(m0,x)
--E 60
)clear all
--S 61 of 764
t0:=(b+2*c*x)/((d+e*x)^(3/2)*(a+b*x+c*x^2)^(3/2))
--R
--R
--R (1)
--R 2c x + b
--R ------------------------------------------------------------------------
--R +--------------+
--R 3 2 +-------+ | 2
--R (c e x + (b e + c d)x + (a e + b d)x + a d)\|e x + d \|c x + b x + a
--R Type: Expression(Integer)
--E 61
--S 62 of 764
--r0:=-2*((b^2-4*a*c)*(c*d-b*e)-c*(b^2-4*a*c)*e*x)/((b^2-4*a*c)*(c*d^2-_
-- b*d*e+a*e^2)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2))+4*e^2*(2*c*d-b*e)*_
-- sqrt(a+b*x+c*x^2)/((c*d^2-b*d*e+a*e^2)^2*sqrt(d+e*x))-_
-- elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))*sqrt(2)*(3*c^2*d^2+b*e^2*(b-sqrt(b^2-4*a*c))-_
-- c*e*(3*b*d+a*e-2*d*sqrt(b^2-4*a*c)))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/_
-- ((c*d^2-b*d*e+a*e^2)^2*sqrt(c)*sqrt(a+b*x+c*x^2))+(2*c*d-b*e)*_
-- elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/((c*d^2-b*d*e+a*e^2)^2*sqrt(c)*_
-- sqrt(a+b*x+c*x^2))
--E 62
--S 63 of 764
--d0:=D(m0,x)
--E 63
)clear all
--S 64 of 764
t0:=(b+2*c*x)*(d+e*x)^(7/2)/(a+b*x+c*x^2)^(5/2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3
--R 2c e x + (b e + 6c d e )x + (3b d e + 6c d e)x + (3b d e + 2c d )x
--R +
--R 3
--R b d
--R *
--R +-------+
--R \|e x + d
--R /
--R +--------------+
--R 2 4 3 2 2 2 | 2
--R (c x + 2b c x + (2a c + b )x + 2a b x + a )\|c x + b x + a
--R Type: Expression(Integer)
--E 64
--S 65 of 764
--r0:=-2/3*(d+e*x)^(7/2)/(a+b*x+c*x^2)^(3/2)-14/3*e*(d+e*x)^(3/2)*(b*d-_
-- 2*a*e+(2*c*d-b*e)*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+14/3*e^2*(2*c*d-_
-- b*e)*sqrt(d+e*x)*sqrt(a+b*x+c*x^2)/(c*(b^2-4*a*c))+7/3*e*_
-- elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))*sqrt(2)*(c*(b*c*d^2+b^2*d*e-8*a*c*d*e+a*b*e^2)-_
-- (c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*(b-sqrt(b^2-4*a*c)))*sqrt(2*c*d-_
-- e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-_
-- b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c))))/(c^(5/2)*(b^2-4*a*c)*sqrt(a+b*x+c*x^2))-_
-- 7/3*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*elliptic_e(asin(sqrt(2)*_
-- sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*_
-- (b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-_
-- e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+_
-- e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^(5/2)*_
-- (b^2-4*a*c)*sqrt(a+b*x+c*x^2))
--E 65
--S 66 of 764
--d0:=D(m0,x)
--E 66
)clear all
--S 67 of 764
t0:=(b+2*c*x)*(d+e*x)^(5/2)/(a+b*x+c*x^2)^(5/2)
--R
--R
--R 2 3 2 2 2 2 +-------+
--R (2c e x + (b e + 4c d e)x + (2b d e + 2c d )x + b d )\|e x + d
--R (1) ------------------------------------------------------------------
--R +--------------+
--R 2 4 3 2 2 2 | 2
--R (c x + 2b c x + (2a c + b )x + 2a b x + a )\|c x + b x + a
--R Type: Expression(Integer)
--E 67
--S 68 of 764
--r0:=-2/3*(d+e*x)^(5/2)/(a+b*x+c*x^2)^(3/2)-10/3*e*(b*d-2*a*e+(2*c*d-_
-- b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+5/3*e*_
-- elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c))))*(b*e*(b-sqrt(b^2-4*a*c))+c*(-4*a*e+2*d*_
-- sqrt(b^2-4*a*c)))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-_
-- sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+_
-- sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/(c^(3/2)*(b^2-_
-- 4*a*c)*sqrt(2)*sqrt(a+b*x+c*x^2))-5/3*(2*c*d-b*e)*elliptic_e(_
-- asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*(2*c*d-_
-- e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-_
-- b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(c^(3/2)*_
-- (b^2-4*a*c)*sqrt(2)*sqrt(a+b*x+c*x^2))
--E 68
--S 69 of 764
--d0:=D(m0,x)
--E 69
)clear all
--S 70 of 764
t0:=(b+2*c*x)*(d+e*x)^(3/2)/(a+b*x+c*x^2)^(5/2)
--R
--R
--R 2 +-------+
--R (2c e x + (b e + 2c d)x + b d)\|e x + d
--R (1) ---------------------------------------------------------------
--R +--------------+
--R 2 4 3 2 2 2 | 2
--R (c x + 2b c x + (2a c + b )x + 2a b x + a )\|c x + b x + a
--R Type: Expression(Integer)
--E 70
--S 71 of 764
--r0:=-2/3*(d+e*x)^(3/2)/(a+b*x+c*x^2)^(3/2)-2*e*(b+2*c*x)*sqrt(d+e*x)/_
-- ((b^2-4*a*c)*sqrt(a+b*x+c*x^2))+e*elliptic_f(asin(sqrt(2)*sqrt(c)*_
-- sqrt(d+e*x)/sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c)))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*sqrt(2*c*d-_
-- e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-_
-- b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-_
-- e*(b+sqrt(b^2-4*a*c))))/(sqrt(c)*sqrt(b^2-4*a*c)*sqrt(a+b*x+c*x^2))-_
-- elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+_
-- sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-_
-- sqrt(b^2-4*a*c))))*sqrt(2)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/((b^2-4*a*c)*sqrt(c)*sqrt(a+b*x+c*x^2))
--E 71
--S 72 of 764
--d0:=D(m0,x)
--E 72
)clear all
--S 73 of 764
t0:=(b+2*c*x)*sqrt(d+e*x)/(a+b*x+c*x^2)^(5/2)
--R
--R
--R +-------+
--R (2c x + b)\|e x + d
--R (1) ---------------------------------------------------------------
--R +--------------+
--R 2 4 3 2 2 2 | 2
--R (c x + 2b c x + (2a c + b )x + 2a b x + a )\|c x + b x + a
--R Type: Expression(Integer)
--E 73
--S 74 of 764
--r0:=-2/3*sqrt(d+e*x)/(a+b*x+c*x^2)^(3/2)-2/3*e*(b*c*d-b^2*e+2*a*c*e+c*_
-- (2*c*d-b*e)*x)*sqrt(d+e*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)*_
-- sqrt(a+b*x+c*x^2))+1/3*e*elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))*(b*e*(b-sqrt(b^2-4*a*c))+c*(-4*a*e+_
-- 2*d*sqrt(b^2-4*a*c)))*sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))/_
-- ((b^2-4*a*c)*(c*d^2-e*(b*d-a*e))*sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))-_
-- 1/3*(2*c*d-b*e)*elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b-sqrt(b^2-4*a*c))))*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/(2*c*d-e*(b+sqrt(b^2-4*a*c))))*_
-- sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/((b^2-4*a*c)*(c*d^2-e*(b*d-a*e))*_
-- sqrt(2)*sqrt(c)*sqrt(a+b*x+c*x^2))
--E 74
--S 75 of 764
--d0:=D(m0,x)
--E 75
)clear all
--S 76 of 764
t0:=(b+2*c*x)/((a+b*x+c*x^2)^(5/2)*sqrt(d+e*x))
--R
--R
--R (1)
--R 2c x + b
--R -------------------------------------------------------------------------
--R +--------------+
--R 2 4 3 2 2 2 +-------+ | 2
--R (c x + 2b c x + (2a c + b )x + 2a b x + a )\|e x + d \|c x + b x + a
--R Type: Expression(Integer)
--E 76
--S 77 of 764
--r0:=-2/3*((b^2-4*a*c)*(c*d-b*e)-c*(b^2-4*a*c)*e*x)*sqrt(d+e*x)/((b^2-_
-- 4*a*c)*(c*d^2-b*d*e+a*e^2)*(a+b*x+c*x^2)^(3/2))-2/3*e*(3*b^2*c*d*e-_
-- 8*a*c^2*d*e-2*b^3*e^2-b*c*(c*d^2-7*a*e^2)-2*c*(c^2*d^2+b^2*e^2-_
-- c*e*(b*d+3*a*e))*x)*sqrt(d+e*x)/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*_
-- sqrt(a+b*x+c*x^2))-1/3*e*elliptic_f(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/_
-- sqrt(2*c*d-e*(b-sqrt(b^2-4*a*c)))),(2*c*d-e*(b-sqrt(b^2-4*a*c)))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2)*(c*(b*c*d^2+b^2*d*e-8*a*c*d*e+_
-- a*b*e^2)-(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*(b-sqrt(b^2-4*a*c)))*_
-- sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/(2*c*d-b*e+e*sqrt(b^2-4*a*c)))*_
-- sqrt(2*c*d-b*e+e*sqrt(b^2-4*a*c))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))/((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*_
-- sqrt(c)*sqrt(a+b*x+c*x^2))+1/3*(c^2*d^2+b^2*e^2-c*e*(b*d+3*a*e))*_
-- elliptic_e(asin(sqrt(2)*sqrt(c)*sqrt(d+e*x)/sqrt(2*c*d-e*(b+sqrt(b^2-_
-- 4*a*c)))),(2*c*d-e*(b+sqrt(b^2-4*a*c)))/(2*c*d-e*(b-sqrt(b^2-4*a*c))))*_
-- sqrt(2)*(2*c*d-e*(b-sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x-sqrt(b^2-4*a*c))/_
-- (2*c*d-b*e+e*sqrt(b^2-4*a*c)))*sqrt(-e*(b+2*c*x+sqrt(b^2-4*a*c))/_
-- (2*c*d-e*(b+sqrt(b^2-4*a*c))))*sqrt(2*c*d-e*(b+sqrt(b^2-4*a*c)))/_
-- ((b^2-4*a*c)*(c*d^2-b*d*e+a*e^2)^2*sqrt(c)*sqrt(a+b*x+c*x^2))
--E 77
--S 78 of 764
--d0:=D(m0,x)
--E 78
)clear all
--S 79 of 764
t0:=(A+B*x)*(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R 2 3 2 2 2 2 m
--R (1) (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )(e x + d)
--R Type: Expression(Integer)
--E 79
--S 80 of 764
r0:=-(b*d-a*e)^2*(B*d-A*e)*(d+e*x)^(1+m)/(e^4*(1+m))+(b*d-a*e)*(3*b*B*d-_
2*A*b*e-a*B*e)*(d+e*x)^(2+m)/(e^4*(2+m))-b*(3*b*B*d-A*b*e-2*a*B*e)*_
(d+e*x)^(3+m)/(e^4*(3+m))+b^2*B*(d+e*x)^(4+m)/(e^4*(4+m))
--R
--R
--R (2)
--R 2 3 2 2 2 2 m + 4
--R (B b m + 6B b m + 11B b m + 6B b )(e x + d)
--R +
--R 2 2 3 2 2 2
--R ((A b + 2B a b)e - 3B b d)m + ((7A b + 14B a b)e - 21B b d)m
--R +
--R 2 2 2 2
--R ((14A b + 28B a b)e - 42B b d)m + (8A b + 16B a b)e - 24B b d
--R *
--R m + 3
--R (e x + d)
--R +
--R 2 2 2 2 2 3
--R ((2A a b + B a )e + (- 2A b - 4B a b)d e + 3B b d )m
--R +
--R 2 2 2 2 2 2
--R ((16A a b + 8B a )e + (- 16A b - 32B a b)d e + 24B b d )m
--R +
--R 2 2 2 2 2
--R ((38A a b + 19B a )e + (- 38A b - 76B a b)d e + 57B b d )m
--R +
--R 2 2 2 2 2
--R (24A a b + 12B a )e + (- 24A b - 48B a b)d e + 36B b d
--R *
--R m + 2
--R (e x + d)
--R +
--R 2 3 2 2 2 2 2 3 3
--R (A a e + (- 2A a b - B a )d e + (A b + 2B a b)d e - B b d )m
--R +
--R 2 3 2 2 2 2
--R 9A a e + (- 18A a b - 9B a )d e + (9A b + 18B a b)d e
--R +
--R 2 3
--R - 9B b d
--R *
--R 2
--R m
--R +
--R 2 3 2 2 2 2
--R 26A a e + (- 52A a b - 26B a )d e + (26A b + 52B a b)d e
--R +
--R 2 3
--R - 26B b d
--R *
--R m
--R +
--R 2 3 2 2 2 2 2 3
--R 24A a e + (- 48A a b - 24B a )d e + (24A b + 48B a b)d e - 24B b d
--R *
--R m + 1
--R (e x + d)
--R /
--R 4 4 4 3 4 2 4 4
--R e m + 10e m + 35e m + 50e m + 24e
--R Type: Expression(Integer)
--E 80
--S 81 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 81
)clear all
--S 82 of 764
t0:=(A+B*x)*(d+e*x)^5*(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 2 5 8 2 5 2 4 7
--R B b e x + ((A b + 2B a b)e + 5B b d e )x
--R +
--R 2 5 2 4 2 2 3 6
--R ((2A a b + B a )e + (5A b + 10B a b)d e + 10B b d e )x
--R +
--R 2 5 2 4 2 2 3 2 3 2 5
--R (A a e + (10A a b + 5B a )d e + (10A b + 20B a b)d e + 10B b d e )x
--R +
--R 2 4 2 2 3 2 3 2 2 4 4
--R (5A a d e + (20A a b + 10B a )d e + (10A b + 20B a b)d e + 5B b d e)x
--R +
--R 2 2 3 2 3 2 2 4 2 5 3
--R (10A a d e + (20A a b + 10B a )d e + (5A b + 10B a b)d e + B b d )x
--R +
--R 2 3 2 2 4 2 5 2
--R (10A a d e + (10A a b + 5B a )d e + (A b + 2B a b)d )x
--R +
--R 2 4 2 5 2 5
--R (5A a d e + (2A a b + B a )d )x + A a d
--R Type: Polynomial(Integer)
--E 82
--S 83 of 764
r0:=-1/6*(b*d-a*e)^2*(B*d-A*e)*(d+e*x)^6/e^4+1/7*(b*d-a*e)*(3*b*B*d-2*A*b*e-_
a*B*e)*(d+e*x)^7/e^4-1/8*b*(3*b*B*d-A*b*e-2*a*B*e)*(d+e*x)^8/e^4+_
1/9*b^2*B*(d+e*x)^9/e^4
--R
--R
--R (2)
--R 1 2 9 9 1 2 1 9 5 2 8 8
--R - B b e x + ((- A b + - B a b)e + - B b d e )x
--R 9 8 4 8
--R +
--R 2 1 2 9 5 2 10 8 10 2 2 7 7
--R ((- A a b + - B a )e + (- A b + -- B a b)d e + -- B b d e )x
--R 7 7 7 7 7
--R +
--R 1 2 9 5 5 2 8 5 2 10 2 7
--R - A a e + (- A a b + - B a )d e + (- A b + -- B a b)d e
--R 6 3 6 3 3
--R +
--R 5 2 3 6
--R - B b d e
--R 3
--R *
--R 6
--R x
--R +
--R 2 8 2 2 7 2 3 6 2 4 5 5
--R (A a d e + (4A a b + 2B a )d e + (2A b + 4B a b)d e + B b d e )x
--R +
--R 5 2 2 7 5 2 3 6 5 2 5 4 5
--R - A a d e + (5A a b + - B a )d e + (- A b + - B a b)d e
--R 2 2 4 2
--R +
--R 1 2 5 4
--R - B b d e
--R 4
--R *
--R 4
--R x
--R +
--R 10 2 3 6 10 5 2 4 5 1 2 2 5 4 3
--R (-- A a d e + (-- A a b + - B a )d e + (- A b + - B a b)d e )x
--R 3 3 3 3 3
--R +
--R 5 2 4 5 1 2 5 4 2 2 5 4 1 2 6 3
--R (- A a d e + (A a b + - B a )d e )x + A a d e x + - A a d e
--R 2 2 6
--R +
--R 1 1 2 7 2 1 2 1 8 1 2 9
--R (- -- A a b - -- B a )d e + (--- A b + -- B a b)d e - --- B b d
--R 21 42 168 84 504
--R /
--R 4
--R e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 83
--S 84 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 84
)clear all
--S 85 of 764
t0:=(A+B*x)*(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 2 4 7 2 4 2 3 6
--R B b e x + ((A b + 2B a b)e + 4B b d e )x
--R +
--R 2 4 2 3 2 2 2 5
--R ((2A a b + B a )e + (4A b + 8B a b)d e + 6B b d e )x
--R +
--R 2 4 2 3 2 2 2 2 3 4
--R (A a e + (8A a b + 4B a )d e + (6A b + 12B a b)d e + 4B b d e)x
--R +
--R 2 3 2 2 2 2 3 2 4 3
--R (4A a d e + (12A a b + 6B a )d e + (4A b + 8B a b)d e + B b d )x
--R +
--R 2 2 2 2 3 2 4 2
--R (6A a d e + (8A a b + 4B a )d e + (A b + 2B a b)d )x
--R +
--R 2 3 2 4 2 4
--R (4A a d e + (2A a b + B a )d )x + A a d
--R Type: Polynomial(Integer)
--E 85
--S 86 of 764
r0:=-1/5*(b*d-a*e)^2*(B*d-A*e)*(d+e*x)^5/e^4+1/6*(b*d-a*e)*(3*b*B*d-2*A*b*e-_
a*B*e)*(d+e*x)^6/e^4-1/7*b*(3*b*B*d-A*b*e-2*a*B*e)*(d+e*x)^7/e^4+_
1/8*b^2*B*(d+e*x)^8/e^4
--R
--R
--R (2)
--R 1 2 8 8 1 2 2 8 4 2 7 7
--R - B b e x + ((- A b + - B a b)e + - B b d e )x
--R 8 7 7 7
--R +
--R 1 1 2 8 2 2 4 7 2 2 6 6
--R ((- A a b + - B a )e + (- A b + - B a b)d e + B b d e )x
--R 3 6 3 3
--R +
--R 1 2 8 8 4 2 7 6 2 12 2 6
--R - A a e + (- A a b + - B a )d e + (- A b + -- B a b)d e
--R 5 5 5 5 5
--R +
--R 4 2 3 5
--R - B b d e
--R 5
--R *
--R 5
--R x
--R +
--R 2 7 3 2 2 6 2 3 5 1 2 4 4 4
--R (A a d e + (3A a b + - B a )d e + (A b + 2B a b)d e + - B b d e )x
--R 2 4
--R +
--R 2 2 6 8 4 2 3 5 1 2 2 4 4 3
--R (2A a d e + (- A a b + - B a )d e + (- A b + - B a b)d e )x
--R 3 3 3 3
--R +
--R 2 3 5 1 2 4 4 2 2 4 4 1 2 5 3
--R (2A a d e + (A a b + - B a )d e )x + A a d e x + - A a d e
--R 2 5
--R +
--R 1 1 2 6 2 1 2 2 7 1 2 8
--R (- -- A a b - -- B a )d e + (--- A b + --- B a b)d e - --- B b d
--R 15 30 105 105 280
--R /
--R 4
--R e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 86
--S 87 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 87
)clear all
--S 88 of 764
t0:=(A+B*x)*(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 2 3 6 2 3 2 2 5
--R B b e x + ((A b + 2B a b)e + 3B b d e )x
--R +
--R 2 3 2 2 2 2 4
--R ((2A a b + B a )e + (3A b + 6B a b)d e + 3B b d e)x
--R +
--R 2 3 2 2 2 2 2 3 3
--R (A a e + (6A a b + 3B a )d e + (3A b + 6B a b)d e + B b d )x
--R +
--R 2 2 2 2 2 3 2
--R (3A a d e + (6A a b + 3B a )d e + (A b + 2B a b)d )x
--R +
--R 2 2 2 3 2 3
--R (3A a d e + (2A a b + B a )d )x + A a d
--R Type: Polynomial(Integer)
--E 88
--S 89 of 764
r0:=-1/4*(b*d-a*e)^2*(B*d-A*e)*(d+e*x)^4/e^4+1/5*(b*d-a*e)*(3*b*B*d-2*A*b*e-_
a*B*e)*(d+e*x)^5/e^4-1/6*b*(3*b*B*d-A*b*e-2*a*B*e)*(d+e*x)^6/e^4+_
1/7*b^2*B*(d+e*x)^7/e^4
--R
--R
--R (2)
--R 1 2 7 7 1 2 1 7 1 2 6 6
--R - B b e x + ((- A b + - B a b)e + - B b d e )x
--R 7 6 3 2
--R +
--R 2 1 2 7 3 2 6 6 3 2 2 5 5
--R ((- A a b + - B a )e + (- A b + - B a b)d e + - B b d e )x
--R 5 5 5 5 5
--R +
--R 1 2 7 3 3 2 6 3 2 3 2 5
--R - A a e + (- A a b + - B a )d e + (- A b + - B a b)d e
--R 4 2 4 4 2
--R +
--R 1 2 3 4
--R - B b d e
--R 4
--R *
--R 4
--R x
--R +
--R 2 6 2 2 5 1 2 2 3 4 3
--R (A a d e + (2A a b + B a )d e + (- A b + - B a b)d e )x
--R 3 3
--R +
--R 3 2 2 5 1 2 3 4 2 2 3 4 1 2 4 3
--R (- A a d e + (A a b + - B a )d e )x + A a d e x + - A a d e
--R 2 2 4
--R +
--R 1 1 2 5 2 1 2 1 6 1 2 7
--R (- -- A a b - -- B a )d e + (-- A b + -- B a b)d e - --- B b d
--R 10 20 60 30 140
--R /
--R 4
--R e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 89
--S 90 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 90
)clear all
--S 91 of 764
t0:=(A+B*x)*(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 2 2 5 2 2 2 4
--R B b e x + ((A b + 2B a b)e + 2B b d e)x
--R +
--R 2 2 2 2 2 3
--R ((2A a b + B a )e + (2A b + 4B a b)d e + B b d )x
--R +
--R 2 2 2 2 2 2
--R (A a e + (4A a b + 2B a )d e + (A b + 2B a b)d )x
--R +
--R 2 2 2 2 2
--R (2A a d e + (2A a b + B a )d )x + A a d
--R Type: Polynomial(Integer)
--E 91
--S 92 of 764
r0:=1/3*(A*b-a*B)*(b*d-a*e)^2*(a+b*x)^3/b^4+1/4*(b*d-a*e)*(b*B*d+2*A*b*e-_
3*a*B*e)*(a+b*x)^4/b^4+1/5*e*(2*b*B*d+A*b*e-3*a*B*e)*(a+b*x)^5/b^4+_
1/6*B*e^2*(a+b*x)^6/b^4
--R
--R
--R (2)
--R 1 6 2 6 1 6 2 5 2 2 6 5
--R - B b e x + ((- A b + - B a b )e + - B b d e)x
--R 6 5 5 5
--R +
--R 1 5 1 2 4 2 1 6 5 1 6 2 4
--R ((- A a b + - B a b )e + (- A b + B a b )d e + - B b d )x
--R 2 4 2 4
--R +
--R 1 2 4 2 4 5 2 2 4 1 6 2 5 2 3
--R (- A a b e + (- A a b + - B a b )d e + (- A b + - B a b )d )x
--R 3 3 3 3 3
--R +
--R 2 4 5 1 2 4 2 2 2 4 2
--R (A a b d e + (A a b + - B a b )d )x + A a b d x
--R 2
--R +
--R 1 5 1 6 2 1 4 2 1 5
--R (-- A a b - -- B a )e + (- - A a b + -- B a b)d e
--R 30 60 6 15
--R +
--R 1 3 3 1 4 2 2
--R (- A a b - -- B a b )d
--R 3 12
--R /
--R 4
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 92
--S 93 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 93
)clear all
--S 94 of 764
t0:=(A+B*x)*(d+e*x)*(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 2 4 2 2 3
--R B b e x + ((A b + 2B a b)e + B b d)x
--R +
--R 2 2 2 2 2 2
--R ((2A a b + B a )e + (A b + 2B a b)d)x + (A a e + (2A a b + B a )d)x + A a d
--R Type: Polynomial(Integer)
--E 94
--S 95 of 764
r0:=1/3*(A*b-a*B)*(b*d-a*e)*(a+b*x)^3/b^3+1/4*(b*B*d+A*b*e-2*a*B*e)*_
(a+b*x)^4/b^3+1/5*B*e*(a+b*x)^5/b^3
--R
--R
--R (2)
--R 1 5 5 1 5 1 4 1 5 4
--R - B b e x + ((- A b + - B a b )e + - B b d)x
--R 5 4 2 4
--R +
--R 2 4 1 2 3 1 5 2 4 3
--R ((- A a b + - B a b )e + (- A b + - B a b )d)x
--R 3 3 3 3
--R +
--R 1 2 3 4 1 2 3 2 2 3
--R (- A a b e + (A a b + - B a b )d)x + A a b d x
--R 2 2
--R +
--R 1 4 1 5 1 3 2 1 4
--R (- -- A a b + -- B a )e + (- A a b - -- B a b)d
--R 12 30 3 12
--R /
--R 3
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 95
--S 96 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 96
)clear all
--S 97 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R 2 3 2 2 2 2
--R (1) B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R Type: Polynomial(Integer)
--E 97
--S 98 of 764
r0:=1/3*(A*b-a*B)*(a+b*x)^3/b^2+1/4*B*(a+b*x)^4/b^2
--R
--R
--R (2)
--R 1 4 4 1 4 2 3 3 3 1 2 2 2 2 2
--R - B b x + (- A b + - B a b )x + (A a b + - B a b )x + A a b x
--R 4 3 3 2
--R +
--R 1 3 1 4
--R - A a b - -- B a
--R 3 12
--R /
--R 2
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 98
--S 99 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 99
)clear all
--S 100 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)
--R
--R
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R (1) ----------------------------------------------------
--R e x + d
--R Type: Fraction(Polynomial(Integer))
--E 100
--S 101 of 764
r0:=b*(b*d-a*e)*(B*d-A*e)*x/e^3-1/2*(B*d-A*e)*(a+b*x)^2/e^2+_
1/3*B*(a+b*x)^3/(b*e)-(b*d-a*e)^2*(B*d-A*e)*log(d+e*x)/e^4
--R
--R
--R (2)
--R 2 3 2 2 2 3 2 2
--R 6A a b e + (- 12A a b - 6B a b)d e + (6A b + 12B a b )d e
--R +
--R 3 3
--R - 6B b d
--R *
--R log(e x + d)
--R +
--R 3 3 3 3 2 3 3 2 2
--R 2B b e x + ((3A b + 6B a b )e - 3B b d e )x
--R +
--R 2 2 3 3 2 2 3 2
--R ((12A a b + 6B a b)e + (- 6A b - 12B a b )d e + 6B b d e)x
--R +
--R 2 3 3 2 2
--R (3A a b + 2B a )e - 3B a b d e
--R /
--R 4
--R 6b e
--R Type: Expression(Integer)
--E 101
--S 102 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 102
)clear all
--S 103 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^2
--R
--R
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R (1) ----------------------------------------------------
--R 2 2 2
--R e x + 2d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 103
--S 104 of 764
r0:=-b*(2*b*B*d-A*b*e-2*a*B*e)*x/e^3+1/2*b^2*B*x^2/e^2+(b*d-a*e)^2*_
(B*d-A*e)/(e^4*(d+e*x))+(b*d-a*e)*(3*b*B*d-2*A*b*e-a*B*e)*log(d+e*x)/e^4
--R
--R
--R (2)
--R 2 3 2 2 2 2
--R ((4A a b + 2B a )e + (- 4A b - 8B a b)d e + 6B b d e)x
--R +
--R 2 2 2 2 2 3
--R (4A a b + 2B a )d e + (- 4A b - 8B a b)d e + 6B b d
--R *
--R log(e x + d)
--R +
--R 2 3 3 2 3 2 2 2
--R B b e x + ((2A b + 4B a b)e - 3B b d e )x
--R +
--R 2 2 2 2 2 3 2 2
--R ((2A b + 4B a b)d e - 4B b d e)x - 2A a e + (4A a b + 2B a )d e
--R +
--R 2 2 2 3
--R (- 2A b - 4B a b)d e + 2B b d
--R /
--R 5 4
--R 2e x + 2d e
--R Type: Expression(Integer)
--E 104
--S 105 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 105
)clear all
--S 106 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^3
--R
--R
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R (1) ----------------------------------------------------
--R 3 3 2 2 2 3
--R e x + 3d e x + 3d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 106
--S 107 of 764
r0:=b^2*B*x/e^3+1/2*(b*d-a*e)^2*(B*d-A*e)/(e^4*(d+e*x)^2)-(b*d-a*e)*_
(3*b*B*d-2*A*b*e-a*B*e)/(e^4*(d+e*x))-b*(3*b*B*d-A*b*e-2*a*B*e)*_
log(d+e*x)/e^4
--R
--R
--R (2)
--R 2 3 2 2 2
--R ((2A b + 4B a b)e - 6B b d e )x
--R +
--R 2 2 2 2 2 2 2 3
--R ((4A b + 8B a b)d e - 12B b d e)x + (2A b + 4B a b)d e - 6B b d
--R *
--R log(e x + d)
--R +
--R 2 3 3 2 2 2
--R 2B b e x + 4B b d e x
--R +
--R 2 3 2 2 2 2 2 3
--R ((- 4A a b - 2B a )e + (4A b + 8B a b)d e - 4B b d e)x - A a e
--R +
--R 2 2 2 2 2 3
--R (- 2A a b - B a )d e + (3A b + 6B a b)d e - 5B b d
--R /
--R 6 2 5 2 4
--R 2e x + 4d e x + 2d e
--R Type: Expression(Integer)
--E 107
--S 108 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 108
)clear all
--S 109 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^4
--R
--R
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R (1) ----------------------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R e x + 4d e x + 6d e x + 4d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 109
--S 110 of 764
r0:=-1/3*(B*d-A*e)*(a+b*x)^3/(e*(b*d-a*e)*(d+e*x)^3)-1/2*B*(b*d-a*e)^2/_
(e^4*(d+e*x)^2)+2*b*B*(b*d-a*e)/(e^4*(d+e*x))+b^2*B*log(d+e*x)/e^4
--R
--R
--R (2)
--R 2 4 3 3 3 2 3 3 2 2 2
--R (6B a b e - 6B b d e )x + (18B a b d e - 18B b d e )x
--R +
--R 2 2 2 3 3 2 3 3 4
--R (18B a b d e - 18B b d e)x + 6B a b d e - 6B b d
--R *
--R log(e x + d)
--R +
--R 3 4 3 3 3
--R (- 2A b e + 2B b d e )x
--R +
--R 2 2 4 2 3 3 2 2 2
--R ((- 6A a b - 12B a b)e + 30B a b d e - 12B b d e )x
--R +
--R 2 3 4 2 3 2 2 2 3 3
--R ((- 6A a b - 3B a )e - 9B a b d e + 39B a b d e - 21B b d e)x
--R +
--R 3 4 3 3 2 2 2 2 3 3 4
--R - 2A a e - B a d e - 3B a b d e + 15B a b d e - 9B b d
--R /
--R 8 7 3 7 2 6 2 2 6 3 5
--R (6a e - 6b d e )x + (18a d e - 18b d e )x + (18a d e - 18b d e )x
--R +
--R 3 5 4 4
--R 6a d e - 6b d e
--R Type: Expression(Integer)
--E 110
--S 111 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 111
)clear all
--S 112 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^5
--R
--R
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R (1) ----------------------------------------------------
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R e x + 5d e x + 10d e x + 10d e x + 5d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 112
--S 113 of 764
r0:=-1/4*(B*d-A*e)*(a+b*x)^3/(e*(b*d-a*e)*(d+e*x)^4)+1/12*(3*b*B*d+A*b*e-_
4*a*B*e)*(a+b*x)^3/(e*(b*d-a*e)^2*(d+e*x)^3)
--R
--R
--R (2)
--R 1 4 1 3 1 4 4
--R ((-- A b - - B a b )e + - B b d)x
--R 12 3 4
--R +
--R 2 2 1 4 2 3 3
--R (- B a b e + (- A b + - B a b )d)x
--R 3 3
--R +
--R 1 2 2 3 3 1 2 2 2
--R ((- - A a b - B a b)e + (A a b + - B a b )d)x
--R 2 2
--R +
--R 2 3 1 4 2 2 1 4 1 3 1 4
--R ((- - A a b - - B a )e + A a b d)x - - A a e + (- A a b - -- B a )d
--R 3 3 4 3 12
--R /
--R 2 6 5 2 2 4 4 2 5 2 4 2 3 3 3
--R (a e - 2a b d e + b d e )x + (4a d e - 8a b d e + 4b d e )x
--R +
--R 2 2 4 3 3 2 4 2 2 2 3 3 4 2 2 5
--R (6a d e - 12a b d e + 6b d e )x + (4a d e - 8a b d e + 4b d e)x
--R +
--R 2 4 2 5 2 6
--R a d e - 2a b d e + b d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 113
--S 114 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 114
)clear all
--S 115 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^6
--R
--R
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R (1) -------------------------------------------------------------
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R e x + 6d e x + 15d e x + 20d e x + 15d e x + 6d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 115
--S 116 of 764
r0:=1/5*(b*d-a*e)^2*(B*d-A*e)/(e^4*(d+e*x)^5)-1/4*(b*d-a*e)*(3*b*B*d-2*A*b*e-_
a*B*e)/(e^4*(d+e*x)^4)+1/3*b*(3*b*B*d-A*b*e-2*a*B*e)/(e^4*(d+e*x)^3)-_
1/2*b^2*B/(e^4*(d+e*x)^2)
--R
--R
--R (2)
--R 1 2 3 3 1 2 2 3 1 2 2 2
--R - - B b e x + ((- - A b - - B a b)e - - B b d e )x
--R 2 3 3 2
--R +
--R 1 1 2 3 1 2 1 2 1 2 2
--R ((- - A a b - - B a )e + (- - A b - - B a b)d e - - B b d e)x
--R 2 4 6 3 4
--R +
--R 1 2 3 1 1 2 2 1 2 1 2
--R - - A a e + (- -- A a b - -- B a )d e + (- -- A b - -- B a b)d e
--R 5 10 20 30 15
--R +
--R 1 2 3
--R - -- B b d
--R 20
--R /
--R 9 5 8 4 2 7 3 3 6 2 4 5 5 4
--R e x + 5d e x + 10d e x + 10d e x + 5d e x + d e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 116
--S 117 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 117
)clear all
--S 118 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^7
--R
--R
--R (1)
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R ------------------------------------------------------------------------
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R e x + 7d e x + 21d e x + 35d e x + 35d e x + 21d e x + 7d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 118
--S 119 of 764
r0:=1/6*(b*d-a*e)^2*(B*d-A*e)/(e^4*(d+e*x)^6)-1/5*(b*d-a*e)*(3*b*B*d-2*A*b*e-_
a*B*e)/(e^4*(d+e*x)^5)+1/4*b*(3*b*B*d-A*b*e-2*a*B*e)/(e^4*(d+e*x)^4)-_
1/3*b^2*B/(e^4*(d+e*x)^3)
--R
--R
--R (2)
--R 1 2 3 3 1 2 1 3 1 2 2 2
--R - - B b e x + ((- - A b - - B a b)e - - B b d e )x
--R 3 4 2 4
--R +
--R 2 1 2 3 1 2 1 2 1 2 2
--R ((- - A a b - - B a )e + (- -- A b - - B a b)d e - -- B b d e)x
--R 5 5 10 5 10
--R +
--R 1 2 3 1 1 2 2 1 2 1 2
--R - - A a e + (- -- A a b - -- B a )d e + (- -- A b - -- B a b)d e
--R 6 15 30 60 30
--R +
--R 1 2 3
--R - -- B b d
--R 60
--R /
--R 10 6 9 5 2 8 4 3 7 3 4 6 2 5 5 6 4
--R e x + 6d e x + 15d e x + 20d e x + 15d e x + 6d e x + d e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 119
--S 120 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 120
)clear all
--S 121 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^8
--R
--R
--R (1)
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R /
--R 8 8 7 7 2 6 6 3 5 5 4 4 4 5 3 3 6 2 2
--R e x + 8d e x + 28d e x + 56d e x + 70d e x + 56d e x + 28d e x
--R +
--R 7 8
--R 8d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 121
--S 122 of 764
r0:=1/7*(b*d-a*e)^2*(B*d-A*e)/(e^4*(d+e*x)^7)-1/6*(b*d-a*e)*(3*b*B*d-2*A*b*e-_
a*B*e)/(e^4*(d+e*x)^6)+1/5*b*(3*b*B*d-A*b*e-2*a*B*e)/(e^4*(d+e*x)^5)-_
1/4*b^2*B/(e^4*(d+e*x)^4)
--R
--R
--R (2)
--R 1 2 3 3 1 2 2 3 3 2 2 2
--R - - B b e x + ((- - A b - - B a b)e - -- B b d e )x
--R 4 5 5 20
--R +
--R 1 1 2 3 1 2 2 2 1 2 2
--R ((- - A a b - - B a )e + (- -- A b - -- B a b)d e - -- B b d e)x
--R 3 6 15 15 20
--R +
--R 1 2 3 1 1 2 2 1 2 2 2
--R - - A a e + (- -- A a b - -- B a )d e + (- --- A b - --- B a b)d e
--R 7 21 42 105 105
--R +
--R 1 2 3
--R - --- B b d
--R 140
--R /
--R 11 7 10 6 2 9 5 3 8 4 4 7 3 5 6 2 6 5
--R e x + 7d e x + 21d e x + 35d e x + 35d e x + 21d e x + 7d e x
--R +
--R 7 4
--R d e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 122
--S 123 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 123
)clear all
--S 124 of 764
t0:=(A+B*x)*(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R m
--R (e x + d)
--R Type: Expression(Integer)
--E 124
--S 125 of 764
r0:=-(b*d-a*e)^4*(B*d-A*e)*(d+e*x)^(1+m)/(e^6*(1+m))+(b*d-a*e)^3*(5*b*B*d-_
4*A*b*e-a*B*e)*(d+e*x)^(2+m)/(e^6*(2+m))-2*b*(b*d-a*e)^2*(5*b*B*d-_
3*A*b*e-2*a*B*e)*(d+e*x)^(3+m)/(e^6*(3+m))+2*b^2*(b*d-a*e)*(5*b*B*d-_
2*A*b*e-3*a*B*e)*(d+e*x)^(4+m)/(e^6*(4+m))-b^3*(5*b*B*d-A*b*e-4*a*B*e)*_
(d+e*x)^(5+m)/(e^6*(5+m))+b^4*B*(d+e*x)^(6+m)/(e^6*(6+m))
--R
--R
--R (2)
--R 4 5 4 4 4 3 4 2 4 4
--R (B b m + 15B b m + 85B b m + 225B b m + 274B b m + 120B b )
--R *
--R m + 6
--R (e x + d)
--R +
--R 4 3 4 5 4 3 4 4
--R ((A b + 4B a b )e - 5B b d)m + ((16A b + 64B a b )e - 80B b d)m
--R +
--R 4 3 4 3
--R ((95A b + 380B a b )e - 475B b d)m
--R +
--R 4 3 4 2
--R ((260A b + 1040B a b )e - 1300B b d)m
--R +
--R 4 3 4 4 3
--R ((324A b + 1296B a b )e - 1620B b d)m + (144A b + 576B a b )e
--R +
--R 4
--R - 720B b d
--R *
--R m + 5
--R (e x + d)
--R +
--R 3 2 2 2 4 3 4 2 5
--R ((4A a b + 6B a b )e + (- 4A b - 16B a b )d e + 10B b d )m
--R +
--R 3 2 2 2 4 3 4 2 4
--R ((68A a b + 102B a b )e + (- 68A b - 272B a b )d e + 170B b d )m
--R +
--R 3 2 2 2 4 3
--R (428A a b + 642B a b )e + (- 428A b - 1712B a b )d e
--R +
--R 4 2
--R 1070B b d
--R *
--R 3
--R m
--R +
--R 3 2 2 2 4 3
--R (1228A a b + 1842B a b )e + (- 1228A b - 4912B a b )d e
--R +
--R 4 2
--R 3070B b d
--R *
--R 2
--R m
--R +
--R 3 2 2 2 4 3
--R (1584A a b + 2376B a b )e + (- 1584A b - 6336B a b )d e
--R +
--R 4 2
--R 3960B b d
--R *
--R m
--R +
--R 3 2 2 2 4 3 4 2
--R (720A a b + 1080B a b )e + (- 720A b - 2880B a b )d e + 1800B b d
--R *
--R m + 4
--R (e x + d)
--R +
--R 2 2 3 3 3 2 2 2
--R (6A a b + 4B a b)e + (- 12A a b - 18B a b )d e
--R +
--R 4 3 2 4 3
--R (6A b + 24B a b )d e - 10B b d
--R *
--R 5
--R m
--R +
--R 2 2 3 3 3 2 2 2
--R (108A a b + 72B a b)e + (- 216A a b - 324B a b )d e
--R +
--R 4 3 2 4 3
--R (108A b + 432B a b )d e - 180B b d
--R *
--R 4
--R m
--R +
--R 2 2 3 3 3 2 2 2
--R (726A a b + 484B a b)e + (- 1452A a b - 2178B a b )d e
--R +
--R 4 3 2 4 3
--R (726A b + 2904B a b )d e - 1210B b d
--R *
--R 3
--R m
--R +
--R 2 2 3 3 3 2 2 2
--R (2232A a b + 1488B a b)e + (- 4464A a b - 6696B a b )d e
--R +
--R 4 3 2 4 3
--R (2232A b + 8928B a b )d e - 3720B b d
--R *
--R 2
--R m
--R +
--R 2 2 3 3 3 2 2 2
--R (3048A a b + 2032B a b)e + (- 6096A a b - 9144B a b )d e
--R +
--R 4 3 2 4 3
--R (3048A b + 12192B a b )d e - 5080B b d
--R *
--R m
--R +
--R 2 2 3 3 3 2 2 2
--R (1440A a b + 960B a b)e + (- 2880A a b - 4320B a b )d e
--R +
--R 4 3 2 4 3
--R (1440A b + 5760B a b )d e - 2400B b d
--R *
--R m + 3
--R (e x + d)
--R +
--R 3 4 4 2 2 3 3
--R (4A a b + B a )e + (- 12A a b - 8B a b)d e
--R +
--R 3 2 2 2 2 4 3 3 4 4
--R (12A a b + 18B a b )d e + (- 4A b - 16B a b )d e + 5B b d
--R *
--R 5
--R m
--R +
--R 3 4 4 2 2 3 3
--R (76A a b + 19B a )e + (- 228A a b - 152B a b)d e
--R +
--R 3 2 2 2 2 4 3 3 4 4
--R (228A a b + 342B a b )d e + (- 76A b - 304B a b )d e + 95B b d
--R *
--R 4
--R m
--R +
--R 3 4 4 2 2 3 3
--R (548A a b + 137B a )e + (- 1644A a b - 1096B a b)d e
--R +
--R 3 2 2 2 2 4 3 3
--R (1644A a b + 2466B a b )d e + (- 548A b - 2192B a b )d e
--R +
--R 4 4
--R 685B b d
--R *
--R 3
--R m
--R +
--R 3 4 4 2 2 3 3
--R (1844A a b + 461B a )e + (- 5532A a b - 3688B a b)d e
--R +
--R 3 2 2 2 2 4 3 3
--R (5532A a b + 8298B a b )d e + (- 1844A b - 7376B a b )d e
--R +
--R 4 4
--R 2305B b d
--R *
--R 2
--R m
--R +
--R 3 4 4 2 2 3 3
--R (2808A a b + 702B a )e + (- 8424A a b - 5616B a b)d e
--R +
--R 3 2 2 2 2 4 3 3
--R (8424A a b + 12636B a b )d e + (- 2808A b - 11232B a b )d e
--R +
--R 4 4
--R 3510B b d
--R *
--R m
--R +
--R 3 4 4 2 2 3 3
--R (1440A a b + 360B a )e + (- 4320A a b - 2880B a b)d e
--R +
--R 3 2 2 2 2 4 3 3
--R (4320A a b + 6480B a b )d e + (- 1440A b - 5760B a b )d e
--R +
--R 4 4
--R 1800B b d
--R *
--R m + 2
--R (e x + d)
--R +
--R 4 5 3 4 4 2 2 3 2 3
--R A a e + (- 4A a b - B a )d e + (6A a b + 4B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (- 4A a b - 6B a b )d e + (A b + 4B a b )d e - B b d
--R *
--R 5
--R m
--R +
--R 4 5 3 4 4 2 2 3 2 3
--R 20A a e + (- 80A a b - 20B a )d e + (120A a b + 80B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (- 80A a b - 120B a b )d e + (20A b + 80B a b )d e - 20B b d
--R *
--R 4
--R m
--R +
--R 4 5 3 4 4
--R 155A a e + (- 620A a b - 155B a )d e
--R +
--R 2 2 3 2 3 3 2 2 3 2
--R (930A a b + 620B a b)d e + (- 620A a b - 930B a b )d e
--R +
--R 4 3 4 4 5
--R (155A b + 620B a b )d e - 155B b d
--R *
--R 3
--R m
--R +
--R 4 5 3 4 4
--R 580A a e + (- 2320A a b - 580B a )d e
--R +
--R 2 2 3 2 3 3 2 2 3 2
--R (3480A a b + 2320B a b)d e + (- 2320A a b - 3480B a b )d e
--R +
--R 4 3 4 4 5
--R (580A b + 2320B a b )d e - 580B b d
--R *
--R 2
--R m
--R +
--R 4 5 3 4 4
--R 1044A a e + (- 4176A a b - 1044B a )d e
--R +
--R 2 2 3 2 3 3 2 2 3 2
--R (6264A a b + 4176B a b)d e + (- 4176A a b - 6264B a b )d e
--R +
--R 4 3 4 4 5
--R (1044A b + 4176B a b )d e - 1044B b d
--R *
--R m
--R +
--R 4 5 3 4 4
--R 720A a e + (- 2880A a b - 720B a )d e
--R +
--R 2 2 3 2 3 3 2 2 3 2
--R (4320A a b + 2880B a b)d e + (- 2880A a b - 4320B a b )d e
--R +
--R 4 3 4 4 5
--R (720A b + 2880B a b )d e - 720B b d
--R *
--R m + 1
--R (e x + d)
--R /
--R 6 6 6 5 6 4 6 3 6 2 6 6
--R e m + 21e m + 175e m + 735e m + 1624e m + 1764e m + 720e
--R Type: Expression(Integer)
--E 125
--S 126 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 126
)clear all
--S 127 of 764
t0:=(A+B*x)*(d+e*x)^7*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 7 12 4 3 7 4 6 11
--R B b e x + ((A b + 4B a b )e + 7B b d e )x
--R +
--R 3 2 2 7 4 3 6 4 2 5 10
--R ((4A a b + 6B a b )e + (7A b + 28B a b )d e + 21B b d e )x
--R +
--R 2 2 3 7 3 2 2 6
--R (6A a b + 4B a b)e + (28A a b + 42B a b )d e
--R +
--R 4 3 2 5 4 3 4
--R (21A b + 84B a b )d e + 35B b d e
--R *
--R 9
--R x
--R +
--R 3 4 7 2 2 3 6
--R (4A a b + B a )e + (42A a b + 28B a b)d e
--R +
--R 3 2 2 2 5 4 3 3 4 4 4 3
--R (84A a b + 126B a b )d e + (35A b + 140B a b )d e + 35B b d e
--R *
--R 8
--R x
--R +
--R 4 7 3 4 6 2 2 3 2 5
--R A a e + (28A a b + 7B a )d e + (126A a b + 84B a b)d e
--R +
--R 3 2 2 3 4 4 3 4 3 4 5 2
--R (140A a b + 210B a b )d e + (35A b + 140B a b )d e + 21B b d e
--R *
--R 7
--R x
--R +
--R 4 6 3 4 2 5 2 2 3 3 4
--R 7A a d e + (84A a b + 21B a )d e + (210A a b + 140B a b)d e
--R +
--R 3 2 2 4 3 4 3 5 2 4 6
--R (140A a b + 210B a b )d e + (21A b + 84B a b )d e + 7B b d e
--R *
--R 6
--R x
--R +
--R 4 2 5 3 4 3 4 2 2 3 4 3
--R 21A a d e + (140A a b + 35B a )d e + (210A a b + 140B a b)d e
--R +
--R 3 2 2 5 2 4 3 6 4 7
--R (84A a b + 126B a b )d e + (7A b + 28B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 4 3 4 3 4 4 3 2 2 3 5 2
--R 35A a d e + (140A a b + 35B a )d e + (126A a b + 84B a b)d e
--R +
--R 3 2 2 6 4 3 7
--R (28A a b + 42B a b )d e + (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 4 3 3 4 5 2 2 2 3 6
--R 35A a d e + (84A a b + 21B a )d e + (42A a b + 28B a b)d e
--R +
--R 3 2 2 7
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 5 2 3 4 6 2 2 3 7 2
--R (21A a d e + (28A a b + 7B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 6 3 4 7 4 7
--R (7A a d e + (4A a b + B a )d )x + A a d
--R Type: Polynomial(Integer)
--E 127
--S 128 of 764
r0:=-1/8*(b*d-a*e)^4*(B*d-A*e)*(d+e*x)^8/e^6+1/9*(b*d-a*e)^3*(5*b*B*d-_
4*A*b*e-a*B*e)*(d+e*x)^9/e^6-1/5*b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-_
2*a*B*e)*(d+e*x)^10/e^6+2/11*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)*_
(d+e*x)^11/e^6-1/12*b^3*(5*b*B*d-A*b*e-4*a*B*e)*(d+e*x)^12/e^6+_
1/13*b^4*B*(d+e*x)^13/e^6
--R
--R
--R (2)
--R 1 4 13 13 1 4 1 3 13 7 4 12 12
--R -- B b e x + ((-- A b + - B a b )e + -- B b d e )x
--R 13 12 3 12
--R +
--R 4 3 6 2 2 13 7 4 28 3 12
--R (-- A a b + -- B a b )e + (-- A b + -- B a b )d e
--R 11 11 11 11
--R +
--R 21 4 2 11
--R -- B b d e
--R 11
--R *
--R 11
--R x
--R +
--R 3 2 2 2 3 13 14 3 21 2 2 12
--R (- A a b + - B a b)e + (-- A a b + -- B a b )d e
--R 5 5 5 5
--R +
--R 21 4 42 3 2 11 7 4 3 10
--R (-- A b + -- B a b )d e + - B b d e
--R 10 5 2
--R *
--R 10
--R x
--R +
--R 4 3 1 4 13 14 2 2 28 3 12
--R (- A a b + - B a )e + (-- A a b + -- B a b)d e
--R 9 9 3 9
--R +
--R 28 3 2 2 2 11 35 4 140 3 3 10 35 4 4 9
--R (-- A a b + 14B a b )d e + (-- A b + --- B a b )d e + -- B b d e
--R 3 9 9 9
--R *
--R 9
--R x
--R +
--R 1 4 13 7 3 7 4 12 63 2 2 21 3 2 11
--R - A a e + (- A a b + - B a )d e + (-- A a b + -- B a b)d e
--R 8 2 8 4 2
--R +
--R 35 3 105 2 2 3 10 35 4 35 3 4 9 21 4 5 8
--R (-- A a b + --- B a b )d e + (-- A b + -- B a b )d e + -- B b d e
--R 2 4 8 2 8
--R *
--R 8
--R x
--R +
--R 4 12 3 4 2 11 2 2 3 3 10
--R A a d e + (12A a b + 3B a )d e + (30A a b + 20B a b)d e
--R +
--R 3 2 2 4 9 4 3 5 8 4 6 7
--R (20A a b + 30B a b )d e + (3A b + 12B a b )d e + B b d e
--R *
--R 7
--R x
--R +
--R 7 4 2 11 70 3 35 4 3 10 2 2 70 3 4 9
--R - A a d e + (-- A a b + -- B a )d e + (35A a b + -- B a b)d e
--R 2 3 6 3
--R +
--R 3 2 2 5 8 7 4 14 3 6 7 1 4 7 6
--R (14A a b + 21B a b )d e + (- A b + -- B a b )d e + - B b d e
--R 6 3 6
--R *
--R 6
--R x
--R +
--R 4 3 10 3 4 4 9 126 2 2 84 3 5 8
--R 7A a d e + (28A a b + 7B a )d e + (--- A a b + -- B a b)d e
--R 5 5
--R +
--R 28 3 42 2 2 6 7 1 4 4 3 7 6
--R (-- A a b + -- B a b )d e + (- A b + - B a b )d e
--R 5 5 5 5
--R *
--R 5
--R x
--R +
--R 35 4 4 9 3 21 4 5 8 21 2 2 3 6 7
--R -- A a d e + (21A a b + -- B a )d e + (-- A a b + 7B a b)d e
--R 4 4 2
--R +
--R 3 3 2 2 7 6
--R (A a b + - B a b )d e
--R 2
--R *
--R 4
--R x
--R +
--R 4 5 8 28 3 7 4 6 7 2 2 4 3 7 6 3
--R (7A a d e + (-- A a b + - B a )d e + (2A a b + - B a b)d e )x
--R 3 3 3
--R +
--R 7 4 6 7 3 1 4 7 6 2 4 7 6 1 4 8 5
--R (- A a d e + (2A a b + - B a )d e )x + A a d e x + - A a d e
--R 2 2 8
--R +
--R 1 3 1 4 9 4 1 2 2 1 3 10 3
--R (- -- A a b - -- B a )d e + (-- A a b + -- B a b)d e
--R 18 72 60 90
--R +
--R 1 3 1 2 2 11 2 1 4 1 3 12
--R (- --- A a b - --- B a b )d e + (---- A b + --- B a b )d e
--R 330 220 3960 990
--R +
--R 1 4 13
--R - ----- B b d
--R 10296
--R /
--R 6
--R e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 128
--S 129 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 129
)clear all
--S 130 of 764
t0:=(A+B*x)*(d+e*x)^6*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 6 11 4 3 6 4 5 10
--R B b e x + ((A b + 4B a b )e + 6B b d e )x
--R +
--R 3 2 2 6 4 3 5 4 2 4 9
--R ((4A a b + 6B a b )e + (6A b + 24B a b )d e + 15B b d e )x
--R +
--R 2 2 3 6 3 2 2 5
--R (6A a b + 4B a b)e + (24A a b + 36B a b )d e
--R +
--R 4 3 2 4 4 3 3
--R (15A b + 60B a b )d e + 20B b d e
--R *
--R 8
--R x
--R +
--R 3 4 6 2 2 3 5
--R (4A a b + B a )e + (36A a b + 24B a b)d e
--R +
--R 3 2 2 2 4 4 3 3 3 4 4 2
--R (60A a b + 90B a b )d e + (20A b + 80B a b )d e + 15B b d e
--R *
--R 7
--R x
--R +
--R 4 6 3 4 5 2 2 3 2 4
--R A a e + (24A a b + 6B a )d e + (90A a b + 60B a b)d e
--R +
--R 3 2 2 3 3 4 3 4 2 4 5
--R (80A a b + 120B a b )d e + (15A b + 60B a b )d e + 6B b d e
--R *
--R 6
--R x
--R +
--R 4 5 3 4 2 4 2 2 3 3 3
--R 6A a d e + (60A a b + 15B a )d e + (120A a b + 80B a b)d e
--R +
--R 3 2 2 4 2 4 3 5 4 6
--R (60A a b + 90B a b )d e + (6A b + 24B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 4 2 4 3 4 3 3 2 2 3 4 2
--R 15A a d e + (80A a b + 20B a )d e + (90A a b + 60B a b)d e
--R +
--R 3 2 2 5 4 3 6
--R (24A a b + 36B a b )d e + (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 3 3 3 4 4 2 2 2 3 5
--R 20A a d e + (60A a b + 15B a )d e + (36A a b + 24B a b)d e
--R +
--R 3 2 2 6
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 4 2 3 4 5 2 2 3 6 2
--R (15A a d e + (24A a b + 6B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 5 3 4 6 4 6
--R (6A a d e + (4A a b + B a )d )x + A a d
--R Type: Polynomial(Integer)
--E 130
--S 131 of 764
r0:=-1/7*(b*d-a*e)^4*(B*d-A*e)*(d+e*x)^7/e^6+1/8*(b*d-a*e)^3*(5*b*B*d-_
4*A*b*e-a*B*e)*(d+e*x)^8/e^6-2/9*b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-_
2*a*B*e)*(d+e*x)^9/e^6+1/5*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)*_
(d+e*x)^10/e^6-1/11*b^3*(5*b*B*d-A*b*e-4*a*B*e)*(d+e*x)^11/e^6+_
1/12*b^4*B*(d+e*x)^12/e^6
--R
--R
--R (2)
--R 1 4 12 12 1 4 4 3 12 6 4 11 11
--R -- B b e x + ((-- A b + -- B a b )e + -- B b d e )x
--R 12 11 11 11
--R +
--R 2 3 3 2 2 12 3 4 12 3 11 3 4 2 10 10
--R ((- A a b + - B a b )e + (- A b + -- B a b )d e + - B b d e )x
--R 5 5 5 5 2
--R +
--R 2 2 2 4 3 12 8 3 2 2 11
--R (- A a b + - B a b)e + (- A a b + 4B a b )d e
--R 3 9 3
--R +
--R 5 4 20 3 2 10 20 4 3 9
--R (- A b + -- B a b )d e + -- B b d e
--R 3 3 9
--R *
--R 9
--R x
--R +
--R 1 3 1 4 12 9 2 2 3 11
--R (- A a b + - B a )e + (- A a b + 3B a b)d e
--R 2 8 2
--R +
--R 15 3 45 2 2 2 10 5 4 3 3 9 15 4 4 8
--R (-- A a b + -- B a b )d e + (- A b + 10B a b )d e + -- B b d e
--R 2 4 2 8
--R *
--R 8
--R x
--R +
--R 1 4 12 24 3 6 4 11 90 2 2 60 3 2 10
--R - A a e + (-- A a b + - B a )d e + (-- A a b + -- B a b)d e
--R 7 7 7 7 7
--R +
--R 80 3 120 2 2 3 9 15 4 60 3 4 8 6 4 5 7
--R (-- A a b + --- B a b )d e + (-- A b + -- B a b )d e + - B b d e
--R 7 7 7 7 7
--R *
--R 7
--R x
--R +
--R 4 11 3 5 4 2 10 2 2 40 3 3 9
--R A a d e + (10A a b + - B a )d e + (20A a b + -- B a b)d e
--R 2 3
--R +
--R 3 2 2 4 8 4 3 5 7 1 4 6 6
--R (10A a b + 15B a b )d e + (A b + 4B a b )d e + - B b d e
--R 6
--R *
--R 6
--R x
--R +
--R 4 2 10 3 4 3 9 2 2 3 4 8
--R 3A a d e + (16A a b + 4B a )d e + (18A a b + 12B a b)d e
--R +
--R 24 3 36 2 2 5 7 1 4 4 3 6 6
--R (-- A a b + -- B a b )d e + (- A b + - B a b )d e
--R 5 5 5 5
--R *
--R 5
--R x
--R +
--R 4 3 9 3 15 4 4 8 2 2 3 5 7
--R 5A a d e + (15A a b + -- B a )d e + (9A a b + 6B a b)d e
--R 4
--R +
--R 3 3 2 2 6 6
--R (A a b + - B a b )d e
--R 2
--R *
--R 4
--R x
--R +
--R 4 4 8 3 4 5 7 2 2 4 3 6 6 3
--R (5A a d e + (8A a b + 2B a )d e + (2A a b + - B a b)d e )x
--R 3
--R +
--R 4 5 7 3 1 4 6 6 2 4 6 6 1 4 7 5
--R (3A a d e + (2A a b + - B a )d e )x + A a d e x + - A a d e
--R 2 7
--R +
--R 1 3 1 4 8 4 1 2 2 1 3 9 3
--R (- -- A a b - -- B a )d e + (-- A a b + -- B a b)d e
--R 14 56 42 63
--R +
--R 1 3 1 2 2 10 2 1 4 2 3 11
--R (- --- A a b - --- B a b )d e + (---- A b + ---- B a b )d e
--R 210 140 2310 1155
--R +
--R 1 4 12
--R - ---- B b d
--R 5544
--R /
--R 6
--R e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 131
--S 132 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 132
)clear all
--S 133 of 764
t0:=(A+B*x)*(d+e*x)^5*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 5 10 4 3 5 4 4 9
--R B b e x + ((A b + 4B a b )e + 5B b d e )x
--R +
--R 3 2 2 5 4 3 4 4 2 3 8
--R ((4A a b + 6B a b )e + (5A b + 20B a b )d e + 10B b d e )x
--R +
--R 2 2 3 5 3 2 2 4
--R (6A a b + 4B a b)e + (20A a b + 30B a b )d e
--R +
--R 4 3 2 3 4 3 2
--R (10A b + 40B a b )d e + 10B b d e
--R *
--R 7
--R x
--R +
--R 3 4 5 2 2 3 4
--R (4A a b + B a )e + (30A a b + 20B a b)d e
--R +
--R 3 2 2 2 3 4 3 3 2 4 4
--R (40A a b + 60B a b )d e + (10A b + 40B a b )d e + 5B b d e
--R *
--R 6
--R x
--R +
--R 4 5 3 4 4 2 2 3 2 3
--R A a e + (20A a b + 5B a )d e + (60A a b + 40B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (40A a b + 60B a b )d e + (5A b + 20B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 4 4 3 4 2 3 2 2 3 3 2
--R 5A a d e + (40A a b + 10B a )d e + (60A a b + 40B a b)d e
--R +
--R 3 2 2 4 4 3 5
--R (20A a b + 30B a b )d e + (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 2 3 3 4 3 2 2 2 3 4
--R 10A a d e + (40A a b + 10B a )d e + (30A a b + 20B a b)d e
--R +
--R 3 2 2 5
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 3 2 3 4 4 2 2 3 5 2
--R (10A a d e + (20A a b + 5B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 4 3 4 5 4 5
--R (5A a d e + (4A a b + B a )d )x + A a d
--R Type: Polynomial(Integer)
--E 133
--S 134 of 764
r0:=-1/6*(b*d-a*e)^4*(B*d-A*e)*(d+e*x)^6/e^6+1/7*(b*d-a*e)^3*(5*b*B*d-_
4*A*b*e-a*B*e)*(d+e*x)^7/e^6-1/4*b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-_
2*a*B*e)*(d+e*x)^8/e^6+2/9*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)*_
(d+e*x)^9/e^6-1/10*b^3*(5*b*B*d-A*b*e-4*a*B*e)*(d+e*x)^10/e^6+_
1/11*b^4*B*(d+e*x)^11/e^6
--R
--R
--R (2)
--R 1 4 11 11 1 4 2 3 11 1 4 10 10
--R -- B b e x + ((-- A b + - B a b )e + - B b d e )x
--R 11 10 5 2
--R +
--R 4 3 2 2 2 11 5 4 20 3 10 10 4 2 9 9
--R ((- A a b + - B a b )e + (- A b + -- B a b )d e + -- B b d e )x
--R 9 3 9 9 9
--R +
--R 3 2 2 1 3 11 5 3 15 2 2 10
--R (- A a b + - B a b)e + (- A a b + -- B a b )d e
--R 4 2 2 4
--R +
--R 5 4 3 2 9 5 4 3 8
--R (- A b + 5B a b )d e + - B b d e
--R 4 4
--R *
--R 8
--R x
--R +
--R 4 3 1 4 11 30 2 2 20 3 10
--R (- A a b + - B a )e + (-- A a b + -- B a b)d e
--R 7 7 7 7
--R +
--R 40 3 60 2 2 2 9 10 4 40 3 3 8 5 4 4 7
--R (-- A a b + -- B a b )d e + (-- A b + -- B a b )d e + - B b d e
--R 7 7 7 7 7
--R *
--R 7
--R x
--R +
--R 1 4 11 10 3 5 4 10 2 2 20 3 2 9
--R - A a e + (-- A a b + - B a )d e + (10A a b + -- B a b)d e
--R 6 3 6 3
--R +
--R 20 3 2 2 3 8 5 4 10 3 4 7 1 4 5 6
--R (-- A a b + 10B a b )d e + (- A b + -- B a b )d e + - B b d e
--R 3 6 3 6
--R *
--R 6
--R x
--R +
--R 4 10 3 4 2 9 2 2 3 3 8
--R A a d e + (8A a b + 2B a )d e + (12A a b + 8B a b)d e
--R +
--R 3 2 2 4 7 1 4 4 3 5 6
--R (4A a b + 6B a b )d e + (- A b + - B a b )d e
--R 5 5
--R *
--R 5
--R x
--R +
--R 5 4 2 9 3 5 4 3 8 15 2 2 3 4 7
--R - A a d e + (10A a b + - B a )d e + (-- A a b + 5B a b)d e
--R 2 2 2
--R +
--R 3 3 2 2 5 6
--R (A a b + - B a b )d e
--R 2
--R *
--R 4
--R x
--R +
--R 10 4 3 8 20 3 5 4 4 7 2 2 4 3 5 6 3
--R (-- A a d e + (-- A a b + - B a )d e + (2A a b + - B a b)d e )x
--R 3 3 3 3
--R +
--R 5 4 4 7 3 1 4 5 6 2 4 5 6 1 4 6 5
--R (- A a d e + (2A a b + - B a )d e )x + A a d e x + - A a d e
--R 2 2 6
--R +
--R 2 3 1 4 7 4 1 2 2 1 3 8 3
--R (- -- A a b - -- B a )d e + (-- A a b + -- B a b)d e
--R 21 42 28 42
--R +
--R 1 3 1 2 2 9 2 1 4 1 3 10
--R (- --- A a b - -- B a b )d e + (---- A b + --- B a b )d e
--R 126 84 1260 315
--R +
--R 1 4 11
--R - ---- B b d
--R 2772
--R /
--R 6
--R e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 134
--S 135 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 135
)clear all
--S 136 of 764
t0:=(A+B*x)*(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 4 9 4 3 4 4 3 8
--R B b e x + ((A b + 4B a b )e + 4B b d e )x
--R +
--R 3 2 2 4 4 3 3 4 2 2 7
--R ((4A a b + 6B a b )e + (4A b + 16B a b )d e + 6B b d e )x
--R +
--R 2 2 3 4 3 2 2 3
--R (6A a b + 4B a b)e + (16A a b + 24B a b )d e
--R +
--R 4 3 2 2 4 3
--R (6A b + 24B a b )d e + 4B b d e
--R *
--R 6
--R x
--R +
--R 3 4 4 2 2 3 3
--R (4A a b + B a )e + (24A a b + 16B a b)d e
--R +
--R 3 2 2 2 2 4 3 3 4 4
--R (24A a b + 36B a b )d e + (4A b + 16B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 4 4 3 4 3 2 2 3 2 2
--R A a e + (16A a b + 4B a )d e + (36A a b + 24B a b)d e
--R +
--R 3 2 2 3 4 3 4
--R (16A a b + 24B a b )d e + (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 3 3 4 2 2 2 2 3 3
--R 4A a d e + (24A a b + 6B a )d e + (24A a b + 16B a b)d e
--R +
--R 3 2 2 4
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 2 2 3 4 3 2 2 3 4 2
--R (6A a d e + (16A a b + 4B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 3 3 4 4 4 4
--R (4A a d e + (4A a b + B a )d )x + A a d
--R Type: Polynomial(Integer)
--E 136
--S 137 of 764
r0:=1/5*(A*b-a*B)*(b*d-a*e)^4*(a+b*x)^5/b^6+1/6*(b*d-a*e)^3*(b*B*d+4*A*b*e-_
5*a*B*e)*(a+b*x)^6/b^6+2/7*e*(b*d-a*e)^2*(2*b*B*d+3*A*b*e-5*a*B*e)*_
(a+b*x)^7/b^6+1/4*e^2*(b*d-a*e)*(3*b*B*d+2*A*b*e-5*a*B*e)*(a+b*x)^8/_
b^6+1/9*e^3*(4*b*B*d+A*b*e-5*a*B*e)*(a+b*x)^9/b^6+1/10*B*e^4*_
(a+b*x)^10/b^6
--R
--R
--R (2)
--R 1 10 4 10 1 10 4 9 4 4 10 3 9
--R -- B b e x + ((- A b + - B a b )e + - B b d e )x
--R 10 9 9 9
--R +
--R 1 9 3 2 8 4 1 10 9 3 3 10 2 2 8
--R ((- A a b + - B a b )e + (- A b + 2B a b )d e + - B b d e )x
--R 2 4 2 4
--R +
--R 6 2 8 4 3 7 4 16 9 24 2 8 3
--R (- A a b + - B a b )e + (-- A a b + -- B a b )d e
--R 7 7 7 7
--R +
--R 6 10 24 9 2 2 4 10 3
--R (- A b + -- B a b )d e + - B b d e
--R 7 7 7
--R *
--R 7
--R x
--R +
--R 2 3 7 1 4 6 4 2 8 8 3 7 3
--R (- A a b + - B a b )e + (4A a b + - B a b )d e
--R 3 6 3
--R +
--R 9 2 8 2 2 2 10 8 9 3 1 10 4
--R (4A a b + 6B a b )d e + (- A b + - B a b )d e + - B b d
--R 3 3 6
--R *
--R 6
--R x
--R +
--R 1 4 6 4 16 3 7 4 4 6 3 36 2 8 24 3 7 2 2
--R - A a b e + (-- A a b + - B a b )d e + (-- A a b + -- B a b )d e
--R 5 5 5 5 5
--R +
--R 16 9 24 2 8 3 1 10 4 9 4
--R (-- A a b + -- B a b )d e + (- A b + - B a b )d
--R 5 5 5 5
--R *
--R 5
--R x
--R +
--R 4 6 3 3 7 3 4 6 2 2 2 8 3 7 3
--R A a b d e + (6A a b + - B a b )d e + (6A a b + 4B a b )d e
--R 2
--R +
--R 9 3 2 8 4
--R (A a b + - B a b )d
--R 2
--R *
--R 4
--R x
--R +
--R 4 6 2 2 16 3 7 4 4 6 3 2 8 4 3 7 4 3
--R (2A a b d e + (-- A a b + - B a b )d e + (2A a b + - B a b )d )x
--R 3 3 3
--R +
--R 4 6 3 3 7 1 4 6 4 2 4 6 4
--R (2A a b d e + (2A a b + - B a b )d )x + A a b d x
--R 2
--R +
--R 1 9 1 10 4 1 8 2 2 9 3
--R (--- A a b - ---- B a )e + (- -- A a b + --- B a b)d e
--R 630 1260 70 315
--R +
--R 2 7 3 3 8 2 2 2 2 6 4 4 7 3 3
--R (-- A a b - --- B a b )d e + (- -- A a b + --- B a b )d e
--R 35 140 15 105
--R +
--R 1 5 5 1 6 4 4
--R (- A a b - -- B a b )d
--R 5 30
--R /
--R 6
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 137
--S 138 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 138
)clear all
--S 139 of 764
t0:=(A+B*x)*(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 3 8 4 3 3 4 2 7
--R B b e x + ((A b + 4B a b )e + 3B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 6
--R ((4A a b + 6B a b )e + (3A b + 12B a b )d e + 3B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (6A a b + 4B a b)e + (12A a b + 18B a b )d e
--R +
--R 4 3 2 4 3
--R (3A b + 12B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 3 4 3 2 2 3 2 3 2 2 2
--R (4A a b + B a )e + (18A a b + 12B a b)d e + (12A a b + 18B a b )d e
--R +
--R 4 3 3
--R (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 3 3 4 2 2 2 3 2
--R A a e + (12A a b + 3B a )d e + (18A a b + 12B a b)d e
--R +
--R 3 2 2 3
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 2 3 4 2 2 2 3 3 2
--R (3A a d e + (12A a b + 3B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 2 3 4 3 4 3
--R (3A a d e + (4A a b + B a )d )x + A a d
--R Type: Polynomial(Integer)
--E 139
--S 140 of 764
r0:=1/5*(A*b-a*B)*(b*d-a*e)^3*(a+b*x)^5/b^5+1/6*(b*d-a*e)^2*(b*B*d+3*A*b*e-_
4*a*B*e)*(a+b*x)^6/b^5+3/7*e*(b*d-a*e)*(b*B*d+A*b*e-2*a*B*e)*_
(a+b*x)^7/b^5+1/8*e^2*(3*b*B*d+A*b*e-4*a*B*e)*(a+b*x)^8/b^5+_
1/9*B*e^3*(a+b*x)^9/b^5
--R
--R
--R (2)
--R 1 9 3 9 1 9 1 8 3 3 9 2 8
--R - B b e x + ((- A b + - B a b )e + - B b d e )x
--R 9 8 2 8
--R +
--R 4 8 6 2 7 3 3 9 12 8 2 3 9 2 7
--R ((- A a b + - B a b )e + (- A b + -- B a b )d e + - B b d e)x
--R 7 7 7 7 7
--R +
--R 2 7 2 3 6 3 8 2 7 2
--R (A a b + - B a b )e + (2A a b + 3B a b )d e
--R 3
--R +
--R 1 9 8 2 1 9 3
--R (- A b + 2B a b )d e + - B b d
--R 2 6
--R *
--R 6
--R x
--R +
--R 4 3 6 1 4 5 3 18 2 7 12 3 6 2
--R (- A a b + - B a b )e + (-- A a b + -- B a b )d e
--R 5 5 5 5
--R +
--R 12 8 18 2 7 2 1 9 4 8 3
--R (-- A a b + -- B a b )d e + (- A b + - B a b )d
--R 5 5 5 5
--R *
--R 5
--R x
--R +
--R 1 4 5 3 3 6 3 4 5 2 9 2 7 3 6 2
--R - A a b e + (3A a b + - B a b )d e + (- A a b + 3B a b )d e
--R 4 4 2
--R +
--R 8 3 2 7 3
--R (A a b + - B a b )d
--R 2
--R *
--R 4
--R x
--R +
--R 4 5 2 3 6 4 5 2 2 7 4 3 6 3 3
--R (A a b d e + (4A a b + B a b )d e + (2A a b + - B a b )d )x
--R 3
--R +
--R 3 4 5 2 3 6 1 4 5 3 2 4 5 3
--R (- A a b d e + (2A a b + - B a b )d )x + A a b d x
--R 2 2
--R +
--R 1 8 1 9 3 1 7 2 3 8 2
--R (- --- A a b + --- B a )e + (-- A a b - --- B a b)d e
--R 280 630 35 280
--R +
--R 1 6 3 1 7 2 2 1 5 4 1 6 3 3
--R (- -- A a b + -- B a b )d e + (- A a b - -- B a b )d
--R 10 35 5 30
--R /
--R 5
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 140
--S 141 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 141
)clear all
--S 142 of 764
t0:=(A+B*x)*(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 2 7 4 3 2 4 6
--R B b e x + ((A b + 4B a b )e + 2B b d e)x
--R +
--R 3 2 2 2 4 3 4 2 5
--R ((4A a b + 6B a b )e + (2A b + 8B a b )d e + B b d )x
--R +
--R 2 2 3 2 3 2 2 4 3 2 4
--R ((6A a b + 4B a b)e + (8A a b + 12B a b )d e + (A b + 4B a b )d )x
--R +
--R 3 4 2 2 2 3 3 2 2 2 3
--R ((4A a b + B a )e + (12A a b + 8B a b)d e + (4A a b + 6B a b )d )x
--R +
--R 4 2 3 4 2 2 3 2 2
--R (A a e + (8A a b + 2B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 3 4 2 4 2
--R (2A a d e + (4A a b + B a )d )x + A a d
--R Type: Polynomial(Integer)
--E 142
--S 143 of 764
r0:=1/5*(A*b-a*B)*(b*d-a*e)^2*(a+b*x)^5/b^4+1/6*(b*d-a*e)*(b*B*d+2*A*b*e-_
3*a*B*e)*(a+b*x)^6/b^4+1/7*e*(2*b*B*d+A*b*e-3*a*B*e)*(a+b*x)^7/b^4+_
1/8*B*e^2*(a+b*x)^8/b^4
--R
--R
--R (2)
--R 1 8 2 8 1 8 4 7 2 2 8 7
--R - B b e x + ((- A b + - B a b )e + - B b d e)x
--R 8 7 7 7
--R +
--R 2 7 2 6 2 1 8 4 7 1 8 2 6
--R ((- A a b + B a b )e + (- A b + - B a b )d e + - B b d )x
--R 3 3 3 6
--R +
--R 6 2 6 4 3 5 2 8 7 12 2 6
--R (- A a b + - B a b )e + (- A a b + -- B a b )d e
--R 5 5 5 5
--R +
--R 1 8 4 7 2
--R (- A b + - B a b )d
--R 5 5
--R *
--R 5
--R x
--R +
--R 3 5 1 4 4 2 2 6 3 5
--R (A a b + - B a b )e + (3A a b + 2B a b )d e
--R 4
--R +
--R 7 3 2 6 2
--R (A a b + - B a b )d
--R 2
--R *
--R 4
--R x
--R +
--R 1 4 4 2 8 3 5 2 4 4 2 6 4 3 5 2 3
--R (- A a b e + (- A a b + - B a b )d e + (2A a b + - B a b )d )x
--R 3 3 3 3
--R +
--R 4 4 3 5 1 4 4 2 2 4 4 2
--R (A a b d e + (2A a b + - B a b )d )x + A a b d x
--R 2
--R +
--R 1 7 1 8 2 1 6 2 2 7
--R (--- A a b - --- B a )e + (- -- A a b + --- B a b)d e
--R 105 280 15 105
--R +
--R 1 5 3 1 6 2 2
--R (- A a b - -- B a b )d
--R 5 30
--R /
--R 4
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 143
--S 144 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 144
)clear all
--S 145 of 764
t0:=(A+B*x)*(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 6 4 3 4 5
--R B b e x + ((A b + 4B a b )e + B b d)x
--R +
--R 3 2 2 4 3 4
--R ((4A a b + 6B a b )e + (A b + 4B a b )d)x
--R +
--R 2 2 3 3 2 2 3
--R ((6A a b + 4B a b)e + (4A a b + 6B a b )d)x
--R +
--R 3 4 2 2 3 2 4 3 4
--R ((4A a b + B a )e + (6A a b + 4B a b)d)x + (A a e + (4A a b + B a )d)x
--R +
--R 4
--R A a d
--R Type: Polynomial(Integer)
--E 145
--S 146 of 764
r0:=1/5*(A*b-a*B)*(b*d-a*e)*(a+b*x)^5/b^3+1/6*(b*B*d+A*b*e-2*a*B*e)*_
(a+b*x)^6/b^3+1/7*B*e*(a+b*x)^7/b^3
--R
--R
--R (2)
--R 1 7 7 1 7 2 6 1 7 6
--R - B b e x + ((- A b + - B a b )e + - B b d)x
--R 7 6 3 6
--R +
--R 4 6 6 2 5 1 7 4 6 5
--R ((- A a b + - B a b )e + (- A b + - B a b )d)x
--R 5 5 5 5
--R +
--R 3 2 5 3 4 6 3 2 5 4
--R ((- A a b + B a b )e + (A a b + - B a b )d)x
--R 2 2
--R +
--R 4 3 4 1 4 3 2 5 4 3 4 3
--R ((- A a b + - B a b )e + (2A a b + - B a b )d)x
--R 3 3 3
--R +
--R 1 4 3 3 4 1 4 3 2 4 3
--R (- A a b e + (2A a b + - B a b )d)x + A a b d x
--R 2 2
--R +
--R 1 6 1 7 1 5 2 1 6
--R (- -- A a b + --- B a )e + (- A a b - -- B a b)d
--R 30 105 5 30
--R /
--R 3
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 146
--S 147 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 147
)clear all
--S 148 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3 2 2 3 2
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x + (6A a b + 4B a b)x
--R +
--R 3 4 4
--R (4A a b + B a )x + A a
--R Type: Polynomial(Integer)
--E 148
--S 149 of 764
r0:=1/5*(A*b-a*B)*(a+b*x)^5/b^2+1/6*B*(a+b*x)^6/b^2
--R
--R
--R (2)
--R 1 6 6 1 6 4 5 5 5 3 2 4 4
--R - B b x + (- A b + - B a b )x + (A a b + - B a b )x
--R 6 5 5 2
--R +
--R 2 4 4 3 3 3 3 3 1 4 2 2 4 2 1 5
--R (2A a b + - B a b )x + (2A a b + - B a b )x + A a b x + - A a b
--R 3 2 5
--R +
--R 1 6
--R - -- B a
--R 30
--R /
--R 2
--R b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 149
--S 150 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 150
)clear all
--S 151 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R e x + d
--R Type: Fraction(Polynomial(Integer))
--E 151
--S 152 of 764
r0:=b*(b*d-a*e)^3*(B*d-A*e)*x/e^5-1/2*(b*d-a*e)^2*(B*d-A*e)*(a+b*x)^2/e^4+_
1/3*(b*d-a*e)*(B*d-A*e)*(a+b*x)^3/e^3-1/4*(B*d-A*e)*(a+b*x)^4/e^2+_
1/5*B*(a+b*x)^5/(b*e)-(b*d-a*e)^4*(B*d-A*e)*log(d+e*x)/e^6
--R
--R
--R (2)
--R 4 5 3 2 4 4
--R 60A a b e + (- 240A a b - 60B a b)d e
--R +
--R 2 3 3 2 2 3 4 2 3 3 2
--R (360A a b + 240B a b )d e + (- 240A a b - 360B a b )d e
--R +
--R 5 4 4 5 5
--R (60A b + 240B a b )d e - 60B b d
--R *
--R log(e x + d)
--R +
--R 5 5 5 5 4 5 5 4 4
--R 12B b e x + ((15A b + 60B a b )e - 15B b d e )x
--R +
--R 4 2 3 5 5 4 4 5 2 3 3
--R ((80A a b + 120B a b )e + (- 20A b - 80B a b )d e + 20B b d e )x
--R +
--R 2 3 3 2 5 4 2 3 4
--R (180A a b + 120B a b )e + (- 120A a b - 180B a b )d e
--R +
--R 5 4 2 3 5 3 2
--R (30A b + 120B a b )d e - 30B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 5 2 3 3 2 4
--R (240A a b + 60B a b)e + (- 360A a b - 240B a b )d e
--R +
--R 4 2 3 2 3 5 4 3 2 5 4
--R (240A a b + 360B a b )d e + (- 60A b - 240B a b )d e + 60B b d e
--R *
--R x
--R +
--R 4 5 5 3 2 4 4
--R (65A a b + 12B a )e + (- 80A a b - 65B a b)d e
--R +
--R 2 3 3 2 2 3 2 3 3 2
--R (30A a b + 80B a b )d e - 30B a b d e
--R /
--R 6
--R 60b e
--R Type: Expression(Integer)
--E 152
--S 153 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 153
)clear all
--S 154 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^2
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 2 2 2
--R e x + 2d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 154
--S 155 of 764
r0:=-2*b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-2*a*B*e)*x/e^5+(b*d-a*e)^4*_
(B*d-A*e)/(e^6*(d+e*x))+b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)*_
(d+e*x)^2/e^6-1/3*b^3*(5*b*B*d-A*b*e-4*a*B*e)*(d+e*x)^3/e^6+1/4*b^4*_
B*(d+e*x)^4/e^6+(b*d-a*e)^3*(5*b*B*d-4*A*b*e-a*B*e)*log(d+e*x)/e^6
--R
--R
--R (2)
--R 3 4 5 2 2 3 4
--R (48A a b + 12B a )e + (- 144A a b - 96B a b)d e
--R +
--R 3 2 2 2 3 4 3 3 2
--R (144A a b + 216B a b )d e + (- 48A b - 192B a b )d e
--R +
--R 4 4
--R 60B b d e
--R *
--R x
--R +
--R 3 4 4 2 2 3 2 3
--R (48A a b + 12B a )d e + (- 144A a b - 96B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (144A a b + 216B a b )d e + (- 48A b - 192B a b )d e + 60B b d
--R *
--R log(e x + d)
--R +
--R 4 5 5 4 3 5 4 4 4
--R 3B b e x + ((4A b + 16B a b )e - 5B b d e )x
--R +
--R 3 2 2 5 4 3 4 4 2 3 3
--R ((24A a b + 36B a b )e + (- 8A b - 32B a b )d e + 10B b d e )x
--R +
--R 2 2 3 5 3 2 2 4
--R (72A a b + 48B a b)e + (- 72A a b - 108B a b )d e
--R +
--R 4 3 2 3 4 3 2
--R (24A b + 96B a b )d e - 30B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 4 3 2 2 2 3
--R (72A a b + 48B a b)d e + (- 72A a b - 108B a b )d e
--R +
--R 4 3 3 2 4 4
--R (16A b + 64B a b )d e - 5B b d e
--R *
--R x
--R +
--R 4 5 3 4 4 2 2 3 2 3
--R - 12A a e + (48A a b + 12B a )d e + (- 72A a b - 48B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (72A a b + 108B a b )d e + (- 32A b - 128B a b )d e + 55B b d
--R /
--R 7 6
--R 12e x + 12d e
--R Type: Expression(Integer)
--E 155
--S 156 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 156
)clear all
--S 157 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^3
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 3 3 2 2 2 3
--R e x + 3d e x + 3d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 157
--S 158 of 764
r0:=2*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)*x/e^5+1/2*(b*d-a*e)^4*_
(B*d-A*e)/(e^6*(d+e*x)^2)-(b*d-a*e)^3*(5*b*B*d-4*A*b*e-a*B*e)/(e^6*_
(d+e*x))-1/2*b^3*(5*b*B*d-A*b*e-4*a*B*e)*(d+e*x)^2/e^6+1/3*b^4*B*_
(d+e*x)^3/e^6-2*b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-2*a*B*e)*log(d+e*x)/e^6
--R
--R
--R (2)
--R 2 2 3 5 3 2 2 4
--R (36A a b + 24B a b)e + (- 72A a b - 108B a b )d e
--R +
--R 4 3 2 3 4 3 2
--R (36A b + 144B a b )d e - 60B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 4 3 2 2 2 3
--R (72A a b + 48B a b)d e + (- 144A a b - 216B a b )d e
--R +
--R 4 3 3 2 4 4
--R (72A b + 288B a b )d e - 120B b d e
--R *
--R x
--R +
--R 2 2 3 2 3 3 2 2 3 2
--R (36A a b + 24B a b)d e + (- 72A a b - 108B a b )d e
--R +
--R 4 3 4 4 5
--R (36A b + 144B a b )d e - 60B b d
--R *
--R log(e x + d)
--R +
--R 4 5 5 4 3 5 4 4 4
--R 2B b e x + ((3A b + 12B a b )e - 5B b d e )x
--R +
--R 3 2 2 5 4 3 4 4 2 3 3
--R ((24A a b + 36B a b )e + (- 12A b - 48B a b )d e + 20B b d e )x
--R +
--R 3 2 2 4 4 3 2 3 4 3 2 2
--R ((48A a b + 72B a b )d e + (- 30A b - 120B a b )d e + 50B b d e )x
--R +
--R 3 4 5 2 2 3 4
--R (- 24A a b - 6B a )e + (72A a b + 48B a b)d e
--R +
--R 3 2 2 2 3 4 3 3 2 4 4
--R (- 48A a b - 72B a b )d e + (12A b + 48B a b )d e - 20B b d e
--R *
--R x
--R +
--R 4 5 3 4 4 2 2 3 2 3
--R - 3A a e + (- 12A a b - 3B a )d e + (54A a b + 36B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (- 60A a b - 90B a b )d e + (24A b + 96B a b )d e - 40B b d
--R /
--R 8 2 7 2 6
--R 6e x + 12d e x + 6d e
--R Type: Expression(Integer)
--E 158
--S 159 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 159
)clear all
--S 160 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^4
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 4 4 3 3 2 2 2 3 4
--R e x + 4d e x + 6d e x + 4d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 160
--S 161 of 764
r0:=-b^3*(4*b*B*d-A*b*e-4*a*B*e)*x/e^5+1/2*b^4*B*x^2/e^4+1/3*(b*d-a*e)^4*_
(B*d-A*e)/(e^6*(d+e*x)^3)-1/2*(b*d-a*e)^3*(5*b*B*d-4*A*b*e-a*B*e)/_
(e^6*(d+e*x)^2)+2*b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-2*a*B*e)/(e^6*(d+_
e*x))+2*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)*log(d+e*x)/e^6
--R
--R
--R (2)
--R 3 2 2 5 4 3 4 4 2 3 3
--R ((24A a b + 36B a b )e + (- 24A b - 96B a b )d e + 60B b d e )x
--R +
--R 3 2 2 4 4 3 2 3
--R (72A a b + 108B a b )d e + (- 72A b - 288B a b )d e
--R +
--R 4 3 2
--R 180B b d e
--R *
--R 2
--R x
--R +
--R 3 2 2 2 3 4 3 3 2
--R (72A a b + 108B a b )d e + (- 72A b - 288B a b )d e
--R +
--R 4 4
--R 180B b d e
--R *
--R x
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (24A a b + 36B a b )d e + (- 24A b - 96B a b )d e + 60B b d
--R *
--R log(e x + d)
--R +
--R 4 5 5 4 3 5 4 4 4
--R 3B b e x + ((6A b + 24B a b )e - 15B b d e )x
--R +
--R 4 3 4 4 2 3 3
--R ((18A b + 72B a b )d e - 63B b d e )x
--R +
--R 2 2 3 5 3 2 2 4
--R (- 36A a b - 24B a b)e + (72A a b + 108B a b )d e
--R +
--R 4 3 2 3 4 3 2
--R (- 18A b - 72B a b )d e - 9B b d e
--R *
--R 2
--R x
--R +
--R 3 4 5 2 2 3 4
--R (- 12A a b - 3B a )e + (- 36A a b - 24B a b)d e
--R +
--R 3 2 2 2 3 4 3 3 2 4 4
--R (108A a b + 162B a b )d e + (- 54A b - 216B a b )d e + 81B b d e
--R *
--R x
--R +
--R 4 5 3 4 4 2 2 3 2 3
--R - 2A a e + (- 4A a b - B a )d e + (- 12A a b - 8B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (44A a b + 66B a b )d e + (- 26A b - 104B a b )d e + 47B b d
--R /
--R 9 3 8 2 2 7 3 6
--R 6e x + 18d e x + 18d e x + 6d e
--R Type: Expression(Integer)
--E 161
--S 162 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 162
)clear all
--S 163 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^5
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R e x + 5d e x + 10d e x + 10d e x + 5d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 163
--S 164 of 764
r0:=b^4*B*x/e^5+1/4*(b*d-a*e)^4*(B*d-A*e)/(e^6*(d+e*x)^4)-1/3*(b*d-a*e)^3*_
(5*b*B*d-4*A*b*e-a*B*e)/(e^6*(d+e*x)^3)+b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-_
2*a*B*e)/(e^6*(d+e*x)^2)-2*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)/_
(e^6*(d+e*x))-b^3*(5*b*B*d-A*b*e-4*a*B*e)*log(d+e*x)/e^6
--R
--R
--R (2)
--R 4 3 5 4 4 4
--R ((12A b + 48B a b )e - 60B b d e )x
--R +
--R 4 3 4 4 2 3 3
--R ((48A b + 192B a b )d e - 240B b d e )x
--R +
--R 4 3 2 3 4 3 2 2
--R ((72A b + 288B a b )d e - 360B b d e )x
--R +
--R 4 3 3 2 4 4 4 3 4
--R ((48A b + 192B a b )d e - 240B b d e)x + (12A b + 48B a b )d e
--R +
--R 4 5
--R - 60B b d
--R *
--R log(e x + d)
--R +
--R 4 5 5 4 4 4
--R 12B b e x + 48B b d e x
--R +
--R 3 2 2 5 4 3 4 4 2 3 3
--R ((- 48A a b - 72B a b )e + (48A b + 192B a b )d e - 48B b d e )x
--R +
--R 2 2 3 5 3 2 2 4
--R (- 36A a b - 24B a b)e + (- 72A a b - 108B a b )d e
--R +
--R 4 3 2 3 4 3 2
--R (108A b + 432B a b )d e - 252B b d e
--R *
--R 2
--R x
--R +
--R 3 4 5 2 2 3 4
--R (- 16A a b - 4B a )e + (- 24A a b - 16B a b)d e
--R +
--R 3 2 2 2 3 4 3 3 2 4 4
--R (- 48A a b - 72B a b )d e + (88A b + 352B a b )d e - 248B b d e
--R *
--R x
--R +
--R 4 5 3 4 4 2 2 3 2 3
--R - 3A a e + (- 4A a b - B a )d e + (- 6A a b - 4B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (- 12A a b - 18B a b )d e + (25A b + 100B a b )d e - 77B b d
--R /
--R 10 4 9 3 2 8 2 3 7 4 6
--R 12e x + 48d e x + 72d e x + 48d e x + 12d e
--R Type: Expression(Integer)
--E 164
--S 165 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 165
)clear all
--S 166 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^6
