% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% The Project Gutenberg EBook of The Integration of Functions of a Single %
% Variable, by G. H. Hardy                                                %
%                                                                         %
% 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.org                          %
%                                                                         %
%                                                                         %
% Title: The Integration of Functions of a Single Variable                %
%                                                                         %
% Author: G. H. Hardy                                                     %
%                                                                         %
% Editor: P. Hall                                                         %
%         F. Smithies                                                     %
%                                                                         %
% Release Date: September 23, 2012 [EBook #38993]                         %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: ISO-8859-1                                      %
%                                                                         %
% *** START OF THIS PROJECT GUTENBERG EBOOK INTEGRATION OF FUNCTIONS OF ONE VARIABLE ***
%                                                                         %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %

\def\ebook{38993}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%                                                                  %%
%% Packages and substitutions:                                      %%
%%                                                                  %%
%% book:     Required.                                              %%
%% inputenc: Standard DP encoding. Required.                        %%
%%                                                                  %%
%% ifthen:   Logical conditionals. Required.                        %%
%%                                                                  %%
%% amsmath:  AMS mathematics enhancements. Required.                %%
%% amssymb:  Additional mathematical symbols. Required.             %%
%%                                                                  %%
%% alltt:    Fixed-width font environment. Required.                %%
%%                                                                  %%
%% footmisc: Extended footnote capabilities. Required.              %%
%% perpage:  Start footnote numbering on each page. Required.       %%
%%                                                                  %%
%% fancyhdr: Enhanced running headers and footers. Required.        %%
%%                                                                  %%
%% graphicx: Standard interface for graphics inclusion. Required.   %%
%%                                                                  %%
%% geometry: Enhanced page layout package. Required.                %%
%% hyperref: Hypertext embellishments for pdf output. Required.     %%
%%                                                                  %%
%%                                                                  %%
%% Producer's Comments:                                             %%
%%                                                                  %%
%%   Changes are noted in this file in three ways.                  %%
%%   1. \DPnote{} for in-line `placeholder' notes.                  %%
%%   2. \DPtypo{}{} for typographical corrections, showing original %%
%%      and replacement text side-by-side.                          %%
%%   3. [** PP: Note]s for lengthier or stylistic comments.         %%
%%                                                                  %%
%%                                                                  %%
%% Compilation Flags:                                               %%
%%                                                                  %%
%%   The following behavior may be controlled by boolean flags.     %%
%%                                                                  %%
%%   ForPrinting (false by default):                                %%
%%   Compile a screen-optimized PDF file. Set to false for print-   %%
%%   optimized file (pages cropped, one-sided, blue hyperlinks).    %%
%%                                                                  %%
%%                                                                  %%
%% Things to Check:                                                 %%
%%                                                                  %%
%%                                                                  %%
%% Spellcheck: .................................. OK                %%
%% Smoothreading pool: ......................... yes                %%
%%                                                                  %%
%% PDF pages: 86 (if ForPrinting set to true)                       %%
%%                                                                  %%
%% Command block:                                                   %%
%%                                                                  %%
%%     pdflatex x2 #(Run pdflatex two times)                        %%
%%                                                                  %%
%%                                                                  %%
%% September 2012: pglatex.                                         %%
%%   Compile this project with:                                     %%
%%   pdflatex 38993-t.tex ..... TWO times                           %%
%%                                                                  %%
%%   pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian)       %%
%%                                                                  %%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\listfiles

\documentclass[12pt]{book}
\usepackage[latin1]{inputenc}[2008/03/30]
\usepackage{ifthen}[2001/05/26]
\usepackage{amsmath}[2000/07/18]
\usepackage{amssymb}[2009/06/22]
\usepackage{graphicx}[1999/02/16]
\usepackage{alltt}[1997/06/16]   %% boilerplate, credits, license

\usepackage{yfonts}[2003/01/08]
\usepackage[perpage,symbol]{footmisc}[2011/06/06]

% for running heads
\usepackage{fancyhdr}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%% Interlude:  Set up PRINTING (default) or SCREEN VIEWING %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% ForPrinting=true (default)           false
% Asymmetric margins                   Symmetric margins
% Black hyperlinks                     Blue hyperlinks
% Start Preface, ToC, etc. recto       No blank verso pages
%
% Chapter-like ``Sections'' start both recto and verso in the scanned
% book. This behavior has been retained.
\newboolean{ForPrinting}

%% UNCOMMENT the next line for a PRINT-OPTIMIZED VERSION of the text %%
%\setboolean{ForPrinting}{true}

%% Initialize values to ForPrinting=false
\newcommand{\Margins}{hmarginratio=1:1}     % Symmetric margins
\newcommand{\HLinkColor}{blue}              % Hyperlink color
\newcommand{\PDFPageLayout}{SinglePage}
\newcommand{\TransNote}{Transcriber's Note}
\newcommand{\TransNoteCommon}{%
{\Loosen Minor typographical corrections and presentational changes have
been made without comment.}
  \bigskip
}

\newcommand{\TransNoteText}{%
  \TransNoteCommon

  This PDF file is optimized for screen viewing, but may easily be
  recompiled for printing. Please see the preamble of the \LaTeX\
  source file for instructions.
}
%% Re-set if ForPrinting=true
\ifthenelse{\boolean{ForPrinting}}{%
  \renewcommand{\Margins}{hmarginratio=2:3} % Asymmetric margins
  \renewcommand{\HLinkColor}{black}         % Hyperlink color
  \renewcommand{\PDFPageLayout}{TwoPageRight}
  \renewcommand{\TransNote}{Transcriber's Note}
  \renewcommand{\TransNoteText}{%
    \TransNoteCommon

    This PDF file is optimized for printing, but may easily be
    recompiled for screen viewing. Please see the preamble of the
    \LaTeX\ source file for instructions.
  }
}{% If ForPrinting=false, don't skip to recto
  \renewcommand{\cleardoublepage}{\clearpage}
}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%  End of PRINTING/SCREEN VIEWING code; back to packages  %%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\usepackage[body={5.25in,8.4in},\Margins]{geometry}[2010/09/12]

\providecommand{\ebook}{00000}    % Overridden during white-washing
\usepackage[pdftex,
  hyperfootnotes=false,
  pdftitle={The Project Gutenberg eBook \#\ebook: The Integration of Functions of a Single Variable},
  pdfauthor={Hardy, G. H.},
  pdfkeywords={Brenda Lewis, Anna Hall,
               Project Gutenberg Online Distributed Proofreading Team,
               The Internet Archive/American Libraries},
  pdfstartview=Fit,    % default value
  pdfstartpage=1,      % default value
  pdfpagemode=UseNone, % default value
  bookmarks=true,      % default value
  linktocpage=false,   % default value
  pdfpagelayout=\PDFPageLayout,
  pdfdisplaydoctitle,
  pdfpagelabels=true,
  bookmarksopen=true,
  bookmarksopenlevel=1,
  colorlinks=true,
  linkcolor=\HLinkColor]{hyperref}[2007/02/07]

% Re-crop screen-formatted version, accommodating wide displays
\ifthenelse{\boolean{ForPrinting}}
  {}
  {\hypersetup{pdfpagescrop= 85 80 527 765}}


%%%% Fixed-width environment to format PG boilerplate %%%%
\newenvironment{PGtext}{%
\begin{alltt}
\fontsize{8.1}{9}\ttfamily\selectfont}%
{\end{alltt}}

%% No hrule in page header
\renewcommand{\headrulewidth}{0pt}

% Running heads
\newcommand{\SetRunningHeads}[1]{%
  \setlength{\headheight}{30pt}
  \fancyhead[CEO]{\textit{\MakeUppercase{#1}}}

  \ifthenelse{\boolean{ForPrinting}}
             {\fancyhead[RO,LE]{\thepage}}
             {\fancyhead[R]{\thepage}}
}

% Running heads
\newcommand{\FlushRunningHeads}{\fancyhf{}}

\newcommand{\BookMark}[2]{\phantomsection\pdfbookmark[#1]{#2}{#2}}

%% Major document divisions %%
\newcommand{\PGBoilerPlate}{%
  \pagenumbering{Alph}
  \pagestyle{empty}
  \BookMark{0}{PG Boilerplate.}
}
\newcommand{\FrontMatter}{%
  \cleardoublepage
  \frontmatter
  \pagestyle{empty}
}

\newcommand{\MainMatter}{%
  \FlushRunningHeads
  \mainmatter
  \pagestyle{fancy}
  \fancyhf{}
  \thispagestyle{empty}
}

\newcommand{\BackMatter}{%
  \FlushRunningHeads
  \backmatter
  \BookMark{-1}{Back Matter.}
}

\newcommand{\PGLicense}{%
  \FlushRunningHeads
  \pagenumbering{alph}
  \clearpage
}

% Sectioning
\newcommand{\Section}[1]{%
 \FlushRunningHeads
 \SetRunningHeads{#1}
 \section*{\Heading #1}
 \phantomsection\label{#1}%
}

\newcommand{\Paragraph}[2]{\paragraph*{\indent #2}
 \phantomsection\label{#1 para #2}%
 \hypertarget{#1 para #2}{}%
}

\newcommand{\Appendix}[2]{%
  \clearpage%
  \section*{\centering\large APPENDIX #1}
  \subsection*{\centering\small\MakeUppercase{#2}}
  \phantomsection\label{Appendix #1}%
  \hypertarget{Appendix #1}{}%
}

\newcommand{\Heading}{\centering\normalfont\bfseries}
\newcommand{\Contents}[1]{%
 \section*{\Heading #1}
}

% Bibliography
\newcommand{\Author}[1]{\subsection*{\centering\normalsize #1}}

\newcommand{\Bibitem}[1][]{%
  \noindent\makebox[\parindent][l]{#1}\hangindent2\parindent\ignorespaces%
}

% Misc. formatting
\newcommand{\Item}[1]{\makebox[2em][l]{\upshape#1}}
\newcommand{\Loosen}{\spaceskip 0.375em plus 0.75em minus 0.25em}

\newenvironment{Result}{\itshape}{\par\medskip\normalfont}

% Conveniences
\newcommand{\ia}{\textit{a}}
\newcommand{\ib}{\textit{b}}
\newcommand{\ic}{\textit{c}}
\newcommand{\id}{\textit{d}}

\newcommand{\OEuvres}{\textit{\OE{}uvres}}
\newcommand{\ibid}{\textit{ibid}}

% Math
\DeclareMathOperator{\ArcSin}{arc\,sin}
\DeclareMathOperator{\ArcTan}{arc\,tan}

\renewcommand{\arcsin}{\ArcSin}
\renewcommand{\arctan}{\ArcTan}

\DeclareMathOperator{\argcosh}{arg\,cosh}
\DeclareMathOperator{\argsinh}{arg\,sinh}
\DeclareMathOperator{\argtanh}{arg\,tanh}
\DeclareMathOperator{\cosec}{cosec}
\DeclareMathOperator{\sn}{sn}

\DeclareMathOperator{\Li}{\textit{Li}\,}
\DeclareMathOperator{\Ci}{\textit{Ci}\,}
\DeclareMathOperator{\Si}{\textit{Si}\,}

\DeclareInputText{176}{\ifmmode{{}^\circ}\else\textdegree\fi}
\DeclareInputText{183}{\ifmmode\cdot\else\textperiodcentered\fi}

\newcommand{\FF}{\textgoth{F}}
\newcommand{\dd}{\partial}

\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}

\newcommand{\Tag}[1]{\tag*{\ensuremath{#1}}}
\newcommand{\Eq}[1]{\ensuremath{#1}}

% For corrections.
\newcommand{\DPtypo}[2]{#2}
\newcommand{\DPnote}[1]{}

\hfuzz=0.5pt
%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}
\PGBoilerPlate
{\small
\begin{PGtext}
The Project Gutenberg EBook of The Integration of Functions of a Single
Variable, by G. H. Hardy

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.org


Title: The Integration of Functions of a Single Variable

Author: G. H. Hardy

Editor: P. Hall
        F. Smithies

Release Date: September 23, 2012 [EBook #38993]

Language: English

Character set encoding: ISO-8859-1

*** START OF THIS PROJECT GUTENBERG EBOOK INTEGRATION OF FUNCTIONS OF ONE VARIABLE ***
\end{PGtext}
\clearpage
\begin{PGtext}
Produced by Brenda Lewis, Anna Hall and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive/American Libraries.)
\end{PGtext}}
\vfill

\BookMark{0}{Transcriber's Note}
\subsection*{\centering\normalfont\scshape
\normalsize\MakeLowercase{\TransNote}}

\TransNoteText
\clearpage

%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%%

\FrontMatter

%% -----File: 001.png---
\begin{center}
{\large Cambridge Tracts in Mathematics\\
and Mathematical Physics\\}

\smallskip

\textsc{General Editors\\
P.~HALL, F.R.S. and F.~SMITHIES, Ph.D.}

\bigskip

No.~2

\smallskip

{\LARGE THE\\
INTEGRATION OF FUNCTIONS\\
OF A SINGLE VARIABLE\\}

\smallskip

{\large BY}

{\Large G.~H. HARDY}
\vfill
{\includegraphics[width=1in]{./images/device.pdf}} %Publisher's device

{\large CAMBRIDGE UNIVERSITY PRESS\\}
%% -----File: 002.png---
%[Blank Page]
%% -----File: 003.png---
\newpage
{\Large Cambridge Tracts in Mathematics\\
and Mathematical Physics\\}

\medskip

\textsc{General Editors\\
P.~HALL, F.R.S. and F.~SMITHIES, Ph.D.}

\vfill

{\large No.~2}

\bigskip

{\large The Integration of Functions of a\\
Single Variable}

\vfill
%% -----File: 004.png---
%[Blank Page]
%% -----File: 005.png---
\newpage
{\large THE\\}
{\LARGE INTEGRATION OF FUNCTIONS\\
OF A SINGLE VARIABLE\\}

\vfill

BY

\medskip

{\large G. H. HARDY}

\vfill

SECOND EDITION

\vfill

{\large CAMBRIDGE\\
AT THE UNIVERSITY PRESS\\
1966\\}

\bigskip
%% -----File: 006.png---
\newpage
PUBLISHED BY

THE SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS

Bentley House, 200 Euston Road, London, N.W. 1\\
American Branch: 32 East 57th Street, New York, N.Y. 10022

\vfill

\begin{tabular}{l l}
  \textit{First Edition} & 1905 \\
  \textit{Second Edition} & 1916 \\
  \textit{Reprinted} & 1928 \\
    & 1958 \\
    & 1966 \\
\end{tabular}

\vfill

\textit{First printed in Great Britain at the University Press, Cambridge
Reprinted by offset-litho by Jarrold \& Sons Ltd., Norwich}
\end{center}
%% -----File: 007.png---
\newpage
\Section{PREFACE}

This tract has been long out of print, and there is still some
demand for it. I did not publish a second edition before,
because I intended to incorporate its contents in a larger treatise on
the subject which I had arranged to write in collaboration with
Dr~Bromwich. Four or five years have passed, and it seems very
doubtful whether either of us will ever find the time to carry out
our intention. I have therefore decided to republish the tract.