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R e x + 6d e x + 15d e x + 20d e x + 15d e x + 6d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 166
--S 167 of 764
r0:=-1/5*(B*d-A*e)*(a+b*x)^5/(e*(b*d-a*e)*(d+e*x)^5)-1/4*B*(b*d-a*e)^4/_
(e^6*(d+e*x)^4)+4/3*b*B*(b*d-a*e)^3/(e^6*(d+e*x)^3)-3*b^2*B*(b*d-_
a*e)^2/(e^6*(d+e*x)^2)+4*b^3*B*(b*d-a*e)/(e^6*(d+e*x))+b^4*B*_
log(d+e*x)/e^6
--R
--R
--R (2)
--R 4 6 5 5 5 4 5 5 2 4 4
--R (60B a b e - 60B b d e )x + (300B a b d e - 300B b d e )x
--R +
--R 4 2 4 5 3 3 3 4 3 3 5 4 2 2
--R (600B a b d e - 600B b d e )x + (600B a b d e - 600B b d e )x
--R +
--R 4 4 2 5 5 4 5 5 6
--R (300B a b d e - 300B b d e)x + 60B a b d e - 60B b d
--R *
--R log(e x + d)
--R +
--R 5 6 5 5 5
--R (- 12A b e + 12B b d e )x
--R +
--R 4 2 3 6 4 5 5 2 4 4
--R ((- 60A a b - 240B a b )e + 540B a b d e - 240B b d e )x
--R +
--R 2 3 3 2 6 2 3 5 4 2 4
--R (- 120A a b - 180B a b )e - 300B a b d e + 1380B a b d e
--R +
--R 5 3 3
--R - 780B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 3 2 5 2 3 2 4
--R (- 120A a b - 80B a b)e - 100B a b d e - 300B a b d e
--R +
--R 4 3 3 5 4 2
--R 1580B a b d e - 980B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 4 5 3 2 2 4 2 3 3 3
--R (- 60A a b - 15B a )e - 25B a b d e - 50B a b d e - 150B a b d e
--R +
--R 4 4 2 5 5
--R 865B a b d e - 565B b d e
--R *
--R x
--R +
--R 5 6 5 5 4 2 4 3 2 3 3 2 3 4 2
--R - 12A a e - 3B a d e - 5B a b d e - 10B a b d e - 30B a b d e
--R +
--R 4 5 5 6
--R 185B a b d e - 125B b d
--R /
--R 12 11 5 11 2 10 4
--R (60a e - 60b d e )x + (300a d e - 300b d e )x
--R +
--R 2 10 3 9 3 3 9 4 8 2
--R (600a d e - 600b d e )x + (600a d e - 600b d e )x
--R +
--R 4 8 5 7 5 7 6 6
--R (300a d e - 300b d e )x + 60a d e - 60b d e
--R Type: Expression(Integer)
--E 167
--S 168 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 168
)clear all
--S 169 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^7
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R e x + 7d e x + 21d e x + 35d e x + 35d e x + 21d e x + 7d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 169
--S 170 of 764
r0:=-1/6*(B*d-A*e)*(a+b*x)^5/(e*(b*d-a*e)*(d+e*x)^6)+1/30*(5*b*B*d+A*b*e-_
6*a*B*e)*(a+b*x)^5/(e*(b*d-a*e)^2*(d+e*x)^5)
--R
--R
--R (2)
--R 1 6 1 5 1 6 6
--R ((-- A b - - B a b )e + - B b d)x
--R 30 5 6
--R +
--R 2 4 1 6 4 5 5
--R (- B a b e + (- A b + - B a b )d)x
--R 5 5
--R +
--R 1 2 4 3 3 5 3 2 4 4
--R ((- - A a b - 2B a b )e + (A a b + - B a b )d)x
--R 2 2
--R +
--R 4 3 3 4 2 2 4 4 3 3 3
--R ((- - A a b - 2B a b )e + (2A a b + - B a b )d)x
--R 3 3
--R +
--R 3 4 2 5 3 3 1 4 2 2
--R ((- - A a b - B a b)e + (2A a b + - B a b )d)x
--R 2 2
--R +
--R 4 5 1 6 4 2 1 6 1 5 1 6
--R ((- - A a b - - B a )e + A a b d)x - - A a e + (- A a b - -- B a )d
--R 5 5 6 5 30
--R /
--R 2 8 7 2 2 6 6 2 7 2 6 2 3 5 5
--R (a e - 2a b d e + b d e )x + (6a d e - 12a b d e + 6b d e )x
--R +
--R 2 2 6 3 5 2 4 4 4
--R (15a d e - 30a b d e + 15b d e )x
--R +
--R 2 3 5 4 4 2 5 3 3
--R (20a d e - 40a b d e + 20b d e )x
--R +
--R 2 4 4 5 3 2 6 2 2 2 5 3 6 2 2 7
--R (15a d e - 30a b d e + 15b d e )x + (6a d e - 12a b d e + 6b d e)x
--R +
--R 2 6 2 7 2 8
--R a d e - 2a b d e + b d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 170
--S 171 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 171
)clear all
--S 172 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^8
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 8 8 7 7 2 6 6 3 5 5 4 4 4 5 3 3 6 2 2
--R e x + 8d e x + 28d e x + 56d e x + 70d e x + 56d e x + 28d e x
--R +
--R 7 8
--R 8d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 172
--S 173 of 764
r0:=-1/7*(B*d-A*e)*(a+b*x)^5/(e*(b*d-a*e)*(d+e*x)^7)+1/42*(5*b*B*d+2*A*b*e-_
7*a*B*e)*(a+b*x)^5/(e*(b*d-a*e)^2*(d+e*x)^6)+1/210*b*(5*b*B*d+2*A*b*e-_
7*a*B*e)*(a+b*x)^5/(e*(b*d-a*e)^3*(d+e*x)^5)
--R
--R
--R (2)
--R 1 7 1 6 2 1 7 7
--R ((- --- A b + -- B a b )e - -- B b d e)x
--R 105 30 42
--R +
--R 1 7 7 6 1 7 2 6
--R ((- -- A b + -- B a b )d e - - B b d )x
--R 15 30 6
--R +
--R 1 3 4 2 3 2 5 1 7 4 6 2 5
--R (- - B a b e + - B a b d e + (- - A b - - B a b )d )x
--R 2 2 5 5
--R +
--R 1 3 4 4 4 3 2 2 5 19 3 4
--R (- - A a b - - B a b )e + (A a b + -- B a b )d e
--R 3 3 6
--R +
--R 6 3 2 5 2
--R (- A a b - - B a b )d
--R 2
--R *
--R 4
--R x
--R +
--R 4 3 3 5 2 2 8 3 4 19 4 3
--R (- A a b - - B a b )e + (- A a b + -- B a b )d e
--R 2 3 6
--R +
--R 2 5 4 3 4 2
--R (- 2A a b - - B a b )d
--R 3
--R *
--R 3
--R x
--R +
--R 6 5 2 4 6 2 4 3 3 5 2
--R (- - A a b - - B a b)e + (3A a b + - B a b )d e
--R 5 5 2
--R +
--R 3 4 1 4 3 2
--R (- 2A a b - - B a b )d
--R 2
--R *
--R 2
--R x
--R +
--R 2 6 1 7 2 8 5 2 7 6 4 3 2
--R ((- - A a b - - B a )e + (- A a b + -- B a b)d e - A a b d )x
--R 3 6 5 30
--R +
--R 1 7 2 1 6 1 7 1 5 2 1 6 2
--R - - A a e + (- A a b - -- B a )d e + (- - A a b + -- B a b)d
--R 7 3 42 5 30
--R /
--R 3 10 2 9 2 2 8 3 3 7 7
--R (a e - 3a b d e + 3a b d e - b d e )x
--R +
--R 3 9 2 2 8 2 3 7 3 4 6 6
--R (7a d e - 21a b d e + 21a b d e - 7b d e )x
--R +
--R 3 2 8 2 3 7 2 4 6 3 5 5 5
--R (21a d e - 63a b d e + 63a b d e - 21b d e )x
--R +
--R 3 3 7 2 4 6 2 5 5 3 6 4 4
--R (35a d e - 105a b d e + 105a b d e - 35b d e )x
--R +
--R 3 4 6 2 5 5 2 6 4 3 7 3 3
--R (35a d e - 105a b d e + 105a b d e - 35b d e )x
--R +
--R 3 5 5 2 6 4 2 7 3 3 8 2 2
--R (21a d e - 63a b d e + 63a b d e - 21b d e )x
--R +
--R 3 6 4 2 7 3 2 8 2 3 9 3 7 3 2 8 2
--R (7a d e - 21a b d e + 21a b d e - 7b d e)x + a d e - 3a b d e
--R +
--R 2 9 3 10
--R 3a b d e - b d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 173
--S 174 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 174
)clear all
--S 175 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^9
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 9 9 8 8 2 7 7 3 6 6 4 5 5 5 4 4 6 3 3
--R e x + 9d e x + 36d e x + 84d e x + 126d e x + 126d e x + 84d e x
--R +
--R 7 2 2 8 9
--R 36d e x + 9d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 175
--S 176 of 764
r0:=-1/8*(B*d-A*e)*(a+b*x)^5/(e*(b*d-a*e)*(d+e*x)^8)+1/56*(5*b*B*d+3*A*b*e-_
8*a*B*e)*(a+b*x)^5/(e*(b*d-a*e)^2*(d+e*x)^7)+1/168*b*(5*b*B*d+3*A*b*e-_
8*a*B*e)*(a+b*x)^5/(e*(b*d-a*e)^3*(d+e*x)^6)+1/840*b^2*(5*b*B*d+_
3*A*b*e-8*a*B*e)*(a+b*x)^5/(e*(b*d-a*e)^4*(d+e*x)^5)
--R
--R
--R (2)
--R 1 8 1 7 3 1 8 2 8
--R ((--- A b - --- B a b )e + --- B b d e )x
--R 280 105 168
--R +
--R 1 8 8 7 2 1 8 2 7
--R ((-- A b - --- B a b )d e + -- B b d e)x
--R 35 105 21
--R +
--R 1 8 4 7 2 1 8 3 6
--R ((-- A b - -- B a b )d e + - B b d )x
--R 10 15 6
--R +
--R 1 4 4 3 4 3 5 2 2 6 2 1 8 4 7 3 5
--R (- - B a b e + - B a b d e - 2B a b d e + (- A b + - B a b )d )x
--R 3 3 5 5
--R +
--R 1 4 4 5 3 3 3 5 43 4 4 2
--R (- - A a b - B a b )e + (A a b + -- B a b )d e
--R 4 12
--R +
--R 3 2 6 13 3 5 2 7 3 2 6 3
--R (- - A a b - -- B a b )d e + (A a b + - B a b )d
--R 2 3 2
--R *
--R 4
--R x
--R +
--R 4 5 3 6 6 2 3 4 4 5 3 2
--R (- - A a b - - B a b )e + (3A a b + 4B a b )d e
--R 5 5
--R +
--R 3 5 13 4 4 2 2 6 4 3 5 3
--R (- 4A a b - -- B a b )d e + (2A a b + - B a b )d
--R 3 3
--R *
--R 3
--R x
--R +
--R 6 2 2 7 3 18 5 3 31 6 2 2
--R (- A a b - - B a b)e + (-- A a b + -- B a b )d e
--R 3 5 15
--R +
--R 9 4 4 5 3 2 3 5 1 4 4 3
--R (- - A a b - 2B a b )d e + (2A a b + - B a b )d
--R 2 2
--R *
--R 2
--R x
--R +
--R 4 7 1 8 3 6 2 8 7 2
--R (- - A a b - - B a )e + (2A a b + -- B a b)d e
--R 7 7 21
--R +
--R 12 5 3 4 6 2 2 4 4 3
--R (- -- A a b - -- B a b )d e + A a b d
--R 5 15
--R *
--R x
--R +
--R 1 8 3 3 7 1 8 2 1 6 2 1 7 2
--R - - A a e + (- A a b - -- B a )d e + (- - A a b + -- B a b)d e
--R 8 7 56 2 21
--R +
--R 1 5 3 1 6 2 3
--R (- A a b - -- B a b )d
--R 5 30
--R /
--R 4 12 3 11 2 2 2 10 3 3 9 4 4 8 8
--R (a e - 4a b d e + 6a b d e - 4a b d e + b d e )x
--R +
--R 4 11 3 2 10 2 2 3 9 3 4 8 4 5 7 7
--R (8a d e - 32a b d e + 48a b d e - 32a b d e + 8b d e )x
--R +
--R 4 2 10 3 3 9 2 2 4 8 3 5 7 4 6 6 6
--R (28a d e - 112a b d e + 168a b d e - 112a b d e + 28b d e )x
--R +
--R 4 3 9 3 4 8 2 2 5 7 3 6 6 4 7 5 5
--R (56a d e - 224a b d e + 336a b d e - 224a b d e + 56b d e )x
--R +
--R 4 4 8 3 5 7 2 2 6 6 3 7 5 4 8 4 4
--R (70a d e - 280a b d e + 420a b d e - 280a b d e + 70b d e )x
--R +
--R 4 5 7 3 6 6 2 2 7 5 3 8 4 4 9 3 3
--R (56a d e - 224a b d e + 336a b d e - 224a b d e + 56b d e )x
--R +
--R 4 6 6 3 7 5 2 2 8 4 3 9 3 4 10 2 2
--R (28a d e - 112a b d e + 168a b d e - 112a b d e + 28b d e )x
--R +
--R 4 7 5 3 8 4 2 2 9 3 3 10 2 4 11 4 8 4
--R (8a d e - 32a b d e + 48a b d e - 32a b d e + 8b d e)x + a d e
--R +
--R 3 9 3 2 2 10 2 3 11 4 12
--R - 4a b d e + 6a b d e - 4a b d e + b d
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 176
--S 177 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 177
)clear all
--S 178 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^10
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R e x + 10d e x + 45d e x + 120d e x + 210d e x + 252d e x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210d e x + 120d e x + 45d e x + 10d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 178
--S 179 of 764
r0:=1/9*(b*d-a*e)^4*(B*d-A*e)/(e^6*(d+e*x)^9)-1/8*(b*d-a*e)^3*(5*b*B*d-_
4*A*b*e-a*B*e)/(e^6*(d+e*x)^8)+2/7*b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-_
2*a*B*e)/(e^6*(d+e*x)^7)-1/3*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)/_
(e^6*(d+e*x)^6)+1/5*b^3*(5*b*B*d-A*b*e-4*a*B*e)/(e^6*(d+e*x)^5)-_
1/4*b^4*B/(e^6*(d+e*x)^4)
--R
--R
--R (2)
--R 1 4 5 5 1 4 4 3 5 1 4 4 4
--R - - B b e x + ((- - A b - - B a b )e - - B b d e )x
--R 4 5 5 4
--R +
--R 2 3 2 2 5 2 4 8 3 4 1 4 2 3 3
--R ((- - A a b - B a b )e + (- -- A b - -- B a b )d e - - B b d e )x
--R 3 15 15 6
--R +
--R 6 2 2 4 3 5 2 3 3 2 2 4
--R (- - A a b - - B a b)e + (- - A a b - - B a b )d e
--R 7 7 7 7
--R +
--R 2 4 8 3 2 3 1 4 3 2
--R (- -- A b - -- B a b )d e - -- B b d e
--R 35 35 14
--R *
--R 2
--R x
--R +
--R 1 3 1 4 5 3 2 2 1 3 4
--R (- - A a b - - B a )e + (- -- A a b - - B a b)d e
--R 2 8 14 7
--R +
--R 1 3 3 2 2 2 3 1 4 2 3 3 2
--R (- -- A a b - -- B a b )d e + (- -- A b - -- B a b )d e
--R 14 28 70 35
--R +
--R 1 4 4
--R - -- B b d e
--R 56
--R *
--R x
--R +
--R 1 4 5 1 3 1 4 4 1 2 2 1 3 2 3
--R - - A a e + (- -- A a b - -- B a )d e + (- -- A a b - -- B a b)d e
--R 9 18 72 42 63
--R +
--R 1 3 1 2 2 3 2 1 4 2 3 4 1 4 5
--R (- --- A a b - -- B a b )d e + (- --- A b - --- B a b )d e - --- B b d
--R 126 84 630 315 504
--R /
--R 15 9 14 8 2 13 7 3 12 6 4 11 5 5 10 4
--R e x + 9d e x + 36d e x + 84d e x + 126d e x + 126d e x
--R +
--R 6 9 3 7 8 2 8 7 9 6
--R 84d e x + 36d e x + 9d e x + d e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 179
--S 180 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 180
)clear all
--S 181 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^11
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 11 11 10 10 2 9 9 3 8 8 4 7 7 5 6 6
--R e x + 11d e x + 55d e x + 165d e x + 330d e x + 462d e x
--R +
--R 6 5 5 7 4 4 8 3 3 9 2 2 10 11
--R 462d e x + 330d e x + 165d e x + 55d e x + 11d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 181
--S 182 of 764
r0:=1/10*(b*d-a*e)^4*(B*d-A*e)/(e^6*(d+e*x)^10)-1/9*(b*d-a*e)^3*(5*b*B*d-_
4*A*b*e-a*B*e)/(e^6*(d+e*x)^9)+1/4*b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-_
2*a*B*e)/(e^6*(d+e*x)^8)-2/7*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)/_
(e^6*(d+e*x)^7)+1/6*b^3*(5*b*B*d-A*b*e-4*a*B*e)/(e^6*(d+e*x)^6)-_
1/5*b^4*B/(e^6*(d+e*x)^5)
--R
--R
--R (2)
--R 1 4 5 5 1 4 2 3 5 1 4 4 4
--R - - B b e x + ((- - A b - - B a b )e - - B b d e )x
--R 5 6 3 6
--R +
--R 4 3 6 2 2 5 2 4 8 3 4 2 4 2 3 3
--R ((- - A a b - - B a b )e + (- -- A b - -- B a b )d e - -- B b d e )x
--R 7 7 21 21 21
--R +
--R 3 2 2 1 3 5 3 3 9 2 2 4
--R (- - A a b - - B a b)e + (- -- A a b - -- B a b )d e
--R 4 2 14 28
--R +
--R 1 4 1 3 2 3 1 4 3 2
--R (- -- A b - - B a b )d e - -- B b d e
--R 28 7 28
--R *
--R 2
--R x
--R +
--R 4 3 1 4 5 1 2 2 1 3 4
--R (- - A a b - - B a )e + (- - A a b - - B a b)d e
--R 9 9 6 9
--R +
--R 1 3 1 2 2 2 3 1 4 2 3 3 2
--R (- -- A a b - -- B a b )d e + (- --- A b - -- B a b )d e
--R 21 14 126 63
--R +
--R 1 4 4
--R - --- B b d e
--R 126
--R *
--R x
--R +
--R 1 4 5 2 3 1 4 4 1 2 2 1 3 2 3
--R - -- A a e + (- -- A a b - -- B a )d e + (- -- A a b - -- B a b)d e
--R 10 45 90 60 90
--R +
--R 1 3 1 2 2 3 2 1 4 1 3 4
--R (- --- A a b - --- B a b )d e + (- ---- A b - --- B a b )d e
--R 210 140 1260 315
--R +
--R 1 4 5
--R - ---- B b d
--R 1260
--R /
--R 16 10 15 9 2 14 8 3 13 7 4 12 6 5 11 5
--R e x + 10d e x + 45d e x + 120d e x + 210d e x + 252d e x
--R +
--R 6 10 4 7 9 3 8 8 2 9 7 10 6
--R 210d e x + 120d e x + 45d e x + 10d e x + d e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 182
--S 183 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 183
)clear all
--S 184 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^12
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 12 12 11 11 2 10 10 3 9 9 4 8 8 5 7 7
--R e x + 12d e x + 66d e x + 220d e x + 495d e x + 792d e x
--R +
--R 6 6 6 7 5 5 8 4 4 9 3 3 10 2 2 11 12
--R 924d e x + 792d e x + 495d e x + 220d e x + 66d e x + 12d e x + d
--R Type: Fraction(Polynomial(Integer))
--E 184
--S 185 of 764
r0:=1/11*(b*d-a*e)^4*(B*d-A*e)/(e^6*(d+e*x)^11)-1/10*(b*d-a*e)^3*(5*b*B*d-_
4*A*b*e-a*B*e)/(e^6*(d+e*x)^10)+2/9*b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-_
2*a*B*e)/(e^6*(d+e*x)^9)-1/4*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)/_
(e^6*(d+e*x)^8)+1/7*b^3*(5*b*B*d-A*b*e-4*a*B*e)/(e^6*(d+e*x)^7)-_
1/6*b^4*B/(e^6*(d+e*x)^6)
--R
--R
--R (2)
--R 1 4 5 5 1 4 4 3 5 5 4 4 4
--R - - B b e x + ((- - A b - - B a b )e - -- B b d e )x
--R 6 7 7 42
--R +
--R 1 3 3 2 2 5 1 4 2 3 4 5 4 2 3 3
--R ((- - A a b - - B a b )e + (- -- A b - - B a b )d e - -- B b d e )x
--R 2 4 14 7 84
--R +
--R 2 2 2 4 3 5 1 3 1 2 2 4
--R (- - A a b - - B a b)e + (- - A a b - - B a b )d e
--R 3 9 6 4
--R +
--R 1 4 2 3 2 3 5 4 3 2
--R (- -- A b - -- B a b )d e - --- B b d e
--R 42 21 252
--R *
--R 2
--R x
--R +
--R 2 3 1 4 5 2 2 2 4 3 4
--R (- - A a b - -- B a )e + (- -- A a b - -- B a b)d e
--R 5 10 15 45
--R +
--R 1 3 1 2 2 2 3 1 4 2 3 3 2
--R (- -- A a b - -- B a b )d e + (- --- A b - --- B a b )d e
--R 30 20 210 105
--R +
--R 1 4 4
--R - --- B b d e
--R 252
--R *
--R x
--R +
--R 1 4 5 2 3 1 4 4
--R - -- A a e + (- -- A a b - --- B a )d e
--R 11 55 110
--R +
--R 2 2 2 4 3 2 3 1 3 1 2 2 3 2
--R (- --- A a b - --- B a b)d e + (- --- A a b - --- B a b )d e
--R 165 495 330 220
--R +
--R 1 4 2 3 4 1 4 5
--R (- ---- A b - ---- B a b )d e - ---- B b d
--R 2310 1155 2772
--R /
--R 17 11 16 10 2 15 9 3 14 8 4 13 7 5 12 6
--R e x + 11d e x + 55d e x + 165d e x + 330d e x + 462d e x
--R +
--R 6 11 5 7 10 4 8 9 3 9 8 2 10 7 11 6
--R 462d e x + 330d e x + 165d e x + 55d e x + 11d e x + d e
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 185
--S 186 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 186
)clear all
--S 187 of 764
t0:=(A+B*x)*(d+e*x)^4/(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B e x + (A e + 4B d e )x + (4A d e + 6B d e )x
--R +
--R 2 2 3 2 3 4 4
--R (6A d e + 4B d e)x + (4A d e + B d )x + A d
--R /
--R 2 2 2
--R b x + 2a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 187
--S 188 of 764
r0:=2*e*(b*d-a*e)^2*(2*b*B*d+3*A*b*e-5*a*B*e)*x/b^5-(A*b-a*B)*(b*d-a*e)^4/_
(b^6*(a+b*x))+e^2*(b*d-a*e)*(3*b*B*d+2*A*b*e-5*a*B*e)*(a+b*x)^2/b^6+_
1/3*e^3*(4*b*B*d+A*b*e-5*a*B*e)*(a+b*x)^3/b^6+1/4*B*e^4*(a+b*x)^4/b^6+_
(b*d-a*e)^3*(b*B*d+4*A*b*e-5*a*B*e)*log(a+b*x)/b^6
--R
--R
--R (2)
--R 3 2 4 4 2 3 3 2 3
--R (- 48A a b + 60B a b)e + (144A a b - 192B a b )d e
--R +
--R 4 2 3 2 2 5 4 3 5 4
--R (- 144A a b + 216B a b )d e + (48A b - 96B a b )d e + 12B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 48A a b + 60B a )e + (144A a b - 192B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3 4 4
--R (- 144A a b + 216B a b )d e + (48A a b - 96B a b )d e + 12B a b d
--R *
--R log(b x + a)
--R +
--R 5 4 5 5 4 4 5 3 4
--R 3B b e x + ((4A b - 5B a b )e + 16B b d e )x
--R +
--R 4 2 3 4 5 4 3 5 2 2 3
--R ((- 8A a b + 10B a b )e + (24A b - 32B a b )d e + 36B b d e )x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (24A a b - 30B a b )e + (- 72A a b + 96B a b )d e
--R +
--R 5 4 2 2 5 3
--R (72A b - 108B a b )d e + 48B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (16A a b - 5B a b)e + (- 72A a b + 64B a b )d e
--R +
--R 4 2 3 2 2 4 3
--R (72A a b - 108B a b )d e + 48B a b d e
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 32A a b + 55B a )e + (72A a b - 128B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (- 72A a b + 108B a b )d e + (48A a b - 48B a b )d e
--R +
--R 5 4 4
--R (- 12A b + 12B a b )d
--R /
--R 7 6
--R 12b x + 12a b
--R Type: Expression(Integer)
--E 188
--S 189 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 189
)clear all
--S 190 of 764
t0:=(A+B*x)*(d+e*x)^3/(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x + A d
--R ----------------------------------------------------------------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 190
--S 191 of 764
r0:=3*e*(b*d-a*e)*(b*B*d+A*b*e-2*a*B*e)*x/b^4-(A*b-a*B)*(b*d-a*e)^3/(b^5*_
(a+b*x))+1/2*e^2*(3*b*B*d+A*b*e-4*a*B*e)*(a+b*x)^2/b^5+1/3*B*e^3*_
(a+b*x)^3/b^5+(b*d-a*e)^2*(b*B*d+3*A*b*e-4*a*B*e)*log(a+b*x)/b^5
--R
--R
--R (2)
--R 2 2 3 3 3 2 2 2
--R (18A a b - 24B a b)e + (- 36A a b + 54B a b )d e
--R +
--R 4 3 2 4 3
--R (18A b - 36B a b )d e + 6B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (18A a b - 24B a )e + (- 36A a b + 54B a b)d e
--R +
--R 3 2 2 2 3 3
--R (18A a b - 36B a b )d e + 6B a b d
--R *
--R log(b x + a)
--R +
--R 4 3 4 4 3 3 4 2 3
--R 2B b e x + ((3A b - 4B a b )e + 9B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 2
--R ((- 9A a b + 12B a b )e + (18A b - 27B a b )d e + 18B b d e)x
--R +
--R 2 2 3 3 3 2 2 2 3 2
--R ((- 9A a b + 8B a b)e + (18A a b - 27B a b )d e + 18B a b d e)x
--R +
--R 3 4 3 2 2 3 2
--R (9A a b - 16B a )e + (- 18A a b + 27B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (18A a b - 18B a b )d e + (- 6A b + 6B a b )d
--R /
--R 6 5
--R 6b x + 6a b
--R Type: Expression(Integer)
--E 191
--S 192 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 192
)clear all
--S 193 of 764
t0:=(A+B*x)*(d+e*x)^2/(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R 2 3 2 2 2 2
--R B e x + (A e + 2B d e)x + (2A d e + B d )x + A d
--R (1) ----------------------------------------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 193
--S 194 of 764
r0:=e*(2*b*B*d+A*b*e-2*a*B*e)*x/b^3+1/2*B*e^2*x^2/b^2-(A*b-a*B)*_
(b*d-a*e)^2/(b^4*(a+b*x))+(b*d-a*e)*(b*B*d+2*A*b*e-3*a*B*e)*log(a+b*x)/b^4
--R
--R
--R (2)
--R 2 2 2 3 2 3 2
--R ((- 4A a b + 6B a b)e + (4A b - 8B a b )d e + 2B b d )x
--R +
--R 2 3 2 2 2 2 2
--R (- 4A a b + 6B a )e + (4A a b - 8B a b)d e + 2B a b d
--R *
--R log(b x + a)
--R +
--R 3 2 3 3 2 2 3 2
--R B b e x + ((2A b - 3B a b )e + 4B b d e)x
--R +
--R 2 2 2 2 2 3 2
--R ((2A a b - 4B a b)e + 4B a b d e)x + (- 2A a b + 2B a )e
--R +
--R 2 2 3 2 2
--R (4A a b - 4B a b)d e + (- 2A b + 2B a b )d
--R /
--R 5 4
--R 2b x + 2a b
--R Type: Expression(Integer)
--E 194
--S 195 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 195
)clear all
--S 196 of 764
t0:=(A+B*x)*(d+e*x)/(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R 2
--R B e x + (A e + B d)x + A d
--R (1) ---------------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 196
--S 197 of 764
r0:=B*e*x/b^2-(A*b-a*B)*(b*d-a*e)/(b^3*(a+b*x))+(b*B*d+A*b*e-2*a*B*e)*_
log(a+b*x)/b^3
--R
--R
--R (2)
--R 2 2 2
--R (((A b - 2B a b)e + B b d)x + (A a b - 2B a )e + B a b d)log(b x + a)
--R +
--R 2 2 2 2
--R B b e x + B a b e x + (A a b - B a )e + (- A b + B a b)d
--R /
--R 4 3
--R b x + a b
--R Type: Expression(Integer)
--E 197
--S 198 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 198
)clear all
--S 199 of 764
t0:=(A+B*x)/(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R B x + A
--R (1) ------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 199
--S 200 of 764
r0:=(-A*b+a*B)/(b^2*(a+b*x))+B*log(a+b*x)/b^2
--R
--R
--R (B b x + B a)log(b x + a) - A b + B a
--R (2) -------------------------------------
--R 3 2
--R b x + a b
--R Type: Expression(Integer)
--E 200
--S 201 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 201
)clear all
--S 202 of 764
t0:=(A+B*x)/((d+e*x)*(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R B x + A
--R (1) -------------------------------------------------
--R 2 3 2 2 2 2
--R b e x + (2a b e + b d)x + (a e + 2a b d)x + a d
--R Type: Fraction(Polynomial(Integer))
--E 202
--S 203 of 764
r0:=(-A*b+a*B)/(b*(b*d-a*e)*(a+b*x))+(B*d-A*e)*log(a+b*x)/(b*d-a*e)^2-_
(B*d-A*e)*log(d+e*x)/(b*d-a*e)^2
--R
--R
--R (2)
--R 2 2
--R ((A b e - B b d)x + A a b e - B a b d)log(e x + d)
--R +
--R 2 2 2
--R ((- A b e + B b d)x - A a b e + B a b d)log(b x + a) + (A a b - B a )e
--R +
--R 2
--R (- A b + B a b)d
--R /
--R 2 2 2 3 4 2 3 2 2 2 3 2
--R (a b e - 2a b d e + b d )x + a b e - 2a b d e + a b d
--R Type: Expression(Integer)
--E 203
--S 204 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 204
)clear all
--S 205 of 764
t0:=(A+B*x)/((d+e*x)^2*(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (1)
--R B x + A
--R /
--R 2 2 4 2 2 3 2 2 2 2 2
--R b e x + (2a b e + 2b d e)x + (a e + 4a b d e + b d )x
--R +
--R 2 2 2 2
--R (2a d e + 2a b d )x + a d
--R Type: Fraction(Polynomial(Integer))
--E 205
--S 206 of 764
r0:=(-A*b+a*B)/((b*d-a*e)^2*(a+b*x))+(B*d-A*e)/((b*d-a*e)^2*(d+e*x))+_
(b*B*d-2*A*b*e+a*B*e)*log(a+b*x)/(b*d-a*e)^3-(b*B*d-2*A*b*e+a*B*e)*_
log(d+e*x)/(b*d-a*e)^3
--R
--R
--R (2)
--R 2 2 2 2
--R ((- 2A b + B a b)e + B b d e)x
--R +
--R 2 2 2 2 2
--R ((- 2A a b + B a )e + (- 2A b + 2B a b)d e + B b d )x
--R +
--R 2 2
--R (- 2A a b + B a )d e + B a b d
--R *
--R log(e x + d)
--R +
--R 2 2 2 2
--R ((2A b - B a b)e - B b d e)x
--R +
--R 2 2 2 2 2
--R ((2A a b - B a )e + (2A b - 2B a b)d e - B b d )x
--R +
--R 2 2
--R (2A a b - B a )d e - B a b d
--R *
--R log(b x + a)
--R +
--R 2 2 2 2 2 2 2 2
--R ((- 2A a b + B a )e + 2A b d e - B b d )x - A a e + 2B a d e
--R +
--R 2 2
--R (A b - 2B a b)d
--R /
--R 3 4 2 2 3 3 2 2 4 3 2
--R (a b e - 3a b d e + 3a b d e - b d e)x
--R +
--R 4 4 3 3 3 3 4 4 4 3 3 2 2 2 2 3
--R (a e - 2a b d e + 2a b d e - b d )x + a d e - 3a b d e + 3a b d e
--R +
--R 3 4
--R - a b d
--R Type: Expression(Integer)
--E 206
--S 207 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 207
)clear all
--S 208 of 764
t0:=(A+B*x)/((d+e*x)^3*(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (1)
--R B x + A
--R /
--R 2 3 5 3 2 2 4 2 3 2 2 2 3
--R b e x + (2a b e + 3b d e )x + (a e + 6a b d e + 3b d e)x
--R +
--R 2 2 2 2 3 2 2 2 3 2 3
--R (3a d e + 6a b d e + b d )x + (3a d e + 2a b d )x + a d
--R Type: Fraction(Polynomial(Integer))
--E 208
--S 209 of 764
r0:=-b*(A*b-a*B)/((b*d-a*e)^3*(a+b*x))+1/2*(B*d-A*e)/((b*d-a*e)^2*(d+e*x)^2)+_
(b*B*d-2*A*b*e+a*B*e)/((b*d-a*e)^3*(d+e*x))+b*(b*B*d-3*A*b*e+2*a*B*e)*_
log(a+b*x)/(b*d-a*e)^4-b*(b*B*d-3*A*b*e+2*a*B*e)*log(d+e*x)/(b*d-a*e)^4
--R
--R
--R (2)
--R 3 2 3 3 2 3
--R ((6A b - 4B a b )e - 2B b d e )x
--R +
--R 2 2 3 3 2 2 3 2 2
--R ((6A a b - 4B a b)e + (12A b - 10B a b )d e - 4B b d e)x
--R +
--R 2 2 2 3 2 2 3 3
--R ((12A a b - 8B a b)d e + (6A b - 8B a b )d e - 2B b d )x
--R +
--R 2 2 2 2 3
--R (6A a b - 4B a b)d e - 2B a b d
--R *
--R log(e x + d)
--R +
--R 3 2 3 3 2 3
--R ((- 6A b + 4B a b )e + 2B b d e )x
--R +
--R 2 2 3 3 2 2 3 2 2
--R ((- 6A a b + 4B a b)e + (- 12A b + 10B a b )d e + 4B b d e)x
--R +
--R 2 2 2 3 2 2 3 3
--R ((- 12A a b + 8B a b)d e + (- 6A b + 8B a b )d e + 2B b d )x
--R +
--R 2 2 2 2 3
--R (- 6A a b + 4B a b)d e + 2B a b d
--R *
--R log(b x + a)
--R +
--R 2 2 3 3 2 2 3 2 2
--R ((6A a b - 4B a b)e + (- 6A b + 2B a b )d e + 2B b d e)x
--R +
--R 2 3 3 2 2 2 3 2 2
--R (3A a b - 2B a )e + (6A a b - 5B a b)d e + (- 9A b + 4B a b )d e
--R +
--R 3 3
--R 3B b d
--R *
--R x
--R +
--R 3 3 2 3 2 2 2 2
--R - A a e + (6A a b - B a )d e + (- 3A a b - 4B a b)d e
--R +
--R 3 2 3
--R (- 2A b + 5B a b )d
--R /
--R 4 6 3 2 5 2 3 2 4 4 3 3 5 4 2 3
--R (2a b e - 8a b d e + 12a b d e - 8a b d e + 2b d e )x
--R +
--R 5 6 4 5 3 2 2 4 2 3 3 3 4 4 2 5 5 2
--R (2a e - 4a b d e - 4a b d e + 16a b d e - 14a b d e + 4b d e)x
--R +
--R 5 5 4 2 4 3 2 3 3 2 3 4 2 4 5 5 6
--R (4a d e - 14a b d e + 16a b d e - 4a b d e - 4a b d e + 2b d )x
--R +
--R 5 2 4 4 3 3 3 2 4 2 2 3 5 4 6
--R 2a d e - 8a b d e + 12a b d e - 8a b d e + 2a b d
--R Type: Expression(Integer)
--E 209
--S 210 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 210
)clear all
--S 211 of 764
t0:=(A+B*x)*(d+e*x)^4/(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B e x + (A e + 4B d e )x + (4A d e + 6B d e )x
--R +
--R 2 2 3 2 3 4 4
--R (6A d e + 4B d e)x + (4A d e + B d )x + A d
--R /
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 211
--S 212 of 764
r0:=e^3*(4*b*B*d+A*b*e-4*a*B*e)*x/b^5+1/2*B*e^4*x^2/b^4-1/3*(A*b-a*B)*_
(b*d-a*e)^4/(b^6*(a+b*x)^3)-1/2*(b*d-a*e)^3*(b*B*d+4*A*b*e-5*a*B*e)/_
(b^6*(a+b*x)^2)-2*e*(b*d-a*e)^2*(2*b*B*d+3*A*b*e-5*a*B*e)/(b^6*(a+b*x))+_
2*e^2*(b*d-a*e)*(3*b*B*d+2*A*b*e-5*a*B*e)*log(a+b*x)/b^6
--R
--R
--R (2)
--R 4 2 3 4 5 4 3 5 2 2 3
--R ((- 24A a b + 60B a b )e + (24A b - 96B a b )d e + 36B b d e )x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (- 72A a b + 180B a b )e + (72A a b - 288B a b )d e
--R +
--R 4 2 2
--R 108B a b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (- 72A a b + 180B a b)e + (72A a b - 288B a b )d e
--R +
--R 2 3 2 2
--R 108B a b d e
--R *
--R x
--R +
--R 4 5 4 3 2 4 3 3 2 2 2
--R (- 24A a b + 60B a )e + (24A a b - 96B a b)d e + 36B a b d e
--R *
--R log(b x + a)
--R +
--R 5 4 5 5 4 4 5 3 4
--R 3B b e x + ((6A b - 15B a b )e + 24B b d e )x
--R +
--R 4 2 3 4 4 3 3
--R ((18A a b - 63B a b )e + 72B a b d e )x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (- 18A a b - 9B a b )e + (72A a b - 72B a b )d e
--R +
--R 5 4 2 2 5 3
--R (- 36A b + 108B a b )d e - 24B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (- 54A a b + 81B a b)e + (108A a b - 216B a b )d e
--R +
--R 4 2 3 2 2 5 4 3 5 4
--R (- 36A a b + 162B a b )d e + (- 12A b - 24B a b )d e - 3B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 26A a b + 47B a )e + (44A a b - 104B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (- 12A a b + 66B a b )d e + (- 4A a b - 8B a b )d e
--R +
--R 5 4 4
--R (- 2A b - B a b )d
--R /
--R 9 3 8 2 2 7 3 6
--R 6b x + 18a b x + 18a b x + 6a b
--R Type: Expression(Integer)
--E 212
--S 213 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 213
)clear all
--S 214 of 764
t0:=(A+B*x)*(d+e*x)^3/(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x + A d
--R ----------------------------------------------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 214
--S 215 of 764
r0:=B*e^3*x/b^4-1/3*(A*b-a*B)*(b*d-a*e)^3/(b^5*(a+b*x)^3)-1/2*(b*d-a*e)^2*_
(b*B*d+3*A*b*e-4*a*B*e)/(b^5*(a+b*x)^2)-3*e*(b*d-a*e)*(b*B*d+A*b*e-_
2*a*B*e)/(b^5*(a+b*x))+e^2*(3*b*B*d+A*b*e-4*a*B*e)*log(a+b*x)/b^5
--R
--R
--R (2)
--R 4 3 3 4 2 3
--R ((6A b - 24B a b )e + 18B b d e )x
--R +
--R 3 2 2 3 3 2 2
--R ((18A a b - 72B a b )e + 54B a b d e )x
--R +
--R 2 2 3 3 2 2 2 3 4 3
--R ((18A a b - 72B a b)e + 54B a b d e )x + (6A a b - 24B a )e
--R +
--R 3 2
--R 18B a b d e
--R *
--R log(b x + a)
--R +
--R 4 3 4 3 3 3
--R 6B b e x + 18B a b e x
--R +
--R 3 2 2 3 4 3 2 4 2 2
--R ((18A a b - 18B a b )e + (- 18A b + 54B a b )d e - 18B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (27A a b - 54B a b)e + (- 18A a b + 81B a b )d e
--R +
--R 4 3 2 4 3
--R (- 9A b - 18B a b )d e - 3B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (11A a b - 26B a )e + (- 6A a b + 33B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (- 3A a b - 6B a b )d e + (- 2A b - B a b )d
--R /
--R 8 3 7 2 2 6 3 5
--R 6b x + 18a b x + 18a b x + 6a b
--R Type: Expression(Integer)
--E 215
--S 216 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 216
)clear all
--S 217 of 764
t0:=(A+B*x)*(d+e*x)^2/(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R 2 3 2 2 2 2
--R B e x + (A e + 2B d e)x + (2A d e + B d )x + A d
--R (1) ----------------------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 217
--S 218 of 764
r0:=-1/2*B*(b*d-a*e)^2/(b^4*(a+b*x)^2)-2*B*e*(b*d-a*e)/(b^4*(a+b*x))-_
1/3*(A*b-a*B)*(d+e*x)^3/(b*(b*d-a*e)*(a+b*x)^3)+B*e^2*log(a+b*x)/b^4
--R
--R
--R (2)
--R 3 3 4 2 3 2 2 3 3 2 2
--R (6B a b e - 6B b d e )x + (18B a b e - 18B a b d e )x
--R +
--R 3 3 2 2 2 4 3 3 2
--R (18B a b e - 18B a b d e )x + 6B a e - 6B a b d e
--R *
--R log(b x + a)
--R +
--R 4 3 3 3
--R (2A b - 2B a b )e x
--R +
--R 2 2 3 4 3 2 4 2 2
--R (12B a b e + (6A b - 30B a b )d e + 12B b d e)x
--R +
--R 3 3 2 2 2 4 3 2 4 3 4 3
--R (21B a b e - 39B a b d e + (6A b + 9B a b )d e + 3B b d )x + 9B a e
--R +
--R 3 2 2 2 2 4 3 3
--R - 15B a b d e + 3B a b d e + (2A b + B a b )d
--R /
--R 7 8 3 2 6 7 2 3 5 2 6 4 4
--R (6a b e - 6b d)x + (18a b e - 18a b d)x + (18a b e - 18a b d)x + 6a b e
--R +
--R 3 5
--R - 6a b d
--R Type: Expression(Integer)
--E 218
--S 219 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 219
)clear all
--S 220 of 764
t0:=(A+B*x)*(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R 2
--R B e x + (A e + B d)x + A d
--R (1) --------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 220
--S 221 of 764
r0:=-1/3*(A*b-a*B)*(b*d-a*e)/(b^3*(a+b*x)^3)+1/2*(-b*B*d-A*b*e+2*a*B*e)/_
(b^3*(a+b*x)^2)-B*e/(b^3*(a+b*x))
--R
--R
--R (2)
--R 2 2 1 2 1 2 1 1 2
--R - B b e x + ((- - A b - B a b)e - - B b d)x + (- - A a b - - B a )e
--R 2 2 6 3
--R +
--R 1 2 1
--R (- - A b - - B a b)d
--R 3 6
--R /
--R 6 3 5 2 2 4 3 3
--R b x + 3a b x + 3a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 221
--S 222 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 222
)clear all
--S 223 of 764
t0:=(A+B*x)/(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R B x + A
--R (1) --------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Fraction(Polynomial(Integer))
--E 223
--S 224 of 764
r0:=1/3*(-A*b+a*B)/(b^2*(a+b*x)^3)-1/2*B/(b^2*(a+b*x)^2)
--R
--R
--R 1 1 1
--R - - B b x - - A b - - B a
--R 2 3 6
--R (2) ------------------------------
--R 5 3 4 2 2 3 3 2
--R b x + 3a b x + 3a b x + a b
--R Type: Fraction(Polynomial(Fraction(Integer)))
--E 224
--S 225 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 225
)clear all
--S 226 of 764
t0:=(A+B*x)/((d+e*x)*(a^2+2*a*b*x+b^2*x^2)^2)
--R
--R
--R (1)
--R B x + A
--R /
--R 4 5 3 4 4 2 2 3 3 3 2 2 2
--R b e x + (4a b e + b d)x + (6a b e + 4a b d)x + (4a b e + 6a b d)x
--R +
--R 4 3 4
--R (a e + 4a b d)x + a d
--R Type: Fraction(Polynomial(Integer))
--E 226
--S 227 of 764
r0:=1/3*(-A*b+a*B)/(b*(b*d-a*e)*(a+b*x)^3)+1/2*(-B*d+A*e)/((b*d-a*e)^2*_
(a+b*x)^2)+e*(B*d-A*e)/((b*d-a*e)^3*(a+b*x))+e^2*(B*d-A*e)*log(a+b*x)/_
(b*d-a*e)^4-e^2*(B*d-A*e)*log(d+e*x)/(b*d-a*e)^4
--R
--R
--R (2)
--R 4 3 4 2 3 3 3 3 2 2
--R (6A b e - 6B b d e )x + (18A a b e - 18B a b d e )x
--R +
--R 2 2 3 2 2 2 3 3 3 2
--R (18A a b e - 18B a b d e )x + 6A a b e - 6B a b d e
--R *
--R log(e x + d)
--R +
--R 4 3 4 2 3 3 3 3 2 2
--R (- 6A b e + 6B b d e )x + (- 18A a b e + 18B a b d e )x
--R +
--R 2 2 3 2 2 2 3 3 3 2
--R (- 18A a b e + 18B a b d e )x - 6A a b e + 6B a b d e
--R *
--R log(b x + a)
--R +
--R 3 3 4 3 2 4 2 2
--R (6A a b e + (- 6A b - 6B a b )d e + 6B b d e)x
--R +
--R 2 2 3 3 2 2 2 4 3 2
--R 15A a b e + (- 18A a b - 15B a b )d e + (3A b + 18B a b )d e
--R +
--R 4 3
--R - 3B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2 3 2 2 2
--R (11A a b - 2B a )e + (- 18A a b - 3B a b)d e + (9A a b + 6B a b )d e
--R +
--R 4 3 3
--R (- 2A b - B a b )d
--R /
--R 4 4 4 3 5 3 2 6 2 2 7 3 8 4 3
--R (6a b e - 24a b d e + 36a b d e - 24a b d e + 6b d )x
--R +
--R 5 3 4 4 4 3 3 5 2 2 2 6 3 7 4 2
--R (18a b e - 72a b d e + 108a b d e - 72a b d e + 18a b d )x
--R +
--R 6 2 4 5 3 3 4 4 2 2 3 5 3 2 6 4 7 4
--R (18a b e - 72a b d e + 108a b d e - 72a b d e + 18a b d )x + 6a b e
--R +
--R 6 2 3 5 3 2 2 4 4 3 3 5 4
--R - 24a b d e + 36a b d e - 24a b d e + 6a b d
--R Type: Expression(Integer)
--E 227
--S 228 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 228
)clear all
--S 229 of 764
t0:=(A+B*x)/((d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^2)
--R
--R
--R (1)
--R B x + A
--R /
--R 4 2 6 3 2 4 5 2 2 2 3 4 2 4
--R b e x + (4a b e + 2b d e)x + (6a b e + 8a b d e + b d )x
--R +
--R 3 2 2 2 3 2 3 4 2 3 2 2 2 2
--R (4a b e + 12a b d e + 4a b d )x + (a e + 8a b d e + 6a b d )x
--R +
--R 4 3 2 4 2
--R (2a d e + 4a b d )x + a d
--R Type: Fraction(Polynomial(Integer))
--E 229
--S 230 of 764
r0:=1/3*(-A*b+a*B)/((b*d-a*e)^2*(a+b*x)^3)+1/2*(-b*B*d+2*A*b*e-a*B*e)/_
((b*d-a*e)^3*(a+b*x)^2)+e*(2*b*B*d-3*A*b*e+a*B*e)/((b*d-a*e)^4*(a+b*x))+_
e^2*(B*d-A*e)/((b*d-a*e)^4*(d+e*x))+e^2*(3*b*B*d-4*A*b*e+a*B*e)*_
log(a+b*x)/(b*d-a*e)^5-e^2*(3*b*B*d-4*A*b*e+a*B*e)*log(d+e*x)/(b*d-a*e)^5
--R
--R
--R (2)
--R 4 3 4 4 3 4
--R ((- 24A b + 6B a b )e + 18B b d e )x
--R +
--R 3 2 2 4 4 3 3
--R (- 72A a b + 18B a b )e + (- 24A b + 60B a b )d e
--R +
--R 4 2 2
--R 18B b d e
--R *
--R 3
--R x
--R +
--R 2 2 3 4 3 2 2 3
--R (- 72A a b + 18B a b)e + (- 72A a b + 72B a b )d e
--R +
--R 3 2 2
--R 54B a b d e
--R *
--R 2
--R x
--R +
--R 3 4 4 2 2 3 3 2 2 2 2
--R ((- 24A a b + 6B a )e + (- 72A a b + 36B a b)d e + 54B a b d e )x
--R +
--R 3 4 3 3 2 2
--R (- 24A a b + 6B a )d e + 18B a b d e
--R *
--R log(e x + d)
--R +
--R 4 3 4 4 3 4
--R ((24A b - 6B a b )e - 18B b d e )x
--R +
--R 3 2 2 4 4 3 3 4 2 2 3
--R ((72A a b - 18B a b )e + (24A b - 60B a b )d e - 18B b d e )x
--R +
--R 2 2 3 4 3 2 2 3 3 2 2 2
--R ((72A a b - 18B a b)e + (72A a b - 72B a b )d e - 54B a b d e )x
--R +
--R 3 4 4 2 2 3 3 2 2 2 2
--R ((24A a b - 6B a )e + (72A a b - 36B a b)d e - 54B a b d e )x
--R +
--R 3 4 3 3 2 2
--R (24A a b - 6B a )d e - 18B a b d e
--R *
--R log(b x + a)
--R +
--R 3 2 2 4 4 3 3 4 2 2 3
--R ((- 24A a b + 6B a b )e + (24A b + 12B a b )d e - 18B b d e )x
--R +
--R 2 2 3 4 3 2 2 3
--R (- 60A a b + 15B a b)e + (48A a b + 33B a b )d e
--R +
--R 4 3 2 2 4 3
--R (12A b - 39B a b )d e - 9B b d e
--R *
--R 2
--R x
--R +
--R 3 4 4 2 2 3 3
--R (- 44A a b + 11B a )e + (12A a b + 30B a b)d e
--R +
--R 3 2 2 2 2 4 3 3 4 4
--R (36A a b - 18B a b )d e + (- 4A b - 26B a b )d e + 3B b d
--R *
--R x
--R +
--R 4 4 3 4 3 2 2 3 2 2
--R - 6A a e + (- 20A a b + 17B a )d e + (36A a b - 9B a b)d e
--R +
--R 3 2 2 3 4 3 4
--R (- 12A a b - 9B a b )d e + (2A b + B a b )d
--R /
--R 5 3 6 4 4 5 3 5 2 4 2 6 3 3 7 4 2 8 5 4
--R (6a b e - 30a b d e + 60a b d e - 60a b d e + 30a b d e - 6b d e)x
--R +
--R 6 2 6 5 3 5 4 4 2 4 3 5 3 3 2 6 4 2
--R 18a b e - 84a b d e + 150a b d e - 120a b d e + 30a b d e
--R +
--R 7 5 8 6
--R 12a b d e - 6b d
--R *
--R 3
--R x
--R +
--R 7 6 6 2 5 5 3 2 4 3 5 4 2 2 6 5
--R 18a b e - 72a b d e + 90a b d e - 90a b d e + 72a b d e
--R +
--R 7 6
--R - 18a b d
--R *
--R 2
--R x
--R +
--R 8 6 7 5 6 2 2 4 5 3 3 3 4 4 4 2
--R 6a e - 12a b d e - 30a b d e + 120a b d e - 150a b d e
--R +
--R 3 5 5 2 6 6
--R 84a b d e - 18a b d
--R *
--R x
--R +
--R 8 5 7 2 4 6 2 3 3 5 3 4 2 4 4 5 3 5 6
--R 6a d e - 30a b d e + 60a b d e - 60a b d e + 30a b d e - 6a b d
--R Type: Expression(Integer)
--E 230
--S 231 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 231
)clear all
--S 232 of 764
t0:=(A+B*x)*(d+e*x)^m*sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R +------------------+
--R m | 2 2 2
--R (1) (B x + A)(e x + d) \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 232
--S 233 of 764
r0:=(A*b*e*(3+m)-B*(2*b*d+a*e*(1+m)))*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^2*(2+m)*(3+m))-(b*d-a*e)*(A*b*e*(3+m)-B*(2*b*d+_
a*e*(1+m)))*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3*(1+m)*_
(2+m)*(3+m)*(a+b*x))+B*(a+b*x)*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e*(3+m))
--R
--R
--R (2)
--R 2 2 2 2 2
--R (B b e m + 3B b e m + 2B b e )x
--R +
--R 2 2 2 2
--R (A b + B a)e m + ((4A b + 4B a)e - 2B b d e)m + (3A b + 3B a)e
--R +
--R - 2B b d e
--R *
--R x
--R +
--R 2 2 2 2
--R A a e m + (5A a e + (- A b - B a)d e)m + 6A a e + (- 3A b - 3B a)d e
--R +
--R 2
--R 2B b d
--R *
--R +------------------+
--R m + 1 | 2 2 2
--R (e x + d) \|b x + 2a b x + a
--R /
--R 3 3 3 2 3 3 3 3 3 2 3 3
--R (b e m + 6b e m + 11b e m + 6b e )x + a e m + 6a e m + 11a e m + 6a e
--R Type: Expression(Integer)
--E 233
--S 234 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 234
)clear all
--S 235 of 764
t0:=(A+B*x)*(d+e*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B e x + (A e + 4B d e )x + (4A d e + 6B d e )x
--R +
--R 2 2 3 2 3 4 4
--R (6A d e + 4B d e)x + (4A d e + B d )x + A d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 235
--S 236 of 764
r0:=-1/42*(2*b*B*d-7*A*b*e+5*a*B*e)*(d+e*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(b*e^2)+1/210*(b*d-a*e)*(2*b*B*d-7*A*b*e+5*a*B*e)*(d+e*x)^5*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(b*e^3*(a+b*x))+1/7*B*(a+b*x)*(d+e*x)^5*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(b*e)
--R
--R
--R (2)
--R 7 7 7 6 6
--R 30B b e x + ((35A b + 35B a)e + 140B b d e )x
--R +
--R 7 6 2 5 5
--R (42A a e + (168A b + 168B a)d e + 252B b d e )x
--R +
--R 6 2 5 3 4 4
--R (210A a d e + (315A b + 315B a)d e + 210B b d e )x
--R +
--R 2 5 3 4 4 3 3
--R (420A a d e + (280A b + 280B a)d e + 70B b d e )x
--R +
--R 3 4 4 3 2 4 3 5 2
--R (420A a d e + (105A b + 105B a)d e )x + 210A a d e x + 42A a d e
--R +
--R 6 7
--R (- 7A b - 7B a)d e + 2B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3 3
--R 210b e x + 210a e
--R Type: Expression(Integer)
--E 236
--S 237 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 237
)clear all
--S 238 of 764
t0:=(A+B*x)*(d+e*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x
--R +
--R 3
--R A d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 238
--S 239 of 764
r0:=-1/15*(b*B*d-3*A*b*e+2*a*B*e)*(d+e*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(b*e^2)+1/60*(b*d-a*e)*(b*B*d-3*A*b*e+2*a*B*e)*(d+e*x)^4*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(b*e^3*(a+b*x))+1/6*B*(a+b*x)*(d+e*x)^4*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(b*e)
--R
--R
--R (2)
--R 6 6 6 5 5
--R 10B b e x + ((12A b + 12B a)e + 36B b d e )x
--R +
--R 6 5 2 4 4
--R (15A a e + (45A b + 45B a)d e + 45B b d e )x
--R +
--R 5 2 4 3 3 3
--R (60A a d e + (60A b + 60B a)d e + 20B b d e )x
--R +
--R 2 4 3 3 2 3 3 4 2
--R (90A a d e + (30A b + 30B a)d e )x + 60A a d e x + 15A a d e
--R +
--R 5 6
--R (- 3A b - 3B a)d e + B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3 3
--R 60b e x + 60a e
--R Type: Expression(Integer)
--E 239
--S 240 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 240
)clear all
--S 241 of 764
t0:=(A+B*x)*(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B e x + (A e + 2B d e)x + (2A d e + B d )x + A d )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 241
--S 242 of 764
r0:=-1/20*(2*b*B*d-5*A*b*e+3*a*B*e)*(d+e*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(b*e^2)+1/60*(b*d-a*e)*(2*b*B*d-5*A*b*e+3*a*B*e)*(d+e*x)^3*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(b*e^3*(a+b*x))+1/5*B*(a+b*x)*(d+e*x)^3*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(b*e)
--R
--R
--R (2)
--R 5 5 5 4 4
--R 12B b e x + ((15A b + 15B a)e + 30B b d e )x
--R +
--R 5 4 2 3 3
--R (20A a e + (40A b + 40B a)d e + 20B b d e )x
--R +
--R 4 2 3 2 2 3 3 2
--R (60A a d e + (30A b + 30B a)d e )x + 60A a d e x + 20A a d e
--R +
--R 4 5
--R (- 5A b - 5B a)d e + 2B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3 3
--R 60b e x + 60a e
--R Type: Expression(Integer)
--E 242
--S 243 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 243
)clear all
--S 244 of 764
t0:=(A+B*x)*(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R +------------------+
--R 2 | 2 2 2
--R (1) (B e x + (A e + B d)x + A d)\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 244
--S 245 of 764
r0:=1/6*(2*A*b*e-B*(b*d+a*e))*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b^3+1/4*(b*d-_
a*e)*(2*A*b*e-B*(b*d+a*e))*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b^3*e)+_
1/4*B*(a+b*x)*(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e)
--R
--R
--R (2)
--R 3 3 3 2 3 2
--R 3B b e x + ((4A b + B a b )e + 4B b d)x
--R +
--R 2 2 3 2 2 3
--R ((2A a b - B a b)e + (6A b + 2B a b )d)x + (- 2A a b + B a )e
--R +
--R 2 2
--R (6A a b - 2B a b)d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3
--R 12b
--R Type: Expression(Integer)
--E 245
--S 246 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 246
)clear all
--S 247 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R +------------------+
--R | 2 2 2
--R (1) (B x + A)\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 247
--S 248 of 764
r0:=1/3*B*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b^2+1/2*(A*b-a*B)*(a+b*x)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/b^2
--R
--R
--R +------------------+
--R 2 2 2 2 | 2 2 2
--R (2B b x + (3A b + B a b)x + 3A a b - B a )\|b x + 2a b x + a
--R (2) -----------------------------------------------------------------
--R 2
--R 6b
--R Type: Expression(Integer)
--E 248
--S 249 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 249
)clear all
--S 250 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)
--R
--R
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R (1) ------------------------------
--R e x + d
--R Type: Expression(Integer)
--E 250
--S 251 of 764
r0:=-(B*d-A*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^2+1/2*B*(a+b*x)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(b*e)+(b*d-a*e)*(B*d-A*e)*log(d+e*x)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(e^3*(a+b*x))
--R
--R
--R (2)
--R 2 2 2 2 2 2 2
--R (2A a b e + (- 2A b - 2B a b)d e + 2B b d )log(e x + d) + B b e x
--R +
--R 2 2 2 2 2
--R ((2A b + 2B a b)e - 2B b d e)x + (2A a b + B a )e - 2B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 2 3 3
--R 2b e x + 2a b e
--R Type: Expression(Integer)
--E 251
--S 252 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 252
)clear all
--S 253 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^2
--R
--R
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R (1) ------------------------------
--R 2 2 2
--R e x + 2d e x + d
--R Type: Expression(Integer)
--E 253
--S 254 of 764
r0:=(2*b*B*d-A*b*e-a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^2*(b*d-a*e))-_
(B*d-A*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e*(b*d-a*e)*(d+e*x))-_
(2*b*B*d-A*b*e-a*B*e)*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(a+b*x))
--R
--R
--R (2)
--R 2 2
--R (((A b + B a)e - 2B b d e)x + (A b + B a)d e - 2B b d )log(e x + d)
--R +
--R 2 2 2 2
--R B b e x + ((- A b + B a)e + 2B b d e)x - A a e + 2B a d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 4 2 4 3 3
--R b e x + (a e + b d e )x + a d e
--R Type: Expression(Integer)
--E 254
--S 255 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 255
)clear all
--S 256 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^3
--R
--R
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R (1) ------------------------------
--R 3 3 2 2 2 3
--R e x + 3d e x + 3d e x + d
--R Type: Expression(Integer)
--E 256
--S 257 of 764
r0:=-1/2*(B*d-A*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e*(b*d-a*e)*_
(d+e*x)^2)+B*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(a+b*x)*_
(d+e*x))+b*B*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(a+b*x))
--R
--R
--R (2)
--R 3 2 2 2 2 2 2 2
--R (2B a b e - 2B b d e )x + (4B a b d e - 4B b d e)x + 2B a b d e
--R +
--R 2 3
--R - 2B b d
--R *
--R log(e x + d)
--R +
--R 2 3 2 2 2
--R (- A b e + B b d e )x
--R +
--R 2 3 2 2 2 2 3 2 2
--R ((- 2A a b - 2B a )e + 6B a b d e - 2B b d e)x - A a e - B a d e
--R +
--R 2 2 3
--R 4B a b d e - 2B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 6 2 5 3 2 6 5 2 2 4 2
--R (2a b e - 2b d e )x + (2a e + 2a b d e - 4b d e )x
--R +
--R 2 5 2 4 2 3 3 2 2 4 3 3
--R (4a d e - 2a b d e - 2b d e )x + 2a d e - 2a b d e
--R Type: Expression(Integer)
--E 257
--S 258 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 258
)clear all
--S 259 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^4
--R
--R
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R (1) --------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R e x + 4d e x + 6d e x + 4d e x + d
--R Type: Expression(Integer)
--E 259
--S 260 of 764
r0:=1/3*(B*d-A*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/((b*d-a*e)^2*(d+e*x)^3)+_
1/2*(A*b-a*B)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)^2*(d+e*x)^2)
--R
--R
--R (2)
--R 2 2 2
--R ((A b - 3B a b)e + 2B b d)x
--R +
--R 2 2 2 2
--R ((- A a b - 3B a )e + (3A b + B a b)d)x - 2A a e + (3A a b - B a )d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 2 5 4 2 2 3 3 2 4 2 3 2 3 2 2
--R (6a e - 12a b d e + 6b d e )x + (18a d e - 36a b d e + 18b d e )x
--R +
--R 2 2 3 3 2 2 4 2 3 2 4 2 5
--R (18a d e - 36a b d e + 18b d e)x + 6a d e - 12a b d e + 6b d
--R Type: Expression(Integer)
--E 260
--S 261 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 261
)clear all
--S 262 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^5
--R
--R
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R (1) --------------------------------------------------
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R e x + 5d e x + 10d e x + 10d e x + 5d e x + d
--R Type: Expression(Integer)
--E 262
--S 263 of 764
r0:=-1/4*(B*d-A*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e*(b*d-a*e)*_
(d+e*x)^4)-1/4*(b*B*d+A*b*e-2*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(e^2*(b*d-a*e)*(d+e*x)^3)-1/12*(b*B*d+A*b*e-2*a*B*e)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(e^3*(a+b*x)*(d+e*x)^3)
--R
--R
--R (2)
--R 2 2 2 2
--R - 6B b e x + ((- 4A b - 4B a)e - 4B b d e)x - 3A a e
--R +
--R 2
--R (- A b - B a)d e - B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 7 5 7 6 4 6 2 5 3
--R 12b e x + (12a e + 48b d e )x + (48a d e + 72b d e )x
--R +
--R 2 5 3 4 2 3 4 4 3 4 3
--R (72a d e + 48b d e )x + (48a d e + 12b d e )x + 12a d e
--R Type: Expression(Integer)
--E 263
--S 264 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 264
)clear all
--S 265 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^6
--R
--R
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R (1) -------------------------------------------------------------
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R e x + 6d e x + 15d e x + 20d e x + 15d e x + 6d e x + d
--R Type: Expression(Integer)
--E 265
--S 266 of 764
r0:=-1/5*(B*d-A*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e*(b*d-a*e)*_
(d+e*x)^5)-1/15*(2*b*B*d+3*A*b*e-5*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(e^2*(b*d-a*e)*(d+e*x)^4)-1/60*(2*b*B*d+3*A*b*e-5*a*B*e)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(e^3*(a+b*x)*(d+e*x)^4)
--R
--R
--R (2)
--R 2 2 2 2
--R - 20B b e x + ((- 15A b - 15B a)e - 10B b d e)x - 12A a e
--R +
--R 2
--R (- 3A b - 3B a)d e - 2B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 8 6 8 7 5 7 2 6 4
--R 60b e x + (60a e + 300b d e )x + (300a d e + 600b d e )x
--R +
--R 2 6 3 5 3 3 5 4 4 2
--R (600a d e + 600b d e )x + (600a d e + 300b d e )x
--R +
--R 4 4 5 3 5 3
--R (300a d e + 60b d e )x + 60a d e
--R Type: Expression(Integer)
--E 266
--S 267 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 267
)clear all
--S 268 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^7
--R
--R
--R (1)
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R ------------------------------------------------------------------------
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R e x + 7d e x + 21d e x + 35d e x + 35d e x + 21d e x + 7d e x + d
--R Type: Expression(Integer)
--E 268
--S 269 of 764
r0:=-1/6*(B*d-A*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e*(b*d-a*e)*_
(d+e*x)^6)-1/12*(b*B*d+2*A*b*e-3*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(e^2*(b*d-a*e)*(d+e*x)^5)-1/60*(b*B*d+2*A*b*e-3*a*B*e)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(e^3*(a+b*x)*(d+e*x)^5)
--R
--R
--R (2)
--R 2 2 2 2
--R - 15B b e x + ((- 12A b - 12B a)e - 6B b d e)x - 10A a e
--R +
--R 2
--R (- 2A b - 2B a)d e - B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 9 7 9 8 6 8 2 7 5
--R 60b e x + (60a e + 360b d e )x + (360a d e + 900b d e )x
--R +
--R 2 7 3 6 4 3 6 4 5 3
--R (900a d e + 1200b d e )x + (1200a d e + 900b d e )x
--R +
--R 4 5 5 4 2 5 4 6 3 6 3
--R (900a d e + 360b d e )x + (360a d e + 60b d e )x + 60a d e
--R Type: Expression(Integer)
--E 269
--S 270 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 270
)clear all
--S 271 of 764
t0:=(A+B*x)*(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 2 3 2 2 2 2 m
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )(e x + d)
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 271
--S 272 of 764
r0:=(A*b*e*(5+m)-B*(4*b*d+a*e*(1+m)))*(d+e*x)^(1+m)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e^2*(4+m)*(5+m))+B*(a+b*x)*(d+e*x)^(1+m)*(a^2+_
2*a*b*x+b^2*x^2)^(3/2)/(b*e*(5+m))+6*(b*d-a*e)^2*(A*b*e*(5+m)-_
B*(4*b*d+a*e*(1+m)))*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^4*_
(2+m)*(3+m)*(4+m)*(5+m))-6*(b*d-a*e)^3*(A*b*e*(5+m)-B*(4*b*d+a*e*_
(1+m)))*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^5*(1+m)*(2+m)*_
(3+m)*(4+m)*(5+m)*(a+b*x))-3*(b*d-a*e)*(A*b*e*(5+m)-B*(4*b*d+a*e*_
(1+m)))*(a+b*x)*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3*_
(3+m)*(4+m)*(5+m))
--R
--R
--R (2)
--R 3 4 4 3 4 3 3 4 2 3 4 3 4 4
--R (B b e m + 10B b e m + 35B b e m + 50B b e m + 24B b e )x
--R +
--R 3 2 4 4 3 2 4 3 3 3
--R (A b + 3B a b )e m + ((11A b + 33B a b )e - 4B b d e )m
--R +
--R 3 2 4 3 3 2
--R ((41A b + 123B a b )e - 24B b d e )m
--R +
--R 3 2 4 3 3 3 2 4
--R ((61A b + 183B a b )e - 44B b d e )m + (30A b + 90B a b )e
--R +
--R 3 3
--R - 24B b d e
--R *
--R 3
--R x
--R +
--R 2 2 4 4
--R (3A a b + 3B a b)e m
--R +
--R 2 2 4 3 2 3 3
--R ((36A a b + 36B a b)e + (- 3A b - 9B a b )d e )m
--R +
--R 2 2 4 3 2 3
--R (147A a b + 147B a b)e + (- 24A b - 72B a b )d e
--R +
--R 3 2 2
--R 12B b d e
--R *
--R 2
--R m
--R +
--R 2 2 4 3 2 3
--R (234A a b + 234B a b)e + (- 51A b - 153B a b )d e
--R +
--R 3 2 2
--R 36B b d e
--R *
--R m
--R +
--R 2 2 4 3 2 3 3 2 2
--R (120A a b + 120B a b)e + (- 30A b - 90B a b )d e + 24B b d e
--R *
--R 2
--R x
--R +
--R 2 3 4 4
--R (3A a b + B a )e m
--R +
--R 2 3 4 2 2 3 3
--R ((39A a b + 13B a )e + (- 6A a b - 6B a b)d e )m
--R +
--R 2 3 4 2 2 3
--R (177A a b + 59B a )e + (- 60A a b - 60B a b)d e
--R +
--R 3 2 2 2
--R (6A b + 18B a b )d e
--R *
--R 2
--R m
--R +
--R 2 3 4 2 2 3
--R (321A a b + 107B a )e + (- 174A a b - 174B a b)d e
--R +
--R 3 2 2 2 3 3
--R (36A b + 108B a b )d e - 24B b d e
--R *
--R m
--R +
--R 2 3 4 2 2 3
--R (180A a b + 60B a )e + (- 120A a b - 120B a b)d e
--R +
--R 3 2 2 2 3 3
--R (30A b + 90B a b )d e - 24B b d e
--R *
--R x
--R +
--R 3 4 4 3 4 2 3 3 3
--R A a e m + (14A a e + (- 3A a b - B a )d e )m
--R +
--R 3 4 2 3 3 2 2 2 2 2
--R (71A a e + (- 36A a b - 12B a )d e + (6A a b + 6B a b)d e )m
--R +
--R 3 4 2 3 3 2 2 2 2
--R 154A a e + (- 141A a b - 47B a )d e + (54A a b + 54B a b)d e
--R +
--R 3 2 3
--R (- 6A b - 18B a b )d e
--R *
--R m
--R +
--R 3 4 2 3 3 2 2 2 2
--R 120A a e + (- 180A a b - 60B a )d e + (120A a b + 120B a b)d e
--R +
--R 3 2 3 3 4
--R (- 30A b - 90B a b )d e + 24B b d
--R *
--R +------------------+
--R m + 1 | 2 2 2
--R (e x + d) \|b x + 2a b x + a
--R /
--R 5 5 5 4 5 3 5 2 5 5 5 5
--R (b e m + 15b e m + 85b e m + 225b e m + 274b e m + 120b e )x + a e m
--R +
--R 5 4 5 3 5 2 5 5
--R 15a e m + 85a e m + 225a e m + 274a e m + 120a e
--R Type: Expression(Integer)
--E 272
--S 273 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 273
)clear all
--S 274 of 764
t0:=(A+B*x)*(d+e*x)^5*(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 2 5 8 2 5 2 4 7
--R B b e x + ((A b + 2B a b)e + 5B b d e )x
--R +
--R 2 5 2 4 2 2 3 6
--R ((2A a b + B a )e + (5A b + 10B a b)d e + 10B b d e )x
--R +
--R 2 5 2 4 2 2 3 2 3 2 5
--R (A a e + (10A a b + 5B a )d e + (10A b + 20B a b)d e + 10B b d e )x
--R +
--R 2 4 2 2 3 2 3 2
--R 5A a d e + (20A a b + 10B a )d e + (10A b + 20B a b)d e
--R +
--R 2 4
--R 5B b d e
--R *
--R 4
--R x
--R +
--R 2 2 3 2 3 2 2 4 2 5 3
--R (10A a d e + (20A a b + 10B a )d e + (5A b + 10B a b)d e + B b d )x
--R +
--R 2 3 2 2 4 2 5 2
--R (10A a d e + (10A a b + 5B a )d e + (A b + 2B a b)d )x
--R +
--R 2 4 2 5 2 5
--R (5A a d e + (2A a b + B a )d )x + A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 274
--S 275 of 764
r0:=-1/45*(2*b*B*d-5*A*b*e+3*a*B*e)*(d+e*x)^6*(a^2+2*a*b*x+b^2*x^2)^(3/2)/_
(b*e^2)+1/10*B*(a+b*x)*(d+e*x)^6*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e)-_
1/420*(b*d-a*e)^2*(2*b*B*d-5*A*b*e+3*a*B*e)*(d+e*x)^6*sqrt(a^2+2*a*b*_
x+b^2*x^2)/(b*e^4)+1/2520*(b*d-a*e)^3*(2*b*B*d-5*A*b*e+3*a*B*e)*_
(d+e*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^5*(a+b*x))+1/120*(b*d-a*e)*_
(2*b*B*d-5*A*b*e+3*a*B*e)*(a+b*x)*(d+e*x)^6*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^3)
--R
--R
--R (2)
--R 3 10 10 3 2 10 3 9 9
--R 252B b e x + ((280A b + 840B a b )e + 1400B b d e )x
--R +
--R 2 2 10 3 2 9
--R (945A a b + 945B a b)e + (1575A b + 4725B a b )d e
--R +
--R 3 2 8
--R 3150B b d e
--R *
--R 8
--R x
--R +
--R 2 3 10 2 2 9
--R (1080A a b + 360B a )e + (5400A a b + 5400B a b)d e
--R +
--R 3 2 2 8 3 3 7
--R (3600A b + 10800B a b )d e + 3600B b d e
--R *
--R 7
--R x
--R +
--R 3 10 2 3 9
--R 420A a e + (6300A a b + 2100B a )d e
--R +
--R 2 2 2 8 3 2 3 7
--R (12600A a b + 12600B a b)d e + (4200A b + 12600B a b )d e
--R +
--R 3 4 6
--R 2100B b d e
--R *
--R 6
--R x
--R +
--R 3 9 2 3 2 8
--R 2520A a d e + (15120A a b + 5040B a )d e
--R +
--R 2 2 3 7 3 2 4 6
--R (15120A a b + 15120B a b)d e + (2520A b + 7560B a b )d e
--R +
--R 3 5 5
--R 504B b d e
--R *
--R 5
--R x
--R +
--R 3 2 8 2 3 3 7
--R 6300A a d e + (18900A a b + 6300B a )d e
--R +
--R 2 2 4 6 3 2 5 5
--R (9450A a b + 9450B a b)d e + (630A b + 1890B a b )d e
--R *
--R 4
--R x
--R +
--R 3 3 7 2 3 4 6
--R 8400A a d e + (12600A a b + 4200B a )d e
--R +
--R 2 2 5 5
--R (2520A a b + 2520B a b)d e
--R *
--R 3
--R x
--R +
--R 3 4 6 2 3 5 5 2 3 5 5
--R (6300A a d e + (3780A a b + 1260B a )d e )x + 2520A a d e x
--R +
--R 3 6 4 2 3 7 3 2 2 8 2
--R 420A a d e + (- 180A a b - 60B a )d e + (45A a b + 45B a b)d e
--R +
--R 3 2 9 3 10
--R (- 5A b - 15B a b )d e + 2B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 5 5
--R 2520b e x + 2520a e
--R Type: Expression(Integer)
--E 275
--S 276 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 276
)clear all
--S 277 of 764
t0:=(A+B*x)*(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 2 4 7 2 4 2 3 6
--R B b e x + ((A b + 2B a b)e + 4B b d e )x
--R +
--R 2 4 2 3 2 2 2 5
--R ((2A a b + B a )e + (4A b + 8B a b)d e + 6B b d e )x
--R +
--R 2 4 2 3 2 2 2 2 3 4
--R (A a e + (8A a b + 4B a )d e + (6A b + 12B a b)d e + 4B b d e)x
--R +
--R 2 3 2 2 2 2 3 2 4 3
--R (4A a d e + (12A a b + 6B a )d e + (4A b + 8B a b)d e + B b d )x
--R +
--R 2 2 2 2 3 2 4 2
--R (6A a d e + (8A a b + 4B a )d e + (A b + 2B a b)d )x
--R +
--R 2 3 2 4 2 4
--R (4A a d e + (2A a b + B a )d )x + A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 277
--S 278 of 764
r0:=-1/72*(4*b*B*d-9*A*b*e+5*a*B*e)*(d+e*x)^5*(a^2+2*a*b*x+b^2*x^2)^(3/2)/_
(b*e^2)+1/9*B*(a+b*x)*(d+e*x)^5*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e)-_
1/504*(b*d-a*e)^2*(4*b*B*d-9*A*b*e+5*a*B*e)*(d+e*x)^5*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(b*e^4)+1/2520*(b*d-a*e)^3*(4*b*B*d-9*A*b*e+5*a*B*e)*_
(d+e*x)^5*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^5*(a+b*x))+1/168*(b*d-a*e)*_
(4*b*B*d-9*A*b*e+5*a*B*e)*(a+b*x)*(d+e*x)^5*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^3)
--R
--R
--R (2)
--R 3 9 9 3 2 9 3 8 8
--R 280B b e x + ((315A b + 945B a b )e + 1260B b d e )x
--R +
--R 2 2 9 3 2 8
--R (1080A a b + 1080B a b)e + (1440A b + 4320B a b )d e
--R +
--R 3 2 7
--R 2160B b d e
--R *
--R 7
--R x
--R +
--R 2 3 9 2 2 8
--R (1260A a b + 420B a )e + (5040A a b + 5040B a b)d e
--R +
--R 3 2 2 7 3 3 6
--R (2520A b + 7560B a b )d e + 1680B b d e
--R *
--R 6
--R x
--R +
--R 3 9 2 3 8
--R 504A a e + (6048A a b + 2016B a )d e
--R +
--R 2 2 2 7 3 2 3 6
--R (9072A a b + 9072B a b)d e + (2016A b + 6048B a b )d e
--R +
--R 3 4 5
--R 504B b d e
--R *
--R 5
--R x
--R +
--R 3 8 2 3 2 7
--R 2520A a d e + (11340A a b + 3780B a )d e
--R +
--R 2 2 3 6 3 2 4 5
--R (7560A a b + 7560B a b)d e + (630A b + 1890B a b )d e
--R *
--R 4
--R x
--R +
--R 3 2 7 2 3 3 6
--R 5040A a d e + (10080A a b + 3360B a )d e
--R +
--R 2 2 4 5
--R (2520A a b + 2520B a b)d e
--R *
--R 3
--R x
--R +
--R 3 3 6 2 3 4 5 2 3 4 5
--R (5040A a d e + (3780A a b + 1260B a )d e )x + 2520A a d e x
--R +
--R 3 5 4 2 3 6 3 2 2 7 2
--R 504A a d e + (- 252A a b - 84B a )d e + (72A a b + 72B a b)d e
--R +
--R 3 2 8 3 9
--R (- 9A b - 27B a b )d e + 4B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 5 5
--R 2520b e x + 2520a e
--R Type: Expression(Integer)
--E 278
--S 279 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 279
)clear all
--S 280 of 764
t0:=(A+B*x)*(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 2 3 6 2 3 2 2 5
--R B b e x + ((A b + 2B a b)e + 3B b d e )x
--R +
--R 2 3 2 2 2 2 4
--R ((2A a b + B a )e + (3A b + 6B a b)d e + 3B b d e)x
--R +
--R 2 3 2 2 2 2 2 3 3
--R (A a e + (6A a b + 3B a )d e + (3A b + 6B a b)d e + B b d )x
--R +
--R 2 2 2 2 2 3 2
--R (3A a d e + (6A a b + 3B a )d e + (A b + 2B a b)d )x
--R +
--R 2 2 2 3 2 3
--R (3A a d e + (2A a b + B a )d )x + A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 280
--S 281 of 764
r0:=1/14*(2*A*b*e-B*(b*d+a*e))*(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^(3/2)/_
(b*e^2)+1/8*B*(a+b*x)*(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e)+_
1/70*(b*d-a*e)^2*(2*A*b*e-B*(b*d+a*e))*(d+e*x)^4*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^4)-1/280*(b*d-a*e)^3*(2*A*b*e-B*(b*d+a*e))*(d+e*x)^4*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^5*(a+b*x))-1/28*(b*d-a*e)*(2*A*b*e-_
B*(b*d+a*e))*(a+b*x)*(d+e*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3)
--R
--R
--R (2)
--R 3 8 8 3 2 8 3 7 7
--R 35B b e x + ((40A b + 120B a b )e + 120B b d e )x
--R +
--R 2 2 8 3 2 7 3 2 6 6
--R ((140A a b + 140B a b)e + (140A b + 420B a b )d e + 140B b d e )x
--R +
--R 2 3 8 2 2 7
--R (168A a b + 56B a )e + (504A a b + 504B a b)d e
--R +
--R 3 2 2 6 3 3 5
--R (168A b + 504B a b )d e + 56B b d e
--R *
--R 5
--R x
--R +
--R 3 8 2 3 7 2 2 2 6
--R 70A a e + (630A a b + 210B a )d e + (630A a b + 630B a b)d e
--R +
--R 3 2 3 5
--R (70A b + 210B a b )d e
--R *
--R 4
--R x
--R +
--R 3 7 2 3 2 6 2 2 3 5 3
--R (280A a d e + (840A a b + 280B a )d e + (280A a b + 280B a b)d e )x
--R +
--R 3 2 6 2 3 3 5 2 3 3 5 3 4 4
--R (420A a d e + (420A a b + 140B a )d e )x + 280A a d e x + 70A a d e
--R +
--R 2 3 5 3 2 2 6 2
--R (- 42A a b - 14B a )d e + (14A a b + 14B a b)d e
--R +
--R 3 2 7 3 8
--R (- 2A b - 6B a b )d e + B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 5 5
--R 280b e x + 280a e
--R Type: Expression(Integer)
--E 281
--S 282 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 282
)clear all
--S 283 of 764
t0:=(A+B*x)*(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 2 2 5 2 2 2 4
--R B b e x + ((A b + 2B a b)e + 2B b d e)x
--R +
--R 2 2 2 2 2 3
--R ((2A a b + B a )e + (2A b + 4B a b)d e + B b d )x
--R +
--R 2 2 2 2 2 2
--R (A a e + (4A a b + 2B a )d e + (A b + 2B a b)d )x
--R +
--R 2 2 2 2 2
--R (2A a d e + (2A a b + B a )d )x + A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 283
--S 284 of 764
r0:=-1/84*(b*d-a*e)^2*(4*b*B*d-7*A*b*e+3*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b^4*e)-1/42*(4*b*B*d-7*A*b*e+3*a*B*e)*(a+b*x)*(d+_
e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b^2*e)+1/7*B*(a+b*x)*(d+e*x)^3*_
(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e)-1/105*(b*d-a*e)*(4*b*B*d-7*A*b*e+_
3*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^4
--R
--R
--R (2)
--R 6 2 6 6 5 2 6 5
--R 60B b e x + ((70A b + 150B a b )e + 140B b d e)x
--R +
--R 5 2 4 2 6 5 6 2 4
--R ((182A a b + 102B a b )e + (168A b + 364B a b )d e + 84B b d )x
--R +
--R 2 4 3 3 2 5 2 4
--R (133A a b + 3B a b )e + (462A a b + 266B a b )d e
--R +
--R 6 5 2
--R (105A b + 231B a b )d
--R *
--R 3
--R x
--R +
--R 3 3 4 2 2 2 4 3 3
--R (7A a b - 3B a b )e + (378A a b + 14B a b )d e
--R +
--R 5 2 4 2
--R (315A a b + 189B a b )d
--R *
--R 2
--R x
--R +
--R 4 2 5 2 3 3 4 2
--R (- 7A a b + 3B a b)e + (42A a b - 14B a b )d e
--R +
--R 2 4 3 3 2
--R (315A a b + 21B a b )d
--R *
--R x
--R +
--R 5 6 2 4 2 5 3 3 4 2 2
--R (7A a b - 3B a )e + (- 42A a b + 14B a b)d e + (105A a b - 21B a b )d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 4
--R 420b
--R Type: Expression(Integer)
--E 284
--S 285 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 285
)clear all
--S 286 of 764
t0:=(A+B*x)*(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 2 4 2 2 3
--R B b e x + ((A b + 2B a b)e + B b d)x
--R +
--R 2 2 2 2 2
--R ((2A a b + B a )e + (A b + 2B a b)d)x + (A a e + (2A a b + B a )d)x
--R +
--R 2
--R A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 286
--S 287 of 764
r0:=-1/12*(b*d-a*e)*(2*b*B*d-3*A*b*e+a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b^3*e)+1/6*B*(a+b*x)*(d+e*x)^2*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e)-1/15*(2*b*B*d-3*A*b*e+a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/b^3
--R
--R
--R (2)
--R 5 5 5 4 5 4
--R 10B b e x + ((12A b + 26B a b )e + 12B b d)x
--R +
--R 4 2 3 5 4 3
--R ((33A a b + 19B a b )e + (15A b + 33B a b )d)x
--R +
--R 2 3 3 2 4 2 3 2
--R ((27A a b + B a b )e + (45A a b + 27B a b )d)x
--R +
--R 3 2 4 2 3 3 2 4 5
--R ((3A a b - B a b)e + (45A a b + 3B a b )d)x + (- 3A a b + B a )e
--R +
--R 3 2 4
--R (15A a b - 3B a b)d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3
--R 60b
--R Type: Expression(Integer)
--E 287
--S 288 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 288
)clear all
--S 289 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 289
--S 290 of 764
r0:=1/4*(A*b-a*B)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/b^2+_
1/5*B*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^2
--R
--R
--R (2)
--R 4 4 4 3 3 3 2 2 2
--R 4B b x + (5A b + 11B a b )x + (15A a b + 9B a b )x
--R +
--R 2 2 3 3 4
--R (15A a b + B a b)x + 5A a b - B a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 2
--R 20b
--R Type: Expression(Integer)
--E 290
--S 291 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 291
)clear all
--S 292 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R e x + d
--R Type: Expression(Integer)
--E 292
--S 293 of 764
r0:=-1/3*(B*d-A*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/e^2+1/4*B*(a+b*x)*_
(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e)-(b*d-a*e)^2*(B*d-A*e)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/e^4+1/2*(b*d-a*e)*(B*d-A*e)*(a+b*x)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/e^3+(b*d-a*e)^3*(B*d-A*e)*log(d+e*x)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(e^5*(a+b*x))
--R
--R
--R (2)
--R 3 4 2 2 3 3 3 2 2 2 2
--R 12A a b e + (- 36A a b - 12B a b)d e + (36A a b + 36B a b )d e
--R +
--R 4 3 3 4 4
--R (- 12A b - 36B a b )d e + 12B b d
--R *
--R log(e x + d)
--R +
--R 4 4 4 4 3 4 4 3 3
--R 3B b e x + ((4A b + 12B a b )e - 4B b d e )x
--R +
--R 3 2 2 4 4 3 3 4 2 2 2
--R ((18A a b + 18B a b )e + (- 6A b - 18B a b )d e + 6B b d e )x
--R +
--R 2 2 3 4 3 2 2 3