The new edition differs from the first in one important point
only. In the first edition I reproduced a proof of Abel's which
Mr~J.~E. Littlewood afterwards discovered to be invalid. The
correction of this error has led me to rewrite a few sections (\hyperlink{5 para 11.}{pp.~36--41}
of the present edition) completely. The proof which I give now is
due to Mr~H.~T.~J. Norton. I am also indebted to Mr~Norton,
and to Mr~S. Pollard, for many other criticisms of a less important
character.

\begin{flushright}
G. H. H.
\end{flushright}

\textit{January} 1916.
%% -----File: 008.png---
%[Blank Page]
%% -----File: 009.png---
\newpage
\Contents{CONTENTS}

\hfill \textsc{page}
\begin{description}
\item[I.] Introduction \hfill \pageref{I. Introduction}
\item[II.] Elementary functions and their classification \hfill \pageref{II. Elementary functions and their classification}
\item[III.] The integration of elementary functions. Summary of results \hfill \pageref{III. The integration of elementary functions. Summary of results}
\item[IV.] The integration of rational functions \hfill \pageref{IV. Rational functions}
\begin{description}
\item[1--3.] The method of partial fractions \hfill \pageref{4 para 1.}
\item[4.] Hermite's method of integration \hfill \pageref{4 para 4.}
\item[5.] Particular problems of integration \hfill \pageref{4 para 5.}
\item[6.] The limitations of the methods of integration \hfill \pageref{4 para 6.}
\item[7.] Conclusion \hfill \pageref{4 para 7.}
\end{description}
\item[V.] The integration of algebraical functions \hfill \pageref{V. Algebraical Functions}
\begin{description}
\item[1.] Algebraical functions \hfill \pageref{5 para 1.}
\item[2.] Integration by rationalisation. Integrals associated with
conics \hfill \pageref{5 para 2.}
\item[3--6.] The integral $\int R\{x, \sqrt{ax^{2} + 2bx + c}\}\,dx$ \hfill \pageref{5 para 3.}
\item[7.] Unicursal plane curves \hfill \pageref{5 para 7.}
\item[8.] Particular cases \hfill \pageref{5 para 8.}
\item[9.] Unicursal curves in space \hfill \pageref{5 para 9.}
\item[10.] Integrals of algebraical functions in general \hfill \pageref{5 para 10.}
\item[11--14.] The general form of the integral of an algebraical function.
Integrals which are themselves algebraical \hfill \pageref{5 para 11.}
\item[15.] Discussion of a particular case \hfill \pageref{5 para 15.}
\item[16.] The transcendence of $e^{x}$ and $\log x$ \hfill \pageref{5 para 16.}
\item[17.] Laplace's principle \hfill \pageref{5 para 17.}
\item[18.] The general form of the integral of an algebraical function
(\emph{continued}). Integrals expressible by algebraical functions
and logarithms \hfill \pageref{5 para 18.}
%% -----File: 010.png---
\item[19.] Elliptic and pseudo-elliptic integrals. Binomial integrals \hfill \pageref{5 para 19.}
\item[20.] Curves of deficiency~$1$. The plane cubic \hfill \pageref{5 para 20.}
\item[21.] Degenerate Abelian integrals \hfill \pageref{5 para 21.}
\item[22.] The classification of elliptic integrals \hfill \pageref{5 para 22.}
\end{description}
\item[VI.] The  integration of transcendental functions \hfill \pageref{VI. Transcendental functions}
\begin{description}
\item[1.] Preliminary \hfill \pageref{6 para 1.}
\item[2.] The integral $\int R (e^{ax}, e^{bx}, \dots, e^{kx})\,dx$ \hfill \pageref{6 para 2.}
\item[3.] The integral $\int P(x, e^{ax}, e^{bx}, \dots)\,dx$ \hfill \pageref{6 para 3.}
\item[4.] The integral $\int e^{x} R(x)\, dx$. The logarithm-integral \hfill \pageref{6 para 5.}
\item[5.] Liouville's general theorem \hfill \pageref{6 para 5.}
\item[6.] The integral $\int \log x R(x)\, dx$ \hfill \pageref{6 para 6.}
\item[7.] Conclusion \hfill \pageref{6 para 7.}
\end{description}
\item[Appendix I.] Bibliography \hfill \pageref{Appendix I}
\item[Appendix II.] On Abel's proof of the theorem of v., §~11 \hfill \pageref{Appendix II}
\end{description}
%% -----File: 011.png---
\MainMatter
\cleardoublepage
\Section{THE INTEGRATION OF FUNCTIONS OF
A SINGLE VARIABLE}

\Section{I. Introduction}

The problem considered in the following pages is what is sometimes
called the problem of `indefinite integration' or of `finding a function
whose differential coefficient is a given function'. These descriptions
are vague and in some ways misleading; and it is necessary to define
our problem more precisely before we proceed further.

Let us suppose for the moment that $f(x)$~is a real continuous
function of the real variable~$x$. We wish to determine a function~$y$
whose differential coefficient is~$f(x)$, or to solve the equation
\[
\frac{dy}{dx} = f(x).
\Tag{(1)}
\]
A little reflection shows that this problem may be analysed into a
number of parts.

We wish, first, to know whether such a function as~$y$ necessarily
exists, whether the equation~\Eq{(1)} has always a solution; whether the
solution, if it exists, is unique; and what relations hold between
different solutions, if there are more than one. The answers to these
questions are contained in that part of the theory of functions of a
real variable which deals with `definite integrals'. The definite
integral
\[
y = \int_{a}^{x} f(t)\,dt,
\Tag{(2)}
\]
which is defined as the limit of a certain sum, is a solution of the
equation~\Eq{(1)}. Further
\[
y + C,
\Tag{(3)}
\]
where $C$~is an arbitrary constant, is also a solution, and all solutions of~\Eq{(1)}
are of the form~\Eq{(3)}.
%% -----File: 012.png---

These results we shall take for granted. The questions with which
we shall be concerned are of a quite different character. They are
questions as to the functional form of~$y$ when $f(x)$~is a function of
some stated form. It is sometimes said that the problem of indefinite
integration is that of `finding an actual expression for~$y$ when $f(x)$~is
given'. This statement is however still lacking in precision. The theory
of definite integrals provides us not only with a proof of the existence
of a solution, but also with an expression for it, an expression in the
form of a limit. The problem of indefinite integration can be stated
precisely only when we introduce sweeping restrictions as to the classes
of functions and the modes of expression which we are considering.

Let us suppose that $f(x)$ belongs to some special class of functions~$\FF$.
Then we may ask whether $y$~is itself a member of~$\FF$, or can be
expressed, according to some simple standard mode of expression, in
terms of functions which are members of~$\FF$. To take a trivial
example, we might suppose that $\FF$~is the class of polynomials with
rational coefficients: the answer would then be that $y$~is in all cases
itself a member of~$\FF$.

The range and difficulty of our problem will depend upon our
choice of (1)~a class of functions and (2)~a standard `mode of expression'.
We shall, for the purposes of this tract, take $\FF$ to be the
class of \emph{elementary functions}, a class which will be defined precisely in
the next section, and our mode of expression to be that of \emph{explicit
expression in finite terms}, \textit{i.e.}\ by formulae which do not involve passages
to a limit.

One or two more preliminary remarks are needed. The subject-matter
of the tract forms a chapter in the `integral calculus'\footnotemark,
  \footnotetext{Euler, the first systematic writer on the `integral calculus', defined it in
  a manner which identifies it with the theory of differential equations: `calculus
  integralis est methodus, ex data differentialium relatione inveniendi relationem
  ipsarum quantitatum' (\textit{Institutiones calculi integralis}, p.~1). We are concerned
  only with the special equation~\Eq{(1)}, but all the remarks we have made may be
  generalised so as to apply to the wider theory.}%
but
does not depend in any way on any direct theory of integration. Such
an equation as
\[
y = \int f(x)\, dx
\Tag{(4)}
\]
is to be regarded as merely another way of writing~\Eq{(1)}: the integral
sign is used merely on grounds of technical convenience, and might
be eliminated throughout without any substantial change in the
argument.
%% -----File: 013.png---

The variable~$x$ is in general supposed to be complex. But the tract
should be intelligible to a reader who is not acquainted with the theory
of analytic functions and who regards~$x$ as real and the functions of~$x$
which occur as real or complex functions of a real variable.

The functions with which we shall be dealing will always be such
as are regular except for certain special values of~$x$. These values of~$x$
we shall simply ignore. The meaning of such an equation as
\[
\int \frac{dx}{x} = \log x
\]
is in no way affected by the fact that $1/x$~and~$\log x$ have infinities for
$x = 0$.

\Section{II. Elementary functions and their classification}

An \emph{elementary function} is a member of the class of functions which
comprises

\Item{(i)} rational functions,

\Item{(ii)} algebraical functions, explicit or implicit,

\Item{(iii)} the exponential function~$e^{x}$,

\Item{(iv)} the logarithmic function~$\log x$,

\Item{(v)} all functions which can be defined by means of any finite
combination of the symbols proper to the preceding four classes of
functions.

A few remarks and examples may help to elucidate this definition.

\Paragraph{2}{1.} A \emph{rational function} is a function defined by means of any finite
combination of the elementary operations of addition, multiplication,
and division, operating on the variable~$x$.

It is shown in elementary algebra that any rational function of~$x$
may be expressed in the form
\[
f(x) = \frac{a_{0}x^{m} + a_{1}x^{m-1} + \dots + a_{m}}
            {b_{0}x^{n} + b_{1}x^{n-1} + \dots + b_{n}},
\]
where $m$~and~$n$ are positive integers, the $a$'s~and~$b$'s are constants, and
the numerator and denominator have no common factor. We shall
adopt this expression as the standard form of a rational function. It
is hardly necessary to remark that it is in no way involved in the
%% -----File: 014.png---
definition of a rational function that these constants should be rational
or algebraical\footnote
  {An algebraical number is a number which is the root of an algebraical equation
  whose coefficients are integral. It is known that there are numbers (such as
  $e$~and~$\pi$) which are not roots of any such equation. See, for example, Hobson's
  \textit{Squaring the circle} (Cambridge, 1913).}
or real \emph{numbers}. Thus
\[
\frac{x^{2} + x + i \sqrt{2}}{x\sqrt{2} - e}
\]
is a rational function.

\Paragraph{2}{2.} An \emph{explicit algebraical function} is a function defined by means
of any finite combination of the four elementary operations and any
finite number of operations of root extraction. Thus
\[
\frac{\sqrt{1+x} - \sqrt[3]{1-x}}
     {\sqrt{1+x} + \sqrt[3]{1-x}},\quad
\sqrt{x +\sqrt{x + \sqrt{x}}},\quad
\left(\frac{x^{2} + x + i \sqrt{2}}{x\sqrt{2} - e}\right)^{\frac{2}{3}}
\]
are explicit algebraical functions. And so is~$x^{m/n}$ (\textit{i.e.}~$\sqrt[n]{x^{m}}$) for any
integral values of $m$~and~$n$. On the other hand
\[
x^{\sqrt{2}},\quad x^{1 + i}
\]
are not algebraical functions at all, but transcendental functions, as
irrational or complex powers are defined by the aid of exponentials
and logarithms.

Any explicit algebraical function of~$x$ satisfies an equation
\[
P_{0}y^{n} + P_{1}y^{n-1} + \dots + P_{n} = 0
\]
whose coefficients are polynomials in~$x$. Thus, for example, the
function
\[
y = \sqrt{x} +\sqrt{x +\sqrt{x}}
\]
satisfies the equation
\[
y^{4} - (4y^{2} + 4y + 1)x = 0.
\]
The converse is not true, since it has been proved that in general
equations of degree higher than the fourth have no roots which are
explicit algebraical functions of their coefficients. A simple example
is given by the equation
\[
y^{5} - y - x = 0.
\]
We are thus led to consider a more general class of functions, \emph{implicit}
algebraical functions, which includes the class of explicit algebraical
functions.
%% -----File: 015.png---

\Paragraph{2}{3.} An \emph{algebraical function} of~$x$ is a function which satisfies an
equation
\[
P_{0}y^{n} + P_{1}y^{n-1} + \dots + P_{n} = 0
\Tag{(1)}
\]
whose coefficients are polynomials in~$x$.

Let us denote by~$P(x, y)$ a polynomial such as occurs on the left-hand
side of~\Eq{(1)}. Then there are two possibilities as regards any
particular polynomial~$P(x, y)$. Either it is possible to express~$P(x, y)$
as the product of two polynomials of the same type, neither of which
is a mere constant, or it is not. In the first case $P(x, y)$~is said to
be \emph{reducible}, in the second \emph{irreducible}. Thus
\[
y^{4} - x^{2} = (y^{2} + x)(y^{2} - x)
\]
is reducible, while both $y^{2} + x$ and $y^{2} - x$ are irreducible.

The equation~\Eq{(1)} is said to be reducible or irreducible according as
its left-hand side is reducible or irreducible. A reducible equation can
always be replaced by the logical alternative of a number of irreducible
equations. Reducible equations are therefore of subsidiary importance
only; and we shall always suppose that the equation~\Eq{(1)} is irreducible.

An algebraical function of~$x$ is regular except at a finite number
of points which are \emph{poles} or \emph{branch points} of the function. Let $D$~be
any closed simply connected domain in the plane of~$x$ which does
not include any branch point. Then there are $n$~and only~$n$ distinct
functions which are one-valued in~$D$ and satisfy the equation~\Eq{(1)}.
These $n$~functions will be called the \emph{roots} of~\Eq{(1)} in~$D$. Thus if we
write
\[
x = r (\cos \theta + i \sin \theta),
\]
where $-\pi < \theta \leq \pi$, then the roots of
\[
y^{2} - x = 0,
\]
in the domain
\[
0 < r_{1} \leq r \leq r_{2},\quad
-\pi < -\pi + \delta \leq \theta \leq \pi - \delta < \pi,
\]
are $\sqrt{x}$~and~$-\sqrt{x}$, where
\[
\sqrt{x} = \sqrt{r} (\cos \tfrac{1}{2} \theta + i \sin \tfrac{1}{2} \theta).
\]

The relations which hold between the different roots of~\Eq{(1)} are of
the greatest importance in the theory of functions\footnotemark.
  \footnotetext{For fuller information the reader may be referred to Appell and Goursat's
  \textit{Théorie des fonctions algébriques}.}%
For our present
purposes we require only the two which follow.

\Item{(i)} Any symmetric polynomial in the roots~$y_{1}$, $y_{2}$,~\dots,~$y_{n}$ of~\Eq{(1)} is
a rational function of~$x$.
%% -----File: 016.png---

\Item{(ii)} Any symmetric polynomial in~$y_{2}$, $y_{3}$, \dots, $y_{n}$ is a polynomial in~$y_{1}$
with coefficients which are rational functions of~$x$.

The first proposition follows directly from the equations
\[
\sum y_{1} y_{2} \dots y_{s} = (-1)^{s}(P_{n-s} / P_{0}) \quad   (s=1, 2, \dots, n).
\]
To prove the second we observe that
\[
\sum_{2, 3, \dots} y_{2} y_{3} \dots y_{s} = \sum_{1, 2, \dots} y_{1} y_{2} \dots y_{s-1} - y_{1}\sum_{2, 3, \dots} y_{2} y_{3} \dots y_{s-1}\DPtypo{.}{,}
\]
so that the theorem is true for $\sum y_{2} y_{3} \dots y_{s}$ if it is true for $\sum y_{2} y_{3} \dots y_{s-1}$.
It is certainly true for
\[
y_{2} + y_{3} + \dots + y_{n} = (y_{1} + y_{2} + \dots + y_{n}) - y_{1}.
\]
It is therefore true for $\sum y_{2} y_{3} \dots y_{s}$, and so for any symmetric polynomial in
$y_{2}, y_{3}, \dots, y_{n}$.

\Paragraph{2}{4.} Elementary functions which are not rational or algebraical are
called \emph{elementary transcendental functions} or elementary transcendents.
They include all the remaining functions which are of ordinary occurrence
in elementary analysis.

The trigonometrical (or circular) and hyperbolic functions, direct
and inverse, may all be expressed in terms of exponential or logarithmic
functions by means of the ordinary formulae of elementary trigonometry.
Thus, for example,
\begin{gather*}
\sin x = \frac{e^{ix} - e^{-ix}}{2i},\qquad  \sinh x = \frac{e^{x} - e^{-x}}{2},\\
\arctan x =\frac{1}{2i} \log \left(\frac{1 + ix}{1 - ix}\right),\qquad  \argtanh x
=\frac{1}{2}\log \left(\frac{1 + x}{1 - x}\right).
\end{gather*}
There was therefore no need to specify them particularly in our
definition.

The elementary transcendents have been further classified in a
manner first indicated by Liouville\footnote{`Mémoire sur la classification des transcendantes, et sur l'impossibilité
d'exprimer les racines de certaines équations en fonction finie explicite des
coefficients', \textit{Journal de mathématiques}, ser.~1, vol.~2, 1837, pp.~56--104; `Suite du
mémoire\dots', \ibid.\ vol.~3, 1838, pp. 523--546.}. According to him a function is
a transcendent \emph{of the first order} if the signs of exponentiation or of
the taking of logarithms which occur in the formula which defines
it apply only to rational or algebraical functions. For example
\[
x e^{-x^{2}},\  e^{x^{2}} + e^{x}\sqrt{\log x}
\]
are of the first order; and so is
\[
\arctan \frac{y}{\sqrt{1 + x^{2}}} ,
\]
%% -----File: 017.png---
where $y$ is defined by the equation
\[
y^{5} - y - x = 0;
\]
and so is the function $y$ defined by the equation
\[
y^{5} - y - e^{x} \log x = 0.
\]

An elementary transcendent \emph{of the second order} is one defined by
a formula in which the exponentiations and takings of logarithms are
applied to rational or algebraical functions or to transcendents of the
first order. This class of functions includes many of great interest and
importance, of which the simplest are
\[
e^{e^{x}},\: \log \log x.
\]
It also includes irrational and complex powers of $x$, since, \textit{e.g.},
\[
x^{\sqrt 2} = e^{\sqrt 2 \log x},\quad x^{1+i} = e^{(1+i) \log x};
\]
the function
\[
x^{x} = e^{x \log x};
\]
and the logarithms of the circular functions.

It is of course presupposed in the definition of a transcendent of the
second kind that the function in question is incapable of expression as
one of the first kind or as a rational or algebraical function. The
function
\[
e^{\log R(x)},
\]
where $R(x)$ is rational, is not a transcendent of the second kind, since
it can be expressed in the simpler form $R(x)$.

It is obvious that we can in this way proceed to define transcendents
of the $n$th order for all values of $n$. Thus
\[
\log \log \log x,\; \log \log \log \log x, \dots
\]
are of the third, fourth, \dots\ orders.

Of course a similar classification of algebraical functions can be and
has been made. Thus we may say that
\[
\sqrt x,\; \sqrt {x + \sqrt x},\; \sqrt{ x + \sqrt {x + \sqrt x}}, \dots
\]
are algebraical functions of the first, second, third, \dots\ orders. But
the fact that there is a general theory of algebraical equations and
therefore of \emph{implicit} algebraical functions has deprived this classification
of most of its importance. There is no such general theory
of elementary transcendental equations\footnote
  {The natural generalisations of the theory of algebraical equations are to be
  found in parts of the theory of differential equations. See Königsberger,
  `Bemerkungen zu Liouville's Classificirung der Transcendenten', \textit{Math.\ Annalen},
  vol.~28, 1886, pp.~483--492.},
and therefore we shall not
%% -----File: 018.png---
rank as `elementary' functions defined by transcendental equations
such as
\[
y = x \log y,
\]
but incapable (as Liouville has shown that in this case $y$~is incapable)
of explicit expression in finite terms.

\Paragraph{2}{5.} The preceding analysis of elementary transcendental functions
rests on the following theorems:

\Item{(\ia)} $e^{x}$ is not an algebraical function of~$x$;

\Item{(\ib)} $\log x$ is not an algebraical function of~$x$;

\Item{(\ic)} $\log x$ is not expressible in finite terms by means of signs of
exponentiation and of algebraical operations, explicit or implicit\footnotemark;
  \footnotetext{For example, $\log x$~cannot be equal to~$e^{y}$, where $y$~is an algebraical function
  of $x$.}

\Item{(\id)} transcendental functions of the first, second, third,~\dots\ orders
actually exist.

A proof of the first two theorems will be given later, but limitations
of space will prevent us from giving detailed proofs of the third and
fourth. Liouville has given interesting extensions of some of these
theorems: he has proved, for example, that no equation of the form
\[
Ae^{\alpha p} + Be^{\beta p} + \dots + Re^{\rho p} = S,
\]
where $p$, $A$, $B$,~\dots, $R$,~$S$ are algebraical functions of~$x$, and $\alpha$, $\beta$,~\dots, $\rho$
different constants, can hold for all values of~$x$.

\Section{III. The integration of elementary functions.
Summary of results}

In the following pages we shall be concerned exclusively with the
problem of the integration of elementary functions. We shall endeavour
to give as complete an account as the space at our disposal permits of
the progress which has been made by mathematicians towards the
solution of the two following problems:

\Item{(i)} \begin{Result}if $f(x)$ is an elementary function, how can we determine
whether its integral is also an elementary function?\end{Result}

\Item{(ii)} \begin{Result}if the integral is an elementary function, how can we find it?\end{Result}

It would be unreasonable to expect complete answers to these
questions. But sufficient has been done to give us a tolerably complete
insight into the nature of the answers, and to ensure that it
%% -----File: 019.png---
shall not be difficult to find the complete answers in any particular
case which is at all likely to occur in elementary analysis or in its
applications.

It will probably be well for us at this point to summarise the
principal results which have been obtained.

\Paragraph{3}{1.} The integral of a rational function (\hyperlink{4 para 1.}{\textsc{iv.}})\ is \emph{always} an elementary
function. It is either rational or the sum of a rational function and
of a finite number of constant multiples of logarithms of rational
functions (\hyperlink{4 para 1.}{\textsc{iv.},~1}).

If certain constants which are the roots of an algebraical equation
are treated as known then the form of the integral can always be
determined completely. But as the roots of such equations are not in
general capable of explicit expression in finite terms, it is not in
general possible to express the integral in an absolutely explicit form
(\hyperlink{4 para 2.}{\textsc{iv.};~2,~3}).

We can always determine, by means of a finite number of
the elementary operations of addition, multiplication, and division,
whether the integral is rational or not. If it is rational, we can
determine it completely by means of such operations; if not, we
can determine its rational part (\hyperlink{4 para 4.}{\textsc{iv.};~4,~5}).

The solution of the problem in the case of rational functions may
therefore be said to be complete; for the difficulty with regard to the
explicit solution of algebraical equations is one not of inadequate
knowledge but of proved impossibility (\hyperlink{4 para 6.}{\textsc{iv.},~6}).

\Paragraph{3}{2.} The integral of an algebraical function (\hyperlink{5 para 1.}{\textsc{v.}}), explicit or implicit,
may or may not be elementary.

If $y$ is an algebraical function of~$x$ then the integral $\int y\, dx$, or, more
generally, the integral
\[
\int R(x,y)\, dx,
\]
where $R$~denotes a rational function, is, if an elementary function,
either algebraical or the sum of an algebraical function and of a finite
number of constant multiples of logarithms of algebraical functions.
All algebraical functions which occur in the integral are \emph{rational
functions of $x$~and~$y$} (\hyperlink{5 para 11.}{\textsc{v.}; 11--14},\hyperlink{5 para 18.}{~18}).

These theorems give a precise statement of a general principle
enunciated by Laplace\footnotemark:
  \footnotetext{\textit{Théorie analytique des probabilités}, p.~7.}
`\emph{l'intégrale d'une fonction différentielle
%% -----File: 020.png---
(algébrique) ne peut contenir d'autres quantités \DPtypo{radicaux}{radicales} que celles
qui entrent dans cette fonction}'; and, we may add, cannot contain
\emph{exponentials} at all. Thus it is impossible that
\[
\int \frac{dx}{\sqrt {1 + x^{2}}}
\]
should contain $e^{x}$ or~$\sqrt {1 - x}$: the appearance of these functions in
the integral could only be apparent, and they could be eliminated
before differentiation. Laplace's principle really rests on the fact, of
which it is easy enough to convince oneself by a little reflection
and the consideration of a few particular cases (though to give a
rigorous proof is of course quite another matter), that \emph{differentiation
will not eliminate exponentials or algebraical irrationalities}. Nor, we
may add, will it eliminate logarithms except when they occur in the
simple form
\[
A \log \phi (x),
\]
where $A$~is a constant, and this is why logarithms can only occur
in this form in the integrals of rational or algebraical functions.

We have thus a general knowledge of the form of the integral
of an algebraical function~$y$, when it is itself an elementary
function. Whether this is so or not of course depends on the nature
of the equation $f(x, y)=0$ which defines~$y$. If this equation, when
interpreted as that of a curve in the plane~$(x,y)$, represents a \emph{unicursal}
curve, \textit{i.e.}~a curve which has the maximum number of double points
possible for a curve of its degree, or whose \emph{deficiency} is zero, then
$x$~and~$y$ can be expressed simultaneously as rational functions of a third
variable~$t$, and the integral can be reduced by a substitution to that
of a rational function (\hyperlink{5 para 2.}{\textsc{v.};~2}, \hyperlink{5 para 7.}{7--9}). In this case, therefore, the integral
is always an elementary function. But this condition, though sufficient,
is not necessary. It is in general true that, when $f(x, y)=0$ is not
unicursal, the integral is not an elementary function but a new
transcendent; and we are able to classify these transcendents according
to the deficiency of the curve. If, for example, the deficiency is unity,
then the integral is in general a transcendent of the kind known as
\emph{elliptic integrals}, whose characteristic is that they can be transformed
into integrals containing no other irrationality than the square root of
a polynomial of the third or fourth degree (\hyperlink{5 para 20.}{\textsc{v.},~20}). But there are infinitely
many cases in which the integral can be expressed by algebraical
functions and logarithms. Similarly there are infinitely many cases
in which integrals associated with curves whose deficiency is greater
%% -----File: 021.png---
than unity are in reality reducible to elliptic integrals. Such abnormal
cases have formed the subject of many exceedingly interesting
researches, but no general method has been devised by which we can
always tell, after a finite series of operations, whether any given
integral is really elementary, or elliptic, or belongs to a higher order
of transcendents.

When $f(x, y) = 0$ is unicursal we can carry out the integration
completely in exactly the same sense as in the case of rational functions.
In particular, if the integral is \emph{algebraical} then it can be found by
means of elementary operations which are always practicable. And
it has been shown, more generally, that we can always determine by
means of such operations whether the integral of any given algebraical
function is algebraical or not, and evaluate the integral when it is
algebraical. And although the general problem of determining whether
any given integral is an elementary function, and calculating it if it
is one, has not been solved, the solution in the particular case in which
the deficiency of the curve $f(x, y) = 0$ is unity is as complete as it is
reasonable to expect any possible solution to be.

\Paragraph{3}{3.} The theory of the integration of transcendental functions
(\hyperlink{6 para 1.}{\textsc{vi.}})\ is naturally much less complete, and the number of classes
of such functions for which general methods of integration exist is
very small. These few classes are, however, of extreme importance
in applications (\hyperlink{6 para 2.}{\textsc{vi.};~2,~3}).

There is a general theorem concerning the form of an integral of
a transcendental function, when it is itself an elementary function,
which is quite analogous to those already stated for rational and
algebraical functions. The general statement of this theorem will be
found in \hyperlink{6 para 5.}{\textsc{vi.},~§5}; it shows, for instance, that the integral of a rational
function of $x$,~$e^{x}$ and~$\log x$ is either a rational function of those
functions or the sum of such a rational function and of a finite
number of constant multiples of logarithms of similar functions.
From this general theorem may be deduced a number of more precise
results concerning integrals of more special forms, such as
\[
\int y e^{x}\, dx,\quad \int y \log x\, dx,
\]
where $y$~is an algebraical function of~$x$ (\hyperlink{6 para 4.}{\textsc{vi.};~4},\hyperlink{6 para 6.}{~6}).
%% -----File: 022.png---

\Section{IV. Rational functions}

\Paragraph{4}{1.} It is proved in treatises on algebra\footnote
  {See, \textit{e.g.}, Weber's \textit{Traité d'algèbre supérieure} (French translation by J.~Griess,
  Paris, 1898), vol.~1, pp.~61--64, 143--149, 350--353; or Chrystal's \textit{Algebra}, vol.~1,
  pp.~151--162.}
that any polynomial
\[
Q(x) = b_{0}x^{n} + b_{1}x^{n-1} + \dots + b_{n}
\]
can be expressed in the form
\[
b_{0} (x - \alpha_{1})^{n_{1}}
      (x - \alpha_{2})^{n_{2}} \dots
      (x - \alpha_{r})^{n_{r}},
\]
where $n_{1}$, $n_{2}$,~\dots\ are positive integers whose sum is~$n$, and $\alpha_{1}$, $\alpha_{2}$,~\dots\ are
constants; and that any rational function~$R(x)$, whose denominator
is~$Q(x)$, may be expressed in the form
\[
A_{0}x^{p} + A_{1}x^{p-1} + \dots + A_{p}
  + \sum_{s=1}^{r} \left\{
    \frac{\beta_{s, 1}}{x - \alpha_{s}}
  + \frac{\beta_{s, 2}}{(x - \alpha_{s})^{2}} + \dots
  + \frac{\beta_{s, n_{s}}}{(x - \alpha_{s})^{n_{s}}}
  \right\},
\]
where $A_{0}$, $A_{1}$,~\dots, $\beta_{s, 1}$,~\dots\ are also constants. It follows that
\begin{multline*}
\int R(x)\, dx
  = A_{0} \frac{x^{p+1}}{p + 1}
  + A_{1} \frac{x^{p}}{p} + \dots + A_{p} x + C \\
  + \sum_{s=1}^{r} \left\{
    \beta_{s, 1} \log(x - \alpha_{s})
  - \frac{\beta_{s, 2}}{x - \alpha_{s}} - \dots
  - \frac{\beta_{s, n_{s}}}{(n_{s} - 1)(x - \alpha_{s})^{n_{s} - 1}}
  \right\}.
\end{multline*}
From this we conclude that \emph{the integral of any rational function is an
elementary function which is rational save for the possible presence
of logarithms of rational functions}. In particular the integral will be
\emph{rational} if each of the numbers~$\beta_{s, 1}$ is zero: this condition is evidently
necessary and sufficient. A necessary but not sufficient condition is
that $Q(x)$~should contain no simple factors.

The integral of the general rational function may be expressed in
a very simple and elegant form by means of symbols of differentiation.
We may suppose for simplicity that the degree of~$P(x)$ is less than
that of~$Q(x)$; this can of course always be ensured by subtracting
a polynomial from~$R(x)$. Then
\begin{align*}
R(x) &= \frac{P(x)}{Q(x)} \\
     &= \frac{1}{(n_{1} - 1)! (n_{2} - 1)! \dots (n_{r} - 1)!}\:
        \frac{\dd^{n - r}}{\dd \alpha_{1}^{n_{1} - 1}
                           \dd \alpha_{2}^{n_{2} - 1} \dots
                           \dd \alpha_{r}^{n_{r} - 1}}\:
        \frac{P(x)}{Q_{0}(x)},
\end{align*}
where
\[
Q_{0}(x) = b_{0} (x - \alpha_{1}) (x - \alpha_{2}) \dots (x - \alpha_{r}).
\]
Now
\[
\frac{P(x)}{Q_{0}(x)}
  = \varpi_{0}(x)
  + \sum_{s=1}^{r} \frac{P(\alpha_{s})}{(x - \alpha_{s}) Q'_{0}(\alpha_{s})},
\]
%% -----File: 023.png---
where $\varpi_{0}(x)$~is a polynomial; and so
{\small \begin{multline*}
\int R(x)\,dx =\\
 \frac{1}{(n_{1}-1)! \dots(n_{r}-1)!}\:
 \frac{\dd^{n-r}}{\dd \alpha_{1}^{n_{1}-1}\dots \dd \alpha_{r}^{n_{r}-1}}
 \left[ \Pi_{0}(x) + \sum_{s=1}^{r} \frac{P(\alpha_{s})}{Q_{0}'(\alpha_{s})} \log (x - \alpha_{s}) \right],
\end{multline*}}%
where
\[
\Pi_{0}(x) = \int \varpi_{0}(x)\,dx.
\]
But
\[
\Pi(x) = \frac{\dd^{n - r} \Pi_{0}(x)}
              {\dd \alpha_{1}^{n_{1} - 1}
               \dd \alpha_{2}^{n_{2} - 1}\dots
               \dd \alpha_{r}^{n_{r} - 1}}
\]
is also a polynomial, and the integral contains no polynomial term,
since the degree of~$P(x)$ is less than that of~$Q(x)$. Thus $\Pi(x)$~must
vanish identically, so that
\begin{multline*}
\int R(x)\,dx =\\
  \frac{1}{(n_{1}-1)!\dots(n_{r}-1)!}\:
  \frac{\dd^{n - r}}{\dd \alpha_{1}^{n_{1}-1}\dots\dd \alpha_{r}^{n_{r}-1}}
  \left[\sum_{s=1}^{r} \frac{P(\alpha_{s})}{Q_{0}'(\alpha_{s})} \log (x - \alpha_{s}) \right].
\end{multline*}

For example
\[
\int\frac{dx}{\{(x-a)(x-b)\}^{2}}
  = \frac{\dd^{2}}{\dd a\, \dd b}\left\{
      \frac{1}{a-b} \log \left(\frac{x-a}{x-b}\right)
    \right\}.
\]

That $\Pi_{0}(x)$~is annihilated by the partial differentiations performed on it
may be verified directly as follows. We obtain~$\Pi_{0}(x)$ by picking out from
the expansion
\[
\frac{P(x)}{x^{r}}
  \left(1 + \frac{\alpha_{1}}{x} + \frac{\alpha_{1}^{2}}{x^{2}} + \dots\right)
  \left(1 + \frac{\alpha_{2}}{x} + \frac{\alpha_{2}^{2}}{x^{2}} + \dots\right) \dots \dots
\]
the terms which involve positive powers of~$x$. Any such term is of the form
\[
Ax^{\nu-r-s_{1}-s_{2}-\dots} \alpha_{1}^{s_{1}} \alpha_{2}^{s_{2}} \dots,
\]
where
\[
s_{1} + s_{2} + \dots \leq \nu - r \leq m - r,
\]
$m$~being the degree of~$P$. It follows that
\[
s_{1} + s_{2} + \dots < n - r = (m_{1} - 1) + (m_{2} - 1) + \dots;
\]
so that at least one of $s_{1}$, $s_{2}$,~\dots\ must be less than the corresponding one of
$m_{1} - 1$, $m_{2} - 1$,~\dots.