--R (36A a b + 12B a b)e + (- 36A a b - 36B a b )d e
--R +
--R 4 3 2 2 4 3
--R (12A b + 36B a b )d e - 12B b d e
--R *
--R x
--R +
--R 3 4 4 2 2 3 3
--R (22A a b + 3B a )e + (- 30A a b - 22B a b)d e
--R +
--R 3 2 2 2 2 3 3
--R (12A a b + 30B a b )d e - 12B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 2 5 5
--R 12b e x + 12a b e
--R Type: Expression(Integer)
--E 293
--S 294 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 294
)clear all
--S 295 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^2
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R 2 2 2
--R e x + 2d e x + d
--R Type: Expression(Integer)
--E 295
--S 296 of 764
r0:=1/3*(4*b*B*d-3*A*b*e-a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e^2*(b*d-a*e))-_
(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*(d+e*x))+_
(b*d-a*e)*(4*b*B*d-3*A*b*e-a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^4-_
1/2*(4*b*B*d-3*A*b*e-a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^3-_
(b*d-a*e)^2*(4*b*B*d-3*A*b*e-a*B*e)*log(d+e*x)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(e^5*(a+b*x))
--R
--R
--R (2)
--R 2 3 4 2 2 3
--R (18A a b + 6B a )e + (- 36A a b - 36B a b)d e
--R +
--R 3 2 2 2 3 3
--R (18A b + 54B a b )d e - 24B b d e
--R *
--R x
--R +
--R 2 3 3 2 2 2 2
--R (18A a b + 6B a )d e + (- 36A a b - 36B a b)d e
--R +
--R 3 2 3 3 4
--R (18A b + 54B a b )d e - 24B b d
--R *
--R log(e x + d)
--R +
--R 3 4 4 3 2 4 3 3 3
--R 2B b e x + ((3A b + 9B a b )e - 4B b d e )x
--R +
--R 2 2 4 3 2 3 3 2 2 2
--R ((18A a b + 18B a b)e + (- 9A b - 27B a b )d e + 12B b d e )x
--R +
--R 2 3 4 2 2 3
--R (9A a b + 11B a )e + (18A a b - 6B a b)d e
--R +
--R 3 2 2 2 3 3
--R (- 18A b - 30B a b )d e + 24B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2
--R - 6A a e + (27A a b + 17B a )d e + (- 18A a b - 42B a b)d e
--R +
--R 2 3
--R 24B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 6 2 6 5 5
--R 6b e x + (6a e + 6b d e )x + 6a d e
--R Type: Expression(Integer)
--E 296
--S 297 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 297
)clear all
--S 298 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^3
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R 3 3 2 2 2 3
--R e x + 3d e x + 3d e x + d
--R Type: Expression(Integer)
--E 298
--S 299 of 764
r0:=-1/2*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^2)-(2*b*B*d-A*b*e-a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e^2*_
(b*d-a*e)*(d+e*x))-3*b*(2*b*B*d-A*b*e-a*B*e)*sqrt(a^2+2*a*b*x+b^2*_
x^2)/e^4+3/2*b*(2*b*B*d-A*b*e-a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*_
x^2)/(e^3*(b*d-a*e))+3*b*(b*d-a*e)*(2*b*B*d-A*b*e-a*B*e)*log(d+e*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x))
--R
--R
--R (2)
--R 2 2 4 3 2 3 3 2 2 2
--R ((6A a b + 6B a b)e + (- 6A b - 18B a b )d e + 12B b d e )x
--R +
--R 2 2 3 3 2 2 2 3 3
--R ((12A a b + 12B a b)d e + (- 12A b - 36B a b )d e + 24B b d e)x
--R +
--R 2 2 2 2 3 2 3 3 4
--R (6A a b + 6B a b)d e + (- 6A b - 18B a b )d e + 12B b d
--R *
--R log(e x + d)
--R +
--R 3 4 4 3 2 4 3 3 3
--R B b e x + ((2A b + 6B a b )e - 4B b d e )x
--R +
--R 2 2 4 3 2 3 3 2 2 2
--R ((- 3A a b + 3B a b)e + (9A b + 15B a b )d e - 18B b d e )x
--R +
--R 2 3 4 2 2 3
--R (- 6A a b - 2B a )e + (6A a b + 18B a b)d e
--R +
--R 3 2 2 2 3 3
--R (6A b - 6B a b )d e - 12B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2 2 3
--R - A a e + (- 3A a b - B a )d e + (6A a b + 12B a b)d e - 12B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 7 3 7 6 2 6 2 5 2 5
--R 2b e x + (2a e + 4b d e )x + (4a d e + 2b d e )x + 2a d e
--R Type: Expression(Integer)
--E 299
--S 300 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 300
)clear all
--S 301 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^4
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R e x + 4d e x + 6d e x + 4d e x + d
--R Type: Expression(Integer)
--E 301
--S 302 of 764
r0:=-1/3*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^3)-1/6*(4*b*B*d-A*b*e-3*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/_
(e^2*(b*d-a*e)*(d+e*x)^2)+b^2*(4*b*B*d-A*b*e-3*a*B*e)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(e^4*(b*d-a*e))-1/2*b*(4*b*B*d-A*b*e-3*a*B*e)*(a+b*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(b*d-a*e)*(d+e*x))-b^2*(4*b*B*d-A*b*e-_
3*a*B*e)*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x))
--R
--R
--R (2)
--R 3 2 4 3 3 3
--R ((6A b + 18B a b )e - 24B b d e )x
--R +
--R 3 2 3 3 2 2 2
--R ((18A b + 54B a b )d e - 72B b d e )x
--R +
--R 3 2 2 2 3 3 3 2 3
--R ((18A b + 54B a b )d e - 72B b d e)x + (6A b + 18B a b )d e
--R +
--R 3 4
--R - 24B b d
--R *
--R log(e x + d)
--R +
--R 3 4 4 3 2 4 3 3 3
--R 6B b e x + ((- 11A b - 9B a b )e + 44B b d e )x
--R +
--R 2 2 4 3 2 3 3 2 2 2
--R ((- 18A a b - 18B a b)e + (- 15A b + 27B a b )d e + 60B b d e )x
--R +
--R 2 3 4 2 2 3
--R (- 9A a b - 3B a )e + (- 18A a b - 18B a b)d e
--R +
--R 3 2 2 2 3 3
--R (- 6A b + 54B a b )d e + 24B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2
--R - 2A a e + (- 3A a b - B a )d e + (- 6A a b - 6B a b)d e
--R +
--R 2 3
--R 24B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 8 4 8 7 3 7 2 6 2
--R 6b e x + (6a e + 18b d e )x + (18a d e + 18b d e )x
--R +
--R 2 6 3 5 3 5
--R (18a d e + 6b d e )x + 6a d e
--R Type: Expression(Integer)
--E 302
--S 303 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 303
)clear all
--S 304 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^5
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R e x + 5d e x + 10d e x + 10d e x + 5d e x + d
--R Type: Expression(Integer)
--E 304
--S 305 of 764
r0:=-1/4*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^4)-1/3*B*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e^2*(d+e*x)^3)-_
1/2*b*B*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(d+e*x)^2)+b^2*B*_
(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x)*(d+e*x))+b^3*B*_
log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x))
--R
--R
--R (2)
--R 3 5 4 4 4 3 4 4 2 3 3
--R (12B a b e - 12B b d e )x + (48B a b d e - 48B b d e )x
--R +
--R 3 2 3 4 3 2 2 3 3 2 4 4
--R (72B a b d e - 72B b d e )x + (48B a b d e - 48B b d e)x
--R +
--R 3 4 4 5
--R 12B a b d e - 12B b d
--R *
--R log(e x + d)
--R +
--R 4 3 5 4 4 4
--R ((- 3A b - 10B a b )e + 13B b d e )x
--R +
--R 3 2 2 5 3 4 4 2 3 3
--R ((- 12A a b - 36B a b )e + 44B a b d e + 4B b d e )x
--R +
--R 2 2 3 5 2 2 4 3 2 3
--R (- 18A a b - 18B a b)e - 36B a b d e + 102B a b d e
--R +
--R 4 3 2
--R - 30B b d e
--R *
--R 2
--R x
--R +
--R 3 4 5 3 4 2 2 2 3 3 3 2
--R (- 12A a b - 4B a )e - 8B a b d e - 24B a b d e + 84B a b d e
--R +
--R 4 4
--R - 36B b d e
--R *
--R x
--R +
--R 4 5 4 4 3 2 3 2 2 3 2 3 4 4 5
--R - 3A a e - B a d e - 2B a b d e - 6B a b d e + 24B a b d e - 12B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 10 2 9 5 2 10 9 2 2 8 4
--R (12a b e - 12b d e )x + (12a e + 36a b d e - 48b d e )x
--R +
--R 2 9 2 8 2 3 7 3
--R (48a d e + 24a b d e - 72b d e )x
--R +
--R 2 2 8 3 7 2 4 6 2
--R (72a d e - 24a b d e - 48b d e )x
--R +
--R 2 3 7 4 6 2 5 5 2 4 6 5 5
--R (48a d e - 36a b d e - 12b d e )x + 12a d e - 12a b d e
--R Type: Expression(Integer)
--E 305
--S 306 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 306
)clear all
--S 307 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^6
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R e x + 6d e x + 15d e x + 20d e x + 15d e x + 6d e x + d
--R Type: Expression(Integer)
--E 307
--S 308 of 764
r0:=1/4*(A*b-a*B)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/((b*d-a*e)^2*_
(d+e*x)^4)+1/5*(B*d-A*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/((b*d-a*e)^2*_
(d+e*x)^5)
--R
--R
--R (2)
--R 4 3 4 4
--R ((A b - 5B a b )e + 4B b d)x
--R +
--R 3 2 2 4 3 3
--R ((- A a b - 15B a b )e + (5A b + 11B a b )d)x
--R +
--R 2 2 3 3 2 2 2
--R ((- 9A a b - 15B a b)e + (15A a b + 9B a b )d)x
--R +
--R 3 4 2 2 3 4 3 4
--R ((- 11A a b - 5B a )e + (15A a b + B a b)d)x - 4A a e + (5A a b - B a )d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 2 7 6 2 2 5 5
--R (20a e - 40a b d e + 20b d e )x
--R +
--R 2 6 2 5 2 3 4 4
--R (100a d e - 200a b d e + 100b d e )x
--R +
--R 2 2 5 3 4 2 4 3 3
--R (200a d e - 400a b d e + 200b d e )x
--R +
--R 2 3 4 4 3 2 5 2 2
--R (200a d e - 400a b d e + 200b d e )x
--R +
--R 2 4 3 5 2 2 6 2 5 2 6 2 7
--R (100a d e - 200a b d e + 100b d e)x + 20a d e - 40a b d e + 20b d
--R Type: Expression(Integer)
--E 308
--S 309 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 309
)clear all
--S 310 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^7
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R e x + 7d e x + 21d e x + 35d e x + 35d e x + 21d e x + 7d e x + d
--R Type: Expression(Integer)
--E 310
--S 311 of 764
r0:=-1/6*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^6)+1/12*(2*b*B*d+A*b*e-3*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e*(b*d-a*e)^2*(d+e*x)^5)+1/60*(2*b*B*d+A*b*e-_
3*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/((b*d-a*e)^3*(d+e*x)^5)
--R
--R
--R (2)
--R 5 4 2 5 5
--R ((- A b + 3B a b )e - 2B b d e)x
--R +
--R 4 2 3 2 5 4 5 2 4
--R ((A a b - 3B a b )e + (- 6A b + 20B a b )d e - 12B b d )x
--R +
--R 2 3 3 2 2 4 2 3
--R (- A a b - 27B a b )e + (6A a b + 70B a b )d e
--R +
--R 5 4 2
--R (- 15A b - 33B a b )d
--R *
--R 3
--R x
--R +
--R 3 2 4 2 2 3 3 2
--R (- 19A a b - 33B a b)e + (54A a b + 70B a b )d e
--R +
--R 4 2 3 2
--R (- 45A a b - 27B a b )d
--R *
--R 2
--R x
--R +
--R 4 5 2 3 2 4
--R (- 26A a b - 12B a )e + (66A a b + 20B a b)d e
--R +
--R 2 3 3 2 2
--R (- 45A a b - 3B a b )d
--R *
--R x
--R +
--R 5 2 4 5 3 2 4 2
--R - 10A a e + (24A a b - 2B a )d e + (- 15A a b + 3B a b)d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3 9 2 8 2 2 7 3 3 6 6
--R (60a e - 180a b d e + 180a b d e - 60b d e )x
--R +
--R 3 8 2 2 7 2 3 6 3 4 5 5
--R (360a d e - 1080a b d e + 1080a b d e - 360b d e )x
--R +
--R 3 2 7 2 3 6 2 4 5 3 5 4 4
--R (900a d e - 2700a b d e + 2700a b d e - 900b d e )x
--R +
--R 3 3 6 2 4 5 2 5 4 3 6 3 3
--R (1200a d e - 3600a b d e + 3600a b d e - 1200b d e )x
--R +
--R 3 4 5 2 5 4 2 6 3 3 7 2 2
--R (900a d e - 2700a b d e + 2700a b d e - 900b d e )x
--R +
--R 3 5 4 2 6 3 2 7 2 3 8 3 6 3
--R (360a d e - 1080a b d e + 1080a b d e - 360b d e)x + 60a d e
--R +
--R 2 7 2 2 8 3 9
--R - 180a b d e + 180a b d e - 60b d
--R Type: Expression(Integer)
--E 311
--S 312 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 312
)clear all
--S 313 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^8
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R /
--R 8 8 7 7 2 6 6 3 5 5 4 4 4 5 3 3 6 2 2
--R e x + 8d e x + 28d e x + 56d e x + 70d e x + 56d e x + 28d e x
--R +
--R 7 8
--R 8d e x + d
--R Type: Expression(Integer)
--E 313
--S 314 of 764
r0:=-1/7*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^7)+1/42*(4*b*B*d+3*A*b*e-7*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e*(b*d-a*e)^2*(d+e*x)^6)+1/84*b*(4*b*B*d+3*A*b*e-_
7*a*B*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)^3*_
(d+e*x)^5)+1/420*b*(4*b*B*d+3*A*b*e-7*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/((b*d-a*e)^4*(d+e*x)^5)
--R
--R
--R (2)
--R 6 5 3 6 2 6
--R ((3A b - 7B a b )e + 4B b d e )x
--R +
--R 5 2 4 3 6 5 2 6 2 5
--R ((- 3A a b + 7B a b )e + (21A b - 53B a b )d e + 28B b d e)x
--R +
--R 2 4 3 3 3 5 2 4 2
--R (3A a b - 7B a b )e + (- 21A a b + 53B a b )d e
--R +
--R 6 5 2 6 3
--R (63A b - 175B a b )d e + 84B b d
--R *
--R 4
--R x
--R +
--R 3 3 4 2 3 2 4 3 3 2
--R (- 3A a b - 133B a b )e + (21A a b + 507B a b )d e
--R +
--R 5 2 4 2 6 5 3
--R (- 63A a b - 665B a b )d e + (105A b + 231B a b )d
--R *
--R 3
--R x
--R +
--R 4 2 5 3 3 3 4 2 2
--R (- 102A a b - 182B a b)e + (399A a b + 613B a b )d e
--R +
--R 2 4 3 3 2 5 2 4 3
--R (- 567A a b - 665B a b )d e + (315A a b + 189B a b )d
--R *
--R 2
--R x
--R +
--R 5 6 3 4 2 5 2
--R (- 150A a b - 70B a )e + (546A a b + 206B a b)d e
--R +
--R 3 3 4 2 2 2 4 3 3 3
--R (- 693A a b - 175B a b )d e + (315A a b + 21B a b )d
--R *
--R x
--R +
--R 6 3 5 6 2 4 2 5 2
--R - 60A a e + (210A a b - 10B a )d e + (- 252A a b + 28B a b)d e
--R +
--R 3 3 4 2 3
--R (105A a b - 21B a b )d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 4 11 3 10 2 2 2 9 3 3 8 4 4 7 7
--R (420a e - 1680a b d e + 2520a b d e - 1680a b d e + 420b d e )x
--R +
--R 4 10 3 2 9 2 2 3 8 3 4 7
--R 2940a d e - 11760a b d e + 17640a b d e - 11760a b d e
--R +
--R 4 5 6
--R 2940b d e
--R *
--R 6
--R x
--R +
--R 4 2 9 3 3 8 2 2 4 7 3 5 6
--R 8820a d e - 35280a b d e + 52920a b d e - 35280a b d e
--R +
--R 4 6 5
--R 8820b d e
--R *
--R 5
--R x
--R +
--R 4 3 8 3 4 7 2 2 5 6 3 6 5
--R 14700a d e - 58800a b d e + 88200a b d e - 58800a b d e
--R +
--R 4 7 4
--R 14700b d e
--R *
--R 4
--R x
--R +
--R 4 4 7 3 5 6 2 2 6 5 3 7 4
--R 14700a d e - 58800a b d e + 88200a b d e - 58800a b d e
--R +
--R 4 8 3
--R 14700b d e
--R *
--R 3
--R x
--R +
--R 4 5 6 3 6 5 2 2 7 4 3 8 3
--R 8820a d e - 35280a b d e + 52920a b d e - 35280a b d e
--R +
--R 4 9 2
--R 8820b d e
--R *
--R 2
--R x
--R +
--R 4 6 5 3 7 4 2 2 8 3 3 9 2
--R 2940a d e - 11760a b d e + 17640a b d e - 11760a b d e
--R +
--R 4 10
--R 2940b d e
--R *
--R x
--R +
--R 4 7 4 3 8 3 2 2 9 2 3 10 4 11
--R 420a d e - 1680a b d e + 2520a b d e - 1680a b d e + 420b d
--R Type: Expression(Integer)
--E 314
--S 315 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 315
)clear all
--S 316 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^9
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R /
--R 9 9 8 8 2 7 7 3 6 6 4 5 5 5 4 4 6 3 3
--R e x + 9d e x + 36d e x + 84d e x + 126d e x + 126d e x + 84d e x
--R +
--R 7 2 2 8 9
--R 36d e x + 9d e x + d
--R Type: Expression(Integer)
--E 316
--S 317 of 764
r0:=-1/8*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^8)-1/14*(b*B*d+A*b*e-2*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/_
(e^2*(b*d-a*e)*(d+e*x)^7)-1/28*b*(b*B*d+A*b*e-2*a*B*e)*(a+b*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(b*d-a*e)*(d+e*x)^6)-1/56*b^2*_
(b*B*d+A*b*e-2*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(b*d-a*e)*_
(d+e*x)^5)-1/280*b^2*(b*B*d+A*b*e-2*a*B*e)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(e^5*(a+b*x)*(d+e*x)^5)
--R
--R
--R (2)
--R 3 4 4 3 2 4 3 3 3
--R - 70B b e x + ((- 56A b - 168B a b )e - 56B b d e )x
--R +
--R 2 2 4 3 2 3 3 2 2 2
--R ((- 140A a b - 140B a b)e + (- 28A b - 84B a b )d e - 28B b d e )x
--R +
--R 2 3 4 2 2 3
--R (- 120A a b - 40B a )e + (- 40A a b - 40B a b)d e
--R +
--R 3 2 2 2 3 3
--R (- 8A b - 24B a b )d e - 8B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2
--R - 35A a e + (- 15A a b - 5B a )d e + (- 5A a b - 5B a b)d e
--R +
--R 3 2 3 3 4
--R (- A b - 3B a b )d e - B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 13 9 13 12 8 12 2 11 7
--R 280b e x + (280a e + 2240b d e )x + (2240a d e + 7840b d e )x
--R +
--R 2 11 3 10 6 3 10 4 9 5
--R (7840a d e + 15680b d e )x + (15680a d e + 19600b d e )x
--R +
--R 4 9 5 8 4 5 8 6 7 3
--R (19600a d e + 15680b d e )x + (15680a d e + 7840b d e )x
--R +
--R 6 7 7 6 2 7 6 8 5 8 5
--R (7840a d e + 2240b d e )x + (2240a d e + 280b d e )x + 280a d e
--R Type: Expression(Integer)
--E 317
--S 318 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 318
)clear all
--S 319 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^10
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R /
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R e x + 10d e x + 45d e x + 120d e x + 210d e x + 252d e x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210d e x + 120d e x + 45d e x + 10d e x + d
--R Type: Expression(Integer)
--E 319
--S 320 of 764
r0:=-1/9*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^9)-1/72*(4*b*B*d+5*A*b*e-9*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e^2*(b*d-a*e)*(d+e*x)^8)-1/168*b*(4*b*B*d+5*A*b*e-_
9*a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(b*d-a*e)*(d+e*x)^7)-_
1/420*b^2*(4*b*B*d+5*A*b*e-9*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(e^4*(b*d-a*e)*(d+e*x)^6)-1/2520*b^2*(4*b*B*d+5*A*b*e-9*a*B*e)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x)*(d+e*x)^6)
--R
--R
--R (2)
--R 3 4 4 3 2 4 3 3 3
--R - 504B b e x + ((- 420A b - 1260B a b )e - 336B b d e )x
--R +
--R 2 2 4 3 2 3
--R (- 1080A a b - 1080B a b)e + (- 180A b - 540B a b )d e
--R +
--R 3 2 2
--R - 144B b d e
--R *
--R 2
--R x
--R +
--R 2 3 4 2 2 3
--R (- 945A a b - 315B a )e + (- 270A a b - 270B a b)d e
--R +
--R 3 2 2 2 3 3
--R (- 45A b - 135B a b )d e - 36B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2
--R - 280A a e + (- 105A a b - 35B a )d e + (- 30A a b - 30B a b)d e
--R +
--R 3 2 3 3 4
--R (- 5A b - 15B a b )d e - 4B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 14 10 14 13 9
--R 2520b e x + (2520a e + 22680b d e )x
--R +
--R 13 2 12 8 2 12 3 11 7
--R (22680a d e + 90720b d e )x + (90720a d e + 211680b d e )x
--R +
--R 3 11 4 10 6 4 10 5 9 5
--R (211680a d e + 317520b d e )x + (317520a d e + 317520b d e )x
--R +
--R 5 9 6 8 4 6 8 7 7 3
--R (317520a d e + 211680b d e )x + (211680a d e + 90720b d e )x
--R +
--R 7 7 8 6 2 8 6 9 5 9 5
--R (90720a d e + 22680b d e )x + (22680a d e + 2520b d e )x + 2520a d e
--R Type: Expression(Integer)
--E 320
--S 321 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 321
)clear all
--S 322 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^11
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R /
--R 11 11 10 10 2 9 9 3 8 8 4 7 7 5 6 6
--R e x + 11d e x + 55d e x + 165d e x + 330d e x + 462d e x
--R +
--R 6 5 5 7 4 4 8 3 3 9 2 2 10 11
--R 462d e x + 330d e x + 165d e x + 55d e x + 11d e x + d
--R Type: Expression(Integer)
--E 322
--S 323 of 764
r0:=-1/10*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^10)-1/45*(2*b*B*d+3*A*b*e-5*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e^2*(b*d-a*e)*(d+e*x)^9)-1/120*b*(2*b*B*d+3*A*b*e-_
5*a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(b*d-a*e)*(d+e*x)^8)-_
1/360*b^2*(2*b*B*d+3*A*b*e-5*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(e^4*(b*d-a*e)*(d+e*x)^7)-1/2520*b^2*(2*b*B*d+3*A*b*e-5*a*B*e)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x)*(d+e*x)^7)
--R
--R
--R (2)
--R 3 4 4 3 2 4 3 3 3
--R - 420B b e x + ((- 360A b - 1080B a b )e - 240B b d e )x
--R +
--R 2 2 4 3 2 3
--R (- 945A a b - 945B a b)e + (- 135A b - 405B a b )d e
--R +
--R 3 2 2
--R - 90B b d e
--R *
--R 2
--R x
--R +
--R 2 3 4 2 2 3
--R (- 840A a b - 280B a )e + (- 210A a b - 210B a b)d e
--R +
--R 3 2 2 2 3 3
--R (- 30A b - 90B a b )d e - 20B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2
--R - 252A a e + (- 84A a b - 28B a )d e + (- 21A a b - 21B a b)d e
--R +
--R 3 2 3 3 4
--R (- 3A b - 9B a b )d e - 2B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 15 11 15 14 10
--R 2520b e x + (2520a e + 25200b d e )x
--R +
--R 14 2 13 9 2 13 3 12 8
--R (25200a d e + 113400b d e )x + (113400a d e + 302400b d e )x
--R +
--R 3 12 4 11 7 4 11 5 10 6
--R (302400a d e + 529200b d e )x + (529200a d e + 635040b d e )x
--R +
--R 5 10 6 9 5 6 9 7 8 4
--R (635040a d e + 529200b d e )x + (529200a d e + 302400b d e )x
--R +
--R 7 8 8 7 3 8 7 9 6 2
--R (302400a d e + 113400b d e )x + (113400a d e + 25200b d e )x
--R +
--R 9 6 10 5 10 5
--R (25200a d e + 2520b d e )x + 2520a d e
--R Type: Expression(Integer)
--E 323
--S 324 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 324
)clear all
--S 325 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^12
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R /
--R 12 12 11 11 2 10 10 3 9 9 4 8 8 5 7 7
--R e x + 12d e x + 66d e x + 220d e x + 495d e x + 792d e x
--R +
--R 6 6 6 7 5 5 8 4 4 9 3 3 10 2 2 11 12
--R 924d e x + 792d e x + 495d e x + 220d e x + 66d e x + 12d e x + d
--R Type: Expression(Integer)
--E 325
--S 326 of 764
r0:=-1/11*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^11)-1/110*(4*b*B*d+7*A*b*e-11*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e^2*(b*d-a*e)*(d+e*x)^10)-1/330*b*(4*b*B*d+7*A*b*e-_
11*a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(b*d-a*e)*(d+e*x)^9)-_
1/1155*b^2*(4*b*B*d+7*A*b*e-11*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(e^4*(b*d-a*e)*(d+e*x)^8)-1/9240*b^2*(4*b*B*d+7*A*b*e-11*a*B*e)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x)*(d+e*x)^8)
--R
--R
--R (2)
--R 3 4 4 3 2 4 3 3 3
--R - 1320B b e x + ((- 1155A b - 3465B a b )e - 660B b d e )x
--R +
--R 2 2 4 3 2 3
--R (- 3080A a b - 3080B a b)e + (- 385A b - 1155B a b )d e
--R +
--R 3 2 2
--R - 220B b d e
--R *
--R 2
--R x
--R +
--R 2 3 4 2 2 3
--R (- 2772A a b - 924B a )e + (- 616A a b - 616B a b)d e
--R +
--R 3 2 2 2 3 3
--R (- 77A b - 231B a b )d e - 44B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2
--R - 840A a e + (- 252A a b - 84B a )d e + (- 56A a b - 56B a b)d e
--R +
--R 3 2 3 3 4
--R (- 7A b - 21B a b )d e - 4B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 16 12 16 15 11
--R 9240b e x + (9240a e + 101640b d e )x
--R +
--R 15 2 14 10 2 14 3 13 9
--R (101640a d e + 508200b d e )x + (508200a d e + 1524600b d e )x
--R +
--R 3 13 4 12 8 4 12 5 11 7
--R (1524600a d e + 3049200b d e )x + (3049200a d e + 4268880b d e )x
--R +
--R 5 11 6 10 6 6 10 7 9 5
--R (4268880a d e + 4268880b d e )x + (4268880a d e + 3049200b d e )x
--R +
--R 7 9 8 8 4 8 8 9 7 3
--R (3049200a d e + 1524600b d e )x + (1524600a d e + 508200b d e )x
--R +
--R 9 7 10 6 2 10 6 11 5
--R (508200a d e + 101640b d e )x + (101640a d e + 9240b d e )x
--R +
--R 11 5
--R 9240a d e
--R Type: Expression(Integer)
--E 326
--S 327 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 327
)clear all
--S 328 of 764
t0:=(A+B*x)*(d+e*x)^m*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R m | 2 2 2
--R (e x + d) \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 328
--S 329 of 764
r0:=20*(b*d-a*e)^2*(A*b*e*(7+m)-B*(6*b*d+a*e*(1+m)))*(d+e*x)^(1+m)*_
(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e^4*(4+m)*(5+m)*(6+m)*(7+m))-_
5*(b*d-a*e)*(A*b*e*(7+m)-B*(6*b*d+a*e*(1+m)))*(a+b*x)*(d+e*x)^(1+m)*_
(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e^3*(5+m)*(6+m)*(7+m))+(A*b*e*(7+m)-_
B*(6*b*d+a*e*(1+m)))*(d+e*x)^(1+m)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/_
(b*e^2*(6+m)*(7+m))+B*(a+b*x)*(d+e*x)^(1+m)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b*e*(7+m))+120*(b*d-a*e)^4*(A*b*e*(7+m)-B*(6*b*d+_
a*e*(1+m)))*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^6*_
(2+m)*(3+m)*(4+m)*(5+m)*(6+m)*(7+m))-120*(b*d-a*e)^5*(A*b*e*(7+m)-_
B*(6*b*d+a*e*(1+m)))*(d+e*x)^(1+m)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(b*e^7*(1+m)*(2+m)*(3+m)*(4+m)*(5+m)*(6+m)*(7+m)*(a+b*x))-60*(b*d-_
a*e)^3*(A*b*e*(7+m)-B*(6*b*d+a*e*(1+m)))*(a+b*x)*(d+e*x)^(1+m)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^5*(3+m)*(4+m)*(5+m)*(6+m)*(7+m))
--R
--R
--R (2)
--R 5 6 6 5 6 5 5 6 4 5 6 3 5 6 2
--R B b e m + 21B b e m + 175B b e m + 735B b e m + 1624B b e m
--R +
--R 5 6 5 6
--R 1764B b e m + 720B b e
--R *
--R 6
--R x
--R +
--R 5 4 6 6 5 4 6 5 5 5
--R (A b + 5B a b )e m + ((22A b + 110B a b )e - 6B b d e )m
--R +
--R 5 4 6 5 5 4
--R ((190A b + 950B a b )e - 90B b d e )m
--R +
--R 5 4 6 5 5 3
--R ((820A b + 4100B a b )e - 510B b d e )m
--R +
--R 5 4 6 5 5 2
--R ((1849A b + 9245B a b )e - 1350B b d e )m
--R +
--R 5 4 6 5 5
--R ((2038A b + 10190B a b )e - 1644B b d e )m
--R +
--R 5 4 6 5 5
--R (840A b + 4200B a b )e - 720B b d e
--R *
--R 5
--R x
--R +
--R 4 2 3 6 6
--R (5A a b + 10B a b )e m
--R +
--R 4 2 3 6 5 4 5 5
--R ((115A a b + 230B a b )e + (- 5A b - 25B a b )d e )m
--R +
--R 4 2 3 6 5 4 5
--R (1035A a b + 2070B a b )e + (- 85A b - 425B a b )d e
--R +
--R 5 2 4
--R 30B b d e
--R *
--R 4
--R m
--R +
--R 4 2 3 6 5 4 5
--R (4625A a b + 9250B a b )e + (- 525A b - 2625B a b )d e
--R +
--R 5 2 4
--R 300B b d e
--R *
--R 3
--R m
--R +
--R 4 2 3 6 5 4 5
--R (10720A a b + 21440B a b )e + (- 1475A b - 7375B a b )d e
--R +
--R 5 2 4
--R 1050B b d e
--R *
--R 2
--R m
--R +
--R 4 2 3 6 5 4 5
--R (12060A a b + 24120B a b )e + (- 1870A b - 9350B a b )d e
--R +
--R 5 2 4
--R 1500B b d e
--R *
--R m
--R +
--R 4 2 3 6 5 4 5
--R (5040A a b + 10080B a b )e + (- 840A b - 4200B a b )d e
--R +
--R 5 2 4
--R 720B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 6 6
--R (10A a b + 10B a b )e m
--R +
--R 2 3 3 2 6 4 2 3 5 5
--R ((240A a b + 240B a b )e + (- 20A a b - 40B a b )d e )m
--R +
--R 2 3 3 2 6 4 2 3 5
--R (2260A a b + 2260B a b )e + (- 380A a b - 760B a b )d e
--R +
--R 5 4 2 4
--R (20A b + 100B a b )d e
--R *
--R 4
--R m
--R +
--R 2 3 3 2 6 4 2 3 5
--R (10560A a b + 10560B a b )e + (- 2620A a b - 5240B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (260A b + 1300B a b )d e - 120B b d e
--R *
--R 3
--R m
--R +
--R 2 3 3 2 6
--R (25450A a b + 25450B a b )e
--R +
--R 4 2 3 5 5 4 2 4
--R (- 8020A a b - 16040B a b )d e + (1060A b + 5300B a b )d e
--R +
--R 5 3 3
--R - 720B b d e
--R *
--R 2
--R m
--R +
--R 2 3 3 2 6
--R (29520A a b + 29520B a b )e
--R +
--R 4 2 3 5 5 4 2 4
--R (- 10800A a b - 21600B a b )d e + (1660A b + 8300B a b )d e
--R +
--R 5 3 3
--R - 1320B b d e
--R *
--R m
--R +
--R 2 3 3 2 6 4 2 3 5
--R (12600A a b + 12600B a b )e + (- 5040A a b - 10080B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (840A b + 4200B a b )d e - 720B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 6
--R (10A a b + 5B a b)e m
--R +
--R 3 2 4 6 2 3 3 2 5 5
--R ((250A a b + 125B a b)e + (- 30A a b - 30B a b )d e )m
--R +
--R 3 2 4 6 2 3 3 2 5
--R (2470A a b + 1235B a b)e + (- 630A a b - 630B a b )d e
--R +
--R 4 2 3 2 4
--R (60A a b + 120B a b )d e
--R *
--R 4
--R m
--R +
--R 3 2 4 6 2 3 3 2 5
--R (12190A a b + 6095B a b)e + (- 4890A a b - 4890B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (960A a b + 1920B a b )d e + (- 60A b - 300B a b )d e
--R *
--R 3
--R m
--R +
--R 3 2 4 6
--R (31120A a b + 15560B a b)e
--R +
--R 2 3 3 2 5
--R (- 17010A a b - 17010B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (4980A a b + 9960B a b )d e + (- 600A b - 3000B a b )d e
--R +
--R 5 4 2
--R 360B b d e
--R *
--R 2
--R m
--R +
--R 3 2 4 6
--R (37960A a b + 18980B a b)e
--R +
--R 2 3 3 2 5
--R (- 25320A a b - 25320B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (9120A a b + 18240B a b )d e + (- 1380A b - 6900B a b )d e
--R +
--R 5 4 2
--R 1080B b d e
--R *
--R m
--R +
--R 3 2 4 6 2 3 3 2 5
--R (16800A a b + 8400B a b)e + (- 12600A a b - 12600B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (5040A a b + 10080B a b )d e + (- 840A b - 4200B a b )d e
--R +
--R 5 4 2
--R 720B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 6
--R (5A a b + B a )e m
--R +
--R 4 5 6 3 2 4 5 5
--R ((130A a b + 26B a )e + (- 20A a b - 10B a b)d e )m
--R +
--R 4 5 6 3 2 4 5
--R (1350A a b + 270B a )e + (- 460A a b - 230B a b)d e
--R +
--R 2 3 3 2 2 4
--R (60A a b + 60B a b )d e
--R *
--R 4
--R m
--R +
--R 4 5 6 3 2 4 5
--R (7100A a b + 1420B a )e + (- 4020A a b - 2010B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (1140A a b + 1140B a b )d e + (- 120A a b - 240B a b )d e
--R *
--R 3
--R m
--R +
--R 4 5 6 3 2 4 5
--R (19645A a b + 3929B a )e + (- 16340A a b - 8170B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (7500A a b + 7500B a b )d e + (- 1680A a b - 3360B a b )d e
--R +
--R 5 4 4 2
--R (120A b + 600B a b )d e
--R *
--R 2
--R m
--R +
--R 4 5 6 3 2 4 5
--R (26370A a b + 5274B a )e + (- 29560A a b - 14780B a b)d e
--R +
--R 2 3 3 2 2 4
--R (19020A a b + 19020B a b )d e
--R +
--R 4 2 3 3 3 5 4 4 2
--R (- 6600A a b - 13200B a b )d e + (960A b + 4800B a b )d e
--R +
--R 5 5
--R - 720B b d e
--R *
--R m
--R +
--R 4 5 6 3 2 4 5
--R (12600A a b + 2520B a )e + (- 16800A a b - 8400B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (12600A a b + 12600B a b )d e + (- 5040A a b - 10080B a b )d e
--R +
--R 5 4 4 2 5 5
--R (840A b + 4200B a b )d e - 720B b d e
--R *
--R x
--R +
--R 5 6 6 5 6 4 5 5 5
--R A a e m + (27A a e + (- 5A a b - B a )d e )m
--R +
--R 5 6 4 5 5 3 2 4 2 4 4
--R (295A a e + (- 125A a b - 25B a )d e + (20A a b + 10B a b)d e )m
--R +
--R 5 6 4 5 5
--R 1665A a e + (- 1225A a b - 245B a )d e
--R +
--R 3 2 4 2 4 2 3 3 2 3 3
--R (440A a b + 220B a b)d e + (- 60A a b - 60B a b )d e
--R *
--R 3
--R m
--R +
--R 5 6 4 5 5
--R 5104A a e + (- 5875A a b - 1175B a )d e
--R +
--R 3 2 4 2 4 2 3 3 2 3 3
--R (3580A a b + 1790B a b)d e + (- 1080A a b - 1080B a b )d e
--R +
--R 4 2 3 4 2
--R (120A a b + 240B a b )d e
--R *
--R 2
--R m
--R +
--R 5 6 4 5 5
--R 8028A a e + (- 13770A a b - 2754B a )d e
--R +
--R 3 2 4 2 4 2 3 3 2 3 3
--R (12760A a b + 6380B a b)d e + (- 6420A a b - 6420B a b )d e
--R +
--R 4 2 3 4 2 5 4 5
--R (1560A a b + 3120B a b )d e + (- 120A b - 600B a b )d e
--R *
--R m
--R +
--R 5 6 4 5 5
--R 5040A a e + (- 12600A a b - 2520B a )d e
--R +
--R 3 2 4 2 4 2 3 3 2 3 3
--R (16800A a b + 8400B a b)d e + (- 12600A a b - 12600B a b )d e
--R +
--R 4 2 3 4 2 5 4 5 5 6
--R (5040A a b + 10080B a b )d e + (- 840A b - 4200B a b )d e + 720B b d
--R *
--R +------------------+
--R m + 1 | 2 2 2
--R (e x + d) \|b x + 2a b x + a
--R /
--R 7 7 7 6 7 5 7 4 7 3 7 2
--R b e m + 28b e m + 322b e m + 1960b e m + 6769b e m + 13132b e m
--R +
--R 7 7
--R 13068b e m + 5040b e
--R *
--R x
--R +
--R 7 7 7 6 7 5 7 4 7 3 7 2
--R a e m + 28a e m + 322a e m + 1960a e m + 6769a e m + 13132a e m
--R +
--R 7 7
--R 13068a e m + 5040a e
--R Type: Expression(Integer)
--E 329
--S 330 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 330
)clear all
--S 331 of 764
t0:=(A+B*x)*(d+e*x)^6*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 6 11 4 3 6 4 5 10
--R B b e x + ((A b + 4B a b )e + 6B b d e )x
--R +
--R 3 2 2 6 4 3 5 4 2 4 9
--R ((4A a b + 6B a b )e + (6A b + 24B a b )d e + 15B b d e )x
--R +
--R 2 2 3 6 3 2 2 5
--R (6A a b + 4B a b)e + (24A a b + 36B a b )d e
--R +
--R 4 3 2 4 4 3 3
--R (15A b + 60B a b )d e + 20B b d e
--R *
--R 8
--R x
--R +
--R 3 4 6 2 2 3 5
--R (4A a b + B a )e + (36A a b + 24B a b)d e
--R +
--R 3 2 2 2 4 4 3 3 3 4 4 2
--R (60A a b + 90B a b )d e + (20A b + 80B a b )d e + 15B b d e
--R *
--R 7
--R x
--R +
--R 4 6 3 4 5 2 2 3 2 4
--R A a e + (24A a b + 6B a )d e + (90A a b + 60B a b)d e
--R +
--R 3 2 2 3 3 4 3 4 2 4 5
--R (80A a b + 120B a b )d e + (15A b + 60B a b )d e + 6B b d e
--R *
--R 6
--R x
--R +
--R 4 5 3 4 2 4 2 2 3 3 3
--R 6A a d e + (60A a b + 15B a )d e + (120A a b + 80B a b)d e
--R +
--R 3 2 2 4 2 4 3 5 4 6
--R (60A a b + 90B a b )d e + (6A b + 24B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 4 2 4 3 4 3 3 2 2 3 4 2
--R 15A a d e + (80A a b + 20B a )d e + (90A a b + 60B a b)d e
--R +
--R 3 2 2 5 4 3 6
--R (24A a b + 36B a b )d e + (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 3 3 3 4 4 2 2 2 3 5
--R 20A a d e + (60A a b + 15B a )d e + (36A a b + 24B a b)d e
--R +
--R 3 2 2 6
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 4 2 3 4 5 2 2 3 6 2
--R (15A a d e + (24A a b + 6B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 5 3 4 6 4 6
--R (6A a d e + (4A a b + B a )d )x + A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 331
--S 332 of 764
r0:=-1/858*(b*d-a*e)^2*(6*b*B*d-13*A*b*e+7*a*B*e)*(d+e*x)^7*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e^4)+5/1716*(b*d-a*e)*(6*b*B*d-13*A*b*e+7*a*B*e)*_
(a+b*x)*(d+e*x)^7*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e^3)-1/156*(6*b*B*d-_
13*A*b*e+7*a*B*e)*(d+e*x)^7*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b*e^2)+_
1/13*B*(a+b*x)*(d+e*x)^7*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b*e)-_
1/10296*(b*d-a*e)^4*(6*b*B*d-13*A*b*e+7*a*B*e)*(d+e*x)^7*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(b*e^6)+1/72072*(b*d-a*e)^5*(6*b*B*d-13*A*b*e+_
7*a*B*e)*(d+e*x)^7*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^7*(a+b*x))+_
1/2574*(b*d-a*e)^3*(6*b*B*d-13*A*b*e+7*a*B*e)*(a+b*x)*(d+e*x)^7*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^5)
--R
--R
--R (2)
--R 5 13 13 5 4 13 5 12 12
--R 5544B b e x + ((6006A b + 30030B a b )e + 36036B b d e )x
--R +
--R 4 2 3 13 5 4 12
--R (32760A a b + 65520B a b )e + (39312A b + 196560B a b )d e
--R +
--R 5 2 11
--R 98280B b d e
--R *
--R 11
--R x
--R +
--R 2 3 3 2 13 4 2 3 12
--R (72072A a b + 72072B a b )e + (216216A a b + 432432B a b )d e
--R +
--R 5 4 2 11 5 3 10
--R (108108A b + 540540B a b )d e + 144144B b d e
--R *
--R 10
--R x
--R +
--R 3 2 4 13 2 3 3 2 12
--R (80080A a b + 40040B a b)e + (480480A a b + 480480B a b )d e
--R +
--R 4 2 3 2 11
--R (600600A a b + 1201200B a b )d e
--R +
--R 5 4 3 10 5 4 9
--R (160160A b + 800800B a b )d e + 120120B b d e
--R *
--R 9
--R x
--R +
--R 4 5 13 3 2 4 12
--R (45045A a b + 9009B a )e + (540540A a b + 270270B a b)d e
--R +
--R 2 3 3 2 2 11
--R (1351350A a b + 1351350B a b )d e
--R +
--R 4 2 3 3 10
--R (900900A a b + 1801800B a b )d e
--R +
--R 5 4 4 9 5 5 8
--R (135135A b + 675675B a b )d e + 54054B b d e
--R *
--R 8
--R x
--R +
--R 5 13 4 5 12
--R 10296A a e + (308880A a b + 61776B a )d e
--R +
--R 3 2 4 2 11
--R (1544400A a b + 772200B a b)d e
--R +
--R 2 3 3 2 3 10
--R (2059200A a b + 2059200B a b )d e
--R +
--R 4 2 3 4 9 5 4 5 8
--R (772200A a b + 1544400B a b )d e + (61776A b + 308880B a b )d e
--R +
--R 5 6 7
--R 10296B b d e
--R *
--R 7
--R x
--R +
--R 5 12 4 5 2 11
--R 72072A a d e + (900900A a b + 180180B a )d e
--R +
--R 3 2 4 3 10
--R (2402400A a b + 1201200B a b)d e
--R +
--R 2 3 3 2 4 9
--R (1801800A a b + 1801800B a b )d e
--R +
--R 4 2 3 5 8 5 4 6 7
--R (360360A a b + 720720B a b )d e + (12012A b + 60060B a b )d e
--R *
--R 6
--R x
--R +
--R 5 2 11 4 5 3 10
--R 216216A a d e + (1441440A a b + 288288B a )d e
--R +
--R 3 2 4 4 9
--R (2162160A a b + 1081080B a b)d e
--R +
--R 2 3 3 2 5 8 4 2 3 6 7
--R (864864A a b + 864864B a b )d e + (72072A a b + 144144B a b )d e
--R *
--R 5
--R x
--R +
--R 5 3 10 4 5 4 9
--R 360360A a d e + (1351350A a b + 270270B a )d e
--R +
--R 3 2 4 5 8 2 3 3 2 6 7
--R (1081080A a b + 540540B a b)d e + (180180A a b + 180180B a b )d e
--R *
--R 4
--R x
--R +
--R 5 4 9 4 5 5 8
--R 360360A a d e + (720720A a b + 144144B a )d e
--R +
--R 3 2 4 6 7
--R (240240A a b + 120120B a b)d e
--R *
--R 3
--R x
--R +
--R 5 5 8 4 5 6 7 2 5 6 7
--R (216216A a d e + (180180A a b + 36036B a )d e )x + 72072A a d e x
--R +
--R 5 7 6 4 5 8 5
--R 10296A a d e + (- 6435A a b - 1287B a )d e
--R +
--R 3 2 4 9 4 2 3 3 2 10 3
--R (2860A a b + 1430B a b)d e + (- 858A a b - 858B a b )d e
--R +
--R 4 2 3 11 2 5 4 12 5 13
--R (156A a b + 312B a b )d e + (- 13A b - 65B a b )d e + 6B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 7 7
--R 72072b e x + 72072a e
--R Type: Expression(Integer)
--E 332
--S 333 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 333
)clear all
--S 334 of 764
t0:=(A+B*x)*(d+e*x)^5*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 5 10 4 3 5 4 4 9
--R B b e x + ((A b + 4B a b )e + 5B b d e )x
--R +
--R 3 2 2 5 4 3 4 4 2 3 8
--R ((4A a b + 6B a b )e + (5A b + 20B a b )d e + 10B b d e )x
--R +
--R 2 2 3 5 3 2 2 4
--R (6A a b + 4B a b)e + (20A a b + 30B a b )d e
--R +
--R 4 3 2 3 4 3 2
--R (10A b + 40B a b )d e + 10B b d e
--R *
--R 7
--R x
--R +
--R 3 4 5 2 2 3 4
--R (4A a b + B a )e + (30A a b + 20B a b)d e
--R +
--R 3 2 2 2 3 4 3 3 2 4 4
--R (40A a b + 60B a b )d e + (10A b + 40B a b )d e + 5B b d e
--R *
--R 6
--R x
--R +
--R 4 5 3 4 4 2 2 3 2 3
--R A a e + (20A a b + 5B a )d e + (60A a b + 40B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (40A a b + 60B a b )d e + (5A b + 20B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 4 4 3 4 2 3 2 2 3 3 2
--R 5A a d e + (40A a b + 10B a )d e + (60A a b + 40B a b)d e
--R +
--R 3 2 2 4 4 3 5
--R (20A a b + 30B a b )d e + (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 2 3 3 4 3 2 2 2 3 4
--R 10A a d e + (40A a b + 10B a )d e + (30A a b + 20B a b)d e
--R +
--R 3 2 2 5
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 3 2 3 4 4 2 2 3 5 2
--R (10A a d e + (20A a b + 5B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 4 3 4 5 4 5
--R (5A a d e + (4A a b + B a )d )x + A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 334
--S 335 of 764
r0:=1/99*(b*d-a*e)^2*(2*A*b*e-B*(b*d+a*e))*(d+e*x)^6*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e^4)-1/44*(b*d-a*e)*(2*A*b*e-B*(b*d+a*e))*(a+b*x)*_
(d+e*x)^6*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e^3)+1/22*(2*A*b*e-B*(b*d+_
a*e))*(d+e*x)^6*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b*e^2)+1/12*B*(a+b*x)*_
(d+e*x)^6*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b*e)+1/924*(b*d-a*e)^4*(2*A*b*e-_
B*(b*d+a*e))*(d+e*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^6)-_
1/5544*(b*d-a*e)^5*(2*A*b*e-B*(b*d+a*e))*(d+e*x)^6*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^7*(a+b*x))-1/264*(b*d-a*e)^3*(2*A*b*e-B*(b*d+a*e))*_
(a+b*x)*(d+e*x)^6*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^5)
--R
--R
--R (2)
--R 5 12 12 5 4 12 5 11 11
--R 462B b e x + ((504A b + 2520B a b )e + 2520B b d e )x
--R +
--R 4 2 3 12 5 4 11
--R (2772A a b + 5544B a b )e + (2772A b + 13860B a b )d e
--R +
--R 5 2 10
--R 5544B b d e
--R *
--R 10
--R x
--R +
--R 2 3 3 2 12 4 2 3 11
--R (6160A a b + 6160B a b )e + (15400A a b + 30800B a b )d e
--R +
--R 5 4 2 10 5 3 9
--R (6160A b + 30800B a b )d e + 6160B b d e
--R *
--R 9
--R x
--R +
--R 3 2 4 12 2 3 3 2 11
--R (6930A a b + 3465B a b)e + (34650A a b + 34650B a b )d e
--R +
--R 4 2 3 2 10 5 4 3 9
--R (34650A a b + 69300B a b )d e + (6930A b + 34650B a b )d e
--R +
--R 5 4 8
--R 3465B b d e
--R *
--R 8
--R x
--R +
--R 4 5 12 3 2 4 11
--R (3960A a b + 792B a )e + (39600A a b + 19800B a b)d e
--R +
--R 2 3 3 2 2 10 4 2 3 3 9
--R (79200A a b + 79200B a b )d e + (39600A a b + 79200B a b )d e
--R +
--R 5 4 4 8 5 5 7
--R (3960A b + 19800B a b )d e + 792B b d e
--R *
--R 7
--R x
--R +
--R 5 12 4 5 11
--R 924A a e + (23100A a b + 4620B a )d e
--R +
--R 3 2 4 2 10 2 3 3 2 3 9
--R (92400A a b + 46200B a b)d e + (92400A a b + 92400B a b )d e
--R +
--R 4 2 3 4 8 5 4 5 7
--R (23100A a b + 46200B a b )d e + (924A b + 4620B a b )d e
--R *
--R 6
--R x
--R +
--R 5 11 4 5 2 10
--R 5544A a d e + (55440A a b + 11088B a )d e
--R +
--R 3 2 4 3 9 2 3 3 2 4 8
--R (110880A a b + 55440B a b)d e + (55440A a b + 55440B a b )d e
--R +
--R 4 2 3 5 7
--R (5544A a b + 11088B a b )d e
--R *
--R 5
--R x
--R +
--R 5 2 10 4 5 3 9
--R 13860A a d e + (69300A a b + 13860B a )d e
--R +
--R 3 2 4 4 8 2 3 3 2 5 7
--R (69300A a b + 34650B a b)d e + (13860A a b + 13860B a b )d e
--R *
--R 4
--R x
--R +
--R 5 3 9 4 5 4 8
--R 18480A a d e + (46200A a b + 9240B a )d e
--R +
--R 3 2 4 5 7
--R (18480A a b + 9240B a b)d e
--R *
--R 3
--R x
--R +
--R 5 4 8 4 5 5 7 2 5 5 7
--R (13860A a d e + (13860A a b + 2772B a )d e )x + 5544A a d e x
--R +
--R 5 6 6 4 5 7 5 3 2 4 8 4
--R 924A a d e + (- 660A a b - 132B a )d e + (330A a b + 165B a b)d e
--R +
--R 2 3 3 2 9 3 4 2 3 10 2
--R (- 110A a b - 110B a b )d e + (22A a b + 44B a b )d e
--R +
--R 5 4 11 5 12
--R (- 2A b - 10B a b )d e + B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 7 7
--R 5544b e x + 5544a e
--R Type: Expression(Integer)
--E 335
--S 336 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 336
)clear all
--S 337 of 764
t0:=(A+B*x)*(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 4 9 4 3 4 4 3 8
--R B b e x + ((A b + 4B a b )e + 4B b d e )x
--R +
--R 3 2 2 4 4 3 3 4 2 2 7
--R ((4A a b + 6B a b )e + (4A b + 16B a b )d e + 6B b d e )x
--R +
--R 2 2 3 4 3 2 2 3
--R (6A a b + 4B a b)e + (16A a b + 24B a b )d e
--R +
--R 4 3 2 2 4 3
--R (6A b + 24B a b )d e + 4B b d e
--R *
--R 6
--R x
--R +
--R 3 4 4 2 2 3 3
--R (4A a b + B a )e + (24A a b + 16B a b)d e
--R +
--R 3 2 2 2 2 4 3 3 4 4
--R (24A a b + 36B a b )d e + (4A b + 16B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 4 4 3 4 3 2 2 3 2 2
--R A a e + (16A a b + 4B a )d e + (36A a b + 24B a b)d e
--R +
--R 3 2 2 3 4 3 4
--R (16A a b + 24B a b )d e + (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 3 3 4 2 2 2 2 3 3
--R 4A a d e + (24A a b + 6B a )d e + (24A a b + 16B a b)d e
--R +
--R 3 2 2 4
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 2 2 3 4 3 2 2 3 4 2
--R (6A a d e + (16A a b + 4B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 3 3 4 4 4 4
--R (4A a d e + (4A a b + B a )d )x + A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 337
--S 338 of 764
r0:=-1/1980*(b*d-a*e)^4*(6*b*B*d-11*A*b*e+5*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b^6*e)-1/660*(b*d-a*e)^2*(6*b*B*d-11*A*b*e+5*a*B*e)*_
(a+b*x)*(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b^4*e)-2/495*(b*d-a*e)*_
(6*b*B*d-11*A*b*e+5*a*B*e)*(a+b*x)*(d+e*x)^3*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b^3*e)-1/110*(6*b*B*d-11*A*b*e+5*a*B*e)*(a+b*x)*_
(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b^2*e)+1/11*B*(a+b*x)*(d+_
e*x)^5*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b*e)-1/2310*(b*d-a*e)^3*(6*b*B*d-_
11*A*b*e+5*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(7/2)/b^6
--R
--R
--R (2)
--R 10 4 10 10 9 4 10 3 9
--R 1260B b e x + ((1386A b + 5670B a b )e + 5544B b d e )x
--R +
--R 9 2 8 4 10 9 3
--R (6314A a b + 9730B a b )e + (6160A b + 25256B a b )d e
--R +
--R 10 2 2
--R 9240B b d e
--R *
--R 8
--R x
--R +
--R 2 8 3 7 4 9 2 8 3
--R (11011A a b + 7595B a b )e + (28490A a b + 44044B a b )d e
--R +
--R 10 9 2 2 10 3
--R (10395A b + 42735B a b )d e + 6930B b d e
--R *
--R 7
--R x
--R +
--R 3 7 4 6 4 2 8 3 7 3
--R (8789A a b + 2305B a b )e + (50710A a b + 35156B a b )d e
--R +
--R 9 2 8 2 2 10 9 3
--R (49005A a b + 76065B a b )d e + (7920A b + 32670B a b )d e
--R +
--R 10 4
--R 1980B b d
--R *
--R 6
--R x
--R +
--R 4 6 5 5 4 3 7 4 6 3
--R (2761A a b + 5B a b )e + (41690A a b + 11044B a b )d e
--R +
--R 2 8 3 7 2 2 9 2 8 3
--R (89595A a b + 62535B a b )d e + (38280A a b + 59730B a b )d e
--R +
--R 10 9 4
--R (2310A b + 9570B a b )d
--R *
--R 5
--R x
--R +
--R 5 5 6 4 4 4 6 5 5 3
--R (11A a b - 5B a b )e + (13750A a b + 44B a b )d e
--R +
--R 3 7 4 6 2 2 2 8 3 7 3
--R (76725A a b + 20625B a b )d e + (72600A a b + 51150B a b )d e
--R +
--R 9 2 8 4
--R (11550A a b + 18150B a b )d
--R *
--R 4
--R x
--R +
--R 6 4 7 3 4 5 5 6 4 3
--R (- 11A a b + 5B a b )e + (110A a b - 44B a b )d e
--R +
--R 4 6 5 5 2 2 3 7 4 6 3
--R (27225A a b + 165B a b )d e + (66000A a b + 18150B a b )d e
--R +
--R 2 8 3 7 4
--R (23100A a b + 16500B a b )d
--R *
--R 3
--R x
--R +
--R 7 3 8 2 4 6 4 7 3 3
--R (11A a b - 5B a b )e + (- 110A a b + 44B a b )d e
--R +
--R 5 5 6 4 2 2 4 6 5 5 3
--R (495A a b - 165B a b )d e + (26400A a b + 330B a b )d e
--R +
--R 3 7 4 6 4
--R (23100A a b + 6600B a b )d
--R *
--R 2
--R x
--R +
--R 8 2 9 4 7 3 8 2 3
--R (- 11A a b + 5B a b)e + (110A a b - 44B a b )d e
--R +
--R 6 4 7 3 2 2 5 5 6 4 3
--R (- 495A a b + 165B a b )d e + (1320A a b - 330B a b )d e
--R +
--R 4 6 5 5 4
--R (11550A a b + 330B a b )d
--R *
--R x
--R +
--R 9 10 4 8 2 9 3
--R (11A a b - 5B a )e + (- 110A a b + 44B a b)d e
--R +
--R 7 3 8 2 2 2 6 4 7 3 3
--R (495A a b - 165B a b )d e + (- 1320A a b + 330B a b )d e
--R +
--R 5 5 6 4 4
--R (2310A a b - 330B a b )d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 6
--R 13860b
--R Type: Expression(Integer)
--E 338
--S 339 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 339
)clear all
--S 340 of 764
t0:=(A+B*x)*(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 3 8 4 3 3 4 2 7
--R B b e x + ((A b + 4B a b )e + 3B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 6
--R ((4A a b + 6B a b )e + (3A b + 12B a b )d e + 3B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (6A a b + 4B a b)e + (12A a b + 18B a b )d e
--R +
--R 4 3 2 4 3
--R (3A b + 12B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 3 4 3 2 2 3 2
--R (4A a b + B a )e + (18A a b + 12B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (12A a b + 18B a b )d e + (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 3 3 4 2 2 2 3 2
--R A a e + (12A a b + 3B a )d e + (18A a b + 12B a b)d e
--R +
--R 3 2 2 3
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 2 3 4 2 2 2 3 3 2
--R (3A a d e + (12A a b + 3B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 2 3 4 3 4 3
--R (3A a d e + (4A a b + B a )d )x + A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 340
--S 341 of 764
r0:=-1/360*(b*d-a*e)^3*(3*b*B*d-5*A*b*e+2*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b^5*e)-1/120*(b*d-a*e)*(3*b*B*d-5*A*b*e+2*a*B*e)*_
(a+b*x)*(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b^3*e)-1/45*(3*b*B*d-_
5*A*b*e+2*a*B*e)*(a+b*x)*(d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b^2*e)+_
1/10*B*(a+b*x)*(d+e*x)^4*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b*e)-_
1/420*(b*d-a*e)^2*(3*b*B*d-5*A*b*e+2*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(7/2)/b^5
--R
--R
--R (2)
--R 9 3 9 9 8 3 9 2 8
--R 252B b e x + ((280A b + 1148B a b )e + 840B b d e )x
--R +
--R 8 2 7 3 9 8 2
--R (1295A a b + 2002B a b )e + (945A b + 3885B a b )d e
--R +
--R 9 2
--R 945B b d e
--R *
--R 7
--R x
--R +
--R 2 7 3 6 3 8 2 7 2
--R (2305A a b + 1598B a b )e + (4455A a b + 6915B a b )d e
--R +
--R 9 8 2 9 3
--R (1080A b + 4455B a b )d e + 360B b d
--R *
--R 6
--R x
--R +
--R 3 6 4 5 3 2 7 3 6 2
--R (1895A a b + 502B a b )e + (8145A a b + 5685B a b )d e
--R +
--R 8 2 7 2 9 8 3
--R (5220A a b + 8145B a b )d e + (420A b + 1740B a b )d
--R *
--R 5
--R x
--R +
--R 4 5 5 4 3 3 6 4 5 2
--R (625A a b + 2B a b )e + (6975A a b + 1875B a b )d e
--R +
--R 2 7 3 6 2 8 2 7 3
--R (9900A a b + 6975B a b )d e + (2100A a b + 3300B a b )d
--R *
--R 4
--R x
--R +
--R 5 4 6 3 3 4 5 5 4 2
--R (5A a b - 2B a b )e + (2475A a b + 15B a b )d e
--R +
--R 3 6 4 5 2 2 7 3 6 3
--R (9000A a b + 2475B a b )d e + (4200A a b + 3000B a b )d
--R *
--R 3
--R x
--R +
--R 6 3 7 2 3 5 4 6 3 2
--R (- 5A a b + 2B a b )e + (45A a b - 15B a b )d e
--R +
--R 4 5 5 4 2 3 6 4 5 3
--R (3600A a b + 45B a b )d e + (4200A a b + 1200B a b )d
--R *
--R 2
--R x
--R +
--R 7 2 8 3 6 3 7 2 2
--R (5A a b - 2B a b)e + (- 45A a b + 15B a b )d e
--R +
--R 5 4 6 3 2 4 5 5 4 3
--R (180A a b - 45B a b )d e + (2100A a b + 60B a b )d
--R *
--R x
--R +
--R 8 9 3 7 2 8 2
--R (- 5A a b + 2B a )e + (45A a b - 15B a b)d e
--R +
--R 6 3 7 2 2 5 4 6 3 3
--R (- 180A a b + 45B a b )d e + (420A a b - 60B a b )d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 5
--R 2520b
--R Type: Expression(Integer)
--E 341
--S 342 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 342
)clear all
--S 343 of 764
t0:=(A+B*x)*(d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 2 7 4 3 2 4 6
--R B b e x + ((A b + 4B a b )e + 2B b d e)x
--R +
--R 3 2 2 2 4 3 4 2 5
--R ((4A a b + 6B a b )e + (2A b + 8B a b )d e + B b d )x
--R +
--R 2 2 3 2 3 2 2 4 3 2 4
--R ((6A a b + 4B a b)e + (8A a b + 12B a b )d e + (A b + 4B a b )d )x
--R +
--R 3 4 2 2 2 3 3 2 2 2 3
--R ((4A a b + B a )e + (12A a b + 8B a b)d e + (4A a b + 6B a b )d )x
--R +
--R 4 2 3 4 2 2 3 2 2
--R (A a e + (8A a b + 2B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 3 4 2 4 2
--R (2A a d e + (4A a b + B a )d )x + A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 343
--S 344 of 764
r0:=-1/72*(b*d-a*e)^2*(2*b*B*d-3*A*b*e+a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b^4*e)-1/24*(2*b*B*d-3*A*b*e+a*B*e)*(a+b*x)*(d+_
e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b^2*e)+1/9*B*(a+b*x)*(d+e*x)^3*_
(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b*e)-1/84*(b*d-a*e)*(2*b*B*d-3*A*b*e+_
a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(7/2)/b^4
--R
--R
--R (2)
--R 8 2 8 8 7 2 8 7
--R 56B b e x + ((63A b + 259B a b )e + 126B b d e)x
--R +
--R 7 2 6 2 8 7 8 2 6
--R ((297A a b + 461B a b )e + (144A b + 594B a b )d e + 72B b d )x
--R +
--R 2 6 3 5 2 7 2 6
--R (543A a b + 379B a b )e + (696A a b + 1086B a b )d e
--R +
--R 8 7 2
--R (84A b + 348B a b )d
--R *
--R 5
--R x
--R +
--R 3 5 4 4 2 2 6 3 5
--R (465A a b + 125B a b )e + (1320A a b + 930B a b )d e
--R +
--R 7 2 6 2
--R (420A a b + 660B a b )d
--R *
--R 4
--R x
--R +
--R 4 4 5 3 2 3 5 4 4
--R (165A a b + B a b )e + (1200A a b + 330B a b )d e
--R +
--R 2 6 3 5 2
--R (840A a b + 600B a b )d
--R *
--R 3
--R x
--R +
--R 5 3 6 2 2 4 4 5 3
--R (3A a b - B a b )e + (480A a b + 6B a b )d e
--R +
--R 3 5 4 4 2
--R (840A a b + 240B a b )d
--R *
--R 2
--R x
--R +
--R 6 2 7 2 5 3 6 2
--R (- 3A a b + B a b)e + (24A a b - 6B a b )d e
--R +
--R 4 4 5 3 2
--R (420A a b + 12B a b )d
--R *
--R x
--R +
--R 7 8 2 6 2 7 5 3 6 2 2
--R (3A a b - B a )e + (- 24A a b + 6B a b)d e + (84A a b - 12B a b )d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 4
--R 504b
--R Type: Expression(Integer)
--E 344
--S 345 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 345
)clear all
--S 346 of 764
t0:=(A+B*x)*(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 6 4 3 4 5
--R B b e x + ((A b + 4B a b )e + B b d)x
--R +
--R 3 2 2 4 3 4
--R ((4A a b + 6B a b )e + (A b + 4B a b )d)x
--R +
--R 2 2 3 3 2 2 3
--R ((6A a b + 4B a b)e + (4A a b + 6B a b )d)x
--R +
--R 3 4 2 2 3 2 4 3 4
--R ((4A a b + B a )e + (6A a b + 4B a b)d)x + (A a e + (4A a b + B a )d)x
--R +
--R 4
--R A a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 346
--S 347 of 764
r0:=-1/24*(b*d-a*e)*(3*b*B*d-4*A*b*e+a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b^3*e)+1/8*B*(a+b*x)*(d+e*x)^2*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b*e)-1/28*(3*b*B*d-4*A*b*e+a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(7/2)/b^3
--R
--R
--R (2)
--R 7 7 7 6 7 6
--R 21B b e x + ((24A b + 99B a b )e + 24B b d)x
--R +
--R 6 2 5 7 6 5
--R ((116A a b + 181B a b )e + (28A b + 116B a b )d)x
--R +
--R 2 5 3 4 6 2 5 4
--R ((220A a b + 155B a b )e + (140A a b + 220B a b )d)x
--R +
--R 3 4 4 3 2 5 3 4 3
--R ((200A a b + 55B a b )e + (280A a b + 200B a b )d)x
--R +
--R 4 3 5 2 3 4 4 3 2
--R ((80A a b + B a b )e + (280A a b + 80B a b )d)x
--R +
--R 5 2 6 4 3 5 2 6 7
--R ((4A a b - B a b)e + (140A a b + 4B a b )d)x + (- 4A a b + B a )e
--R +
--R 5 2 6
--R (28A a b - 4B a b)d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3
--R 168b
--R Type: Expression(Integer)
--E 347
--S 348 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 348
)clear all
--S 349 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 349
--S 350 of 764
r0:=1/6*(A*b-a*B)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/b^2+_
1/7*B*(a^2+2*a*b*x+b^2*x^2)^(7/2)/b^2
--R
--R
--R (2)
--R 6 6 6 5 5 5 2 4 4
--R 6B b x + (7A b + 29B a b )x + (35A a b + 55B a b )x
--R +
--R 2 4 3 3 3 3 3 4 2 2 4 2 5
--R (70A a b + 50B a b )x + (70A a b + 20B a b )x + (35A a b + B a b)x
--R +
--R 5 6
--R 7A a b - B a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 2
--R 42b
--R Type: Expression(Integer)
--E 350
--S 351 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 351
)clear all
--S 352 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R e x + d
--R Type: Expression(Integer)
--E 352
--S 353 of 764
r0:=-1/3*(b*d-a*e)^2*(B*d-A*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/e^4+_
1/4*(b*d-a*e)*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/e^3-_
1/5*(B*d-A*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/e^2+1/6*B*(a+b*x)*_
(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b*e)-(b*d-a*e)^4*(B*d-A*e)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/e^6+1/2*(b*d-a*e)^3*(B*d-A*e)*(a+b*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/e^5+(b*d-a*e)^5*(B*d-A*e)*log(d+e*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^7*(a+b*x))
--R
--R
--R (2)
--R 5 6 4 2 5 5
--R 60A a b e + (- 300A a b - 60B a b)d e
--R +
--R 3 3 4 2 2 4 2 4 3 3 3 3
--R (600A a b + 300B a b )d e + (- 600A a b - 600B a b )d e
--R +
--R 5 2 4 4 2 6 5 5 6 6
--R (300A a b + 600B a b )d e + (- 60A b - 300B a b )d e + 60B b d
--R *
--R log(e x + d)
--R +
--R 6 6 6 6 5 6 6 5 5
--R 10B b e x + ((12A b + 60B a b )e - 12B b d e )x
--R +
--R 5 2 4 6 6 5 5 6 2 4 4
--R ((75A a b + 150B a b )e + (- 15A b - 75B a b )d e + 15B b d e )x
--R +
--R 2 4 3 3 6 5 2 4 5
--R (200A a b + 200B a b )e + (- 100A a b - 200B a b )d e
--R +
--R 6 5 2 4 6 3 3
--R (20A b + 100B a b )d e - 20B b d e
--R *
--R 3
--R x
--R +
--R 3 3 4 2 6 2 4 3 3 5
--R (300A a b + 150B a b )e + (- 300A a b - 300B a b )d e
--R +
--R 5 2 4 2 4 6 5 3 3 6 4 2
--R (150A a b + 300B a b )d e + (- 30A b - 150B a b )d e + 30B b d e
--R *
--R 2
--R x
--R +
--R 4 2 5 6 3 3 4 2 5
--R (300A a b + 60B a b)e + (- 600A a b - 300B a b )d e
--R +
--R 2 4 3 3 2 4 5 2 4 3 3
--R (600A a b + 600B a b )d e + (- 300A a b - 600B a b )d e
--R +
--R 6 5 4 2 6 5
--R (60A b + 300B a b )d e - 60B b d e
--R *
--R x
--R +
--R 5 6 6 4 2 5 5
--R (137A a b + 10B a )e + (- 385A a b - 137B a b)d e
--R +
--R 3 3 4 2 2 4 2 4 3 3 3 3
--R (470A a b + 385B a b )d e + (- 270A a b - 470B a b )d e
--R +
--R 5 2 4 4 2 5 5
--R (60A a b + 270B a b )d e - 60B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 2 7 7
--R 60b e x + 60a b e
--R Type: Expression(Integer)
--E 353
--S 354 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 354
)clear all
--S 355 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^2
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 2 2 2
--R e x + 2d e x + d
--R Type: Expression(Integer)
--E 355
--S 356 of 764
r0:=1/3*(b*d-a*e)*(6*b*B*d-5*A*b*e-a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/e^4-_
1/4*(6*b*B*d-5*A*b*e-a*B*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/e^3+_
1/5*(6*b*B*d-5*A*b*e-a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e^2*(b*d-a*e))-_
(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*(d+e*x))+_
(b*d-a*e)^3*(6*b*B*d-5*A*b*e-a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^6-_
1/2*(b*d-a*e)^2*(6*b*B*d-5*A*b*e-a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/e^5-(b*d-a*e)^4*(6*b*B*d-5*A*b*e-a*B*e)*log(d+e*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^7*(a+b*x))
--R
--R
--R (2)
--R 4 5 6 3 2 4 5
--R (300A a b + 60B a )e + (- 1200A a b - 600B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (1800A a b + 1800B a b )d e + (- 1200A a b - 2400B a b )d e
--R +
--R 5 4 4 2 5 5
--R (300A b + 1500B a b )d e - 360B b d e
--R *
--R x
--R +
--R 4 5 5 3 2 4 2 4
--R (300A a b + 60B a )d e + (- 1200A a b - 600B a b)d e
--R +
--R 2 3 3 2 3 3 4 2 3 4 2
--R (1800A a b + 1800B a b )d e + (- 1200A a b - 2400B a b )d e
--R +
--R 5 4 5 5 6
--R (300A b + 1500B a b )d e - 360B b d
--R *
--R log(e x + d)
--R +
--R 5 6 6 5 4 6 5 5 5
--R 12B b e x + ((15A b + 75B a b )e - 18B b d e )x
--R +
--R 4 2 3 6 5 4 5 5 2 4 4
--R ((100A a b + 200B a b )e + (- 25A b - 125B a b )d e + 30B b d e )x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (300A a b + 300B a b )e + (- 200A a b - 400B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (50A b + 250B a b )d e - 60B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (600A a b + 300B a b)e + (- 900A a b - 900B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (600A a b + 1200B a b )d e + (- 150A b - 750B a b )d e
--R +
--R 5 4 2
--R 180B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (325A a b + 137B a )e + (- 100A a b - 410B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (- 750A a b - 30B a b )d e + (900A a b + 1080B a b )d e
--R +
--R 5 4 4 2 5 5
--R (- 300A b - 1140B a b )d e + 360B b d e
--R *
--R x
--R +
--R 5 6 4 5 5 3 2 4 2 4
--R - 60A a e + (625A a b + 197B a )d e + (- 1300A a b - 1010B a b)d e
--R +
--R 2 3 3 2 3 3 4 2 3 4 2
--R (1050A a b + 1770B a b )d e + (- 300A a b - 1320B a b )d e
--R +
--R 4 5
--R 360B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 8 2 8 7 7
--R 60b e x + (60a e + 60b d e )x + 60a d e
--R Type: Expression(Integer)
--E 356
--S 357 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 357
)clear all
--S 358 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^3
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3 3 2 2 2 3
--R e x + 3d e x + 3d e x + d
--R Type: Expression(Integer)
--E 358
--S 359 of 764
r0:=-5/3*b*(3*b*B*d-2*A*b*e-a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/e^4+_
5/4*b*(3*b*B*d-2*A*b*e-a*B*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/_
(e^3*(b*d-a*e))-1/2*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/_
(e*(b*d-a*e)*(d+e*x)^2)-(3*b*B*d-2*A*b*e-a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(e^2*(b*d-a*e)*(d+e*x))-5*b*(b*d-a*e)^2*(3*b*B*d-_
2*A*b*e-a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^6+5/2*b*(b*d-a*e)*_
(3*b*B*d-2*A*b*e-a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^5+5*b*_
(b*d-a*e)^3*(3*b*B*d-2*A*b*e-a*B*e)*log(d+e*x)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(e^7*(a+b*x))
--R
--R
--R (2)
--R 3 2 4 6 2 3 3 2 5
--R (120A a b + 60B a b)e + (- 360A a b - 360B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (360A a b + 720B a b )d e + (- 120A b - 600B a b )d e
--R +
--R 5 4 2
--R 180B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 5 2 3 3 2 2 4
--R (240A a b + 120B a b)d e + (- 720A a b - 720B a b )d e
--R +
--R 4 2 3 3 3 5 4 4 2
--R (720A a b + 1440B a b )d e + (- 240A b - 1200B a b )d e
--R +
--R 5 5
--R 360B b d e
--R *
--R x
--R +
--R 3 2 4 2 4 2 3 3 2 3 3
--R (120A a b + 60B a b)d e + (- 360A a b - 360B a b )d e
--R +
--R 4 2 3 4 2 5 4 5 5 6
--R (360A a b + 720B a b )d e + (- 120A b - 600B a b )d e + 180B b d
--R *
--R log(e x + d)
--R +
--R 5 6 6 5 4 6 5 5 5
--R 3B b e x + ((4A b + 20B a b )e - 6B b d e )x
--R +
--R 4 2 3 6 5 4 5 5 2 4 4
--R ((30A a b + 60B a b )e + (- 10A b - 50B a b )d e + 15B b d e )x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (120A a b + 120B a b )e + (- 120A a b - 240B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (40A b + 200B a b )d e - 60B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (40A a b + 65B a b)e + (240A a b + 60B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (- 420A a b - 570B a b )d e + (180A b + 720B a b )d e
--R +
--R 5 4 2
--R - 270B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (- 60A a b - 12B a )e + (320A a b + 250B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (- 240A a b - 600B a b )d e + (- 120A a b + 300B a b )d e
--R +
--R 5 4 4 2 5 5
--R (120A b + 240B a b )d e - 180B b d e
--R *
--R x
--R +
--R 5 6 4 5 5 3 2 4 2 4
--R - 6A a e + (- 30A a b - 6B a )d e + (220A a b + 155B a b)d e
--R +
--R 2 3 3 2 3 3 4 2 3 4 2
--R (- 300A a b - 480B a b )d e + (120A a b + 510B a b )d e
--R +
--R 4 5
--R - 180B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 9 3 9 8 2 8 2 7 2 7
--R 12b e x + (12a e + 24b d e )x + (24a d e + 12b d e )x + 12a d e
--R Type: Expression(Integer)
--E 359
--S 360 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 360
)clear all
--S 361 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^4
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 4 4 3 3 2 2 2 3 4
--R e x + 4d e x + 6d e x + 4d e x + d
--R Type: Expression(Integer)
--E 361
--S 362 of 764
r0:=10/3*b^2*(2*b*B*d-A*b*e-a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e^4*(b*d-_
a*e))-5/2*b*(2*b*B*d-A*b*e-a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e^3*(b*d-a*e)*(d+e*x))-1/3*(B*d-A*e)*(a+b*x)*(a^2+_
2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*(d+e*x)^3)-1/2*(2*b*B*d-A*b*e-_
a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e^2*(b*d-a*e)*(d+e*x)^2)+_
10*b^2*(b*d-a*e)*(2*b*B*d-A*b*e-a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^6-_
5*b^2*(2*b*B*d-A*b*e-a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^5-_
10*b^2*(b*d-a*e)^2*(2*b*B*d-A*b*e-a*B*e)*log(d+e*x)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(e^7*(a+b*x))
--R
--R
--R (2)
--R 2 3 3 2 6 4 2 3 5
--R (60A a b + 60B a b )e + (- 120A a b - 240B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (60A b + 300B a b )d e - 120B b d e
--R *
--R 3
--R x
--R +
--R 2 3 3 2 5 4 2 3 2 4
--R (180A a b + 180B a b )d e + (- 360A a b - 720B a b )d e
--R +
--R 5 4 3 3 5 4 2
--R (180A b + 900B a b )d e - 360B b d e
--R *
--R 2
--R x
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (180A a b + 180B a b )d e + (- 360A a b - 720B a b )d e
--R +
--R 5 4 4 2 5 5
--R (180A b + 900B a b )d e - 360B b d e
--R *
--R x
--R +
--R 2 3 3 2 3 3 4 2 3 4 2
--R (60A a b + 60B a b )d e + (- 120A a b - 240B a b )d e
--R +
--R 5 4 5 5 6
--R (60A b + 300B a b )d e - 120B b d
--R *
--R log(e x + d)
--R +
--R 5 6 6 5 4 6 5 5 5
--R 2B b e x + ((3A b + 15B a b )e - 6B b d e )x
--R +
--R 4 2 3 6 5 4 5 5 2 4 4
--R ((30A a b + 60B a b )e + (- 15A b - 75B a b )d e + 30B b d e )x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (- 20A a b + 20B a b )e + (160A a b + 200B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (- 110A b - 430B a b )d e + 220B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (- 60A a b - 30B a b)e + (120A a b + 240B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (120A a b - 120B a b )d e + (- 150A b - 390B a b )d e
--R +
--R 5 4 2
--R 300B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (- 15A a b - 3B a )e + (- 60A a b - 30B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (210A a b + 330B a b )d e + (- 60A a b - 480B a b )d e
--R +
--R 5 4 4 2 5 5
--R (- 60A b + 60B a b )d e + 120B b d e
--R *
--R x
--R +
--R 5 6 4 5 5 3 2 4 2 4
--R - 2A a e + (- 5A a b - B a )d e + (- 20A a b - 10B a b)d e
--R +
--R 2 3 3 2 3 3 4 2 3 4 2 4 5
--R (90A a b + 130B a b )d e + (- 60A a b - 240B a b )d e + 120B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 10 4 10 9 3 9 2 8 2
--R 6b e x + (6a e + 18b d e )x + (18a d e + 18b d e )x
--R +
--R 2 8 3 7 3 7
--R (18a d e + 6b d e )x + 6a d e
--R Type: Expression(Integer)
--E 362
--S 363 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 363
)clear all
--S 364 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^5
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 5 5 4 4 2 3 3 3 2 2 4 5
--R e x + 5d e x + 10d e x + 10d e x + 5d e x + d
--R Type: Expression(Integer)
--E 364
--S 365 of 764
r0:=-5/12*b*(3*b*B*d-A*b*e-2*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e^3*(b*d-a*e)*(d+e*x)^2)-5/3*b^2*(3*b*B*d-A*b*e-_
2*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e^4*(b*d-a*e)*(d+e*x))-_
1/4*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*_
(d+e*x)^4)-1/6*(3*b*B*d-A*b*e-2*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/_
(e^2*(b*d-a*e)*(d+e*x)^3)-5*b^3*(3*b*B*d-A*b*e-2*a*B*e)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/e^6+5/2*b^3*(3*b*B*d-A*b*e-2*a*B*e)*(a+b*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(b*d-a*e))+5*b^3*(b*d-a*e)*(3*b*B*d-_
A*b*e-2*a*B*e)*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^7*(a+b*x))
--R
--R
--R (2)
--R 4 2 3 6 5 4 5
--R (60A a b + 120B a b )e + (- 60A b - 300B a b )d e
--R +
--R 5 2 4
--R 180B b d e
--R *
--R 4
--R x
--R +
--R 4 2 3 5 5 4 2 4
--R (240A a b + 480B a b )d e + (- 240A b - 1200B a b )d e
--R +
--R 5 3 3
--R 720B b d e
--R *
--R 3
--R x
--R +
--R 4 2 3 2 4 5 4 3 3
--R (360A a b + 720B a b )d e + (- 360A b - 1800B a b )d e
--R +
--R 5 4 2
--R 1080B b d e
--R *
--R 2
--R x
--R +
--R 4 2 3 3 3 5 4 4 2
--R (240A a b + 480B a b )d e + (- 240A b - 1200B a b )d e
--R +
--R 5 5
--R 720B b d e
--R *
--R x
--R +
--R 4 2 3 4 2 5 4 5 5 6
--R (60A a b + 120B a b )d e + (- 60A b - 300B a b )d e + 180B b d
--R *
--R log(e x + d)
--R +
--R 5 6 6 5 4 6 5 5 5
--R 6B b e x + ((12A b + 60B a b )e - 36B b d e )x
--R +
--R 4 2 3 6 5 4 5 5 2 4 4
--R ((- 65A a b - 40B a b )e + (125A b + 445B a b )d e - 375B b d e )x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (- 120A a b - 120B a b )e + (- 20A a b + 320B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (260A b + 580B a b )d e - 780B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (- 60A a b - 30B a b)e + (- 180A a b - 180B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3 5 4 2