It has been assumed above that if
\[
F(x, \alpha) = \int f(x, \alpha)\,dx,
\]
then
\[
\frac{\dd F}{\dd \alpha} = \int \frac{\dd f}{\dd \alpha}\,dx.
\]
%% -----File: 024.png---
The first equation means that $f = \dfrac{\dd F}{\dd x}$ and the second that $\dfrac{\dd f}{\dd \alpha} = \dfrac{\dd^{2} F}{\dd x\,\dd \alpha}$. As it
follows from the first that $\dfrac{\dd f}{\dd \alpha} = \dfrac{\dd^{2} F}{\dd \alpha\, \dd x}$, what has really been assumed is that
\[
\frac{\dd^{2} F}{\dd \alpha\, \dd x} = \frac{\dd^{2} F}{\dd x\, \dd \alpha}.
\]
It is known that this equation is always true for $x = x_{0}$, $\alpha = \alpha_{0}$ if a circle
can be drawn in the plane of~$(x, \alpha)$ whose centre is~$(x_{0}, \alpha_{0})$ and within which
the differential coefficients are continuous.

\Paragraph{4}{2.} It appears from~§1 that the integral of a rational function is
in general composed of two parts, one of which is a rational function
and the other a function of the form
\[
\sum A \log(x - \alpha).
\Tag{(1)}
\]
We may call these two functions the \emph{rational part} and the \emph{transcendental
part} of the integral. It is evidently of great importance to
show that the `transcendental part' of the integral is really transcendental
and cannot be expressed, wholly or in part, as a rational or
algebraical function.

We are not yet in a position to prove this completely\footnotemark;
  \footnotetext{The proof will be completed in \hyperlink{5 para 16.}{v.,~16.}}
but we can
take the first step in this direction by showing that \emph{no sum of the
form~\Eq{(1)} can be rational, unless every~$A$ is zero}.

Suppose, if possible, that
\[
\sum A \log(x - \alpha) = \frac{P(x)}{Q(x)},
\Tag{(2)}
\]
where $P$~and~$Q$ are polynomials without common factor. Then
\[
\sum \frac{A}{x - \alpha} = \frac{P'Q - PQ'}{Q^{2}}.
\Tag{(3)}
\]

Suppose now that $(x - p)^{r}$ is a factor of~$Q$. Then $P'Q - PQ'$ is
divisible by~$(x-p)^{r-1}$ and by no higher power of~$x-p$. Thus the
right-hand side of~\Eq{(3)}, when expressed in its lowest terms, has a factor
$(x-p)^{r+1}$ in its denominator. On the other hand the left-hand side,
when expressed as a rational fraction in its lowest terms, has no
repeated factor in its denominator. Hence $r=0$, and so $Q$~is a constant.
We may therefore replace~\Eq{(2)} by
\[
\sum A \log (x - \alpha) = P(x),
\]
and \Eq{(3)}~by
\[
\sum \frac{A}{x - \alpha} = P'(x).
\]
Multiplying by~$x-\alpha$, and making $x$~tend to~$\alpha$, we see that $A = 0$.
%% -----File: 025.png---

\Paragraph{4}{3.} The method of~§1 gives a complete solution of the problem if
the roots of $Q(x)=0$ can be determined; and in practice this is
usually the case. But this case, though it is the one which occurs
most frequently in practice, is from a theoretical point of view an
exceedingly special case. The roots of $Q(x)=0$ are not in general
explicit algebraical functions of the coefficients, and cannot as a rule
be determined in any explicit form. The method of partial fractions
is therefore subject to serious limitations. For example, we cannot
determine, by the method of decomposition into partial fractions, such
an integral as
\[
\int \frac{4x^{9} + 21x^{6} + 2x^{3} - 3x^{2} - 3}
          {(x^{7} - x + 1)^{2}}\, dx,
\]
or even determine whether the integral is rational or not, although it
is in reality a very simple function. A high degree of importance
therefore attaches to the further problem of determining the integral
of a given rational function so far as possible in an absolutely explicit
form and by means of operations which are always practicable.

It is easy to see that a complete solution of this problem cannot be
looked for.

{\small Suppose for example that $P(x)$~reduces to unity, and that $Q(x)=0$ is
an equation of the fifth degree, whose roots~$\alpha_{1}$, $\alpha_{2}$,~\dots\ $\alpha_{\DPtypo{6}{5}}$ are all distinct and
not capable of explicit algebraical expression.

Then
\begin{align*}
\int R(x) \,dx
  &= \sum_{1}^{5} \frac{\log(x - \alpha_{s})}{Q'(\alpha_{s})}\\
  &= \log \prod_{1}^{5} \left\{(x - \alpha_{s})^{1/Q'(\alpha_{s})}\right\},
\end{align*}
and it is only if at least two of the numbers~$Q'(\alpha_{s})$ are commensurable that
any two or more of the factors $(x - \alpha_{s})^{1/Q'(\alpha_{s})}$ can be associated so as to give
a single term of the type~$A \log S(x)$, where $S(x)$~is rational. In general this
will not be the case, and so it will not be possible to express the integral in
any finite form which does not explicitly involve the roots. A more precise
result in this connection will be proved later~(§6).}

\Paragraph{4}{4.} The first and most important part of the problem has been
solved by Hermite, who has shown that the \emph{rational part} of the
integral can always be determined without a knowledge of the roots of~$Q(x)$,
and indeed without the performance of any operations other
than those of elementary algebra\footnotemark.
  \footnotetext{The following account of Hermite's method is taken in substance from
  Goursat's \textit{Cours d'analyse mathématique} (first edition), t.~1, pp.~238--241.}
%% -----File: 026.png---

Hermite's method depends upon a fundamental theorem in
elementary algebra\footnote{See Chrystal's \textit{Algebra}, vol.~1, pp.~119 \textit{et~seq.}}
which is also of great importance in the ordinary
theory of partial fractions, viz.:

\begin{Result}
`If $X_{1}$~and~$X_{2}$ are two polynomials in~$x$ which have no common
factor, and $X_{3}$~any third polynomial, then we can determine two polynomials
$A_{1}$,~$A_{2}$, such that
\[
A_{1}X_{1} + A_{2}X_{2} = X_{3}.\text{'}
\]
\end{Result}

Suppose that
\[
Q(x) = Q_{1} Q_{2}^{2} Q_{3}^{3}\dots Q_{t}^{t},
\]
$Q_{1}$,~\dots\ denoting polynomials which have only simple roots and of
which no two have any common factor. We can always determine
$Q_{1}$,~\dots\ by elementary methods, as is shown in the elements of the
theory of equations\footnotemark.
  \footnotetext{See, for example, Hardy, \textit{A course of pure mathematics} (2nd~edition), p.~208.}

We can determine $B$~and~$A_{1}$ so that
\[
B Q_{1} + A_{1} Q_{2}^{2} Q_{3}^{3}\dots Q_{t}^{t} = P,
\]
and therefore so that
\[
R(x)
  = \frac{P}{Q}
  = \frac{A_{1}}{Q_{1}} + \frac{B}{Q_{2}^{2} Q_{3}^{3}\dots Q_{t}^{t}}.
\]
By a repetition of this process we can express~$R(x)$ in the form
\[
\frac{A_{1}}{Q_{1}}
  + \frac{A_{2}}{Q_{2}^{2}} + \dots
  + \frac{A_{t}}{Q_{t}^{t}},
\]
and the problem of the integration of~$R(x)$ is reduced to that of the
integration of a function
\[
\frac{A}{Q^{\nu}},
\]
where $Q$~is a polynomial whose roots are all distinct. Since this is so,
$Q$~and its derived function~$Q'$ have no common factor: we can therefore
determine $C$~and~$D$ so that
\[
CQ + DQ' = A.
\]
Hence
\begin{align*}
\int \frac{A}{Q^{\nu}}\,dx
  &= \int \frac{CQ + DQ'}{Q^{\nu}}\,dx\\
  &= \int \frac{C}{Q^{\nu-1}}\,dx
   - \frac{1}{\nu - 1} \int D \frac{d}{dx} \left(\frac{1}{Q^{\nu-1}}\right) dx\\
  &= - \frac{D}{(\nu - 1) Q^{\nu - 1}} + \int \frac{E}{Q^{\nu - 1}}\,dx,
\end{align*}
where
\[
E = C + \frac{D'}{\nu - 1}.
\]
%% -----File: 027.png---
Proceeding in this way, and reducing by unity at each step the power
of~$1/Q$ which figures under the sign of integration, we ultimately
arrive at an equation
\[
\int \frac{A}{Q^{\nu}}\,dx = R_{\nu}(x) + \int \frac{S}{Q}\,dx,
\]
where $R_{\nu}$~is a rational function and $S$~a polynomial.

The integral on the right-hand side has no rational part, since all
the roots of~$Q$ are simple~(§2). Thus the rational part of $\int R(x)\,dx$ is
\[
R_{2}(x) + R_{3}(x) + \dots + R_{t}(x),
\]
and it has been determined without the need of any calculations other
than those involved in the addition, multiplication and division of
polynomials\footnotemark.
  \footnotetext{The operation of forming the derived function of a given polynomial can of
  course be effected by a combination of these operations.}

\Paragraph{4}{5.} \Item{(i)} Let us consider, for example, the integral
\[
\int \frac{4x^{9} + 21x^{6} + 2x^{3} - 3x^{2} - 3}{(x^{7} - x + 1)^{2}}\,dx,
\]
mentioned above~(§3). We require polynomials $A_{1}$,~$A_{2}$ such that
\[
A_{1}X_{1} + A_{2}X_{2} = X_{3},
\Tag{(1)}
\]
where
\[
X_{1} = x^{7} - x + 1,\quad
X_{2} = 7x^{6} - 1,\quad
X_{3} = 4x^{9} + 21x^{6} + 2x^{3} - 3x^{2} - 3.
\]

In general, if the degrees of $X_{1}$~and~$X_{2}$ are $m_{1}$~and~$m_{2}$, and that of~$X_{3}$
does not exceed $m_{1} + m_{2} - 1$, we can suppose that the degrees of $A_{1}$~and~$A_{2}$ do
not exceed $m_{2} - 1$ and $m_{1} - 1$ respectively. For we know that polynomials
$B_{1}$~and~$B_{2}$ exist such that
\[
B_{1} X_{1} + B_{2} X_{2} = X_{3}.
\]
If $B_{1}$~is of degree not exceeding $m_{2} - 1$, we take $A_{1} = B_{1}$, and if it is of higher
degree we write
\[
B_{1} = L_{1} X_{2} + A_{1},
\]
where $A_{1}$~is of degree not exceeding $m_{2} - 1$. Similarly we write
\[
B_{2} = L_{2} X_{1} + A_{2}.
\]
We have then
\[
(L_{1} + L_{2}) X_{1} X_{2} + A_{1} X_{1} + A_{2} X_{2} = X_{3}.
\]
In this identity $L_{1}$~or~$L_{2}$ or both may vanish identically, and in any case we
see, by equating to zero the coefficients of the powers of~$x$ higher than the
$(m_{1} + m_{2} - 1)$th, that $L_{1} + L_{2}$ vanishes identically. Thus $X_{3}$~is expressed in
the form required.

The actual determination of the coefficients in $A_{1}$~and~$A_{2}$ is most easily
performed by equating coefficients. We have then $m_{1} + m_{2}$~linear equations
%% -----File: 028.png---
in the same number of unknowns. These equations must be consistent,
since we know that a solution exists\footnotemark.
  \footnotetext{It is easy to show that the solution is also unique.}

If $X_{3}$~is of degree higher than $m_{1} + m_{2} - 1$, we must divide it by~$X_{1}X_{2}$ and
express the remainder in the form required.

In this case we may suppose $A_{1}$~of degree~$5$ and $A_{2}$~of degree~$6$, and we
find that
\[
A_{1} = -3x^{2},\quad A_{2} = x^{3} + 3.
\]
Thus the rational part of the integral is
\[
- \frac{x^{3} + 3}{x^{7} - x + 1},
\]
and, since $-3x^{2} + (x^{3} + 3)'=0$, there is no transcendental part.

\Item{(ii)} The following problem is instructive: \emph{to find the conditions that
\[
\int \frac{\alpha x^{2} + 2 \beta x + \gamma}{(Ax^{2} + 2Bx + C)^{2}}\, dx
\]
may be rational, and to determine the integral when it is rational.}

We shall suppose that $Ax^{2} + 2Bx + C$ is not a perfect square, as if it were
the integral would certainly be rational. We can determine $p$,~$q$ and~$r$
so that
\[
p(Ax^{2} + 2Bx + C) + 2(qx + r)(Ax + B) = \alpha x^{2} + 2 \beta x + \gamma,
\]
and the integral becomes
\begin{multline*}
p \int\frac{dx}{Ax^{2} + 2Bx + C}
  - \int(qx + r)\, \frac{d}{dx} \left(\frac{1}{Ax^{2} + 2Bx + C}\right) dx\\
  = -\frac{qx + r}{Ax^{2} + 2Bx + C}
    + (p + q) \int\frac{dx}{Ax^{2} + 2Bx + C}.
\end{multline*}
The condition that the integral should be rational is therefore $p + q = 0$.

Equating coefficients we find
\[
A(p + 2q) = \alpha, \quad
B(p + q) + Ar = \beta, \quad
Cp + 2Br = \gamma.
\]
Hence we deduce
\[
p = -\frac{\alpha}{A}, \quad
q =  \frac{\alpha}{A}, \quad
r =  \frac{\beta}{A},
\]
and $A \gamma + C \alpha = 2B \beta$. The condition required is therefore that the two quadratics
$\alpha x^{2} + 2 \beta x + \gamma$ and $Ax^{2} + 2Bx + C$ should be harmonically related, and in this
case
\[
\int \frac{\alpha x^{2} + 2 \beta x + \gamma}{(Ax^{2} + 2Bx + C)^{2}}\,dx
  = -\frac{\alpha x + \beta}{A (Ax^{2} + 2Bx + C)}.
\]

\Item{(iii)} Another method of solution of this problem is as follows. If we write
\[
Ax^{2} + 2Bx + C = A(x - \lambda)(x - \mu),
\]
and use the bilinear substitution
\[
x = \frac{\lambda y + \mu}{y + 1},
\]
then the integral is reduced to one of the form
\[
\int \frac{ay^{2} + 2by + c}{y^{2}}\,dy,
\]
%% -----File: 029.png---
and is rational if and only if $b = 0$. But this is the condition that the
quadratic $ay^{2} + 2by + c$, corresponding to $\alpha x^{2} + 2 \beta x + \gamma$, should be harmonically
related to the degenerate quadratic~$y$, corresponding to $Ax^{2} + 2Bx + C$. The
result now follows from the fact that harmonic relations are not changed by
bilinear transformation.

It is not difficult to show, by an adaptation of this method, that
\[
\int\frac{(\alpha x^{2} + 2 \beta x + \gamma)
          (\alpha_{1} x^{2} + 2 \beta_{1} x + \gamma_{1}) \dots
          (\alpha_{n} x^{2} + 2 \beta_{n} x + \gamma_{n})}
         {(Ax^{2} + 2Bx + C)^{n+2}}\, dx
\]
is rational if all the quadratics are harmonically related to any one of those
in the numerator. This condition is sufficient but not necessary.

\Item{(iv)} As a further example of the use of the method~(ii) the reader may
show that \begin{Result}the necessary and sufficient condition that
\[
\int\frac{f(x)}{\{ F(x) \}^{2}}\,dx,
\]
where $f$~and~$F$ are polynomials with no common factor, and $F$~has no repeated
factor, should be rational, is that $f'F' - fF''$ should be divisible by~$F$.\end{Result}

\Paragraph{4}{6.} It appears from the preceding paragraphs that we can always
find the rational part of the integral, and can find the complete integral
if we can find the roots of $Q(x) = 0$. The question is naturally
suggested as to the maximum of information which can be obtained
about the logarithmic part of the integral in the general case in which
the factors of the denominator cannot be determined explicitly. For
there are polynomials which, although they cannot be completely resolved
into such factors, can nevertheless be partially resolved. For example
\begin{multline*}
x^{14} - 2x^{8} - 2x^{7} - x^{4} - 2x^{3} + 2x + 1
  = (x^{7} + x^{2} - 1) (x^{7} - x^{2} - 2x - 1),\\
\shoveleft{x^{14} - 2x^{8} - 2x^{7} - 2x^{4} - 4x^{3} - x^{2} + 2x + 1}\\
  = \{x^{7} + x^{2} \sqrt{2} + x (\sqrt{2} - 1) - 1\}
    \{x^{7} - x^{2} \sqrt{2} - x (\sqrt{2} + 1) - 1\}.
\end{multline*}
The factors of the first polynomial have rational coefficients: in the
language of the theory of equations, the polynomial is \emph{reducible in the
rational domain}. The second polynomial is reducible in the domain
formed by the \emph{adjunction} of the single irrational~$\sqrt{2}$ to the rational
domain\footnotemark.
  \footnotetext{See Cajori, \textit{An introduction to the modern theory of equations} (Macmillan,
  1904); Mathews, \textit{Algebraic equations} (\textit{Cambridge tracts in mathematics}, no.~6),
  pp.~6--7.}

We may suppose that every possible decomposition of~$Q(x)$ of this
nature has been made, so that
\[
Q = Q_{1} Q_{2}\dots Q_{t}.
\]
%% -----File: 030.png---
Then we can resolve~$R(x)$ into a sum of partial fractions of the type
\[
\int\frac{P_{\nu}}{Q_{\nu}}\, dx,
\]
and so we need only consider integrals of the type
\[
\int\frac{P}{Q}\, dx,
\]
where no further resolution of~$Q$ is possible or, in technical language,
\emph{$Q$~is irreducible by the adjunction of any algebraical irrationality}.

Suppose that this integral can be evaluated in a form involving only
constants which can be expressed explicitly in terms of the constants
which occur in~$P/Q$. It must be of the form
\[
A_{1} \log X_{1} + \dots + A_{k} \log X_{k},
\Tag{(1)}
\]
where the~$A$'s are constants and the~$X$'s polynomials. We can
suppose that no~$X$ has any repeated factor~$\xi^{m}$, where $\xi$~is a polynomial.
For such a factor could be determined rationally in terms of the coefficients
of~$X$, and the expression~\Eq{(1)} could then be modified by
taking out the factor~$\xi^{m}$ from~$X$ and inserting a new term $mA \log \xi$.
And for similar reasons we can suppose that no two~$X$'s have any
factor in common.

Now
\[
\frac{P}{Q}
  = A_{1}\frac{X'_{1}}{X_{1}}
  + A_{2}\frac{X'_{2}}{X_{2}} + \dots
  + A_{k}\frac{X'_{k}}{X_{k}},
\]
or
\[
P X_{1} X_{2} \dots X_{k}
  = Q \sum A_{\nu} X_{1} \dots X_{\nu-1} X'_{\nu} X_{\nu+1} \dots X_{k}.
\]
All the terms under the sign of summation are divisible by~$X_{1}$ save the
first, which is prime to~$X_{1}$. Hence $Q$~must be divisible by~$X_{1}$: and
similarly, of course, by $X_{2}$, $X_{3}$,~\dots, $X_{k}$. But, since $P$~is prime to~$Q$,
$X_{1} X_{2} \dots X_{k}$ is divisible by~$Q$. Thus $Q$~must be a constant multiple of
$X_{1} X_{2} \dots X_{k}$. But $Q$~is \textit{ex~hypothesi} not resoluble into factors which
contain only explicit algebraical irrationalities. Hence all the~$X$'s
save one must reduce to constants, and so $P$~must be a constant
multiple of~$Q'$, and
\[
\int\frac{P}{Q}\, dx = A \log Q,
\]
where $A$~is a constant. Unless this is the case the integral cannot be
expressed in a form involving only constants expressed explicitly in
terms of the constants which occur in $P$~and~$Q$.

{\small Thus, for instance, the integral
\[
\int\frac{dx}{x^{5} + ax + b}
\]
%% -----File: 031.png---
cannot, except in special cases\footnotemark,
  \footnotetext{The equation $x^{5} + ax + b = 0$ is soluble by radicals in certain cases. See
  Mathews, \textit{l.c.}, pp.~52~\textit{et~seq.}}
be expressed in a form involving only
constants expressed explicitly in terms of $a$~and~$b$; and the integral
\[
\int \frac{5x^{4} + c}{x^{5} + ax + b}\, dx
\]
can in general be so expressed if and only if $c = a$. We thus confirm an
inference made before~(§3) in a less accurate way.

Before quitting this part of our subject we may consider one further
problem: \emph{under what circumstances is
\[
\int R(x)\, dx = A \log R_{1}(x)
\]
where $A$~is a constant and $R_{1}$~rational?} Since the integral has no rational
part, it is clear that $Q(x)$~must have only simple factors, and that the degree
of~$P(x)$ must be less than that of~$Q(x)$. We may therefore use the formula
\[
\int R(x)\,dx
  = \log \prod_{1}^{r} \left\{
      (x - \alpha_{s})^{P(\alpha_{s})/Q'(\alpha_{s})}
  \right\}.
\]
The necessary and sufficient condition is that all the numbers $P(\alpha_{s})/Q'(\alpha_{s})$
should be commensurable. If~\textit{e.g.}
\[
R(x) = \frac{x - \gamma}{(x - \alpha) (x - \beta)},
\]
{\Loosen then $(\alpha - \gamma)/(\alpha - \beta)$ and $(\beta - \gamma)/(\beta - \alpha)$ must be commensurable, \textit{i.e.}\ $(\alpha - \gamma)/(\beta - \gamma)$
must be a rational number. If the denominator is given we can find all the
values of~$\gamma$ which are admissible: for $\gamma = (\alpha q - \beta p)/(q - p)$, where $p$~and~$q$ are
integers.}

\Paragraph{4}{7.} Our discussion of the integration of rational functions is now
complete. It has been throughout of a theoretical character. We
have not attempted to consider what are the simplest and quickest
methods for the actual calculation of the types of integral which occur
most commonly in practice. This problem lies outside our present
range: the reader may consult

O. Stolz, \textit{Grundzüge der \DPtypo{Differential- und Integralrechnung}{Differential- und~Integralrechnung}}, vol.~1,
ch.~7:

J. Tannery, \textit{Leçons d'algèbre et d'analyse,} vol.~2, ch.~18:

Ch.-J. de~la Vallée-Poussin, \textit{Cours d'analyse,} ed.~3, vol.~1, ch.~5:

T.~J.~I'A. Bromwich, \textit{Elementary integrals} (Bowes and Bowes,
1911):

G.~H. Hardy, \textit{A course of pure mathematics,} ed.~2, ch.~6.
%% -----File: 032.png---

\Section{V. Algebraical Functions}

\Paragraph{5}{1.} We shall now consider the integrals of algebraical functions,
explicit or implicit. The theory of the integration of such functions is
far more extensive and difficult than that of rational functions, and
we can give here only a brief account of a few of the most important
results and of the most obvious of their applications.

If $y_{1}$, $y_{2}$,~\dots, $y_{n}$ are algebraical functions of~$x$, then any algebraical
function~$z$ of $x$,~$y_{1}$, $y_{2}$,~\dots, $y_{n}$ is an algebraical function of~$x$. This is
obvious if we confine ourselves to \emph{explicit} algebraical functions. In
the general case we have a number of equations of the type
\[
P_{\nu, 0}(x) y_{\nu}^{m_{\nu}}
  + P_{\nu, 1}(x) y_{\nu}^{m_{\nu}-1} + \dots
  + P_{\nu, m_{\nu}}(x) = 0 \quad(\nu = 1, 2,\dots, n),
\]
and
\[
P_{0}(x, y_{1},\dots, y_{n}) z^{m} + \dots
  + P_{m}(x, y_{1},\dots, y_{n}) = 0,
\]
where the~$P$'s represent polynomials in their arguments. The elimination
of $y_{1}$, $y_{2}$,~\dots, $y_{n}$ between these equations gives an equation in~$z$
whose coefficients are polynomials in $x$~only.

The importance of this from our present point of view lies in the
fact that we may consider the standard algebraical integral under any
of the forms
\[
\int y\,dx,
\]
where $f(x, y) = 0$;
\[
\int R(x, y)\,dx,
\]
where $f(x, y) = 0$ and $R$~is rational; or
\[
\int R(x, y_{1},\dots, y_{n})\,dx,
\]
where $f_{1}(x, \DPtypo{y}{y_1}) = 0$,~\dots, $f_{n}(x, y_{n}) = 0$. It is, for example, much more
convenient to treat such an irrational as
\[
\frac{x - \sqrt{x+1} - \sqrt{x-1}}
     {1 + \sqrt{x+1} + \sqrt{x-1}}
\]
as a rational function of $x$,~$y_{1}$,~$y_{2}$, where $y_{1} = \sqrt{x+1}$, $y_{2} = \sqrt{x-1}$,
$y_{1}^{2} = x + 1$, $y_{2}^{2} = x- 1$, than as a rational function of $x$~and~$y$, where
\begin{gather*}
y = \sqrt{x+1} + \sqrt{x-1}, \\
y^{4} - 4xy^{2} + 4 = 0.
\end{gather*}
To treat it as a simple irrational~$y$, so that our fundamental equation is
\[
(x - y)^{4} - 4x(x - y)^{2} (1 + y)^{2} + 4(1 + y)^{4} = 0
\]
is evidently the least convenient course of all.
%% -----File: 033.png---

Before we proceed to consider the general form of the integral of an
algebraical function we shall consider one most important case in which
the integral can be at once reduced to that of a rational function, and
is therefore always an elementary function itself.

\Paragraph{5}{2.} The class of integrals alluded to immediately above is that
covered by the following theorem.

\begin{Result}
If there is a variable~$t$ connected with $x$ and $y$ (or $y_{1}$, $y_{2}$, \dots, $y_{n}$)
by rational relations
\[
x = R_{1}(t),\quad y = R_{2}(t)
\]
(or $y_{1} = R_{2}^{(1)}(t)$, $y_{2} = R_{2}^{(2)}(t)$, \dots), then the integral
\[
\int R(x, y)\, dx
\]
(or $\int R(x, y_{1}, \dots, y_{n})\, dx$) is an elementary function.
\end{Result}

The truth of this proposition follows immediately from the
equations
\begin{gather*}
R(x, y) = R\{R_{1}(t), R_{2}(t)\} = S(t), \\
\frac{dx}{dt} = R_{1}'(t) = T(t), \\
\int R(x, y)\, dx = \int S(t) T(t)\, dt = \int U(t)\, dt,
\end{gather*}
where all the capital letters denote rational functions.

The most important case of this theorem is that in which $x$ and $y$
are connected by the general quadratic relation
\[
(a, b, c, f, g, h \between x, y, 1)^{2} = 0.
\]
The integral can then be made rational in an infinite number of ways\DPtypo{}{.}
For suppose that $(\xi, \eta)$ is any point on the conic, and that
\[
(y - \eta) = t(x - \xi)
\]
is any line through the point. If we eliminate $y$ between these
equations, we obtain an equation of the second degree in $x$, say
\[
T_{0}x^{2} + 2T_{1}x + T_{2} = 0,
\]
where $T_{0}$, $T_{1}$, $T_{2}$ are polynomials in $t$. But one root of this equation
must be $\xi$, which is independent of $t$; and when we divide by $x - \xi$ we
obtain an equation of the \emph{first} degree for the abscissa of the variable
point of intersection, in which the coefficients are again polynomials
in $t$. Hence this abscissa is a rational function of $t$; the ordinate of
the point is also a rational function of $t$, and as $t$ varies this point
%% -----File: 034.png---
coincides with every point of the conic in turn. In fact the equation
of the conic may be written in the form
\[
au^{2} + 2huv + bv^{2} + 2(a\xi + h\eta + g)u + 2(h\xi + b\eta + f)v = 0,
\]
where $u = x - \xi$, $v = y - \eta$, and the other point of intersection of the line
$v = tu$ and the conic is given by
\begin{gather*}
x = \xi - \frac{2\{a\xi + h\eta + g + t(h\xi + b\eta + f)\}}{a + 2ht +bt^{2}}, \\
y = \eta - \frac{2t\{a\xi + h\eta + g + t(h\xi + b\eta + f)\}}{a + 2ht +bt^{2}}.
\end{gather*}

An alternative method is to write
\[
ax^{2} + 2hxy + by^{2} = b(y - \mu x)(y - \mu' x),
\]
so that $y - \mu x = 0$ and $y - \mu' x = 0$ are parallel to the asymptotes of
the conic, and to put
\[
y - \mu x = t.
\]
Then
\[
y - \mu' x = - \frac{2gx + 2fy + c}{bt};
\]
and from these two equations we can calculate $x$ and $y$ as rational
functions of $t$. The principle of this method is of course the same as
that of the former method: $(\xi, \eta)$ is now at infinity, and the pencil of
lines through $(\xi, \eta)$ is replaced by a pencil parallel to an asymptote.

The most important case is that in which $b = -1$, $f = h = 0$, so that
\[
y^{2} = ax^{2} + 2gx + c.
\]
The integral is then made rational by the substitution
\[
x = \xi - \frac{2(a\xi + g - t\eta)}{a - t^{2}},\quad y = \eta - \frac{2t(a\xi + g - t\eta)}{a - t^{2}}
\]
where $\xi$, $\eta$ are any numbers such that
\[
\eta^{2} = a\xi^{2} + 2g\xi + c.
\]
We may for instance suppose that $\xi = 0$, $\eta = \sqrt c$; or that $\eta = 0$, while $\xi$
is a root of the equation $a\xi^{2} + 2g\xi + c = 0$. Or again the integral is
made rational by putting $y - x \sqrt a = t$, when
\[
x = - \frac{t^{2} - c}{2(t \sqrt a - g)},\quad y = \frac{(t^{2} + c) \sqrt a - 2gt}{2(t \sqrt a - g)}.
\]

\Paragraph{5}{3.} We shall now consider in more detail the problem of the calculation of
\[
\int R(x, y)\, dx,
\]
where
\[
y = \sqrt X = \sqrt{ax^{2} + 2bx + c}\;\footnote
  {We now write $b$ for $g$ for the sake of symmetry in notation.}
\]
%% -----File: 035.png---
The most interesting case is that in which $a$,~$b$,~$c$ and the constants which
occur in~$R$ are real, and we shall confine our attention to this case.

Let
\[
R(x, y) = \frac{P(x, y)}{Q(x, y)},
\]
where $P$~and~$Q$ are polynomials. Then, by means of the equation
\[
y^{2} = ax^{2} + 2bx + c,
\]
$R(x, y)$ may be reduced to the form
\[
\frac{A + B \sqrt{X}}{C + D \sqrt{X}}
  = \frac{(A + B \sqrt{X})(C - D \sqrt{X})}{C^{2} - D^{2}X},
\]
where $A$, $B$, $C$,~$D$ are polynomials in~$x$; and so to the form $M + N \sqrt{X}$, where
$M$~and~$N$ are rational, or (what is the same thing) the form
\[
P + \frac{Q}{\sqrt{X}},
\]
where $P$~and~$Q$ are rational. The rational part may be integrated by the
methods of \hyperlink{6 para 1.}{section~\textsc{iv.}}, and the integral
\[
\int \frac{Q}{\sqrt{X}}\, dx
\]
may be reduced to the sum of a number of integrals of the forms
\[
\int \frac{x^{r}}{\sqrt{X}}\, dx,\quad
\int \frac{dx}{(x - p)^{r} \sqrt{X}},\quad
\int \frac{\xi x + \eta}{(\alpha x^{2} + 2\beta x + \gamma)^{r} \sqrt{X}}\, dx,
\Tag{(1)}
\]
where $p$, $\xi$,~$\eta$, $\alpha$,~$\beta$,~$\gamma$ are real constants and $r$~a positive integer. The result
is generally required in an explicitly real form: and, as further progress
depends on transformations involving~$p$ (or $\alpha$,~$\beta$,~$\gamma$), it is generally not
advisable to break up a quadratic factor $\alpha x^{2} + 2\beta x + \gamma$ into its constituent
linear factors when these factors are complex.

All of the integrals~\Eq{(1)} may be reduced, by means of elementary formulae
of reduction\footnotemark,
  \footnotetext{See, for example, Bromwich, \textit{l.c.}, pp.~16~\textit{et~seq.}}
to dependence upon three fundamental integrals, viz.
\[
\int \frac{dx}{\sqrt{X}},\quad
\int \frac{dx}{(x - p) \sqrt{X}},\quad
\int \frac{\xi x + \eta}{(\alpha x^{2} + 2\beta x + \gamma) \sqrt{X}}\, dx.
\Tag{(2)}
\]

\Paragraph{5}{4.} The first of these integrals may be reduced, by a substitution of the
type $x = t + k$, to one or other of the three standard forms
\[
\int \frac{dt}{\sqrt{m^{2} - t^{2}}},\quad
\int \frac{dt}{\sqrt{t^{2} + m^{2}}},\quad
\int \frac{dt}{\sqrt{t^{2} - m^{2}}},
\]
where $m > 0$. These integrals may be rationalised by the substitutions
\[
t = \frac{2mu}{1 + u^{2}},\quad
t = \frac{2mu}{1 - u^{2}},\quad
t = \frac{m(1 + u^{2})}{2u};
\]
but it is simpler to use the transcendental substitutions
\[
t = m \sin \phi,\quad
t = m \sinh \phi,\quad
t = m \cosh \phi.
\]
%% -----File: 036.png---
These last substitutions are generally the most convenient for the reduction
of an integral which contains one or other of the irrationalities
\[
\sqrt{m^{2} - t^{2}},\quad
\sqrt{t^{2} + m^{2}},\quad
\sqrt{t^{2} - m^{2}},
\]
though the alternative substitutions
\[
t = m \tanh \phi,\quad
t = m \tan \phi,\quad
t = m \sec \phi
\]
are often useful.