--R (150A a b + 840B a b )d e + (210A b - 30B a b )d e - 630B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (- 20A a b - 4B a )e + (- 40A a b - 20B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (- 120A a b - 120B a b )d e + (180A a b + 720B a b )d e
--R +
--R 5 4 4 2 5 5
--R (60A b - 420B a b )d e - 180B b d e
--R *
--R x
--R +
--R 5 6 4 5 5 3 2 4 2 4
--R - 3A a e + (- 5A a b - B a )d e + (- 10A a b - 5B a b)d e
--R +
--R 2 3 3 2 3 3 4 2 3 4 2 4 5
--R (- 30A a b - 30B a b )d e + (60A a b + 210B a b )d e - 180B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 11 5 11 10 4 10 2 9 3
--R 12b e x + (12a e + 48b d e )x + (48a d e + 72b d e )x
--R +
--R 2 9 3 8 2 3 8 4 7 4 7
--R (72a d e + 48b d e )x + (48a d e + 12b d e )x + 12a d e
--R Type: Expression(Integer)
--E 365
--S 366 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 366
)clear all
--S 367 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^6
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R e x + 6d e x + 15d e x + 20d e x + 15d e x + 6d e x + d
--R Type: Expression(Integer)
--E 367
--S 368 of 764
r0:=-1/12*b*(6*b*B*d-A*b*e-5*a*B*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/_
(e^3*(b*d-a*e)*(d+e*x)^3)-1/6*b^2*(6*b*B*d-A*b*e-5*a*B*e)*(a^2+_
2*a*b*x+b^2*x^2)^(3/2)/(e^4*(b*d-a*e)*(d+e*x)^2)-1/5*(B*d-A*e)*_
(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*(d+e*x)^5)-_
1/20*(6*b*B*d-A*b*e-5*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e^2*_
(b*d-a*e)*(d+e*x)^4)+b^4*(6*b*B*d-A*b*e-5*a*B*e)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(e^6*(b*d-a*e))-1/2*b^3*(6*b*B*d-A*b*e-5*a*B*e)*(a+b*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(b*d-a*e)*(d+e*x))-b^4*(6*b*B*d-A*b*e-_
5*a*B*e)*log(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^7*(a+b*x))
--R
--R
--R (2)
--R 5 4 6 5 5 5
--R ((60A b + 300B a b )e - 360B b d e )x
--R +
--R 5 4 5 5 2 4 4
--R ((300A b + 1500B a b )d e - 1800B b d e )x
--R +
--R 5 4 2 4 5 3 3 3
--R ((600A b + 3000B a b )d e - 3600B b d e )x
--R +
--R 5 4 3 3 5 4 2 2
--R ((600A b + 3000B a b )d e - 3600B b d e )x
--R +
--R 5 4 4 2 5 5
--R ((300A b + 1500B a b )d e - 1800B b d e)x
--R +
--R 5 4 5 5 6
--R (60A b + 300B a b )d e - 360B b d
--R *
--R log(e x + d)
--R +
--R 5 6 6 5 4 6 5 5 5
--R 60B b e x + ((- 137A b - 325B a b )e + 822B b d e )x
--R +
--R 4 2 3 6 5 4 5
--R (- 300A a b - 600B a b )e + (- 385A b - 125B a b )d e
--R +
--R 5 2 4
--R 2310B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (- 300A a b - 300B a b )e + (- 600A a b - 1200B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (- 470A b + 1250B a b )d e + 2820B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (- 200A a b - 100B a b)e + (- 300A a b - 300B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (- 600A a b - 1200B a b )d e + (- 270A b + 2250B a b )d e
--R +
--R 5 4 2
--R 1620B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (- 75A a b - 15B a )e + (- 100A a b - 50B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (- 150A a b - 150B a b )d e + (- 300A a b - 600B a b )d e
--R +
--R 5 4 4 2 5 5
--R (- 60A b + 1500B a b )d e + 360B b d e
--R *
--R x
--R +
--R 5 6 4 5 5 3 2 4 2 4
--R - 12A a e + (- 15A a b - 3B a )d e + (- 20A a b - 10B a b)d e
--R +
--R 2 3 3 2 3 3 4 2 3 4 2 4 5
--R (- 30A a b - 30B a b )d e + (- 60A a b - 120B a b )d e + 360B a b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 12 6 12 11 5 11 2 10 4
--R 60b e x + (60a e + 300b d e )x + (300a d e + 600b d e )x
--R +
--R 2 10 3 9 3 3 9 4 8 2
--R (600a d e + 600b d e )x + (600a d e + 300b d e )x
--R +
--R 4 8 5 7 5 7
--R (300a d e + 60b d e )x + 60a d e
--R Type: Expression(Integer)
--E 368
--S 369 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 369
)clear all
--S 370 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^7
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 7 7 6 6 2 5 5 3 4 4 4 3 3 5 2 2 6 7
--R e x + 7d e x + 21d e x + 35d e x + 35d e x + 21d e x + 7d e x + d
--R Type: Expression(Integer)
--E 370
--S 371 of 764
r0:=-1/4*b*B*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e^3*(d+e*x)^4)-_
1/3*b^2*B*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e^4*(d+e*x)^3)-1/6*(B*d-A*e)*_
(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*(d+e*x)^6)-_
1/5*B*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e^2*(d+e*x)^5)-1/2*b^3*B*(a+b*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(d+e*x)^2)+b^4*B*(b*d-a*e)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(e^7*(a+b*x)*(d+e*x))+b^5*B*log(d+e*x)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(e^7*(a+b*x))
--R
--R
--R (2)
--R 5 7 6 6 6 5 6 6 2 5 5
--R (60B a b e - 60B b d e )x + (360B a b d e - 360B b d e )x
--R +
--R 5 2 5 6 3 4 4 5 3 4 6 4 3 3
--R (900B a b d e - 900B b d e )x + (1200B a b d e - 1200B b d e )x
--R +
--R 5 4 3 6 5 2 2 5 5 2 6 6
--R (900B a b d e - 900B b d e )x + (360B a b d e - 360B b d e)x
--R +
--R 5 6 6 7
--R 60B a b d e - 60B b d
--R *
--R log(e x + d)
--R +
--R 6 5 7 6 6 6
--R ((- 10A b - 77B a b )e + 87B b d e )x
--R +
--R 5 2 4 7 5 6 6 2 5 5
--R ((- 60A a b - 300B a b )e + 198B a b d e + 162B b d e )x
--R +
--R 2 4 3 3 7 2 4 6 5 2 5
--R (- 150A a b - 300B a b )e - 450B a b d e + 945B a b d e
--R +
--R 6 3 4
--R - 45B b d e
--R *
--R 4
--R x
--R +
--R 3 3 4 2 7 3 3 6 2 4 2 5
--R (- 200A a b - 200B a b )e - 200B a b d e - 600B a b d e
--R +
--R 5 3 4 6 4 3
--R 1660B a b d e - 460B b d e
--R *
--R 3
--R x
--R +
--R 4 2 5 7 4 2 6 3 3 2 5
--R (- 150A a b - 75B a b)e - 75B a b d e - 150B a b d e
--R +
--R 2 4 3 4 5 4 3 6 5 2
--R - 450B a b d e + 1470B a b d e - 570B b d e
--R *
--R 2
--R x
--R +
--R 5 6 7 5 6 4 2 2 5 3 3 3 4
--R (- 60A a b - 12B a )e - 18B a b d e - 30B a b d e - 60B a b d e
--R +
--R 2 4 4 3 5 5 2 6 6
--R - 180B a b d e + 660B a b d e - 300B b d e
--R *
--R x
--R +
--R 6 7 6 6 5 2 5 4 2 3 4 3 3 4 3
--R - 10A a e - 2B a d e - 3B a b d e - 5B a b d e - 10B a b d e
--R +
--R 2 4 5 2 5 6 6 7
--R - 30B a b d e + 120B a b d e - 60B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 14 2 13 7 2 14 13 2 2 12 6
--R (60a b e - 60b d e )x + (60a e + 300a b d e - 360b d e )x
--R +
--R 2 13 2 12 2 3 11 5
--R (360a d e + 540a b d e - 900b d e )x
--R +
--R 2 2 12 3 11 2 4 10 4
--R (900a d e + 300a b d e - 1200b d e )x
--R +
--R 2 3 11 4 10 2 5 9 3
--R (1200a d e - 300a b d e - 900b d e )x
--R +
--R 2 4 10 5 9 2 6 8 2
--R (900a d e - 540a b d e - 360b d e )x
--R +
--R 2 5 9 6 8 2 7 7 2 6 8 7 7
--R (360a d e - 300a b d e - 60b d e )x + 60a d e - 60a b d e
--R Type: Expression(Integer)
--E 371
--S 372 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 372
)clear all
--S 373 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^8
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 8 8 7 7 2 6 6 3 5 5 4 4 4 5 3 3 6 2 2
--R e x + 8d e x + 28d e x + 56d e x + 70d e x + 56d e x + 28d e x
--R +
--R 7 8
--R 8d e x + d
--R Type: Expression(Integer)
--E 373
--S 374 of 764
r0:=1/6*(A*b-a*B)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/((b*d-a*e)^2*_
(d+e*x)^6)+1/7*(B*d-A*e)*(a^2+2*a*b*x+b^2*x^2)^(7/2)/((b*d-a*e)^2*_
(d+e*x)^7)
--R
--R
--R (2)
--R 6 5 6 6
--R ((A b - 7B a b )e + 6B b d)x
--R +
--R 5 2 4 6 5 5
--R ((- A a b - 35B a b )e + (7A b + 29B a b )d)x
--R +
--R 2 4 3 3 5 2 4 4
--R ((- 20A a b - 70B a b )e + (35A a b + 55B a b )d)x
--R +
--R 3 3 4 2 2 4 3 3 3
--R ((- 50A a b - 70B a b )e + (70A a b + 50B a b )d)x
--R +
--R 4 2 5 3 3 4 2 2
--R ((- 55A a b - 35B a b)e + (70A a b + 20B a b )d)x
--R +
--R 5 6 4 2 5 6 5 6
--R ((- 29A a b - 7B a )e + (35A a b + B a b)d)x - 6A a e + (7A a b - B a )d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 2 9 8 2 2 7 7
--R (42a e - 84a b d e + 42b d e )x
--R +
--R 2 8 2 7 2 3 6 6
--R (294a d e - 588a b d e + 294b d e )x
--R +
--R 2 2 7 3 6 2 4 5 5
--R (882a d e - 1764a b d e + 882b d e )x
--R +
--R 2 3 6 4 5 2 5 4 4
--R (1470a d e - 2940a b d e + 1470b d e )x
--R +
--R 2 4 5 5 4 2 6 3 3
--R (1470a d e - 2940a b d e + 1470b d e )x
--R +
--R 2 5 4 6 3 2 7 2 2
--R (882a d e - 1764a b d e + 882b d e )x
--R +
--R 2 6 3 7 2 2 8 2 7 2 8 2 9
--R (294a d e - 588a b d e + 294b d e)x + 42a d e - 84a b d e + 42b d
--R Type: Expression(Integer)
--E 374
--S 375 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 375
)clear all
--S 376 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^9
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 9 9 8 8 2 7 7 3 6 6 4 5 5 5 4 4 6 3 3
--R e x + 9d e x + 36d e x + 84d e x + 126d e x + 126d e x + 84d e x
--R +
--R 7 2 2 8 9
--R 36d e x + 9d e x + d
--R Type: Expression(Integer)
--E 376
--S 377 of 764
r0:=-1/8*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*_
(d+e*x)^8)+1/24*(3*b*B*d+A*b*e-4*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(e*(b*d-a*e)^2*(d+e*x)^7)+1/168*(3*b*B*d+A*b*e-_
4*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(7/2)/((b*d-a*e)^3*(d+e*x)^7)
--R
--R
--R (2)
--R 7 6 2 7 7
--R ((- A b + 4B a b )e - 3B b d e)x
--R +
--R 6 2 5 2 7 6 7 2 6
--R ((A a b - 4B a b )e + (- 8A b + 35B a b )d e - 24B b d )x
--R +
--R 2 5 3 4 2 6 2 5
--R (- A a b - 80B a b )e + (8A a b + 217B a b )d e
--R +
--R 7 6 2
--R (- 28A b - 116B a b )d
--R *
--R 5
--R x
--R +
--R 3 4 4 3 2 2 5 3 4
--R (- 55A a b - 200B a b )e + (160A a b + 455B a b )d e
--R +
--R 6 2 5 2
--R (- 140A a b - 220B a b )d
--R *
--R 4
--R x
--R +
--R 4 3 5 2 2 3 4 4 3
--R (- 155A a b - 220B a b )e + (400A a b + 455B a b )d e
--R +
--R 2 5 3 4 2
--R (- 280A a b - 200B a b )d
--R *
--R 3
--R x
--R +
--R 5 2 6 2 4 3 5 2
--R (- 181A a b - 116B a b)e + (440A a b + 217B a b )d e
--R +
--R 3 4 4 3 2
--R (- 280A a b - 80B a b )d
--R *
--R 2
--R x
--R +
--R 6 7 2 5 2 6
--R (- 99A a b - 24B a )e + (232A a b + 35B a b)d e
--R +
--R 4 3 5 2 2
--R (- 140A a b - 4B a b )d
--R *
--R x
--R +
--R 7 2 6 7 5 2 6 2
--R - 21A a e + (48A a b - 3B a )d e + (- 28A a b + 4B a b)d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3 11 2 10 2 2 9 3 3 8 8
--R (168a e - 504a b d e + 504a b d e - 168b d e )x
--R +
--R 3 10 2 2 9 2 3 8 3 4 7 7
--R (1344a d e - 4032a b d e + 4032a b d e - 1344b d e )x
--R +
--R 3 2 9 2 3 8 2 4 7 3 5 6 6
--R (4704a d e - 14112a b d e + 14112a b d e - 4704b d e )x
--R +
--R 3 3 8 2 4 7 2 5 6 3 6 5 5
--R (9408a d e - 28224a b d e + 28224a b d e - 9408b d e )x
--R +
--R 3 4 7 2 5 6 2 6 5 3 7 4 4
--R (11760a d e - 35280a b d e + 35280a b d e - 11760b d e )x
--R +
--R 3 5 6 2 6 5 2 7 4 3 8 3 3
--R (9408a d e - 28224a b d e + 28224a b d e - 9408b d e )x
--R +
--R 3 6 5 2 7 4 2 8 3 3 9 2 2
--R (4704a d e - 14112a b d e + 14112a b d e - 4704b d e )x
--R +
--R 3 7 4 2 8 3 2 9 2 3 10 3 8 3
--R (1344a d e - 4032a b d e + 4032a b d e - 1344b d e)x + 168a d e
--R +
--R 2 9 2 2 10 3 11
--R - 504a b d e + 504a b d e - 168b d
--R Type: Expression(Integer)
--E 377
--S 378 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 378
)clear all
--S 379 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^10
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 10 10 9 9 2 8 8 3 7 7 4 6 6 5 5 5
--R e x + 10d e x + 45d e x + 120d e x + 210d e x + 252d e x
--R +
--R 6 4 4 7 3 3 8 2 2 9 10
--R 210d e x + 120d e x + 45d e x + 10d e x + d
--R Type: Expression(Integer)
--E 379
--S 380 of 764
r0:=-1/9*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*_
(d+e*x)^9)+1/24*(2*b*B*d+A*b*e-3*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(e*(b*d-a*e)^2*(d+e*x)^8)+1/72*b*(2*b*B*d+A*b*e-_
3*a*B*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)^3*_
(d+e*x)^7)+1/504*b*(2*b*B*d+A*b*e-3*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(7/2)/((b*d-a*e)^4*(d+e*x)^7)
--R
--R
--R (2)
--R 8 7 3 8 2 8
--R ((A b - 3B a b )e + 2B b d e )x
--R +
--R 7 2 6 3 8 7 2 8 2 7
--R ((- A a b + 3B a b )e + (9A b - 29B a b )d e + 18B b d e)x
--R +
--R 2 6 3 5 3 7 2 6 2
--R (A a b - 3B a b )e + (- 9A a b + 29B a b )d e
--R +
--R 8 7 2 8 3
--R (36A b - 126B a b )d e + 72B b d
--R *
--R 6
--R x
--R +
--R 3 5 4 4 3 2 6 3 5 2
--R (- A a b - 165B a b )e + (9A a b + 643B a b )d e
--R +
--R 7 2 6 2 8 7 3
--R (- 36A a b - 882B a b )d e + (84A b + 348B a b )d
--R *
--R 5
--R x
--R +
--R 4 4 5 3 3 3 5 4 4 2
--R (- 125A a b - 465B a b )e + (495A a b + 1625B a b )d e
--R +
--R 2 6 3 5 2 7 2 6 3
--R (- 720A a b - 1890B a b )d e + (420A a b + 660B a b )d
--R *
--R 4
--R x
--R +
--R 5 3 6 2 3 4 4 5 3 2
--R (- 379A a b - 543B a b )e + (1395A a b + 1777B a b )d e
--R +
--R 3 5 4 4 2 2 6 3 5 3
--R (- 1800A a b - 1890B a b )d e + (840A a b + 600B a b )d
--R *
--R 3
--R x
--R +
--R 6 2 7 3 5 3 6 2 2
--R (- 461A a b - 297B a b)e + (1629A a b + 911B a b )d e
--R +
--R 4 4 5 3 2 3 5 4 4 3
--R (- 1980A a b - 882B a b )d e + (840A a b + 240B a b )d
--R *
--R 2
--R x
--R +
--R 7 8 3 6 2 7 2
--R (- 259A a b - 63B a )e + (891A a b + 169B a b)d e
--R +
--R 5 3 6 2 2 4 4 5 3 3
--R (- 1044A a b - 126B a b )d e + (420A a b + 12B a b )d
--R *
--R x
--R +
--R 8 3 7 8 2 6 2 7 2
--R - 56A a e + (189A a b - 7B a )d e + (- 216A a b + 18B a b)d e
--R +
--R 5 3 6 2 3
--R (84A a b - 12B a b )d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 4 13 3 12 2 2 2 11 3 3 10 4 4 9 9
--R (504a e - 2016a b d e + 3024a b d e - 2016a b d e + 504b d e )x
--R +
--R 4 12 3 2 11 2 2 3 10 3 4 9
--R 4536a d e - 18144a b d e + 27216a b d e - 18144a b d e
--R +
--R 4 5 8
--R 4536b d e
--R *
--R 8
--R x
--R +
--R 4 2 11 3 3 10 2 2 4 9 3 5 8
--R 18144a d e - 72576a b d e + 108864a b d e - 72576a b d e
--R +
--R 4 6 7
--R 18144b d e
--R *
--R 7
--R x
--R +
--R 4 3 10 3 4 9 2 2 5 8 3 6 7
--R 42336a d e - 169344a b d e + 254016a b d e - 169344a b d e
--R +
--R 4 7 6
--R 42336b d e
--R *
--R 6
--R x
--R +
--R 4 4 9 3 5 8 2 2 6 7 3 7 6
--R 63504a d e - 254016a b d e + 381024a b d e - 254016a b d e
--R +
--R 4 8 5
--R 63504b d e
--R *
--R 5
--R x
--R +
--R 4 5 8 3 6 7 2 2 7 6 3 8 5
--R 63504a d e - 254016a b d e + 381024a b d e - 254016a b d e
--R +
--R 4 9 4
--R 63504b d e
--R *
--R 4
--R x
--R +
--R 4 6 7 3 7 6 2 2 8 5 3 9 4
--R 42336a d e - 169344a b d e + 254016a b d e - 169344a b d e
--R +
--R 4 10 3
--R 42336b d e
--R *
--R 3
--R x
--R +
--R 4 7 6 3 8 5 2 2 9 4 3 10 3
--R 18144a d e - 72576a b d e + 108864a b d e - 72576a b d e
--R +
--R 4 11 2
--R 18144b d e
--R *
--R 2
--R x
--R +
--R 4 8 5 3 9 4 2 2 10 3 3 11 2
--R 4536a d e - 18144a b d e + 27216a b d e - 18144a b d e
--R +
--R 4 12
--R 4536b d e
--R *
--R x
--R +
--R 4 9 4 3 10 3 2 2 11 2 3 12 4 13
--R 504a d e - 2016a b d e + 3024a b d e - 2016a b d e + 504b d
--R Type: Expression(Integer)
--E 380
--S 381 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 381
)clear all
--S 382 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^11
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 11 11 10 10 2 9 9 3 8 8 4 7 7 5 6 6
--R e x + 11d e x + 55d e x + 165d e x + 330d e x + 462d e x
--R +
--R 6 5 5 7 4 4 8 3 3 9 2 2 10 11
--R 462d e x + 330d e x + 165d e x + 55d e x + 11d e x + d
--R Type: Expression(Integer)
--E 382
--S 383 of 764
r0:=-1/10*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*_
(d+e*x)^10)+1/45*(3*b*B*d+2*A*b*e-5*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(e*(b*d-a*e)^2*(d+e*x)^9)+1/120*b*(3*b*B*d+2*A*b*e-_
5*a*B*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)^3*(d+_
e*x)^8)+1/360*b^2*(3*b*B*d+2*A*b*e-5*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(e*(b*d-a*e)^4*(d+e*x)^7)+1/2520*b^2*(3*b*B*d+2*A*b*e-_
5*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(7/2)/((b*d-a*e)^5*(d+e*x)^7)
--R
--R
--R (2)
--R 9 8 4 9 3 9
--R ((- 2A b + 5B a b )e - 3B b d e )x
--R +
--R 8 2 7 4 9 8 3 9 2 2 8
--R ((2A a b - 5B a b )e + (- 20A b + 53B a b )d e - 30B b d e )x
--R +
--R 2 7 3 6 4 8 2 7 3
--R (- 2A a b + 5B a b )e + (20A a b - 53B a b )d e
--R +
--R 9 8 2 2 9 3
--R (- 90A b + 255B a b )d e - 135B b d e
--R *
--R 7
--R x
--R +
--R 3 6 4 5 4 2 7 3 6 3
--R (2A a b - 5B a b )e + (- 20A a b + 53B a b )d e
--R +
--R 8 2 7 2 2 9 8 3 9 4
--R (90A a b - 255B a b )d e + (- 240A b + 735B a b )d e - 360B b d
--R *
--R 6
--R x
--R +
--R 4 5 5 4 4 3 6 4 5 3
--R (- 2A a b - 625B a b )e + (20A a b + 3097B a b )d e
--R +
--R 2 7 3 6 2 2 8 2 7 3
--R (- 90A a b - 6045B a b )d e + (240A a b + 5565B a b )d e
--R +
--R 9 8 4
--R (- 420A b - 1740B a b )d
--R *
--R 5
--R x
--R +
--R 5 4 6 3 4 4 5 5 4 3
--R (- 502A a b - 1895B a b )e + (2500A a b + 8747B a b )d e
--R +
--R 3 6 4 5 2 2 2 7 3 6 3
--R (- 4950A a b - 15375B a b )d e + (4800A a b + 12075B a b )d e
--R +
--R 8 2 7 4
--R (- 2100A a b - 3300B a b )d
--R *
--R 4
--R x
--R +
--R 6 3 7 2 4 5 4 6 3 3
--R (- 1598A a b - 2305B a b )e + (7580A a b + 10153B a b )d e
--R +
--R 4 5 5 4 2 2 3 6 4 5 3
--R (- 13950A a b - 16755B a b )d e + (12000A a b + 12075B a b )d e
--R +
--R 2 7 3 6 4
--R (- 4200A a b - 3000B a b )d
--R *
--R 3
--R x
--R +
--R 7 2 8 4 6 3 7 2 3
--R (- 2002A a b - 1295B a b)e + (9220A a b + 5447B a b )d e
--R +
--R 5 4 6 3 2 2 4 5 5 4 3
--R (- 16290A a b - 8445B a b )d e + (13200A a b + 5565B a b )d e
--R +
--R 3 6 4 5 4
--R (- 4200A a b - 1200B a b )d
--R *
--R 2
--R x
--R +
--R 8 9 4 7 2 8 3
--R (- 1148A a b - 280B a )e + (5180A a b + 1078B a b)d e
--R +
--R 6 3 7 2 2 2 5 4 6 3 3
--R (- 8910A a b - 1455B a b )d e + (6960A a b + 735B a b )d e
--R +
--R 4 5 5 4 4
--R (- 2100A a b - 60B a b )d
--R *
--R x
--R +
--R 9 4 8 9 3 7 2 8 2 2
--R - 252A a e + (1120A a b - 28B a )d e + (- 1890A a b + 105B a b)d e
--R +
--R 6 3 7 2 3 5 4 6 3 4
--R (1440A a b - 135B a b )d e + (- 420A a b + 60B a b )d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 5 15 4 14 3 2 2 13 2 3 3 12
--R 2520a e - 12600a b d e + 25200a b d e - 25200a b d e
--R +
--R 4 4 11 5 5 10
--R 12600a b d e - 2520b d e
--R *
--R 10
--R x
--R +
--R 5 14 4 2 13 3 2 3 12 2 3 4 11
--R 25200a d e - 126000a b d e + 252000a b d e - 252000a b d e
--R +
--R 4 5 10 5 6 9
--R 126000a b d e - 25200b d e
--R *
--R 9
--R x
--R +
--R 5 2 13 4 3 12 3 2 4 11 2 3 5 10
--R 113400a d e - 567000a b d e + 1134000a b d e - 1134000a b d e
--R +
--R 4 6 9 5 7 8
--R 567000a b d e - 113400b d e
--R *
--R 8
--R x
--R +
--R 5 3 12 4 4 11 3 2 5 10 2 3 6 9
--R 302400a d e - 1512000a b d e + 3024000a b d e - 3024000a b d e
--R +
--R 4 7 8 5 8 7
--R 1512000a b d e - 302400b d e
--R *
--R 7
--R x
--R +
--R 5 4 11 4 5 10 3 2 6 9 2 3 7 8
--R 529200a d e - 2646000a b d e + 5292000a b d e - 5292000a b d e
--R +
--R 4 8 7 5 9 6
--R 2646000a b d e - 529200b d e
--R *
--R 6
--R x
--R +
--R 5 5 10 4 6 9 3 2 7 8 2 3 8 7
--R 635040a d e - 3175200a b d e + 6350400a b d e - 6350400a b d e
--R +
--R 4 9 6 5 10 5
--R 3175200a b d e - 635040b d e
--R *
--R 5
--R x
--R +
--R 5 6 9 4 7 8 3 2 8 7 2 3 9 6
--R 529200a d e - 2646000a b d e + 5292000a b d e - 5292000a b d e
--R +
--R 4 10 5 5 11 4
--R 2646000a b d e - 529200b d e
--R *
--R 4
--R x
--R +
--R 5 7 8 4 8 7 3 2 9 6 2 3 10 5
--R 302400a d e - 1512000a b d e + 3024000a b d e - 3024000a b d e
--R +
--R 4 11 4 5 12 3
--R 1512000a b d e - 302400b d e
--R *
--R 3
--R x
--R +
--R 5 8 7 4 9 6 3 2 10 5 2 3 11 4
--R 113400a d e - 567000a b d e + 1134000a b d e - 1134000a b d e
--R +
--R 4 12 3 5 13 2
--R 567000a b d e - 113400b d e
--R *
--R 2
--R x
--R +
--R 5 9 6 4 10 5 3 2 11 4 2 3 12 3
--R 25200a d e - 126000a b d e + 252000a b d e - 252000a b d e
--R +
--R 4 13 2 5 14
--R 126000a b d e - 25200b d e
--R *
--R x
--R +
--R 5 10 5 4 11 4 3 2 12 3 2 3 13 2
--R 2520a d e - 12600a b d e + 25200a b d e - 25200a b d e
--R +
--R 4 14 5 15
--R 12600a b d e - 2520b d
--R Type: Expression(Integer)
--E 383
--S 384 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 384
)clear all
--S 385 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^12
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 12 12 11 11 2 10 10 3 9 9 4 8 8 5 7 7
--R e x + 12d e x + 66d e x + 220d e x + 495d e x + 792d e x
--R +
--R 6 6 6 7 5 5 8 4 4 9 3 3 10 2 2 11 12
--R 924d e x + 792d e x + 495d e x + 220d e x + 66d e x + 12d e x + d
--R Type: Expression(Integer)
--E 385
--S 386 of 764
r0:=-1/198*b*(6*b*B*d+5*A*b*e-11*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e^3*(b*d-a*e)*(d+e*x)^9)-1/396*b^2*(6*b*B*d+5*A*b*e-_
11*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e^4*(b*d-a*e)*(d+e*x)^8)-_
1/11*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*_
(d+e*x)^11)-1/110*(6*b*B*d+5*A*b*e-11*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(e^2*(b*d-a*e)*(d+e*x)^10)-1/924*b^3*(6*b*B*d+5*A*b*e-_
11*a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(b*d-a*e)*_
(d+e*x)^7)-1/2310*b^4*(6*b*B*d+5*A*b*e-11*a*B*e)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(e^6*(b*d-a*e)*(d+e*x)^6)-1/13860*b^4*(6*b*B*d+5*A*b*e-_
11*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^7*(a+b*x)*(d+e*x)^6)
--R
--R
--R (2)
--R 5 6 6 5 4 6 5 5 5
--R - 2772B b e x + ((- 2310A b - 11550B a b )e - 2772B b d e )x
--R +
--R 4 2 3 6 5 4 5
--R (- 9900A a b - 19800B a b )e + (- 1650A b - 8250B a b )d e
--R +
--R 5 2 4
--R - 1980B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (- 17325A a b - 17325B a b )e + (- 4950A a b - 9900B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (- 825A b - 4125B a b )d e - 990B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (- 15400A a b - 7700B a b)e + (- 5775A a b - 5775B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (- 1650A a b - 3300B a b )d e + (- 275A b - 1375B a b )d e
--R +
--R 5 4 2
--R - 330B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (- 6930A a b - 1386B a )e + (- 3080A a b - 1540B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (- 1155A a b - 1155B a b )d e + (- 330A a b - 660B a b )d e
--R +
--R 5 4 4 2 5 5
--R (- 55A b - 275B a b )d e - 66B b d e
--R *
--R x
--R +
--R 5 6 4 5 5
--R - 1260A a e + (- 630A a b - 126B a )d e
--R +
--R 3 2 4 2 4 2 3 3 2 3 3
--R (- 280A a b - 140B a b)d e + (- 105A a b - 105B a b )d e
--R +
--R 4 2 3 4 2 5 4 5 5 6
--R (- 30A a b - 60B a b )d e + (- 5A b - 25B a b )d e - 6B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 18 12 18 17 11
--R 13860b e x + (13860a e + 152460b d e )x
--R +
--R 17 2 16 10 2 16 3 15 9
--R (152460a d e + 762300b d e )x + (762300a d e + 2286900b d e )x
--R +
--R 3 15 4 14 8 4 14 5 13 7
--R (2286900a d e + 4573800b d e )x + (4573800a d e + 6403320b d e )x
--R +
--R 5 13 6 12 6 6 12 7 11 5
--R (6403320a d e + 6403320b d e )x + (6403320a d e + 4573800b d e )x
--R +
--R 7 11 8 10 4 8 10 9 9 3
--R (4573800a d e + 2286900b d e )x + (2286900a d e + 762300b d e )x
--R +
--R 9 9 10 8 2 10 8 11 7
--R (762300a d e + 152460b d e )x + (152460a d e + 13860b d e )x
--R +
--R 11 7
--R 13860a d e
--R Type: Expression(Integer)
--E 386
--S 387 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 387
)clear all
--S 388 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^13
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 13 13 12 12 2 11 11 3 10 10 4 9 9 5 8 8
--R e x + 13d e x + 78d e x + 286d e x + 715d e x + 1287d e x
--R +
--R 6 7 7 7 6 6 8 5 5 9 4 4 10 3 3 11 2 2
--R 1716d e x + 1716d e x + 1287d e x + 715d e x + 286d e x + 78d e x
--R +
--R 12 13
--R 13d e x + d
--R Type: Expression(Integer)
--E 388
--S 389 of 764
r0:=-1/44*b*(b*B*d+A*b*e-2*a*B*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/_
(e^3*(b*d-a*e)*(d+e*x)^10)-1/99*b^2*(b*B*d+A*b*e-2*a*B*e)*(a^2+_
2*a*b*x+b^2*x^2)^(3/2)/(e^4*(b*d-a*e)*(d+e*x)^9)-1/12*(B*d-A*e)*_
(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*(d+e*x)^12)-_
1/22*(b*B*d+A*b*e-2*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e^2*(b*d-_
a*e)*(d+e*x)^11)-1/264*b^3*(b*B*d+A*b*e-2*a*B*e)*(a+b*x)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(e^5*(b*d-a*e)*(d+e*x)^8)-1/792*b^4*(b*B*d+A*b*e-_
2*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^6*(b*d-a*e)*(d+e*x)^7)-_
1/5544*b^4*(b*B*d+A*b*e-2*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^7*_
(a+b*x)*(d+e*x)^7)
--R
--R
--R (2)
--R 5 6 6 5 4 6 5 5 5
--R - 924B b e x + ((- 792A b - 3960B a b )e - 792B b d e )x
--R +
--R 4 2 3 6 5 4 5
--R (- 3465A a b - 6930B a b )e + (- 495A b - 2475B a b )d e
--R +
--R 5 2 4
--R - 495B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (- 6160A a b - 6160B a b )e + (- 1540A a b - 3080B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (- 220A b - 1100B a b )d e - 220B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (- 5544A a b - 2772B a b)e + (- 1848A a b - 1848B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (- 462A a b - 924B a b )d e + (- 66A b - 330B a b )d e
--R +
--R 5 4 2
--R - 66B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (- 2520A a b - 504B a )e + (- 1008A a b - 504B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (- 336A a b - 336B a b )d e + (- 84A a b - 168B a b )d e
--R +
--R 5 4 4 2 5 5
--R (- 12A b - 60B a b )d e - 12B b d e
--R *
--R x
--R +
--R 5 6 4 5 5 3 2 4 2 4
--R - 462A a e + (- 210A a b - 42B a )d e + (- 84A a b - 42B a b)d e
--R +
--R 2 3 3 2 3 3 4 2 3 4 2
--R (- 28A a b - 28B a b )d e + (- 7A a b - 14B a b )d e
--R +
--R 5 4 5 5 6
--R (- A b - 5B a b )d e - B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 19 13 19 18 12
--R 5544b e x + (5544a e + 66528b d e )x
--R +
--R 18 2 17 11 2 17 3 16 10
--R (66528a d e + 365904b d e )x + (365904a d e + 1219680b d e )x
--R +
--R 3 16 4 15 9 4 15 5 14 8
--R (1219680a d e + 2744280b d e )x + (2744280a d e + 4390848b d e )x
--R +
--R 5 14 6 13 7 6 13 7 12 6
--R (4390848a d e + 5122656b d e )x + (5122656a d e + 4390848b d e )x
--R +
--R 7 12 8 11 5 8 11 9 10 4
--R (4390848a d e + 2744280b d e )x + (2744280a d e + 1219680b d e )x
--R +
--R 9 10 10 9 3 10 9 11 8 2
--R (1219680a d e + 365904b d e )x + (365904a d e + 66528b d e )x
--R +
--R 11 8 12 7 12 7
--R (66528a d e + 5544b d e )x + 5544a d e
--R Type: Expression(Integer)
--E 389
--S 390 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 390
)clear all
--S 391 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^14
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 14 14 13 13 2 12 12 3 11 11 4 10 10
--R e x + 14d e x + 91d e x + 364d e x + 1001d e x
--R +
--R 5 9 9 6 8 8 7 7 7 8 6 6 9 5 5
--R 2002d e x + 3003d e x + 3432d e x + 3003d e x + 2002d e x
--R +
--R 10 4 4 11 3 3 12 2 2 13 14
--R 1001d e x + 364d e x + 91d e x + 14d e x + d
--R Type: Expression(Integer)
--E 391
--S 392 of 764
r0:=-5/1716*b*(6*b*B*d+7*A*b*e-13*a*B*e)*(a+b*x)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e^3*(b*d-a*e)*(d+e*x)^11)-1/858*b^2*(6*b*B*d+7*A*b*e-_
13*a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e^4*(b*d-a*e)*(d+e*x)^10)-_
1/13*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*_
(d+e*x)^13)-1/156*(6*b*B*d+7*A*b*e-13*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(e^2*(b*d-a*e)*(d+e*x)^12)-1/2574*b^3*(6*b*B*d+_
7*A*b*e-13*a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(b*d-_
a*e)*(d+e*x)^9)-1/9009*b^4*(6*b*B*d+7*A*b*e-13*a*B*e)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(e^6*(b*d-a*e)*(d+e*x)^8)-1/72072*b^4*(6*b*B*d+_
7*A*b*e-13*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^7*(a+b*x)*(d+e*x)^8)
--R
--R
--R (2)
--R 5 6 6 5 4 6 5 5 5
--R - 10296B b e x + ((- 9009A b - 45045B a b )e - 7722B b d e )x
--R +
--R 4 2 3 6 5 4 5
--R (- 40040A a b - 80080B a b )e + (- 5005A b - 25025B a b )d e
--R +
--R 5 2 4
--R - 4290B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (- 72072A a b - 72072B a b )e + (- 16016A a b - 32032B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (- 2002A b - 10010B a b )d e - 1716B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (- 65520A a b - 32760B a b)e + (- 19656A a b - 19656B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (- 4368A a b - 8736B a b )d e + (- 546A b - 2730B a b )d e
--R +
--R 5 4 2
--R - 468B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (- 30030A a b - 6006B a )e + (- 10920A a b - 5460B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (- 3276A a b - 3276B a b )d e + (- 728A a b - 1456B a b )d e
--R +
--R 5 4 4 2 5 5
--R (- 91A b - 455B a b )d e - 78B b d e
--R *
--R x
--R +
--R 5 6 4 5 5
--R - 5544A a e + (- 2310A a b - 462B a )d e
--R +
--R 3 2 4 2 4 2 3 3 2 3 3
--R (- 840A a b - 420B a b)d e + (- 252A a b - 252B a b )d e
--R +
--R 4 2 3 4 2 5 4 5 5 6
--R (- 56A a b - 112B a b )d e + (- 7A b - 35B a b )d e - 6B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 20 14 20 19 13
--R 72072b e x + (72072a e + 936936b d e )x
--R +
--R 19 2 18 12
--R (936936a d e + 5621616b d e )x
--R +
--R 2 18 3 17 11
--R (5621616a d e + 20612592b d e )x
--R +
--R 3 17 4 16 10
--R (20612592a d e + 51531480b d e )x
--R +
--R 4 16 5 15 9
--R (51531480a d e + 92756664b d e )x
--R +
--R 5 15 6 14 8
--R (92756664a d e + 123675552b d e )x
--R +
--R 6 14 7 13 7
--R (123675552a d e + 123675552b d e )x
--R +
--R 7 13 8 12 6
--R (123675552a d e + 92756664b d e )x
--R +
--R 8 12 9 11 5
--R (92756664a d e + 51531480b d e )x
--R +
--R 9 11 10 10 4
--R (51531480a d e + 20612592b d e )x
--R +
--R 10 10 11 9 3
--R (20612592a d e + 5621616b d e )x
--R +
--R 11 9 12 8 2 12 8 13 7
--R (5621616a d e + 936936b d e )x + (936936a d e + 72072b d e )x
--R +
--R 13 7
--R 72072a d e
--R Type: Expression(Integer)
--E 392
--S 393 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 393
)clear all
--S 394 of 764
t0:=(A+B*x)*(d+e*x)^3/sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x + A d
--R ----------------------------------------------------------------------------
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 394
--S 395 of 764
r0:=1/2*(A*b-a*B)*(b*d-a*e)*(a+b*x)*(d+e*x)^2/(b^3*sqrt(a^2+2*a*b*x+_
b^2*x^2))+1/3*(A*b-a*B)*(a+b*x)*(d+e*x)^3/(b^2*sqrt(a^2+2*a*b*x+_
b^2*x^2))+1/4*B*(a+b*x)*(d+e*x)^4/(b*e*sqrt(a^2+2*a*b*x+b^2*x^2))+_
(A*b-a*B)*(b*d-a*e)^3*(a+b*x)*log(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+_
b^2*x^2))+(A*b-a*B)*e*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5
--R
--R
--R (2)
--R 3 2 4 4 2 3 3 2 3
--R (- 12A a b + 12B a b)e + (36A a b - 36B a b )d e
--R +
--R 4 2 3 2 2 5 4 3
--R (- 36A a b + 36B a b )d e + (12A b - 12B a b )d e
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 12A a b + 12B a )e + (36A a b - 36B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (- 36A a b + 36B a b )d e + (12A a b - 12B a b )d e
--R *
--R log(b x + a)
--R +
--R 5 4 5 5 4 4 5 3 4
--R 3B b e x + ((4A b - B a b )e + 12B b d e )x
--R +
--R 4 2 3 4 5 4 3 5 2 2 3
--R ((- 2A a b + 2B a b )e + (18A b - 6B a b )d e + 18B b d e )x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (6A a b - 6B a b )e + (- 18A a b + 18B a b )d e
--R +
--R 5 4 2 2 5 3
--R (36A b - 18B a b )d e + 12B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (24A a b - 24B a b)e + (- 60A a b + 60B a b )d e
--R +
--R 4 2 3 2 2 5 4 3 5 4
--R (42A a b - 42B a b )d e + (10A b + 2B a b )d e + 3B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (12A a b - 12B a )e + (- 24A a b + 24B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3 4 4
--R (6A a b - 6B a b )d e + (10A a b - 10B a b )d e + 3B a b d
--R /
--R +------------------+
--R 5 | 2 2 2
--R 12b e\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 395
--S 396 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 396
)clear all
--S 397 of 764
t0:=(A+B*x)*(d+e*x)^2/sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R 2 3 2 2 2 2
--R B e x + (A e + 2B d e)x + (2A d e + B d )x + A d
--R (1) ----------------------------------------------------
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 397
--S 398 of 764
r0:=1/2*(A*b-a*B)*(a+b*x)*(d+e*x)^2/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+_
1/3*B*(a+b*x)*(d+e*x)^3/(b*e*sqrt(a^2+2*a*b*x+b^2*x^2))+(A*b-a*B)*_
(b*d-a*e)^2*(a+b*x)*log(a+b*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))+_
(A*b-a*B)*e*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/b^4
--R
--R
--R (2)
--R 2 2 3 3 3 2 2 2
--R (6A a b - 6B a b)e + (- 12A a b + 12B a b )d e
--R +
--R 4 3 2
--R (6A b - 6B a b )d e
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (6A a b - 6B a )e + (- 12A a b + 12B a b)d e
--R +
--R 3 2 2 2
--R (6A a b - 6B a b )d e
--R *
--R log(b x + a)
--R +
--R 4 3 4 4 3 3 4 2 3
--R 2B b e x + ((3A b - B a b )e + 6B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 2
--R ((- 3A a b + 3B a b )e + (12A b - 6B a b )d e + 6B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (- 12A a b + 12B a b)e + (18A a b - 18B a b )d e
--R +
--R 4 3 2 4 3
--R (3A b + 3B a b )d e + 2B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2 3 2 2 2
--R (- 6A a b + 6B a )e + (6A a b - 6B a b)d e + (3A a b - 3B a b )d e
--R +
--R 3 3
--R 2B a b d
--R /
--R +------------------+
--R 4 | 2 2 2
--R 6b e\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 398
--S 399 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 399
)clear all
--S 400 of 764
t0:=(A+B*x)*(d+e*x)/sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R 2
--R B e x + (A e + B d)x + A d
--R (1) ---------------------------
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 400
--S 401 of 764
r0:=1/2*B*(a+b*x)*(d+e*x)^2/(b*e*sqrt(a^2+2*a*b*x+b^2*x^2))+(A*b-a*B)*_
(b*d-a*e)*(a+b*x)*log(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))+(A*b-_
a*B)*e*sqrt(a^2+2*a*b*x+b^2*x^2)/b^3
--R
--R
--R (2)
--R 2 2 2 3 2
--R ((- 2A a b + 2B a b)e + (2A b - 2B a b )d e)x
--R +
--R 2 3 2 2 2
--R (- 2A a b + 2B a )e + (2A a b - 2B a b)d e
--R *
--R log(b x + a)
--R +
--R 3 2 3 3 2 2 3 2
--R B b e x + ((2A b - B a b )e + 2B b d e)x
--R +
--R 2 2 2 2 3 2 2 3 2
--R ((4A a b - 4B a b)e + 2B a b d e + B b d )x + (2A a b - 2B a )e
--R +
--R 2 2
--R B a b d
--R /
--R +------------------+
--R 3 | 2 2 2
--R 2b e\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 401
--S 402 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 402
)clear all
--S 403 of 764
t0:=(A+B*x)/sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R B x + A
--R (1) ---------------------
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 403
--S 404 of 764
r0:=(A*b-a*B)*(a+b*x)*log(a+b*x)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+_
B*sqrt(a^2+2*a*b*x+b^2*x^2)/b^2
--R
--R
--R (2)
--R 2 2 2 2 2
--R ((A b - B a b)x + A a b - B a )log(b x + a) + B b x + 2B a b x + B a
--R -----------------------------------------------------------------------
--R +------------------+
--R 2 | 2 2 2
--R b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 404
--S 405 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 405
)clear all
--S 406 of 764
t0:=(A+B*x)/((d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R B x + A
--R (1) ------------------------------
--R +------------------+
--R | 2 2 2
--R (e x + d)\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 406
--S 407 of 764
r0:=(A*b-a*B)*(a+b*x)*log(a+b*x)/(b*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))+_
(B*d-A*e)*(a+b*x)*log(d+e*x)/(e*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 2 2
--R ((A b e - B b d)x + A a b e - B a b d)log(e x + d)
--R +
--R 2 2
--R ((- A b + B a b)e x + (- A a b + B a )e)log(b x + a)
--R /
--R +------------------+
--R 2 2 | 2 2 2
--R (a b e - b d e)\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 407
--S 408 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 408
)clear all
--S 409 of 764
t0:=(A+B*x)/((d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R B x + A
--R (1) -----------------------------------------
--R +------------------+
--R 2 2 2 | 2 2 2
--R (e x + 2d e x + d )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 409
--S 410 of 764
r0:=-(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))+_
(A*b-a*B)*(a+b*x)*log(a+b*x)/((b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-_
(A*b-a*B)*(a+b*x)*log(d+e*x)/((b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 2 2 2 2 2 2
--R (- A b + B a b)e x + ((- A a b + B a )e + (- A b + B a b)d e)x
--R +
--R 2
--R (- A a b + B a )d e
--R *
--R log(e x + d)
--R +
--R 2 2 2 2 2 2
--R (A b - B a b)e x + ((A a b - B a )e + (A b - B a b)d e)x
--R +
--R 2
--R (A a b - B a )d e
--R *
--R log(b x + a)
--R +
--R 2 2 2 2 2 2 2
--R (- A a b e + (A b + B a b)d e - B b d )x - A a e + (A a b + B a )d e
--R +
--R 2
--R - B a b d
--R /
--R 2 4 3 2 2 2 2 3 2 2 2 3
--R ((a e - 2a b d e + b d e )x + a d e - 2a b d e + b d e)
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 410
--S 411 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 411
)clear all
--S 412 of 764
t0:=(A+B*x)/((d+e*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R B x + A
--R (1) ---------------------------------------------------
--R +------------------+
--R 3 3 2 2 2 3 | 2 2 2
--R (e x + 3d e x + 3d e x + d )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 412
--S 413 of 764
r0:=-1/2*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+_
b*(A*b-a*B)*(a+b*x)*log(a+b*x)/((b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-_
b*(A*b-a*B)*(a+b*x)*log(d+e*x)/((b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))-_
(A*b-a*B)*e*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)^3*(d+e*x))
--R
--R
--R (2)
--R 3 2 3 3
--R (2A b - 2B a b )e x
--R +
--R 2 2 3 3 2 2 2
--R ((2A a b - 2B a b)e + (4A b - 4B a b )d e )x
--R +
--R 2 2 2 3 2 2
--R ((4A a b - 4B a b)d e + (2A b - 2B a b )d e)x
--R +
--R 2 2 2
--R (2A a b - 2B a b)d e
--R *
--R log(e x + d)
--R +
--R 3 2 3 3
--R (- 2A b + 2B a b )e x
--R +
--R 2 2 3 3 2 2 2
--R ((- 2A a b + 2B a b)e + (- 4A b + 4B a b )d e )x
--R +
--R 2 2 2 3 2 2
--R ((- 4A a b + 4B a b)d e + (- 2A b + 2B a b )d e)x
--R +
--R 2 2 2
--R (- 2A a b + 2B a b)d e
--R *
--R log(b x + a)
--R +
--R 3 2 3 3 2 2 3 3 2 2 2
--R (2A b - 2B a b )e x + ((4A a b - 4B a b)e + (2A b - 2B a b )d e )x
--R +
--R 2 3 3 2 2 2 3 2 2
--R (A a b - 2B a )e + (6A a b - 3B a b)d e + (- A b - 2B a b )d e
--R +
--R 3 3
--R B b d
--R *
--R x
--R +
--R 3 3 2 3 2 2 2 2 2 3
--R - A a e + (4A a b - B a )d e + (- A a b - 2B a b)d e + B a b d
--R /
--R 3 6 2 5 2 2 4 3 3 3 2
--R (2a e - 6a b d e + 6a b d e - 2b d e )x
--R +
--R 3 5 2 2 4 2 3 3 3 4 2 3 2 4 2 3 3
--R (4a d e - 12a b d e + 12a b d e - 4b d e )x + 2a d e - 6a b d e
--R +
--R 2 4 2 3 5
--R 6a b d e - 2b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 413
--S 414 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 414
)clear all
--S 415 of 764
t0:=(A+B*x)/((d+e*x)^4*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R B x + A
--R (1) -------------------------------------------------------------
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (e x + 4d e x + 6d e x + 4d e x + d )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 415
--S 416 of 764
r0:=-1/3*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+_
1/2*(A*b-a*B)*(a+b*x)/((b*d-a*e)^2*(d+e*x)^2*sqrt(a^2+2*a*b*x+b^2*x^2))+_
b^2*(A*b-a*B)*(a+b*x)*log(a+b*x)/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))-_
b^2*(A*b-a*B)*(a+b*x)*log(d+e*x)/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))-_
b*(A*b-a*B)*e*sqrt(a^2+2*a*b*x+b^2*x^2)/((b*d-a*e)^4*(d+e*x))
--R
--R
--R (2)
--R 4 3 4 4
--R (- 6A b + 6B a b )e x
--R +
--R 3 2 2 4 4 3 3 3
--R ((- 6A a b + 6B a b )e + (- 18A b + 18B a b )d e )x
--R +
--R 3 2 2 3 4 3 2 2 2
--R ((- 18A a b + 18B a b )d e + (- 18A b + 18B a b )d e )x
--R +
--R 3 2 2 2 2 4 3 3
--R ((- 18A a b + 18B a b )d e + (- 6A b + 6B a b )d e)x
--R +
--R 3 2 2 3
--R (- 6A a b + 6B a b )d e
--R *
--R log(e x + d)
--R +
--R 4 3 4 4
--R (6A b - 6B a b )e x
--R +
--R 3 2 2 4 4 3 3 3
--R ((6A a b - 6B a b )e + (18A b - 18B a b )d e )x
--R +
--R 3 2 2 3 4 3 2 2 2
--R ((18A a b - 18B a b )d e + (18A b - 18B a b )d e )x
--R +
--R 3 2 2 2 2 4 3 3
--R ((18A a b - 18B a b )d e + (6A b - 6B a b )d e)x
--R +
--R 3 2 2 3
--R (6A a b - 6B a b )d e
--R *
--R log(b x + a)
--R +
--R 4 3 4 4
--R (- 6A b + 6B a b )e x
--R +
--R 3 2 2 4 4 3 3 3
--R ((- 12A a b + 12B a b )e + (- 12A b + 12B a b )d e )x
--R +
--R 2 2 3 4 3 2 2 3
--R (- 3A a b + 3B a b)e + (- 30A a b + 30B a b )d e
--R +
--R 4 3 2 2
--R (- 3A b + 3B a b )d e
--R *
--R 2
--R x
--R +
--R 3 4 4 2 2 3 3
--R (A a b - 3B a )e + (- 9A a b + 17B a b)d e
--R +
--R 3 2 2 2 2 4 3 3 4 4
--R (- 21A a b + 9B a b )d e + (5A b + 3B a b )d e - 2B b d
--R *
--R x
--R +
--R 4 4 3 4 3 2 2 3 2 2
--R - 2A a e + (9A a b - B a )d e + (- 18A a b + 6B a b)d e
--R +
--R 3 2 2 3 3 4
--R (5A a b + 3B a b )d e - 2B a b d
--R /
--R 4 8 3 7 2 2 2 6 3 3 5 4 4 4 3
--R (6a e - 24a b d e + 36a b d e - 24a b d e + 6b d e )x
--R +
--R 4 7 3 2 6 2 2 3 5 3 4 4 4 5 3 2
--R (18a d e - 72a b d e + 108a b d e - 72a b d e + 18b d e )x
--R +
--R 4 2 6 3 3 5 2 2 4 4 3 5 3 4 6 2
--R (18a d e - 72a b d e + 108a b d e - 72a b d e + 18b d e )x
--R +
--R 4 3 5 3 4 4 2 2 5 3 3 6 2 4 7
--R 6a d e - 24a b d e + 36a b d e - 24a b d e + 6b d e
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 416
--S 417 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 417
)clear all
--S 418 of 764
t0:=(A+B*x)*(d+e*x)^4/(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B e x + (A e + 4B d e )x + (4A d e + 6B d e )x
--R +
--R 2 2 3 2 3 4 4
--R (6A d e + 4B d e)x + (4A d e + B d )x + A d
--R /
--R +------------------+
--R 2 2 2 | 2 2 2
--R (b x + 2a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 418
--S 419 of 764
r0:=-1/6*(2*b*B*d+3*A*b*e-5*a*B*e)*(a+b*x)*(d+e*x)^4/(b^2*e*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+1/3*B*(a+b*x)*(d+e*x)^5/(b*e*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+e*(2*b*B*d+3*A*b*e-5*a*B*e)*(a+b*x)*(d+e*x)^2/(b^4*_
sqrt(a^2+2*a*b*x+b^2*x^2))-2/3*(2*b*B*d+3*A*b*e-5*a*B*e)*(d+e*x)^3/_
(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))+2*e*(b*d-a*e)^2*(2*b*B*d+3*A*b*e-_
5*a*B*e)*(a+b*x)*log(a+b*x)/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))+_
2*e^2*(b*d-a*e)*(2*b*B*d+3*A*b*e-5*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/b^6
--R
--R
--R (2)
--R 2 3 3 2 4 4 2 3 3
--R (36A a b - 60B a b )e + (- 72A a b + 144B a b )d e
--R +
--R 5 4 2 2 5 3
--R (36A b - 108B a b )d e + 24B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (72A a b - 120B a b)e + (- 144A a b + 288B a b )d e
--R +
--R 4 2 3 2 2 4 3
--R (72A a b - 216B a b )d e + 48B a b d e
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (36A a b - 60B a )e + (- 72A a b + 144B a b)d e
--R +
--R 2 3 3 2 2 2 2 3 3
--R (36A a b - 108B a b )d e + 24B a b d e
--R *
--R log(b x + a)
--R +
--R 5 4 5 5 4 4 5 3 4
--R 2B b e x + ((3A b - 5B a b )e + 12B b d e )x
--R +
--R 4 2 3 4 5 4 3 5 2 2 3
--R ((- 12A a b + 20B a b )e + (24A b - 48B a b )d e + 36B b d e )x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (- 90A a b + 150B a b )e + (144A a b - 300B a b )d e
--R +
--R 5 4 2 2 5 3
--R (- 36A b + 156B a b )d e - 4B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (- 108A a b + 180B a b)e + (144A a b - 312B a b )d e
--R +
--R 2 3 2 2 5 4 3 5 4
--R 96B a b d e + (- 24A b + 40B a b )d e - 6B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 36A a b + 60B a )e + (36A a b - 84B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (18A a b - 6B a b )d e + (- 12A a b + 32B a b )d e
--R +
--R 5 4 4
--R (- 3A b - 3B a b )d
--R /
--R +------------------+
--R 7 6 | 2 2 2
--R (6b x + 6a b )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 419
--S 420 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 420
)clear all
--S 421 of 764
t0:=(A+B*x)*(d+e*x)^3/(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x + A d
--R ----------------------------------------------------------------------------
--R +------------------+
--R 2 2 2 | 2 2 2
--R (b x + 2a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 421
--S 422 of 764
r0:=-1/2*(b*B*d+A*b*e-2*a*B*e)*(a+b*x)*(d+e*x)^3/(b^2*e*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+1/2*B*(a+b*x)*(d+e*x)^4/(b*e*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))-3/2*(b*B*d+A*b*e-2*a*B*e)*(d+e*x)^2/(b^3*sqrt(a^2+_
2*a*b*x+b^2*x^2))+3*e*(b*d-a*e)*(b*B*d+A*b*e-2*a*B*e)*(a+b*x)*_
log(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))+3*e^2*(b*B*d+A*b*e-_
2*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/b^5
--R
--R
--R (2)
--R 3 2 2 3 4 3 2 4 2 2
--R ((- 6A a b + 12B a b )e + (6A b - 18B a b )d e + 6B b d e)x
--R +
--R 2 2 3 3 3 2 2 2 3 2
--R ((- 12A a b + 24B a b)e + (12A a b - 36B a b )d e + 12B a b d e)x
--R +
--R 3 4 3 2 2 3 2 2 2 2
--R (- 6A a b + 12B a )e + (6A a b - 18B a b)d e + 6B a b d e
--R *
--R log(b x + a)
--R +
--R 4 3 4 4 3 3 4 2 3
--R B b e x + ((2A b - 4B a b )e + 6B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 2
--R ((15A a b - 30B a b )e + (- 9A b + 33B a b )d e - 3B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (18A a b - 36B a b)e + (- 6A a b + 30B a b )d e
--R +
--R 4 3 2 4 3
--R (- 6A b + 6B a b )d e - 2B b d
--R *
--R x
--R +
--R 3 4 3 3 2 3 2 2 2
--R (6A a b - 12B a )e + 6B a b d e + (- 3A a b + 6B a b )d e
--R +
--R 4 3 3
--R (- A b - B a b )d
--R /
--R +------------------+
--R 6 5 | 2 2 2
--R (2b x + 2a b )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 422
--S 423 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 423
)clear all
--S 424 of 764
t0:=(A+B*x)*(d+e*x)^2/(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R 2 3 2 2 2 2
--R B e x + (A e + 2B d e)x + (2A d e + B d )x + A d
--R (1) ----------------------------------------------------
--R +------------------+
--R 2 2 2 | 2 2 2
--R (b x + 2a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 424
--S 425 of 764
r0:=-1/2*(2*b*B*d+A*b*e-3*a*B*e)*(a+b*x)*(d+e*x)^2/(b^2*e*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+B*(a+b*x)*(d+e*x)^3/(b*e*(a^2+2*a*b*x+b^2*x^2)^(3/2))-_
(2*b*B*d+A*b*e-3*a*B*e)*(d+e*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))+_
e*(2*b*B*d+A*b*e-3*a*B*e)*(a+b*x)*log(a+b*x)/(b^4*sqrt(a^2+2*a*b*x+_
b^2*x^2))
--R
--R
--R (2)
--R 3 2 2 3 2
--R ((2A b - 6B a b )e + 4B b d e)x
--R +
--R 2 2 2 2 2 3 2 2
--R ((4A a b - 12B a b)e + 8B a b d e)x + (2A a b - 6B a )e + 4B a b d e
--R *
--R log(b x + a)
--R +
--R 3 2 3 3 2 2 2
--R 2B b e x + (- 3A b + 9B a b )e x
--R +
--R 2 2 2 3 2 3 2
--R ((- 2A a b + 6B a b)e + (- 4A b + 8B a b )d e - 2B b d )x
--R +
--R 2 2 3 2 2
--R (- 2A a b + 6B a b)d e + (- A b - B a b )d
--R /
--R +------------------+
--R 5 4 | 2 2 2
--R (2b x + 2a b )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 425
--S 426 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 426
)clear all
--S 427 of 764
t0:=(A+B*x)*(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R 2
--R B e x + (A e + B d)x + A d
--R (1) -----------------------------------------
--R +------------------+
--R 2 2 2 | 2 2 2
--R (b x + 2a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 427
--S 428 of 764
r0:=-1/2*(A*b-a*B)*(a+b*x)*(d+e*x)^2/(b*(b*d-a*e)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))-B*(b*d-a*e)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))+_
B*e*(a+b*x)*log(a+b*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2)),-1/2*B*(a+_
b*x)*(d+e*x)^2/(b*e*(a^2+2*a*b*x+b^2*x^2)^(3/2))+1/2*(B*d-A*e)*(a+_
b*x)*(d+e*x)^2/(e*(b*d-a*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2))-B*(d+e*x)/_
(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+B*e*(a+b*x)*log(a+b*x)/(b^3*_
sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R [
--R 2 2 3 2 2 2 2 3 2
--R (2B a b e - 2B b d e)x + (4B a b e - 4B a b d e)x + 2B a e
--R +
--R 2
--R - 2B a b d e
--R *
--R log(b x + a)
--R +
--R 3 2 2 2 2 2 3 2 3 2
--R (A b - B a b )e x + (2B a b e + (2A b - 6B a b )d e + 2B b d )x
--R +
--R 3 2 2 3 2 2
--R 2B a e - 4B a b d e + (A b + B a b )d
--R /
--R +------------------+
--R 4 5 2 3 4 | 2 2 2
--R ((2a b e - 2b d)x + 2a b e - 2a b d)\|b x + 2a b x + a
--R ,
--R
--R 2 2 3 2 2 2 2 3 2
--R (2B a b e - 2B b d e)x + (4B a b e - 4B a b d e)x + 2B a e
--R +
--R 2
--R - 2B a b d e
--R *
--R log(b x + a)
--R +
--R 3 2 2 3 2
--R ((A b - 3B a b )e + 2B b d e)x
--R +
--R 2 2 3 2 3 2 2
--R (- 2B a b e + (2A b - 2B a b )d e + 2B b d )x - 2B a b d e
--R +
--R 3 2 2
--R (A b + B a b )d
--R /
--R +------------------+
--R 4 5 2 3 4 | 2 2 2
--R ((2a b e - 2b d)x + 2a b e - 2a b d)\|b x + 2a b x + a
--R ]
--R Type: Tuple(Expression(Integer))
--E 428
--S 429 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 429
)clear all
--S 430 of 764
t0:=(A+B*x)/(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R B x + A
--R (1) -----------------------------------------
--R +------------------+
--R 2 2 2 | 2 2 2
--R (b x + 2a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 430
--S 431 of 764
r0:=-1/2*(a+b*x)*(A+B*x)^2/((A*b-a*B)*(a^2+2*a*b*x+b^2*x^2)^(3/2))
--R
--R
--R 2 2 2
--R - B x - 2A B x - A
--R (2) ---------------------------------------------------------
--R +------------------+
--R 2 2 | 2 2 2
--R ((2A b - 2B a b)x + 2A a b - 2B a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 431
--S 432 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 432
)clear all
--S 433 of 764
t0:=(A+B*x)/((d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2))
--R
--R
--R (1)
--R B x + A
--R ------------------------------------------------------------------------
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (b e x + (2a b e + b d)x + (a e + 2a b d)x + a d)\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 433
--S 434 of 764
r0:=-1/2*(A*b-a*B)*(a+b*x)/(b*(b*d-a*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2))+_
(-B*d+A*e)/((b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2))-e*(B*d-A*e)*_
(a+b*x)*log(a+b*x)/((b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))+_
e*(B*d-A*e)*(a+b*x)*log(d+e*x)/((b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 3 2 3 2 2 2 2 2 2
--R (2A b e - 2B b d e)x + (4A a b e - 4B a b d e)x + 2A a b e
--R +
--R 2
--R - 2B a b d e
--R *
--R log(e x + d)
--R +
--R 3 2 3 2 2 2 2 2 2
--R (- 2A b e + 2B b d e)x + (- 4A a b e + 4B a b d e)x - 2A a b e
--R +
--R 2
--R 2B a b d e
--R *
--R log(b x + a)
--R +
--R 2 2 3 2 3 2 2 3 2
--R (2A a b e + (- 2A b - 2B a b )d e + 2B b d )x + (3A a b - B a )e
--R +
--R 2 3 2 2
--R - 4A a b d e + (A b + B a b )d
--R /
--R 3 2 3 2 3 2 4 2 5 3 4 3 3 2 2
--R (2a b e - 6a b d e + 6a b d e - 2b d )x + 2a b e - 6a b d e
--R +
--R 2 3 2 4 3
--R 6a b d e - 2a b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 434
--S 435 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 435
)clear all
--S 436 of 764
t0:=(A+B*x)/((d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(3/2))
--R
--R
--R (1)
--R B x + A
--R /
--R 2 2 4 2 2 3 2 2 2 2 2
--R b e x + (2a b e + 2b d e)x + (a e + 4a b d e + b d )x
--R +
--R 2 2 2 2
--R (2a d e + 2a b d )x + a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 436
--S 437 of 764
r0:=1/2*(2*b*B*d-3*A*b*e+a*B*e)*(a+b*x)/(e*(b*d-a*e)^2*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))-(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+(-2*b*B*d+3*A*b*e-a*B*e)/((b*d-a*e)^3*sqrt(a^2+_
2*a*b*x+b^2*x^2))-e*(2*b*B*d-3*A*b*e+a*B*e)*(a+b*x)*log(a+b*x)/_
((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))+e*(2*b*B*d-3*A*b*e+a*B*e)*_
(a+b*x)*log(d+e*x)/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 3 2 3 3 2 3
--R ((- 6A b + 2B a b )e + 4B b d e )x
--R +
--R 2 2 3 3 2 2 3 2 2
--R ((- 12A a b + 4B a b)e + (- 6A b + 10B a b )d e + 4B b d e)x
--R +
--R 2 3 3 2 2 2 2 2
--R ((- 6A a b + 2B a )e + (- 12A a b + 8B a b)d e + 8B a b d e)x
--R +
--R 2 3 2 2 2
--R (- 6A a b + 2B a )d e + 4B a b d e
--R *
--R log(e x + d)
--R +
--R 3 2 3 3 2 3
--R ((6A b - 2B a b )e - 4B b d e )x
--R +
--R 2 2 3 3 2 2 3 2 2
--R ((12A a b - 4B a b)e + (6A b - 10B a b )d e - 4B b d e)x
--R +
--R 2 3 3 2 2 2 2 2
--R ((6A a b - 2B a )e + (12A a b - 8B a b)d e - 8B a b d e)x
--R +
--R 2 3 2 2 2
--R (6A a b - 2B a )d e - 4B a b d e
--R *
--R log(b x + a)
--R +
--R 2 2 3 3 2 2 3 2 2
--R ((- 6A a b + 2B a b)e + (6A b + 2B a b )d e - 4B b d e)x
--R +
--R 2 3 3 2 2 2 3 2 2
--R (- 9A a b + 3B a )e + (6A a b + 4B a b)d e + (3A b - 5B a b )d e
--R +
--R 3 3
--R - 2B b d
--R *
--R x
--R +
--R 3 3 2 3 2 2 2 2
--R - 2A a e + (- 3A a b + 5B a )d e + (6A a b - 4B a b)d e
--R +
--R 3 2 3
--R (- A b - B a b )d
--R /
--R 4 5 3 2 4 2 3 2 3 4 3 2 5 4 2
--R (2a b e - 8a b d e + 12a b d e - 8a b d e + 2b d e)x
--R +
--R 5 5 4 4 3 2 2 3 2 3 3 2 4 4 5 5
--R (2a e - 6a b d e + 4a b d e + 4a b d e - 6a b d e + 2b d )x
--R +
--R 5 4 4 2 3 3 2 3 2 2 3 4 4 5
--R 2a d e - 8a b d e + 12a b d e - 8a b d e + 2a b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 437
--S 438 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 438
)clear all
--S 439 of 764
t0:=(A+B*x)/((d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^(3/2))
--R
--R
--R (1)
--R B x + A
--R /
--R 2 3 5 3 2 2 4 2 3 2 2 2 3
--R b e x + (2a b e + 3b d e )x + (a e + 6a b d e + 3b d e)x
--R +
--R 2 2 2 2 3 2 2 2 3 2 3
--R (3a d e + 6a b d e + b d )x + (3a d e + 2a b d )x + a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 439
--S 440 of 764
r0:=-1/2*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)^2*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+1/2*(b*B*d-2*A*b*e+a*B*e)*(a+b*x)/(e*(b*d-a*e)^2*_
(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2))-3/2*(b*B*d-2*A*b*e+a*B*e)/_
((b*d-a*e)^3*(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-3*b*e*(b*B*d-_
2*A*b*e+a*B*e)*(a+b*x)*log(a+b*x)/((b*d-a*e)^5*sqrt(a^2+2*a*b*x+_
b^2*x^2))+3*b*e*(b*B*d-2*A*b*e+a*B*e)*(a+b*x)*log(d+e*x)/((b*d-a*e)^5*_
sqrt(a^2+2*a*b*x+b^2*x^2))+3*e^2*(b*B*d-2*A*b*e+a*B*e)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/((b*d-a*e)^5*(d+e*x))
--R
--R
--R (2)
--R 4 3 4 4 3 4
--R ((12A b - 6B a b )e - 6B b d e )x
--R +
--R 3 2 2 4 4 3 3 4 2 2 3
--R ((24A a b - 12B a b )e + (24A b - 24B a b )d e - 12B b d e )x
--R +
--R 2 2 3 4 3 2 2 3
--R (12A a b - 6B a b)e + (48A a b - 30B a b )d e
--R +
--R 4 3 2 2 4 3
--R (12A b - 30B a b )d e - 6B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 3 3 2 2 2 2 3 3
--R ((24A a b - 12B a b)d e + (24A a b - 24B a b )d e - 12B a b d e)x
--R +
--R 2 2 3 2 2 2 2 3
--R (12A a b - 6B a b)d e - 6B a b d e
--R *
--R log(e x + d)
--R +
--R 4 3 4 4 3 4
--R ((- 12A b + 6B a b )e + 6B b d e )x
--R +
--R 3 2 2 4 4 3 3
--R (- 24A a b + 12B a b )e + (- 24A b + 24B a b )d e
--R +
--R 4 2 2
--R 12B b d e
--R *
--R 3
--R x
--R +
--R 2 2 3 4 3 2 2 3
--R (- 12A a b + 6B a b)e + (- 48A a b + 30B a b )d e
--R +
--R 4 3 2 2 4 3
--R (- 12A b + 30B a b )d e + 6B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 3 3 2 2 2 2
--R (- 24A a b + 12B a b)d e + (- 24A a b + 24B a b )d e
--R +
--R 3 3
--R 12B a b d e
--R *
--R x
--R +
--R 2 2 3 2 2 2 2 3
--R (- 12A a b + 6B a b)d e + 6B a b d e
--R *
--R log(b x + a)
--R +
--R 4 3 4 4 3 4
--R ((12A b - 6B a b )e - 6B b d e )x
--R +
--R 3 2 2 4 4 3 3 4 2 2 3
--R ((36A a b - 18B a b )e + (12A b - 24B a b )d e - 6B b d e )x
--R +
--R 2 2 3 4 3 2 2 3
--R (30A a b - 15B a b)e + (48A a b - 39B a b )d e
--R +
--R 4 3 2 2 4 3
--R (- 6A b - 21B a b )d e + 3B b d e
--R *
--R 2
--R x
--R +
--R 3 4 4 2 2 3 3 2 2 2 2
--R (4A a b - 2B a )e + (48A a b - 26B a b)d e - 24B a b d e
--R +
--R 4 3 3 4 4
--R (- 4A b + 2B a b )d e + 2B b d
--R *
--R x
--R +
--R 4 4 3 4 3 2 2 3 2 2
--R - A a e + (8A a b - B a )d e + (12A a b - 15B a b)d e
--R +
--R 3 2 2 3 4 3 4
--R (- 8A a b + 3B a b )d e + (A b + B a b )d
--R /
--R 5 7 4 2 6 3 3 2 5 2 4 3 4 5 4 3
--R 2a b e - 10a b d e + 20a b d e - 20a b d e + 10a b d e
--R +
--R 6 5 2
--R - 2b d e
--R *
--R 3
--R x
--R +
--R 6 7 5 6 3 3 3 4 2 4 4 3 5 5 2 6 6 2
--R (2a e - 6a b d e + 20a b d e - 30a b d e + 18a b d e - 4b d e)x
--R +
--R 6 6 5 2 5 4 2 3 4 3 3 4 3 5 6 6 7
--R (4a d e - 18a b d e + 30a b d e - 20a b d e + 6a b d e - 2b d )x
--R +
--R 6 2 5 5 3 4 4 2 4 3 3 3 5 2 2 4 6 5 7
--R 2a d e - 10a b d e + 20a b d e - 20a b d e + 10a b d e - 2a b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 440
--S 441 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 441
)clear all
--S 442 of 764
t0:=(A+B*x)*(d+e*x)^5/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 5 6 5 4 5 4 2 3 4
--R B e x + (A e + 5B d e )x + (5A d e + 10B d e )x
--R +
--R 2 3 3 2 3 3 2 4 2 4 5 5
--R (10A d e + 10B d e )x + (10A d e + 5B d e)x + (5A d e + B d )x + A d
--R /
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 442
--S 443 of 764
r0:=-1/4*(2*b*B*d+A*b*e-3*a*B*e)*(a+b*x)*(d+e*x)^5/(b^2*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))+1/2*B*(a+b*x)*(d+e*x)^6/(b*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))-5/6*e*(2*b*B*d+A*b*e-3*a*B*e)*(a+b*x)*(d+e*x)^3/_
(b^4*(a^2+2*a*b*x+b^2*x^2)^(3/2))-5/12*(2*b*B*d+A*b*e-3*a*B*e)*_
(d+e*x)^4/(b^3*(a^2+2*a*b*x+b^2*x^2)^(3/2))-5/2*e^2*(2*b*B*d+A*b*e-_
3*a*B*e)*(d+e*x)^2/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))+5*e^3*(b*d-a*e)*_
(2*b*B*d+A*b*e-3*a*B*e)*(a+b*x)*log(a+b*x)/(b^7*sqrt(a^2+2*a*b*x+_
b^2*x^2))+5*e^4*(2*b*B*d+A*b*e-3*a*B*e)*sqrt(a^2+2*a*b*x+b^2*x^2)/b^7
--R
--R
--R (2)
--R 5 2 4 5 6 5 4
--R (- 60A a b + 180B a b )e + (60A b - 300B a b )d e
--R +
--R 6 2 3
--R 120B b d e
--R *
--R 4
--R x
--R +
--R 2 4 3 3 5 5 2 4 4
--R (- 240A a b + 720B a b )e + (240A a b - 1200B a b )d e
--R +
--R 5 2 3
--R 480B a b d e
--R *
--R 3
--R x
--R +
--R 3 3 4 2 5 2 4 3 3 4
--R (- 360A a b + 1080B a b )e + (360A a b - 1800B a b )d e
--R +
--R 2 4 2 3
--R 720B a b d e
--R *
--R 2
--R x
--R +
--R 4 2 5 5 3 3 4 2 4
--R (- 240A a b + 720B a b)e + (240A a b - 1200B a b )d e
--R +
--R 3 3 2 3
--R 480B a b d e
--R *
--R x
--R +
--R 5 6 5 4 2 5 4 4 2 2 3
--R (- 60A a b + 180B a )e + (60A a b - 300B a b)d e + 120B a b d e
--R *
--R log(b x + a)
--R +
--R 6 5 6 6 5 5 6 4 5
--R 6B b e x + ((12A b - 36B a b )e + 60B b d e )x
--R +
--R 5 2 4 5 6 5 4 6 2 3 4
--R ((185A a b - 555B a b )e + (- 125A b + 745B a b )d e - 160B b d e )x
--R +
--R 2 4 3 3 5 5 2 4 4
--R (500A a b - 1500B a b )e + (- 260A a b + 1780B a b )d e
--R +
--R 6 5 2 3 6 3 2
--R (- 120A b - 160B a b )d e - 120B b d e
--R *
--R 3
--R x
--R +
--R 3 3 4 2 5 2 4 3 3 4
--R (570A a b - 1710B a b )e + (- 210A a b + 1770B a b )d e
--R +
--R 5 2 4 2 3 6 5 3 2 6 4
--R (- 180A a b + 120B a b )d e + (- 60A b - 180B a b )d e - 30B b d e
--R *
--R 2
--R x
--R +
--R 4 2 5 5 3 3 4 2 4
--R (300A a b - 900B a b)e + (- 60A a b + 780B a b )d e
--R +
--R 2 4 3 3 2 3 5 2 4 3 2
--R (- 120A a b + 240B a b )d e + (- 40A a b - 120B a b )d e
--R +
--R 6 5 4 6 5
--R (- 20A b - 20B a b )d e - 4B b d
--R *
--R x
--R +
--R 5 6 5 5 4 3 3 4 2 2 3
--R (60A a b - 180B a )e + 120B a b d e + (- 30A a b + 90B a b )d e
--R +
--R 2 4 3 3 3 2 5 2 4 4
--R (- 10A a b - 30B a b )d e + (- 5A a b - 5B a b )d e
--R +
--R 6 5 5
--R (- 3A b - B a b )d
--R /
--R +------------------+
--R 10 3 9 2 2 8 3 7 | 2 2 2
--R (12b x + 36a b x + 36a b x + 12a b )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 443
--S 444 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 444
)clear all
--S 445 of 764
t0:=(A+B*x)*(d+e*x)^4/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B e x + (A e + 4B d e )x + (4A d e + 6B d e )x
--R +
--R 2 2 3 2 3 4 4
--R (6A d e + 4B d e)x + (4A d e + B d )x + A d
--R /
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 445
--S 446 of 764
r0:=-1/4*(4*b*B*d+A*b*e-5*a*B*e)*(a+b*x)*(d+e*x)^4/(b^2*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))+B*(a+b*x)*(d+e*x)^5/(b*e*(a^2+2*a*b*x+b^2*x^2)^(5/2))-_
1/2*e*(4*b*B*d+A*b*e-5*a*B*e)*(a+b*x)*(d+e*x)^2/(b^4*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))-1/3*(4*b*B*d+A*b*e-5*a*B*e)*(d+e*x)^3/(b^3*(a^2+_
2*a*b*x+b^2*x^2)^(3/2))-e^2*(4*b*B*d+A*b*e-5*a*B*e)*(d+e*x)/(b^5*_
sqrt(a^2+2*a*b*x+b^2*x^2))+e^3*(4*b*B*d+A*b*e-5*a*B*e)*(a+b*x)*_
log(a+b*x)/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 4 4 5 3 4
--R ((12A b - 60B a b )e + 48B b d e )x
--R +
--R 4 2 3 4 4 3 3
--R ((48A a b - 240B a b )e + 192B a b d e )x
--R +
--R 2 3 3 2 4 2 3 3 2
--R ((72A a b - 360B a b )e + 288B a b d e )x
--R +
--R 3 2 4 4 3 2 3 4 5 4
--R ((48A a b - 240B a b)e + 192B a b d e )x + (12A a b - 60B a )e
--R +
--R 4 3
--R 48B a b d e
--R *
--R log(b x + a)
--R +
--R 5 4 5 5 4 4 5 3 4
--R 12B b e x + ((- 25A b + 125B a b )e - 40B b d e )x
--R +
--R 4 2 3 4 5 4 3 5 2 2 3
--R ((- 52A a b + 260B a b )e + (- 48A b + 32B a b )d e - 72B b d e )x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (- 42A a b + 210B a b )e + (- 72A a b + 192B a b )d e
--R +
--R 5 4 2 2 5 3
--R (- 36A b - 108B a b )d e - 24B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (- 12A a b + 60B a b)e + (- 48A a b + 192B a b )d e
--R +
--R 4 2 3 2 2 5 4 3 5 4
--R (- 24A a b - 72B a b )d e + (- 16A b - 16B a b )d e - 4B b d
--R *
--R x
--R +
--R 3 2 4 3 2 3 3 2 2 2
--R (- 12A a b + 60B a b)d e + (- 6A a b - 18B a b )d e
--R +
--R 4 2 3 3 5 4 4
--R (- 4A a b - 4B a b )d e + (- 3A b - B a b )d
--R /
--R +------------------+
--R 9 3 8 2 2 7 3 6 | 2 2 2
--R (12b x + 36a b x + 36a b x + 12a b )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 446
--S 447 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 447
)clear all
--S 448 of 764
t0:=(A+B*x)*(d+e*x)^3/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x + A d
--R ----------------------------------------------------------------------------
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 448
--S 449 of 764
r0:=-1/4*(A*b-a*B)*(a+b*x)*(d+e*x)^4/(b*(b*d-a*e)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))-1/2*B*e*(a+b*x)*(d+e*x)^2/(b^3*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))-1/3*B*(d+e*x)^3/(b^2*(a^2+2*a*b*x+b^2*x^2)^(3/2))-_
B*e^2*(d+e*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))+B*e^3*(a+b*x)*_
log(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2)),-1/4*B*(a+b*x)*_
(d+e*x)^4/(b*e*(a^2+2*a*b*x+b^2*x^2)^(5/2))+1/4*(B*d-A*e)*(a+b*x)*_
(d+e*x)^4/(e*(b*d-a*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2))-1/2*B*e*(a+b*x)*_
(d+e*x)^2/(b^3*(a^2+2*a*b*x+b^2*x^2)^(3/2))-1/3*B*(d+e*x)^3/(b^2*_
(a^2+2*a*b*x+b^2*x^2)^(3/2))-B*e^2*(d+e*x)/(b^4*sqrt(a^2+2*a*b*x+_
b^2*x^2))+B*e^3*(a+b*x)*log(a+b*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R [
--R 4 4 5 3 4 2 3 4 4 3 3
--R (12B a b e - 12B b d e )x + (48B a b e - 48B a b d e )x
--R +
--R 3 2 4 2 3 3 2 4 4 3 2 3
--R (72B a b e - 72B a b d e )x + (48B a b e - 48B a b d e )x
--R +
--R 5 4 4 3
--R 12B a e - 12B a b d e
--R *
--R log(b x + a)
--R +
--R 5 4 4 5 3 4
--R ((3A b - 25B a b )e + 22B b d e )x
--R +
--R 2 3 4 5 4 3 5 2 2 3
--R (- 52B a b e + (12A b + 4B a b )d e + 36B b d e )x
--R +
--R 3 2 4 2 3 3 5 4 2 2 5 3 2
--R (- 42B a b e - 30B a b d e + (18A b + 36B a b )d e + 18B b d e)x
--R +
--R 4 4 3 2 3 2 3 2 2 5 4 3
--R - 12B a b e - 36B a b d e + 24B a b d e + (12A b + 8B a b )d e
--R +
--R 5 4
--R 4B b d
--R *
--R x
--R +
--R 4 3 3 2 2 2 2 3 3 5 4 4
--R - 12B a b d e + 6B a b d e + 2B a b d e + (3A b + B a b )d
--R /
--R 8 9 3 2 7 8 2 3 6 2 7
--R (12a b e - 12b d)x + (36a b e - 36a b d)x + (36a b e - 36a b d)x
--R +
--R 4 5 3 6
--R 12a b e - 12a b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R ,
--R
--R 4 4 5 3 4 2 3 4 4 3 3
--R (12B a b e - 12B b d e )x + (48B a b e - 48B a b d e )x
--R +
--R 3 2 4 2 3 3 2 4 4 3 2 3
--R (72B a b e - 72B a b d e )x + (48B a b e - 48B a b d e )x
--R +
--R 5 4 4 3
--R 12B a e - 12B a b d e
--R *
--R log(b x + a)
--R +
--R 5 4 4 5 3 4
--R ((3A b - 25B a b )e + 22B b d e )x
--R +
--R 2 3 4 5 4 3 5 2 2 3
--R (- 52B a b e + (12A b + 4B a b )d e + 36B b d e )x
--R +
--R 3 2 4 2 3 3 5 4 2 2 5 3 2
--R (- 42B a b e - 30B a b d e + (18A b + 36B a b )d e + 18B b d e)x
--R +
--R 4 4 3 2 3 2 3 2 2 5 4 3
--R - 12B a b e - 36B a b d e + 24B a b d e + (12A b + 8B a b )d e
--R +
--R 5 4
--R 4B b d
--R *
--R x
--R +
--R 4 3 3 2 2 2 2 3 3 5 4 4
--R - 12B a b d e + 6B a b d e + 2B a b d e + (3A b + B a b )d
--R /
--R 8 9 3 2 7 8 2 3 6 2 7
--R (12a b e - 12b d)x + (36a b e - 36a b d)x + (36a b e - 36a b d)x
--R +
--R 4 5 3 6
--R 12a b e - 12a b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R ]
--R Type: Tuple(Expression(Integer))
--E 449
--S 450 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 450
)clear all
--S 451 of 764
t0:=(A+B*x)*(d+e*x)^2/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R 2 3 2 2 2 2
--R B e x + (A e + 2B d e)x + (2A d e + B d )x + A d
--R (1) -------------------------------------------------------------
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 451
--S 452 of 764
r0:=-1/4*(A*b-a*B)*(a+b*x)*(d+e*x)^4/((b*d-a*e)^2*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))-1/3*(B*d-A*e)*(d+e*x)^3/((b*d-a*e)^2*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))
--R
--R
--R (2)
--R 4 3 4 4 3 2 2 3
--R ((A b + 3B a)e - 4B b d e )x + (4A a e + 8B a d e - 12B b d e )x
--R +
--R 3 2 2 3 2
--R (12A a d e + (- 6A b + 6B a)d e - 12B b d e)x
--R +
--R 2 2 3 4 3 4
--R (12A a d e - 8A b d e - 4B b d )x + 4A a d e + (- 3A b - B a)d
--R /
--R 2 3 2 4 5 2 3 3 2 2 2 3 4 2 2
--R (12a b e - 24a b d e + 12b d )x + (36a b e - 72a b d e + 36a b d )x
--R +
--R 4 2 3 2 2 3 2 5 2 4 3 2 2
--R (36a b e - 72a b d e + 36a b d )x + 12a e - 24a b d e + 12a b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 452
--S 453 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 453
)clear all
--S 454 of 764
t0:=(A+B*x)*(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R 2
--R B e x + (A e + B d)x + A d
--R (1) -------------------------------------------------------------
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 454
--S 455 of 764
r0:=1/4*(b*d-a*e)*(2*b*B*d-A*b*e-a*B*e)*(a+b*x)/(b^3*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))-1/2*B*(a+b*x)*(d+e*x)^2/(b*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))+1/3*(2*b*B*d-A*b*e-a*B*e)/(b^3*(a^2+2*a*b*x+b^2*x^2)^(3/2))
--R
--R
--R (2)
--R 2 2 2 2 2
--R - 6B b e x + ((- 4A b - 4B a b)e - 4B b d)x + (- A a b - B a )e
--R +
--R 2
--R (- 3A b - B a b)d
--R /
--R +------------------+
--R 6 3 5 2 2 4 3 3 | 2 2 2
--R (12b x + 36a b x + 36a b x + 12a b )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 455
--S 456 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 456
)clear all
--S 457 of 764
t0:=(A+B*x)/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R B x + A
--R (1) -------------------------------------------------------------
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 457
--S 458 of 764
r0:=-1/4*(A*b-a*B)*(a+b*x)/(b^2*(a^2+2*a*b*x+b^2*x^2)^(5/2))-_
1/3*B/(b^2*(a^2+2*a*b*x+b^2*x^2)^(3/2))
--R
--R
--R - 4B b x - 3A b - B a
--R (2) -----------------------------------------------------------
--R +------------------+
--R 5 3 4 2 2 3 3 2 | 2 2 2
--R (12b x + 36a b x + 36a b x + 12a b )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 458
--S 459 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 459
)clear all
--S 460 of 764
t0:=(A+B*x)/((d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2))
--R
--R
--R (1)
--R B x + A
--R /
--R 4 5 3 4 4 2 2 3 3 3 2 2 2
--R b e x + (4a b e + b d)x + (6a b e + 4a b d)x + (4a b e + 6a b d)x
--R +
--R 4 3 4
--R (a e + 4a b d)x + a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 460
--S 461 of 764
r0:=-1/4*(A*b-a*B)*(a+b*x)/(b*(b*d-a*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2))+_
1/3*(-B*d+A*e)/((b*d-a*e)^2*(a^2+2*a*b*x+b^2*x^2)^(3/2))+_
1/2*e*(B*d-A*e)*(a+b*x)/((b*d-a*e)^3*(a^2+2*a*b*x+b^2*x^2)^(3/2))-_
e^2*(B*d-A*e)/((b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))-e^3*(B*d-A*e)*_
(a+b*x)*log(a+b*x)/((b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2))+e^3*_
(B*d-A*e)*(a+b*x)*log(d+e*x)/((b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 4 5 3 4 4 4 4 3 3
--R (12A b e - 12B b d e )x + (48A a b e - 48B a b d e )x
--R +
--R 2 3 4 2 3 3 2 3 2 4 3 2 3
--R (72A a b e - 72B a b d e )x + (48A a b e - 48B a b d e )x
--R +
--R 4 4 4 3
--R 12A a b e - 12B a b d e
--R *
--R log(e x + d)
--R +
--R 5 4 5 3 4 4 4 4 3 3
--R (- 12A b e + 12B b d e )x + (- 48A a b e + 48B a b d e )x
--R +
--R 2 3 4 2 3 3 2 3 2 4 3 2 3
--R (- 72A a b e + 72B a b d e )x + (- 48A a b e + 48B a b d e )x
--R +
--R 4 4 4 3
--R - 12A a b e + 12B a b d e
--R *
--R log(b x + a)
--R +
--R 4 4 5 4 3 5 2 2 3
--R (12A a b e + (- 12A b - 12B a b )d e + 12B b d e )x
--R +
--R 2 3 4 4 2 3 3 5 4 2 2
--R 42A a b e + (- 48A a b - 42B a b )d e + (6A b + 48B a b )d e
--R +
--R 5 3
--R - 6B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 2 3 3 2 3 4 2 3 2 2
--R 52A a b e + (- 72A a b - 52B a b )d e + (24A a b + 72B a b )d e
--R +
--R 5 4 3 5 4
--R (- 4A b - 24B a b )d e + 4B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (25A a b - 3B a )e + (- 48A a b - 10B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3 5 4 4
--R (36A a b + 18B a b )d e + (- 16A a b - 6B a b )d e + (3A b + B a b )d
--R /
--R 5 4 5 4 5 4 3 6 2 3 2 7 3 2 8 4
--R 12a b e - 60a b d e + 120a b d e - 120a b d e + 60a b d e
--R +
--R 9 5
--R - 12b d
--R *
--R 3
--R x
--R +
--R 6 3 5 5 4 4 4 5 2 3 3 6 3 2 2 7 4
--R 36a b e - 180a b d e + 360a b d e - 360a b d e + 180a b d e
--R +
--R 8 5
--R - 36a b d
--R *
--R 2
--R x
--R +
--R 7 2 5 6 3 4 5 4 2 3 4 5 3 2 3 6 4
--R 36a b e - 180a b d e + 360a b d e - 360a b d e + 180a b d e
--R +
--R 2 7 5
--R - 36a b d
--R *
--R x
--R +
--R 8 5 7 2 4 6 3 2 3 5 4 3 2 4 5 4 3 6 5
--R 12a b e - 60a b d e + 120a b d e - 120a b d e + 60a b d e - 12a b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 461
--S 462 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 462
)clear all
--S 463 of 764
t0:=(A+B*x)/((d+e*x)^2*(a^2+2*a*b*x+b^2*x^2)^(5/2))
--R
--R
--R (1)
--R B x + A
--R /
--R 4 2 6 3 2 4 5 2 2 2 3 4 2 4
--R b e x + (4a b e + 2b d e)x + (6a b e + 8a b d e + b d )x
--R +
--R 3 2 2 2 3 2 3 4 2 3 2 2 2 2
--R (4a b e + 12a b d e + 4a b d )x + (a e + 8a b d e + 6a b d )x
--R +
--R 4 3 2 4 2
--R (2a d e + 4a b d )x + a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 463
--S 464 of 764
r0:=1/4*(4*b*B*d-5*A*b*e+a*B*e)*(a+b*x)/(e*(b*d-a*e)^2*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))-(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)*(a^2+_
2*a*b*x+b^2*x^2)^(5/2))+1/3*(-4*b*B*d+5*A*b*e-a*B*e)/((b*d-a*e)^3*_
(a^2+2*a*b*x+b^2*x^2)^(3/2))+1/2*e*(4*b*B*d-5*A*b*e+a*B*e)*(a+b*x)/_
((b*d-a*e)^4*(a^2+2*a*b*x+b^2*x^2)^(3/2))-e^2*(4*b*B*d-5*A*b*e+_
a*B*e)/((b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2))-e^3*(4*b*B*d-_
5*A*b*e+a*B*e)*(a+b*x)*log(a+b*x)/((b*d-a*e)^6*sqrt(a^2+2*a*b*x+_
b^2*x^2))+e^3*(4*b*B*d-5*A*b*e+a*B*e)*(a+b*x)*log(d+e*x)/_
((b*d-a*e)^6*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 4 5 5 4 5
--R ((- 60A b + 12B a b )e + 48B b d e )x
--R +
--R 4 2 3 5 5 4 4
--R (- 240A a b + 48B a b )e + (- 60A b + 204B a b )d e
--R +
--R 5 2 3
--R 48B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 5 4 2 3 4
--R (- 360A a b + 72B a b )e + (- 240A a b + 336B a b )d e
--R +
--R 4 2 3
--R 192B a b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 5 2 3 3 2 4
--R (- 240A a b + 48B a b)e + (- 360A a b + 264B a b )d e
--R +
--R 2 3 2 3
--R 288B a b d e
--R *
--R 2
--R x
--R +
--R 4 5 5 3 2 4 4
--R (- 60A a b + 12B a )e + (- 240A a b + 96B a b)d e
--R +
--R 3 2 2 3
--R 192B a b d e
--R *
--R x
--R +
--R 4 5 4 4 2 3
--R (- 60A a b + 12B a )d e + 48B a b d e
--R *
--R log(e x + d)
--R +
--R 5 4 5 5 4 5
--R ((60A b - 12B a b )e - 48B b d e )x
--R +
--R 4 2 3 5 5 4 4 5 2 3 4
--R ((240A a b - 48B a b )e + (60A b - 204B a b )d e - 48B b d e )x
--R +
--R 2 3 3 2 5 4 2 3 4
--R (360A a b - 72B a b )e + (240A a b - 336B a b )d e
--R +
--R 4 2 3
--R - 192B a b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 5 2 3 3 2 4
--R (240A a b - 48B a b)e + (360A a b - 264B a b )d e
--R +
--R 2 3 2 3
--R - 288B a b d e
--R *
--R 2
--R x
--R +
--R 4 5 5 3 2 4 4 3 2 2 3
--R ((60A a b - 12B a )e + (240A a b - 96B a b)d e - 192B a b d e )x
--R +
--R 4 5 4 4 2 3
--R (60A a b - 12B a )d e - 48B a b d e
--R *
--R log(b x + a)
--R +
--R 4 2 3 5 5 4 4 5 2 3 4
--R ((- 60A a b + 12B a b )e + (60A b + 36B a b )d e - 48B b d e )x
--R +
--R 2 3 3 2 5 4 2 3 4
--R (- 210A a b + 42B a b )e + (180A a b + 132B a b )d e
--R +
--R 5 4 2 3 5 3 2
--R (30A b - 150B a b )d e - 24B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 5 2 3 3 2 4
--R (- 260A a b + 52B a b)e + (150A a b + 178B a b )d e
--R +
--R 4 2 3 2 3 5 4 3 2 5 4
--R (120A a b - 144B a b )d e + (- 10A b - 94B a b )d e + 8B b d e
--R *
--R 2
--R x
--R +
--R 4 5 5 3 2 4 4
--R (- 125A a b + 25B a )e + (- 20A a b + 104B a b)d e
--R +
--R 2 3 3 2 2 3 4 2 3 3 2
--R (180A a b - 20B a b )d e + (- 40A a b - 136B a b )d e
--R +
--R 5 4 4 5 5
--R (5A b + 31B a b )d e - 4B b d
--R *
--R x
--R +
--R 5 5 4 5 4 3 2 4 2 3
--R - 12A a e + (- 65A a b + 37B a )d e + (120A a b - 8B a b)d e
--R +
--R 2 3 3 2 3 2 4 2 3 4
--R (- 60A a b - 36B a b )d e + (20A a b + 8B a b )d e
--R +
--R 5 4 5
--R (- 3A b - B a b )d
--R /
--R 6 3 7 5 4 6 4 5 2 5 3 6 3 4 2 7 4 3
--R 12a b e - 72a b d e + 180a b d e - 240a b d e + 180a b d e
--R +
--R 8 5 2 9 6
--R - 72a b d e + 12b d e
--R *
--R 4
--R x
--R +
--R 7 2 7 6 3 6 5 4 2 5 4 5 3 4 3 6 4 3
--R 36a b e - 204a b d e + 468a b d e - 540a b d e + 300a b d e
--R +
--R 2 7 5 2 8 6 9 7
--R - 36a b d e - 36a b d e + 12b d
--R *
--R 3
--R x
--R +
--R 8 7 7 2 6 6 3 2 5 5 4 3 4 4 5 4 3
--R 36a b e - 180a b d e + 324a b d e - 180a b d e - 180a b d e
--R +
--R 3 6 5 2 2 7 6 8 7
--R 324a b d e - 180a b d e + 36a b d
--R *
--R 2
--R x
--R +
--R 9 7 8 6 7 2 2 5 6 3 3 4 5 4 4 3
--R 12a e - 36a b d e - 36a b d e + 300a b d e - 540a b d e
--R +
--R 4 5 5 2 3 6 6 2 7 7
--R 468a b d e - 204a b d e + 36a b d
--R *
--R x
--R +
--R 9 6 8 2 5 7 2 3 4 6 3 4 3 5 4 5 2
--R 12a d e - 72a b d e + 180a b d e - 240a b d e + 180a b d e
--R +
--R 4 5 6 3 6 7
--R - 72a b d e + 12a b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 464
--S 465 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 465
)clear all
--S 466 of 764
t0:=(A+B*x)/((d+e*x)^3*(a^2+2*a*b*x+b^2*x^2)^(5/2))
--R
--R
--R (1)
--R B x + A
--R /
--R 4 3 7 3 3 4 2 6 2 2 3 3 2 4 2 5
--R b e x + (4a b e + 3b d e )x + (6a b e + 12a b d e + 3b d e)x
--R +
--R 3 3 2 2 2 3 2 4 3 4
--R (4a b e + 18a b d e + 12a b d e + b d )x
--R +
--R 4 3 3 2 2 2 2 3 3 3
--R (a e + 12a b d e + 18a b d e + 4a b d )x
--R +
--R 4 2 3 2 2 2 3 2 4 2 3 3 4 3
--R (3a d e + 12a b d e + 6a b d )x + (3a d e + 4a b d )x + a d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 466
--S 467 of 764
r0:=-1/2*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)^2*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))+1/4*(2*b*B*d-3*A*b*e+a*B*e)*(a+b*x)/(e*(b*d-a*e)^2*_
(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2))-5/12*(2*b*B*d-3*A*b*e+a*B*e)/_
((b*d-a*e)^3*(d+e*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2))+5/6*e*(2*b*B*d-_
3*A*b*e+a*B*e)*(a+b*x)/((b*d-a*e)^4*(d+e*x)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))-5/2*e^2*(2*b*B*d-3*A*b*e+a*B*e)/((b*d-a*e)^5*(d+_
e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-5*b*e^3*(2*b*B*d-3*A*b*e+a*B*e)*_
(a+b*x)*log(a+b*x)/((b*d-a*e)^7*sqrt(a^2+2*a*b*x+b^2*x^2))+5*b*e^3*_
(2*b*B*d-3*A*b*e+a*B*e)*(a+b*x)*log(d+e*x)/((b*d-a*e)^7*sqrt(a^2+_
2*a*b*x+b^2*x^2))+5*e^4*(2*b*B*d-3*A*b*e+a*B*e)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/((b*d-a*e)^7*(d+e*x))
--R
--R
--R (2)
--R 6 5 6 6 5 6
--R ((180A b - 60B a b )e - 120B b d e )x
--R +
--R 5 2 4 6 6 5 5
--R (720A a b - 240B a b )e + (360A b - 600B a b )d e
--R +
--R 6 2 4
--R - 240B b d e
--R *
--R 5
--R x
--R +
--R 2 4 3 3 6 5 2 4 5
--R (1080A a b - 360B a b )e + (1440A a b - 1200B a b )d e
--R +
--R 6 5 2 4 6 3 3
--R (180A b - 1020B a b )d e - 120B b d e
--R *
--R 4
--R x
--R +
--R 3 3 4 2 6 2 4 3 3 5
--R (720A a b - 240B a b )e + (2160A a b - 1200B a b )d e
--R +
--R 5 2 4 2 4 5 3 3
--R (720A a b - 1680B a b )d e - 480B a b d e
--R *
--R 3
--R x
--R +
--R 4 2 5 6 3 3 4 2 5
--R (180A a b - 60B a b)e + (1440A a b - 600B a b )d e
--R +
--R 2 4 3 3 2 4 2 4 3 3
--R (1080A a b - 1320B a b )d e - 720B a b d e
--R *
--R 2
--R x
--R +
--R 4 2 5 5 3 3 4 2 2 4
--R (360A a b - 120B a b)d e + (720A a b - 480B a b )d e
--R +
--R 3 3 3 3
--R - 480B a b d e
--R *
--R x
--R +
--R 4 2 5 2 4 4 2 3 3
--R (180A a b - 60B a b)d e - 120B a b d e
--R *
--R log(e x + d)
--R +
--R 6 5 6 6 5 6
--R ((- 180A b + 60B a b )e + 120B b d e )x
--R +
--R 5 2 4 6 6 5 5
--R (- 720A a b + 240B a b )e + (- 360A b + 600B a b )d e
--R +
--R 6 2 4
--R 240B b d e
--R *
--R 5
--R x
--R +
--R 2 4 3 3 6 5 2 4 5
--R (- 1080A a b + 360B a b )e + (- 1440A a b + 1200B a b )d e
--R +
--R 6 5 2 4 6 3 3
--R (- 180A b + 1020B a b )d e + 120B b d e
--R *
--R 4
--R x
--R +
--R 3 3 4 2 6 2 4 3 3 5
--R (- 720A a b + 240B a b )e + (- 2160A a b + 1200B a b )d e
--R +
--R 5 2 4 2 4 5 3 3
--R (- 720A a b + 1680B a b )d e + 480B a b d e
--R *
--R 3
--R x
--R +
--R 4 2 5 6 3 3 4 2 5
--R (- 180A a b + 60B a b)e + (- 1440A a b + 600B a b )d e
--R +
--R 2 4 3 3 2 4 2 4 3 3
--R (- 1080A a b + 1320B a b )d e + 720B a b d e
--R *
--R 2
--R x
--R +
--R 4 2 5 5 3 3 4 2 2 4
--R (- 360A a b + 120B a b)d e + (- 720A a b + 480B a b )d e
--R +
--R 3 3 3 3
--R 480B a b d e
--R *
--R x
--R +
--R 4 2 5 2 4 4 2 3 3
--R (- 180A a b + 60B a b)d e + 120B a b d e
--R *
--R log(b x + a)
--R +
--R 6 5 6 6 5 6
--R ((180A b - 60B a b )e - 120B b d e )x
--R +
--R 5 2 4 6 6 5 5 6 2 4 5
--R ((900A a b - 300B a b )e + (180A b - 660B a b )d e - 120B b d e )x
--R +
--R 2 4 3 3 6 5 2 4 5
--R (1710A a b - 570B a b )e + (1080A a b - 1500B a b )d e
--R +
--R 6 5 2 4 6 3 3
--R (- 90A b - 690B a b )d e + 60B b d e
--R *
--R 4
--R x
--R +
--R 3 3 4 2 6 2 4 3 3 5
--R (1500A a b - 500B a b )e + (2340A a b - 1780B a b )d e
--R +
--R 5 2 4 2 4 6 5 3 3
--R (- 180A a b - 1500B a b )d e + (- 60A b + 140B a b )d e
--R +
--R 6 4 2
--R 40B b d e
--R *
--R 3
--R x
--R +
--R 4 2 5 6 3 3 4 2 5
--R (555A a b - 185B a b)e + (2280A a b - 1130B a b )d e
--R +
--R 2 4 3 3 2 4 5 2 4 3 3
--R (90A a b - 1550B a b )d e + (- 240A a b + 20B a b )d e
--R +
--R 6 5 4 2 6 5
--R (15A b + 155B a b )d e - 10B b d e
--R *
--R 2
--R x
--R +
--R 5 6 6 4 2 5 5
--R (36A a b - 12B a )e + (930A a b - 334B a b)d e
--R +
--R 3 3 4 2 2 4 2 4 3 3 3 3
--R (420A a b - 760B a b )d e + (- 360A a b - 160B a b )d e
--R +
--R 5 2 4 4 2 6 5 5 6 6
--R (60A a b + 220B a b )d e + (- 6A b - 38B a b )d e + 4B b d
--R *
--R x
--R +
--R 6 6 5 6 5 4 2 5 2 4
--R - 6A a e + (72A a b - 6B a )d e + (285A a b - 185B a b)d e
--R +
--R 3 3 4 2 3 3 2 4 3 3 4 2
--R (- 240A a b - 40B a b )d e + (90A a b + 60B a b )d e
--R +
--R 5 2 4 5 6 5 6
--R (- 24A a b - 10B a b )d e + (3A b + B a b )d
--R /
--R 7 3 9 6 4 8 5 5 2 7 4 6 3 6 3 7 4 5
--R 12a b e - 84a b d e + 252a b d e - 420a b d e + 420a b d e
--R +
--R 2 8 5 4 9 6 3 10 7 2
--R - 252a b d e + 84a b d e - 12b d e
--R *
--R 5
--R x
--R +
--R 8 2 9 7 3 8 6 4 2 7 5 5 3 6 4 6 4 5
--R 36a b e - 228a b d e + 588a b d e - 756a b d e + 420a b d e
--R +
--R 3 7 5 4 2 8 6 3 9 7 2 10 8
--R 84a b d e - 252a b d e + 132a b d e - 24b d e
--R *
--R 4
--R x
--R +
--R 9 9 8 2 8 7 3 2 7 6 4 3 6 5 5 4 5
--R 36a b e - 180a b d e + 264a b d e + 168a b d e - 1008a b d e
--R +
--R 4 6 5 4 3 7 6 3 2 8 7 2 9 8 10 9
--R 1344a b d e - 840a b d e + 216a b d e + 12a b d e - 12b d
--R *
--R 3
--R x
--R +
--R 10 9 9 8 8 2 2 7 7 3 3 6 6 4 4 5
--R 12a e - 12a b d e - 216a b d e + 840a b d e - 1344a b d e
--R +
--R 5 5 5 4 4 6 6 3 3 7 7 2 2 8 8 9 9
--R 1008a b d e - 168a b d e - 264a b d e + 180a b d e - 36a b d
--R *
--R 2
--R x
--R +
--R 10 8 9 2 7 8 2 3 6 7 3 4 5 6 4 5 4
--R 24a d e - 132a b d e + 252a b d e - 84a b d e - 420a b d e
--R +
--R 5 5 6 3 4 6 7 2 3 7 8 2 8 9
--R 756a b d e - 588a b d e + 228a b d e - 36a b d
--R *
--R x
--R +
--R 10 2 7 9 3 6 8 2 4 5 7 3 5 4 6 4 6 3
--R 12a d e - 84a b d e + 252a b d e - 420a b d e + 420a b d e
--R +
--R 5 5 7 2 4 6 8 3 7 9
--R - 252a b d e + 84a b d e - 12a b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 467
--S 468 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 468
)clear all
--S 469 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 2 3 6 2 3 2 2 5
--R B b e x + ((A b + 2B a b)e + 3B b d e )x
--R +
--R 2 3 2 2 2 2 4
--R ((2A a b + B a )e + (3A b + 6B a b)d e + 3B b d e)x
--R +
--R 2 3 2 2 2 2 2 3 3
--R (A a e + (6A a b + 3B a )d e + (3A b + 6B a b)d e + B b d )x
--R +
--R 2 2 2 2 2 3 2
--R (3A a d e + (6A a b + 3B a )d e + (A b + 2B a b)d )x
--R +
--R 2 2 2 3 2 3
--R (3A a d e + (2A a b + B a )d )x + A a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 469
--S 470 of 764
r0:=-2/9*(b*d-a*e)^2*(B*d-A*e)*(d+e*x)^(9/2)/e^4+2/11*(b*d-a*e)*_
(3*b*B*d-2*A*b*e-a*B*e)*(d+e*x)^(11/2)/e^4-2/13*b*(3*b*B*d-A*b*e-_
2*a*B*e)*(d+e*x)^(13/2)/e^4+2/15*b^2*B*(d+e*x)^(15/2)/e^4
--R
--R
--R (2)
--R 2 7 7 2 7 2 6 6
--R 858B b e x + ((990A b + 1980B a b)e + 3036B b d e )x
--R +
--R 2 7 2 6
--R (2340A a b + 1170B a )e + (3600A b + 7200B a b)d e
--R +
--R 2 2 5
--R 3708B b d e
--R *
--R 5
--R x
--R +
--R 2 7 2 6
--R 1430A a e + (8840A a b + 4420B a )d e
--R +
--R 2 2 5 2 3 4
--R (4580A b + 9160B a b)d e + 1600B b d e
--R *
--R 4
--R x
--R +
--R 2 6 2 2 5
--R 5720A a d e + (11960A a b + 5980B a )d e
--R +
--R 2 3 4 2 4 3
--R (2120A b + 4240B a b)d e + 10B b d e
--R *
--R 3
--R x
--R +
--R 2 2 5 2 3 4 2 4 3
--R 8580A a d e + (6240A a b + 3120B a )d e + (30A b + 60B a b)d e
--R +
--R 2 5 2
--R - 12B b d e
--R *
--R 2
--R x
--R +
--R 2 3 4 2 4 3 2 5 2
--R 5720A a d e + (260A a b + 130B a )d e + (- 40A b - 80B a b)d e
--R +
--R 2 6
--R 16B b d e
--R *
--R x
--R +
--R 2 4 3 2 5 2 2 6
--R 1430A a d e + (- 520A a b - 260B a )d e + (80A b + 160B a b)d e
--R +
--R 2 7
--R - 32B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 4
--R 6435e
--R Type: Expression(Integer)
--E 470
--S 471 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 471
)clear all
--S 472 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 2 2 5 2 2 2 4
--R B b e x + ((A b + 2B a b)e + 2B b d e)x
--R +
--R 2 2 2 2 2 3
--R ((2A a b + B a )e + (2A b + 4B a b)d e + B b d )x
--R +
--R 2 2 2 2 2 2
--R (A a e + (4A a b + 2B a )d e + (A b + 2B a b)d )x
--R +
--R 2 2 2 2 2
--R (2A a d e + (2A a b + B a )d )x + A a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 472
--S 473 of 764
r0:=-2/7*(b*d-a*e)^2*(B*d-A*e)*(d+e*x)^(7/2)/e^4+2/9*(b*d-a*e)*_
(3*b*B*d-2*A*b*e-a*B*e)*(d+e*x)^(9/2)/e^4-2/11*b*(3*b*B*d-A*b*e-_
2*a*B*e)*(d+e*x)^(11/2)/e^4+2/13*b^2*B*(d+e*x)^(13/2)/e^4
--R
--R
--R (2)
--R 2 6 6 2 6 2 5 5
--R 1386B b e x + ((1638A b + 3276B a b)e + 3402B b d e )x
--R +
--R 2 6 2 5
--R (4004A a b + 2002B a )e + (4186A b + 8372B a b)d e
--R +
--R 2 2 4
--R 2226B b d e
--R *
--R 4
--R x
--R +
--R 2 6 2 5
--R 2574A a e + (10868A a b + 5434B a )d e
--R +
--R 2 2 4 2 3 3
--R (2938A b + 5876B a b)d e + 30B b d e
--R *
--R 3
--R x
--R +
--R 2 5 2 2 4 2 3 3
--R 7722A a d e + (8580A a b + 4290B a )d e + (78A b + 156B a b)d e
--R +
--R 2 4 2
--R - 36B b d e
--R *
--R 2
--R x
--R +
--R 2 2 4 2 3 3
--R 7722A a d e + (572A a b + 286B a )d e
--R +
--R 2 4 2 2 5
--R (- 104A b - 208B a b)d e + 48B b d e
--R *
--R x
--R +
--R 2 3 3 2 4 2 2 5
--R 2574A a d e + (- 1144A a b - 572B a )d e + (208A b + 416B a b)d e
--R +
--R 2 6
--R - 96B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 4
--R 9009e
--R Type: Expression(Integer)
--E 473
--S 474 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 474
)clear all
--S 475 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 2 4 2 2 3
--R B b e x + ((A b + 2B a b)e + B b d)x
--R +
--R 2 2 2 2 2
--R ((2A a b + B a )e + (A b + 2B a b)d)x + (A a e + (2A a b + B a )d)x
--R +
--R 2
--R A a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 475
--S 476 of 764
r0:=-2/5*(b*d-a*e)^2*(B*d-A*e)*(d+e*x)^(5/2)/e^4+2/7*(b*d-a*e)*_
(3*b*B*d-2*A*b*e-a*B*e)*(d+e*x)^(7/2)/e^4-2/9*b*(3*b*B*d-A*b*e-_
2*a*B*e)*(d+e*x)^(9/2)/e^4+2/11*b^2*B*(d+e*x)^(11/2)/e^4
--R
--R
--R (2)
--R 2 5 5 2 5 2 4 4
--R 630B b e x + ((770A b + 1540B a b)e + 840B b d e )x
--R +
--R 2 5 2 4 2 2 3 3
--R ((1980A a b + 990B a )e + (1100A b + 2200B a b)d e + 30B b d e )x
--R +
--R 2 5 2 4 2 2 3
--R 1386A a e + (3168A a b + 1584B a )d e + (66A b + 132B a b)d e
--R +
--R 2 3 2
--R - 36B b d e
--R *
--R 2
--R x
--R +
--R 2 4 2 2 3 2 3 2
--R 2772A a d e + (396A a b + 198B a )d e + (- 88A b - 176B a b)d e
--R +
--R 2 4
--R 48B b d e
--R *
--R x
--R +
--R 2 2 3 2 3 2 2 4
--R 1386A a d e + (- 792A a b - 396B a )d e + (176A b + 352B a b)d e
--R +
--R 2 5
--R - 96B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 4
--R 3465e
--R Type: Expression(Integer)
--E 476
--S 477 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 477
)clear all
--S 478 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)*sqrt(d+e*x)
--R
--R
--R 2 3 2 2 2 2 +-------+
--R (1) (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|e x + d
--R Type: Expression(Integer)
--E 478
--S 479 of 764
r0:=-2/3*(b*d-a*e)^2*(B*d-A*e)*(d+e*x)^(3/2)/e^4+2/5*(b*d-a*e)*_
(3*b*B*d-2*A*b*e-a*B*e)*(d+e*x)^(5/2)/e^4-2/7*b*(3*b*B*d-A*b*e-_
2*a*B*e)*(d+e*x)^(7/2)/e^4+2/9*b^2*B*(d+e*x)^(9/2)/e^4
--R
--R
--R (2)
--R 2 4 4 2 4 2 3 3
--R 70B b e x + ((90A b + 180B a b)e + 10B b d e )x
--R +
--R 2 4 2 3 2 2 2 2
--R ((252A a b + 126B a )e + (18A b + 36B a b)d e - 12B b d e )x
--R +
--R 2 4 2 3 2 2 2
--R 210A a e + (84A a b + 42B a )d e + (- 24A b - 48B a b)d e
--R +
--R 2 3
--R 16B b d e
--R *
--R x
--R +
--R 2 3 2 2 2 2 3
--R 210A a d e + (- 168A a b - 84B a )d e + (48A b + 96B a b)d e
--R +
--R 2 4
--R - 32B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 4
--R 315e
--R Type: Expression(Integer)
--E 479
--S 480 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 480
)clear all
--S 481 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/sqrt(d+e*x)
--R
--R
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R (1) ----------------------------------------------------
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 481
--S 482 of 764
r0:=2/3*(b*d-a*e)*(3*b*B*d-2*A*b*e-a*B*e)*(d+e*x)^(3/2)/e^4-2/5*b*_
(3*b*B*d-A*b*e-2*a*B*e)*(d+e*x)^(5/2)/e^4+2/7*b^2*B*(d+e*x)^(7/2)/e^4-_
2*(b*d-a*e)^2*(B*d-A*e)*sqrt(d+e*x)/e^4
--R
--R
--R (2)
--R 2 3 3 2 3 2 2 2
--R 30B b e x + ((42A b + 84B a b)e - 36B b d e )x
--R +
--R 2 3 2 2 2 2
--R ((140A a b + 70B a )e + (- 56A b - 112B a b)d e + 48B b d e)x
--R +
--R 2 3 2 2 2 2
--R 210A a e + (- 280A a b - 140B a )d e + (112A b + 224B a b)d e
--R +
--R 2 3
--R - 96B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 4
--R 105e
--R Type: Expression(Integer)
--E 482
--S 483 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 483
)clear all
--S 484 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(3/2)
--R
--R
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R (1) ----------------------------------------------------
--R +-------+
--R (e x + d)\|e x + d
--R Type: Expression(Integer)
--E 484
--S 485 of 764
r0:=-2/3*b*(3*b*B*d-A*b*e-2*a*B*e)*(d+e*x)^(3/2)/e^4+2/5*b^2*B*_
(d+e*x)^(5/2)/e^4+2*(b*d-a*e)^2*(B*d-A*e)/(e^4*sqrt(d+e*x))+_
2*(b*d-a*e)*(3*b*B*d-2*A*b*e-a*B*e)*sqrt(d+e*x)/e^4
--R
--R
--R (2)
--R 2 3 3 2 3 2 2 2
--R 6B b e x + ((10A b + 20B a b)e - 12B b d e )x
--R +
--R 2 3 2 2 2 2 2 3
--R ((60A a b + 30B a )e + (- 40A b - 80B a b)d e + 48B b d e)x - 30A a e
--R +
--R 2 2 2 2 2 3
--R (120A a b + 60B a )d e + (- 80A b - 160B a b)d e + 96B b d
--R /
--R 4 +-------+
--R 15e \|e x + d
--R Type: Expression(Integer)
--E 485
--S 486 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 486
)clear all
--S 487 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(5/2)
--R
--R
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R (1) ----------------------------------------------------
--R 2 2 2 +-------+
--R (e x + 2d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 487
--S 488 of 764
r0:=2/3*(b*d-a*e)^2*(B*d-A*e)/(e^4*(d+e*x)^(3/2))+2/3*b^2*B*_
(d+e*x)^(3/2)/e^4-2*(b*d-a*e)*(3*b*B*d-2*A*b*e-a*B*e)/(e^4*_
sqrt(d+e*x))-2*b*(3*b*B*d-A*b*e-2*a*B*e)*sqrt(d+e*x)/e^4
--R
--R
--R (2)
--R 2 3 3 2 3 2 2 2
--R 2B b e x + ((6A b + 12B a b)e - 12B b d e )x
--R +
--R 2 3 2 2 2 2 2 3
--R ((- 12A a b - 6B a )e + (24A b + 48B a b)d e - 48B b d e)x - 2A a e
--R +
--R 2 2 2 2 2 3
--R (- 8A a b - 4B a )d e + (16A b + 32B a b)d e - 32B b d
--R /
--R 5 4 +-------+
--R (3e x + 3d e )\|e x + d
--R Type: Expression(Integer)
--E 488
--S 489 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 489
)clear all
--S 490 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(7/2)
--R
--R
--R 2 3 2 2 2 2
--R B b x + (A b + 2B a b)x + (2A a b + B a )x + A a
--R (1) ----------------------------------------------------
--R 3 3 2 2 2 3 +-------+
--R (e x + 3d e x + 3d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 490
--S 491 of 764
r0:=2/5*(b*d-a*e)^2*(B*d-A*e)/(e^4*(d+e*x)^(5/2))-2/3*(b*d-a*e)*_
(3*b*B*d-2*A*b*e-a*B*e)/(e^4*(d+e*x)^(3/2))+2*b*(3*b*B*d-A*b*e-_
2*a*B*e)/(e^4*sqrt(d+e*x))+2*b^2*B*sqrt(d+e*x)/e^4
--R
--R
--R (2)
--R 2 3 3 2 3 2 2 2
--R 30B b e x + ((- 30A b - 60B a b)e + 180B b d e )x
--R +
--R 2 3 2 2 2 2
--R ((- 20A a b - 10B a )e + (- 40A b - 80B a b)d e + 240B b d e)x
--R +
--R 2 3 2 2 2 2 2 3
--R - 6A a e + (- 8A a b - 4B a )d e + (- 16A b - 32B a b)d e + 96B b d
--R /
--R 6 2 5 2 4 +-------+
--R (15e x + 30d e x + 15d e )\|e x + d
--R Type: Expression(Integer)
--E 491
--S 492 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 492
)clear all
--S 493 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 3 8 4 3 3 4 2 7
--R B b e x + ((A b + 4B a b )e + 3B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 6
--R ((4A a b + 6B a b )e + (3A b + 12B a b )d e + 3B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (6A a b + 4B a b)e + (12A a b + 18B a b )d e
--R +
--R 4 3 2 4 3
--R (3A b + 12B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 3 4 3 2 2 3 2
--R (4A a b + B a )e + (18A a b + 12B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (12A a b + 18B a b )d e + (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 3 3 4 2 2 2 3 2
--R A a e + (12A a b + 3B a )d e + (18A a b + 12B a b)d e
--R +
--R 3 2 2 3
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 2 3 4 2 2 2 3 3 2
--R (3A a d e + (12A a b + 3B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 2 3 4 3 4 3
--R (3A a d e + (4A a b + B a )d )x + A a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 493
--S 494 of 764
r0:=-2/9*(b*d-a*e)^4*(B*d-A*e)*(d+e*x)^(9/2)/e^6+2/11*(b*d-a*e)^3*_
(5*b*B*d-4*A*b*e-a*B*e)*(d+e*x)^(11/2)/e^6-4/13*b*(b*d-a*e)^2*_
(5*b*B*d-3*A*b*e-2*a*B*e)*(d+e*x)^(13/2)/e^6+4/15*b^2*(b*d-a*e)*_
(5*b*B*d-2*A*b*e-3*a*B*e)*(d+e*x)^(15/2)/e^6-2/17*b^3*(5*b*B*d-_
A*b*e-4*a*B*e)*(d+e*x)^(17/2)/e^6+2/19*b^4*B*(d+e*x)^(19/2)/e^6
--R
--R
--R (2)
--R 4 9 9 4 3 9 4 8 8
--R 218790B b e x + ((244530A b + 978120B a b )e + 746460B b d e )x
--R +
--R 3 2 2 9
--R (1108536A a b + 1662804B a b )e
--R +
--R 4 3 8 4 2 7
--R (847704A b + 3390816B a b )d e + 866580B b d e
--R *
--R 7
--R x
--R +
--R 2 2 3 9
--R (1918620A a b + 1279080B a b)e
--R +
--R 3 2 2 8
--R (3922512A a b + 5883768B a b )d e
--R +
--R 4 3 2 7 4 3 6
--R (1005708A b + 4022832B a b )d e + 345840B b d e
--R *
--R 6
--R x
--R +
--R 3 4 9 2 2 3 8
--R (1511640A a b + 377910B a )e + (6976800A a b + 4651200B a b)d e
--R +
--R 3 2 2 2 7
--R (4790736A a b + 7186104B a b )d e
--R +
--R 4 3 3 6 4 4 5
--R (414504A b + 1658016B a b )d e + 630B b d e
--R *
--R 5
--R x
--R +
--R 4 9 3 4 8
--R 461890A a e + (5710640A a b + 1427660B a )d e
--R +
--R 2 2 3 2 7
--R (8876040A a b + 5917360B a b)d e
--R +
--R 3 2 2 3 6 4 3 4 5
--R (2067200A a b + 3100800B a b )d e + (1330A b + 5320B a b )d e
--R +
--R 4 5 4
--R - 700B b d e
--R *
--R 4
--R x
--R +
--R 4 8 3 4 2 7
--R 1847560A a d e + (7726160A a b + 1931540B a )d e
--R +
--R 2 2 3 3 6
--R (4108560A a b + 2739040B a b)d e
--R +
--R 3 2 2 4 5 4 3 5 4
--R (12920A a b + 19380B a b )d e + (- 1520A b - 6080B a b )d e
--R +
--R 4 6 3
--R 800B b d e
--R *
--R 3
--R x
--R +
--R 4 2 7 3 4 3 6
--R 2771340A a d e + (4031040A a b + 1007760B a )d e
--R +
--R 2 2 3 4 5 3 2 2 5 4
--R (58140A a b + 38760B a b)d e + (- 15504A a b - 23256B a b )d e
--R +
--R 4 3 6 3 4 7 2
--R (1824A b + 7296B a b )d e - 960B b d e
--R *
--R 2
--R x
--R +
--R 4 3 6 3 4 4 5
--R 1847560A a d e + (167960A a b + 41990B a )d e
--R +
--R 2 2 3 5 4 3 2 2 6 3
--R (- 77520A a b - 51680B a b)d e + (20672A a b + 31008B a b )d e
--R +
--R 4 3 7 2 4 8
--R (- 2432A b - 9728B a b )d e + 1280B b d e
--R *
--R x
--R +
--R 4 4 5 3 4 5 4
--R 461890A a d e + (- 335920A a b - 83980B a )d e
--R +
--R 2 2 3 6 3 3 2 2 7 2
--R (155040A a b + 103360B a b)d e + (- 41344A a b - 62016B a b )d e
--R +
--R 4 3 8 4 9
--R (4864A b + 19456B a b )d e - 2560B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 6
--R 2078505e
--R Type: Expression(Integer)
--E 494
--S 495 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 495
)clear all
--S 496 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 2 7 4 3 2 4 6
--R B b e x + ((A b + 4B a b )e + 2B b d e)x
--R +
--R 3 2 2 2 4 3 4 2 5
--R ((4A a b + 6B a b )e + (2A b + 8B a b )d e + B b d )x
--R +
--R 2 2 3 2 3 2 2 4 3 2 4
--R ((6A a b + 4B a b)e + (8A a b + 12B a b )d e + (A b + 4B a b )d )x
--R +
--R 3 4 2 2 2 3 3 2 2 2 3
--R ((4A a b + B a )e + (12A a b + 8B a b)d e + (4A a b + 6B a b )d )x
--R +
--R 4 2 3 4 2 2 3 2 2
--R (A a e + (8A a b + 2B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 3 4 2 4 2
--R (2A a d e + (4A a b + B a )d )x + A a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 496
--S 497 of 764
r0:=-2/7*(b*d-a*e)^4*(B*d-A*e)*(d+e*x)^(7/2)/e^6+2/9*(b*d-a*e)^3*_
(5*b*B*d-4*A*b*e-a*B*e)*(d+e*x)^(9/2)/e^6-4/11*b*(b*d-a*e)^2*_
(5*b*B*d-3*A*b*e-2*a*B*e)*(d+e*x)^(11/2)/e^6+4/13*b^2*(b*d-a*e)*_
(5*b*B*d-2*A*b*e-3*a*B*e)*(d+e*x)^(13/2)/e^6-2/15*b^3*(5*b*B*d-A*b*e-_
4*a*B*e)*(d+e*x)^(15/2)/e^6+2/17*b^4*B*(d+e*x)^(17/2)/e^6
--R
--R
--R (2)
--R 4 8 8 4 3 8 4 7 7
--R 90090B b e x + ((102102A b + 408408B a b )e + 210210B b d e )x
--R +
--R 3 2 2 8 4 3 7
--R (471240A a b + 706860B a b )e + (243474A b + 973896B a b )d e
--R +
--R 4 2 6
--R 127050B b d e
--R *
--R 6
--R x
--R +
--R 2 2 3 8
--R (835380A a b + 556920B a b)e
--R +
--R 3 2 2 7
--R (1156680A a b + 1735020B a b )d e
--R +
--R 4 3 2 6 4 3 5
--R (152082A b + 608328B a b )d e + 630B b d e
--R *
--R 5
--R x
--R +
--R 3 4 8 2 2 3 7
--R (680680A a b + 170170B a )e + (2134860A a b + 1423240B a b)d e
--R +
--R 3 2 2 2 6 4 3 3 5
--R (756840A a b + 1135260B a b )d e + (1190A b + 4760B a b )d e
--R +
--R 4 4 4
--R - 700B b d e
--R *
--R 4
--R x
--R +
--R 4 8 3 4 7
--R 218790A a e + (1847560A a b + 461890B a )d e
--R +
--R 2 2 3 2 6 3 2 2 3 5
--R (1498380A a b + 998920B a b)d e + (10200A a b + 15300B a b )d e
--R +
--R 4 3 4 4 4 5 3
--R (- 1360A b - 5440B a b )d e + 800B b d e
--R *
--R 3
--R x
--R +
--R 4 7 3 4 2 6
--R 656370A a d e + (1458600A a b + 364650B a )d e
--R +
--R 2 2 3 3 5 3 2 2 4 4
--R (39780A a b + 26520B a b)d e + (- 12240A a b - 18360B a b )d e
--R +
--R 4 3 5 3 4 6 2
--R (1632A b + 6528B a b )d e - 960B b d e
--R *
--R 2
--R x
--R +
--R 4 2 6 3 4 3 5
--R 656370A a d e + (97240A a b + 24310B a )d e
--R +
--R 2 2 3 4 4 3 2 2 5 3
--R (- 53040A a b - 35360B a b)d e + (16320A a b + 24480B a b )d e
--R +
--R 4 3 6 2 4 7
--R (- 2176A b - 8704B a b )d e + 1280B b d e
--R *
--R x
--R +
--R 4 3 5 3 4 4 4
--R 218790A a d e + (- 194480A a b - 48620B a )d e
--R +
--R 2 2 3 5 3 3 2 2 6 2
--R (106080A a b + 70720B a b)d e + (- 32640A a b - 48960B a b )d e
--R +
--R 4 3 7 4 8
--R (4352A b + 17408B a b )d e - 2560B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 6
--R 765765e
--R Type: Expression(Integer)
--E 497
--S 498 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 498
)clear all
--S 499 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 6 4 3 4 5
--R B b e x + ((A b + 4B a b )e + B b d)x
--R +
--R 3 2 2 4 3 4
--R ((4A a b + 6B a b )e + (A b + 4B a b )d)x
--R +
--R 2 2 3 3 2 2 3
--R ((6A a b + 4B a b)e + (4A a b + 6B a b )d)x
--R +
--R 3 4 2 2 3 2 4 3 4
--R ((4A a b + B a )e + (6A a b + 4B a b)d)x + (A a e + (4A a b + B a )d)x
--R +
--R 4
--R A a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 499
--S 500 of 764
r0:=-2/5*(b*d-a*e)^4*(B*d-A*e)*(d+e*x)^(5/2)/e^6+2/7*(b*d-a*e)^3*_
(5*b*B*d-4*A*b*e-a*B*e)*(d+e*x)^(7/2)/e^6-4/9*b*(b*d-a*e)^2*(5*b*B*d-_
3*A*b*e-2*a*B*e)*(d+e*x)^(9/2)/e^6+4/11*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-_
3*a*B*e)*(d+e*x)^(11/2)/e^6-2/13*b^3*(5*b*B*d-A*b*e-4*a*B*e)*_
(d+e*x)^(13/2)/e^6+2/15*b^4*B*(d+e*x)^(15/2)/e^6
--R
--R
--R (2)
--R 4 7 7 4 3 7 4 6 6
--R 6006B b e x + ((6930A b + 27720B a b )e + 7392B b d e )x
--R +
--R 3 2 2 7 4 3 6
--R (32760A a b + 49140B a b )e + (8820A b + 35280B a b )d e
--R +
--R 4 2 5
--R 126B b d e
--R *
--R 5
--R x
--R +
--R 2 2 3 7 3 2 2 6
--R (60060A a b + 40040B a b)e + (43680A a b + 65520B a b )d e
--R +
--R 4 3 2 5 4 3 4
--R (210A b + 840B a b )d e - 140B b d e
--R *
--R 4
--R x
--R +
--R 3 4 7 2 2 3 6
--R (51480A a b + 12870B a )e + (85800A a b + 57200B a b)d e
--R +
--R 3 2 2 2 5 4 3 3 4
--R (1560A a b + 2340B a b )d e + (- 240A b - 960B a b )d e
--R +
--R 4 4 3
--R 160B b d e
--R *
--R 3
--R x
--R +
--R 4 7 3 4 6
--R 18018A a e + (82368A a b + 20592B a )d e
--R +
--R 2 2 3 2 5 3 2 2 3 4
--R (5148A a b + 3432B a b)d e + (- 1872A a b - 2808B a b )d e
--R +
--R 4 3 4 3 4 5 2
--R (288A b + 1152B a b )d e - 192B b d e
--R *
--R 2
--R x
--R +
--R 4 6 3 4 2 5
--R 36036A a d e + (10296A a b + 2574B a )d e
--R +
--R 2 2 3 3 4 3 2 2 4 3
--R (- 6864A a b - 4576B a b)d e + (2496A a b + 3744B a b )d e
--R +
--R 4 3 5 2 4 6
--R (- 384A b - 1536B a b )d e + 256B b d e
--R *
--R x
--R +
--R 4 2 5 3 4 3 4
--R 18018A a d e + (- 20592A a b - 5148B a )d e
--R +
--R 2 2 3 4 3 3 2 2 5 2
--R (13728A a b + 9152B a b)d e + (- 4992A a b - 7488B a b )d e
--R +
--R 4 3 6 4 7
--R (768A b + 3072B a b )d e - 512B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 6
--R 45045e
--R Type: Expression(Integer)
--E 500
--S 501 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 501
)clear all
--S 502 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2*sqrt(d+e*x)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 502
--S 503 of 764
r0:=-2/3*(b*d-a*e)^4*(B*d-A*e)*(d+e*x)^(3/2)/e^6+2/5*(b*d-a*e)^3*_
(5*b*B*d-4*A*b*e-a*B*e)*(d+e*x)^(5/2)/e^6-4/7*b*(b*d-a*e)^2*_
(5*b*B*d-3*A*b*e-2*a*B*e)*(d+e*x)^(7/2)/e^6+4/9*b^2*(b*d-a*e)*_
(5*b*B*d-2*A*b*e-3*a*B*e)*(d+e*x)^(9/2)/e^6-2/11*b^3*_
(5*b*B*d-A*b*e-4*a*B*e)*(d+e*x)^(11/2)/e^6+2/13*b^4*B*(d+e*x)^(13/2)/e^6
--R
--R
--R (2)
--R 4 6 6 4 3 6 4 5 5
--R 6930B b e x + ((8190A b + 32760B a b )e + 630B b d e )x
--R +
--R 3 2 2 6 4 3 5
--R (40040A a b + 60060B a b )e + (910A b + 3640B a b )d e
--R +
--R 4 2 4
--R - 700B b d e
--R *
--R 4
--R x
--R +
--R 2 2 3 6 3 2 2 5
--R (77220A a b + 51480B a b)e + (5720A a b + 8580B a b )d e
--R +
--R 4 3 2 4 4 3 3
--R (- 1040A b - 4160B a b )d e + 800B b d e
--R *
--R 3
--R x
--R +
--R 3 4 6 2 2 3 5
--R (72072A a b + 18018B a )e + (15444A a b + 10296B a b)d e
--R +
--R 3 2 2 2 4 4 3 3 3
--R (- 6864A a b - 10296B a b )d e + (1248A b + 4992B a b )d e
--R +
--R 4 4 2
--R - 960B b d e
--R *
--R 2
--R x
--R +
--R 4 6 3 4 5
--R 30030A a e + (24024A a b + 6006B a )d e
--R +
--R 2 2 3 2 4 3 2 2 3 3
--R (- 20592A a b - 13728B a b)d e + (9152A a b + 13728B a b )d e
--R +
--R 4 3 4 2 4 5
--R (- 1664A b - 6656B a b )d e + 1280B b d e
--R *
--R x
--R +
--R 4 5 3 4 2 4
--R 30030A a d e + (- 48048A a b - 12012B a )d e
--R +
--R 2 2 3 3 3 3 2 2 4 2
--R (41184A a b + 27456B a b)d e + (- 18304A a b - 27456B a b )d e
--R +
--R 4 3 5 4 6
--R (3328A b + 13312B a b )d e - 2560B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 6
--R 45045e
--R Type: Expression(Integer)
--E 503
--S 504 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 504
)clear all
--S 505 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/sqrt(d+e*x)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 505
--S 506 of 764
r0:=2/3*(b*d-a*e)^3*(5*b*B*d-4*A*b*e-a*B*e)*(d+e*x)^(3/2)/e^6-4/5*b*_
(b*d-a*e)^2*(5*b*B*d-3*A*b*e-2*a*B*e)*(d+e*x)^(5/2)/e^6+4/7*b^2*_
(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)*(d+e*x)^(7/2)/e^6-2/9*b^3*_
(5*b*B*d-A*b*e-4*a*B*e)*(d+e*x)^(9/2)/e^6+2/11*b^4*B*(d+e*x)^(11/2)/e^6-_
2*(b*d-a*e)^4*(B*d-A*e)*sqrt(d+e*x)/e^6
--R
--R
--R (2)
--R 4 5 5 4 3 5 4 4 4
--R 630B b e x + ((770A b + 3080B a b )e - 700B b d e )x
--R +
--R 3 2 2 5 4 3 4
--R (3960A a b + 5940B a b )e + (- 880A b - 3520B a b )d e
--R +
--R 4 2 3
--R 800B b d e
--R *
--R 3
--R x
--R +
--R 2 2 3 5 3 2 2 4
--R (8316A a b + 5544B a b)e + (- 4752A a b - 7128B a b )d e
--R +
--R 4 3 2 3 4 3 2
--R (1056A b + 4224B a b )d e - 960B b d e
--R *
--R 2
--R x
--R +
--R 3 4 5 2 2 3 4
--R (9240A a b + 2310B a )e + (- 11088A a b - 7392B a b)d e
--R +
--R 3 2 2 2 3 4 3 3 2
--R (6336A a b + 9504B a b )d e + (- 1408A b - 5632B a b )d e
--R +
--R 4 4
--R 1280B b d e
--R *
--R x
--R +
--R 4 5 3 4 4
--R 6930A a e + (- 18480A a b - 4620B a )d e
--R +
--R 2 2 3 2 3 3 2 2 3 2
--R (22176A a b + 14784B a b)d e + (- 12672A a b - 19008B a b )d e
--R +
--R 4 3 4 4 5
--R (2816A b + 11264B a b )d e - 2560B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 6
--R 3465e
--R Type: Expression(Integer)
--E 506
--S 507 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 507
)clear all
--S 508 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^(3/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R +-------+
--R (e x + d)\|e x + d
--R Type: Expression(Integer)
--E 508
--S 509 of 764
r0:=-4/3*b*(b*d-a*e)^2*(5*b*B*d-3*A*b*e-2*a*B*e)*(d+e*x)^(3/2)/e^6+_
4/5*b^2*(b*d-a*e)*(5*b*B*d-2*A*b*e-3*a*B*e)*(d+e*x)^(5/2)/e^6-_
2/7*b^3*(5*b*B*d-A*b*e-4*a*B*e)*(d+e*x)^(7/2)/e^6+2/9*b^4*B*_
(d+e*x)^(9/2)/e^6+2*(b*d-a*e)^4*(B*d-A*e)/(e^6*sqrt(d+e*x))+_
2*(b*d-a*e)^3*(5*b*B*d-4*A*b*e-a*B*e)*sqrt(d+e*x)/e^6
--R
--R
--R (2)
--R 4 5 5 4 3 5 4 4 4
--R 70B b e x + ((90A b + 360B a b )e - 100B b d e )x
--R +
--R 3 2 2 5 4 3 4 4 2 3 3
--R ((504A a b + 756B a b )e + (- 144A b - 576B a b )d e + 160B b d e )x
--R +
--R 2 2 3 5 3 2 2 4
--R (1260A a b + 840B a b)e + (- 1008A a b - 1512B a b )d e
--R +
--R 4 3 2 3 4 3 2
--R (288A b + 1152B a b )d e - 320B b d e
--R *
--R 2
--R x
--R +
--R 3 4 5 2 2 3 4
--R (2520A a b + 630B a )e + (- 5040A a b - 3360B a b)d e
--R +
--R 3 2 2 2 3 4 3 3 2
--R (4032A a b + 6048B a b )d e + (- 1152A b - 4608B a b )d e
--R +
--R 4 4
--R 1280B b d e
--R *
--R x
--R +
--R 4 5 3 4 4
--R - 630A a e + (5040A a b + 1260B a )d e
--R +
--R 2 2 3 2 3 3 2 2 3 2
--R (- 10080A a b - 6720B a b)d e + (8064A a b + 12096B a b )d e
--R +
--R 4 3 4 4 5
--R (- 2304A b - 9216B a b )d e + 2560B b d
--R /
--R 6 +-------+
--R 315e \|e x + d
--R Type: Expression(Integer)
--E 509
--S 510 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 510
)clear all
--S 511 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^(5/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 2 2 2 +-------+
--R (e x + 2d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 511
--S 512 of 764
r0:=2/3*(b*d-a*e)^4*(B*d-A*e)/(e^6*(d+e*x)^(3/2))+4/3*b^2*(b*d-a*e)*_
(5*b*B*d-2*A*b*e-3*a*B*e)*(d+e*x)^(3/2)/e^6-2/5*b^3*(5*b*B*d-A*b*e-_
4*a*B*e)*(d+e*x)^(5/2)/e^6+2/7*b^4*B*(d+e*x)^(7/2)/e^6-2*(b*d-_
a*e)^3*(5*b*B*d-4*A*b*e-a*B*e)/(e^6*sqrt(d+e*x))-4*b*(b*d-a*e)^2*_
(5*b*B*d-3*A*b*e-2*a*B*e)*sqrt(d+e*x)/e^6
--R
--R
--R (2)
--R 4 5 5 4 3 5 4 4 4
--R 30B b e x + ((42A b + 168B a b )e - 60B b d e )x
--R +
--R 3 2 2 5 4 3 4 4 2 3 3
--R ((280A a b + 420B a b )e + (- 112A b - 448B a b )d e + 160B b d e )x
--R +
--R 2 2 3 5 3 2 2 4
--R (1260A a b + 840B a b)e + (- 1680A a b - 2520B a b )d e
--R +
--R 4 3 2 3 4 3 2
--R (672A b + 2688B a b )d e - 960B b d e
--R *
--R 2
--R x
--R +
--R 3 4 5 2 2 3 4
--R (- 840A a b - 210B a )e + (5040A a b + 3360B a b)d e
--R +
--R 3 2 2 2 3 4 3 3 2
--R (- 6720A a b - 10080B a b )d e + (2688A b + 10752B a b )d e
--R +
--R 4 4
--R - 3840B b d e
--R *
--R x
--R +
--R 4 5 3 4 4 2 2 3 2 3
--R - 70A a e + (- 560A a b - 140B a )d e + (3360A a b + 2240B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (- 4480A a b - 6720B a b )d e + (1792A b + 7168B a b )d e - 2560B b d
--R /
--R 7 6 +-------+
--R (105e x + 105d e )\|e x + d
--R Type: Expression(Integer)
--E 512
--S 513 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 513
)clear all
--S 514 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^2/(d+e*x)^(7/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R /
--R 3 3 2 2 2 3 +-------+
--R (e x + 3d e x + 3d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 514
--S 515 of 764
r0:=2/5*(b*d-a*e)^4*(B*d-A*e)/(e^6*(d+e*x)^(5/2))-2/3*(b*d-a*e)^3*_
(5*b*B*d-4*A*b*e-a*B*e)/(e^6*(d+e*x)^(3/2))-2/3*b^3*(5*b*B*d-A*b*e-_
4*a*B*e)*(d+e*x)^(3/2)/e^6+2/5*b^4*B*(d+e*x)^(5/2)/e^6+4*b*(b*d-a*e)^2*_
(5*b*B*d-3*A*b*e-2*a*B*e)/(e^6*sqrt(d+e*x))+4*b^2*(b*d-a*e)*_
(5*b*B*d-2*A*b*e-3*a*B*e)*sqrt(d+e*x)/e^6
--R
--R
--R (2)
--R 4 5 5 4 3 5 4 4 4
--R 6B b e x + ((10A b + 40B a b )e - 20B b d e )x
--R +
--R 3 2 2 5 4 3 4 4 2 3 3
--R ((120A a b + 180B a b )e + (- 80A b - 320B a b )d e + 160B b d e )x
--R +
--R 2 2 3 5 3 2 2 4
--R (- 180A a b - 120B a b)e + (720A a b + 1080B a b )d e
--R +
--R 4 3 2 3 4 3 2
--R (- 480A b - 1920B a b )d e + 960B b d e
--R *
--R 2
--R x
--R +
--R 3 4 5 2 2 3 4
--R (- 40A a b - 10B a )e + (- 240A a b - 160B a b)d e
--R +
--R 3 2 2 2 3 4 3 3 2
--R (960A a b + 1440B a b )d e + (- 640A b - 2560B a b )d e
--R +
--R 4 4
--R 1280B b d e
--R *
--R x
--R +
--R 4 5 3 4 4 2 2 3 2 3
--R - 6A a e + (- 16A a b - 4B a )d e + (- 96A a b - 64B a b)d e
--R +
--R 3 2 2 3 2 4 3 4 4 5
--R (384A a b + 576B a b )d e + (- 256A b - 1024B a b )d e + 512B b d
--R /
--R 8 2 7 2 6 +-------+
--R (15e x + 30d e x + 15d e )\|e x + d
--R Type: Expression(Integer)
--E 515
--S 516 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 516
)clear all
--S 517 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^3
--R
--R
--R (1)
--R 6 3 10 6 5 3 6 2 9
--R B b e x + ((A b + 6B a b )e + 3B b d e )x
--R +
--R 5 2 4 3 6 5 2 6 2 8
--R ((6A a b + 15B a b )e + (3A b + 18B a b )d e + 3B b d e)x
--R +
--R 2 4 3 3 3 5 2 4 2
--R (15A a b + 20B a b )e + (18A a b + 45B a b )d e
--R +
--R 6 5 2 6 3
--R (3A b + 18B a b )d e + B b d
--R *
--R 7
--R x
--R +
--R 3 3 4 2 3 2 4 3 3 2
--R (20A a b + 15B a b )e + (45A a b + 60B a b )d e
--R +
--R 5 2 4 2 6 5 3
--R (18A a b + 45B a b )d e + (A b + 6B a b )d
--R *
--R 6
--R x
--R +
--R 4 2 5 3 3 3 4 2 2
--R (15A a b + 6B a b)e + (60A a b + 45B a b )d e
--R +
--R 2 4 3 3 2 5 2 4 3
--R (45A a b + 60B a b )d e + (6A a b + 15B a b )d
--R *
--R 5
--R x
--R +
--R 5 6 3 4 2 5 2
--R (6A a b + B a )e + (45A a b + 18B a b)d e
--R +
--R 3 3 4 2 2 2 4 3 3 3
--R (60A a b + 45B a b )d e + (15A a b + 20B a b )d
--R *
--R 4
--R x
--R +
--R 6 3 5 6 2 4 2 5 2
--R A a e + (18A a b + 3B a )d e + (45A a b + 18B a b)d e
--R +
--R 3 3 4 2 3
--R (20A a b + 15B a b )d
--R *
--R 3
--R x
--R +
--R 6 2 5 6 2 4 2 5 3 2
--R (3A a d e + (18A a b + 3B a )d e + (15A a b + 6B a b)d )x
--R +
--R 6 2 5 6 3 6 3
--R (3A a d e + (6A a b + B a )d )x + A a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 517
--S 518 of 764
r0:=-2/9*(b*d-a*e)^6*(B*d-A*e)*(d+e*x)^(9/2)/e^8+2/11*(b*d-a*e)^5*_
(7*b*B*d-6*A*b*e-a*B*e)*(d+e*x)^(11/2)/e^8-6/13*b*(b*d-a*e)^4*_
(7*b*B*d-5*A*b*e-2*a*B*e)*(d+e*x)^(13/2)/e^8+2/3*b^2*(b*d-a*e)^3*_
(7*b*B*d-4*A*b*e-3*a*B*e)*(d+e*x)^(15/2)/e^8-10/17*b^3*(b*d-a*e)^2*_
(7*b*B*d-3*A*b*e-4*a*B*e)*(d+e*x)^(17/2)/e^8+6/19*b^4*(b*d-a*e)*_
(7*b*B*d-2*A*b*e-5*a*B*e)*(d+e*x)^(19/2)/e^8-2/21*b^5*(7*b*B*d-A*b*e-_
6*a*B*e)*(d+e*x)^(21/2)/e^8+2/23*b^6*B*(d+e*x)^(23/2)/e^8
--R
--R
--R (2)
--R 6 11 11
--R 5819814B b e x
--R +
--R 6 5 11 6 10 10
--R ((6374082A b + 38244492B a b )e + 19399380B b d e )x
--R +
--R 5 2 4 11
--R (42270228A a b + 105675570B a b )e
--R +
--R 6 5 10 6 2 9
--R (21470592A b + 128823552B a b )d e + 21849828B b d e
--R *
--R 9
--R x
--R +
--R 2 4 3 3 11
--R (118107990A a b + 157477320B a b )e
--R +
--R 5 2 4 10
--R (144216072A a b + 360540180B a b )d e
--R +
--R 6 5 2 9 6 3 8
--R (24509628A b + 147057768B a b )d e + 8360352B b d e
--R *
--R 8
--R x
--R +
--R 3 3 4 2 11
--R (178474296A a b + 133855722B a b )e
--R +
--R 2 4 3 3 10
--R (409441032A a b + 545921376B a b )d e
--R +
--R 5 2 4 2 9
--R (167423256A a b + 418558140B a b )d e
--R +
--R 6 5 3 8 6 4 7
--R (9551256A b + 57307536B a b )d e + 6006B b d e
--R *
--R 7
--R x
--R +
--R 4 2 5 11
--R (154448910A a b + 61779564B a b)e
--R +
--R 3 3 4 2 10
--R (631524432A a b + 473643324B a b )d e
--R +
--R 2 4 3 3 2 9
--R (485756964A a b + 647675952B a b )d e
--R +
--R 5 2 4 3 8
--R (66816288A a b + 167040720B a b )d e
--R +
--R 6 5 4 7 6 5 6
--R (10626A b + 63756B a b )d e - 6468B b d e
--R *
--R 6
--R x
--R +
--R 5 6 11
--R (73012212A a b + 12168702B a )e
--R +
--R 4 2 5 10
--R (561632400A a b + 224652960B a b)d e
--R +
--R 3 3 4 2 2 9
--R (771308496A a b + 578481372B a b )d e
--R +
--R 2 4 3 3 3 8
--R (200205432A a b + 266940576B a b )d e
--R +
--R 5 2 4 4 7 6 5 5 6
--R (121716A a b + 304290B a b )d e + (- 11592A b - 69552B a b )d e
--R +
--R 6 6 5
--R 7056B b d e
--R *
--R 5
--R x
--R +
--R 6 11 5 6 10
--R 14872858A a e + (275823912A a b + 45970652B a )d e
--R +
--R 4 2 5 2 9
--R (714521220A a b + 285808488B a b)d e
--R +
--R 3 3 4 2 3 8
--R (332819200A a b + 249614400B a b )d e
--R +
--R 2 4 3 3 4 7
--R (642390A a b + 856520B a b )d e
--R +
--R 5 2 4 5 6 6 5 6 5
--R (- 135240A a b - 338100B a b )d e + (12880A b + 77280B a b )d e
--R +
--R 6 7 4
--R - 7840B b d e
--R *
--R 4
--R x
--R +
--R 6 10 5 6 2 9
--R 59491432A a d e + (373173528A a b + 62195588B a )d e
--R +
--R 4 2 5 3 8
--R (330739080A a b + 132295632B a b)d e
--R +
--R 3 3 4 2 4 7
--R (2080120A a b + 1560090B a b )d e
--R +
--R 2 4 3 3 5 6
--R (- 734160A a b - 978880B a b )d e
--R +
--R 5 2 4 6 5 6 5 7 4
--R (154560A a b + 386400B a b )d e + (- 14720A b - 88320B a b )d e
--R +
--R 6 8 3
--R 8960B b d e
--R *
--R 3
--R x
--R +
--R 6 2 9 5 6 3 8
--R 89237148A a d e + (194699232A a b + 32449872B a )d e
--R +
--R 4 2 5 4 7
--R (4680270A a b + 1872108B a b)d e
--R +
--R 3 3 4 2 5 6
--R (- 2496144A a b - 1872108B a b )d e
--R +
--R 2 4 3 3 6 5
--R (880992A a b + 1174656B a b )d e
--R +
--R 5 2 4 7 4
--R (- 185472A a b - 463680B a b )d e
--R +
--R 6 5 8 3 6 9 2
--R (17664A b + 105984B a b )d e - 10752B b d e
--R *
--R 2
--R x
--R +
--R 6 3 8 5 6 4 7
--R 59491432A a d e + (8112468A a b + 1352078B a )d e
--R +
--R 4 2 5 5 6
--R (- 6240360A a b - 2496144B a b)d e
--R +
--R 3 3 4 2 6 5
--R (3328192A a b + 2496144B a b )d e
--R +
--R 2 4 3 3 7 4
--R (- 1174656A a b - 1566208B a b )d e
--R +
--R 5 2 4 8 3
--R (247296A a b + 618240B a b )d e
--R +
--R 6 5 9 2 6 10
--R (- 23552A b - 141312B a b )d e + 14336B b d e
--R *
--R x
--R +
--R 6 4 7 5 6 5 6
--R 14872858A a d e + (- 16224936A a b - 2704156B a )d e
--R +
--R 4 2 5 6 5
--R (12480720A a b + 4992288B a b)d e
--R +
--R 3 3 4 2 7 4
--R (- 6656384A a b - 4992288B a b )d e
--R +
--R 2 4 3 3 8 3
--R (2349312A a b + 3132416B a b )d e
--R +
--R 5 2 4 9 2 6 5 10
--R (- 494592A a b - 1236480B a b )d e + (47104A b + 282624B a b )d e
--R +
--R 6 11
--R - 28672B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 8
--R 66927861e
--R Type: Expression(Integer)
--E 518
--S 519 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 519
)clear all
--S 520 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^3
--R
--R
--R (1)
--R 6 2 9 6 5 2 6 8
--R B b e x + ((A b + 6B a b )e + 2B b d e)x
--R +
--R 5 2 4 2 6 5 6 2 7
--R ((6A a b + 15B a b )e + (2A b + 12B a b )d e + B b d )x
--R +
--R 2 4 3 3 2 5 2 4
--R (15A a b + 20B a b )e + (12A a b + 30B a b )d e
--R +
--R 6 5 2
--R (A b + 6B a b )d
--R *
--R 6
--R x
--R +
--R 3 3 4 2 2 2 4 3 3
--R (20A a b + 15B a b )e + (30A a b + 40B a b )d e
--R +
--R 5 2 4 2
--R (6A a b + 15B a b )d
--R *
--R 5
--R x
--R +
--R 4 2 5 2 3 3 4 2
--R (15A a b + 6B a b)e + (40A a b + 30B a b )d e
--R +
--R 2 4 3 3 2
--R (15A a b + 20B a b )d
--R *
--R 4
--R x
--R +
--R 5 6 2 4 2 5 3 3 4 2 2 3
--R ((6A a b + B a )e + (30A a b + 12B a b)d e + (20A a b + 15B a b )d )x
--R +
--R 6 2 5 6 4 2 5 2 2
--R (A a e + (12A a b + 2B a )d e + (15A a b + 6B a b)d )x
--R +
--R 6 5 6 2 6 2
--R (2A a d e + (6A a b + B a )d )x + A a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 520
--S 521 of 764
r0:=-2/7*(b*d-a*e)^6*(B*d-A*e)*(d+e*x)^(7/2)/e^8+2/9*(b*d-a*e)^5*_
(7*b*B*d-6*A*b*e-a*B*e)*(d+e*x)^(9/2)/e^8-6/11*b*(b*d-a*e)^4*_
(7*b*B*d-5*A*b*e-2*a*B*e)*(d+e*x)^(11/2)/e^8+10/13*b^2*(b*d-a*e)^3*_
(7*b*B*d-4*A*b*e-3*a*B*e)*(d+e*x)^(13/2)/e^8-2/3*b^3*(b*d-a*e)^2*_
(7*b*B*d-3*A*b*e-4*a*B*e)*(d+e*x)^(15/2)/e^8+6/17*b^4*(b*d-a*e)*_
(7*b*B*d-2*A*b*e-5*a*B*e)*(d+e*x)^(17/2)/e^8-2/19*b^5*_
(7*b*B*d-A*b*e-6*a*B*e)*(d+e*x)^(19/2)/e^8+2/21*b^6*B*(d+e*x)^(21/2)/e^8
--R
--R
--R (2)
--R 6 10 10 6 5 10 6 9 9
--R 277134B b e x + ((306306A b + 1837836B a b )e + 627198B b d e )x
--R +
--R 5 2 4 10
--R (2054052A a b + 5135130B a b )e
--R +
--R 6 5 9 6 2 8
--R (702702A b + 4216212B a b )d e + 362934B b d e
--R *
--R 8
--R x
--R +
--R 2 4 3 3 10
--R (5819814A a b + 7759752B a b )e
--R +
--R 5 2 4 9
--R (4792788A a b + 11981970B a b )d e
--R +
--R 6 5 2 8 6 3 7
--R (414414A b + 2486484B a b )d e + 858B b d e
--R *
--R 7
--R x
--R +
--R 3 3 4 2 10
--R (8953560A a b + 6715170B a b )e
--R +
--R 2 4 3 3 9
--R (13878018A a b + 18504024B a b )d e
--R +
--R 5 2 4 2 8 6 5 3 7
--R (2896740A a b + 7241850B a b )d e + (1386A b + 8316B a b )d e
--R +
--R 6 4 6
--R - 924B b d e
--R *
--R 6
--R x
--R +
--R 4 2 5 10
--R (7936110A a b + 3174444B a b)e
--R +
--R 3 3 4 2 9
--R (21976920A a b + 16482690B a b )d e
--R +
--R 2 4 3 3 2 8
--R (8668674A a b + 11558232B a b )d e
--R +
--R 5 2 4 3 7 6 5 4 6
--R (14364A a b + 35910B a b )d e + (- 1512A b - 9072B a b )d e
--R +
--R 6 5 5
--R 1008B b d e
--R *
--R 5
--R x
--R +
--R 5 6 10
--R (3879876A a b + 646646B a )e
--R +
--R 4 2 5 9
--R (20281170A a b + 8112468B a b)d e
--R +
--R 3 3 4 2 2 8
--R (14379960A a b + 10784970B a b )d e
--R +
--R 2 4 3 3 3 7 5 2 4 4 6
--R (67830A a b + 90440B a b )d e + (- 15960A a b - 39900B a b )d e
--R +
--R 6 5 5 5 6 6 4
--R (1680A b + 10080B a b )d e - 1120B b d e
--R *
--R 4
--R x
--R +
--R 6 10 5 6 9
--R 831402A a e + (10531092A a b + 1755182B a )d e
--R +
--R 4 2 5 2 8
--R (14234610A a b + 5693844B a b)d e
--R +
--R 3 3 4 2 3 7
--R (193800A a b + 145350B a b )d e
--R +
--R 2 4 3 3 4 6
--R (- 77520A a b - 103360B a b )d e
--R +
--R 5 2 4 5 5 6 5 6 4
--R (18240A a b + 45600B a b )d e + (- 1920A b - 11520B a b )d e
--R +
--R 6 7 3
--R 1280B b d e
--R *
--R 3
--R x
--R +
--R 6 9 5 6 2 8
--R 2494206A a d e + (8314020A a b + 1385670B a )d e
--R +
--R 4 2 5 3 7
--R (377910A a b + 151164B a b)d e
--R +
--R 3 3 4 2 4 6
--R (- 232560A a b - 174420B a b )d e
--R +
--R 2 4 3 3 5 5
--R (93024A a b + 124032B a b )d e
--R +
--R 5 2 4 6 4 6 5 7 3
--R (- 21888A a b - 54720B a b )d e + (2304A b + 13824B a b )d e
--R +
--R 6 8 2
--R - 1536B b d e
--R *
--R 2
--R x
--R +
--R 6 2 8 5 6 3 7
--R 2494206A a d e + (554268A a b + 92378B a )d e
--R +
--R 4 2 5 4 6
--R (- 503880A a b - 201552B a b)d e
--R +
--R 3 3 4 2 5 5
--R (310080A a b + 232560B a b )d e
--R +
--R 2 4 3 3 6 4
--R (- 124032A a b - 165376B a b )d e
--R +
--R 5 2 4 7 3 6 5 8 2
--R (29184A a b + 72960B a b )d e + (- 3072A b - 18432B a b )d e
--R +
--R 6 9
--R 2048B b d e
--R *
--R x
--R +
--R 6 3 7 5 6 4 6
--R 831402A a d e + (- 1108536A a b - 184756B a )d e
--R +
--R 4 2 5 5 5 3 3 4 2 6 4
--R (1007760A a b + 403104B a b)d e + (- 620160A a b - 465120B a b )d e
--R +
--R 2 4 3 3 7 3 5 2 4 8 2
--R (248064A a b + 330752B a b )d e + (- 58368A a b - 145920B a b )d e
--R +
--R 6 5 9 6 10
--R (6144A b + 36864B a b )d e - 4096B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 8
--R 2909907e
--R Type: Expression(Integer)
--E 521
--S 522 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 522
)clear all
--S 523 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^3
--R
--R
--R (1)
--R 6 8 6 5 6 7
--R B b e x + ((A b + 6B a b )e + B b d)x
--R +
--R 5 2 4 6 5 6
--R ((6A a b + 15B a b )e + (A b + 6B a b )d)x
--R +
--R 2 4 3 3 5 2 4 5
--R ((15A a b + 20B a b )e + (6A a b + 15B a b )d)x
--R +
--R 3 3 4 2 2 4 3 3 4
--R ((20A a b + 15B a b )e + (15A a b + 20B a b )d)x
--R +
--R 4 2 5 3 3 4 2 3
--R ((15A a b + 6B a b)e + (20A a b + 15B a b )d)x
--R +
--R 5 6 4 2 5 2 6 5 6
--R ((6A a b + B a )e + (15A a b + 6B a b)d)x + (A a e + (6A a b + B a )d)x
--R +
--R 6
--R A a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 523
--S 524 of 764
r0:=-2/5*(b*d-a*e)^6*(B*d-A*e)*(d+e*x)^(5/2)/e^8+2/7*(b*d-a*e)^5*_
(7*b*B*d-6*A*b*e-a*B*e)*(d+e*x)^(7/2)/e^8-2/3*b*(b*d-a*e)^4*_
(7*b*B*d-5*A*b*e-2*a*B*e)*(d+e*x)^(9/2)/e^8+10/11*b^2*(b*d-a*e)^3*_
(7*b*B*d-4*A*b*e-3*a*B*e)*(d+e*x)^(11/2)/e^8-10/13*b^3*(b*d-a*e)^2*_
(7*b*B*d-3*A*b*e-4*a*B*e)*(d+e*x)^(13/2)/e^8+2/5*b^4*(b*d-a*e)*_
(7*b*B*d-2*A*b*e-5*a*B*e)*(d+e*x)^(15/2)/e^8-2/17*b^5*_
(7*b*B*d-A*b*e-6*a*B*e)*(d+e*x)^(17/2)/e^8+2/19*b^6*B*(d+e*x)^(19/2)/e^8
--R
--R
--R (2)
--R 6 9 9 6 5 9 6 8 8
--R 510510B b e x + ((570570A b + 3423420B a b )e + 600600B b d e )x
--R +
--R 5 2 4 9
--R (3879876A a b + 9699690B a b )e
--R +
--R 6 5 8 6 2 7
--R (684684A b + 4108104B a b )d e + 6006B b d e
--R *
--R 7
--R x
--R +
--R 2 4 3 3 9
--R (11191950A a b + 14922600B a b )e
--R +
--R 5 2 4 8 6 5 2 7
--R (4775232A a b + 11938080B a b )d e + (8778A b + 52668B a b )d e
--R +
--R 6 3 6
--R - 6468B b d e
--R *
--R 6
--R x
--R +
--R 3 3 4 2 9
--R (17635800A a b + 13226850B a b )e
--R +
--R 2 4 3 3 8
--R (14244300A a b + 18992400B a b )d e
--R +
--R 5 2 4 2 7 6 5 3 6
--R (81396A a b + 203490B a b )d e + (- 9576A b - 57456B a b )d e
--R +
--R 6 4 5
--R 7056B b d e
--R *
--R 5
--R x
--R +
--R 4 2 5 9
--R (16166150A a b + 6466460B a b)e
--R +
--R 3 3 4 2 8
--R (23514400A a b + 17635800B a b )d e
--R +
--R 2 4 3 3 2 7
--R (339150A a b + 452200B a b )d e
--R +
--R 5 2 4 3 6 6 5 4 5
--R (- 90440A a b - 226100B a b )d e + (10640A b + 63840B a b )d e
--R +
--R 6 5 4
--R - 7840B b d e
--R *
--R 4
--R x
--R +
--R 5 6 9
--R (8314020A a b + 1385670B a )e
--R +
--R 4 2 5 8
--R (23094500A a b + 9237800B a b)d e
--R +
--R 3 3 4 2 2 7
--R (839800A a b + 629850B a b )d e
--R +
--R 2 4 3 3 3 6
--R (- 387600A a b - 516800B a b )d e
--R +
--R 5 2 4 4 5 6 5 5 4
--R (103360A a b + 258400B a b )d e + (- 12160A b - 72960B a b )d e
--R +
--R 6 6 3
--R 8960B b d e
--R *
--R 3
--R x
--R +
--R 6 9 5 6 8
--R 1939938A a e + (13302432A a b + 2217072B a )d e
--R +
--R 4 2 5 2 7
--R (1385670A a b + 554268B a b)d e
--R +
--R 3 3 4 2 3 6
--R (- 1007760A a b - 755820B a b )d e
--R +
--R 2 4 3 3 4 5
--R (465120A a b + 620160B a b )d e
--R +
--R 5 2 4 5 4 6 5 6 3
--R (- 124032A a b - 310080B a b )d e + (14592A b + 87552B a b )d e
--R +
--R 6 7 2
--R - 10752B b d e
--R *
--R 2
--R x
--R +
--R 6 8 5 6 2 7
--R 3879876A a d e + (1662804A a b + 277134B a )d e
--R +
--R 4 2 5 3 6
--R (- 1847560A a b - 739024B a b)d e
--R +
--R 3 3 4 2 4 5
--R (1343680A a b + 1007760B a b )d e
--R +
--R 2 4 3 3 5 4
--R (- 620160A a b - 826880B a b )d e
--R +
--R 5 2 4 6 3
--R (165376A a b + 413440B a b )d e
--R +
--R 6 5 7 2 6 8
--R (- 19456A b - 116736B a b )d e + 14336B b d e
--R *
--R x
--R +
--R 6 2 7 5 6 3 6
--R 1939938A a d e + (- 3325608A a b - 554268B a )d e
--R +
--R 4 2 5 4 5
--R (3695120A a b + 1478048B a b)d e
--R +
--R 3 3 4 2 5 4
--R (- 2687360A a b - 2015520B a b )d e
--R +
--R 2 4 3 3 6 3
--R (1240320A a b + 1653760B a b )d e
--R +
--R 5 2 4 7 2 6 5 8
--R (- 330752A a b - 826880B a b )d e + (38912A b + 233472B a b )d e
--R +
--R 6 9
--R - 28672B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 8
--R 4849845e
--R Type: Expression(Integer)
--E 524
--S 525 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 525
)clear all
--S 526 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^3*sqrt(d+e*x)
--R
--R
--R (1)
--R 6 7 6 5 6 5 2 4 5
--R B b x + (A b + 6B a b )x + (6A a b + 15B a b )x
--R +
--R 2 4 3 3 4 3 3 4 2 3 4 2 5 2
--R (15A a b + 20B a b )x + (20A a b + 15B a b )x + (15A a b + 6B a b)x
--R +
--R 5 6 6
--R (6A a b + B a )x + A a
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 526
--S 527 of 764
r0:=-2/3*(b*d-a*e)^6*(B*d-A*e)*(d+e*x)^(3/2)/e^8+2/5*(b*d-a*e)^5*_
(7*b*B*d-6*A*b*e-a*B*e)*(d+e*x)^(5/2)/e^8-6/7*b*(b*d-a*e)^4*_
(7*b*B*d-5*A*b*e-2*a*B*e)*(d+e*x)^(7/2)/e^8+10/9*b^2*(b*d-a*e)^3*_
(7*b*B*d-4*A*b*e-3*a*B*e)*(d+e*x)^(9/2)/e^8-10/11*b^3*(b*d-a*e)^2*_
(7*b*B*d-3*A*b*e-4*a*B*e)*(d+e*x)^(11/2)/e^8+6/13*b^4*(b*d-a*e)*_
(7*b*B*d-2*A*b*e-5*a*B*e)*(d+e*x)^(13/2)/e^8-2/15*b^5*_
(7*b*B*d-A*b*e-6*a*B*e)*(d+e*x)^(15/2)/e^8+2/17*b^6*B*(d+e*x)^(17/2)/e^8
--R
--R
--R (2)
--R 6 8 8 6 5 8 6 7 7
--R 90090B b e x + ((102102A b + 612612B a b )e + 6006B b d e )x
--R +
--R 5 2 4 8 6 5 7
--R (706860A a b + 1767150B a b )e + (7854A b + 47124B a b )d e
--R +
--R 6 2 6
--R - 6468B b d e
--R *
--R 6
--R x
--R +
--R 2 4 3 3 8
--R (2088450A a b + 2784600B a b )e
--R +
--R 5 2 4 7 6 5 2 6
--R (64260A a b + 160650B a b )d e + (- 8568A b - 51408B a b )d e
--R +
--R 6 3 5
--R 7056B b d e
--R *
--R 5
--R x
--R +
--R 3 3 4 2 8
--R (3403400A a b + 2552550B a b )e
--R +
--R 2 4 3 3 7
--R (232050A a b + 309400B a b )d e
--R +
--R 5 2 4 2 6 6 5 3 5
--R (- 71400A a b - 178500B a b )d e + (9520A b + 57120B a b )d e
--R +
--R 6 4 4
--R - 7840B b d e
--R *
--R 4
--R x
--R +
--R 4 2 5 8
--R (3281850A a b + 1312740B a b)e
--R +
--R 3 3 4 2 7
--R (486200A a b + 364650B a b )d e
--R +
--R 2 4 3 3 2 6
--R (- 265200A a b - 353600B a b )d e
--R +
--R 5 2 4 3 5 6 5 4 4
--R (81600A a b + 204000B a b )d e + (- 10880A b - 65280B a b )d e
--R +
--R 6 5 3
--R 8960B b d e
--R *
--R 3
--R x
--R +
--R 5 6 8 4 2 5 7
--R (1837836A a b + 306306B a )e + (656370A a b + 262548B a b)d e
--R +
--R 3 3 4 2 2 6
--R (- 583440A a b - 437580B a b )d e
--R +
--R 2 4 3 3 3 5
--R (318240A a b + 424320B a b )d e
--R +
--R 5 2 4 4 4 6 5 5 3
--R (- 97920A a b - 244800B a b )d e + (13056A b + 78336B a b )d e
--R +
--R 6 6 2
--R - 10752B b d e
--R *
--R 2
--R x
--R +
--R 6 8 5 6 7
--R 510510A a e + (612612A a b + 102102B a )d e
--R +
--R 4 2 5 2 6
--R (- 875160A a b - 350064B a b)d e
--R +
--R 3 3 4 2 3 5
--R (777920A a b + 583440B a b )d e
--R +
--R 2 4 3 3 4 4
--R (- 424320A a b - 565760B a b )d e
--R +
--R 5 2 4 5 3
--R (130560A a b + 326400B a b )d e
--R +
--R 6 5 6 2 6 7
--R (- 17408A b - 104448B a b )d e + 14336B b d e
--R *
--R x
--R +
--R 6 7 5 6 2 6
--R 510510A a d e + (- 1225224A a b - 204204B a )d e
--R +
--R 4 2 5 3 5
--R (1750320A a b + 700128B a b)d e
--R +
--R 3 3 4 2 4 4
--R (- 1555840A a b - 1166880B a b )d e
--R +
--R 2 4 3 3 5 3
--R (848640A a b + 1131520B a b )d e
--R +
--R 5 2 4 6 2 6 5 7
--R (- 261120A a b - 652800B a b )d e + (34816A b + 208896B a b )d e
--R +
--R 6 8
--R - 28672B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 8
--R 765765e
--R Type: Expression(Integer)
--E 527
--S 528 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 528
)clear all
--S 529 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^3/sqrt(d+e*x)
--R
--R
--R (1)
--R 6 7 6 5 6 5 2 4 5
--R B b x + (A b + 6B a b )x + (6A a b + 15B a b )x
--R +
--R 2 4 3 3 4 3 3 4 2 3 4 2 5 2
--R (15A a b + 20B a b )x + (20A a b + 15B a b )x + (15A a b + 6B a b)x
--R +
--R 5 6 6
--R (6A a b + B a )x + A a
--R /
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 529
--S 530 of 764
r0:=2/3*(b*d-a*e)^5*(7*b*B*d-6*A*b*e-a*B*e)*(d+e*x)^(3/2)/e^8-6/5*b*_
(b*d-a*e)^4*(7*b*B*d-5*A*b*e-2*a*B*e)*(d+e*x)^(5/2)/e^8+10/7*b^2*_
(b*d-a*e)^3*(7*b*B*d-4*A*b*e-3*a*B*e)*(d+e*x)^(7/2)/e^8-10/9*b^3*_
(b*d-a*e)^2*(7*b*B*d-3*A*b*e-4*a*B*e)*(d+e*x)^(9/2)/e^8+6/11*b^4*_
(b*d-a*e)*(7*b*B*d-2*A*b*e-5*a*B*e)*(d+e*x)^(11/2)/e^8-2/13*b^5*_
(7*b*B*d-A*b*e-6*a*B*e)*(d+e*x)^(13/2)/e^8+2/15*b^6*B*_
(d+e*x)^(15/2)/e^8-2*(b*d-a*e)^6*(B*d-A*e)*sqrt(d+e*x)/e^8
--R
--R
--R (2)
--R 6 7 7 6 5 7 6 6 6
--R 6006B b e x + ((6930A b + 41580B a b )e - 6468B b d e )x
--R +
--R 5 2 4 7 6 5 6
--R (49140A a b + 122850B a b )e + (- 7560A b - 45360B a b )d e
--R +
--R 6 2 5
--R 7056B b d e
--R *
--R 5
--R x
--R +
--R 2 4 3 3 7
--R (150150A a b + 200200B a b )e
--R +
--R 5 2 4 6 6 5 2 5
--R (- 54600A a b - 136500B a b )d e + (8400A b + 50400B a b )d e
--R +
--R 6 3 4
--R - 7840B b d e
--R *
--R 4
--R x
--R +
--R 3 3 4 2 7
--R (257400A a b + 193050B a b )e
--R +
--R 2 4 3 3 6
--R (- 171600A a b - 228800B a b )d e
--R +
--R 5 2 4 2 5 6 5 3 4
--R (62400A a b + 156000B a b )d e + (- 9600A b - 57600B a b )d e
--R +
--R 6 4 3
--R 8960B b d e
--R *
--R 3
--R x
--R +
--R 4 2 5 7
--R (270270A a b + 108108B a b)e
--R +
--R 3 3 4 2 6
--R (- 308880A a b - 231660B a b )d e
--R +
--R 2 4 3 3 2 5
--R (205920A a b + 274560B a b )d e
--R +
--R 5 2 4 3 4 6 5 4 3
--R (- 74880A a b - 187200B a b )d e + (11520A b + 69120B a b )d e
--R +
--R 6 5 2
--R - 10752B b d e
--R *
--R 2
--R x
--R +
--R 5 6 7 4 2 5 6
--R (180180A a b + 30030B a )e + (- 360360A a b - 144144B a b)d e
--R +
--R 3 3 4 2 2 5
--R (411840A a b + 308880B a b )d e
--R +
--R 2 4 3 3 3 4
--R (- 274560A a b - 366080B a b )d e
--R +
--R 5 2 4 4 3 6 5 5 2
--R (99840A a b + 249600B a b )d e + (- 15360A b - 92160B a b )d e
--R +
--R 6 6
--R 14336B b d e
--R *
--R x
--R +
--R 6 7 5 6 6
--R 90090A a e + (- 360360A a b - 60060B a )d e
--R +
--R 4 2 5 2 5 3 3 4 2 3 4
--R (720720A a b + 288288B a b)d e + (- 823680A a b - 617760B a b )d e
--R +
--R 2 4 3 3 4 3 5 2 4 5 2
--R (549120A a b + 732160B a b )d e + (- 199680A a b - 499200B a b )d e
--R +
--R 6 5 6 6 7
--R (30720A b + 184320B a b )d e - 28672B b d
--R *
--R +-------+
--R \|e x + d
--R /
--R 8
--R 45045e
--R Type: Expression(Integer)
--E 530
--S 531 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 531
)clear all
--S 532 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^(3/2)
--R
--R
--R (1)
--R 6 7 6 5 6 5 2 4 5
--R B b x + (A b + 6B a b )x + (6A a b + 15B a b )x
--R +
--R 2 4 3 3 4 3 3 4 2 3 4 2 5 2
--R (15A a b + 20B a b )x + (20A a b + 15B a b )x + (15A a b + 6B a b)x
--R +
--R 5 6 6
--R (6A a b + B a )x + A a
--R /
--R +-------+
--R (e x + d)\|e x + d
--R Type: Expression(Integer)
--E 532
--S 533 of 764
r0:=-2*b*(b*d-a*e)^4*(7*b*B*d-5*A*b*e-2*a*B*e)*(d+e*x)^(3/2)/e^8+2*b^2*_
(b*d-a*e)^3*(7*b*B*d-4*A*b*e-3*a*B*e)*(d+e*x)^(5/2)/e^8-10/7*b^3*_
(b*d-a*e)^2*(7*b*B*d-3*A*b*e-4*a*B*e)*(d+e*x)^(7/2)/e^8+2/3*b^4*_
(b*d-a*e)*(7*b*B*d-2*A*b*e-5*a*B*e)*(d+e*x)^(9/2)/e^8-2/11*b^5*_
(7*b*B*d-A*b*e-6*a*B*e)*(d+e*x)^(11/2)/e^8+2/13*b^6*B*_
(d+e*x)^(13/2)/e^8+2*(b*d-a*e)^6*(B*d-A*e)/(e^8*sqrt(d+e*x))+_
2*(b*d-a*e)^5*(7*b*B*d-6*A*b*e-a*B*e)*sqrt(d+e*x)/e^8
--R
--R
--R (2)
--R 6 7 7 6 5 7 6 6 6
--R 462B b e x + ((546A b + 3276B a b )e - 588B b d e )x
--R +
--R 5 2 4 7 6 5 6
--R (4004A a b + 10010B a b )e + (- 728A b - 4368B a b )d e
--R +
--R 6 2 5
--R 784B b d e
--R *
--R 5
--R x
--R +
--R 2 4 3 3 7 5 2 4 6
--R (12870A a b + 17160B a b )e + (- 5720A a b - 14300B a b )d e
--R +
--R 6 5 2 5 6 3 4
--R (1040A b + 6240B a b )d e - 1120B b d e
--R *
--R 4
--R x
--R +
--R 3 3 4 2 7 2 4 3 3 6
--R (24024A a b + 18018B a b )e + (- 20592A a b - 27456B a b )d e
--R +
--R 5 2 4 2 5 6 5 3 4
--R (9152A a b + 22880B a b )d e + (- 1664A b - 9984B a b )d e
--R +
--R 6 4 3
--R 1792B b d e
--R *
--R 3
--R x
--R +
--R 4 2 5 7 3 3 4 2 6
--R (30030A a b + 12012B a b)e + (- 48048A a b - 36036B a b )d e
--R +
--R 2 4 3 3 2 5 5 2 4 3 4
--R (41184A a b + 54912B a b )d e + (- 18304A a b - 45760B a b )d e