It has been pointed out by Dr~Bromwich that the forms usually given in
text-books for these three standard integrals, viz.
\[
\arcsin \frac{t}{m},\quad
\argsinh \frac{t}{m},\quad
\argcosh \frac{t}{m}
\]
are not quite accurate. It is obvious, for example, that the first two of these
functions are odd functions of~$m$, while the corresponding integrals are even
functions. The correct formulae are
\[
\arcsin \frac{t}{|m|},\quad
\argsinh \frac{t}{|m|} = \log \frac{t + \sqrt{t^{2} + m^{2}}}{|m|}
\]
and
\[
±\argcosh \frac{|t|}{|m|}
  = \log \left|\frac{t + \sqrt{t^{2} - m^{2}}}{m}\right|,
\]
where the ambiguous sign is the same as that of~$t$. It is in some ways more
convenient to use the equivalent forms
\[
\arctan \frac{t}{\sqrt{m^{2} - t^{2}}},\quad
\argtanh \frac{t}{\sqrt{t^{2} + m^{2}}},\quad
\argtanh \frac{t}{\sqrt{t^{2} - m^{2}}}.
\]

\Paragraph{5}{5.} The integral
\[
\int \frac{dx}{(x - p) \sqrt{X}}
\]
may be evaluated in a variety of ways.

If $p$~is a root of the equation $X = 0$, then $X$~may be written in the form
$a(x - p)(x - q)$, and the value of the integral is given by one or other of the
formulae
\begin{gather*}
\int \frac{dx}{(x - p) \sqrt{(x - p)(x - q)}}
  = \frac{2}{q - p} \sqrt{\frac{x - q}{x - p}}, \\
\int \frac{dx}{(x - p)^{5/2}} = -\frac{2}{3(x - p)^{3/2}}.
\end{gather*}
We may therefore suppose that $p$~is not a root of $X = 0$.

\Item{(i)} We may follow the general method described above, taking
\[
\xi = p,\quad
\eta = \sqrt{ap^{2} + 2bp + c}\footnotemark.
\]
\footnotetext{Cf.\ Jordan, \textit{Cours d'analyse}, ed.~2, vol.~2, p.~21.}%
Eliminating~$y$ from the equations
\[
y^{2} = ax^{2} + 2bx + c,\quad
y - \eta = t(x - \xi),
\]
and dividing by $x - \xi$, we obtain
\[
t^{2}(x - \xi) + 2\eta t - a(x + \xi) - 2b = 0,
\]
and so
\[
-\frac{2\, dt}{t^{2} - a}
  = \frac{dx}{t(x - \xi) + \eta}
  = \frac{dx}{y}.
\]
%% -----File: 037.png---

Hence
\[
\int \frac{dx}{(x - \xi)y} = -2 \int \frac{dt}{(x - \xi)(t^{2} - a)}.
\]
But
\[
(t^{2} - a)(x - \xi) = 2a\xi + 2b - 2\eta t;
\]
and so
\begin{align*}
\int \frac{dx}{(x - p)y}
   &= - \int \frac{dt}{a\xi + b - \eta t}
   = \frac{1}{\eta} \log(a\xi + b - \eta t)\\
  &= \frac{1}{\sqrt{ap^{2} + 2bp + c}} \log\{t \sqrt{ap^{2} + 2bp + c} - ap -b\}.
\end{align*}
If $ap^{2} + 2bp + c < 0$ the transformation is imaginary.

Suppose, \textit{e.g.}, (\ia)~$y = \sqrt{x + 1}$, $p = 0$, or (\ib)~$y = \sqrt{x - 1}$, $p = 0$. We find

\Item{(\ia)}
\[
\int \frac{dx}{x \sqrt{x + 1}} = \log(t - \tfrac{1}{2}),
\]
where
\[
t^{2}x + 2t - 1 = 0,
\]
or
\[
t = \frac{-1 + \sqrt{x + 1}}{x};
\]
and

\Item{(\ib)}
\[
\int \frac{dx}{x \sqrt{x - 1}} = -i \log(it - \tfrac{1}{2}),
\]
where
\[
t^{2}x + 2it - 1 = 0.
\]
Neither of these results is expressed in the simplest form, the second in
particular being very inconvenient.

\Item{(ii)} The most straightforward method of procedure is to use the
substitution
\[
x - p = \frac{1}{t}.
\]
We then obtain
\[
\int \frac{dx}{(x - p)y}
  = \int \frac{dt}{\sqrt{a_{1}t^{2} + 2b_{1}t + c_{1}}},
\]
where $a_{1}$,~$b_{1}$,~$c_{1}$ are certain simple functions of $a$,~$b$,~$c$, and~$p$. The further
reduction of this integral has been discussed already.

\Item{(iii)} A third method of integration is that adopted by Sir~G. Greenhill\footnotemark,
  \footnotetext{A.~G. Greenhill, \textit{A chapter in the integral calculus} (Francis Hodgson, 1888),
  p.~12: \textit{Differential and integral calculus}, p.~399.}%
who uses the transformation
\[
t = \frac{\sqrt{ax^{2} + 2bx + c}}{x - p}\DPtypo{}{.}
\]
It will be found that
\[
\int \frac{dx}{(x - p) \sqrt{X}}
  = \int \frac{dt}{\sqrt{(ap^{2} + 2bp + c)t^{2} + b^{2} - ac}},
\]
which is of one of the three standard forms mentioned in \hyperlink{5 para 4.}{§~4}.
%% -----File: 038.png---

\Paragraph{5}{6.} It remains to consider the integral
\[
\int \frac{\xi x + \eta}{(\alpha x^{2} + 2\beta x + \gamma) \sqrt{X}}\, dx
  = \int \frac{\xi x + \eta}{X_{1} \sqrt{X}}\, dx,
\]
where $\alpha x^{2} + 2\beta x + \gamma$ or~$X_{1}$ is a quadratic with complex linear factors. Here
again there is a choice of methods at our disposal.

We may suppose that $X_{1}$~is not a constant multiple of~$X$. If it is, then
the value of the integral is given by the formula
\[
\int \frac{\xi x + \eta}{(ax^{2} + 2bx + c)^{3/2}}\, dx
  = \frac{\eta(ax + b) - \xi(bx + c)}{\sqrt{(ac - b^{2})(ax^{2} + 2bx + c)}}\footnotemark.
\]
\footnotetext{Bromwich, \textit{l.c.}, p.~16.}

\Item{(i)} The standard method is to use the substitution
\[
x = \frac{\mu t + \nu}{t + 1},
\Tag{(1)}
\]
where $\mu$~and~$\nu$ are so chosen that
\[
a\mu\nu + b(\mu + \nu) + c = 0,\quad
\alpha\mu\nu + \beta(\mu + \nu) + \gamma = 0.
\Tag{(2)}
\]

The values of $\mu$~and~$\nu$ which satisfy these conditions are the roots of the
quadratic
\[
(a\beta - b\alpha)\mu^{2} - (c\alpha - a\gamma)\mu + (b\gamma - c\beta) = 0.
\]
The roots will be real and distinct if
\[
(c\alpha - a\gamma)^{2} > 4(a\beta - b\alpha)(b\gamma - c\beta),
\]
or if
\[
(a\gamma + c\alpha - 2b\beta)^{2} > 4(ac - b^{2})(\alpha\gamma - \beta^{2}).
\Tag{(3)}
\]
Now $\alpha\gamma - \beta^{2} > 0$, so that \Eq{(3)}~is certainly satisfied if $ac - b^{2} < 0$.   But if $ac - b^{2}$
and $\alpha\gamma - \beta^{2}$ are both positive then $a\gamma$~and~$c\alpha$ have the same sign, and
\begin{align*}
(a\gamma + c\alpha - 2b\beta)^{2}
  &\geq (|a\gamma + c\alpha| - 2|b\beta|)^{2} > 4\{\sqrt{ac\alpha\gamma} - |b\beta|\}^{2}\\
  &= 4[(ac - b^{2})(\alpha\gamma - \beta^{2}) + \{|b| \sqrt{\alpha\gamma} - |\beta| \sqrt{ac}\}^{2}]\\
  &\geq 4(ac - b^{2})(\alpha\gamma - \beta^{2}).
\end{align*}
Thus the values of $\mu$~and~$\nu$ are in any case real and distinct.

It will be found, on carrying out the substitution~\Eq{(1)}, that
\[
\int \frac{\xi x + \eta}{X_{1} \sqrt{X}}\, dx
  = H \int \frac{t\,dt}{(\mathbf{A}t^{2} + \mathbf{B}) \sqrt{At^{2} + B}}
  + K \int \frac{dt}{(\mathbf{A}t^{2} + \mathbf{B}) \sqrt{At^{2} + B}},
\]
where $\mathbf{A}$,~$\mathbf{B}$, $A$,~$B$, $H$, and~$K$ are constants. Of these two integrals, the first
is rationalised by the substitution
\[
\frac{1}{\sqrt{At^{2} + B}} = u,
\]
and the second by the substitution
\[
\frac{t}{\sqrt{At^{2} + B}} = v.\footnote
  {The method sketched here is that followed by Stolz (see the references given
on p.~21). Dr~Bromwich's method is different in detail but the same in principle.}
\]

It should be observed that this method fails in the special case in which
%% -----File: 039.png---
$a\beta - b\alpha = 0$. In this case, however, the substitution $ax + b = t$ reduces the
integral to one of the form
\[
\int \frac{Ht + K}{(\mathbf{A}t^{2} + \mathbf{B}) \sqrt{At^{2} + B}} \,dt,
\]
and the reduction may then be completed as before.

\Item{(ii)} An alternative method is to use Sir~G. Greenhill's substitution
\[
t = \sqrt{\frac{X}{\alpha x^{2} + 2\beta x + \gamma}}
  = \sqrt{\frac{X}{X_{1}}}.
\]
If
\[
J = (a\beta - b\alpha)x^{2} - (c\alpha - a\gamma)x + (b\gamma - c\beta),
\]
then
\[
\frac{1}{t}\, \frac{dt}{dx} = \frac{J}{XX_{1}}.
\Tag{(1)}
\]
The maximum and minimum values of~$t$ are given by $J = 0$.

Again
\[
t^{2} - \lambda
  = \frac{(a - \lambda\alpha)x^{2} + 2(b - \lambda\beta)x + (c - \lambda\gamma)}{X_{1}};
\]
and the numerator will be a perfect square if
\[
K = (\alpha\gamma - \beta^{2})\lambda^{2}
  - (a\gamma + c\alpha - 2b\beta)\lambda
  + (ac - b^{2}) = 0.
\]

It will be found by a little calculation that the discriminant of this
quadratic and that of~$J$ differ from one another and from
\[
(\phi - \phi_{1})(\phi - \phi_{1}')(\phi' - \phi_{1})(\phi' - \phi_{1}'),
\]
where $\phi$,~$\phi'$ are the roots of $X = 0$ and $\phi_{1}$,~$\phi_{1}'$ those of $X_{1} = 0$, only by
a constant factor which is always negative. Since $\phi_{1}$~and~$\phi_{1}'$ are conjugate
complex numbers, this product is positive, and so $J = 0$ and $K = 0$ have real
roots\footnotemark.
  \footnotetext{That the roots of $J = 0$ are real has been proved already (\hyperlink{5 para 6.}{p.~28}) in a different
  manner.}%
We denote the roots of the latter by
\[
\lambda_{1}, \lambda_{2} \quad(\lambda_{1} > \lambda_{2}).
\]
Then
\begin{alignat*}{2}
\lambda_{1} - t^{2}
  &=  \frac{\{x \sqrt{\lambda_{1}\alpha - a} + \sqrt{\lambda_{1}\gamma - c}\}^{2}}{X_{1}}
  &&= \frac{(mx + n)^{2}}{X_{1}},
\Tag{(2)}\\
%
t^{2} - \lambda_{2}
  &=  \frac{\{x \sqrt{a - \lambda_{2}\alpha} + \sqrt{c - \lambda_{2}\gamma}\}^{2}}{X_{1}}
  &&= \frac{(m'x + n')^{2}}{X_{1}},
\Tag{(2')}
\end{alignat*}
say. Further, since $t^{2} - \lambda$ can vanish for two equal values of~$x$ only if $\lambda$~is
equal to $\lambda_{1}$~or~$\lambda_{2}$, \textit{i.e.}\ when $t$~is a maximum or a minimum, $J$~can differ from
\[
(mx + n)(m'x + n')
\]
only by a constant factor; and by comparing coefficients and using the
identity
\[
(\lambda_{1}\alpha - a)(a - \lambda_{2}\alpha)
  = \frac{(a\beta - b\alpha)^{2}}{\alpha\gamma - \beta^{2}},
\]
we find that
\[
J = \sqrt{\alpha\gamma - \beta^{2}}(mx + n)(m'x + n').
\Tag{(3)}
\]

Finally, we can write $\xi x + \eta$ in the form
\[
A(mx + n) + B(m'x + n').
\]
%% -----File: 040.png---
{\small
Using equations \Eq{(1)}, \Eq{(2)}, \Eq{(2')}, and \Eq{(3)}, we find that
\begin{align*}
\int \frac{\xi x + \eta}{X_{1} \sqrt{X}}\, dx &= \int \frac{A(mx + n) + B(m'x + n')}{J} \sqrt{X_{1}}\, dt\\
  &= \frac{A}{\sqrt{\alpha\gamma - \beta^{2}}} \int \frac{dt}{\sqrt{\lambda_{1} - t^{2}}} + \frac{B}{\sqrt{\alpha\gamma - \beta^{2}}} \int \frac{dt}{\sqrt{t^{2} - \lambda_{2}}},
\end{align*}
and the integral is reduced to a sum of two standard forms.

This method is very elegant, and has the advantage that the whole work
of transformation is performed in one step. On the other hand it is
somewhat artificial, and it is open to the logical objection that it introduces
the root $\sqrt{X_{1}}$, which, in virtue of Laplace's principle (\hyperlink{3 para 2.}{\textsc{iii}., 2}), cannot really
be involved in the final result\footnotemark.\footnotetext
{The superfluous root may be eliminated from the result by a trivial transformation,
just as $\sqrt{1 + x^{2}}$ may be eliminated from
\[
\arcsin \frac{x}{\sqrt{1 + x^{2}}}
\]
by writing this function in the form $\arctan x$.}}

\Paragraph{5}{7.} We may now proceed to consider the general case to which the
theorem of \hyperlink{4 para 2.}{\textsc{iv}., 2} applies. It will be convenient to recall two well-known
definitions in the theory of algebraical plane curves. A curve
of degree $n$ can have at most $\frac{1}{2} (n - 1)(n - 2)$ double points\footnotemark.\footnotetext
{Salmon, \emph{Higher plane curves}, p.~29.}
If the
actual number of double points is $\nu$, then the number
\[
p = \tfrac{1}{2} (n - 1)(n - 2) - \nu
\]
is called the \emph{deficiency}\footnote{Salmon, \ibid., p.~29. French \emph{genre}, German \emph{Geschlecht}.} of the curve.

If the coordinates $x$, $y$ of the points on a curve can be expressed
\emph{rationally} in terms of a parameter $t$ by means of equations
\[
x = R_{1}(t),\quad y = R_{2}(t),
\]
then we shall say that the curve is \emph{unicursal}. In this case we have
seen that we can always evaluate
\[
\int R(x, y)\, dx
\]
in terms of elementary functions.