--R +
--R 6 5 4 3 6 5 2
--R (3328A b + 19968B a b )d e - 3584B b d e
--R *
--R 2
--R x
--R +
--R 5 6 7 4 2 5 6
--R (36036A a b + 6006B a )e + (- 120120A a b - 48048B a b)d e
--R +
--R 3 3 4 2 2 5
--R (192192A a b + 144144B a b )d e
--R +
--R 2 4 3 3 3 4
--R (- 164736A a b - 219648B a b )d e
--R +
--R 5 2 4 4 3 6 5 5 2
--R (73216A a b + 183040B a b )d e + (- 13312A b - 79872B a b )d e
--R +
--R 6 6
--R 14336B b d e
--R *
--R x
--R +
--R 6 7 5 6 6
--R - 6006A a e + (72072A a b + 12012B a )d e
--R +
--R 4 2 5 2 5 3 3 4 2 3 4
--R (- 240240A a b - 96096B a b)d e + (384384A a b + 288288B a b )d e
--R +
--R 2 4 3 3 4 3 5 2 4 5 2
--R (- 329472A a b - 439296B a b )d e + (146432A a b + 366080B a b )d e
--R +
--R 6 5 6 6 7
--R (- 26624A b - 159744B a b )d e + 28672B b d
--R /
--R 8 +-------+
--R 3003e \|e x + d
--R Type: Expression(Integer)
--E 533
--S 534 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 534
)clear all
--S 535 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^(5/2)
--R
--R
--R (1)
--R 6 7 6 5 6 5 2 4 5
--R B b x + (A b + 6B a b )x + (6A a b + 15B a b )x
--R +
--R 2 4 3 3 4 3 3 4 2 3 4 2 5 2
--R (15A a b + 20B a b )x + (20A a b + 15B a b )x + (15A a b + 6B a b)x
--R +
--R 5 6 6
--R (6A a b + B a )x + A a
--R /
--R 2 2 2 +-------+
--R (e x + 2d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 535
--S 536 of 764
r0:=2/3*(b*d-a*e)^6*(B*d-A*e)/(e^8*(d+e*x)^(3/2))+10/3*b^2*(b*d-a*e)^3*_
(7*b*B*d-4*A*b*e-3*a*B*e)*(d+e*x)^(3/2)/e^8-2*b^3*(b*d-a*e)^2*_
(7*b*B*d-3*A*b*e-4*a*B*e)*(d+e*x)^(5/2)/e^8+6/7*b^4*(b*d-a*e)*_
(7*b*B*d-2*A*b*e-5*a*B*e)*(d+e*x)^(7/2)/e^8-2/9*b^5*(7*b*B*d-A*b*e-_
6*a*B*e)*(d+e*x)^(9/2)/e^8+2/11*b^6*B*(d+e*x)^(11/2)/e^8-2*_
(b*d-a*e)^5*(7*b*B*d-6*A*b*e-a*B*e)/(e^8*sqrt(d+e*x))-6*b*(b*d-a*e)^4*_
(7*b*B*d-5*A*b*e-2*a*B*e)*sqrt(d+e*x)/e^8
--R
--R
--R (2)
--R 6 7 7 6 5 7 6 6 6
--R 126B b e x + ((154A b + 924B a b )e - 196B b d e )x
--R +
--R 5 2 4 7 6 5 6
--R (1188A a b + 2970B a b )e + (- 264A b - 1584B a b )d e
--R +
--R 6 2 5
--R 336B b d e
--R *
--R 5
--R x
--R +
--R 2 4 3 3 7 5 2 4 6
--R (4158A a b + 5544B a b )e + (- 2376A a b - 5940B a b )d e
--R +
--R 6 5 2 5 6 3 4
--R (528A b + 3168B a b )d e - 672B b d e
--R *
--R 4
--R x
--R +
--R 3 3 4 2 7 2 4 3 3 6
--R (9240A a b + 6930B a b )e + (- 11088A a b - 14784B a b )d e
--R +
--R 5 2 4 2 5 6 5 3 4
--R (6336A a b + 15840B a b )d e + (- 1408A b - 8448B a b )d e
--R +
--R 6 4 3
--R 1792B b d e
--R *
--R 3
--R x
--R +
--R 4 2 5 7 3 3 4 2 6
--R (20790A a b + 8316B a b)e + (- 55440A a b - 41580B a b )d e
--R +
--R 2 4 3 3 2 5 5 2 4 3 4
--R (66528A a b + 88704B a b )d e + (- 38016A a b - 95040B a b )d e
--R +
--R 6 5 4 3 6 5 2
--R (8448A b + 50688B a b )d e - 10752B b d e
--R *
--R 2
--R x
--R +
--R 5 6 7 4 2 5 6
--R (- 8316A a b - 1386B a )e + (83160A a b + 33264B a b)d e
--R +
--R 3 3 4 2 2 5
--R (- 221760A a b - 166320B a b )d e
--R +
--R 2 4 3 3 3 4
--R (266112A a b + 354816B a b )d e
--R +
--R 5 2 4 4 3 6 5 5 2
--R (- 152064A a b - 380160B a b )d e + (33792A b + 202752B a b )d e
--R +
--R 6 6
--R - 43008B b d e
--R *
--R x
--R +
--R 6 7 5 6 6
--R - 462A a e + (- 5544A a b - 924B a )d e
--R +
--R 4 2 5 2 5 3 3 4 2 3 4
--R (55440A a b + 22176B a b)d e + (- 147840A a b - 110880B a b )d e
--R +
--R 2 4 3 3 4 3 5 2 4 5 2
--R (177408A a b + 236544B a b )d e + (- 101376A a b - 253440B a b )d e
--R +
--R 6 5 6 6 7
--R (22528A b + 135168B a b )d e - 28672B b d
--R /
--R 9 8 +-------+
--R (693e x + 693d e )\|e x + d
--R Type: Expression(Integer)
--E 536
--S 537 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 537
)clear all
--S 538 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^3/(d+e*x)^(7/2)
--R
--R
--R (1)
--R 6 7 6 5 6 5 2 4 5
--R B b x + (A b + 6B a b )x + (6A a b + 15B a b )x
--R +
--R 2 4 3 3 4 3 3 4 2 3 4 2 5 2
--R (15A a b + 20B a b )x + (20A a b + 15B a b )x + (15A a b + 6B a b)x
--R +
--R 5 6 6
--R (6A a b + B a )x + A a
--R /
--R 3 3 2 2 2 3 +-------+
--R (e x + 3d e x + 3d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 538
--S 539 of 764
r0:=2/5*(b*d-a*e)^6*(B*d-A*e)/(e^8*(d+e*x)^(5/2))-2/3*(b*d-a*e)^5*_
(7*b*B*d-6*A*b*e-a*B*e)/(e^8*(d+e*x)^(3/2))-10/3*b^3*(b*d-a*e)^2*_
(7*b*B*d-3*A*b*e-4*a*B*e)*(d+e*x)^(3/2)/e^8+6/5*b^4*(b*d-a*e)*_
(7*b*B*d-2*A*b*e-5*a*B*e)*(d+e*x)^(5/2)/e^8-2/7*b^5*(7*b*B*d-_
A*b*e-6*a*B*e)*(d+e*x)^(7/2)/e^8+2/9*b^6*B*(d+e*x)^(9/2)/e^8+6*b*_
(b*d-a*e)^4*(7*b*B*d-5*A*b*e-2*a*B*e)/(e^8*sqrt(d+e*x))+10*b^2*_
(b*d-a*e)^3*(7*b*B*d-4*A*b*e-3*a*B*e)*sqrt(d+e*x)/e^8
--R
--R
--R (2)
--R 6 7 7 6 5 7 6 6 6
--R 70B b e x + ((90A b + 540B a b )e - 140B b d e )x
--R +
--R 5 2 4 7 6 5 6
--R (756A a b + 1890B a b )e + (- 216A b - 1296B a b )d e
--R +
--R 6 2 5
--R 336B b d e
--R *
--R 5
--R x
--R +
--R 2 4 3 3 7 5 2 4 6
--R (3150A a b + 4200B a b )e + (- 2520A a b - 6300B a b )d e
--R +
--R 6 5 2 5 6 3 4
--R (720A b + 4320B a b )d e - 1120B b d e
--R *
--R 4
--R x
--R +
--R 3 3 4 2 7 2 4 3 3 6
--R (12600A a b + 9450B a b )e + (- 25200A a b - 33600B a b )d e
--R +
--R 5 2 4 2 5 6 5 3 4
--R (20160A a b + 50400B a b )d e + (- 5760A b - 34560B a b )d e
--R +
--R 6 4 3
--R 8960B b d e
--R *
--R 3
--R x
--R +
--R 4 2 5 7 3 3 4 2 6
--R (- 9450A a b - 3780B a b)e + (75600A a b + 56700B a b )d e
--R +
--R 2 4 3 3 2 5
--R (- 151200A a b - 201600B a b )d e
--R +
--R 5 2 4 3 4 6 5 4 3
--R (120960A a b + 302400B a b )d e + (- 34560A b - 207360B a b )d e
--R +
--R 6 5 2
--R 53760B b d e
--R *
--R 2
--R x
--R +
--R 5 6 7 4 2 5 6
--R (- 1260A a b - 210B a )e + (- 12600A a b - 5040B a b)d e
--R +
--R 3 3 4 2 2 5
--R (100800A a b + 75600B a b )d e
--R +
--R 2 4 3 3 3 4
--R (- 201600A a b - 268800B a b )d e
--R +
--R 5 2 4 4 3 6 5 5 2
--R (161280A a b + 403200B a b )d e + (- 46080A b - 276480B a b )d e
--R +
--R 6 6
--R 71680B b d e
--R *
--R x
--R +
--R 6 7 5 6 6 4 2 5 2 5
--R - 126A a e + (- 504A a b - 84B a )d e + (- 5040A a b - 2016B a b)d e
--R +
--R 3 3 4 2 3 4 2 4 3 3 4 3
--R (40320A a b + 30240B a b )d e + (- 80640A a b - 107520B a b )d e
--R +
--R 5 2 4 5 2 6 5 6
--R (64512A a b + 161280B a b )d e + (- 18432A b - 110592B a b )d e
--R +
--R 6 7
--R 28672B b d
--R /
--R 10 2 9 2 8 +-------+
--R (315e x + 630d e x + 315d e )\|e x + d
--R Type: Expression(Integer)
--E 539
--S 540 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 540
)clear all
--S 541 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)/(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x
--R +
--R 3
--R A d
--R *
--R +-------+
--R \|e x + d
--R /
--R 2 2 2
--R b x + 2a b x + a
--R Type: Expression(Integer)
--E 541
--S 542 of 764
r0:=1/3*(b*d-a*e)*(2*b*B*d+7*A*b*e-9*a*B*e)*(d+e*x)^(3/2)/b^4+_
1/5*(2*b*B*d+7*A*b*e-9*a*B*e)*(d+e*x)^(5/2)/b^3+1/7*(2*b*B*d+7*A*b*e-_
9*a*B*e)*(d+e*x)^(7/2)/(b^2*(b*d-a*e))-(A*b-a*B)*(d+e*x)^(9/2)/(b*_
(b*d-a*e)*(a+b*x))-(b*d-a*e)^(5/2)*(2*b*B*d+7*A*b*e-9*a*B*e)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/b^(11/2)+(b*d-a*e)^2*_
(2*b*B*d+7*A*b*e-9*a*B*e)*sqrt(d+e*x)/b^5
--R
--R
--R (2)
--R 2 2 3 3 3 2 2 2
--R (- 735A a b + 945B a b)e + (1470A a b - 2100B a b )d e
--R +
--R 4 3 2 4 3
--R (- 735A b + 1365B a b )d e - 210B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (- 735A a b + 945B a )e + (1470A a b - 2100B a b)d e
--R +
--R 3 2 2 2 3 3
--R (- 735A a b + 1365B a b )d e - 210B a b d
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 4 4 3 3 4 2 3
--R 30B b e x + ((42A b - 54B a b )e + 132B b d e )x
--R +
--R 3 2 2 3 4 3 2
--R (- 98A a b + 126B a b )e + (224A b - 316B a b )d e
--R +
--R 4 2
--R 244B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 3 3 2 2 2
--R (490A a b - 630B a b)e + (- 1176A a b + 1652B a b )d e
--R +
--R 4 3 2 4 3
--R (812A b - 1380B a b )d e + 352B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (735A a b - 945B a )e + (- 1715A a b + 2415B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (1127A a b - 1939B a b )d e + (- 105A b + 457B a b )d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R 6 5 +-+
--R (105b x + 105a b )\|b
--R Type: Expression(Integer)
--E 542
--S 543 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 543
)clear all
--S 544 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)/(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R 2 3 2 2 2 2 +-------+
--R (B e x + (A e + 2B d e)x + (2A d e + B d )x + A d )\|e x + d
--R (1) ----------------------------------------------------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Expression(Integer)
--E 544
--S 545 of 764
r0:=1/3*(2*b*B*d+5*A*b*e-7*a*B*e)*(d+e*x)^(3/2)/b^3+1/5*(2*b*B*d+5*A*b*e-_
7*a*B*e)*(d+e*x)^(5/2)/(b^2*(b*d-a*e))-(A*b-a*B)*(d+e*x)^(7/2)/(b*_
(b*d-a*e)*(a+b*x))-(b*d-a*e)^(3/2)*(2*b*B*d+5*A*b*e-7*a*B*e)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/b^(9/2)+(b*d-a*e)*(2*b*B*d+_
5*A*b*e-7*a*B*e)*sqrt(d+e*x)/b^4
--R
--R
--R (2)
--R 2 2 2 3 2 3 2
--R ((75A a b - 105B a b)e + (- 75A b + 135B a b )d e - 30B b d )x
--R +
--R 2 3 2 2 2 2 2
--R (75A a b - 105B a )e + (- 75A a b + 135B a b)d e - 30B a b d
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 3 3 2 2 3 2
--R 6B b e x + ((10A b - 14B a b )e + 22B b d e)x
--R +
--R 2 2 2 3 2 3 2
--R ((- 50A a b + 70B a b)e + (70A b - 118B a b )d e + 46B b d )x
--R +
--R 2 3 2 2 2
--R (- 75A a b + 105B a )e + (100A a b - 170B a b)d e
--R +
--R 3 2 2
--R (- 15A b + 61B a b )d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R 5 4 +-+
--R (15b x + 15a b )\|b
--R Type: Expression(Integer)
--E 545
--S 546 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 546
)clear all
--S 547 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)/(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R 2 +-------+
--R (B e x + (A e + B d)x + A d)\|e x + d
--R (1) ---------------------------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Expression(Integer)
--E 547
--S 548 of 764
r0:=1/3*(2*b*B*d+3*A*b*e-5*a*B*e)*(d+e*x)^(3/2)/(b^2*(b*d-a*e))-_
(A*b-a*B)*(d+e*x)^(5/2)/(b*(b*d-a*e)*(a+b*x))-(2*b*B*d+3*A*b*e-_
5*a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-a*e)/b^(7/2)+_
(2*b*B*d+3*A*b*e-5*a*B*e)*sqrt(d+e*x)/b^3
--R
--R
--R (2)
--R 2 2 2
--R (((- 9A b + 15B a b)e - 6B b d)x + (- 9A a b + 15B a )e - 6B a b d)
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 2 2 2 2 2
--R 2B b e x + ((6A b - 10B a b)e + 8B b d)x + (9A a b - 15B a )e
--R +
--R 2
--R (- 3A b + 11B a b)d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R 4 3 +-+
--R (3b x + 3a b )\|b
--R Type: Expression(Integer)
--E 548
--S 549 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 549
)clear all
--S 550 of 764
t0:=(A+B*x)*sqrt(d+e*x)/(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R +-------+
--R (B x + A)\|e x + d
--R (1) -------------------
--R 2 2 2
--R b x + 2a b x + a
--R Type: Expression(Integer)
--E 550
--S 551 of 764
r0:=-(A*b-a*B)*(d+e*x)^(3/2)/(b*(b*d-a*e)*(a+b*x))-(2*b*B*d+A*b*e-3*a*B*e)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(5/2)*sqrt(b*d-a*e))+_
(2*b*B*d+A*b*e-3*a*B*e)*sqrt(d+e*x)/(b^2*(b*d-a*e))
--R
--R
--R (2)
--R 2 2 2
--R (((- A b + 3B a b)e - 2B b d)x + (- A a b + 3B a )e - 2B a b d)
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R +-----------+ +-+ +-------+
--R (2B b x - A b + 3B a)\|- a e + b d \|b \|e x + d
--R /
--R 3 2 +-----------+ +-+
--R (b x + a b )\|- a e + b d \|b
--R Type: Expression(Integer)
--E 551
--S 552 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 552
)clear all
--S 553 of 764
t0:=(A+B*x)/((a^2+2*a*b*x+b^2*x^2)*sqrt(d+e*x))
--R
--R
--R B x + A
--R (1) ------------------------------
--R 2 2 2 +-------+
--R (b x + 2a b x + a )\|e x + d
--R Type: Expression(Integer)
--E 553
--S 554 of 764
r0:=-(2*b*B*d-A*b*e-a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/_
(b^(3/2)*(b*d-a*e)^(3/2))-(A*b-a*B)*sqrt(d+e*x)/(b*(b*d-a*e)*(a+b*x))
--R
--R
--R (2)
--R 2 2 2
--R (((- A b - B a b)e + 2B b d)x + (- A a b - B a )e + 2B a b d)
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R +-----------+ +-+ +-------+
--R (A b - B a)\|- a e + b d \|b \|e x + d
--R /
--R 2 3 2 2 +-----------+ +-+
--R ((a b e - b d)x + a b e - a b d)\|- a e + b d \|b
--R Type: Expression(Integer)
--E 554
--S 555 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 555
)clear all
--S 556 of 764
t0:=(A+B*x)/((d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R B x + A
--R (1) -------------------------------------------------------------
--R 2 3 2 2 2 2 +-------+
--R (b e x + (2a b e + b d)x + (a e + 2a b d)x + a d)\|e x + d
--R Type: Expression(Integer)
--E 556
--S 557 of 764
r0:=-(2*b*B*d-3*A*b*e+a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/_
((b*d-a*e)^(5/2)*sqrt(b))+(2*b*B*d-3*A*b*e+a*B*e)/(b*(b*d-a*e)^2*_
sqrt(d+e*x))+(-A*b+a*B)/(b*(b*d-a*e)*(a+b*x)*sqrt(d+e*x))
--R
--R
--R (2)
--R 2 2 2 +-------+
--R (((3A b - B a b)e - 2B b d)x + (3A a b - B a )e - 2B a b d)\|e x + d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R +-----------+ +-+
--R (((- 3A b + B a)e + 2B b d)x - 2A a e + (- A b + 3B a)d)\|- a e + b d \|b
--R /
--R 2 2 2 3 2 3 2 2 2 2 +-----------+
--R ((a b e - 2a b d e + b d )x + a e - 2a b d e + a b d )\|- a e + b d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R Type: Expression(Integer)
--E 557
--S 558 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 558
)clear all
--S 559 of 764
t0:=(A+B*x)/((d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (1)
--R B x + A
--R /
--R 2 2 4 2 2 3 2 2 2 2 2
--R b e x + (2a b e + 2b d e)x + (a e + 4a b d e + b d )x
--R +
--R 2 2 2 2
--R (2a d e + 2a b d )x + a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 559
--S 560 of 764
r0:=1/3*(2*b*B*d-5*A*b*e+3*a*B*e)/(b*(b*d-a*e)^2*(d+e*x)^(3/2))+(-A*b+a*B)/_
(b*(b*d-a*e)*(a+b*x)*(d+e*x)^(3/2))-(2*b*B*d-5*A*b*e+3*a*B*e)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b)/(b*d-a*e)^(7/2)+_
(2*b*B*d-5*A*b*e+3*a*B*e)/((b*d-a*e)^3*sqrt(d+e*x))
--R
--R
--R (2)
--R 2 2 2 2
--R ((- 15A b + 9B a b)e + 6B b d e)x
--R +
--R 2 2 2 2 2
--R ((- 15A a b + 9B a )e + (- 15A b + 15B a b)d e + 6B b d )x
--R +
--R 2 2
--R (- 15A a b + 9B a )d e + 6B a b d
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 2 2 2 2
--R ((15A b - 9B a b)e - 6B b d e)x
--R +
--R 2 2 2 2 2 2 2
--R ((10A a b - 6B a )e + (20A b - 16B a b)d e - 8B b d )x - 2A a e
--R +
--R 2 2 2
--R (14A a b - 4B a )d e + (3A b - 11B a b)d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 3 4 2 2 3 3 2 2 4 3 2
--R (3a b e - 9a b d e + 9a b d e - 3b d e)x
--R +
--R 4 4 3 3 3 3 4 4 4 3 3 2 2
--R (3a e - 6a b d e + 6a b d e - 3b d )x + 3a d e - 9a b d e
--R +
--R 2 2 3 3 4
--R 9a b d e - 3a b d
--R *
--R +-----------+ +-------+
--R \|- a e + b d \|e x + d
--R Type: Expression(Integer)
--E 560
--S 561 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 561
)clear all
--S 562 of 764
t0:=(A+B*x)/((d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (1)
--R B x + A
--R /
--R 2 3 5 3 2 2 4 2 3 2 2 2 3
--R b e x + (2a b e + 3b d e )x + (a e + 6a b d e + 3b d e)x
--R +
--R 2 2 2 2 3 2 2 2 3 2 3
--R (3a d e + 6a b d e + b d )x + (3a d e + 2a b d )x + a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 562
--S 563 of 764
r0:=1/5*(2*b*B*d-7*A*b*e+5*a*B*e)/(b*(b*d-a*e)^2*(d+e*x)^(5/2))+_
(-A*b+a*B)/(b*(b*d-a*e)*(a+b*x)*(d+e*x)^(5/2))+1/3*(2*b*B*d-7*A*b*e+_
5*a*B*e)/((b*d-a*e)^3*(d+e*x)^(3/2))-b^(3/2)*(2*b*B*d-7*A*b*e+_
5*a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b*d-a*e)^(9/2)+_
b*(2*b*B*d-7*A*b*e+5*a*B*e)/((b*d-a*e)^4*sqrt(d+e*x))
--R
--R
--R (2)
--R 3 2 3 3 2 3
--R ((105A b - 75B a b )e - 30B b d e )x
--R +
--R 2 2 3 3 2 2 3 2 2
--R ((105A a b - 75B a b)e + (210A b - 180B a b )d e - 60B b d e)x
--R +
--R 2 2 2 3 2 2 3 3
--R ((210A a b - 150B a b)d e + (105A b - 135B a b )d e - 30B b d )x
--R +
--R 2 2 2 2 3
--R (105A a b - 75B a b)d e - 30B a b d
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 3 3 2 3
--R ((- 105A b + 75B a b )e + 30B b d e )x
--R +
--R 2 2 3 3 2 2
--R (- 70A a b + 50B a b)e + (- 245A b + 195B a b )d e
--R +
--R 3 2
--R 70B b d e
--R *
--R 2
--R x
--R +
--R 2 3 3 2 2 2
--R (14A a b - 10B a )e + (- 168A a b + 116B a b)d e
--R +
--R 3 2 2 3 3
--R (- 161A b + 163B a b )d e + 46B b d
--R *
--R x
--R +
--R 3 3 2 3 2 2 2 2
--R - 6A a e + (32A a b - 4B a )d e + (- 116A a b + 48B a b)d e
--R +
--R 3 2 3
--R (- 15A b + 61B a b )d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 4 6 3 2 5 2 3 2 4 4 3 3 5 4 2 3
--R (15a b e - 60a b d e + 90a b d e - 60a b d e + 15b d e )x
--R +
--R 5 6 4 5 3 2 2 4 2 3 3 3 4 4 2
--R 15a e - 30a b d e - 30a b d e + 120a b d e - 105a b d e
--R +
--R 5 5
--R 30b d e
--R *
--R 2
--R x
--R +
--R 5 5 4 2 4 3 2 3 3 2 3 4 2 4 5
--R 30a d e - 105a b d e + 120a b d e - 30a b d e - 30a b d e
--R +
--R 5 6
--R 15b d
--R *
--R x
--R +
--R 5 2 4 4 3 3 3 2 4 2 2 3 5 4 6
--R 15a d e - 60a b d e + 90a b d e - 60a b d e + 15a b d
--R *
--R +-----------+ +-------+
--R \|- a e + b d \|e x + d
--R Type: Expression(Integer)
--E 563
--S 564 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 564
)clear all
--S 565 of 764
t0:=(A+B*x)*(d+e*x)^(9/2)/(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B e x + (A e + 4B d e )x + (4A d e + 6B d e )x
--R +
--R 2 2 3 2 3 4 4
--R (6A d e + 4B d e)x + (4A d e + B d )x + A d
--R *
--R +-------+
--R \|e x + d
--R /
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Expression(Integer)
--E 565
--S 566 of 764
r0:=7/8*e^2*(6*b*B*d+5*A*b*e-11*a*B*e)*(d+e*x)^(3/2)/b^5+21/40*e^2*_
(6*b*B*d+5*A*b*e-11*a*B*e)*(d+e*x)^(5/2)/(b^4*(b*d-a*e))-3/8*e*_
(6*b*B*d+5*A*b*e-11*a*B*e)*(d+e*x)^(7/2)/(b^3*(b*d-a*e)*(a+b*x))-_
1/12*(6*b*B*d+5*A*b*e-11*a*B*e)*(d+e*x)^(9/2)/(b^2*(b*d-a*e)*_
(a+b*x)^2)-1/3*(A*b-a*B)*(d+e*x)^(11/2)/(b*(b*d-a*e)*(a+b*x)^3)-_
21/8*e^2*(b*d-a*e)^(3/2)*(6*b*B*d+5*A*b*e-11*a*B*e)*atanh(sqrt(b)*_
sqrt(d+e*x)/sqrt(b*d-a*e))/b^(13/2)+21/8*e^2*(b*d-a*e)*(6*b*B*d+_
5*A*b*e-11*a*B*e)*sqrt(d+e*x)/b^6
--R
--R
--R (2)
--R 4 2 3 4 5 4 3
--R (1575A a b - 3465B a b )e + (- 1575A b + 5355B a b )d e
--R +
--R 5 2 2
--R - 1890B b d e
--R *
--R 3
--R x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (4725A a b - 10395B a b )e + (- 4725A a b + 16065B a b )d e
--R +
--R 4 2 2
--R - 5670B a b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (4725A a b - 10395B a b)e + (- 4725A a b + 16065B a b )d e
--R +
--R 2 3 2 2
--R - 5670B a b d e
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (1575A a b - 3465B a )e + (- 1575A a b + 5355B a b)d e
--R +
--R 3 2 2 2
--R - 1890B a b d e
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 5 5 4 4 5 3 4
--R 48B b e x + ((80A b - 176B a b )e + 336B b d e )x
--R +
--R 4 2 3 4 5 4 3
--R (- 720A a b + 1584B a b )e + (1040A b - 3152B a b )d e
--R +
--R 5 2 2
--R 1728B b d e
--R *
--R 3
--R x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (- 3465A a b + 7623B a b )e + (4770A a b - 14652B a b )d e
--R +
--R 5 4 2 2 5 3
--R (- 825A b + 7539B a b )d e - 510B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (- 4200A a b + 9240B a b)e + (5670A a b - 17514B a b )d e
--R +
--R 4 2 3 2 2 5 4 3
--R (- 900A a b + 8784B a b )d e + (- 250A b - 530B a b )d e
--R +
--R 5 4
--R - 60B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 1575A a b + 3465B a )e + (2100A a b - 6510B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (- 315A a b + 3213B a b )d e + (- 90A a b - 180B a b )d e
--R +
--R 5 4 4
--R (- 40A b - 20B a b )d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R 9 3 8 2 2 7 3 6 +-+
--R (120b x + 360a b x + 360a b x + 120a b )\|b
--R Type: Expression(Integer)
--E 566
--S 567 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 567
)clear all
--S 568 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)/(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x
--R +
--R 3
--R A d
--R *
--R +-------+
--R \|e x + d
--R /
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Expression(Integer)
--E 568
--S 569 of 764
r0:=35/24*e^2*(2*b*B*d+A*b*e-3*a*B*e)*(d+e*x)^(3/2)/(b^4*(b*d-a*e))-_
7/8*e*(2*b*B*d+A*b*e-3*a*B*e)*(d+e*x)^(5/2)/(b^3*(b*d-a*e)*(a+b*x))-_
1/4*(2*b*B*d+A*b*e-3*a*B*e)*(d+e*x)^(7/2)/(b^2*(b*d-a*e)*(a+b*x)^2)-_
1/3*(A*b-a*B)*(d+e*x)^(9/2)/(b*(b*d-a*e)*(a+b*x)^3)-35/8*e^2*(2*b*B*d+_
A*b*e-3*a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-_
a*e)/b^(11/2)+35/8*e^2*(2*b*B*d+A*b*e-3*a*B*e)*sqrt(d+e*x)/b^5
--R
--R
--R (2)
--R 4 3 3 4 2 3
--R ((- 105A b + 315B a b )e - 210B b d e )x
--R +
--R 3 2 2 3 3 2 2
--R ((- 315A a b + 945B a b )e - 630B a b d e )x
--R +
--R 2 2 3 3 2 2 2
--R ((- 315A a b + 945B a b)e - 630B a b d e )x
--R +
--R 3 4 3 3 2
--R (- 105A a b + 315B a )e - 210B a b d e
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 4 4 3 3 4 2 3
--R 16B b e x + ((48A b - 144B a b )e + 160B b d e )x
--R +
--R 3 2 2 3 4 3 2
--R (231A a b - 693B a b )e + (- 87A b + 723B a b )d e
--R +
--R 4 2
--R - 78B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 3 3 2 2 2
--R (280A a b - 840B a b)e + (- 98A a b + 854B a b )d e
--R +
--R 4 3 2 4 3
--R (- 38A b - 82B a b )d e - 12B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (105A a b - 315B a )e + (- 35A a b + 315B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (- 14A a b - 28B a b )d e + (- 8A b - 4B a b )d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R 8 3 7 2 2 6 3 5 +-+
--R (24b x + 72a b x + 72a b x + 24a b )\|b
--R Type: Expression(Integer)
--E 569
--S 570 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 570
)clear all
--S 571 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)/(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R 2 3 2 2 2 2 +-------+
--R (B e x + (A e + 2B d e)x + (2A d e + B d )x + A d )\|e x + d
--R (1) ----------------------------------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Expression(Integer)
--E 571
--S 572 of 764
r0:=-5/24*e*(6*b*B*d+A*b*e-7*a*B*e)*(d+e*x)^(3/2)/(b^3*(b*d-a*e)*(a+b*x))-_
1/12*(6*b*B*d+A*b*e-7*a*B*e)*(d+e*x)^(5/2)/(b^2*(b*d-a*e)*(a+b*x)^2)-_
1/3*(A*b-a*B)*(d+e*x)^(7/2)/(b*(b*d-a*e)*(a+b*x)^3)-5/8*e^2*_
(6*b*B*d+A*b*e-7*a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/_
(b^(9/2)*sqrt(b*d-a*e))+5/8*e^2*(6*b*B*d+A*b*e-7*a*B*e)*sqrt(d+e*x)/_
(b^4*(b*d-a*e))
--R
--R
--R (2)
--R 4 3 3 4 2 3
--R ((- 15A b + 105B a b )e - 90B b d e )x
--R +
--R 3 2 2 3 3 2 2
--R ((- 45A a b + 315B a b )e - 270B a b d e )x
--R +
--R 2 2 3 3 2 2 2
--R ((- 45A a b + 315B a b)e - 270B a b d e )x
--R +
--R 3 4 3 3 2
--R (- 15A a b + 105B a )e - 90B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 3 3 2 2 3 2
--R 48B b e x + ((- 33A b + 231B a b )e - 54B b d e)x
--R +
--R 2 2 2 3 2 3 2
--R ((- 40A a b + 280B a b)e + (- 26A b - 58B a b )d e - 12B b d )x
--R +
--R 2 3 2 2 2
--R (- 15A a b + 105B a )e + (- 10A a b - 20B a b)d e
--R +
--R 3 2 2
--R (- 8A b - 4B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 7 3 6 2 2 5 3 4 +-----------+ +-+
--R (24b x + 72a b x + 72a b x + 24a b )\|- a e + b d \|b
--R Type: Expression(Integer)
--E 572
--S 573 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 573
)clear all
--S 574 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)/(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R 2 +-------+
--R (B e x + (A e + B d)x + A d)\|e x + d
--R (1) ---------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Expression(Integer)
--E 574
--S 575 of 764
r0:=-1/12*(6*b*B*d-A*b*e-5*a*B*e)*(d+e*x)^(3/2)/(b^2*(b*d-a*e)*(a+b*x)^2)-_
1/3*(A*b-a*B)*(d+e*x)^(5/2)/(b*(b*d-a*e)*(a+b*x)^3)-1/8*e^2*(6*b*B*d-_
A*b*e-5*a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(7/2)*_
(b*d-a*e)^(3/2))-1/8*e*(6*b*B*d-A*b*e-5*a*B*e)*sqrt(d+e*x)/(b^3*_
(b*d-a*e)*(a+b*x))
--R
--R
--R (2)
--R 4 3 3 4 2 3
--R ((- 3A b - 15B a b )e + 18B b d e )x
--R +
--R 3 2 2 3 3 2 2
--R ((- 9A a b - 45B a b )e + 54B a b d e )x
--R +
--R 2 2 3 3 2 2 2 3 4 3
--R ((- 9A a b - 45B a b)e + 54B a b d e )x + (- 3A a b - 15B a )e
--R +
--R 3 2
--R 18B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 2 3 2
--R ((3A b - 33B a b )e + 30B b d e)x
--R +
--R 2 2 2 3 2 3 2
--R ((- 8A a b - 40B a b)e + (14A b + 22B a b )d e + 12B b d )x
--R +
--R 2 3 2 2 2 3 2 2
--R (- 3A a b - 15B a )e + (- 2A a b + 8B a b)d e + (8A b + 4B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 6 7 3 2 5 6 2 3 4 2 5
--R (24a b e - 24b d)x + (72a b e - 72a b d)x + (72a b e - 72a b d)x
--R +
--R 4 3 3 4
--R 24a b e - 24a b d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R Type: Expression(Integer)
--E 575
--S 576 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 576
)clear all
--S 577 of 764
t0:=(A+B*x)*sqrt(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^2
--R
--R
--R +-------+
--R (B x + A)\|e x + d
--R (1) --------------------------------------
--R 4 4 3 3 2 2 2 3 4
--R b x + 4a b x + 6a b x + 4a b x + a
--R Type: Expression(Integer)
--E 577
--S 578 of 764
r0:=-1/3*(A*b-a*B)*(d+e*x)^(3/2)/(b*(b*d-a*e)*(a+b*x)^3)+1/8*e^2*(2*b*B*d-_
A*b*e-a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(5/2)*_
(b*d-a*e)^(5/2))-1/4*(2*b*B*d-A*b*e-a*B*e)*sqrt(d+e*x)/(b^2*_
(b*d-a*e)*(a+b*x)^2)-1/8*e*(2*b*B*d-A*b*e-a*B*e)*sqrt(d+e*x)/_
(b^2*(b*d-a*e)^2*(a+b*x))
--R
--R
--R (2)
--R 4 3 3 4 2 3
--R ((- 3A b - 3B a b )e + 6B b d e )x
--R +
--R 3 2 2 3 3 2 2
--R ((- 9A a b - 9B a b )e + 18B a b d e )x
--R +
--R 2 2 3 3 2 2 2 3 4 3
--R ((- 9A a b - 9B a b)e + 18B a b d e )x + (- 3A a b - 3B a )e
--R +
--R 3 2
--R 6B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 2 3 2
--R ((3A b + 3B a b )e - 6B b d e)x
--R +
--R 2 2 2 3 2 3 2
--R ((8A a b - 8B a b)e + (- 2A b + 14B a b )d e - 12B b d )x
--R +
--R 2 3 2 2 2 3 2 2
--R (- 3A a b - 3B a )e + (14A a b + 4B a b)d e + (- 8A b - 4B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 2 5 2 6 7 2 3
--R (24a b e - 48a b d e + 24b d )x
--R +
--R 3 4 2 2 5 6 2 2
--R (72a b e - 144a b d e + 72a b d )x
--R +
--R 4 3 2 3 4 2 5 2 5 2 2 4 3 3 4 2
--R (72a b e - 144a b d e + 72a b d )x + 24a b e - 48a b d e + 24a b d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R Type: Expression(Integer)
--E 578
--S 579 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 579
)clear all
--S 580 of 764
t0:=(A+B*x)/((a^2+2*a*b*x+b^2*x^2)^2*sqrt(d+e*x))
--R
--R
--R B x + A
--R (1) --------------------------------------------------
--R 4 4 3 3 2 2 2 3 4 +-------+
--R (b x + 4a b x + 6a b x + 4a b x + a )\|e x + d
--R Type: Expression(Integer)
--E 580
--S 581 of 764
r0:=-1/8*e^2*(6*b*B*d-5*A*b*e-a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/_
sqrt(b*d-a*e))/(b^(3/2)*(b*d-a*e)^(7/2))-1/3*(A*b-a*B)*sqrt(d+e*x)/_
(b*(b*d-a*e)*(a+b*x)^3)-1/12*(6*b*B*d-5*A*b*e-a*B*e)*sqrt(d+e*x)/_
(b*(b*d-a*e)^2*(a+b*x)^2)+1/8*e*(6*b*B*d-5*A*b*e-a*B*e)*sqrt(d+e*x)/_
(b*(b*d-a*e)^3*(a+b*x))
--R
--R
--R (2)
--R 4 3 3 4 2 3
--R ((- 15A b - 3B a b )e + 18B b d e )x
--R +
--R 3 2 2 3 3 2 2
--R ((- 45A a b - 9B a b )e + 54B a b d e )x
--R +
--R 2 2 3 3 2 2 2 3 4 3
--R ((- 45A a b - 9B a b)e + 54B a b d e )x + (- 15A a b - 3B a )e
--R +
--R 3 2
--R 18B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 2 3 2
--R ((15A b + 3B a b )e - 18B b d e)x
--R +
--R 2 2 2 3 2 3 2
--R ((40A a b + 8B a b)e + (- 10A b - 50B a b )d e + 12B b d )x
--R +
--R 2 3 2 2 2 3 2 2
--R (33A a b - 3B a )e + (- 26A a b - 16B a b)d e + (8A b + 4B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 3 4 3 2 5 2 6 2 7 3 3
--R (24a b e - 72a b d e + 72a b d e - 24b d )x
--R +
--R 4 3 3 3 4 2 2 5 2 6 3 2
--R (72a b e - 216a b d e + 216a b d e - 72a b d )x
--R +
--R 5 2 3 4 3 2 3 4 2 2 5 3 6 3
--R (72a b e - 216a b d e + 216a b d e - 72a b d )x + 24a b e
--R +
--R 5 2 2 4 3 2 3 4 3
--R - 72a b d e + 72a b d e - 24a b d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R Type: Expression(Integer)
--E 581
--S 582 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 582
)clear all
--S 583 of 764
t0:=(A+B*x)/((d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^2)
--R
--R
--R (1)
--R B x + A
--R /
--R 4 5 3 4 4 2 2 3 3 3 2 2 2
--R b e x + (4a b e + b d)x + (6a b e + 4a b d)x + (4a b e + 6a b d)x
--R +
--R 4 3 4
--R (a e + 4a b d)x + a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 583
--S 584 of 764
r0:=-5/8*e^2*(6*b*B*d-7*A*b*e+a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/_
sqrt(b*d-a*e))/((b*d-a*e)^(9/2)*sqrt(b))+5/8*e^2*(6*b*B*d-7*A*b*e+_
a*B*e)/(b*(b*d-a*e)^4*sqrt(d+e*x))+1/3*(-A*b+a*B)/(b*(b*d-a*e)*_
(a+b*x)^3*sqrt(d+e*x))+1/12*(-6*b*B*d+7*A*b*e-a*B*e)/(b*(b*d-a*e)^2*_
(a+b*x)^2*sqrt(d+e*x))+5/24*e*(6*b*B*d-7*A*b*e+a*B*e)/(b*(b*d-a*e)^3*_
(a+b*x)*sqrt(d+e*x))
--R
--R
--R (2)
--R 4 3 3 4 2 3
--R ((105A b - 15B a b )e - 90B b d e )x
--R +
--R 3 2 2 3 3 2 2
--R ((315A a b - 45B a b )e - 270B a b d e )x
--R +
--R 2 2 3 3 2 2 2 3 4 3
--R ((315A a b - 45B a b)e - 270B a b d e )x + (105A a b - 15B a )e
--R +
--R 3 2
--R - 90B a b d e
--R *
--R +-+ +-------+
--R +-------+ \|b \|e x + d
--R \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 3 3 2 3
--R ((- 105A b + 15B a b )e + 90B b d e )x
--R +
--R 2 2 3 3 2 2
--R (- 280A a b + 40B a b)e + (- 35A b + 245B a b )d e
--R +
--R 3 2
--R 30B b d e
--R *
--R 2
--R x
--R +
--R 2 3 3 2 2 2
--R (- 231A a b + 33B a )e + (- 98A a b + 212B a b)d e
--R +
--R 3 2 2 3 3
--R (14A b + 82B a b )d e - 12B b d
--R *
--R x
--R +
--R 3 3 2 3 2 2 2 2
--R - 48A a e + (- 87A a b + 81B a )d e + (38A a b + 28B a b)d e
--R +
--R 3 2 3
--R (- 8A b - 4B a b )d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R /
--R 4 3 4 3 4 3 2 5 2 2 6 3 7 4 3
--R (24a b e - 96a b d e + 144a b d e - 96a b d e + 24b d )x
--R +
--R 5 2 4 4 3 3 3 4 2 2 2 5 3 6 4 2
--R (72a b e - 288a b d e + 432a b d e - 288a b d e + 72a b d )x
--R +
--R 6 4 5 2 3 4 3 2 2 3 4 3 2 5 4
--R (72a b e - 288a b d e + 432a b d e - 288a b d e + 72a b d )x
--R +
--R 7 4 6 3 5 2 2 2 4 3 3 3 4 4
--R 24a e - 96a b d e + 144a b d e - 96a b d e + 24a b d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R Type: Expression(Integer)
--E 584
--S 585 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 585
)clear all
--S 586 of 764
t0:=(A+B*x)/((d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^2)
--R
--R
--R (1)
--R B x + A
--R /
--R 4 2 6 3 2 4 5 2 2 2 3 4 2 4
--R b e x + (4a b e + 2b d e)x + (6a b e + 8a b d e + b d )x
--R +
--R 3 2 2 2 3 2 3 4 2 3 2 2 2 2
--R (4a b e + 12a b d e + 4a b d )x + (a e + 8a b d e + 6a b d )x
--R +
--R 4 3 2 4 2
--R (2a d e + 4a b d )x + a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 586
--S 587 of 764
r0:=35/24*e^2*(2*b*B*d-3*A*b*e+a*B*e)/(b*(b*d-a*e)^4*(d+e*x)^(3/2))+_
1/3*(-A*b+a*B)/(b*(b*d-a*e)*(a+b*x)^3*(d+e*x)^(3/2))+1/4*(-2*b*B*d+_
3*A*b*e-a*B*e)/(b*(b*d-a*e)^2*(a+b*x)^2*(d+e*x)^(3/2))+7/8*e*(2*b*B*d-_
3*A*b*e+a*B*e)/(b*(b*d-a*e)^3*(a+b*x)*(d+e*x)^(3/2))-35/8*e^2*(2*b*B*d-_
3*A*b*e+a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b)/(b*d-_
a*e)^(11/2)+35/8*e^2*(2*b*B*d-3*A*b*e+a*B*e)/((b*d-a*e)^5*sqrt(d+e*x))
--R
--R
--R (2)
--R 4 3 4 4 3 4
--R ((- 315A b + 105B a b )e + 210B b d e )x
--R +
--R 3 2 2 4 4 3 3
--R (- 945A a b + 315B a b )e + (- 315A b + 735B a b )d e
--R +
--R 4 2 2
--R 210B b d e
--R *
--R 3
--R x
--R +
--R 2 2 3 4 3 2 2 3
--R (- 945A a b + 315B a b)e + (- 945A a b + 945B a b )d e
--R +
--R 3 2 2
--R 630B a b d e
--R *
--R 2
--R x
--R +
--R 3 4 4 2 2 3 3
--R (- 315A a b + 105B a )e + (- 945A a b + 525B a b)d e
--R +
--R 2 2 2 2
--R 630B a b d e
--R *
--R x
--R +
--R 3 4 3 3 2 2
--R (- 315A a b + 105B a )d e + 210B a b d e
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 4 4 3 4
--R ((315A b - 105B a b )e - 210B b d e )x
--R +
--R 3 2 2 4 4 3 3
--R (840A a b - 280B a b )e + (420A b - 700B a b )d e
--R +
--R 4 2 2
--R - 280B b d e
--R *
--R 3
--R x
--R +
--R 2 2 3 4 3 2 2 3
--R (693A a b - 231B a b)e + (1134A a b - 840B a b )d e
--R +
--R 4 3 2 2 4 3
--R (63A b - 777B a b )d e - 42B b d e
--R *
--R 2
--R x
--R +
--R 3 4 4 2 2 3 3
--R (144A a b - 48B a )e + (954A a b - 414B a b)d e
--R +
--R 3 2 2 2 2 4 3 3 4 4
--R (180A a b - 696B a b )d e + (- 18A b - 114B a b )d e + 12B b d
--R *
--R x
--R +
--R 4 4 3 4 3 2 2 3 2 2
--R - 16A a e + (208A a b - 32B a )d e + (165A a b - 247B a b)d e
--R +
--R 3 2 2 3 4 3 4
--R (- 50A a b - 40B a b )d e + (8A b + 4B a b )d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 5 3 6 4 4 5 3 5 2 4 2 6 3 3 7 4 2
--R 24a b e - 120a b d e + 240a b d e - 240a b d e + 120a b d e
--R +
--R 8 5
--R - 24b d e
--R *
--R 4
--R x
--R +
--R 6 2 6 5 3 5 4 4 2 4 3 5 3 3 2 6 4 2
--R 72a b e - 336a b d e + 600a b d e - 480a b d e + 120a b d e
--R +
--R 7 5 8 6
--R 48a b d e - 24b d
--R *
--R 3
--R x
--R +
--R 7 6 6 2 5 5 3 2 4 3 5 4 2 2 6 5
--R 72a b e - 288a b d e + 360a b d e - 360a b d e + 288a b d e
--R +
--R 7 6
--R - 72a b d
--R *
--R 2
--R x
--R +
--R 8 6 7 5 6 2 2 4 5 3 3 3 4 4 4 2
--R 24a e - 48a b d e - 120a b d e + 480a b d e - 600a b d e
--R +
--R 3 5 5 2 6 6
--R 336a b d e - 72a b d
--R *
--R x
--R +
--R 8 5 7 2 4 6 2 3 3 5 3 4 2 4 4 5
--R 24a d e - 120a b d e + 240a b d e - 240a b d e + 120a b d e
--R +
--R 3 5 6
--R - 24a b d
--R *
--R +-----------+ +-------+
--R \|- a e + b d \|e x + d
--R Type: Expression(Integer)
--E 587
--S 588 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 588
)clear all
--S 589 of 764
t0:=(A+B*x)/((d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^2)
--R
--R
--R (1)
--R B x + A
--R /
--R 4 3 7 3 3 4 2 6 2 2 3 3 2 4 2 5
--R b e x + (4a b e + 3b d e )x + (6a b e + 12a b d e + 3b d e)x
--R +
--R 3 3 2 2 2 3 2 4 3 4
--R (4a b e + 18a b d e + 12a b d e + b d )x
--R +
--R 4 3 3 2 2 2 2 3 3 3
--R (a e + 12a b d e + 18a b d e + 4a b d )x
--R +
--R 4 2 3 2 2 2 3 2 4 2 3 3 4 3
--R (3a d e + 12a b d e + 6a b d )x + (3a d e + 4a b d )x + a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 589
--S 590 of 764
r0:=21/40*e^2*(6*b*B*d-11*A*b*e+5*a*B*e)/(b*(b*d-a*e)^4*(d+e*x)^(5/2))+_
1/3*(-A*b+a*B)/(b*(b*d-a*e)*(a+b*x)^3*(d+e*x)^(5/2))+1/12*(-6*b*B*d+_
11*A*b*e-5*a*B*e)/(b*(b*d-a*e)^2*(a+b*x)^2*(d+e*x)^(5/2))+3/8*e*(6*b*B*d-_
11*A*b*e+5*a*B*e)/(b*(b*d-a*e)^3*(a+b*x)*(d+e*x)^(5/2))+7/8*e^2*(6*b*B*d-_
11*A*b*e+5*a*B*e)/((b*d-a*e)^5*(d+e*x)^(3/2))-21/8*b^(3/2)*e^2*(6*b*B*d-_
11*A*b*e+5*a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/_
(b*d-a*e)^(13/2)+21/8*b*e^2*(6*b*B*d-11*A*b*e+5*a*B*e)/((b*d-a*e)^6*_
sqrt(d+e*x))
--R
--R
--R (2)
--R 5 4 5 5 4 5
--R ((3465A b - 1575B a b )e - 1890B b d e )x
--R +
--R 4 2 3 5 5 4 4
--R (10395A a b - 4725B a b )e + (6930A b - 8820B a b )d e
--R +
--R 5 2 3
--R - 3780B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 5 4 2 3 4
--R (10395A a b - 4725B a b )e + (20790A a b - 15120B a b )d e
--R +
--R 5 4 2 3 5 3 2
--R (3465A b - 12915B a b )d e - 1890B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 5 2 3 3 2 4
--R (3465A a b - 1575B a b)e + (20790A a b - 11340B a b )d e
--R +
--R 4 2 3 2 3 4 3 2
--R (10395A a b - 16065B a b )d e - 5670B a b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 2 3
--R (6930A a b - 3150B a b)d e + (10395A a b - 8505B a b )d e
--R +
--R 2 3 3 2
--R - 5670B a b d e
--R *
--R x
--R +
--R 3 2 4 2 3 3 2 3 2
--R (3465A a b - 1575B a b)d e - 1890B a b d e
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 5 5 4 5
--R ((- 3465A b + 1575B a b )e + 1890B b d e )x
--R +
--R 4 2 3 5 5 4 4
--R (- 9240A a b + 4200B a b )e + (- 8085A b + 8715B a b )d e
--R +
--R 5 2 3
--R 4410B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 5 4 2 3 4
--R (- 7623A a b + 3465B a b )e + (- 21714A a b + 14028B a b )d e
--R +
--R 5 4 2 3 5 3 2
--R (- 5313A b + 14259B a b )d e + 2898B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 5 2 3 3 2 4
--R (- 1584A a b + 720B a b)e + (- 18117A a b + 9099B a b )d e
--R +
--R 4 2 3 2 3 5 4 3 2
--R (- 14454A a b + 16452B a b )d e + (- 495A b + 8109B a b )d e
--R +
--R 5 4
--R 270B b d e
--R *
--R 2
--R x
--R +
--R 4 5 5 3 2 4 4
--R (176A a b - 80B a )e + (- 3872A a b + 1664B a b)d e
--R +
--R 2 3 3 2 2 3
--R (- 12309A a b + 7707B a b )d e
--R +
--R 4 2 3 3 2 5 4 4
--R (- 1430A a b + 7364B a b )d e + (110A b + 730B a b )d e
--R +
--R 5 5
--R - 60B b d
--R *
--R x
--R +
--R 5 5 4 5 4 3 2 4 2 3
--R - 48A a e + (416A a b - 32B a )d e + (- 2768A a b + 704B a b)d e
--R +
--R 2 3 3 2 3 2 4 2 3 4
--R (- 1335A a b + 2553B a b )d e + (310A a b + 260B a b )d e
--R +
--R 5 4 5
--R (- 40A b - 20B a b )d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 6 3 8 5 4 7 4 5 2 6 3 6 3 5
--R 120a b e - 720a b d e + 1800a b d e - 2400a b d e
--R +
--R 2 7 4 4 8 5 3 9 6 2
--R 1800a b d e - 720a b d e + 120b d e
--R *
--R 5
--R x
--R +
--R 7 2 8 6 3 7 5 4 2 6 4 5 3 5
--R 360a b e - 1920a b d e + 3960a b d e - 3600a b d e
--R +
--R 3 6 4 4 2 7 5 3 8 6 2 9 7
--R 600a b d e + 1440a b d e - 1080a b d e + 240b d e
--R *
--R 4
--R x
--R +
--R 8 8 7 2 7 6 3 2 6 5 4 3 5
--R 360a b e - 1440a b d e + 1200a b d e + 2880a b d e
--R +
--R 4 5 4 4 3 6 5 3 2 7 6 2 9 8
--R - 7200a b d e + 6240a b d e - 2160a b d e + 120b d
--R *
--R 3
--R x
--R +
--R 9 8 7 2 2 6 6 3 3 5 5 4 4 4 4 5 5 3
--R 120a e - 2160a b d e + 6240a b d e - 7200a b d e + 2880a b d e
--R +
--R 3 6 6 2 2 7 7 8 8
--R 1200a b d e - 1440a b d e + 360a b d
--R *
--R 2
--R x
--R +
--R 9 7 8 2 6 7 2 3 5 6 3 4 4
--R 240a d e - 1080a b d e + 1440a b d e + 600a b d e
--R +
--R 5 4 5 3 4 5 6 2 3 6 7 2 7 8
--R - 3600a b d e + 3960a b d e - 1920a b d e + 360a b d
--R *
--R x
--R +
--R 9 2 6 8 3 5 7 2 4 4 6 3 5 3 5 4 6 2
--R 120a d e - 720a b d e + 1800a b d e - 2400a b d e + 1800a b d e
--R +
--R 4 5 7 3 6 8
--R - 720a b d e + 120a b d
--R *
--R +-----------+ +-------+
--R \|- a e + b d \|e x + d
--R Type: Expression(Integer)
--E 590
--S 591 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 591
)clear all
--S 592 of 764
t0:=(A+B*x)*(d+e*x)^(11/2)/(a^2+2*a*b*x+b^2*x^2)^3
--R
--R
--R (1)
--R 5 6 5 4 5 4 2 3 4
--R B e x + (A e + 5B d e )x + (5A d e + 10B d e )x
--R +
--R 2 3 3 2 3 3 2 4 2 4 5 5
--R (10A d e + 10B d e )x + (10A d e + 5B d e)x + (5A d e + B d )x + A d
--R *
--R +-------+
--R \|e x + d
--R /
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a
--R Type: Expression(Integer)
--E 592
--S 593 of 764
r0:=77/128*e^4*(10*b*B*d+3*A*b*e-13*a*B*e)*(d+e*x)^(3/2)/(b^6*(b*d-a*e))-_
231/640*e^3*(10*b*B*d+3*A*b*e-13*a*B*e)*(d+e*x)^(5/2)/(b^5*(b*d-a*e)*_
(a+b*x))-33/320*e^2*(10*b*B*d+3*A*b*e-13*a*B*e)*(d+e*x)^(7/2)/(b^4*_
(b*d-a*e)*(a+b*x)^2)-11/240*e*(10*b*B*d+3*A*b*e-13*a*B*e)*(d+e*x)^(9/2)/_
(b^3*(b*d-a*e)*(a+b*x)^3)-1/40*(10*b*B*d+3*A*b*e-13*a*B*e)*_
(d+e*x)^(11/2)/(b^2*(b*d-a*e)*(a+b*x)^4)-1/5*(A*b-a*B)*(d+e*x)^(13/2)/_
(b*(b*d-a*e)*(a+b*x)^5)-231/128*e^4*(10*b*B*d+3*A*b*e-13*a*B*e)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-a*e)/b^(15/2)+_
231/128*e^4*(10*b*B*d+3*A*b*e-13*a*B*e)*sqrt(d+e*x)/b^7
--R
--R
--R (2)
--R 6 5 5 6 4 5
--R ((- 10395A b + 45045B a b )e - 34650B b d e )x
--R +
--R 5 2 4 5 5 4 4
--R ((- 51975A a b + 225225B a b )e - 173250B a b d e )x
--R +
--R 2 4 3 3 5 2 4 4 3
--R ((- 103950A a b + 450450B a b )e - 346500B a b d e )x
--R +
--R 3 3 4 2 5 3 3 4 2
--R ((- 103950A a b + 450450B a b )e - 346500B a b d e )x
--R +
--R 4 2 5 5 4 2 4
--R ((- 51975A a b + 225225B a b)e - 173250B a b d e )x
--R +
--R 5 6 5 5 4
--R (- 10395A a b + 45045B a )e - 34650B a b d e
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 6 5 6 6 5 5 6 4 5
--R 1280B b e x + ((3840A b - 16640B a b )e + 20480B b d e )x
--R +
--R 5 2 4 5 6 5 4
--R (31845A a b - 137995B a b )e + (- 12645A b + 160945B a b )d e
--R +
--R 6 2 3
--R - 22950B b d e
--R *
--R 4
--R x
--R +
--R 2 4 3 3 5
--R (78210A a b - 338910B a b )e
--R +
--R 5 2 4 4
--R (- 29040A a b + 386540B a b )d e
--R +
--R 6 5 2 3 6 3 2
--R (- 10770A b - 50130B a b )d e - 10300B b d e
--R *
--R 3
--R x
--R +
--R 3 3 4 2 5
--R (88704A a b - 384384B a b )e
--R +
--R 2 4 3 3 4
--R (- 31482A a b + 432102B a b )d e
--R +
--R 5 2 4 2 3
--R (- 12078A a b - 52602B a b )d e
--R +
--R 6 5 3 2 6 4
--R (- 6744A b - 11036B a b )d e - 3280B b d e
--R *
--R 2
--R x
--R +
--R 4 2 5 5
--R (48510A a b - 210210B a b)e
--R +
--R 3 3 4 2 4
--R (- 16632A a b + 233772B a b )d e
--R +
--R 2 4 3 3 2 3
--R (- 6534A a b - 27126B a b )d e
--R +
--R 5 2 4 3 2 6 5 4
--R (- 3696A a b - 5764B a b )d e + (- 2448A b - 1712B a b )d e
--R +
--R 6 5
--R - 480B b d
--R *
--R x
--R +
--R 5 6 5 4 2 5 4
--R (10395A a b - 45045B a )e + (- 3465A a b + 49665B a b)d e
--R +
--R 3 3 4 2 2 3 2 4 3 3 3 2
--R (- 1386A a b - 5544B a b )d e + (- 792A a b - 1188B a b )d e
--R +
--R 5 2 4 4 6 5 5
--R (- 528A a b - 352B a b )d e + (- 384A b - 96B a b )d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R 12 5 11 4 2 10 3 3 9 2 4 8
--R 1920b x + 9600a b x + 19200a b x + 19200a b x + 9600a b x
--R +
--R 5 7
--R 1920a b
--R *
--R +-+
--R \|b
--R Type: Expression(Integer)
--E 593
--S 594 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 594
)clear all
--S 595 of 764
t0:=(A+B*x)*(d+e*x)^(9/2)/(a^2+2*a*b*x+b^2*x^2)^3
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B e x + (A e + 4B d e )x + (4A d e + 6B d e )x
--R +
--R 2 2 3 2 3 4 4
--R (6A d e + 4B d e)x + (4A d e + B d )x + A d
--R *
--R +-------+
--R \|e x + d
--R /
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a
--R Type: Expression(Integer)
--E 595
--S 596 of 764
r0:=-21/128*e^3*(10*b*B*d+A*b*e-11*a*B*e)*(d+e*x)^(3/2)/(b^5*(b*d-a*e)*_
(a+b*x))-21/320*e^2*(10*b*B*d+A*b*e-11*a*B*e)*(d+e*x)^(5/2)/(b^4*_
(b*d-a*e)*(a+b*x)^2)-3/80*e*(10*b*B*d+A*b*e-11*a*B*e)*(d+e*x)^(7/2)/_
(b^3*(b*d-a*e)*(a+b*x)^3)-1/40*(10*b*B*d+A*b*e-11*a*B*e)*_
(d+e*x)^(9/2)/(b^2*(b*d-a*e)*(a+b*x)^4)-1/5*(A*b-a*B)*(d+e*x)^(11/2)/_
(b*(b*d-a*e)*(a+b*x)^5)-63/128*e^4*(10*b*B*d+A*b*e-11*a*B*e)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(13/2)*sqrt(b*d-a*e))+_
63/128*e^4*(10*b*B*d+A*b*e-11*a*B*e)*sqrt(d+e*x)/(b^6*(b*d-a*e))
--R
--R
--R (2)
--R 6 5 5 6 4 5
--R ((- 315A b + 3465B a b )e - 3150B b d e )x
--R +
--R 5 2 4 5 5 4 4
--R ((- 1575A a b + 17325B a b )e - 15750B a b d e )x
--R +
--R 2 4 3 3 5 2 4 4 3
--R ((- 3150A a b + 34650B a b )e - 31500B a b d e )x
--R +
--R 3 3 4 2 5 3 3 4 2
--R ((- 3150A a b + 34650B a b )e - 31500B a b d e )x
--R +
--R 4 2 5 5 4 2 4
--R ((- 1575A a b + 17325B a b)e - 15750B a b d e )x
--R +
--R 5 6 5 5 4
--R (- 315A a b + 3465B a )e - 3150B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 5 5 4 4 5 3 4
--R 1280B b e x + ((- 965A b + 10615B a b )e - 3250B b d e )x
--R +
--R 4 2 3 4 5 4 3
--R (- 2370A a b + 26070B a b )e + (- 1490A b - 7310B a b )d e
--R +
--R 5 2 2
--R - 2100B b d e
--R *
--R 3
--R x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (- 2688A a b + 29568B a b )e + (- 1734A a b - 7806B a b )d e
--R +
--R 5 4 2 2 5 3
--R (- 1368A b - 2292B a b )d e - 880B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (- 1470A a b + 16170B a b)e + (- 966A a b - 4074B a b )d e
--R +
--R 4 2 3 2 2 5 4 3
--R (- 768A a b - 1212B a b )d e + (- 656A b - 464B a b )d e
--R +
--R 5 4
--R - 160B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 315A a b + 3465B a )e + (- 210A a b - 840B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (- 168A a b - 252B a b )d e + (- 144A a b - 96B a b )d e
--R +
--R 5 4 4
--R (- 128A b - 32B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 11 5 10 4 2 9 3 3 8 2 4 7 5 6
--R (640b x + 3200a b x + 6400a b x + 6400a b x + 3200a b x + 640a b )
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R Type: Expression(Integer)
--E 596
--S 597 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 597
)clear all
--S 598 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)/(a^2+2*a*b*x+b^2*x^2)^3
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x
--R +
--R 3
--R A d
--R *
--R +-------+
--R \|e x + d
--R /
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a
--R Type: Expression(Integer)
--E 598
--S 599 of 764
r0:=-7/192*e^2*(10*b*B*d-A*b*e-9*a*B*e)*(d+e*x)^(3/2)/(b^4*(b*d-a*e)*_
(a+b*x)^2)-7/240*e*(10*b*B*d-A*b*e-9*a*B*e)*(d+e*x)^(5/2)/(b^3*_
(b*d-a*e)*(a+b*x)^3)-1/40*(10*b*B*d-A*b*e-9*a*B*e)*(d+e*x)^(7/2)/_
(b^2*(b*d-a*e)*(a+b*x)^4)-1/5*(A*b-a*B)*(d+e*x)^(9/2)/(b*(b*d-a*e)*_
(a+b*x)^5)-7/128*e^4*(10*b*B*d-A*b*e-9*a*B*e)*atanh(sqrt(b)*_
sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(11/2)*(b*d-a*e)^(3/2))-_
7/128*e^3*(10*b*B*d-A*b*e-9*a*B*e)*sqrt(d+e*x)/(b^5*(b*d-a*e)*(a+b*x))
--R
--R
--R (2)
--R 6 5 5 6 4 5
--R ((- 105A b - 945B a b )e + 1050B b d e )x
--R +
--R 5 2 4 5 5 4 4
--R ((- 525A a b - 4725B a b )e + 5250B a b d e )x
--R +
--R 2 4 3 3 5 2 4 4 3
--R ((- 1050A a b - 9450B a b )e + 10500B a b d e )x
--R +
--R 3 3 4 2 5 3 3 4 2
--R ((- 1050A a b - 9450B a b )e + 10500B a b d e )x
--R +
--R 4 2 5 5 4 2 4
--R ((- 525A a b - 4725B a b)e + 5250B a b d e )x
--R +
--R 5 6 5 5 4
--R (- 105A a b - 945B a )e + 1050B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 4 5 3 4
--R ((105A b - 2895B a b )e + 2790B b d e )x
--R +
--R 4 2 3 4 5 4 3
--R (- 790A a b - 7110B a b )e + (1210A b + 3430B a b )d e
--R +
--R 5 2 2
--R 3260B b d e
--R *
--R 3
--R x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (- 896A a b - 8064B a b )e + (- 578A a b + 3758B a b )d e
--R +
--R 5 4 2 2 5 3
--R (2104A b + 1676B a b )d e + 2000B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (- 490A a b - 4410B a b)e + (- 322A a b + 2002B a b )d e
--R +
--R 4 2 3 2 2 5 4 3
--R (- 256A a b + 916B a b )d e + (1488A b + 592B a b )d e
--R +
--R 5 4
--R 480B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 105A a b - 945B a )e + (- 70A a b + 420B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (- 56A a b + 196B a b )d e + (- 48A a b + 128B a b )d e
--R +
--R 5 4 4
--R (384A b + 96B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 10 11 5 2 9 10 4
--R (1920a b e - 1920b d)x + (9600a b e - 9600a b d)x
--R +
--R 3 8 2 9 3 4 7 3 8 2
--R (19200a b e - 19200a b d)x + (19200a b e - 19200a b d)x
--R +
--R 5 6 4 7 6 5 5 6
--R (9600a b e - 9600a b d)x + 1920a b e - 1920a b d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R Type: Expression(Integer)
--E 599
--S 600 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 600
)clear all
--S 601 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)/(a^2+2*a*b*x+b^2*x^2)^3
--R
--R
--R 2 3 2 2 2 2 +-------+
--R (B e x + (A e + 2B d e)x + (2A d e + B d )x + A d )\|e x + d
--R (1) ----------------------------------------------------------------
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a
--R Type: Expression(Integer)
--E 601
--S 602 of 764
r0:=-1/48*e*(10*b*B*d-3*A*b*e-7*a*B*e)*(d+e*x)^(3/2)/(b^3*(b*d-a*e)*_
(a+b*x)^3)-1/40*(10*b*B*d-3*A*b*e-7*a*B*e)*(d+e*x)^(5/2)/(b^2*(b*d-_
a*e)*(a+b*x)^4)-1/5*(A*b-a*B)*(d+e*x)^(7/2)/(b*(b*d-a*e)*(a+b*x)^5)+_
1/128*e^4*(10*b*B*d-3*A*b*e-7*a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/_
sqrt(b*d-a*e))/(b^(9/2)*(b*d-a*e)^(5/2))-1/64*e^2*(10*b*B*d-_
3*A*b*e-7*a*B*e)*sqrt(d+e*x)/(b^4*(b*d-a*e)*(a+b*x)^2)-_
1/128*e^3*(10*b*B*d-3*A*b*e-7*a*B*e)*sqrt(d+e*x)/(b^4*(b*d-a*e)^2*(a+b*x))
--R
--R
--R (2)
--R 6 5 5 6 4 5
--R ((- 45A b - 105B a b )e + 150B b d e )x
--R +
--R 5 2 4 5 5 4 4
--R ((- 225A a b - 525B a b )e + 750B a b d e )x
--R +
--R 2 4 3 3 5 2 4 4 3
--R ((- 450A a b - 1050B a b )e + 1500B a b d e )x
--R +
--R 3 3 4 2 5 3 3 4 2
--R ((- 450A a b - 1050B a b )e + 1500B a b d e )x
--R +
--R 4 2 5 5 4 2 4
--R ((- 225A a b - 525B a b)e + 750B a b d e )x
--R +
--R 5 6 5 5 4
--R (- 45A a b - 105B a )e + 150B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 4 5 3 4
--R ((45A b + 105B a b )e - 150B b d e )x
--R +
--R 4 2 3 4 5 4 3
--R (210A a b - 790B a b )e + (- 30A b + 1790B a b )d e
--R +
--R 5 2 2
--R - 1180B b d e
--R *
--R 3
--R x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (- 384A a b - 896B a b )e + (1398A a b + 702B a b )d e
--R +
--R 5 4 2 2 5 3
--R (- 744A b + 1284B a b )d e - 1360B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (- 210A a b - 490B a b)e + (- 138A a b + 378B a b )d e
--R +
--R 4 2 3 2 2 5 4 3
--R (1536A a b + 204B a b )d e + (- 1008A b + 208B a b )d e
--R +
--R 5 4
--R - 480B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 45A a b - 105B a )e + (- 30A a b + 80B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (- 24A a b + 44B a b )d e + (528A a b + 32B a b )d e
--R +
--R 5 4 4
--R (- 384A b - 96B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 2 9 2 10 11 2 5
--R (1920a b e - 3840a b d e + 1920b d )x
--R +
--R 3 8 2 2 9 10 2 4
--R (9600a b e - 19200a b d e + 9600a b d )x
--R +
--R 4 7 2 3 8 2 9 2 3
--R (19200a b e - 38400a b d e + 19200a b d )x
--R +
--R 5 6 2 4 7 3 8 2 2
--R (19200a b e - 38400a b d e + 19200a b d )x
--R +
--R 6 5 2 5 6 4 7 2 7 4 2 6 5
--R (9600a b e - 19200a b d e + 9600a b d )x + 1920a b e - 3840a b d e
--R +
--R 5 6 2
--R 1920a b d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R Type: Expression(Integer)
--E 602
--S 603 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 603
)clear all
--S 604 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)/(a^2+2*a*b*x+b^2*x^2)^3
--R
--R
--R 2 +-------+
--R (B e x + (A e + B d)x + A d)\|e x + d
--R (1) -------------------------------------------------------------
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a
--R Type: Expression(Integer)
--E 604
--S 605 of 764
r0:=-1/8*(2*b*B*d-A*b*e-a*B*e)*(d+e*x)^(3/2)/(b^2*(b*d-a*e)*(a+b*x)^4)-_
1/5*(A*b-a*B)*(d+e*x)^(5/2)/(b*(b*d-a*e)*(a+b*x)^5)-3/128*e^4*(2*b*B*d-_
A*b*e-a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(7/2)*(b*d-_
a*e)^(7/2))-1/16*e*(2*b*B*d-A*b*e-a*B*e)*sqrt(d+e*x)/(b^3*(b*d-a*e)*_
(a+b*x)^3)-1/64*e^2*(2*b*B*d-A*b*e-a*B*e)*sqrt(d+e*x)/(b^3*(b*d-a*e)^2*_
(a+b*x)^2)+3/128*e^3*(2*b*B*d-A*b*e-a*B*e)*sqrt(d+e*x)/(b^3*_
(b*d-a*e)^3*(a+b*x))
--R
--R
--R (2)
--R 6 5 5 6 4 5
--R ((- 15A b - 15B a b )e + 30B b d e )x
--R +
--R 5 2 4 5 5 4 4
--R ((- 75A a b - 75B a b )e + 150B a b d e )x
--R +
--R 2 4 3 3 5 2 4 4 3
--R ((- 150A a b - 150B a b )e + 300B a b d e )x
--R +
--R 3 3 4 2 5 3 3 4 2
--R ((- 150A a b - 150B a b )e + 300B a b d e )x
--R +
--R 4 2 5 5 4 2 4 5 6 5
--R ((- 75A a b - 75B a b)e + 150B a b d e )x + (- 15A a b - 15B a )e
--R +
--R 5 4
--R 30B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 4 5 3 4
--R ((15A b + 15B a b )e - 30B b d e )x
--R +
--R 4 2 3 4 5 4 3 5 2 2 3
--R ((70A a b + 70B a b )e + (- 10A b - 150B a b )d e + 20B b d e )x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (128A a b - 128B a b )e + (- 46A a b + 466B a b )d e
--R +
--R 5 4 2 2 5 3
--R (8A b - 668B a b )d e + 240B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (- 70A a b - 70B a b)e + (466A a b + 94B a b )d e
--R +
--R 4 2 3 2 2 5 4 3 5 4
--R (- 512A a b + 92B a b )d e + (176A b - 336B a b )d e + 160B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 15A a b - 15B a )e + (- 10A a b + 20B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (248A a b + 12B a b )d e + (- 336A a b - 64B a b )d e
--R +
--R 5 4 4
--R (128A b + 32B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 3 8 3 2 9 2 10 2 11 3 5
--R (640a b e - 1920a b d e + 1920a b d e - 640b d )x
--R +
--R 4 7 3 3 8 2 2 9 2 10 3 4
--R (3200a b e - 9600a b d e + 9600a b d e - 3200a b d )x
--R +
--R 5 6 3 4 7 2 3 8 2 2 9 3 3
--R (6400a b e - 19200a b d e + 19200a b d e - 6400a b d )x
--R +
--R 6 5 3 5 6 2 4 7 2 3 8 3 2
--R (6400a b e - 19200a b d e + 19200a b d e - 6400a b d )x
--R +
--R 7 4 3 6 5 2 5 6 2 4 7 3 8 3 3
--R (3200a b e - 9600a b d e + 9600a b d e - 3200a b d )x + 640a b e
--R +
--R 7 4 2 6 5 2 5 6 3
--R - 1920a b d e + 1920a b d e - 640a b d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R Type: Expression(Integer)
--E 605
--S 606 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 606
)clear all
--S 607 of 764
t0:=(A+B*x)*sqrt(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^3
--R
--R
--R +-------+
--R (B x + A)\|e x + d
--R (1) -------------------------------------------------------------
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6
--R b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a
--R Type: Expression(Integer)
--E 607
--S 608 of 764
r0:=-1/5*(A*b-a*B)*(d+e*x)^(3/2)/(b*(b*d-a*e)*(a+b*x)^5)+1/128*e^4*_
(10*b*B*d-7*A*b*e-3*a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/_
(b^(5/2)*(b*d-a*e)^(9/2))-1/40*(10*b*B*d-7*A*b*e-3*a*B*e)*sqrt(d+e*x)/_
(b^2*(b*d-a*e)*(a+b*x)^4)-1/240*e*(10*b*B*d-7*A*b*e-3*a*B*e)*_
sqrt(d+e*x)/(b^2*(b*d-a*e)^2*(a+b*x)^3)+1/192*e^2*(10*b*B*d-7*A*b*e-_
3*a*B*e)*sqrt(d+e*x)/(b^2*(b*d-a*e)^3*(a+b*x)^2)-1/128*e^3*(10*b*B*d-_
7*A*b*e-3*a*B*e)*sqrt(d+e*x)/(b^2*(b*d-a*e)^4*(a+b*x))
--R
--R
--R (2)
--R 6 5 5 6 4 5
--R ((- 105A b - 45B a b )e + 150B b d e )x
--R +
--R 5 2 4 5 5 4 4
--R ((- 525A a b - 225B a b )e + 750B a b d e )x
--R +
--R 2 4 3 3 5 2 4 4 3
--R ((- 1050A a b - 450B a b )e + 1500B a b d e )x
--R +
--R 3 3 4 2 5 3 3 4 2
--R ((- 1050A a b - 450B a b )e + 1500B a b d e )x
--R +
--R 4 2 5 5 4 2 4
--R ((- 525A a b - 225B a b)e + 750B a b d e )x
--R +
--R 5 6 5 5 4
--R (- 105A a b - 45B a )e + 150B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 4 5 3 4
--R ((105A b + 45B a b )e - 150B b d e )x
--R +
--R 4 2 3 4 5 4 3
--R (490A a b + 210B a b )e + (- 70A b - 730B a b )d e
--R +
--R 5 2 2
--R 100B b d e
--R *
--R 3
--R x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (896A a b + 384B a b )e + (- 322A a b - 1418B a b )d e
--R +
--R 5 4 2 2 5 3
--R (56A b + 484B a b )d e - 80B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (790A a b - 210B a b)e + (- 578A a b + 818B a b )d e
--R +
--R 4 2 3 2 2 5 4 3
--R (256A a b - 2356B a b )d e + (- 48A b + 1808B a b )d e
--R +
--R 5 4
--R - 480B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (- 105A a b - 45B a )e + (1210A a b + 120B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (- 2104A a b - 436B a b )d e + (1488A a b + 352B a b )d e
--R +
--R 5 4 4
--R (- 384A b - 96B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 4 7 4 3 8 3 2 9 2 2 10 3
--R 1920a b e - 7680a b d e + 11520a b d e - 7680a b d e
--R +
--R 11 4
--R 1920b d
--R *
--R 5
--R x
--R +
--R 5 6 4 4 7 3 3 8 2 2 2 9 3
--R 9600a b e - 38400a b d e + 57600a b d e - 38400a b d e
--R +
--R 10 4
--R 9600a b d
--R *
--R 4
--R x
--R +
--R 6 5 4 5 6 3 4 7 2 2 3 8 3
--R 19200a b e - 76800a b d e + 115200a b d e - 76800a b d e
--R +
--R 2 9 4
--R 19200a b d
--R *
--R 3
--R x
--R +
--R 7 4 4 6 5 3 5 6 2 2 4 7 3
--R 19200a b e - 76800a b d e + 115200a b d e - 76800a b d e
--R +
--R 3 8 4
--R 19200a b d
--R *
--R 2
--R x
--R +
--R 8 3 4 7 4 3 6 5 2 2 5 6 3
--R 9600a b e - 38400a b d e + 57600a b d e - 38400a b d e
--R +
--R 4 7 4
--R 9600a b d
--R *
--R x
--R +
--R 9 2 4 8 3 3 7 4 2 2 6 5 3 5 6 4
--R 1920a b e - 7680a b d e + 11520a b d e - 7680a b d e + 1920a b d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R Type: Expression(Integer)
--E 608
--S 609 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 609
)clear all
--S 610 of 764
t0:=(A+B*x)/((a^2+2*a*b*x+b^2*x^2)^3*sqrt(d+e*x))
--R
--R
--R (1)
--R B x + A
--R -------------------------------------------------------------------------
--R 6 6 5 5 2 4 4 3 3 3 4 2 2 5 6 +-------+
--R (b x + 6a b x + 15a b x + 20a b x + 15a b x + 6a b x + a )\|e x + d
--R Type: Expression(Integer)
--E 610
--S 611 of 764
r0:=-7/128*e^4*(10*b*B*d-9*A*b*e-a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/_
sqrt(b*d-a*e))/(b^(3/2)*(b*d-a*e)^(11/2))-1/5*(A*b-a*B)*sqrt(d+e*x)/_
(b*(b*d-a*e)*(a+b*x)^5)-1/40*(10*b*B*d-9*A*b*e-a*B*e)*sqrt(d+e*x)/_
(b*(b*d-a*e)^2*(a+b*x)^4)+7/240*e*(10*b*B*d-9*A*b*e-a*B*e)*_
sqrt(d+e*x)/(b*(b*d-a*e)^3*(a+b*x)^3)-7/192*e^2*(10*b*B*d-9*A*b*e-_
a*B*e)*sqrt(d+e*x)/(b*(b*d-a*e)^4*(a+b*x)^2)+7/128*e^3*(10*b*B*d-_
9*A*b*e-a*B*e)*sqrt(d+e*x)/(b*(b*d-a*e)^5*(a+b*x))
--R
--R
--R (2)
--R 6 5 5 6 4 5
--R ((- 945A b - 105B a b )e + 1050B b d e )x
--R +
--R 5 2 4 5 5 4 4
--R ((- 4725A a b - 525B a b )e + 5250B a b d e )x
--R +
--R 2 4 3 3 5 2 4 4 3
--R ((- 9450A a b - 1050B a b )e + 10500B a b d e )x
--R +
--R 3 3 4 2 5 3 3 4 2
--R ((- 9450A a b - 1050B a b )e + 10500B a b d e )x
--R +
--R 4 2 5 5 4 2 4
--R ((- 4725A a b - 525B a b)e + 5250B a b d e )x
--R +
--R 5 6 5 5 4
--R (- 945A a b - 105B a )e + 1050B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 4 5 3 4
--R ((945A b + 105B a b )e - 1050B b d e )x
--R +
--R 4 2 3 4 5 4 3
--R (4410A a b + 490B a b )e + (- 630A b - 4970B a b )d e
--R +
--R 5 2 2
--R 700B b d e
--R *
--R 3
--R x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (8064A a b + 896B a b )e + (- 2898A a b - 9282B a b )d e
--R +
--R 5 4 2 2 5 3
--R (504A b + 3276B a b )d e - 560B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (7110A a b + 790B a b)e + (- 5202A a b - 8478B a b )d e
--R +
--R 4 2 3 2 2 5 4 3
--R (2304A a b + 6036B a b )d e + (- 432A b - 2608B a b )d e
--R +
--R 5 4
--R 480B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (2895A a b - 105B a )e + (- 4470A a b - 1580B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (4104A a b + 1156B a b )d e + (- 1968A a b - 512B a b )d e
--R +
--R 5 4 4
--R (384A b + 96B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 5 6 5 4 7 4 3 8 2 3 2 9 3 2
--R 1920a b e - 9600a b d e + 19200a b d e - 19200a b d e
--R +
--R 10 4 11 5
--R 9600a b d e - 1920b d
--R *
--R 5
--R x
--R +
--R 6 5 5 5 6 4 4 7 2 3 3 8 3 2
--R 9600a b e - 48000a b d e + 96000a b d e - 96000a b d e
--R +
--R 2 9 4 10 5
--R 48000a b d e - 9600a b d
--R *
--R 4
--R x
--R +
--R 7 4 5 6 5 4 5 6 2 3 4 7 3 2
--R 19200a b e - 96000a b d e + 192000a b d e - 192000a b d e
--R +
--R 3 8 4 2 9 5
--R 96000a b d e - 19200a b d
--R *
--R 3
--R x
--R +
--R 8 3 5 7 4 4 6 5 2 3 5 6 3 2
--R 19200a b e - 96000a b d e + 192000a b d e - 192000a b d e
--R +
--R 4 7 4 3 8 5
--R 96000a b d e - 19200a b d
--R *
--R 2
--R x
--R +
--R 9 2 5 8 3 4 7 4 2 3 6 5 3 2
--R 9600a b e - 48000a b d e + 96000a b d e - 96000a b d e
--R +
--R 5 6 4 4 7 5
--R 48000a b d e - 9600a b d
--R *
--R x
--R +
--R 10 5 9 2 4 8 3 2 3 7 4 3 2
--R 1920a b e - 9600a b d e + 19200a b d e - 19200a b d e
--R +
--R 6 5 4 5 6 5
--R 9600a b d e - 1920a b d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R Type: Expression(Integer)
--E 611
--S 612 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 612
)clear all
--S 613 of 764
t0:=(A+B*x)/((d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^3)
--R
--R
--R (1)
--R B x + A
--R /
--R 6 7 5 6 6 2 4 5 5
--R b e x + (6a b e + b d)x + (15a b e + 6a b d)x
--R +
--R 3 3 2 4 4 4 2 3 3 3 5 4 2 2
--R (20a b e + 15a b d)x + (15a b e + 20a b d)x + (6a b e + 15a b d)x
--R +
--R 6 5 6
--R (a e + 6a b d)x + a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 613
--S 614 of 764
r0:=-63/128*e^4*(10*b*B*d-11*A*b*e+a*B*e)*atanh(sqrt(b)*sqrt(d+e*x)/_
sqrt(b*d-a*e))/((b*d-a*e)^(13/2)*sqrt(b))+63/128*e^4*(10*b*B*d-_
11*A*b*e+a*B*e)/(b*(b*d-a*e)^6*sqrt(d+e*x))+1/5*(-A*b+a*B)/(b*(b*d-_
a*e)*(a+b*x)^5*sqrt(d+e*x))+1/40*(-10*b*B*d+11*A*b*e-a*B*e)/(b*(b*d-_
a*e)^2*(a+b*x)^4*sqrt(d+e*x))+3/80*e*(10*b*B*d-11*A*b*e+a*B*e)/(b*_
(b*d-a*e)^3*(a+b*x)^3*sqrt(d+e*x))-21/320*e^2*(10*b*B*d-11*A*b*e+_
a*B*e)/(b*(b*d-a*e)^4*(a+b*x)^2*sqrt(d+e*x))+21/128*e^3*(10*b*B*d-_
11*A*b*e+a*B*e)/(b*(b*d-a*e)^5*(a+b*x)*sqrt(d+e*x))
--R
--R
--R (2)
--R 6 5 5 6 4 5
--R ((3465A b - 315B a b )e - 3150B b d e )x
--R +
--R 5 2 4 5 5 4 4
--R ((17325A a b - 1575B a b )e - 15750B a b d e )x
--R +
--R 2 4 3 3 5 2 4 4 3
--R ((34650A a b - 3150B a b )e - 31500B a b d e )x
--R +
--R 3 3 4 2 5 3 3 4 2
--R ((34650A a b - 3150B a b )e - 31500B a b d e )x
--R +
--R 4 2 5 5 4 2 4
--R ((17325A a b - 1575B a b)e - 15750B a b d e )x
--R +
--R 5 6 5 5 4
--R (3465A a b - 315B a )e - 3150B a b d e
--R *
--R +-+ +-------+
--R +-------+ \|b \|e x + d
--R \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 5 5 4 5
--R ((- 3465A b + 315B a b )e + 3150B b d e )x
--R +
--R 4 2 3 5 5 4 4
--R (- 16170A a b + 1470B a b )e + (- 1155A b + 14805B a b )d e
--R +
--R 5 2 3
--R 1050B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 5 4 2 3 4
--R (- 29568A a b + 2688B a b )e + (- 5544A a b + 27384B a b )d e
--R +
--R 5 4 2 3 5 3 2
--R (462A b + 4998B a b )d e - 420B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 5 2 3 3 2 4
--R (- 26070A a b + 2370B a b)e + (- 10494A a b + 24654B a b )d e
--R +
--R 4 2 3 2 3 5 4 3 2
--R (2178A a b + 9342B a b )d e + (- 264A b - 1956B a b )d e
--R +
--R 5 4
--R 240B b d e
--R *
--R 2
--R x
--R +
--R 4 5 5 3 2 4 4
--R (- 10615A a b + 965B a )e + (- 9680A a b + 10530B a b)d e
--R +
--R 2 3 3 2 2 3 4 2 3 3 2
--R (4026A a b + 8434B a b )d e + (- 1232A a b - 3548B a b )d e
--R +
--R 5 4 4 5 5
--R (176A b + 1104B a b )d e - 160B b d
--R *
--R x
--R +
--R 5 5 4 5 4
--R - 1280A a e + (- 4215A a b + 2245B a )d e
--R +
--R 3 2 4 2 3 2 3 3 2 3 2
--R (3590A a b + 1760B a b)d e + (- 2248A a b - 732B a b )d e
--R +
--R 4 2 3 4 5 4 5
--R (816A a b + 224B a b )d e + (- 128A b - 32B a b )d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R /
--R 6 5 6 5 6 5 4 7 2 4 3 8 3 3
--R 640a b e - 3840a b d e + 9600a b d e - 12800a b d e
--R +
--R 2 9 4 2 10 5 11 6
--R 9600a b d e - 3840a b d e + 640b d
--R *
--R 5
--R x
--R +
--R 7 4 6 6 5 5 5 6 2 4 4 7 3 3
--R 3200a b e - 19200a b d e + 48000a b d e - 64000a b d e
--R +
--R 3 8 4 2 2 9 5 10 6
--R 48000a b d e - 19200a b d e + 3200a b d
--R *
--R 4
--R x
--R +
--R 8 3 6 7 4 5 6 5 2 4 5 6 3 3
--R 6400a b e - 38400a b d e + 96000a b d e - 128000a b d e
--R +
--R 4 7 4 2 3 8 5 2 9 6
--R 96000a b d e - 38400a b d e + 6400a b d
--R *
--R 3
--R x
--R +
--R 9 2 6 8 3 5 7 4 2 4 6 5 3 3
--R 6400a b e - 38400a b d e + 96000a b d e - 128000a b d e
--R +
--R 5 6 4 2 4 7 5 3 8 6
--R 96000a b d e - 38400a b d e + 6400a b d
--R *
--R 2
--R x
--R +
--R 10 6 9 2 5 8 3 2 4 7 4 3 3
--R 3200a b e - 19200a b d e + 48000a b d e - 64000a b d e
--R +
--R 6 5 4 2 5 6 5 4 7 6
--R 48000a b d e - 19200a b d e + 3200a b d
--R *
--R x
--R +
--R 11 6 10 5 9 2 2 4 8 3 3 3 7 4 4 2
--R 640a e - 3840a b d e + 9600a b d e - 12800a b d e + 9600a b d e
--R +
--R 6 5 5 5 6 6
--R - 3840a b d e + 640a b d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R Type: Expression(Integer)
--E 614
--S 615 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 615
)clear all
--S 616 of 764
t0:=(A+B*x)/((d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^3)
--R
--R
--R (1)
--R B x + A
--R /
--R 6 2 8 5 2 6 7 2 4 2 5 6 2 6
--R b e x + (6a b e + 2b d e)x + (15a b e + 12a b d e + b d )x
--R +
--R 3 3 2 2 4 5 2 5
--R (20a b e + 30a b d e + 6a b d )x
--R +
--R 4 2 2 3 3 2 4 2 4
--R (15a b e + 40a b d e + 15a b d )x
--R +
--R 5 2 4 2 3 3 2 3 6 2 5 4 2 2 2
--R (6a b e + 30a b d e + 20a b d )x + (a e + 12a b d e + 15a b d )x
--R +
--R 6 5 2 6 2
--R (2a d e + 6a b d )x + a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 616
--S 617 of 764
r0:=77/128*e^4*(10*b*B*d-13*A*b*e+3*a*B*e)/(b*(b*d-a*e)^6*(d+e*x)^(3/2))+_
1/5*(-A*b+a*B)/(b*(b*d-a*e)*(a+b*x)^5*(d+e*x)^(3/2))+1/40*(-10*b*B*d+_
13*A*b*e-3*a*B*e)/(b*(b*d-a*e)^2*(a+b*x)^4*(d+e*x)^(3/2))+11/240*e*_
(10*b*B*d-13*A*b*e+3*a*B*e)/(b*(b*d-a*e)^3*(a+b*x)^3*(d+e*x)^(3/2))-_
33/320*e^2*(10*b*B*d-13*A*b*e+3*a*B*e)/(b*(b*d-a*e)^4*(a+b*x)^2*_
(d+e*x)^(3/2))+231/640*e^3*(10*b*B*d-13*A*b*e+3*a*B*e)/(b*(b*d-a*e)^5*_
(a+b*x)*(d+e*x)^(3/2))-231/128*e^4*(10*b*B*d-13*A*b*e+3*a*B*e)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b)/(b*d-a*e)^(15/2)+_
231/128*e^4*(10*b*B*d-13*A*b*e+3*a*B*e)/((b*d-a*e)^7*sqrt(d+e*x))
--R
--R
--R (2)
--R 6 5 6 6 5 6
--R ((- 45045A b + 10395B a b )e + 34650B b d e )x
--R +
--R 5 2 4 6
--R (- 225225A a b + 51975B a b )e
--R +
--R 6 5 5 6 2 4
--R (- 45045A b + 183645B a b )d e + 34650B b d e
--R *
--R 5
--R x
--R +
--R 2 4 3 3 6
--R (- 450450A a b + 103950B a b )e
--R +
--R 5 2 4 5 5 2 4
--R (- 225225A a b + 398475B a b )d e + 173250B a b d e
--R *
--R 4
--R x
--R +
--R 3 3 4 2 6
--R (- 450450A a b + 103950B a b )e
--R +