The fundamental theorem in this part of our subject is

\begin{Result}
`A curve whose deficiency is zero is unicursal, and vice versa'.
\end{Result}

{\Loosen Suppose first that the curve possesses the maximum number of
double points\footnotemark.\footnotetext
  {We suppose in what follows that the singularities of the curve are all ordinary
nodes. The necessary modifications when this is not the case are not difficult to
make. An ordinary multiple point of order~$k$ may be regarded as equivalent to
$\frac{1}{2} k(k - 1)$ ordinary double points. A curve of degree~$n$ which has an ordinary
multiple point of order~$n - 1$, equivalent to $\frac{1}{2} (n - 1)(n - 2)$ ordinary double points,
is therefore unicursal. The theory of higher plane curves abounds in puzzling
particular cases which have to be fitted into the general theory by more or less
obvious conventions, and to give a satisfactory account of a complicated compound
singularity is sometimes by no means easy. In the investigation which follows we
confine ourselves to the simplest case.}
Since
\[
\tfrac{1}{2} (n - 1)(n - 2) + n - 3 = \tfrac{1}{2} (n - 2)(n + 1) - 1,
\]
%% -----File: 041.png---
and $\frac{1}{2} (n - 2)(n + 1)$ points are just sufficient to determine a curve of
degree $n - 2$\footnotemark,
  \footnotetext{Salmon, \textit{l.c.}, p.~16.}
we can draw, through the $\frac{1}{2} (n - 1)(n - 2)$ double points
and $n - 3$~other points chosen arbitrarily on the curve, a simply infinite
set of curves of degree~$n - 2$, which we may suppose to have the
equation
\[
g(x, y) + t\,h(x, y) = 0,
\]
where $t$~is a variable parameter and $g = 0$, $h = 0$ are the equations of
two particular members of the set. Any one of these curves meets
the given curve in $n(n - 2)$~points, of which $(n - 1)(n - 2)$ are accounted
for by the $\frac{1}{2} (n - 1)(n - 2)$ double points, and $n - 3$~by the
other $n - 3$ arbitrarily chosen points. These
\[
(n - 1)(n - 2) + n - 3 = n(n - 2) - 1
\]
points are independent of~$t$; and so there is but \emph{one} point of intersection
which depends on~$t$. The coordinates of this point are given by
\[
g(x, y) + t\,h(x, y) = 0,\quad f(x, y) = 0.
\]
The elimination of~$y$ gives an equation of degree $n(n - 2)$ in~$x$, whose
coefficients are polynomials in~$t$; and but one root of this equation
varies with~$t$. The eliminant is therefore divisible by a factor of
degree $n(n - 2) - 1$ which does not contain~$t$. There remains a simple
equation in~$x$ whose coefficients are polynomials in~$t$. Thus the
$x$-coordinate of the variable point is determined as a rational function
of~$t$, and the $y$-coordinate may be similarly determined.

We may therefore write
\[
x = R_{1}(t),\quad y = R_{2}(t).
\]
If we reduce these fractions to the same denominator, we express the
coordinates in the form
\[
x = \frac{\phi_{1}(t)}{\phi_{3}(t)},\quad
y = \frac{\phi_{2}(t)}{\phi_{3}(t)},
\Tag{(1)}
\]
where $\phi_{1}$,~$\phi_{2}$,~$\phi_{3}$ are polynomials which have no common factor. The
polynomials will in general be of degree~$n$; none of them can be of
%% -----File: 042.png---
higher degree, and one at least must be actually of that degree, since
an arbitrary straight line
\[
\lambda x + \mu y + \nu = 0
\]
must cut the curve in exactly $n$~points\footnotemark.
  \footnotetext{See Niewenglowski's \textit{Cours de géométrie analytique}, vol.~2, p.~103. By way of
  illustration of the remark concerning particular cases in the footnote~(§) to \hyperlink{5 para 7.}{page~30},
  the reader may consider the example given by Niewenglowski in which
  \[
  x = \frac{t^{2}}{t^{2} - 1},\quad
  y = \frac{t^{2} + 1}{t^{2} - 1};
  \]
  equations which appear to represent the straight line $2x = y + 1$ (part of the line
  only, if we consider only real values of~$t$).}

We can now prove the second part of the theorem. If
\[
x : y : 1 :: \phi_{1}(t) : \phi_{2}(t) : \phi_{3}(t),
\]
where $\phi_{1}$,~$\phi_{2}$,~$\phi_{3}$ are polynomials of degree~$n$, then the line
\[
ux + vy + w = 0
\]
will meet the curve in $n$~points whose parameters are given by
\[
u\phi_{1}(t) + v\phi_{2}(t) + w\phi_{3}(t) = 0.
\]

This equation will have a double root~$t_{0}$ if
\begin{align*}
  u\phi_{1}(t_{0}) + v\phi_{2}(t_{0}) + w\phi_{3}(t_{0}) &= 0,\\
  u\phi_{1}'(t_{0}) + v\phi_{2}'(t_{0}) + w\phi_{3}'(t_{0}) &= 0.
\end{align*}
Hence the equation of the tangent at the point~$t_{0}$ is
\[
\begin{vmatrix}
  x  &  y  &  1\\
  \phi_{1}(t_{0}) & \phi_{2}(t_{0}) & \phi_{3}(t_{0})\\
  \phi_{1}'(t_{0})& \phi_{2}'(t_{0})& \phi_{3}'(t_{0})
\end{vmatrix}
= 0.
\Tag{(2)}
\]

If $(x, y)$~is a fixed point, then the equation~\Eq{(2)} may be regarded as
an equation to determine the parameters of the points of contact
of the tangents from~$(x, y)$. Now
\[
\phi_{2}(t_{0})\phi_{3}'(t_{0}) - \phi_{2}'(t_{0})\phi_{3}(t_{0})
\]
is of degree $2n - 2$ in~$t_{0}$, the coefficient of~$t_{0}^{2n-1}$ obviously vanishing.
Hence in general the number of tangents which can be drawn to a
unicursal curve from a fixed point (the \emph{class} of the curve) is~$2n - 2$.
But the class of a curve whose only singular points are $\delta$~nodes is
known\footnote
  {Salmon, \textit{l.c.}, p.~54.}
to be $n(n - 1) - 2\delta$. Hence the number of nodes is
\[
\tfrac{1}{2} \{n(n - 1) - (2n - 2)\} = \tfrac{1}{2} (n - 1)(n - 2).
\]

It is perhaps worth pointing out how the proof which precedes requires
modification if some only of the singular points are nodes and the rest
ordinary cusps. The first part of the proof remains unaltered. The equation~\Eq(2)
%% -----File: 043.png---
must now be regarded as giving the values of~$t$ which correspond to
(\ia)~points at which the tangent passes through~$(x, y)$ and (\ib)~cusps, since any
line through a cusp `cuts the curve in two coincident points'\footnotemark.
  \footnotetext{This means of course that the equation obtained by substituting for $x$~and~$y$,
  in the equation of the line, their parametric expressions in terms of~$t$, has a
  repeated root. This property is possessed by the tangent at an ordinary point and
  by any line through a cusp, but not by any line through a node except the two
  tangents.}
We have
therefore
\[
2n - 2 = m + \kappa,
\]
where $m$~is the class of the curve. But
\[
m = n(n - 1) - 2\delta - 3\kappa,\footnotemark
\]
\footnotetext{Salmon, \textit{l.c.}, p.~65.}%
and so
\[
\delta + \kappa = \tfrac{1}{2}(n - 1)(n - 2).\footnotemark
\]
\footnotetext{I owe this remark to Mr~A.~B. Mayne. Dr~Bromwich has however pointed
  out to me that substantially the same argument is given by Mr~W.~A. Houston, `Note
  on unicursal plane curves', \textit{Messenger of mathematics}, vol.~28, 1899, pp.~187--189.}%


\Paragraph{5}{8.} \Item{(i)} The preceding argument fails if $n < 3$, but we have already
seen that all conics are unicursal. The case next in importance is
that of a cubic with a double point. If the double point is not at
infinity we can, by a change of origin, reduce the equation of the
curve to the form
\[
(ax + by)(cx + dy) = px^{3} + 3qx^{2}y + 3rxy^{2} + sy^{3};
\]
and, by considering the intersections of the curve with the line
$y = tx$, we find
\[
x = \frac{(a + bt)(c + dt)}{p + 3qt + 3rt^{2} + st^{3}},\quad
y = \frac{t(a + bt)(c + dt)}{p + 3qt + 3rt^{2} + st^{3}}.
\]

If the double point is at infinity, the equation of the curve is of the
form
\[
(\alpha x + \beta y)^{2}(\gamma x + \delta y) + \epsilon x + \zeta y + \theta = 0,
\]
the curve having a pair of parallel asymptotes; and, by considering
the intersection of the curve with the line $\alpha x + \beta y = t$, we find
\[
x = -\frac{\delta t^{3} + \zeta t + \beta\theta}
          {(\beta\gamma - \alpha\delta)t^{2} + \epsilon\beta - \alpha\zeta},\quad
y =  \frac{\gamma t^{3} + \epsilon t + \alpha\theta}
          {(\beta\gamma - \alpha\delta)\DPtypo{t^{3}}{t^{2}} + \epsilon\beta - \alpha\zeta}.
\]

\Item{(ii)} The case next in complexity is that of a quartic with three double
points.

\Item{(\ia)} The lemniscate
\[
(x^{2} + y^{2})^{2} = a^{2}(x^{2} - y^{2})
\]
has three double points, the origin and the circular points at infinity. The
circle
\[
x^{2} + y^{2} = t(x - y)
\]
%% -----File: 044.png---
passes through these points and one other fixed point at the origin, as it
touches the curve there. Solving, we find
\[
x = \frac{a^{2}t(t^{2} + a^{2})}{t^{4} + a^{4}},\quad
y = \frac{a^{2}t(t^{2} - a^{2})}{t^{4} + a^{4}}.
\]

\Item{(\ib)} The curve
\[
2ay^{3} - 3a^{2}y^{2} = x^{4} - 2a^{2}x^{2}
\]
has the double points $(0, 0)$, $(a, a)$, $(-a, a)$. Using the auxiliary conic
\[
x^{2} - ay = tx(y - a),
\]
we find
\[
x = \frac{a}{t^{3}}(2 - 3t^{2}),\quad
y = \frac{a}{2t^{4}}(2 - 3t^{2})(2 - t^{2}).
\]

\Item{(iii)} \Item{(\ia)} The curve
\[
y^{n} = x^{n} + ax^{n-1}
\]
has a multiple point of order~$n - 1$ at the origin, and is therefore unicursal.
In this case it is sufficient to consider the intersection of the curve with the
line $y = tx$. This may be harmonised with the general theory by regarding
the curve
\[
y^{n-3}(y - tx) = 0,
\]
as passing through each of the $\frac{1}{2} (n - 1)(n - 2)$ double points collected at the
origin and through $n - 3$~other fixed points collected at the point
\[
x = -a,\quad y = 0.
\]

The curves
\begin{align*}
y^{n} &= x^{n} + ax^{n-1},
\Tag{(1)} \\
y^{n} &= 1 + az,
\Tag{(2)}
\end{align*}
are projectively equivalent, as appears on rendering their equations homogeneous
by the introduction of variables $z$ in~\Eq{(1)} and $x$ in~\Eq{(2)}. We conclude
that \Eq{(2)}~is unicursal, having the maximum number of double points at
infinity. In fact we may put
\[
y = t,\quad az = t^{n} - 1.
\]
The integral
\[
\int R\{z, \sqrt[n]{1 + az}\}\,dz
\]
is accordingly an elementary function.

\Item{(\ib)} The curve
\[
y^{m} = A(x - a)^{\mu} (x - b)^{\nu}
\]
is unicursal if and only if either (i)~$\mu = 0$ or (ii)~$\nu = 0$ or (iii)~$\mu + \nu = m$.
Hence the integral
\[
\int R\{x, (x - a)^{\mu/m} (x - b)^{\nu/n}\}\,dx
\]
is an elementary function, for all forms of~$R$, in these three cases only; of
course it is integrable for special forms of~$R$ in other cases\footnotemark.
  \footnotetext{See Ptaszycki, `Extrait d'une lettre adressée à M.~Hermite', \textit{Bulletin des
  sciences mathématiques}, ser.~2, vol.~12, 1888, pp.~262--270: Appell and Goursat,
  \textit{Théorie des fonctions algébriques}, p.~245.}
%% -----File: 045.png---

\Paragraph{5}{9.} There is a similar theory connected with unicursal curves
in space of any number of dimensions. Consider for example the
integral
\[
\int R\{x, \sqrt{ax + b}, \sqrt{cx + d}\}\,dx.
\]
A linear substitution $\DPtypo{x}{y} = lx + m$ reduces this integral to the form
\[
\int R_{1}\{y, \sqrt{y + 2}, \sqrt{y - 2}\}\,dy;
\]
and this integral can be rationalised by putting
\[
y = t^{2} + \frac{1}{t^{2}},\quad
\sqrt{y + 2} = t + \frac{1}{t},\quad
\sqrt{y - 2} = t - \frac{1}{t}.
\]

The curve whose Cartesian coordinates $\xi$,~$\eta$,~$\zeta$ are given by
\[
\xi : \eta : \zeta : 1 :: t^{4} + 1 : t(t^{2} + 1) : t(t^{2} - 1) : t^{2},
\]
is a unicursal twisted quartic, the intersection of the parabolic cylinders
\[
\xi = \eta^{2} - 2,\quad
\xi = \zeta^{2} + 2.
\]

It is easy to deduce that the integral
\[
\int R\left\{x, \sqrt{\frac{ax + b}{mx + n}}, \sqrt{\frac{cx + d}{mx + n}}\right\} dx
\]
is always an elementary function.

\Paragraph{5}{10.} When the deficiency of the curve $f(x, y) = 0$ is not zero, the
integral
\[
\int R(x, y)\,dx
\]
is in general not an elementary function; and the consideration of
such integrals has consequently introduced a whole series of classes of
new transcendents into analysis. The simplest case is that in which
the deficiency is unity: in this case, as we shall see later on, the
integrals are expressible in terms of elementary functions and certain
new transcendents known as elliptic integrals. When the deficiency
rises above unity the integration necessitates the introduction of new
transcendents of growing complexity.

But there are infinitely many particular cases in which integrals,
associated with curves whose deficiency is unity or greater than unity,
%% -----File: 046.png---
can be expressed in terms of elementary functions, or are even
algebraical themselves. For instance the deficiency of
\[
y^{2} = 1 + x^{3}
\]
is unity. But
\begin{gather*}
\int\frac{x + 1}{x - 2}\, \frac{dx}{\sqrt{1 + x^{3}}}
  = 3\log\frac{(1 + x)^{2} - 3\sqrt{1 + x^{3}}}
              {(1 + x)^{2} + 3\sqrt{1 + x^{3}}},\\
%
\int\frac{2 - x^{3}}{1 + x^{3}}\, \frac{dx}{\sqrt{1 + x^{3}}}
  = \frac{2x}{\sqrt{1 + x^{3}}}.
\end{gather*}
And, before we say anything concerning the new transcendents to
which integrals of this class in general give rise, we shall consider what
has been done in the way of formulating rules to enable us to identify
such cases and to assign the form of the integral when it is an
elementary function. It will be as well to say at once that this
problem has not been solved completely.

\Paragraph{5}{11.} The first general theorem of this character deals with the
case in which the integral is algebraical, and asserts that \begin{Result}if
\[
u = \int y\, dx
\]
is an algebraical function of~$x$, then it is a rational function of~$x$~and~$y$.
\end{Result}

Our proof will be based on the following lemmas.

\begin{Result}
\Item{(1)} If $f(x, y)$ and~$g(x, y)$ are polynomials, and there is no factor
common to all the coefficients of the various powers of~$y$ in~$g(x, y)$; and
\[
f(x, y) = g(x, y)h(x),
\]
where $h(x)$~is a rational function of~$x$; then $h(x)$~is a polynomial.
\end{Result}

Let $h = P/Q$, where $P$~and~$Q$ are polynomials without a common
factor. Then
\[
fQ = gP.
\]
If $x - a$~is a factor of~$Q$, then
\[
g(a, y) = 0
\]
for all values of~$y$; and so all the coefficients of powers of~$y$ in~$g(x, y)$
are divisible by~$x - a$, which is contrary to our hypotheses. Hence
$Q$~is a constant and $h$~a polynomial.

\begin{Result}
\Item{(2)} Suppose that $f(x, y)$~is an irreducible polynomial, and that
$y_{1}$, $y_{2}$,~\dots, $y_{n}$ are the roots of
\[
f(x, y) = 0
\]
%% -----File: 047.png---
in a certain domain~$D$. Suppose further that $\phi(x, y)$~is another
polynomial, and that
\[
\phi(x, y_{1}) = 0.
\]
Then
\[
\phi(x, y_{s}) = 0,
\]
where $y_{s}$~is any one of the roots of~\Eq{(1)}; and
\[
\phi(x, y) = f(x, y) \psi(x, y),
\]
where $\psi(x, y)$ also is a polynomial in $x$~and~$y$.
\end{Result}

Let us determine the highest common factor~$\varpi$ of $f$~and~$\phi$, considered
as polynomials in~$y$, by the ordinary process for the determination
of the highest common factor of two polynomials. This
process depends only on a series of algebraical divisions, and so $\varpi$~is a
polynomial in~$y$ with coefficients rational in~$x$. We have therefore
\begin{gather*}
\varpi(x, y) = \omega(x, y) \lambda(x),
\Tag{(1)}\\
f(x, y) = \omega(x, y) p(x, y) \mu(x) = g(x, y) \mu(x),
\Tag{(2)}\\
\phi(x, y) = \omega(x, y) q(x, y) \nu(x) = h(x, y) \nu(x),
\Tag{(3)}
\end{gather*}
where $\omega$, $p$,~$q$,~$g$, and~$h$ are polynomials and $\lambda$,~$\mu$, and~$\nu$ rational
functions; and evidently we may suppose that neither in~$g$ nor in~$h$
have the coefficients of all powers of~$y$ a common factor. Hence, by
Lemma~(1), $\mu$~and~$\nu$ are polynomials. But $f$~is irreducible, and therefore
$\mu$ and either $\omega$~or~$p$ must be constants. If $\omega$~were a constant,
$\varpi$~would be a function of $x$~only. But this is impossible. For we can
determine polynomials $L$,~$M$ in~$y$, with coefficients rational in~$x$, such
that
\[
Lf + M\phi = \varpi,
\Tag{(4)}
\]
and the left-hand side of~\Eq{(4)} vanishes when we write $y_{1}$ for~$y$. Hence
$p$~is a constant, and so $\omega$~is a constant multiple of~$f$. The truth of
the lemma now follows from~\Eq{(3)}.