--R 2 4 3 3 5 2 4 2 4
--R (- 450450A a b + 450450B a b )d e + 346500B a b d e
--R *
--R 3
--R x
--R +
--R 4 2 5 6
--R (- 225225A a b + 51975B a b)e
--R +
--R 3 3 4 2 5 3 3 2 4
--R (- 450450A a b + 277200B a b )d e + 346500B a b d e
--R *
--R 2
--R x
--R +
--R 5 6 6 4 2 5 5
--R (- 45045A a b + 10395B a )e + (- 225225A a b + 86625B a b)d e
--R +
--R 4 2 2 4
--R 173250B a b d e
--R *
--R x
--R +
--R 5 6 5 5 2 4
--R (- 45045A a b + 10395B a )d e + 34650B a b d e
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 6 5 6 6 5 6
--R ((45045A b - 10395B a b )e - 34650B b d e )x
--R +
--R 5 2 4 6 6 5 5
--R (210210A a b - 48510B a b )e + (60060A b - 175560B a b )d e
--R +
--R 6 2 4
--R - 46200B b d e
--R *
--R 5
--R x
--R +
--R 2 4 3 3 6
--R (384384A a b - 88704B a b )e
--R +
--R 5 2 4 5 6 5 2 4
--R (282282A a b - 360822B a b )d e + (9009A b - 219219B a b )d e
--R +
--R 6 3 3
--R - 6930B b d e
--R *
--R 4
--R x
--R +
--R 3 3 4 2 6
--R (338910A a b - 78210B a b )e
--R +
--R 2 4 3 3 5
--R (520806A a b - 380886B a b )d e
--R +
--R 5 2 4 2 4 6 5 3 3
--R (43758A a b - 410718B a b )d e + (- 2574A b - 33066B a b )d e
--R +
--R 6 4 2
--R 1980B b d e
--R *
--R 3
--R x
--R +
--R 4 2 5 6 3 3 4 2 5
--R (137995A a b - 31845B a b)e + (464750A a b - 213400B a b )d e
--R +
--R 2 4 3 3 2 4
--R (84084A a b - 376904B a b )d e
--R +
--R 5 2 4 3 3 6 5 4 2
--R (- 12298A a b - 61842B a b )d e + (1144A b + 9196B a b )d e
--R +
--R 6 5
--R - 880B b d e
--R *
--R 2
--R x
--R +
--R 5 6 6 4 2 5 5
--R (16640A a b - 3840B a )e + (192790A a b - 57290B a b)d e
--R +
--R 3 3 4 2 2 4
--R (79170A a b - 166570B a b )d e
--R +
--R 2 4 3 3 3 3
--R (- 23114A a b - 55566B a b )d e
--R +
--R 5 2 4 4 2 6 5 5
--R (5408A a b + 16532B a b )d e + (- 624A b - 4016B a b )d e
--R +
--R 6 6
--R 480B b d
--R *
--R x
--R +
--R 6 6 5 6 5
--R - 1280A a e + (24320A a b - 2560B a )d e
--R +
--R 4 2 5 2 4 3 3 4 2 3 3
--R (35595A a b - 33125B a b)d e + (- 21070A a b - 12180B a b )d e
--R +
--R 2 4 3 3 4 2 5 2 4 5
--R (10024A a b + 3556B a b )d e + (- 2928A a b - 832B a b )d e
--R +
--R 6 5 6
--R (384A b + 96B a b )d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 7 5 8 6 6 7 5 7 2 6 4 8 3 5
--R 1920a b e - 13440a b d e + 40320a b d e - 67200a b d e
--R +
--R 3 9 4 4 2 10 5 3 11 6 2 12 7
--R 67200a b d e - 40320a b d e + 13440a b d e - 1920b d e
--R *
--R 6
--R x
--R +
--R 8 4 8 7 5 7 6 6 2 6 5 7 3 5
--R 9600a b e - 65280a b d e + 188160a b d e - 295680a b d e
--R +
--R 4 8 4 4 3 9 5 3 2 10 6 2 11 7
--R 268800a b d e - 134400a b d e + 26880a b d e + 3840a b d e
--R +
--R 12 8
--R - 1920b d
--R *
--R 5
--R x
--R +
--R 9 3 8 8 4 7 7 5 2 6 6 6 3 5
--R 19200a b e - 124800a b d e + 336000a b d e - 470400a b d e
--R +
--R 5 7 4 4 4 8 5 3 3 9 6 2 2 10 7
--R 336000a b d e - 67200a b d e - 67200a b d e + 48000a b d e
--R +
--R 11 8
--R - 9600a b d
--R *
--R 4
--R x
--R +
--R 10 2 8 9 3 7 8 4 2 6 7 5 3 5
--R 19200a b e - 115200a b d e + 268800a b d e - 268800a b d e
--R +
--R 5 7 5 3 4 8 6 2 3 9 7 2 10 8
--R 268800a b d e - 268800a b d e + 115200a b d e - 19200a b d
--R *
--R 3
--R x
--R +
--R 11 8 10 2 7 9 3 2 6 8 4 3 5
--R 9600a b e - 48000a b d e + 67200a b d e + 67200a b d e
--R +
--R 7 5 4 4 6 6 5 3 5 7 6 2 4 8 7
--R - 336000a b d e + 470400a b d e - 336000a b d e + 124800a b d e
--R +
--R 3 9 8
--R - 19200a b d
--R *
--R 2
--R x
--R +
--R 12 8 11 7 10 2 2 6 9 3 3 5
--R 1920a e - 3840a b d e - 26880a b d e + 134400a b d e
--R +
--R 8 4 4 4 7 5 5 3 6 6 6 2 5 7 7
--R - 268800a b d e + 295680a b d e - 188160a b d e + 65280a b d e
--R +
--R 4 8 8
--R - 9600a b d
--R *
--R x
--R +
--R 12 7 11 2 6 10 2 3 5 9 3 4 4
--R 1920a d e - 13440a b d e + 40320a b d e - 67200a b d e
--R +
--R 8 4 5 3 7 5 6 2 6 6 7 5 7 8
--R 67200a b d e - 40320a b d e + 13440a b d e - 1920a b d
--R *
--R +-----------+ +-------+
--R \|- a e + b d \|e x + d
--R Type: Expression(Integer)
--E 617
--S 618 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 618
)clear all
--S 619 of 764
t0:=(A+B*x)/((d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^3)
--R
--R
--R (1)
--R B x + A
--R /
--R 6 3 9 5 3 6 2 8 2 4 3 5 2 6 2 7
--R b e x + (6a b e + 3b d e )x + (15a b e + 18a b d e + 3b d e)x
--R +
--R 3 3 3 2 4 2 5 2 6 3 6
--R (20a b e + 45a b d e + 18a b d e + b d )x
--R +
--R 4 2 3 3 3 2 2 4 2 5 3 5
--R (15a b e + 60a b d e + 45a b d e + 6a b d )x
--R +
--R 5 3 4 2 2 3 3 2 2 4 3 4
--R (6a b e + 45a b d e + 60a b d e + 15a b d )x
--R +
--R 6 3 5 2 4 2 2 3 3 3 3
--R (a e + 18a b d e + 45a b d e + 20a b d )x
--R +
--R 6 2 5 2 4 2 3 2 6 2 5 3 6 3
--R (3a d e + 18a b d e + 15a b d )x + (3a d e + 6a b d )x + a d
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 619
--S 620 of 764
r0:=3003/640*e^4*(2*b*B*d-3*A*b*e+a*B*e)/(b*(b*d-a*e)^6*(d+e*x)^(5/2))+_
1/5*(-A*b+a*B)/(b*(b*d-a*e)*(a+b*x)^5*(d+e*x)^(5/2))+1/8*(-2*b*B*d+_
3*A*b*e-a*B*e)/(b*(b*d-a*e)^2*(a+b*x)^4*(d+e*x)^(5/2))+13/48*e*(2*b*B*d-_
3*A*b*e+a*B*e)/(b*(b*d-a*e)^3*(a+b*x)^3*(d+e*x)^(5/2))-143/192*e^2*_
(2*b*B*d-3*A*b*e+a*B*e)/(b*(b*d-a*e)^4*(a+b*x)^2*(d+e*x)^(5/2))+_
429/128*e^3*(2*b*B*d-3*A*b*e+a*B*e)/(b*(b*d-a*e)^5*(a+b*x)*_
(d+e*x)^(5/2))+1001/128*e^4*(2*b*B*d-3*A*b*e+a*B*e)/((b*d-a*e)^7*_
(d+e*x)^(3/2))-3003/128*b^(3/2)*e^4*(2*b*B*d-3*A*b*e+a*B*e)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b*d-a*e)^(17/2)+_
3003/128*b*e^4*(2*b*B*d-3*A*b*e+a*B*e)/((b*d-a*e)^8*sqrt(d+e*x))
--R
--R
--R (2)
--R 7 6 7 7 6 7
--R ((135135A b - 45045B a b )e - 90090B b d e )x
--R +
--R 6 2 5 7 7 6 6
--R (675675A a b - 225225B a b )e + (270270A b - 540540B a b )d e
--R +
--R 7 2 5
--R - 180180B b d e
--R *
--R 6
--R x
--R +
--R 2 5 3 4 7
--R (1351350A a b - 450450B a b )e
--R +
--R 6 2 5 6
--R (1351350A a b - 1351350B a b )d e
--R +
--R 7 6 2 5 7 3 4
--R (135135A b - 945945B a b )d e - 90090B b d e
--R *
--R 5
--R x
--R +
--R 3 4 4 3 7
--R (1351350A a b - 450450B a b )e
--R +
--R 2 5 3 4 6
--R (2702700A a b - 1801800B a b )d e
--R +
--R 6 2 5 2 5 6 3 4
--R (675675A a b - 2027025B a b )d e - 450450B a b d e
--R *
--R 4
--R x
--R +
--R 4 3 5 2 7
--R (675675A a b - 225225B a b )e
--R +
--R 3 4 4 3 6
--R (2702700A a b - 1351350B a b )d e
--R +
--R 2 5 3 4 2 5 2 5 3 4
--R (1351350A a b - 2252250B a b )d e - 900900B a b d e
--R *
--R 3
--R x
--R +
--R 5 2 6 7
--R (135135A a b - 45045B a b)e
--R +
--R 4 3 5 2 6
--R (1351350A a b - 540540B a b )d e
--R +
--R 3 4 4 3 2 5 3 4 3 4
--R (1351350A a b - 1351350B a b )d e - 900900B a b d e
--R *
--R 2
--R x
--R +
--R 5 2 6 6
--R (270270A a b - 90090B a b)d e
--R +
--R 4 3 5 2 2 5 4 3 3 4
--R (675675A a b - 405405B a b )d e - 450450B a b d e
--R *
--R x
--R +
--R 5 2 6 2 5 5 2 3 4
--R (135135A a b - 45045B a b)d e - 90090B a b d e
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 7 6 7 7 6 7
--R ((- 135135A b + 45045B a b )e + 90090B b d e )x
--R +
--R 6 2 5 7
--R (- 630630A a b + 210210B a b )e
--R +
--R 7 6 6 7 2 5
--R (- 315315A b + 525525B a b )d e + 210210B b d e
--R *
--R 6
--R x
--R +
--R 2 5 3 4 7
--R (- 1153152A a b + 384384B a b )e
--R +
--R 6 2 5 6
--R (- 1477476A a b + 1261260B a b )d e
--R +
--R 7 6 2 5 7 3 4
--R (- 207207A b + 1054053B a b )d e + 138138B b d e
--R *
--R 5
--R x
--R +
--R 3 4 4 3 7
--R (- 1016730A a b + 338910B a b )e
--R +
--R 2 5 3 4 6
--R (- 2715570A a b + 1583010B a b )d e
--R +
--R 6 2 5 2 5
--R (- 978120A a b + 2136420B a b )d e
--R +
--R 7 6 3 4 7 4 3
--R (- 19305A b + 658515B a b )d e + 12870B b d e
--R *
--R 4
--R x
--R +
--R 4 3 5 2 7
--R (- 413985A a b + 137995B a b )e
--R +
--R 3 4 4 3 6
--R (- 2410980A a b + 1079650B a b )d e
--R +
--R 2 5 3 4 2 5
--R (- 1814670A a b + 2212210B a b )d e
--R +
--R 6 2 5 3 4
--R (- 94380A a b + 1241240B a b )d e
--R +
--R 7 6 4 3 7 5 2
--R (4290A b + 61490B a b )d e - 2860B b d e
--R *
--R 3
--R x
--R +
--R 5 2 6 7
--R (- 49920A a b + 16640B a b)e
--R +
--R 4 3 5 2 6
--R (- 992355A a b + 364065B a b )d e
--R +
--R 3 4 4 3 2 5
--R (- 1631760A a b + 1205490B a b )d e
--R +
--R 2 5 3 4 3 4
--R (- 182910A a b + 1148810B a b )d e
--R +
--R 6 2 5 4 3 7 6 5 2
--R (20670A a b + 115050B a b )d e + (- 1560A b - 13260B a b )d e
--R +
--R 7 6
--R 1040B b d e
--R *
--R 2
--R x
--R +
--R 6 7 7 5 2 6 6
--R (3840A a b - 1280B a )e + (- 122880A a b + 38400B a b)d e
--R +
--R 4 3 5 2 2 5
--R (- 685155A a b + 310305B a b )d e
--R +
--R 3 4 4 3 3 4
--R (- 174300A a b + 514870B a b )d e
--R +
--R 2 5 3 4 4 3
--R (39270A a b + 103110B a b )d e
--R +
--R 6 2 5 5 2 7 6 6
--R (- 7440A a b - 23700B a b )d e + (720A b + 4720B a b )d e
--R +
--R 7 7
--R - 480B b d
--R *
--R x
--R +
--R 7 7 6 7 6
--R - 768A a e + (9216A a b - 512B a )d e
--R +
--R 5 2 6 2 5 4 3 5 2 3 4
--R (- 89088A a b + 16384B a b)d e + (- 79905A a b + 100363B a b )d e
--R +
--R 3 4 4 3 4 3 2 5 3 4 5 2
--R (36330A a b + 23240B a b )d e + (- 13944A a b - 5236B a b )d e
--R +
--R 6 2 5 6 7 6 7
--R (3408A a b + 992B a b )d e + (- 384A b - 96B a b )d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 8 5 10 7 6 9 6 7 2 8 5 8 3 7
--R 1920a b e - 15360a b d e + 53760a b d e - 107520a b d e
--R +
--R 4 9 4 6 3 10 5 5 2 11 6 4 12 7 3
--R 134400a b d e - 107520a b d e + 53760a b d e - 15360a b d e
--R +
--R 13 8 2
--R 1920b d e
--R *
--R 7
--R x
--R +
--R 9 4 10 8 5 9 7 6 2 8 6 7 3 7
--R 9600a b e - 72960a b d e + 238080a b d e - 430080a b d e
--R +
--R 5 8 4 6 4 9 5 5 3 10 6 4 2 11 7 3
--R 456960a b d e - 268800a b d e + 53760a b d e + 30720a b d e
--R +
--R 12 8 2 13 9
--R - 21120a b d e + 3840b d e
--R *
--R 6
--R x
--R +
--R 10 3 10 9 4 9 8 5 2 8 7 6 3 7
--R 19200a b e - 134400a b d e + 385920a b d e - 552960a b d e
--R +
--R 6 7 4 6 5 8 5 5 4 9 6 4 3 10 7 3
--R 322560a b d e + 161280a b d e - 403200a b d e + 276480a b d e
--R +
--R 2 11 8 2 12 9 13 10
--R - 80640a b d e + 3840a b d e + 1920b d
--R *
--R 5
--R x
--R +
--R 11 2 10 10 3 9 9 4 2 8 8 5 3 7
--R 19200a b e - 115200a b d e + 240000a b d e - 76800a b d e
--R +
--R 7 6 4 6 6 7 5 5 5 8 6 4
--R - 537600a b d e + 1075200a b d e - 940800a b d e
--R +
--R 4 9 7 3 3 10 8 2 2 11 9 12 10
--R 384000a b d e - 19200a b d e - 38400a b d e + 9600a b d
--R *
--R 4
--R x
--R +
--R 12 10 11 2 9 10 3 2 8 9 4 3 7
--R 9600a b e - 38400a b d e - 19200a b d e + 384000a b d e
--R +
--R 8 5 4 6 7 6 5 5 6 7 6 4 5 8 7 3
--R - 940800a b d e + 1075200a b d e - 537600a b d e - 76800a b d e
--R +
--R 4 9 8 2 3 10 9 2 11 10
--R 240000a b d e - 115200a b d e + 19200a b d
--R *
--R 3
--R x
--R +
--R 13 10 12 9 11 2 2 8 10 3 3 7
--R 1920a e + 3840a b d e - 80640a b d e + 276480a b d e
--R +
--R 9 4 4 6 8 5 5 5 7 6 6 4 6 7 7 3
--R - 403200a b d e + 161280a b d e + 322560a b d e - 552960a b d e
--R +
--R 5 8 8 2 4 9 9 3 10 10
--R 385920a b d e - 134400a b d e + 19200a b d
--R *
--R 2
--R x
--R +
--R 13 9 12 2 8 11 2 3 7 10 3 4 6
--R 3840a d e - 21120a b d e + 30720a b d e + 53760a b d e
--R +
--R 9 4 5 5 8 5 6 4 7 6 7 3 6 7 8 2
--R - 268800a b d e + 456960a b d e - 430080a b d e + 238080a b d e
--R +
--R 5 8 9 4 9 10
--R - 72960a b d e + 9600a b d
--R *
--R x
--R +
--R 13 2 8 12 3 7 11 2 4 6 10 3 5 5
--R 1920a d e - 15360a b d e + 53760a b d e - 107520a b d e
--R +
--R 9 4 6 4 8 5 7 3 7 6 8 2 6 7 9
--R 134400a b d e - 107520a b d e + 53760a b d e - 15360a b d e
--R +
--R 5 8 10
--R 1920a b d
--R *
--R +-----------+ +-------+
--R \|- a e + b d \|e x + d
--R Type: Expression(Integer)
--E 620
--S 621 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 621
)clear all
--S 622 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x
--R +
--R 3
--R A d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 622
--S 623 of 764
r0:=-2/143*(4*b*B*d-13*A*b*e+9*a*B*e)*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^2)+4/1287*(b*d-a*e)*(4*b*B*d-13*A*b*e+9*a*B*e)*_
(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3*(a+b*x))+_
2/13*B*(a+b*x)*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e)
--R
--R
--R (2)
--R 6 6 6 5 5
--R 198B b e x + ((234A b + 234B a)e + 720B b d e )x
--R +
--R 6 5 2 4 4
--R (286A a e + (884A b + 884B a)d e + 916B b d e )x
--R +
--R 5 2 4 3 3 3
--R (1144A a d e + (1196A b + 1196B a)d e + 424B b d e )x
--R +
--R 2 4 3 3 4 2 2
--R (1716A a d e + (624A b + 624B a)d e + 6B b d e )x
--R +
--R 3 3 4 2 5 4 2
--R (1144A a d e + (26A b + 26B a)d e - 8B b d e)x + 286A a d e
--R +
--R 5 6
--R (- 52A b - 52B a)d e + 16B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 3 3
--R 1287b e x + 1287a e
--R Type: Expression(Integer)
--E 623
--S 624 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 624
)clear all
--S 625 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R (1)
--R 2 3 2 2 2 2 +-------+
--R (B e x + (A e + 2B d e)x + (2A d e + B d )x + A d )\|e x + d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 625
--S 626 of 764
r0:=-2/99*(4*b*B*d-11*A*b*e+7*a*B*e)*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^2)+4/693*(b*d-a*e)*(4*b*B*d-11*A*b*e+7*a*B*e)*_
(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3*(a+b*x))+2/11*B*_
(a+b*x)*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e)
--R
--R
--R (2)
--R 5 5 5 4 4
--R 126B b e x + ((154A b + 154B a)e + 322B b d e )x
--R +
--R 5 4 2 3 3
--R (198A a e + (418A b + 418B a)d e + 226B b d e )x
--R +
--R 4 2 3 3 2 2
--R (594A a d e + (330A b + 330B a)d e + 6B b d e )x
--R +
--R 2 3 3 2 4 3 2
--R (594A a d e + (22A b + 22B a)d e - 8B b d e)x + 198A a d e
--R +
--R 4 5
--R (- 44A b - 44B a)d e + 16B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 3 3
--R 693b e x + 693a e
--R Type: Expression(Integer)
--E 626
--S 627 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 627
)clear all
--S 628 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R +------------------+
--R 2 +-------+ | 2 2 2
--R (1) (B e x + (A e + B d)x + A d)\|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 628
--S 629 of 764
r0:=-2/63*(4*b*B*d-9*A*b*e+5*a*B*e)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^2)+4/315*(b*d-a*e)*(4*b*B*d-9*A*b*e+5*a*B*e)*_
(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3*(a+b*x))+2/9*B*_
(a+b*x)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e)
--R
--R
--R (2)
--R 4 4 4 3 3
--R 70B b e x + ((90A b + 90B a)e + 100B b d e )x
--R +
--R 4 3 2 2 2
--R (126A a e + (144A b + 144B a)d e + 6B b d e )x
--R +
--R 3 2 2 3 2 2
--R (252A a d e + (18A b + 18B a)d e - 8B b d e)x + 126A a d e
--R +
--R 3 4
--R (- 36A b - 36B a)d e + 16B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 3 3
--R 315b e x + 315a e
--R Type: Expression(Integer)
--E 629
--S 630 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 630
)clear all
--S 631 of 764
t0:=(A+B*x)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R +------------------+
--R +-------+ | 2 2 2
--R (1) (B x + A)\|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 631
--S 632 of 764
r0:=-2/35*(4*b*B*d-7*A*b*e+3*a*B*e)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^2)+4/105*(b*d-a*e)*(4*b*B*d-7*A*b*e+3*a*B*e)*_
(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3*(a+b*x))+2/7*B*_
(a+b*x)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e)
--R
--R
--R (2)
--R 3 3 3 2 2
--R 30B b e x + ((42A b + 42B a)e + 6B b d e )x
--R +
--R 3 2 2 2
--R (70A a e + (14A b + 14B a)d e - 8B b d e)x + 70A a d e
--R +
--R 2 3
--R (- 28A b - 28B a)d e + 16B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 3 3
--R 105b e x + 105a e
--R Type: Expression(Integer)
--E 632
--S 633 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 633
)clear all
--S 634 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/sqrt(d+e*x)
--R
--R
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R (1) ------------------------------
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 634
--S 635 of 764
r0:=-2/15*(4*b*B*d-5*A*b*e+a*B*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^2)+4/15*(b*d-a*e)*(4*b*B*d-5*A*b*e+a*B*e)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3*(a+b*x))+2/5*B*(a+b*x)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e)
--R
--R
--R (2)
--R 2 2 2 2
--R 6B b e x + ((10A b + 10B a)e - 8B b d e)x + 30A a e
--R +
--R 2
--R (- 20A b - 20B a)d e + 16B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 3 3
--R 15b e x + 15a e
--R Type: Expression(Integer)
--E 635
--S 636 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 636
)clear all
--S 637 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(3/2)
--R
--R
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R (1) ------------------------------
--R +-------+
--R (e x + d)\|e x + d
--R Type: Expression(Integer)
--E 637
--S 638 of 764
r0:=-2*(B*d-A*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e*(b*d-a*e)*_
sqrt(d+e*x))+2/3*(4*b*B*d-3*A*b*e-a*B*e)*sqrt(d+e*x)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/(e^2*(b*d-a*e))-4/3*(4*b*B*d-3*A*b*e-a*B*e)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(a+b*x))
--R
--R
--R (2)
--R 2 2 2 2
--R 2B b e x + ((6A b + 6B a)e - 8B b d e)x - 6A a e
--R +
--R 2
--R (12A b + 12B a)d e - 16B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3 3 +-------+
--R (3b e x + 3a e )\|e x + d
--R Type: Expression(Integer)
--E 638
--S 639 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 639
)clear all
--S 640 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(5/2)
--R
--R
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R (1) ------------------------------
--R 2 2 2 +-------+
--R (e x + 2d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 640
--S 641 of 764
r0:=-2/3*(B*d-A*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e*(b*d-a*e)*_
(d+e*x)^(3/2))+2/3*(4*b*B*d-A*b*e-3*a*B*e)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(e^2*(b*d-a*e)*sqrt(d+e*x))+4/3*(4*b*B*d-A*b*e-3*a*B*e)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(a+b*x)*sqrt(d+e*x))
--R
--R
--R (2)
--R 2 2 2 2
--R 6B b e x + ((- 6A b - 6B a)e + 24B b d e)x - 2A a e
--R +
--R 2
--R (- 4A b - 4B a)d e + 16B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 4 2 4 3 3 +-------+
--R (3b e x + (3a e + 3b d e )x + 3a d e )\|e x + d
--R Type: Expression(Integer)
--E 641
--S 642 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 642
)clear all
--S 643 of 764
t0:=(A+B*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(d+e*x)^(7/2)
--R
--R
--R +------------------+
--R | 2 2 2
--R (B x + A)\|b x + 2a b x + a
--R (1) ----------------------------------------
--R 3 3 2 2 2 3 +-------+
--R (e x + 3d e x + 3d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 643
--S 644 of 764
r0:=-2/5*(B*d-A*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e*(b*d-a*e)*_
(d+e*x)^(5/2))-2/5*(4*b*B*d+A*b*e-5*a*B*e)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(e^2*(b*d-a*e)*(d+e*x)^(3/2))-4/15*(4*b*B*d+A*b*e-5*a*B*e)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(a+b*x)*(d+e*x)^(3/2))
--R
--R
--R (2)
--R 2 2 2 2
--R - 30B b e x + ((- 10A b - 10B a)e - 40B b d e)x - 6A a e
--R +
--R 2
--R (- 4A b - 4B a)d e - 16B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 5 3 5 4 2 4 2 3 2 3
--R (15b e x + (15a e + 30b d e )x + (30a d e + 15b d e )x + 15a d e )
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 644
--S 645 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 645
)clear all
--S 646 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 2 3 6 2 3 2 2 5
--R B b e x + ((A b + 2B a b)e + 3B b d e )x
--R +
--R 2 3 2 2 2 2 4
--R ((2A a b + B a )e + (3A b + 6B a b)d e + 3B b d e)x
--R +
--R 2 3 2 2 2 2 2 3 3
--R (A a e + (6A a b + 3B a )d e + (3A b + 6B a b)d e + B b d )x
--R +
--R 2 2 2 2 2 3 2
--R (3A a d e + (6A a b + 3B a )d e + (A b + 2B a b)d )x
--R +
--R 2 2 2 3 2 3
--R (3A a d e + (2A a b + B a )d )x + A a d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 646
--S 647 of 764
r0:=-2/255*(8*b*B*d-17*A*b*e+9*a*B*e)*(d+e*x)^(9/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e^2)+2/17*B*(a+b*x)*(d+e*x)^(9/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e)-16/12155*(b*d-a*e)^2*(8*b*B*d-17*A*b*e+_
9*a*B*e)*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^4)+32/109395*_
(b*d-a*e)^3*(8*b*B*d-17*A*b*e+9*a*B*e)*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^5*(a+b*x))+4/1105*(b*d-a*e)*(8*b*B*d-17*A*b*e+_
9*a*B*e)*(a+b*x)*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3)
--R
--R
--R (2)
--R 3 8 8 3 2 8 3 7 7
--R 12870B b e x + ((14586A b + 43758B a b )e + 44616B b d e )x
--R +
--R 2 2 8 3 2 7
--R (50490A a b + 50490B a b)e + (51612A b + 154836B a b )d e
--R +
--R 3 2 6
--R 52932B b d e
--R *
--R 6
--R x
--R +
--R 2 3 8 2 2 7
--R (59670A a b + 19890B a )e + (183600A a b + 183600B a b)d e
--R +
--R 3 2 2 6 3 3 5
--R (63036A b + 189108B a b )d e + 21816B b d e
--R *
--R 5
--R x
--R +
--R 3 8 2 3 7
--R 24310A a e + (225420A a b + 75140B a )d e
--R +
--R 2 2 2 6 3 2 3 5
--R (233580A a b + 233580B a b)d e + (27200A b + 81600B a b )d e
--R +
--R 3 4 4
--R 70B b d e
--R *
--R 4
--R x
--R +
--R 3 7 2 3 2 6
--R 97240A a d e + (304980A a b + 101660B a )d e
--R +
--R 2 2 3 5 3 2 4 4
--R (108120A a b + 108120B a b)d e + (170A b + 510B a b )d e
--R +
--R 3 5 3
--R - 80B b d e
--R *
--R 3
--R x
--R +
--R 3 2 6 2 3 3 5
--R 145860A a d e + (159120A a b + 53040B a )d e
--R +
--R 2 2 4 4 3 2 5 3
--R (1530A a b + 1530B a b)d e + (- 204A b - 612B a b )d e
--R +
--R 3 6 2
--R 96B b d e
--R *
--R 2
--R x
--R +
--R 3 3 5 2 3 4 4
--R 97240A a d e + (6630A a b + 2210B a )d e
--R +
--R 2 2 5 3 3 2 6 2
--R (- 2040A a b - 2040B a b)d e + (272A b + 816B a b )d e
--R +
--R 3 7
--R - 128B b d e
--R *
--R x
--R +
--R 3 4 4 2 3 5 3
--R 24310A a d e + (- 13260A a b - 4420B a )d e
--R +
--R 2 2 6 2 3 2 7 3 8
--R (4080A a b + 4080B a b)d e + (- 544A b - 1632B a b )d e + 256B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 5 5
--R 109395b e x + 109395a e
--R Type: Expression(Integer)
--E 647
--S 648 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 648
)clear all
--S 649 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 2 2 5 2 2 2 4
--R B b e x + ((A b + 2B a b)e + 2B b d e)x
--R +
--R 2 2 2 2 2 3
--R ((2A a b + B a )e + (2A b + 4B a b)d e + B b d )x
--R +
--R 2 2 2 2 2 2
--R (A a e + (4A a b + 2B a )d e + (A b + 2B a b)d )x
--R +
--R 2 2 2 2 2
--R (2A a d e + (2A a b + B a )d )x + A a d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 649
--S 650 of 764
r0:=-2/195*(8*b*B*d-15*A*b*e+7*a*B*e)*(d+e*x)^(7/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e^2)+2/15*B*(a+b*x)*(d+e*x)^(7/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e)-16/6435*(b*d-a*e)^2*(8*b*B*d-15*A*b*e+7*a*B*e)*_
(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^4)+32/45045*(b*d-a*e)^3*_
(8*b*B*d-15*A*b*e+7*a*B*e)*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(b*e^5*(a+b*x))+4/715*(b*d-a*e)*(8*b*B*d-15*A*b*e+7*a*B*e)*(a+b*x)*_
(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3)
--R
--R
--R (2)
--R 3 7 7 3 2 7 3 6 6
--R 6006B b e x + ((6930A b + 20790B a b )e + 14322B b d e )x
--R +
--R 2 2 7 3 2 6
--R (24570A a b + 24570B a b)e + (17010A b + 51030B a b )d e
--R +
--R 3 2 5
--R 8946B b d e
--R *
--R 5
--R x
--R +
--R 2 3 7 2 2 6
--R (30030A a b + 10010B a )e + (62790A a b + 62790B a b)d e
--R +
--R 3 2 2 5 3 3 4
--R (11130A b + 33390B a b )d e + 70B b d e
--R *
--R 4
--R x
--R +
--R 3 7 2 3 6
--R 12870A a e + (81510A a b + 27170B a )d e
--R +
--R 2 2 2 5 3 2 3 4
--R (44070A a b + 44070B a b)d e + (150A b + 450B a b )d e
--R +
--R 3 4 3
--R - 80B b d e
--R *
--R 3
--R x
--R +
--R 3 6 2 3 2 5
--R 38610A a d e + (64350A a b + 21450B a )d e
--R +
--R 2 2 3 4 3 2 4 3
--R (1170A a b + 1170B a b)d e + (- 180A b - 540B a b )d e
--R +
--R 3 5 2
--R 96B b d e
--R *
--R 2
--R x
--R +
--R 3 2 5 2 3 3 4
--R 38610A a d e + (4290A a b + 1430B a )d e
--R +
--R 2 2 4 3 3 2 5 2
--R (- 1560A a b - 1560B a b)d e + (240A b + 720B a b )d e
--R +
--R 3 6
--R - 128B b d e
--R *
--R x
--R +
--R 3 3 4 2 3 4 3
--R 12870A a d e + (- 8580A a b - 2860B a )d e
--R +
--R 2 2 5 2 3 2 6 3 7
--R (3120A a b + 3120B a b)d e + (- 480A b - 1440B a b )d e + 256B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 5 5
--R 45045b e x + 45045a e
--R Type: Expression(Integer)
--E 650
--S 651 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 651
)clear all
--S 652 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 2 4 2 2 3
--R B b e x + ((A b + 2B a b)e + B b d)x
--R +
--R 2 2 2 2 2
--R ((2A a b + B a )e + (A b + 2B a b)d)x + (A a e + (2A a b + B a )d)x
--R +
--R 2
--R A a d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 652
--S 653 of 764
r0:=-2/143*(8*b*B*d-13*A*b*e+5*a*B*e)*(d+e*x)^(5/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e^2)+2/13*B*(a+b*x)*(d+e*x)^(5/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e)-16/3003*(b*d-a*e)^2*(8*b*B*d-13*A*b*e+5*a*B*e)*_
(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^4)+32/15015*(b*d-a*e)^3*_
(8*b*B*d-13*A*b*e+5*a*B*e)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(b*e^5*(a+b*x))+4/429*(b*d-a*e)*(8*b*B*d-13*A*b*e+5*a*B*e)*(a+b*x)*_
(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3)
--R
--R
--R (2)
--R 3 6 6 3 2 6 3 5 5
--R 2310B b e x + ((2730A b + 8190B a b )e + 2940B b d e )x
--R +
--R 2 2 6 3 2 5
--R (10010A a b + 10010B a b)e + (3640A b + 10920B a b )d e
--R +
--R 3 2 4
--R 70B b d e
--R *
--R 4
--R x
--R +
--R 2 3 6 2 2 5
--R (12870A a b + 4290B a )e + (14300A a b + 14300B a b)d e
--R +
--R 3 2 2 4 3 3 3
--R (130A b + 390B a b )d e - 80B b d e
--R *
--R 3
--R x
--R +
--R 3 6 2 3 5
--R 6006A a e + (20592A a b + 6864B a )d e
--R +
--R 2 2 2 4 3 2 3 3 3 4 2
--R (858A a b + 858B a b)d e + (- 156A b - 468B a b )d e + 96B b d e
--R *
--R 2
--R x
--R +
--R 3 5 2 3 2 4
--R 12012A a d e + (2574A a b + 858B a )d e
--R +
--R 2 2 3 3 3 2 4 2
--R (- 1144A a b - 1144B a b)d e + (208A b + 624B a b )d e
--R +
--R 3 5
--R - 128B b d e
--R *
--R x
--R +
--R 3 2 4 2 3 3 3
--R 6006A a d e + (- 5148A a b - 1716B a )d e
--R +
--R 2 2 4 2 3 2 5 3 6
--R (2288A a b + 2288B a b)d e + (- 416A b - 1248B a b )d e + 256B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 5 5
--R 15015b e x + 15015a e
--R Type: Expression(Integer)
--E 653
--S 654 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 654
)clear all
--S 655 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x)
--R
--R
--R (1)
--R 2 3 2 2 2 2 +-------+
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|e x + d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 655
--S 656 of 764
r0:=-2/99*(8*b*B*d-11*A*b*e+3*a*B*e)*(d+e*x)^(3/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e^2)+2/11*B*(a+b*x)*(d+e*x)^(3/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e)-16/1155*(b*d-a*e)^2*(8*b*B*d-11*A*b*e+3*a*B*e)*_
(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^4)+32/3465*(b*d-a*e)^3*_
(8*b*B*d-11*A*b*e+3*a*B*e)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(b*e^5*(a+b*x))+4/231*(b*d-a*e)*(8*b*B*d-11*A*b*e+3*a*B*e)*(a+b*x)*_
(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3)
--R
--R
--R (2)
--R 3 5 5 3 2 5 3 4 4
--R 630B b e x + ((770A b + 2310B a b )e + 70B b d e )x
--R +
--R 2 2 5 3 2 4 3 2 3 3
--R ((2970A a b + 2970B a b)e + (110A b + 330B a b )d e - 80B b d e )x
--R +
--R 2 3 5 2 2 4
--R (4158A a b + 1386B a )e + (594A a b + 594B a b)d e
--R +
--R 3 2 2 3 3 3 2
--R (- 132A b - 396B a b )d e + 96B b d e
--R *
--R 2
--R x
--R +
--R 3 5 2 3 4
--R 2310A a e + (1386A a b + 462B a )d e
--R +
--R 2 2 2 3 3 2 3 2 3 4
--R (- 792A a b - 792B a b)d e + (176A b + 528B a b )d e - 128B b d e
--R *
--R x
--R +
--R 3 4 2 3 2 3
--R 2310A a d e + (- 2772A a b - 924B a )d e
--R +
--R 2 2 3 2 3 2 4 3 5
--R (1584A a b + 1584B a b)d e + (- 352A b - 1056B a b )d e + 256B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 5 5
--R 3465b e x + 3465a e
--R Type: Expression(Integer)
--E 656
--S 657 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 657
)clear all
--S 658 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/sqrt(d+e*x)
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 658
--S 659 of 764
r0:=-2/63*(8*b*B*d-9*A*b*e+a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)*_
sqrt(d+e*x)/(b*e^2)+2/9*B*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)*_
sqrt(d+e*x)/(b*e)-16/315*(b*d-a*e)^2*(8*b*B*d-9*A*b*e+a*B*e)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^4)+32/315*(b*d-a*e)^3*_
(8*b*B*d-9*A*b*e+a*B*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(b*e^5*(a+b*x))+4/105*(b*d-a*e)*(8*b*B*d-9*A*b*e+a*B*e)*(a+b*x)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^3)
--R
--R
--R (2)
--R 3 4 4 3 2 4 3 3 3
--R 70B b e x + ((90A b + 270B a b )e - 80B b d e )x
--R +
--R 2 2 4 3 2 3 3 2 2 2
--R ((378A a b + 378B a b)e + (- 108A b - 324B a b )d e + 96B b d e )x
--R +
--R 2 3 4 2 2 3
--R (630A a b + 210B a )e + (- 504A a b - 504B a b)d e
--R +
--R 3 2 2 2 3 3
--R (144A b + 432B a b )d e - 128B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2
--R 630A a e + (- 1260A a b - 420B a )d e + (1008A a b + 1008B a b)d e
--R +
--R 3 2 3 3 4
--R (- 288A b - 864B a b )d e + 256B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 5 5
--R 315b e x + 315a e
--R Type: Expression(Integer)
--E 659
--S 660 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 660
)clear all
--S 661 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^(3/2)
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R +-------+
--R (e x + d)\|e x + d
--R Type: Expression(Integer)
--E 661
--S 662 of 764
r0:=-2*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
sqrt(d+e*x))+2/7*(8*b*B*d-7*A*b*e-a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)*_
sqrt(d+e*x)/(e^2*(b*d-a*e))+16/35*(b*d-a*e)*(8*b*B*d-7*A*b*e-a*B*e)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^4-32/35*(b*d-a*e)^2*(8*b*B*d-_
7*A*b*e-a*B*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x))-_
12/35*(8*b*B*d-7*A*b*e-a*B*e)*(a+b*x)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/e^3
--R
--R
--R (2)
--R 3 4 4 3 2 4 3 3 3
--R 10B b e x + ((14A b + 42B a b )e - 16B b d e )x
--R +
--R 2 2 4 3 2 3 3 2 2 2
--R ((70A a b + 70B a b)e + (- 28A b - 84B a b )d e + 32B b d e )x
--R +
--R 2 3 4 2 2 3
--R (210A a b + 70B a )e + (- 280A a b - 280B a b)d e
--R +
--R 3 2 2 2 3 3
--R (112A b + 336B a b )d e - 128B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2
--R - 70A a e + (420A a b + 140B a )d e + (- 560A a b - 560B a b)d e
--R +
--R 3 2 3 3 4
--R (224A b + 672B a b )d e - 256B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 5 5 +-------+
--R (35b e x + 35a e )\|e x + d
--R Type: Expression(Integer)
--E 662
--S 663 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 663
)clear all
--S 664 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^(5/2)
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R 2 2 2 +-------+
--R (e x + 2d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 664
--S 665 of 764
r0:=-2/3*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^(3/2))-2/3*(8*b*B*d-5*A*b*e-3*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e^2*(b*d-a*e)*sqrt(d+e*x))-16/15*b*(8*b*B*d-5*A*b*e-_
3*a*B*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^4+32/15*b*(b*d-a*e)*_
(8*b*B*d-5*A*b*e-3*a*B*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(e^5*(a+b*x))+4/5*b*(8*b*B*d-5*A*b*e-3*a*B*e)*(a+b*x)*sqrt(d+e*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(b*d-a*e))
--R
--R
--R (2)
--R 3 4 4 3 2 4 3 3 3
--R 6B b e x + ((10A b + 30B a b )e - 16B b d e )x
--R +
--R 2 2 4 3 2 3 3 2 2 2
--R ((90A a b + 90B a b)e + (- 60A b - 180B a b )d e + 96B b d e )x
--R +
--R 2 3 4 2 2 3
--R (- 90A a b - 30B a )e + (360A a b + 360B a b)d e
--R +
--R 3 2 2 2 3 3
--R (- 240A b - 720B a b )d e + 384B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2
--R - 10A a e + (- 60A a b - 20B a )d e + (240A a b + 240B a b)d e
--R +
--R 3 2 3 3 4
--R (- 160A b - 480B a b )d e + 256B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 6 2 6 5 5 +-------+
--R (15b e x + (15a e + 15b d e )x + 15a d e )\|e x + d
--R Type: Expression(Integer)
--E 665
--S 666 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 666
)clear all
--S 667 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(d+e*x)^(7/2)
--R
--R
--R (1)
--R +------------------+
--R 2 3 2 2 2 2 | 2 2 2
--R (B b x + (A b + 2B a b)x + (2A a b + B a )x + A a )\|b x + 2a b x + a
--R ---------------------------------------------------------------------------
--R 3 3 2 2 2 3 +-------+
--R (e x + 3d e x + 3d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 667
--S 668 of 764
r0:=-2/5*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e*(b*d-a*e)*_
(d+e*x)^(5/2))-2/15*(8*b*B*d-3*A*b*e-5*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(e^2*(b*d-a*e)*(d+e*x)^(3/2))-4/5*b*(8*b*B*d-_
3*A*b*e-5*a*B*e)*(a+b*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^3*(b*d-a*e)*_
sqrt(d+e*x))+16/15*b^2*(8*b*B*d-3*A*b*e-5*a*B*e)*sqrt(d+e*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^4*(b*d-a*e))-32/15*b^2*(8*b*B*d-3*A*b*e-_
5*a*B*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^5*(a+b*x))
--R
--R
--R (2)
--R 3 4 4 3 2 4 3 3 3
--R 10B b e x + ((30A b + 90B a b )e - 80B b d e )x
--R +
--R 2 2 4 3 2 3 3 2 2 2
--R ((- 90A a b - 90B a b)e + (180A b + 540B a b )d e - 480B b d e )x
--R +
--R 2 3 4 2 2 3
--R (- 30A a b - 10B a )e + (- 120A a b - 120B a b)d e
--R +
--R 3 2 2 2 3 3
--R (240A b + 720B a b )d e - 640B b d e
--R *
--R x
--R +
--R 3 4 2 3 3 2 2 2 2
--R - 6A a e + (- 12A a b - 4B a )d e + (- 48A a b - 48B a b)d e
--R +
--R 3 2 3 3 4
--R (96A b + 288B a b )d e - 256B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 7 3 7 6 2 6 2 5 2 5
--R (15b e x + (15a e + 30b d e )x + (30a d e + 15b d e )x + 15a d e )
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 668
--S 669 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 669
)clear all
--S 670 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 3 8 4 3 3 4 2 7
--R B b e x + ((A b + 4B a b )e + 3B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 6
--R ((4A a b + 6B a b )e + (3A b + 12B a b )d e + 3B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (6A a b + 4B a b)e + (12A a b + 18B a b )d e
--R +
--R 4 3 2 4 3
--R (3A b + 12B a b )d e + B b d
--R *
--R 5
--R x
--R +
--R 3 4 3 2 2 3 2
--R (4A a b + B a )e + (18A a b + 12B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (12A a b + 18B a b )d e + (A b + 4B a b )d
--R *
--R 4
--R x
--R +
--R 4 3 3 4 2 2 2 3 2
--R A a e + (12A a b + 3B a )d e + (18A a b + 12B a b)d e
--R +
--R 3 2 2 3
--R (4A a b + 6B a b )d
--R *
--R 3
--R x
--R +
--R 4 2 3 4 2 2 2 3 3 2
--R (3A a d e + (12A a b + 3B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 2 3 4 3 4 3
--R (3A a d e + (4A a b + B a )d )x + A a d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 670
--S 671 of 764
r0:=-32/6783*(b*d-a*e)^2*(4*b*B*d-7*A*b*e+3*a*B*e)*(d+e*x)^(9/2)*_
(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e^4)+20/2261*(b*d-a*e)*(4*b*B*d-7*A*b*e+_
3*a*B*e)*(a+b*x)*(d+e*x)^(9/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e^3)-_
2/133*(4*b*B*d-7*A*b*e+3*a*B*e)*(d+e*x)^(9/2)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b*e^2)+2/21*B*(a+b*x)*(d+e*x)^(9/2)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b*e)-256/323323*(b*d-a*e)^4*(4*b*B*d-7*A*b*e+_
3*a*B*e)*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^6)+_
512/2909907*(b*d-a*e)^5*(4*b*B*d-7*A*b*e+3*a*B*e)*(d+e*x)^(9/2)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^7*(a+b*x))+64/29393*(b*d-a*e)^3*_
(4*b*B*d-7*A*b*e+3*a*B*e)*(a+b*x)*(d+e*x)^(9/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^5)
--R
--R
--R (2)
--R 5 10 10 5 4 10 5 9 9
--R 277134B b e x + ((306306A b + 1531530B a b )e + 933504B b d e )x
--R +
--R 4 2 3 10
--R (1711710A a b + 3423420B a b )e
--R +
--R 5 4 9 5 2 8
--R (1045044A b + 5225220B a b )d e + 1065636B b d e
--R *
--R 8
--R x
--R +
--R 2 3 3 2 10
--R (3879876A a b + 3879876B a b )e
--R +
--R 4 2 3 9
--R (5933928A a b + 11867856B a b )d e
--R +
--R 5 4 2 8 5 3 7
--R (1213212A b + 6066060B a b )d e + 415272B b d e
--R *
--R 7
--R x
--R +
--R 3 2 4 10
--R (4476780A a b + 2238390B a b)e
--R +
--R 2 3 3 2 9
--R (13728792A a b + 13728792B a b )d e
--R +
--R 4 2 3 2 8
--R (7039956A a b + 14079912B a b )d e
--R +
--R 5 4 3 7 5 4 6
--R (484176A b + 2420880B a b )d e + 462B b d e
--R *
--R 6
--R x
--R +
--R 4 5 10
--R (2645370A a b + 529074B a )e
--R +
--R 3 2 4 9
--R (16279200A a b + 8139600B a b)d e
--R +
--R 2 3 3 2 2 8
--R (16767576A a b + 16767576B a b )d e
--R +
--R 4 2 3 3 7 5 4 4 6
--R (2901528A a b + 5803056B a b )d e + (882A b + 4410B a b )d e
--R +
--R 5 5 5
--R - 504B b d e
--R *
--R 5
--R x
--R +
--R 5 10 4 5 9
--R 646646A a e + (9993620A a b + 1998724B a )d e
--R +
--R 3 2 4 2 8
--R (20710760A a b + 10355380B a b)d e
--R +
--R 2 3 3 2 3 7
--R (7235200A a b + 7235200B a b )d e
--R +
--R 4 2 3 4 6 5 4 5 5
--R (9310A a b + 18620B a b )d e + (- 980A b - 4900B a b )d e
--R +
--R 5 6 4
--R 560B b d e
--R *
--R 4
--R x
--R +
--R 5 9 4 5 2 8
--R 2586584A a d e + (13520780A a b + 2704156B a )d e
--R +
--R 3 2 4 3 7
--R (9586640A a b + 4793320B a b)d e
--R +
--R 2 3 3 2 4 6 4 2 3 5 5
--R (45220A a b + 45220B a b )d e + (- 10640A a b - 21280B a b )d e
--R +
--R 5 4 6 4 5 7 3
--R (1120A b + 5600B a b )d e - 640B b d e
--R *
--R 3
--R x
--R +
--R 5 2 8 4 5 3 7
--R 3879876A a d e + (7054320A a b + 1410864B a )d e
--R +
--R 3 2 4 4 6 2 3 3 2 5 5
--R (135660A a b + 67830B a b)d e + (- 54264A a b - 54264B a b )d e
--R +
--R 4 2 3 6 4 5 4 7 3
--R (12768A a b + 25536B a b )d e + (- 1344A b - 6720B a b )d e
--R +
--R 5 8 2
--R 768B b d e
--R *
--R 2
--R x
--R +
--R 5 3 7 4 5 4 6
--R 2586584A a d e + (293930A a b + 58786B a )d e
--R +
--R 3 2 4 5 5 2 3 3 2 6 4
--R (- 180880A a b - 90440B a b)d e + (72352A a b + 72352B a b )d e
--R +
--R 4 2 3 7 3 5 4 8 2
--R (- 17024A a b - 34048B a b )d e + (1792A b + 8960B a b )d e
--R +
--R 5 9
--R - 1024B b d e
--R *
--R x
--R +
--R 5 4 6 4 5 5 5
--R 646646A a d e + (- 587860A a b - 117572B a )d e
--R +
--R 3 2 4 6 4 2 3 3 2 7 3
--R (361760A a b + 180880B a b)d e + (- 144704A a b - 144704B a b )d e
--R +
--R 4 2 3 8 2 5 4 9
--R (34048A a b + 68096B a b )d e + (- 3584A b - 17920B a b )d e
--R +
--R 5 10
--R 2048B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 7 7
--R 2909907b e x + 2909907a e
--R Type: Expression(Integer)
--E 671
--S 672 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 672
)clear all
--S 673 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 2 7 4 3 2 4 6
--R B b e x + ((A b + 4B a b )e + 2B b d e)x
--R +
--R 3 2 2 2 4 3 4 2 5
--R ((4A a b + 6B a b )e + (2A b + 8B a b )d e + B b d )x
--R +
--R 2 2 3 2 3 2 2 4 3 2 4
--R ((6A a b + 4B a b)e + (8A a b + 12B a b )d e + (A b + 4B a b )d )x
--R +
--R 3 4 2 2 2 3 3 2 2 2 3
--R ((4A a b + B a )e + (12A a b + 8B a b)d e + (4A a b + 6B a b )d )x
--R +
--R 4 2 3 4 2 2 3 2 2
--R (A a e + (8A a b + 2B a )d e + (6A a b + 4B a b)d )x
--R +
--R 4 3 4 2 4 2
--R (2A a d e + (4A a b + B a )d )x + A a d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 673
--S 674 of 764
r0:=-32/12597*(b*d-a*e)^2*(12*b*B*d-19*A*b*e+7*a*B*e)*(d+e*x)^(7/2)*_
(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e^4)+4/969*(b*d-a*e)*(12*b*B*d-19*A*b*e+_
7*a*B*e)*(a+b*x)*(d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e^3)-_
2/323*(12*b*B*d-19*A*b*e+7*a*B*e)*(d+e*x)^(7/2)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b*e^2)+2/19*B*(a+b*x)*(d+e*x)^(7/2)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b*e)-256/415701*(b*d-a*e)^4*(12*b*B*d-19*A*b*e+_
7*a*B*e)*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^6)+_
512/2909907*(b*d-a*e)^5*(12*b*B*d-19*A*b*e+7*a*B*e)*(d+e*x)^(7/2)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^7*(a+b*x))+64/46189*(b*d-a*e)^3*_
(12*b*B*d-19*A*b*e+7*a*B*e)*(a+b*x)*(d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^5)
--R
--R
--R (2)
--R 5 9 9 5 4 9 5 8 8
--R 306306B b e x + ((342342A b + 1711710B a b )e + 702702B b d e )x
--R +
--R 4 2 3 9
--R (1939938A a b + 3879876B a b )e
--R +
--R 5 4 8 5 2 7
--R (798798A b + 3993990B a b )d e + 414414B b d e
--R *
--R 7
--R x
--R +
--R 2 3 3 2 9
--R (4476780A a b + 4476780B a b )e
--R +
--R 4 2 3 8
--R (4626006A a b + 9252012B a b )d e
--R +
--R 5 4 2 7 5 3 6
--R (482790A b + 2413950B a b )d e + 1386B b d e
--R *
--R 6
--R x
--R +
--R 3 2 4 9
--R (5290740A a b + 2645370B a b)e
--R +
--R 2 3 3 2 8
--R (10988460A a b + 10988460B a b )d e
--R +
--R 4 2 3 2 7 5 4 3 6
--R (2889558A a b + 5779116B a b )d e + (2394A b + 11970B a b )d e
--R +
--R 5 4 5
--R - 1512B b d e
--R *
--R 5
--R x
--R +
--R 4 5 9 3 2 4 8
--R (3233230A a b + 646646B a )e + (13520780A a b + 6760390B a b)d e
--R +
--R 2 3 3 2 2 7
--R (7189980A a b + 7189980B a b )d e
--R +
--R 4 2 3 3 6 5 4 4 5
--R (22610A a b + 45220B a b )d e + (- 2660A b - 13300B a b )d e
--R +
--R 5 5 4
--R 1680B b d e
--R *
--R 4
--R x
--R +
--R 5 9 4 5 8
--R 831402A a e + (8775910A a b + 1755182B a )d e
--R +
--R 3 2 4 2 7
--R (9489740A a b + 4744870B a b)d e
--R +
--R 2 3 3 2 3 6 4 2 3 4 5
--R (96900A a b + 96900B a b )d e + (- 25840A a b - 51680B a b )d e
--R +
--R 5 4 5 4 5 6 3
--R (3040A b + 15200B a b )d e - 1920B b d e
--R *
--R 3
--R x
--R +
--R 5 8 4 5 2 7
--R 2494206A a d e + (6928350A a b + 1385670B a )d e
--R +
--R 3 2 4 3 6
--R (251940A a b + 125970B a b)d e
--R +
--R 2 3 3 2 4 5
--R (- 116280A a b - 116280B a b )d e
--R +
--R 4 2 3 5 4 5 4 6 3
--R (31008A a b + 62016B a b )d e + (- 3648A b - 18240B a b )d e
--R +
--R 5 7 2
--R 2304B b d e
--R *
--R 2
--R x
--R +
--R 5 2 7 4 5 3 6
--R 2494206A a d e + (461890A a b + 92378B a )d e
--R +
--R 3 2 4 4 5
--R (- 335920A a b - 167960B a b)d e
--R +
--R 2 3 3 2 5 4
--R (155040A a b + 155040B a b )d e
--R +
--R 4 2 3 6 3 5 4 7 2
--R (- 41344A a b - 82688B a b )d e + (4864A b + 24320B a b )d e
--R +
--R 5 8
--R - 3072B b d e
--R *
--R x
--R +
--R 5 3 6 4 5 4 5
--R 831402A a d e + (- 923780A a b - 184756B a )d e
--R +
--R 3 2 4 5 4 2 3 3 2 6 3
--R (671840A a b + 335920B a b)d e + (- 310080A a b - 310080B a b )d e
--R +
--R 4 2 3 7 2 5 4 8
--R (82688A a b + 165376B a b )d e + (- 9728A b - 48640B a b )d e
--R +
--R 5 9
--R 6144B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 7 7
--R 2909907b e x + 2909907a e
--R Type: Expression(Integer)
--E 674
--S 675 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 675
)clear all
--S 676 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 6 4 3 4 5
--R B b e x + ((A b + 4B a b )e + B b d)x
--R +
--R 3 2 2 4 3 4
--R ((4A a b + 6B a b )e + (A b + 4B a b )d)x
--R +
--R 2 2 3 3 2 2 3
--R ((6A a b + 4B a b)e + (4A a b + 6B a b )d)x
--R +
--R 3 4 2 2 3 2 4 3 4
--R ((4A a b + B a )e + (6A a b + 4B a b)d)x + (A a e + (4A a b + B a )d)x
--R +
--R 4
--R A a d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 676
--S 677 of 764
r0:=-32/7293*(b*d-a*e)^2*(12*b*B*d-17*A*b*e+5*a*B*e)*(d+e*x)^(5/2)*_
(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e^4)+4/663*(b*d-a*e)*(12*b*B*d-_
17*A*b*e+5*a*B*e)*(a+b*x)*(d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/_
(b*e^3)-2/255*(12*b*B*d-17*A*b*e+5*a*B*e)*(d+e*x)^(5/2)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b*e^2)+2/17*B*(a+b*x)*(d+e*x)^(5/2)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(b*e)-256/153153*(b*d-a*e)^4*(12*b*B*d-17*A*b*e+_
5*a*B*e)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^6)+_
512/765765*(b*d-a*e)^5*(12*b*B*d-17*A*b*e+5*a*B*e)*(d+e*x)^(5/2)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^7*(a+b*x))+64/21879*(b*d-a*e)^3*_
(12*b*B*d-17*A*b*e+5*a*B*e)*(a+b*x)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/(b*e^5)
--R
--R
--R (2)
--R 5 8 8 5 4 8 5 7 7
--R 90090B b e x + ((102102A b + 510510B a b )e + 108108B b d e )x
--R +
--R 4 2 3 8 5 4 7
--R (589050A a b + 1178100B a b )e + (125664A b + 628320B a b )d e
--R +
--R 5 2 6
--R 1386B b d e
--R *
--R 6
--R x
--R +
--R 2 3 3 2 8
--R (1392300A a b + 1392300B a b )e
--R +
--R 4 2 3 7 5 4 2 6
--R (749700A a b + 1499400B a b )d e + (2142A b + 10710B a b )d e
--R +
--R 5 3 5
--R - 1512B b d e
--R *
--R 5
--R x
--R +
--R 3 2 4 8
--R (1701700A a b + 850850B a b)e
--R +
--R 2 3 3 2 7
--R (1856400A a b + 1856400B a b )d e
--R +
--R 4 2 3 2 6 5 4 3 5
--R (17850A a b + 35700B a b )d e + (- 2380A b - 11900B a b )d e
--R +
--R 5 4 4
--R 1680B b d e
--R *
--R 4
--R x
--R +
--R 4 5 8 3 2 4 7
--R (1093950A a b + 218790B a )e + (2431000A a b + 1215500B a b)d e
--R +
--R 2 3 3 2 2 6 4 2 3 3 5
--R (66300A a b + 66300B a b )d e + (- 20400A a b - 40800B a b )d e
--R +
--R 5 4 4 4 5 5 3
--R (2720A b + 13600B a b )d e - 1920B b d e
--R *
--R 3
--R x
--R +
--R 5 8 4 5 7
--R 306306A a e + (1750320A a b + 350064B a )d e
--R +
--R 3 2 4 2 6 2 3 3 2 3 5
--R (145860A a b + 72930B a b)d e + (- 79560A a b - 79560B a b )d e
--R +
--R 4 2 3 4 4 5 4 5 3
--R (24480A a b + 48960B a b )d e + (- 3264A b - 16320B a b )d e
--R +
--R 5 6 2
--R 2304B b d e
--R *
--R 2
--R x
--R +
--R 5 7 4 5 2 6
--R 612612A a d e + (218790A a b + 43758B a )d e
--R +
--R 3 2 4 3 5
--R (- 194480A a b - 97240B a b)d e
--R +
--R 2 3 3 2 4 4
--R (106080A a b + 106080B a b )d e
--R +
--R 4 2 3 5 3 5 4 6 2
--R (- 32640A a b - 65280B a b )d e + (4352A b + 21760B a b )d e
--R +
--R 5 7
--R - 3072B b d e
--R *
--R x
--R +
--R 5 2 6 4 5 3 5
--R 306306A a d e + (- 437580A a b - 87516B a )d e
--R +
--R 3 2 4 4 4 2 3 3 2 5 3
--R (388960A a b + 194480B a b)d e + (- 212160A a b - 212160B a b )d e
--R +
--R 4 2 3 6 2 5 4 7
--R (65280A a b + 130560B a b )d e + (- 8704A b - 43520B a b )d e
--R +
--R 5 8
--R 6144B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 7 7
--R 765765b e x + 765765a e
--R Type: Expression(Integer)
--E 677
--S 678 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 678
)clear all
--S 679 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)*sqrt(d+e*x)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 679
--S 680 of 764
r0:=-32/1287*(b*d-a*e)^2*(4*b*B*d-5*A*b*e+a*B*e)*(d+e*x)^(3/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)/(b*e^4)+4/143*(b*d-a*e)*(4*b*B*d-5*A*b*e+a*B*e)*(a+b*x)*_
(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(b*e^3)-2/65*(4*b*B*d-_
5*A*b*e+a*B*e)*(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b*e^2)+_
2/15*B*(a+b*x)*(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(b*e)-_
256/15015*(b*d-a*e)^4*(4*b*B*d-5*A*b*e+a*B*e)*(d+e*x)^(3/2)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^6)+512/45045*(b*d-a*e)^5*(4*b*B*d-_
5*A*b*e+a*B*e)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^7*_
(a+b*x))+64/3003*(b*d-a*e)^3*(4*b*B*d-5*A*b*e+a*B*e)*(a+b*x)*_
(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^5)
--R
--R
--R (2)
--R 5 7 7 5 4 7 5 6 6
--R 6006B b e x + ((6930A b + 34650B a b )e + 462B b d e )x
--R +
--R 4 2 3 7 5 4 6
--R (40950A a b + 81900B a b )e + (630A b + 3150B a b )d e
--R +
--R 5 2 5
--R - 504B b d e
--R *
--R 5
--R x
--R +
--R 2 3 3 2 7 4 2 3 6
--R (100100A a b + 100100B a b )e + (4550A a b + 9100B a b )d e
--R +
--R 5 4 2 5 5 3 4
--R (- 700A b - 3500B a b )d e + 560B b d e
--R *
--R 4
--R x
--R +
--R 3 2 4 7 2 3 3 2 6
--R (128700A a b + 64350B a b)e + (14300A a b + 14300B a b )d e
--R +
--R 4 2 3 2 5 5 4 3 4
--R (- 5200A a b - 10400B a b )d e + (800A b + 4000B a b )d e
--R +
--R 5 4 3
--R - 640B b d e
--R *
--R 3
--R x
--R +
--R 4 5 7 3 2 4 6
--R (90090A a b + 18018B a )e + (25740A a b + 12870B a b)d e
--R +
--R 2 3 3 2 2 5 4 2 3 3 4
--R (- 17160A a b - 17160B a b )d e + (6240A a b + 12480B a b )d e
--R +
--R 5 4 4 3 5 5 2
--R (- 960A b - 4800B a b )d e + 768B b d e
--R *
--R 2
--R x
--R +
--R 5 7 4 5 6
--R 30030A a e + (30030A a b + 6006B a )d e
--R +
--R 3 2 4 2 5 2 3 3 2 3 4
--R (- 34320A a b - 17160B a b)d e + (22880A a b + 22880B a b )d e
--R +
--R 4 2 3 4 3 5 4 5 2
--R (- 8320A a b - 16640B a b )d e + (1280A b + 6400B a b )d e
--R +
--R 5 6
--R - 1024B b d e
--R *
--R x
--R +
--R 5 6 4 5 2 5
--R 30030A a d e + (- 60060A a b - 12012B a )d e
--R +
--R 3 2 4 3 4 2 3 3 2 4 3
--R (68640A a b + 34320B a b)d e + (- 45760A a b - 45760B a b )d e
--R +
--R 4 2 3 5 2 5 4 6
--R (16640A a b + 33280B a b )d e + (- 2560A b - 12800B a b )d e
--R +
--R 5 7
--R 2048B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 7 7
--R 45045b e x + 45045a e
--R Type: Expression(Integer)
--E 680
--S 681 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 681
)clear all
--S 682 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/sqrt(d+e*x)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 682
--S 683 of 764
r0:=-160/9009*(b*d-a*e)^2*(12*b*B*d-13*A*b*e+a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)*sqrt(d+e*x)/(b*e^4)+20/1287*(b*d-a*e)*(12*b*B*d-_
13*A*b*e+a*B*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x)/_
(b*e^3)-2/143*(12*b*B*d-13*A*b*e+a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)*_
sqrt(d+e*x)/(b*e^2)+2/13*B*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)*_
sqrt(d+e*x)/(b*e)-256/9009*(b*d-a*e)^4*(12*b*B*d-13*A*b*e+a*B*e)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^6)+512/9009*(b*d-a*e)^5*_
(12*b*B*d-13*A*b*e+a*B*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(b*e^7*(a+b*x))+64/3003*(b*d-a*e)^3*(12*b*B*d-13*A*b*e+a*B*e)*(a+b*x)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(b*e^5)
--R
--R
--R (2)
--R 5 6 6 5 4 6 5 5 5
--R 1386B b e x + ((1638A b + 8190B a b )e - 1512B b d e )x
--R +
--R 4 2 3 6 5 4 5
--R (10010A a b + 20020B a b )e + (- 1820A b - 9100B a b )d e
--R +
--R 5 2 4
--R 1680B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (25740A a b + 25740B a b )e + (- 11440A a b - 22880B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (2080A b + 10400B a b )d e - 1920B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (36036A a b + 18018B a b)e + (- 30888A a b - 30888B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (13728A a b + 27456B a b )d e + (- 2496A b - 12480B a b )d e
--R +
--R 5 4 2
--R 2304B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (30030A a b + 6006B a )e + (- 48048A a b - 24024B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (41184A a b + 41184B a b )d e + (- 18304A a b - 36608B a b )d e
--R +
--R 5 4 4 2 5 5
--R (3328A b + 16640B a b )d e - 3072B b d e
--R *
--R x
--R +
--R 5 6 4 5 5
--R 18018A a e + (- 60060A a b - 12012B a )d e
--R +
--R 3 2 4 2 4 2 3 3 2 3 3
--R (96096A a b + 48048B a b)d e + (- 82368A a b - 82368B a b )d e
--R +
--R 4 2 3 4 2 5 4 5
--R (36608A a b + 73216B a b )d e + (- 6656A b - 33280B a b )d e
--R +
--R 5 6
--R 6144B b d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R /
--R 7 7
--R 9009b e x + 9009a e
--R Type: Expression(Integer)
--E 683
--S 684 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 684
)clear all
--S 685 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^(3/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R +-------+
--R (e x + d)\|e x + d
--R Type: Expression(Integer)
--E 685
--S 686 of 764
r0:=-2*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*_
sqrt(d+e*x))+160/693*(b*d-a*e)*(12*b*B*d-11*A*b*e-a*B*e)*(a^2+_
2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x)/e^4-20/99*(12*b*B*d-11*A*b*e-_
a*B*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x)/e^3+2/11*_
(12*b*B*d-11*A*b*e-a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)*sqrt(d+e*x)/_
(e^2*(b*d-a*e))+256/693*(b*d-a*e)^3*(12*b*B*d-11*A*b*e-a*B*e)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^6-512/693*(b*d-a*e)^4*_
(12*b*B*d-11*A*b*e-a*B*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/_
(e^7*(a+b*x))-64/231*(b*d-a*e)^2*(12*b*B*d-11*A*b*e-a*B*e)*(a+b*x)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^5
--R
--R
--R (2)
--R 5 6 6 5 4 6 5 5 5
--R 126B b e x + ((154A b + 770B a b )e - 168B b d e )x
--R +
--R 4 2 3 6 5 4 5
--R (990A a b + 1980B a b )e + (- 220A b - 1100B a b )d e
--R +
--R 5 2 4
--R 240B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (2772A a b + 2772B a b )e + (- 1584A a b - 3168B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (352A b + 1760B a b )d e - 384B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (4620A a b + 2310B a b)e + (- 5544A a b - 5544B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (3168A a b + 6336B a b )d e + (- 704A b - 3520B a b )d e
--R +
--R 5 4 2
--R 768B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (6930A a b + 1386B a )e + (- 18480A a b - 9240B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (22176A a b + 22176B a b )d e + (- 12672A a b - 25344B a b )d e
--R +
--R 5 4 4 2 5 5
--R (2816A b + 14080B a b )d e - 3072B b d e
--R *
--R x
--R +
--R 5 6 4 5 5
--R - 1386A a e + (13860A a b + 2772B a )d e
--R +
--R 3 2 4 2 4 2 3 3 2 3 3
--R (- 36960A a b - 18480B a b)d e + (44352A a b + 44352B a b )d e
--R +
--R 4 2 3 4 2 5 4 5
--R (- 25344A a b - 50688B a b )d e + (5632A b + 28160B a b )d e
--R +
--R 5 6
--R - 6144B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 7 7 +-------+
--R (693b e x + 693a e )\|e x + d
--R Type: Expression(Integer)
--E 686
--S 687 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 687
)clear all
--S 688 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^(5/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 2 2 2 +-------+
--R (e x + 2d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 688
--S 689 of 764
r0:=-2/3*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*_
(d+e*x)^(3/2))-2*(4*b*B*d-3*A*b*e-a*B*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/_
(e^2*(b*d-a*e)*sqrt(d+e*x))-160/63*b*(4*b*B*d-3*A*b*e-a*B*e)*(a^2+_
2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x)/e^4+20/9*b*(4*b*B*d-3*A*b*e-a*B*e)*_
(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x)/(e^3*(b*d-a*e))-_
256/63*b*(b*d-a*e)^2*(4*b*B*d-3*A*b*e-a*B*e)*sqrt(d+e*x)*sqrt(a^2+_
2*a*b*x+b^2*x^2)/e^6+512/63*b*(b*d-a*e)^3*(4*b*B*d-3*A*b*e-a*B*e)*_
sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/(e^7*(a+b*x))+64/21*b*(b*d-_
a*e)*(4*b*B*d-3*A*b*e-a*B*e)*(a+b*x)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+_
b^2*x^2)/e^5
--R
--R
--R (2)
--R 5 6 6 5 4 6 5 5 5
--R 14B b e x + ((18A b + 90B a b )e - 24B b d e )x
--R +
--R 4 2 3 6 5 4 5 5 2 4 4
--R ((126A a b + 252B a b )e + (- 36A b - 180B a b )d e + 48B b d e )x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (420A a b + 420B a b )e + (- 336A a b - 672B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (96A b + 480B a b )d e - 128B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (1260A a b + 630B a b)e + (- 2520A a b - 2520B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (2016A a b + 4032B a b )d e + (- 576A b - 2880B a b )d e
--R +
--R 5 4 2
--R 768B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (- 630A a b - 126B a )e + (5040A a b + 2520B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (- 10080A a b - 10080B a b )d e + (8064A a b + 16128B a b )d e
--R +
--R 5 4 4 2 5 5
--R (- 2304A b - 11520B a b )d e + 3072B b d e
--R *
--R x
--R +
--R 5 6 4 5 5 3 2 4 2 4
--R - 42A a e + (- 420A a b - 84B a )d e + (3360A a b + 1680B a b)d e
--R +
--R 2 3 3 2 3 3 4 2 3 4 2
--R (- 6720A a b - 6720B a b )d e + (5376A a b + 10752B a b )d e
--R +
--R 5 4 5 5 6
--R (- 1536A b - 7680B a b )d e + 2048B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 8 2 8 7 7 +-------+
--R (63b e x + (63a e + 63b d e )x + 63a d e )\|e x + d
--R Type: Expression(Integer)
--E 689
--S 690 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 690
)clear all
--S 691 of 764
t0:=(A+B*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(d+e*x)^(7/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B b x + (A b + 4B a b )x + (4A a b + 6B a b )x
--R +
--R 2 2 3 2 3 4 4
--R (6A a b + 4B a b)x + (4A a b + B a )x + A a
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 3 3 2 2 2 3 +-------+
--R (e x + 3d e x + 3d e x + d )\|e x + d
--R Type: Expression(Integer)
--E 691
--S 692 of 764
r0:=-2/5*(B*d-A*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(5/2)/(e*(b*d-a*e)*_
(d+e*x)^(5/2))-2/15*(12*b*B*d-7*A*b*e-5*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2)/(e^2*(b*d-a*e)*(d+e*x)^(3/2))-4/3*b*(12*b*B*d-_
7*A*b*e-5*a*B*e)*(a+b*x)*(a^2+2*a*b*x+b^2*x^2)^(3/2)/(e^3*(b*d-a*e)*_
sqrt(d+e*x))+32/21*b^2*(12*b*B*d-7*A*b*e-5*a*B*e)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)*sqrt(d+e*x)/(e^4*(b*d-a*e))+256/105*b^2*(b*d-a*e)*_
(12*b*B*d-7*A*b*e-5*a*B*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^6-_
512/105*b^2*(b*d-a*e)^2*(12*b*B*d-7*A*b*e-5*a*B*e)*sqrt(d+e*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2)/(e^7*(a+b*x))-64/35*b^2*(12*b*B*d-7*A*b*_
e-5*a*B*e)*(a+b*x)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2)/e^5
--R
--R
--R (2)
--R 5 6 6 5 4 6 5 5 5
--R 30B b e x + ((42A b + 210B a b )e - 72B b d e )x
--R +
--R 4 2 3 6 5 4 5
--R (350A a b + 700B a b )e + (- 140A b - 700B a b )d e
--R +
--R 5 2 4
--R 240B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 6 4 2 3 5
--R (2100A a b + 2100B a b )e + (- 2800A a b - 5600B a b )d e
--R +
--R 5 4 2 4 5 3 3
--R (1120A b + 5600B a b )d e - 1920B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 6 2 3 3 2 5
--R (- 2100A a b - 1050B a b)e + (12600A a b + 12600B a b )d e
--R +
--R 4 2 3 2 4 5 4 3 3
--R (- 16800A a b - 33600B a b )d e + (6720A b + 33600B a b )d e
--R +
--R 5 4 2
--R - 11520B b d e
--R *
--R 2
--R x
--R +
--R 4 5 6 3 2 4 5
--R (- 350A a b - 70B a )e + (- 2800A a b - 1400B a b)d e
--R +
--R 2 3 3 2 2 4 4 2 3 3 3
--R (16800A a b + 16800B a b )d e + (- 22400A a b - 44800B a b )d e
--R +
--R 5 4 4 2 5 5
--R (8960A b + 44800B a b )d e - 15360B b d e
--R *
--R x
--R +
--R 5 6 4 5 5 3 2 4 2 4
--R - 42A a e + (- 140A a b - 28B a )d e + (- 1120A a b - 560B a b)d e
--R +
--R 2 3 3 2 3 3 4 2 3 4 2
--R (6720A a b + 6720B a b )d e + (- 8960A a b - 17920B a b )d e
--R +
--R 5 4 5 5 6
--R (3584A b + 17920B a b )d e - 6144B b d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R /
--R 9 3 9 8 2 8 2 7
--R 105b e x + (105a e + 210b d e )x + (210a d e + 105b d e )x
--R +
--R 2 7
--R 105a d e
--R *
--R +-------+
--R \|e x + d
--R Type: Expression(Integer)
--E 692
--S 693 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 693
)clear all
--S 694 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)/sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R 2 3 2 2 2 2 +-------+
--R (B e x + (A e + 2B d e)x + (2A d e + B d )x + A d )\|e x + d
--R (1) ----------------------------------------------------------------
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 694
--S 695 of 764
r0:=2/3*(A*b-a*B)*(b*d-a*e)*(a+b*x)*(d+e*x)^(3/2)/(b^3*sqrt(a^2+2*a*b*x+_
b^2*x^2))+2/5*(A*b-a*B)*(a+b*x)*(d+e*x)^(5/2)/(b^2*sqrt(a^2+2*a*b*x+_
b^2*x^2))+2/7*B*(a+b*x)*(d+e*x)^(7/2)/(b*e*sqrt(a^2+2*a*b*x+b^2*x^2))-_
2*(A*b-a*B)*(b*d-a*e)^(5/2)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/_
sqrt(b*d-a*e))/(b^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*(A*b-a*B)*_
(b*d-a*e)^2*(a+b*x)*sqrt(d+e*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 2 2 3 3 3 2 2 2
--R (- 210A a b + 210B a b)e + (420A a b - 420B a b )d e
--R +
--R 4 3 2
--R (- 210A b + 210B a b )d e
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (- 210A a b + 210B a )e + (420A a b - 420B a b)d e
--R +
--R 3 2 2 2
--R (- 210A a b + 210B a b )d e
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 4 4 3 3 4 2 3
--R 30B b e x + ((42A b - 12B a b )e + 90B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 2
--R ((- 28A a b + 28B a b )e + (154A b - 64B a b )d e + 90B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (140A a b - 140B a b)e + (- 336A a b + 336B a b )d e
--R +
--R 4 3 2 4 3
--R (322A b - 232B a b )d e + 30B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (210A a b - 210B a )e + (- 490A a b + 490B a b)d e
--R +
--R 3 2 2 2 3 3
--R (322A a b - 322B a b )d e + 30B a b d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R +------------------+
--R 4 +-+ | 2 2 2
--R 105b e\|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 695
--S 696 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 696
)clear all
--S 697 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)/sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R 2 +-------+
--R (B e x + (A e + B d)x + A d)\|e x + d
--R (1) ---------------------------------------
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 697
--S 698 of 764
r0:=2/3*(A*b-a*B)*(a+b*x)*(d+e*x)^(3/2)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))+_
2/5*B*(a+b*x)*(d+e*x)^(5/2)/(b*e*sqrt(a^2+2*a*b*x+b^2*x^2))-_
2*(A*b-a*B)*(b*d-a*e)^(3/2)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/_
sqrt(b*d-a*e))/(b^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*(A*b-a*B)*_
(b*d-a*e)*(a+b*x)*sqrt(d+e*x)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 2 2 2 3 2
--R ((30A a b - 30B a b)e + (- 30A b + 30B a b )d e)x
--R +
--R 2 3 2 2 2
--R (30A a b - 30B a )e + (- 30A a b + 30B a b)d e
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 3 3 2 2 3 2
--R 6B b e x + ((10A b - 4B a b )e + 12B b d e)x
--R +
--R 2 2 2 3 2 3 2
--R ((- 20A a b + 20B a b)e + (40A b - 28B a b )d e + 6B b d )x
--R +
--R 2 3 2 2 2 2 2
--R (- 30A a b + 30B a )e + (40A a b - 40B a b)d e + 6B a b d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R +------------------+
--R 3 +-+ | 2 2 2
--R 15b e\|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 698
--S 699 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 699
)clear all
--S 700 of 764
t0:=(A+B*x)*sqrt(d+e*x)/sqrt(a^2+2*a*b*x+b^2*x^2)
--R
--R
--R +-------+
--R (B x + A)\|e x + d
--R (1) ---------------------
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 700
--S 701 of 764
r0:=2/3*B*(a+b*x)*(d+e*x)^(3/2)/(b*e*sqrt(a^2+2*a*b*x+b^2*x^2))-_
2*(A*b-a*B)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*_
sqrt(b*d-a*e)/(b^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*(A*b-a*B)*_
(a+b*x)*sqrt(d+e*x)/(b^2*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 2 2 +-----------+
--R ((- 6A b + 6B a b)e x + (- 6A a b + 6B a )e)\|- a e + b d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 2 2 2 2 2
--R 2B b e x + ((6A b - 4B a b)e + 2B b d)x + (6A a b - 6B a )e
--R +
--R 2B a b d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R +------------------+
--R 2 +-+ | 2 2 2
--R 3b e\|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 701
--S 702 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 702
)clear all
--S 703 of 764
t0:=(A+B*x)/(sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R B x + A
--R (1) -------------------------------
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 703
--S 704 of 764
r0:=-2*(A*b-a*B)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/_
(b^(3/2)*sqrt(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*B*(a+b*x)*_
sqrt(d+e*x)/(b*e*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R +-+ +-------+
--R 2 2 \|b \|e x + d
--R ((- 2A b + 2B a b)e x + (- 2A a b + 2B a )e)atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R +-----------+ +-+ +-------+
--R (2B b x + 2B a)\|- a e + b d \|b \|e x + d
--R /
--R +------------------+
--R +-----------+ +-+ | 2 2 2
--R b e\|- a e + b d \|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 704
--S 705 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 705
)clear all
--S 706 of 764
t0:=(A+B*x)/((d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R B x + A
--R (1) ----------------------------------------
--R +------------------+
--R +-------+ | 2 2 2
--R (e x + d)\|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 706
--S 707 of 764
r0:=-2*(A*b-a*B)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/_
((b*d-a*e)^(3/2)*sqrt(b)*sqrt(a^2+2*a*b*x+b^2*x^2))-2*(B*d-A*e)*_
(a+b*x)/(e*(b*d-a*e)*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R +-+ +-------+
--R 2 2 +-------+ \|b \|e x + d
--R ((2A b - 2B a b)e x + (2A a b - 2B a )e)\|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R +-----------+ +-+
--R ((- 2A b e + 2B b d)x - 2A a e + 2B a d)\|- a e + b d \|b
--R /
--R +------------------+
--R 2 +-----------+ +-+ +-------+ | 2 2 2
--R (a e - b d e)\|- a e + b d \|b \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 707
--S 708 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 708
)clear all
--S 709 of 764
t0:=(A+B*x)/((d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R B x + A
--R (1) ---------------------------------------------------
--R +------------------+
--R 2 2 2 +-------+ | 2 2 2
--R (e x + 2d e x + d )\|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 709
--S 710 of 764
r0:=-2/3*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2))-2*(A*b-a*B)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-_
a*e))*sqrt(b)/((b*d-a*e)^(5/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+2*(A*b-_
a*B)*(a+b*x)/((b*d-a*e)^2*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 2 2 2
--R (- 6A b + 6B a b)e x
--R +
--R 2 2 2 2
--R ((- 6A a b + 6B a )e + (- 6A b + 6B a b)d e)x + (- 6A a b + 6B a )d e
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 2 2 2
--R (6A b - 6B a b)e x
--R +
--R 2 2 2 2 2 2 2
--R ((4A a b - 6B a )e + (8A b - 4B a b)d e - 2B b d )x - 2A a e
--R +
--R 2 2
--R (8A a b - 4B a )d e - 2B a b d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 2 4 3 2 2 2 2 3 2 2 2 3
--R ((3a e - 6a b d e + 3b d e )x + 3a d e - 6a b d e + 3b d e)
--R *
--R +------------------+
--R +-----------+ +-------+ | 2 2 2
--R \|- a e + b d \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 710
--S 711 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 711
)clear all
--S 712 of 764
t0:=(A+B*x)/((d+e*x)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R B x + A
--R (1) -------------------------------------------------------------
--R +------------------+
--R 3 3 2 2 2 3 +-------+ | 2 2 2
--R (e x + 3d e x + 3d e x + d )\|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 712
--S 713 of 764
r0:=-2/5*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)^(5/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2))+2/3*(A*b-a*B)*(a+b*x)/((b*d-a*e)^2*(d+e*x)^(3/2)*sqrt(a^2+_
2*a*b*x+b^2*x^2))-2*b^(3/2)*(A*b-a*B)*(a+b*x)*atanh(sqrt(b)*sqrt(d+_
e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+_
2*b*(A*b-a*B)*(a+b*x)/((b*d-a*e)^3*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 3 2 3 3
--R (30A b - 30B a b )e x
--R +
--R 2 2 3 3 2 2 2
--R ((30A a b - 30B a b)e + (60A b - 60B a b )d e )x
--R +
--R 2 2 2 3 2 2
--R ((60A a b - 60B a b)d e + (30A b - 30B a b )d e)x
--R +
--R 2 2 2
--R (30A a b - 30B a b)d e
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 3 3
--R (- 30A b + 30B a b )e x
--R +
--R 2 2 3 3 2 2 2
--R ((- 20A a b + 20B a b)e + (- 70A b + 70B a b )d e )x
--R +
--R 2 3 3 2 2 2
--R (4A a b - 10B a )e + (- 48A a b + 66B a b)d e
--R +
--R 3 2 2 3 3
--R (- 46A b + 28B a b )d e + 6B b d
--R *
--R x
--R +
--R 3 3 2 3 2 2 2 2
--R - 6A a e + (22A a b - 4B a )d e + (- 46A a b + 28B a b)d e
--R +
--R 2 3
--R 6B a b d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 3 6 2 5 2 2 4 3 3 3 2
--R (15a e - 45a b d e + 45a b d e - 15b d e )x
--R +
--R 3 5 2 2 4 2 3 3 3 4 2 3 2 4
--R (30a d e - 90a b d e + 90a b d e - 30b d e )x + 15a d e
--R +
--R 2 3 3 2 4 2 3 5
--R - 45a b d e + 45a b d e - 15b d e
--R *
--R +------------------+
--R +-----------+ +-------+ | 2 2 2
--R \|- a e + b d \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 713
--S 714 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 714
)clear all
--S 715 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)/(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x
--R +
--R 3
--R A d
--R *
--R +-------+
--R \|e x + d
--R /
--R +------------------+
--R 2 2 2 | 2 2 2
--R (b x + 2a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 715
--S 716 of 764
r0:=-1/10*(4*b*B*d+5*A*b*e-9*a*B*e)*(a+b*x)*(d+e*x)^(7/2)/(b^2*e*(a^2+_
2*a*b*x+b^2*x^2)^(3/2))+2/5*B*(a+b*x)*(d+e*x)^(9/2)/(b*e*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+7/12*e*(4*b*B*d+5*A*b*e-9*a*B*e)*(a+b*x)*_
(d+e*x)^(3/2)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))-7/20*(4*b*B*d+5*A*b*e-_
9*a*B*e)*(d+e*x)^(5/2)/(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))-7/4*e*(b*d-_
a*e)^(3/2)*(4*b*B*d+5*A*b*e-9*a*B*e)*(a+b*x)*atanh(sqrt(b)*sqrt(d+_
e*x)/sqrt(b*d-a*e))/(b^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+7/4*e*(b*d-_
a*e)*(4*b*B*d+5*A*b*e-9*a*B*e)*(a+b*x)*sqrt(d+e*x)/(b^5*sqrt(a^2+_
2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 3 2 2 3 4 3 2
--R (525A a b - 945B a b )e + (- 525A b + 1365B a b )d e
--R +
--R 4 2
--R - 420B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 3 3 2 2 2
--R (1050A a b - 1890B a b)e + (- 1050A a b + 2730B a b )d e
--R +
--R 3 2
--R - 840B a b d e
--R *
--R x
--R +
--R 3 4 3 2 2 3 2 2 2 2
--R (525A a b - 945B a )e + (- 525A a b + 1365B a b)d e - 420B a b d e
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 4 4 3 3 4 2 3
--R 24B b e x + ((40A b - 72B a b )e + 128B b d e )x
--R +
--R 3 2 2 3 4 3 2
--R (- 280A a b + 504B a b )e + (400A b - 944B a b )d e
--R +
--R 4 2
--R 464B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 3 3 2 2 2
--R (- 875A a b + 1575B a b)e + (1190A a b - 2842B a b )d e
--R +
--R 4 3 2 4 3
--R (- 195A b + 1303B a b )d e - 60B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (- 525A a b + 945B a )e + (700A a b - 1680B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (- 105A a b + 749B a b )d e + (- 30A b - 30B a b )d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R +------------------+
--R 6 5 +-+ | 2 2 2
--R (60b x + 60a b )\|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 716
--S 717 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 717
)clear all
--S 718 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)/(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R 2 3 2 2 2 2 +-------+
--R (B e x + (A e + 2B d e)x + (2A d e + B d )x + A d )\|e x + d
--R (1) ----------------------------------------------------------------
--R +------------------+
--R 2 2 2 | 2 2 2
--R (b x + 2a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 718
--S 719 of 764
r0:=-1/6*(4*b*B*d+3*A*b*e-7*a*B*e)*(a+b*x)*(d+e*x)^(5/2)/(b^2*e*(a^2+_
2*a*b*x+b^2*x^2)^(3/2))+2/3*B*(a+b*x)*(d+e*x)^(7/2)/(b*e*(a^2+_
2*a*b*x+b^2*x^2)^(3/2))-5/12*(4*b*B*d+3*A*b*e-7*a*B*e)*(d+e*x)^(3/2)/_
(b^3*sqrt(a^2+2*a*b*x+b^2*x^2))-5/4*e*(4*b*B*d+3*A*b*e-7*a*B*e)*_
(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-a*e)/_
(b^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+5/4*e*(4*b*B*d+3*A*b*e-7*a*B*e)*_
(a+b*x)*sqrt(d+e*x)/(b^4*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 3 2 2 3 2
--R ((- 45A b + 105B a b )e - 60B b d e)x
--R +
--R 2 2 2 2 2 3 2
--R ((- 90A a b + 210B a b)e - 120B a b d e)x + (- 45A a b + 105B a )e
--R +
--R 2
--R - 60B a b d e
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 3 3 2 2 3 2
--R 8B b e x + ((24A b - 56B a b )e + 56B b d e)x
--R +
--R 2 2 2 3 2 3 2
--R ((75A a b - 175B a b)e + (- 27A b + 163B a b )d e - 12B b d )x
--R +
--R 2 3 2 2 2
--R (45A a b - 105B a )e + (- 15A a b + 95B a b)d e
--R +
--R 3 2 2
--R (- 6A b - 6B a b )d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R +------------------+
--R 5 4 +-+ | 2 2 2
--R (12b x + 12a b )\|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 719
--S 720 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 720
)clear all
--S 721 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)/(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R 2 +-------+
--R (B e x + (A e + B d)x + A d)\|e x + d
--R (1) -----------------------------------------
--R +------------------+
--R 2 2 2 | 2 2 2
--R (b x + 2a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 721
--S 722 of 764
r0:=-1/2*(4*b*B*d+A*b*e-5*a*B*e)*(a+b*x)*(d+e*x)^(3/2)/(b^2*e*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+2*B*(a+b*x)*(d+e*x)^(5/2)/(b*e*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))-3/4*e*(4*b*B*d+A*b*e-5*a*B*e)*(a+b*x)*atanh(sqrt(b)*_
sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(7/2)*sqrt(b*d-a*e)*sqrt(a^2+2*a*b*x+_
b^2*x^2))-3/4*(4*b*B*d+A*b*e-5*a*B*e)*sqrt(d+e*x)/(b^3*sqrt(a^2+_
2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 3 2 2 3 2
--R ((- 3A b + 15B a b )e - 12B b d e)x
--R +
--R 2 2 2 2 2 3 2
--R ((- 6A a b + 30B a b)e - 24B a b d e)x + (- 3A a b + 15B a )e
--R +
--R 2
--R - 12B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 2 2 2 2 2
--R 8B b e x + ((- 5A b + 25B a b)e - 4B b d)x + (- 3A a b + 15B a )e
--R +
--R 2
--R (- 2A b - 2B a b)d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R +------------------+
--R 4 3 +-----------+ +-+ | 2 2 2
--R (4b x + 4a b )\|- a e + b d \|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 722
--S 723 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 723
)clear all
--S 724 of 764
t0:=(A+B*x)*sqrt(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^(3/2)
--R
--R
--R +-------+
--R (B x + A)\|e x + d
--R (1) -----------------------------------------
--R +------------------+
--R 2 2 2 | 2 2 2
--R (b x + 2a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 724
--S 725 of 764
r0:=-2*B*(a+b*x)*(d+e*x)^(3/2)/(b*e*(a^2+2*a*b*x+b^2*x^2)^(3/2))+_
1/2*(4*b*B*d-A*b*e-3*a*B*e)*(a+b*x)*(d+e*x)^(3/2)/(b*e*(b*d-a*e)*_
(a^2+2*a*b*x+b^2*x^2)^(3/2))-1/4*e*(4*b*B*d-A*b*e-3*a*B*e)*(a+b*x)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(5/2)*(b*d-a*e)^(3/2)*_
sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*(4*b*B*d-A*b*e-3*a*B*e)*sqrt(d+e*x)/_
(b^2*(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 3 2 2 3 2
--R ((- A b - 3B a b )e + 4B b d e)x
--R +
--R 2 2 2 2 2 3 2
--R ((- 2A a b - 6B a b)e + 8B a b d e)x + (- A a b - 3B a )e
--R +
--R 2
--R 4B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 2 2 2 2
--R (((A b - 5B a b)e + 4B b d)x + (- A a b - 3B a )e + (2A b + 2B a b)d)
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R +------------------+
--R 3 4 2 2 3 +-----------+ +-+ | 2 2 2
--R ((4a b e - 4b d)x + 4a b e - 4a b d)\|- a e + b d \|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 725
--S 726 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 726
)clear all
--S 727 of 764
t0:=(A+B*x)/((a^2+2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x))
--R
--R
--R B x + A
--R (1) ---------------------------------------------------
--R +------------------+
--R 2 2 2 +-------+ | 2 2 2
--R (b x + 2a b x + a )\|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 727
--S 728 of 764
r0:=-2/3*B*(a+b*x)*sqrt(d+e*x)/(b*e*(a^2+2*a*b*x+b^2*x^2)^(3/2))+_
1/6*(4*b*B*d-3*A*b*e-a*B*e)*(a+b*x)*sqrt(d+e*x)/(b*e*(b*d-a*e)*_
(a^2+2*a*b*x+b^2*x^2)^(3/2))+1/4*e*(4*b*B*d-3*A*b*e-a*B*e)*(a+b*x)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(3/2)*(b*d-a*e)^(5/2)*_
sqrt(a^2+2*a*b*x+b^2*x^2))-1/4*(4*b*B*d-3*A*b*e-a*B*e)*sqrt(d+e*x)/_
(b*(b*d-a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 3 2 2 3 2
--R ((- 3A b - B a b )e + 4B b d e)x
--R +
--R 2 2 2 2 2 3 2
--R ((- 6A a b - 2B a b)e + 8B a b d e)x + (- 3A a b - B a )e
--R +
--R 2
--R 4B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 2 2 2 2
--R (((3A b + B a b)e - 4B b d)x + (5A a b - B a )e + (- 2A b - 2B a b)d)
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 2 2 2 3 4 2 3 2 2 2 3 2
--R ((4a b e - 8a b d e + 4b d )x + 4a b e - 8a b d e + 4a b d )
--R *
--R +------------------+
--R +-----------+ +-+ | 2 2 2
--R \|- a e + b d \|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 728
)clear all
--S 729 of 764
t0:=(A+B*x)/((d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2))
--R
--R
--R (1)
--R B x + A
--R /
--R 2 3 2 2 2 2 +-------+
--R (b e x + (2a b e + b d)x + (a e + 2a b d)x + a d)\|e x + d
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 729
--S 730 of 764
r0:=-2*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2)*_
sqrt(d+e*x))+1/2*(4*b*B*d-5*A*b*e+a*B*e)*(a+b*x)*sqrt(d+e*x)/(e*_
(b*d-a*e)^2*(a^2+2*a*b*x+b^2*x^2)^(3/2))+3/4*e*(4*b*B*d-5*A*b*e+_
a*B*e)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-_
a*e)^(7/2)*sqrt(b)*sqrt(a^2+2*a*b*x+b^2*x^2))-3/4*(4*b*B*d-5*A*b*e+_
a*B*e)*sqrt(d+e*x)/((b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 3 2 2 3 2
--R ((15A b - 3B a b )e - 12B b d e)x
--R +
--R 2 2 2 2 2 3 2
--R ((30A a b - 6B a b)e - 24B a b d e)x + (15A a b - 3B a )e
--R +
--R 2
--R - 12B a b d e
--R *
--R +-+ +-------+
--R +-------+ \|b \|e x + d
--R \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 2 2 2 2
--R ((- 15A b + 3B a b)e + 12B b d e)x
--R +
--R 2 2 2 2 2 2 2
--R ((- 25A a b + 5B a )e + (- 5A b + 21B a b)d e + 4B b d )x - 8A a e
--R +
--R 2 2 2
--R (- 9A a b + 13B a )d e + (2A b + 2B a b)d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R /
--R 3 3 2 2 2 3 2 4 3 4 3 3 2
--R (4a b e - 12a b d e + 12a b d e - 4b d )x + 4a e - 12a b d e
--R +
--R 2 2 2 3 3
--R 12a b d e - 4a b d
--R *
--R +------------------+
--R +-----------+ +-+ +-------+ | 2 2 2
--R \|- a e + b d \|b \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 730
--S 731 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 731
)clear all
--S 732 of 764
t0:=(A+B*x)/((d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2))
--R
--R
--R (1)
--R B x + A
--R /
--R 2 2 4 2 2 3 2 2 2 2 2
--R b e x + (2a b e + 2b d e)x + (a e + 4a b d e + b d )x
--R +
--R 2 2 2 2
--R (2a d e + 2a b d )x + a d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 732
--S 733 of 764
r0:=-2/3*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)^(3/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+1/6*(4*b*B*d-7*A*b*e+3*a*B*e)*(a+b*x)/(e*(b*d-_
a*e)^2*(a^2+2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x))+5/4*e*(4*b*B*d-_
7*A*b*e+3*a*B*e)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*_
sqrt(b)/((b*d-a*e)^(9/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-5/12*(4*b*B*d-_
7*A*b*e+3*a*B*e)/((b*d-a*e)^3*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-_
5/4*e*(4*b*B*d-7*A*b*e+3*a*B*e)*(a+b*x)/((b*d-a*e)^4*sqrt(d+e*x)*_
sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 3 2 3 3 2 3
--R ((- 105A b + 45B a b )e + 60B b d e )x
--R +
--R 2 2 3 3 2 2
--R (- 210A a b + 90B a b)e + (- 105A b + 165B a b )d e
--R +
--R 3 2
--R 60B b d e
--R *
--R 2
--R x
--R +
--R 2 3 3 2 2 2
--R (- 105A a b + 45B a )e + (- 210A a b + 150B a b)d e
--R +
--R 2 2
--R 120B a b d e
--R *
--R x
--R +
--R 2 3 2 2 2
--R (- 105A a b + 45B a )d e + 60B a b d e
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 3 2 3 3 2 3
--R ((105A b - 45B a b )e - 60B b d e )x
--R +
--R 2 2 3 3 2 2 3 2 2
--R ((175A a b - 75B a b)e + (140A b - 160B a b )d e - 80B b d e)x
--R +
--R 2 3 3 2 2 2
--R (56A a b - 24B a )e + (238A a b - 134B a b)d e
--R +
--R 3 2 2 3 3
--R (21A b - 145B a b )d e - 12B b d
--R *
--R x
--R +
--R 3 3 2 3 2 2 2 2
--R - 8A a e + (80A a b - 16B a )d e + (39A a b - 83B a b)d e
--R +
--R 3 2 3
--R (- 6A b - 6B a b )d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 4 5 3 2 4 2 3 2 3 4 3 2 5 4 2
--R (12a b e - 48a b d e + 72a b d e - 48a b d e + 12b d e)x
--R +
--R 5 5 4 4 3 2 2 3 2 3 3 2 4 4 5 5
--R (12a e - 36a b d e + 24a b d e + 24a b d e - 36a b d e + 12b d )x
--R +
--R 5 4 4 2 3 3 2 3 2 2 3 4 4 5
--R 12a d e - 48a b d e + 72a b d e - 48a b d e + 12a b d
--R *
--R +------------------+
--R +-----------+ +-------+ | 2 2 2
--R \|- a e + b d \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 733
--S 734 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 734
)clear all
--S 735 of 764
t0:=(A+B*x)/((d+e*x)^(7/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2))
--R
--R
--R (1)
--R B x + A
--R /
--R 2 3 5 3 2 2 4 2 3 2 2 2 3
--R b e x + (2a b e + 3b d e )x + (a e + 6a b d e + 3b d e)x
--R +
--R 2 2 2 2 3 2 2 2 3 2 3
--R (3a d e + 6a b d e + b d )x + (3a d e + 2a b d )x + a d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 735
--S 736 of 764
r0:=-2/5*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)^(5/2)*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+1/10*(4*b*B*d-9*A*b*e+5*a*B*e)*(a+b*x)/(e*(b*d-_
a*e)^2*(d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(3/2))-7/20*(4*b*B*d-_
9*A*b*e+5*a*B*e)/((b*d-a*e)^3*(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2))-7/12*e*(4*b*B*d-9*A*b*e+5*a*B*e)*(a+b*x)/((b*d-a*e)^4*_
(d+e*x)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+7/4*b^(3/2)*e*(4*b*B*d-_
9*A*b*e+5*a*B*e)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/_
((b*d-a*e)^(11/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-7/4*b*e*(4*b*B*d-_
9*A*b*e+5*a*B*e)*(a+b*x)/((b*d-a*e)^5*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+_
b^2*x^2))
--R
--R
--R (2)
--R 4 3 4 4 3 4
--R ((945A b - 525B a b )e - 420B b d e )x
--R +
--R 3 2 2 4 4 3 3
--R (1890A a b - 1050B a b )e + (1890A b - 1890B a b )d e
--R +
--R 4 2 2
--R - 840B b d e
--R *
--R 3
--R x
--R +
--R 2 2 3 4 3 2 2 3
--R (945A a b - 525B a b)e + (3780A a b - 2520B a b )d e
--R +
--R 4 3 2 2 4 3
--R (945A b - 2205B a b )d e - 420B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 3 3 2 2 2 2
--R (1890A a b - 1050B a b)d e + (1890A a b - 1890B a b )d e
--R +
--R 3 3
--R - 840B a b d e
--R *
--R x
--R +
--R 2 2 3 2 2 2 2 3
--R (945A a b - 525B a b)d e - 420B a b d e
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 4 4 3 4
--R ((- 945A b + 525B a b )e + 420B b d e )x
--R +
--R 3 2 2 4 4 3 3
--R (- 1575A a b + 875B a b )e + (- 2205A b + 1925B a b )d e
--R +
--R 4 2 2
--R 980B b d e
--R *
--R 3
--R x
--R +
--R 2 2 3 4 3 2 2 3
--R (- 504A a b + 280B a b)e + (- 3717A a b + 2289B a b )d e
--R +
--R 4 3 2 2 4 3
--R (- 1449A b + 2457B a b )d e + 644B b d e
--R *
--R 2
--R x
--R +
--R 3 4 4 2 2 3 3
--R (72A a b - 40B a )e + (- 1224A a b + 648B a b)d e
--R +
--R 3 2 2 2 2 4 3 3
--R (- 2493A a b + 1929B a b )d e + (- 135A b + 1183B a b )d e
--R +
--R 4 4
--R 60B b d
--R *
--R x
--R +
--R 4 4 3 4 3 2 2 3 2 2
--R - 24A a e + (168A a b - 16B a )d e + (- 864A a b + 272B a b)d e
--R +
--R 3 2 2 3 4 3 4
--R (- 255A a b + 659B a b )d e + (30A b + 30B a b )d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 5 7 4 2 6 3 3 2 5 2 4 3 4 5 4 3
--R 60a b e - 300a b d e + 600a b d e - 600a b d e + 300a b d e
--R +
--R 6 5 2
--R - 60b d e
--R *
--R 3
--R x
--R +
--R 6 7 5 6 3 3 3 4 2 4 4 3 5 5 2
--R 60a e - 180a b d e + 600a b d e - 900a b d e + 540a b d e
--R +
--R 6 6
--R - 120b d e
--R *
--R 2
--R x
--R +
--R 6 6 5 2 5 4 2 3 4 3 3 4 3 5 6
--R 120a d e - 540a b d e + 900a b d e - 600a b d e + 180a b d e
--R +
--R 6 7
--R - 60b d
--R *
--R x
--R +
--R 6 2 5 5 3 4 4 2 4 3 3 3 5 2 2 4 6
--R 60a d e - 300a b d e + 600a b d e - 600a b d e + 300a b d e
--R +
--R 5 7
--R - 60a b d
--R *
--R +------------------+
--R +-----------+ +-------+ | 2 2 2
--R \|- a e + b d \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 736
--S 737 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 737
)clear all
--S 738 of 764
t0:=(A+B*x)*(d+e*x)^(11/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 5 6 5 4 5 4 2 3 4
--R B e x + (A e + 5B d e )x + (5A d e + 10B d e )x
--R +
--R 2 3 3 2 3 3 2 4 2 4 5 5
--R (10A d e + 10B d e )x + (10A d e + 5B d e)x + (5A d e + B d )x + A d
--R *
--R +-------+
--R \|e x + d
--R /
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 738
--S 739 of 764
r0:=-1/20*(8*b*B*d+5*A*b*e-13*a*B*e)*(a+b*x)*(d+e*x)^(11/2)/(b^2*e*_
(a^2+2*a*b*x+b^2*x^2)^(5/2))+2/5*B*(a+b*x)*(d+e*x)^(13/2)/(b*e*_
(a^2+2*a*b*x+b^2*x^2)^(5/2))-33/160*e*(8*b*B*d+5*A*b*e-13*a*B*e)*_
(a+b*x)*(d+e*x)^(7/2)/(b^4*(a^2+2*a*b*x+b^2*x^2)^(3/2))-11/120*_
(8*b*B*d+5*A*b*e-13*a*B*e)*(d+e*x)^(9/2)/(b^3*(a^2+2*a*b*x+_
b^2*x^2)^(3/2))+77/64*e^3*(8*b*B*d+5*A*b*e-13*a*B*e)*(a+b*x)*(d+_
e*x)^(3/2)/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))-231/320*e^2*(8*b*B*d+_
5*A*b*e-13*a*B*e)*(d+e*x)^(5/2)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))-_
231/64*e^3*(b*d-a*e)^(3/2)*(8*b*B*d+5*A*b*e-13*a*B*e)*(a+b*x)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(15/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2))+231/64*e^3*(b*d-a*e)*(8*b*B*d+5*A*b*e-13*a*B*e)*(a+b*x)*_
sqrt(d+e*x)/(b^7*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 2 4 5 6 5 4
--R (17325A a b - 45045B a b )e + (- 17325A b + 72765B a b )d e
--R +
--R 6 2 3
--R - 27720B b d e
--R *
--R 4
--R x
--R +
--R 2 4 3 3 5
--R (69300A a b - 180180B a b )e
--R +
--R 5 2 4 4 5 2 3
--R (- 69300A a b + 291060B a b )d e - 110880B a b d e
--R *
--R 3
--R x
--R +
--R 3 3 4 2 5
--R (103950A a b - 270270B a b )e
--R +
--R 2 4 3 3 4 2 4 2 3
--R (- 103950A a b + 436590B a b )d e - 166320B a b d e
--R *
--R 2
--R x
--R +
--R 4 2 5 5
--R (69300A a b - 180180B a b)e
--R +
--R 3 3 4 2 4 3 3 2 3
--R (- 69300A a b + 291060B a b )d e - 110880B a b d e
--R *
--R x
--R +
--R 5 6 5 4 2 5 4
--R (17325A a b - 45045B a )e + (- 17325A a b + 72765B a b)d e
--R +
--R 4 2 2 3
--R - 27720B a b d e
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 6 5 6 6 5 5 6 4 5
--R 384B b e x + ((640A b - 1664B a b )e + 3328B b d e )x
--R +
--R 5 2 4 5 6 5 4
--R (- 7040A a b + 18304B a b )e + (10240A b - 37888B a b )d e
--R +
--R 6 2 3
--R 22144B b d e
--R *
--R 4
--R x
--R +
--R 2 4 3 3 5
--R (- 46035A a b + 119691B a b )e
--R +
--R 5 2 4 4
--R (63910A a b - 239822B a b )d e
--R +
--R 6 5 2 3 6 3 2
--R (- 11475A b + 132091B a b )d e - 10680B b d e
--R *
--R 3
--R x
--R +
--R 3 3 4 2 5
--R (- 84315A a b + 219219B a b )e
--R +
--R 2 4 3 3 4
--R (114840A a b - 433488B a b )d e
--R +
--R 5 2 4 2 3
--R (- 18975A a b + 233079B a b )d e
--R +
--R 6 5 3 2 6 4
--R (- 5150A b - 16970B a b )d e - 2480B b d e
--R *
--R 2
--R x
--R +
--R 4 2 5 5
--R (- 63525A a b + 165165B a b)e
--R +
--R 3 3 4 2 4
--R (85470A a b - 323862B a b )d e
--R +
--R 2 4 3 3 2 3
--R (- 13365A a b + 171501B a b )d e
--R +
--R 5 2 4 3 2 6 5 4
--R (- 3740A a b - 11660B a b )d e + (- 1640A b - 1720B a b )d e
--R +
--R 6 5
--R - 320B b d
--R *
--R x
--R +
--R 5 6 5 4 2 5 4
--R (- 17325A a b + 45045B a )e + (23100A a b - 87780B a b)d e
--R +
--R 3 3 4 2 2 3 2 4 3 3 3 2
--R (- 3465A a b + 45969B a b )d e + (- 990A a b - 2970B a b )d e
--R +
--R 5 2 4 4 6 5 5
--R (- 440A a b - 440B a b )d e + (- 240A b - 80B a b )d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R +------------------+
--R 10 3 9 2 2 8 3 7 +-+ | 2 2 2
--R (960b x + 2880a b x + 2880a b x + 960a b )\|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 739
--S 740 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 740
)clear all
--S 741 of 764
t0:=(A+B*x)*(d+e*x)^(9/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 4 5 4 3 4 3 2 2 3
--R B e x + (A e + 4B d e )x + (4A d e + 6B d e )x
--R +
--R 2 2 3 2 3 4 4
--R (6A d e + 4B d e)x + (4A d e + B d )x + A d
--R *
--R +-------+
--R \|e x + d
--R /
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 741
--S 742 of 764
r0:=-1/12*(8*b*B*d+3*A*b*e-11*a*B*e)*(a+b*x)*(d+e*x)^(9/2)/(b^2*e*(a^2+_
2*a*b*x+b^2*x^2)^(5/2))+2/3*B*(a+b*x)*(d+e*x)^(11/2)/(b*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))-7/32*e*(8*b*B*d+3*A*b*e-11*a*B*e)*(a+b*x)*(d+_
e*x)^(5/2)/(b^4*(a^2+2*a*b*x+b^2*x^2)^(3/2))-1/8*(8*b*B*d+3*A*b*e-_
11*a*B*e)*(d+e*x)^(7/2)/(b^3*(a^2+2*a*b*x+b^2*x^2)^(3/2))-_
35/64*e^2*(8*b*B*d+3*A*b*e-11*a*B*e)*(d+e*x)^(3/2)/(b^5*sqrt(a^2+_
2*a*b*x+b^2*x^2))-105/64*e^3*(8*b*B*d+3*A*b*e-11*a*B*e)*(a+b*x)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b*d-a*e)/(b^(13/2)*_
sqrt(a^2+2*a*b*x+b^2*x^2))+105/64*e^3*(8*b*B*d+3*A*b*e-11*a*B*e)*_
(a+b*x)*sqrt(d+e*x)/(b^6*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 4 4 5 3 4
--R ((- 945A b + 3465B a b )e - 2520B b d e )x
--R +
--R 4 2 3 4 4 3 3
--R ((- 3780A a b + 13860B a b )e - 10080B a b d e )x
--R +
--R 2 3 3 2 4 2 3 3 2
--R ((- 5670A a b + 20790B a b )e - 15120B a b d e )x
--R +
--R 3 2 4 4 3 2 3
--R ((- 3780A a b + 13860B a b)e - 10080B a b d e )x
--R +
--R 4 5 4 4 3
--R (- 945A a b + 3465B a )e - 2520B a b d e
--R *
--R +-+ +-------+
--R +-----------+ \|b \|e x + d
--R \|- a e + b d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 5 5 4 4 5 3 4
--R 128B b e x + ((384A b - 1408B a b )e + 1664B b d e )x
--R +
--R 4 2 3 4 5 4 3
--R (2511A a b - 9207B a b )e + (- 975A b + 10271B a b )d e
--R +
--R 5 2 2
--R - 1320B b d e
--R *
--R 3
--R x
--R +
--R 2 3 3 2 4 4 2 3 3
--R (4599A a b - 16863B a b )e + (- 1665A a b + 18369B a b )d e
--R +
--R 5 4 2 2 5 3
--R (- 630A b - 2130B a b )d e - 400B b d e
--R *
--R 2
--R x
--R +
--R 3 2 4 4 2 3 3 2 3
--R (3465A a b - 12705B a b)e + (- 1197A a b + 13629B a b )d e
--R +
--R 4 2 3 2 2 5 4 3
--R (- 468A a b - 1476B a b )d e + (- 264A b - 280B a b )d e
--R +
--R 5 4
--R - 64B b d
--R *
--R x
--R +
--R 4 5 4 3 2 4 3
--R (945A a b - 3465B a )e + (- 315A a b + 3675B a b)d e
--R +
--R 2 3 3 2 2 2 4 2 3 3
--R (- 126A a b - 378B a b )d e + (- 72A a b - 72B a b )d e
--R +
--R 5 4 4
--R (- 48A b - 16B a b )d
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R /
--R +------------------+
--R 9 3 8 2 2 7 3 6 +-+ | 2 2 2
--R (192b x + 576a b x + 576a b x + 192a b )\|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 742
--S 743 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 743
)clear all
--S 744 of 764
t0:=(A+B*x)*(d+e*x)^(7/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R (1)
--R 3 4 3 2 3 2 2 2 2 3
--R B e x + (A e + 3B d e )x + (3A d e + 3B d e)x + (3A d e + B d )x
--R +
--R 3
--R A d
--R *
--R +-------+
--R \|e x + d
--R /
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 744
--S 745 of 764
r0:=-1/4*(8*b*B*d+A*b*e-9*a*B*e)*(a+b*x)*(d+e*x)^(7/2)/(b^2*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))+2*B*(a+b*x)*(d+e*x)^(9/2)/(b*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))-35/96*e*(8*b*B*d+A*b*e-9*a*B*e)*(a+b*x)*(d+_
e*x)^(3/2)/(b^4*(a^2+2*a*b*x+b^2*x^2)^(3/2))-7/24*(8*b*B*d+A*b*e-_
9*a*B*e)*(d+e*x)^(5/2)/(b^3*(a^2+2*a*b*x+b^2*x^2)^(3/2))-35/64*e^3*_
(8*b*B*d+A*b*e-9*a*B*e)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-_
a*e))/(b^(11/2)*sqrt(b*d-a*e)*sqrt(a^2+2*a*b*x+b^2*x^2))-35/64*e^2*_
(8*b*B*d+A*b*e-9*a*B*e)*sqrt(d+e*x)/(b^5*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 4 4 5 3 4
--R ((- 105A b + 945B a b )e - 840B b d e )x
--R +
--R 4 2 3 4 4 3 3
--R ((- 420A a b + 3780B a b )e - 3360B a b d e )x
--R +
--R 2 3 3 2 4 2 3 3 2
--R ((- 630A a b + 5670B a b )e - 5040B a b d e )x
--R +
--R 3 2 4 4 3 2 3
--R ((- 420A a b + 3780B a b)e - 3360B a b d e )x
--R +
--R 4 5 4 4 3
--R (- 105A a b + 945B a )e - 840B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 4 4 3 3 4 2 3
--R 384B b e x + ((- 279A b + 2511B a b )e - 696B b d e )x
--R +
--R 3 2 2 3 4 3 2
--R (- 511A a b + 4599B a b )e + (- 326A b - 1154B a b )d e
--R +
--R 4 2
--R - 304B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 3 3 2 2 2
--R (- 385A a b + 3465B a b)e + (- 252A a b - 812B a b )d e
--R +
--R 4 3 2 4 3
--R (- 200A b - 216B a b )d e - 64B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (- 105A a b + 945B a )e + (- 70A a b - 210B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (- 56A a b - 56B a b )d e + (- 48A b - 16B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 8 3 7 2 2 6 3 5 +-----------+ +-+
--R (192b x + 576a b x + 576a b x + 192a b )\|- a e + b d \|b
--R *
--R +------------------+
--R | 2 2 2
--R \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 745
--S 746 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 746
)clear all
--S 747 of 764
t0:=(A+B*x)*(d+e*x)^(5/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R 2 3 2 2 2 2 +-------+
--R (B e x + (A e + 2B d e)x + (2A d e + B d )x + A d )\|e x + d
--R (1) ----------------------------------------------------------------
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 747
--S 748 of 764
r0:=1/4*(8*b*B*d-A*b*e-7*a*B*e)*(a+b*x)*(d+e*x)^(5/2)/(b^2*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))-2*B*(a+b*x)*(d+e*x)^(7/2)/(b*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))+5/24*(8*b*B*d-A*b*e-7*a*B*e)*(d+e*x)^(3/2)/(b^3*_
(a^2+2*a*b*x+b^2*x^2)^(3/2))+5/32*e*(8*b*B*d-A*b*e-7*a*B*e)*(a+b*x)*_
(d+e*x)^(3/2)/(b^3*(b*d-a*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2))-5/64*e^3*_
(8*b*B*d-A*b*e-7*a*B*e)*(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-_
a*e))/(b^(9/2)*(b*d-a*e)^(3/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-5/64*e^2*_
(8*b*B*d-A*b*e-7*a*B*e)*sqrt(d+e*x)/(b^4*(b*d-a*e)*sqrt(a^2+2*a*b*x+_
b^2*x^2))
--R
--R
--R (2)
--R 5 4 4 5 3 4
--R ((- 15A b - 105B a b )e + 120B b d e )x
--R +
--R 4 2 3 4 4 3 3
--R ((- 60A a b - 420B a b )e + 480B a b d e )x
--R +
--R 2 3 3 2 4 2 3 3 2
--R ((- 90A a b - 630B a b )e + 720B a b d e )x
--R +
--R 3 2 4 4 3 2 3
--R ((- 60A a b - 420B a b)e + 480B a b d e )x
--R +
--R 4 5 4 4 3
--R (- 15A a b - 105B a )e + 120B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 3 4 2 3
--R ((15A b - 279B a b )e + 264B b d e )x
--R +
--R 3 2 2 3 4 3 2
--R (- 73A a b - 511B a b )e + (118A b + 258B a b )d e
--R +
--R 4 2
--R 208B b d e
--R *
--R 2
--R x
--R +
--R 2 2 3 3 3 2 2 2
--R (- 55A a b - 385B a b)e + (- 36A a b + 188B a b )d e
--R +
--R 4 3 2 4 3
--R (136A b + 88B a b )d e + 64B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (- 15A a b - 105B a )e + (- 10A a b + 50B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (- 8A a b + 24B a b )d e + (48A b + 16B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 7 8 3 2 6 7 2
--R (192a b e - 192b d)x + (576a b e - 576a b d)x
--R +
--R 3 5 2 6 4 4 3 5
--R (576a b e - 576a b d)x + 192a b e - 192a b d
--R *
--R +------------------+
--R +-----------+ +-+ | 2 2 2
--R \|- a e + b d \|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 748
--S 749 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 749
)clear all
--S 750 of 764
t0:=(A+B*x)*(d+e*x)^(3/2)/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R 2 +-------+
--R (B e x + (A e + B d)x + A d)\|e x + d
--R (1) -------------------------------------------------------------
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 750
--S 751 of 764
r0:=1/12*(8*b*B*d-3*A*b*e-5*a*B*e)*(a+b*x)*(d+e*x)^(3/2)/(b^2*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))-2/3*B*(a+b*x)*(d+e*x)^(5/2)/(b*e*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))+1/24*(8*b*B*d-3*A*b*e-5*a*B*e)*sqrt(d+e*x)/(b^3*_
(a^2+2*a*b*x+b^2*x^2)^(3/2))+1/96*e*(8*b*B*d-3*A*b*e-5*a*B*e)*(a+_
b*x)*sqrt(d+e*x)/(b^3*(b*d-a*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2))+_
1/64*e^3*(8*b*B*d-3*A*b*e-5*a*B*e)*(a+b*x)*atanh(sqrt(b)*sqrt(d+_
e*x)/sqrt(b*d-a*e))/(b^(7/2)*(b*d-a*e)^(5/2)*sqrt(a^2+2*a*b*x+_
b^2*x^2))-1/64*e^2*(8*b*B*d-3*A*b*e-5*a*B*e)*sqrt(d+e*x)/(b^3*(b*d-_
a*e)^2*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 4 4 5 3 4
--R ((- 9A b - 15B a b )e + 24B b d e )x
--R +
--R 4 2 3 4 4 3 3
--R ((- 36A a b - 60B a b )e + 96B a b d e )x
--R +
--R 2 3 3 2 4 2 3 3 2
--R ((- 54A a b - 90B a b )e + 144B a b d e )x
--R +
--R 3 2 4 4 3 2 3 4 5 4
--R ((- 36A a b - 60B a b)e + 96B a b d e )x + (- 9A a b - 15B a )e
--R +
--R 4 3
--R 24B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 3 4 2 3
--R ((9A b + 15B a b )e - 24B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 2
--R ((33A a b - 73B a b )e + (- 6A b + 158B a b )d e - 112B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (- 33A a b - 55B a b)e + (132A a b + 52B a b )d e
--R +
--R 4 3 2 4 3
--R (- 72A b + 40B a b )d e - 64B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (- 9A a b - 15B a )e + (- 6A a b + 14B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (72A a b + 8B a b )d e + (- 48A b - 16B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 2 6 2 7 8 2 3
--R (192a b e - 384a b d e + 192b d )x
--R +
--R 3 5 2 2 6 7 2 2
--R (576a b e - 1152a b d e + 576a b d )x
--R +
--R 4 4 2 3 5 2 6 2 5 3 2 4 4
--R (576a b e - 1152a b d e + 576a b d )x + 192a b e - 384a b d e
--R +
--R 3 5 2
--R 192a b d
--R *
--R +------------------+
--R +-----------+ +-+ | 2 2 2
--R \|- a e + b d \|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 751
--S 752 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 752
)clear all
--S 753 of 764
t0:=(A+B*x)*sqrt(d+e*x)/(a^2+2*a*b*x+b^2*x^2)^(5/2)
--R
--R
--R +-------+
--R (B x + A)\|e x + d
--R (1) -------------------------------------------------------------
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|b x + 2a b x + a
--R Type: Expression(Integer)
--E 753
--S 754 of 764
r0:=-2/5*B*(a+b*x)*(d+e*x)^(3/2)/(b*e*(a^2+2*a*b*x+b^2*x^2)^(5/2))+_
1/20*(8*b*B*d-5*A*b*e-3*a*B*e)*(a+b*x)*(d+e*x)^(3/2)/(b*e*(b*d-a*e)*_
(a^2+2*a*b*x+b^2*x^2)^(5/2))-1/24*(8*b*B*d-5*A*b*e-3*a*B*e)*_
sqrt(d+e*x)/(b^2*(b*d-a*e)*(a^2+2*a*b*x+b^2*x^2)^(3/2))-1/96*e*_
(8*b*B*d-5*A*b*e-3*a*B*e)*(a+b*x)*sqrt(d+e*x)/(b^2*(b*d-a*e)^2*_
(a^2+2*a*b*x+b^2*x^2)^(3/2))-1/64*e^3*(8*b*B*d-5*A*b*e-3*a*B*e)*_
(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(5/2)*(b*d-_
a*e)^(7/2)*sqrt(a^2+2*a*b*x+b^2*x^2))+1/64*e^2*(8*b*B*d-5*A*b*e-_
3*a*B*e)*sqrt(d+e*x)/(b^2*(b*d-a*e)^3*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 4 4 5 3 4
--R ((- 15A b - 9B a b )e + 24B b d e )x
--R +
--R 4 2 3 4 4 3 3
--R ((- 60A a b - 36B a b )e + 96B a b d e )x
--R +
--R 2 3 3 2 4 2 3 3 2
--R ((- 90A a b - 54B a b )e + 144B a b d e )x
--R +
--R 3 2 4 4 3 2 3 4 5 4
--R ((- 60A a b - 36B a b)e + 96B a b d e )x + (- 15A a b - 9B a )e
--R +
--R 4 3
--R 24B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 3 4 2 3
--R ((15A b + 9B a b )e - 24B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 2
--R ((55A a b + 33B a b )e + (- 10A b - 94B a b )d e + 16B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (73A a b - 33B a b)e + (- 36A a b + 92B a b )d e
--R +
--R 4 3 2 4 3
--R (8A b - 168B a b )d e + 64B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (- 15A a b - 9B a )e + (118A a b + 18B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (- 136A a b - 40B a b )d e + (48A b + 16B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 3 5 3 2 6 2 7 2 8 3 3
--R (192a b e - 576a b d e + 576a b d e - 192b d )x
--R +
--R 4 4 3 3 5 2 2 6 2 7 3 2
--R (576a b e - 1728a b d e + 1728a b d e - 576a b d )x
--R +
--R 5 3 3 4 4 2 3 5 2 2 6 3 6 2 3
--R (576a b e - 1728a b d e + 1728a b d e - 576a b d )x + 192a b e
--R +
--R 5 3 2 4 4 2 3 5 3
--R - 576a b d e + 576a b d e - 192a b d
--R *
--R +------------------+
--R +-----------+ +-+ | 2 2 2
--R \|- a e + b d \|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 754
--S 755 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 755
)clear all
--S 756 of 764
t0:=(A+B*x)/((a^2+2*a*b*x+b^2*x^2)^(5/2)*sqrt(d+e*x))
--R
--R
--R (1)
--R B x + A
--R -----------------------------------------------------------------------
--R +------------------+
--R 4 4 3 3 2 2 2 3 4 +-------+ | 2 2 2
--R (b x + 4a b x + 6a b x + 4a b x + a )\|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 756
--S 757 of 764
r0:=-2/7*B*(a+b*x)*sqrt(d+e*x)/(b*e*(a^2+2*a*b*x+b^2*x^2)^(5/2))+_
1/28*(8*b*B*d-7*A*b*e-a*B*e)*(a+b*x)*sqrt(d+e*x)/(b*e*(b*d-a*e)*_
(a^2+2*a*b*x+b^2*x^2)^(5/2))-1/24*(8*b*B*d-7*A*b*e-a*B*e)*_
sqrt(d+e*x)/(b*(b*d-a*e)^2*(a^2+2*a*b*x+b^2*x^2)^(3/2))+5/96*e*_
(8*b*B*d-7*A*b*e-a*B*e)*(a+b*x)*sqrt(d+e*x)/(b*(b*d-a*e)^3*(a^2+_
2*a*b*x+b^2*x^2)^(3/2))+5/64*e^3*(8*b*B*d-7*A*b*e-a*B*e)*(a+b*x)*_
atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/(b^(3/2)*(b*d-a*e)^(9/2)*_
sqrt(a^2+2*a*b*x+b^2*x^2))-5/64*e^2*(8*b*B*d-7*A*b*e-a*B*e)*_
sqrt(d+e*x)/(b*(b*d-a*e)^4*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 4 4 5 3 4
--R ((- 105A b - 15B a b )e + 120B b d e )x
--R +
--R 4 2 3 4 4 3 3
--R ((- 420A a b - 60B a b )e + 480B a b d e )x
--R +
--R 2 3 3 2 4 2 3 3 2
--R ((- 630A a b - 90B a b )e + 720B a b d e )x
--R +
--R 3 2 4 4 3 2 3
--R ((- 420A a b - 60B a b)e + 480B a b d e )x
--R +
--R 4 5 4 4 3
--R (- 105A a b - 15B a )e + 120B a b d e
--R *
--R +-+ +-------+
--R \|b \|e x + d
--R atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 3 4 2 3
--R ((105A b + 15B a b )e - 120B b d e )x
--R +
--R 3 2 2 3 4 3 2 4 2 2
--R ((385A a b + 55B a b )e + (- 70A b - 450B a b )d e + 80B b d e)x
--R +
--R 2 2 3 3 3 2 2 2
--R (511A a b + 73B a b)e + (- 252A a b - 620B a b )d e
--R +
--R 4 3 2 4 3
--R (56A b + 296B a b )d e - 64B b d
--R *
--R x
--R +
--R 3 4 3 2 2 3 2
--R (279A a b - 15B a )e + (- 326A a b - 146B a b)d e
--R +
--R 3 2 2 2 4 3 3
--R (200A a b + 72B a b )d e + (- 48A b - 16B a b )d
--R *
--R +-----------+ +-+ +-------+
--R \|- a e + b d \|b \|e x + d
--R /
--R 4 4 4 3 5 3 2 6 2 2 7 3 8 4 3
--R (192a b e - 768a b d e + 1152a b d e - 768a b d e + 192b d )x
--R +
--R 5 3 4 4 4 3 3 5 2 2 2 6 3 7 4 2
--R (576a b e - 2304a b d e + 3456a b d e - 2304a b d e + 576a b d )x
--R +
--R 6 2 4 5 3 3 4 4 2 2 3 5 3 2 6 4
--R (576a b e - 2304a b d e + 3456a b d e - 2304a b d e + 576a b d )x
--R +
--R 7 4 6 2 3 5 3 2 2 4 4 3 3 5 4
--R 192a b e - 768a b d e + 1152a b d e - 768a b d e + 192a b d
--R *
--R +------------------+
--R +-----------+ +-+ | 2 2 2
--R \|- a e + b d \|b \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 757
--S 758 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 758
)clear all
--S 759 of 764
t0:=(A+B*x)/((d+e*x)^(3/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2))
--R
--R
--R (1)
--R B x + A
--R /
--R 4 5 3 4 4 2 2 3 3 3 2 2 2
--R b e x + (4a b e + b d)x + (6a b e + 4a b d)x + (4a b e + 6a b d)x
--R +
--R 4 3 4
--R (a e + 4a b d)x + a d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 759
--S 760 of 764
r0:=-2*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(a^2+2*a*b*x+b^2*x^2)^(5/2)*_
sqrt(d+e*x))+1/4*(8*b*B*d-9*A*b*e+a*B*e)*(a+b*x)*sqrt(d+e*x)/(e*_
(b*d-a*e)^2*(a^2+2*a*b*x+b^2*x^2)^(5/2))-7/24*(8*b*B*d-9*A*b*e+_
a*B*e)*sqrt(d+e*x)/((b*d-a*e)^3*(a^2+2*a*b*x+b^2*x^2)^(3/2))+_
35/96*e*(8*b*B*d-9*A*b*e+a*B*e)*(a+b*x)*sqrt(d+e*x)/((b*d-a*e)^4*_
(a^2+2*a*b*x+b^2*x^2)^(3/2))+35/64*e^3*(8*b*B*d-9*A*b*e+a*B*e)*_
(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))/((b*d-a*e)^(11/2)*_
sqrt(b)*sqrt(a^2+2*a*b*x+b^2*x^2))-35/64*e^2*(8*b*B*d-9*A*b*e+_
a*B*e)*sqrt(d+e*x)/((b*d-a*e)^5*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 4 4 5 3 4
--R ((945A b - 105B a b )e - 840B b d e )x
--R +
--R 4 2 3 4 4 3 3
--R ((3780A a b - 420B a b )e - 3360B a b d e )x
--R +
--R 2 3 3 2 4 2 3 3 2
--R ((5670A a b - 630B a b )e - 5040B a b d e )x
--R +
--R 3 2 4 4 3 2 3
--R ((3780A a b - 420B a b)e - 3360B a b d e )x
--R +
--R 4 5 4 4 3
--R (945A a b - 105B a )e - 840B a b d e
--R *
--R +-+ +-------+
--R +-------+ \|b \|e x + d
--R \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 4 3 4 4 3 4
--R ((- 945A b + 105B a b )e + 840B b d e )x
--R +
--R 3 2 2 4 4 3 3
--R (- 3465A a b + 385B a b )e + (- 315A b + 3115B a b )d e
--R +
--R 4 2 2
--R 280B b d e
--R *
--R 3
--R x
--R +
--R 2 2 3 4 3 2 2 3
--R (- 4599A a b + 511B a b)e + (- 1197A a b + 4221B a b )d e
--R +
--R 4 3 2 2 4 3
--R (126A b + 1050B a b )d e - 112B b d e
--R *
--R 2
--R x
--R +
--R 3 4 4 2 2 3 3
--R (- 2511A a b + 279B a )e + (- 1665A a b + 2417B a b)d e
--R +
--R 3 2 2 2 2 4 3 3 4 4
--R (468A a b + 1428B a b )d e + (- 72A b - 408B a b )d e + 64B b d
--R *
--R x
--R +
--R 4 4 3 4 3 2 2 3 2 2
--R - 384A a e + (- 975A a b + 663B a )d e + (630A a b + 370B a b)d e
--R +
--R 3 2 2 3 4 3 4
--R (- 264A a b - 104B a b )d e + (48A b + 16B a b )d
--R *
--R +-----------+ +-+
--R \|- a e + b d \|b
--R /
--R 5 3 5 4 4 4 3 5 2 3 2 6 3 2 7 4
--R 192a b e - 960a b d e + 1920a b d e - 1920a b d e + 960a b d e
--R +
--R 8 5
--R - 192b d
--R *
--R 3
--R x
--R +
--R 6 2 5 5 3 4 4 4 2 3 3 5 3 2
--R 576a b e - 2880a b d e + 5760a b d e - 5760a b d e
--R +
--R 2 6 4 7 5
--R 2880a b d e - 576a b d
--R *
--R 2
--R x
--R +
--R 7 5 6 2 4 5 3 2 3 4 4 3 2
--R 576a b e - 2880a b d e + 5760a b d e - 5760a b d e
--R +
--R 3 5 4 2 6 5
--R 2880a b d e - 576a b d
--R *
--R x
--R +
--R 8 5 7 4 6 2 2 3 5 3 3 2 4 4 4
--R 192a e - 960a b d e + 1920a b d e - 1920a b d e + 960a b d e
--R +
--R 3 5 5
--R - 192a b d
--R *
--R +------------------+
--R +-----------+ +-+ +-------+ | 2 2 2
--R \|- a e + b d \|b \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 760
--S 761 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 761
)clear all
--S 762 of 764
t0:=(A+B*x)/((d+e*x)^(5/2)*(a^2+2*a*b*x+b^2*x^2)^(5/2))
--R
--R
--R (1)
--R B x + A
--R /
--R 4 2 6 3 2 4 5 2 2 2 3 4 2 4
--R b e x + (4a b e + 2b d e)x + (6a b e + 8a b d e + b d )x
--R +
--R 3 2 2 2 3 2 3 4 2 3 2 2 2 2
--R (4a b e + 12a b d e + 4a b d )x + (a e + 8a b d e + 6a b d )x
--R +
--R 4 3 2 4 2
--R (2a d e + 4a b d )x + a d
--R *
--R +------------------+
--R +-------+ | 2 2 2
--R \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 762
--S 763 of 764
r0:=-2/3*(B*d-A*e)*(a+b*x)/(e*(b*d-a*e)*(d+e*x)^(3/2)*(a^2+2*a*b*x+_
b^2*x^2)^(5/2))+1/12*(8*b*B*d-11*A*b*e+3*a*B*e)*(a+b*x)/(e*(b*d-_
a*e)^2*(a^2+2*a*b*x+b^2*x^2)^(5/2)*sqrt(d+e*x))+1/8*(-8*b*B*d+11*A*b*e-_
3*a*B*e)/((b*d-a*e)^3*(a^2+2*a*b*x+b^2*x^2)^(3/2)*sqrt(d+e*x))+_
7/32*e*(8*b*B*d-11*A*b*e+3*a*B*e)*(a+b*x)/((b*d-a*e)^4*(a^2+2*a*b*x+_
b^2*x^2)^(3/2)*sqrt(d+e*x))+105/64*e^3*(8*b*B*d-11*A*b*e+3*a*B*e)*_
(a+b*x)*atanh(sqrt(b)*sqrt(d+e*x)/sqrt(b*d-a*e))*sqrt(b)/((b*d-_
a*e)^(13/2)*sqrt(a^2+2*a*b*x+b^2*x^2))-35/64*e^2*(8*b*B*d-11*A*b*e+_
3*a*B*e)/((b*d-a*e)^5*sqrt(d+e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))-_
105/64*e^3*(8*b*B*d-11*A*b*e+3*a*B*e)*(a+b*x)/((b*d-a*e)^6*sqrt(d+_
e*x)*sqrt(a^2+2*a*b*x+b^2*x^2))
--R
--R
--R (2)
--R 5 4 5 5 4 5
--R ((- 3465A b + 945B a b )e + 2520B b d e )x
--R +
--R 4 2 3 5 5 4 4
--R (- 13860A a b + 3780B a b )e + (- 3465A b + 11025B a b )d e
--R +
--R 5 2 3
--R 2520B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 5
--R (- 20790A a b + 5670B a b )e
--R +
--R 4 2 3 4 4 2 3
--R (- 13860A a b + 18900B a b )d e + 10080B a b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 5 2 3 3 2 4
--R (- 13860A a b + 3780B a b)e + (- 20790A a b + 15750B a b )d e
--R +
--R 2 3 2 3
--R 15120B a b d e
--R *
--R 2
--R x
--R +
--R 4 5 5 3 2 4 4
--R (- 3465A a b + 945B a )e + (- 13860A a b + 6300B a b)d e
--R +
--R 3 2 2 3
--R 10080B a b d e
--R *
--R x
--R +
--R 4 5 4 4 2 3
--R (- 3465A a b + 945B a )d e + 2520B a b d e
--R *
--R +-+ +-------+
--R +-+ +-------+ \|b \|e x + d
--R \|b \|e x + d atanh(--------------)
--R +-----------+
--R \|- a e + b d
--R +
--R 5 4 5 5 4 5
--R ((3465A b - 945B a b )e - 2520B b d e )x
--R +
--R 4 2 3 5 5 4 4
--R (12705A a b - 3465B a b )e + (4620A b - 10500B a b )d e
--R +
--R 5 2 3
--R - 3360B b d e
--R *
--R 4
--R x
--R +
--R 2 3 3 2 5 4 2 3 4
--R (16863A a b - 4599B a b )e + (17094A a b - 16926B a b )d e
--R +
--R 5 4 2 3 5 3 2
--R (693A b - 12621B a b )d e - 504B b d e
--R *
--R 3
--R x
--R +
--R 3 2 4 5 2 3 3 2 4
--R (9207A a b - 2511B a b)e + (22968A a b - 12960B a b )d e
--R +
--R 4 2 3 2 3 5 4 3 2
--R (2673A a b - 17433B a b )d e + (- 198A b - 1890B a b )d e
--R +
--R 5 4
--R 144B b d e
--R *
--R 2
--R x
--R +
--R 4 5 5 3 2 4 4
--R (1408A a b - 384B a )e + (12782A a b - 4510B a b)d e
--R +
--R 2 3 3 2 2 3 4 2 3 3 2
--R (3795A a b - 10331B a b )d e + (- 748A a b - 2556B a b )d e
--R +
--R 5 4 4 5 5
--R (88A b + 520B a b )d e - 64B b d
--R *
--R x
--R +
--R 5 5 4 5 4
--R - 128A a e + (2048A a b - 256B a )d e
--R +
--R 3 2 4 2 3 2 3 3 2 3 2
--R (2295A a b - 2639B a b)d e + (- 1030A a b - 690B a b )d e
--R +
--R 4 2 3 4 5 4 5
--R (328A a b + 136B a b )d e + (- 48A b - 16B a b )d
--R *
--R +-----------+
--R \|- a e + b d
--R /
--R 6 3 7 5 4 6 4 5 2 5 3 6 3 4
--R 192a b e - 1152a b d e + 2880a b d e - 3840a b d e
--R +
--R 2 7 4 3 8 5 2 9 6
--R 2880a b d e - 1152a b d e + 192b d e
--R *
--R 4
--R x
--R +
--R 7 2 7 6 3 6 5 4 2 5 4 5 3 4
--R 576a b e - 3264a b d e + 7488a b d e - 8640a b d e
--R +
--R 3 6 4 3 2 7 5 2 8 6 9 7
--R 4800a b d e - 576a b d e - 576a b d e + 192b d
--R *
--R 3
--R x
--R +
--R 8 7 7 2 6 6 3 2 5 5 4 3 4
--R 576a b e - 2880a b d e + 5184a b d e - 2880a b d e
--R +
--R 4 5 4 3 3 6 5 2 2 7 6 8 7
--R - 2880a b d e + 5184a b d e - 2880a b d e + 576a b d
--R *
--R 2
--R x
--R +
--R 9 7 8 6 7 2 2 5 6 3 3 4 5 4 4 3
--R 192a e - 576a b d e - 576a b d e + 4800a b d e - 8640a b d e
--R +
--R 4 5 5 2 3 6 6 2 7 7
--R 7488a b d e - 3264a b d e + 576a b d
--R *
--R x
--R +
--R 9 6 8 2 5 7 2 3 4 6 3 4 3 5 4 5 2
--R 192a d e - 1152a b d e + 2880a b d e - 3840a b d e + 2880a b d e
--R +
--R 4 5 6 3 6 7
--R - 1152a b d e + 192a b d
--R *
--R +------------------+
--R +-----------+ +-------+ | 2 2 2
--R \|- a e + b d \|e x + d \|b x + 2a b x + a
--R Type: Expression(Integer)
--E 763
--S 764 of 764
d0:=normalize(t0-D(r0,x))
--R
--R
--R (3) 0
--R Type: Expression(Integer)
--E 764
)spool
)lisp (bye)
\end{chunk}
\eject
\begin{thebibliography}{99}
\bibitem{1} nothing
\end{thebibliography}
\end{document}
|
http://lierdakil.github.io/pandoc-crossref/demo/output-listings.latex
|
github.io
|
CC-MAIN-2022-21
|
application/x-latex
|
application/x-latex
|
crawl-data/CC-MAIN-2022-21/segments/1652662521152.22/warc/CC-MAIN-20220518052503-20220518082503-00019.warc.gz
| 35,356,267 | 2,670 |
This is a demo file for pandoc-crossref. With this filter, you can
cross-reference figures (see
figs.~\ref{fig:figure1}, \ref{fig:figure2}, \ref{fig:figure3}), display
equations (see eq.~\ref{eq:eqn1}), tables (see tbl.~\ref{tbl:table1})
and sections
(secs.~\ref{sec:sec1}, \ref{sec:sec2}, \ref{sec:caption-attr}, \ref{sec:table-capts}, \ref{sec:wrapping-div})
For immediate example, see fig.~\ref{fig:figure0}
\hypertarget{fig:figure0}{%
\begin{figure}
\centering
\includegraphics{img1.jpg}
\caption{A figure}\label{fig:figure0}
\end{figure}
}
There is also support for code blocks, for example,
lsts.~\ref{lst:captionAttr}, \ref{lst:tableCaption}, \ref{lst:wrappingDiv}
It's possible to capitalize reference prefixes, like this:
Fig.~\ref{fig:figure1}.
In case of multiple references, capitalization is determined by first
reference. Figs.~\ref{fig:figure1}, \ref{fig:figure2} is capitalized,
while figs.~\ref{fig:figure2}, \ref{fig:figure1} is not.
It is also possible to mix different references, like
fig.~\ref{fig:figure1}, tbl.~\ref{tbl:table1},
lsts.~\ref{lst:captionAttr}, \ref{lst:tableCaption},
figs.~\ref{fig:figure2}, \ref{fig:figure3}, which will be grouped in
order they are specified. You can even intermix this with regular
citations, although it's not recommended: fig.~\ref{fig:figure1},
tbl.~\ref{tbl:table1}, {[}@unprocessedCitation{]}
You can also have custom chapter reference labels, like
sec.~\ref{sec:custlabs}
Subfigures are supported, see
figs.~\ref{fig:subfigures}, \ref{fig:subfigureB}
\hypertarget{sec:sec1}{%
\section{Chapter 1. Figures}\label{sec:sec1}}
\hypertarget{fig:figure1}{%
\begin{figure}
\centering
\includegraphics{img1.jpg}
\caption{First figure}\label{fig:figure1}
\end{figure}
}
\hypertarget{fig:figure2}{%
\begin{figure}
\centering
\includegraphics{img2.jpg}
\caption{Second figure}\label{fig:figure2}
\end{figure}
}
\hypertarget{fig:figure3}{%
\begin{figure}
\centering
\includegraphics{img3.jpg}
\caption{Third figure}\label{fig:figure3}
\end{figure}
}
\begin{figure}
\centering
\includegraphics{img1.jpg}
\caption{Unlabelled image}
\end{figure}
\begin{figure}
\centering
\subfloat[Subfigure a]{\includegraphics{img1.jpg}}
\subfloat[Subfigure b]{\includegraphics{img1.jpg}\label{fig:subfigureB}}
\caption{Subfigures caption}
\label{fig:subfigures}
\end{figure}
\hypertarget{sec:sec2}{%
\section{Chapter 2. Equations}\label{sec:sec2}}
Display equations are labelled and numbered
\begin{equation} P_i(x) = \sum_i a_i x^i \label{eq:eqn1}\end{equation}
Since 0.1.6.0 those can also appear in the middle of paragraph
\begin{equation}a x^2 + b x^2 + c = 0\label{eq:quadr}\end{equation} like
this.
\hypertarget{sec:chapter-3.-tables}{%
\section{Chapter 3. Tables}\label{sec:chapter-3.-tables}}
\hypertarget{tbl:table1}{}
\begin{longtable}[]{@{}ll@{}}
\caption{\label{tbl:table1}Table example}\tabularnewline
\toprule
First Header & Second Header\tabularnewline
\midrule
\endfirsthead
\toprule
First Header & Second Header\tabularnewline
\midrule
\endhead
Content Cell & Content Cell\tabularnewline
Content Cell & Content Cell\tabularnewline
\bottomrule
\end{longtable}
Table without caption:
\begin{longtable}[]{@{}ll@{}}
\toprule
First Header & Second Header\tabularnewline
\midrule
\endhead
Content Cell & Content Cell\tabularnewline
Content Cell & Content Cell\tabularnewline
\bottomrule
\end{longtable}
\hypertarget{sec:chapter-4.-code-blocks}{%
\section{Chapter 4. Code blocks}\label{sec:chapter-4.-code-blocks}}
There are a couple options for code block labels. Those work only if
code block id starts with \passthrough{\lstinline!lst:!}, e.g.
\passthrough{\lstinline!\{#lst:label\}!}
\hypertarget{sec:caption-attr}{%
\subsection{\texorpdfstring{\texttt{caption}
attribute}{caption attribute}}\label{sec:caption-attr}}
\passthrough{\lstinline!caption!} attribute will be treated as code
block caption. If code block has both id and
\passthrough{\lstinline!caption!} attributes, it will be treated as
numbered code block.
\begin{lstlisting}[language=Haskell, caption={Listing caption}, label=lst:captionAttr]
main :: IO ()
main = putStrLn "Hello World!"
\end{lstlisting}
\pagebreak
\hypertarget{sec:table-capts}{%
\subsection{Table-style captions}\label{sec:table-capts}}
Enabled with \passthrough{\lstinline!codeBlockCaptions!} metadata
option. If code block is immediately adjacent to paragraph, starting
with \passthrough{\lstinline!Listing:!} or \passthrough{\lstinline!:!},
said paragraph will be treated as code block caption.
\begin{lstlisting}[language=Haskell, caption={Listing caption}, label=lst:tableCaption]
main :: IO ()
main = putStrLn "Hello World!"
\end{lstlisting}
\hypertarget{sec:wrapping-div}{%
\subsection{Wrapping div}\label{sec:wrapping-div}}
Wrapping code block without label in a div with id
\passthrough{\lstinline!lst:...!} and class, starting with
\passthrough{\lstinline!listing!}, and adding paragraph before code
block, but inside div, will treat said paragraph as code block caption.
\begin{lstlisting}[language=Haskell, caption={Listing caption}, label=lst:wrappingDiv]
main :: IO ()
main = putStrLn "Hello World!"
\end{lstlisting}
\hypertarget{sec:unnumbered-chapter.}{%
\section*{Unnumbered chapter.}\label{sec:unnumbered-chapter.}}
\addcontentsline{toc}{section}{Unnumbered chapter.}
This chapter doesn't change chapter prefix of referenced elements,
instead keeping number of previous chapter, e.g.
\begin{equation} S(x) = \int_{x_1}^{x_2} a x+b \ \mathrm{d}x \label{eq:eqn2}\end{equation}
\hypertarget{sec:chapter-5.-reference-lists}{%
\section{Chapter 5. Reference
lists}\label{sec:chapter-5.-reference-lists}}
It's also possible to show lists of figures and tables, like this:
\listoffigures
\listoftables
\listoflistings
\hypertarget{sec:appendix-a.-custom-labels}{%
\section{Appendix A. Custom
labels}\label{sec:appendix-a.-custom-labels}}
\hypertarget{sec:custlabs}{%
\subsection{This section will have custom label}\label{sec:custlabs}}
|
http://porocila.imfm.si/2016/mat/clani/pavesic.tex
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imfm.si
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CC-MAIN-2020-29
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| 113,902,667 | 2,346 |
\clan {Petar Pavešić}
%--------------------------------------------------------
% A. objavljene znanstvene monografije
%--------------------------------------------------------
%\begin{skupina}{A}
%\disertacija
% {NASLOV}
% {UNIVERZA}
% {FAKULTETA}
% {ODDELEK}
% {KRAJ} {DRZAVA} {LETO}
%\magisterij
% {NASLOV}
% {UNIVERZA}
% {FAKULTETA}
% {ODDELEK}
% {KRAJ} {DRZAVA} {LETO}
%\monografija
% {AVTORJI}
% {NASLOV}
% {ZALOZBA}
% {KRAJ} {DRZAVA} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% B. raziskovalni clanki sprejeti v objavo v znanstvenih
% revijah in v zbornikih konferenc
%--------------------------------------------------------
%\begin{skupina}{B}
%\sprejetoRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA}
%\sprejetoZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% C. raziskovalni clanki objavljeni v znanstvenih revijah
% in v zbornikih konferenc
%--------------------------------------------------------
\begin{skupina}{C}
\objavljenoRevija
{D.~Franetič, \crta}
{Loop near-rings and unique decompositions of H-spaces}
{Algebr. Geom. Topol.} {16} {2016} {} {3563–3580}
%\objavljenoZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
% {ZBORNIK} {STRANI}
\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% D. urednistvo v znanstvenih revijah in zbornikih
% znanstvenih konferenc
%--------------------------------------------------------
%\begin{skupina}{D}
%\urednikRevija
% {OPIS}
% {REVIJA}
%\urednikZbornik
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% E. organizacija mednarodnih in domacih znanstvenih
% srecanj
%--------------------------------------------------------
%\begin{skupina}{E}
%\organizacija
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% F. vabljena predavanja na tujih ustanovah in
% mednarodnih konferencah
%--------------------------------------------------------
%\begin{skupina}{F}
%\predavanjeUstanova
% {NASLOV}
% {OPIS}
% {USTANOVA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\predavanjeKonferenca
% {NASLOV}
% {OPIS}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
\begin{skupina}{F}
\predavanjeUstanova
% 3.14: {\bf 1}. Mathematical Colloquim, Mathematical Institute of the Serbian Academy of Sciences and Arts, PAVE\v{S}I\'{C}, Petar
%{\it Complexity of the forward kinematic map : Beograd, 14. 6. 2016}. $[$COBISS.SI-ID 17731161$]$\\
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8353/150: Ne najdem podatka {KONFERENCA}
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8353/151: Ne najdem podatka {KRAJ}
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8353/152: Ne najdem podatka {DRZAVA}
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8353/153: Ne najdem podatka {MESEC}
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8353/154: Ne najdem podatka {LETO}
{Complexity of the forward kinematic map}
{seminarsko predavanje}
{Ma\-te\-ma\-tič\-ki institut SANU}
{Beograd} {Srbija} {junij} {2016}
\predavanjeUstanova
% 3.14: %list {\bf 2}. Cannon Seminar, Department of Mathematics, Brigham Young University, PAVE\v{S}I\'{C}, Petar
%{\it Manipulation of complexity of spidery linkages : Provo, 21. 9. 2016}. $[$COBISS.SI-ID 17790041$]$\\
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8359/137: Ne najdem podatka {KONFERENCA}
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8359/138: Ne najdem podatka {KRAJ}
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8359/139: Ne najdem podatka {DRZAVA}
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8359/140: Ne najdem podatka {MESEC}
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8359/141: Ne najdem podatka {LETO}
{CManipulation of complexity of spidery linkages}
{seminarsko predavanje}
{Brigham Young University}
{Provo} {ZDA} {september} {2016}
\end{skupina}
%--------------------------------------------------------
% G. aktivne udelezbe na mednarodnih in domacih
% konferencah
%--------------------------------------------------------
%\begin{skupina}{G}
%\konferenca
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{skupina}
\begin{skupina}{G}
\konferenca
% 3.15: {\bf 3}. Mini-Workshop: Topological Complexity and Related Topics, Mathematisches Forschungsinstitut Oberwolfach, 28 Feb. - 5 Mar. 2016,
%PAVE\v{S}I\'{C}, Petar{\it Remarks on the topological complexity of a map : $[$Oberwolfach, 3. 3. 2016$]$}. $[$COBISS.SI-ID 17634137$]$\\
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8370/181: Ne najdem podatka {NASLOV}
{Remarks on the topological complexity of a map}
{Mini-Workshop: Topological Complexity and Related Topics}
{Mathematisches Forschungsinstitut Oberwolfach} {Nemčija} {februar/marec} {2016}
\konferenca
% 3.15: %list {\bf 4}. Mini-Workshop: Topological Complexity and Related Topics, Mathematisches Forschungsinstitut Oberwolfach, 28 Feb. - 5 Mar. 2016,
%PAVE\v{S}I\'{C}, Petar{\it A topologist's view of kinematic maps : $[$Oberwolfach, 29. 2. 2016$]$}. $[$COBISS.SI-ID 17633881$]$\\
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8377/189: Ne najdem podatka {NASLOV}
{A topologist's view of kinematic maps}
{Mini-Workshop: Topological Complexity and Related Topics}
{Mathematisches Forschungsinstitut Oberwolfach} {Nemčija} {februar/marec} {2016}
\end{skupina}
%--------------------------------------------------------
% H. strokovni clanki
%--------------------------------------------------------
%\begin{skupina}{H}
%\clanekRevija
% {AVTORJI}
% {NASLOV}
% {REVIJA} {LETNIK} {LETO} {STEVILKA} {STRANI}
%\clanekZbornik
% {AVTORJI}
% {NASLOV}
% {KONFERENCA}
% {KRAJ} {DRZAVA} {MESEC} {LETO}
% {ZBORNIK} {STRANI}
%\end{skupina}
% Ni podatkov za to sekcijo
%--------------------------------------------------------
% I. razno
%--------------------------------------------------------
\begin{skupina}{I}
\razno
{\v{C}lan uredni\v{s}kega odbora revije {\it Obzornik za matematiko in fiziko}\phantom{}}
\end{skupina}
%\begin{skupina}{I}
%POZOR: Bibliografija2016.tex > 2016\mat\clani\pavesic.tex 8392/230: Stevilo neopredeljenih zadetkov: 1
%\razno
% Ured: {\bf 5}. {\it Obzornik za matematiko in fiziko}. Pave\v{s}i\'{c}, Petar (\v{c}lan uredni\v{s}kega odbora 2006-). Ljubljana:
%Dru\v{s}tvo matematikov, fizikov in astronomov Slovenije, 1951-. ISSN 0473-7466. $[$COBISS.SI-ID 753412$]$\\
%\end{skupina}
%--------------------------------------------------------
% tuji gosti
%--------------------------------------------------------
%\begin{seznam}
%\gost {IME} {TRAJANJE} {USTANOVA} {KRAJ} {DRZAVA} {MESEC} {LETO} {POVABILO}
%\end{seznam}
%--------------------------------------------------------
% gostovanja
%--------------------------------------------------------
%\begin{seznam}
%\gostovanje {IME} {TRAJANJE} {USTANOVA} {KRAJ} {DRZAVA} {MESEC} {LETO}
%\end{seznam}
|
http://refbase.cvc.uab.es/search.php?sqlQuery=SELECT%20author%2C%20title%2C%20type%2C%20year%2C%20publication%2C%20abbrev_journal%2C%20volume%2C%20issue%2C%20pages%2C%20keywords%2C%20abstract%2C%20thesis%2C%20editor%2C%20publisher%2C%20place%2C%20abbrev_series_title%2C%20series_title%2C%20series_editor%2C%20series_volume%2C%20series_issue%2C%20edition%2C%20language%2C%20author_count%2C%20online_publication%2C%20online_citation%2C%20doi%2C%20serial%20FROM%20refs%20WHERE%20serial%20%3D%201571%20ORDER%20BY%20first_author%2C%20author_count%2C%20author%2C%20year%2C%20title&client=&formType=sqlSearch&submit=Cite&viewType=&showQuery=0&showLinks=1&showRows=15&rowOffset=&wrapResults=1&citeOrder=&citeStyle=Chicago&exportFormat=RIS&exportType=html&exportStylesheet=&citeType=LaTeX&headerMsg=
|
uab.es
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crawl-data/CC-MAIN-2022-40/segments/1664030331677.90/warc/CC-MAIN-20220924151538-20220924181538-00448.warc.gz
| 45,584,225 | 1,350 |
%&LaTeX
\documentclass{article}
\usepackage[utf8]{inputenc}
\usepackage[T1]{fontenc}
\usepackage{textcomp}
\begin{document}
\begin{thebibliography}{1}
\bibitem{1571} Josep Llados, Enric Marti, \& Jordi Regincos. (1993).\textbf{" Interpretaci{\'o}n de dise{\~n}os a mano alzada como t{\'e}cnica de entrada a un sistema CAD en un {\'a}mbito de arquitectura" } In \textit{III National Conference on Computer Graphics}. Granada.
\end{thebibliography}
\end{document}
|
https://gapil.gnulinux.it/gitweb/?p=gapil.git;a=blob_plain;f=trasplayer.tex;hb=406973e35011347c3812c671511ce738378a525b
|
gnulinux.it
|
CC-MAIN-2020-50
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text/x-tex
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text/x-matlab
|
crawl-data/CC-MAIN-2020-50/segments/1606141746320.91/warc/CC-MAIN-20201205044004-20201205074004-00498.warc.gz
| 286,412,729 | 1,881 |
%% tcpprot.tex
%%
%% Copyright (C) 2002-2004 Simone Piccardi. Permission is granted to
%% copy, distribute and/or modify this document under the terms of the GNU Free
%% Documentation License, Version 1.1 or any later version published by the
%% Free Software Foundation; with the Invariant Sections being "Un preambolo",
%% with no Front-Cover Texts, and with no Back-Cover Texts. A copy of the
%% license is included in the section entitled "GNU Free Documentation
%% License".
%%
\chapter{Il livello di trasporto}
\label{cha:transport_layer}
In questa appendice tratteremo i vari protocolli relativi al livello di
trasporto.\footnote{al solito per la definizione dei livelli si faccia
riferimento alle spiegazioni fornite in sez.~\ref{sec:net_protocols}.} In
particolare gran parte del capitolo sarà dedicato al più importante di questi,
il TCP, che è pure il più complesso ed utilizzato su internet.
\section{Il protocollo TCP}
\label{sec:tcp_protocol}
In questa sezione prenderemo in esame i vari aspetti del protocollo TCP, il
protocollo più comunemente usato dalle applicazioni di rete.
\subsection{Gli stati del TCP}
\label{sec:TCP_states}
In sez.~\ref{sec:TCP_connession} abbiamo descritto in dettaglio le modalità con
cui il protocollo TCP avvia e conclude una connessione, ed abbiamo accennato
alla presenza dei vari stati del protocollo. In generale infatti il
funzionamento del protocollo segue una serie di regole, che possono essere
riassunte nel comportamento di una macchina a stati, il cui diagramma di
transizione è riportato in fig.~\ref{fig:TCP_state_diag}.
\begin{figure}[htb]
\centering
\includegraphics[width=10cm]{img/tcp_state_diagram}
\caption{Il diagramma degli stati del TCP.}
\label{fig:TCP_state_diag}
\end{figure}
Il protocollo prevede l'esistenza di 11 diversi stati per una connessione ed
un insieme di regole per le transizioni da uno stato all'altro basate sullo
stato corrente, sull'operazione effettuata dall'applicazione o sul tipo di
segmento ricevuto; i nomi degli stati mostrati in
fig.~\ref{fig:TCP_state_diag} sono gli stessi che vengono riportati del
comando \cmd{netstat} nel campo \textit{State}.
\section{Il protocollo UDP}
\label{sec:udp_protocol}
In questa sezione prenderemo in esame i vari aspetti del protocollo UDP, che
dopo il TCP è il protocollo più usato dalle applicazioni di rete.
%%% Local Variables:
%%% mode: latex
%%% TeX-master: "gapil"
%%% End:
|
https://www.zentralblatt-math.org/matheduc/en/?id=149745&type=tex
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| 1,087,562,460 | 1,146 |
\input zb-basic
\input zb-matheduc
\iteman{ZMATH 1982b.05990}
\itemau{Paetzolt, R. (Staatliche Berufsschule Wiesau (Germany, F.R.))}
\itemti{The computer in teaching classes of apprentices for metal working vocations. Der Computer im Unterricht fuer Metallklassen.}
\itemso{BUS. (May 1981) (no.4) p. 11.}
\itemab
\itemrv{~}
\itemcc{R27 P57}
\itemut{Experience Reports; Part-Time Vocational Schools; Gewerblich-Technische Fachrich; Vocational Drawing; Vocational Mathematics; Basic; Computer Programming; Mathematical Applications; Computers; Automatisierung; ; Erfahrungsbericht; Berufsschule; Gewerblich-Technische Fachrich; Fachzeichnen; Fachmathematik; Basic; Programmieren; Anwendungen der Mathematik; Computer; Automatisierung}
\itemli{}
\end
|
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