It follows from Lemma~(2) that \begin{Result}$y$~cannot satisfy any equation of
degree less than~$n$ whose coefficients are polynomials in~$x$.
\end{Result}

\begin{Result}
\Item{(3)} If $y$~is an algebraical function of~$x$, defined by an equation
\[
f(x, y) = 0
\Tag{(1)}
\]
of degree~$n$, then any rational function $R(x, y)$ of $x$~and~$y$ can be
expressed in the form
\[
R(x, y) = R_{0} + R_{1}y + \dots + R_{n-1}y^{n-1},
\Tag{(2)}
\]
where $R_{0}$, $R_{1}$,~\dots, $R_{n-1}$ are rational functions of~$x$.
\end{Result}
%% -----File: 048.png---

The function~$y$ is one of the $n$~roots of~\Eq{(1)}. Let $y$, $y'$, $y''$,~\dots\ be the
complete system of roots. Then
\begin{align*}
R(x, y) &= \frac{P(x, y)}{Q(x, y)}\\
        &= \frac{P(x, y) Q(x, y') Q(x, y'')\dots}
                {Q(x, y) Q(x, y') Q(x, y'')\dots},
\Tag{(3)}
\end{align*}
where $P$~and~$Q$ are polynomials. The denominator is a polynomial in~$x$
whose coefficients are symmetric polynomials in $y$, $y'$, $y''$,~\dots, and is
therefore, by \hyperlink{2 para 3.}{\textsc{ii.},~§3,~(i)}, a rational function of~$x$. On the other hand
\[
Q(x, y') Q(x, y'')\dots
\]
is a polynomial in~$x$ whose coefficients are symmetric polynomials
in $y'$, $y''$,~\dots, and therefore, by \hyperlink{2 para 3.}{\textsc{ii.},~§3,~(ii)}, polynomials in~$y$ with
coefficients rational in~$x$. Thus the numerator of~\Eq{(3)} is a polynomial
in~$y$ with coefficients rational in~$x$.

It follows that $R (x, y)$~is a polynomial in~$y$ with coefficients rational
in~$x$. From this polynomial we can eliminate, by means of~\Eq{(1)}, all
powers of~$y$ as high as or higher than the~$n$th. Hence $R(x, y)$~is of
the form prescribed by the lemma.

\Paragraph{5}{12.} We proceed now to the proof of our main theorem. We have
\[
\int y\,dx = u
\]
where $u$~is algebraical. Let
\[
f(x, y) = 0,\quad \psi(x, u) = 0
\Tag{(1)}
\]
be the irreducible equations satisfied by $y$~and~$u$, and let us suppose
that they are of degrees $n$~and~$m$ respectively. The first stage in the
proof consists in showing that
\[
m = n.
\]
It will be convenient now to write $y_{1}$,~$u_{1}$ for $y$,~$u$, and to denote by
\[
y_{1},\ y_{2},\ \dots,\ y_{n},\quad
u_{1},\ u_{2},\ \dots,\ u_{m},
\]
the complete systems of roots of the equations~\Eq{(1)}.

We have
\[
\psi(x, u_{1}) = 0,
\]
and so
\[
\chi_{1}
  = \frac{\dd\psi}{\dd x} + \frac{\dd\psi}{\dd u_{1}}\, \frac{du_{1}}{dx}
  = \frac{\dd\psi}{\dd x} + y_{1} \frac{\dd\psi}{\dd u_{1}} = 0.
\]
Now let
\[
\Omega(x, u_{1})
  = \prod_{r=1}^{n} \left(\frac{\dd\psi}{\dd x} + y_{r} \frac{\dd\psi}{\dd u_{1}}\right).
\]
Then $\Omega$~is a polynomial in~$u_{1}$, with coefficients symmetric in $y_{1}$, $y_{2}$,~\dots, $y_{n}$
and therefore rational in~$x$.
%% -----File: 049.png---

The equations $\psi = 0$ and $\Omega = 0$ have a root~$u_{1}$ in common, and the
first equation is irreducible. It follows, by Lemma~(2) of~\hyperlink{5 para 11.}{§11}, that
\[
\Omega(x, u_{s}) = 0
\]
for $s = 1$, $2$,~\dots, $m$.\footnote
  {If $p(x)$~is the least common multiple of the denominators of the coefficients
  of powers of~$u$ in~$\Omega$, then
  \[
  \Omega(x, u) p(x) = \chi(x, u),
  \]
  where $\chi$~is a polynomial. Applying Lemma~(2), we see that $\chi(x, u_{s}) = 0$, and so
  \[
  \Omega(x, u_{s}) = 0.
  \]}
And from this it follows that, when $s$~is given,
we have
\[
\frac{\dd \psi}{\dd x} + y_{r} \frac{\dd \psi}{\dd u_{s}}
\Tag{(2)}
\]
for some value of the suffix~$r$.

But we have also
\[
\frac{\dd \psi}{\dd x} + \frac{\dd \psi}{\dd u_{s}}\, \frac{du_{s}}{dx} = 0;
\Tag{(3)}
\]
and from \Eq{(2)}~and~\Eq{(3)} it follows\footnote
  {It is impossible that $\psi$~and~$\dfrac{\dd \psi}{\dd u}$ should both vanish for $u = u_{s}$, since $\psi$~is
  irreducible.}
that
\[
\frac{du_{s}}{dx} = y_{r},
\Tag{(4)}
\]
\textit{i.e.}\ that \emph{every~$u$ is the integral of some~$y$.}

In the same way we can show that \emph{every~$y$ is the derivative of some~$u$.}
Let
\[
\omega(x, y_{1})
  = \prod_{s=1}^{m}\left(
      \frac{\dd \psi}{\dd x} + y_{1}\frac{\dd \psi}{\dd u_{s}}
  \right).
\]
Then $\omega$~is a polynomial in~$y_{1}$, with coefficients symmetric in $u_{1}$, $u_{2}$,~\dots, $u_{m}$
and therefore rational in~$x$. The equations $f = 0$ and $\omega = 0$ have a
root~$y_{1}$ in common, and so
\[
\omega(x, y_{r}) = 0
\]
for $r = 1$, $2$,~\dots, $n$. From this we deduce that, when $r$~is given, \Eq{(2)}~must
be true for some value of~$s$, and so that the same is true of~\Eq{(4)}.

Now it is impossible that, in~\Eq{(4)}, two different values of~$s$ should
correspond to the same value of~$r$. For this would involve
\[
u_{s} - u_{t} = c
\]
where $s \neq t$ and $c$~is a constant. Hence we should have
\[
\psi(x, u_{s}) = 0,\quad
\psi(x, u_{s} - c) = 0.
\]
%% -----File: 050.png---
Subtracting these equations, we should obtain an equation of degree
$m - 1$ in~$u_{s}$, with coefficients which are polynomials in~$x$; and this is
impossible. In the same way we can prove that two different values of~$r$
cannot correspond to the same value of~$s$.

The equation~\Eq{(4)} therefore establishes a one-one correspondence
between the values of $r$~and~$s$. It follows that
\[
m = n.
\]
It is moreover evident that, by arranging the suffixes properly, we can
make
\[
\frac{du_{r}}{dx} = y_{r}
\Tag{(5)}
\]
for $r = 1$, $2$,~\dots, $n$.

\Paragraph{5}{13.} We have
\[
y_{r} = \frac{du_{r}}{dx}
  = - \frac{\dd \psi}{\dd x} \bigg/ \frac{\dd \psi}{\dd u_{r}}
  = R(x, u_{r}),
\]
where $R$~is a rational function which may, in virtue of Lemma~(3) of~\hyperlink{5 para 11.}{§~11},
be expressed as a polynomial of degree $n - 1$ in~$u_{r}$, with coefficients
rational in~$x$.

The product
\[
\prod_{s \neq r} (z - y_{s})
\]
is a polynomial of degree $n - 1$ in~$z$, with coefficients which are symmetric
polynomials in $y_{1}$, $y_{2}$,~\dots, $y_{r-1}$, $y_{r+1}$,~\dots,~$y_{n}$ and therefore,
by \hyperlink{2 para 3.}{\textsc{ii.},~§3,~(ii)}, polynomials in~$y_{r}$ with coefficients rational in~$x$.
Replacing $y_{r}$ by its expression as a polynomial in~$u_{r}$ obtained above,
and eliminating $u_{r}^{n}$ and all higher powers of~$u_{r}$, we obtain an equation
\[
\prod_{s \neq r} (z - y_{s})
  = \sum_{j=0}^{n-1} \sum_{k=0}^{n-1} S_{j,k}(x) z^{j} u_{r}^{k},
\]
where the $S$'s are rational functions of~$x$ which are, from the method
of their formation, independent of the particular value of~$r$ selected.
We may therefore write
\[
\prod_{s \neq r} (z - y_{s}) = P(x, z, u_{r}),
\]
where $P$~is a polynomial in $z$~and~$u_{r}$ with coefficients rational in~$x$. It
is evident that
\[
P(x, y_{s}, u_{r}) = 0
\]
for every value of $s$ other than~$r$. In particular
\[
P(x, y_{1}, u_{r}) = 0\qquad
(r = 2,\ 3,\ \dots,\DPtypo{}{\ n).}
\]
%% -----File: 051.png---

It follows that the $n-1$~roots of the equation in~$u$
\[
P(x, y_{1}, u) = 0
\]
are $u_{2}$, $u_{3}$,~\dots, $u_{n}$. We have therefore
\begin{align*}
P(x, y_{1}, u)
  &= T_{0}(x, y_{1}) \prod_{2}^{n} (u - u_{r})\\
  &= T_{0}(x, y_{1}) \{u^{n-1} - u^{n-2}(u_{2} + u_{3} + \dots + u_{n}) + \dots\}\\
  &= T_{0}(x, y_{1}) \left[u^{n-1} + u^{n-2}\left\{u_{1} + \frac{B_{1}(x)}{B_{0}(x)}\right\} + \dots\right],
\end{align*}
where $T_{0}(x, y_{1})$ is the coefficient of~$u^{n-1}$ in~$P$, and $B_{0}(x)$ and~$B_{1}(x)$
are the coefficients of $u^{n}$~and~$u^{n-1}$ in~$\psi$. Equating the coefficients of~$u^{n-2}$
on the two sides of this equation, we obtain
\[
u_{1} + \frac{B_{1}(x)}{B_{0}(x)} = \frac{T_{1}(x, y_{1})}{T_{0}(x, y_{1})},
\]
where $T_{1}(x, y_{1})$~is the coefficient of~$u^{n-2}$ in~$P$. Thus the theorem is
proved.

\Paragraph{5}{14.} We can now apply Lemma~(3) of §11; and we arrive at the
final conclusion that \begin{Result}if
\[
\int y\,dx
\]
is algebraical then it can be expressed in the form
\[
R_{0} + R_{1}y + \dots + R_{n-1}y^{n-1},
\]
where $R_{0}$, $R_{1}$,~\dots\ are rational functions of~$x$.
\end{Result}

The most important case is that in which
\[
y = \sqrt[n]{R(x)},
\]
where $R(x)$~is rational. In this case
\begin{align*}
y^{n} &= R(x),
\Tag{(1)}\\
\frac{dy}{dx} &= \frac{R'(x)}{ny^{n-1}}.
\Tag{(2)}
\end{align*}

But
\begin{multline*}
y = R_{0}' + R_{1}'y + \dots + R_{n-1}'y^{n-1} \\
  + \{R_{1} + 2R_{2}y + \dots + (n - 1)R_{n-1}y^{n-2}\}\, \frac{dy}{dx}.
\Tag{(3)}
\end{multline*}
Eliminating $\dfrac{dy}{dx}$ between these equations, we obtain an equation
\[
\varpi(x, y) = 0,
\Tag{(4)}
\]
where $\varpi(x, y)$~is a polynomial. It follows from Lemma~(2) of §11
that this equation must be satisfied by all the roots of~\Eq{(1)}. Thus~\Eq{(4)}
is still true if we replace~$y$ by any other root~$y'$ of~\Eq{(1)}; and as
%% -----File: 052.png---
\Eq{(2)}~is still true when we effect this substitution, it follows that \Eq{(3)}~is
also still true. Integrating, we see that the equation
\[
\int y\,dx = R_{0} + R_{1}y + \dots + R_{n-1}y^{n-1}
\]
is true when $y$~is replaced by~$y'$. We may therefore replace~$y$ by~$\omega y$,
$\omega$~being any primitive $n$th~root of unity. Making this substitution,
and multiplying by~$\omega^{n-1}$, we obtain
\[
\int y\,dx = \omega^{n-1}R_{0} + R_{1}y + \omega R_{2}y + \dots + \omega^{n-2}R_{n-1}y^{n-1};
\]
and on adding the $n$~equations of this type we obtain
\[
\int y\,dx = R_{1}y.
\]
Thus in this case the functions $R_{0}$, $R_{2}$,~\dots, $R_{n-1}$ all disappear.

It has been shown by Liouville\footnote
  {`Premier mémoire sur la détermination des intégrales dont la valeur est
  algébrique', \textit{Journal de l'École Polytechnique}, vol.~14, cahier~22, 1833, pp.~124--148;
  `Second mémoire\dots', \ibid., pp.~149--193.}
that the preceding results enable
us to obtain in all cases, by a finite number of elementary algebraical
operations, a solution of the problem `\emph{to determine whether $\int y\,dx$ is
algebraical, and to find the integral when it is algebraical}'.

{\small \Paragraph{5}{15.} It would take too long to attempt to trace in detail the steps of the
general argument. We shall confine ourselves to a solution of a particular
problem which will give a sufficient illustration of the general nature of the
arguments which must be employed.

We shall determine under what circumstances the integral
\[
\int \frac{dx}{(x - p) \sqrt{ax^{2} + 2bx + c}}
\]
is algebraical. This question might of course be answered by actually
evaluating the integral in the general case and finding when the integral
function reduces to an algebraical function. We are now, however, in a
position to answer it without any such integration.

We shall suppose first that $ax^{2} + 2bx + c$ is not a perfect square. In this
case
\[
y = \frac{1}{\sqrt{X}},
\]
where
\[
X = (x - p)^{2} (ax^{2} + 2bx + c),
\]
and if $\int y\,dx$ is algebraical it must be of the form
\[
\frac{R(x)}{\sqrt{X}}.
\]
Hence
\[
y = \frac{d}{dx}\left(\frac{R}{\sqrt{X}}\right),
\]
or
\[
2X = 2XR' - RX'.
\]
%% -----File: 053.png---

We can now show that $R$~is a polynomial in~$x$. For if $R = U/V$, where $U$
and~$V$ are polynomials, then~$V$, if not a mere constant, must contain a factor
\[
(x - \alpha)^{\mu}\qquad (\mu > 0),
\]
and we can put
\[
R = \frac{U}{W(x - \alpha)^{\mu}},
\]
where $U$~and~$W$ do not contain the factor~$x - \alpha$. Substituting this expression
for~$R$, and reducing, we obtain
\[
\frac{2\mu UWX}{x - \alpha}
  = 2U'WX - 2UW'X - UWX' - 2W^{2}X (x - \alpha)^{\mu}.
\]
Hence $X$~must be divisible by~$x - \alpha$. Suppose then that
\[
X = (x - \alpha)^{k} Y,
\]
where $Y$~is prime to~$x - \alpha$. Substituting in the equation last obtained we
deduce
\[
\frac{(2\mu + k)UWY}{x - \alpha}
  = 2U'WY - 2UW'Y - UWY' - 2W^{2}Y (x - \alpha)^{\mu},
\]
which is obviously impossible, since neither $U$,~$W$, nor~$Y$ is divisible by~$x - \alpha$.
Thus $V$~must be a constant. Hence
\[
\int \frac{dx}{(x - p) \sqrt{ax^{2} + 2bx + c}}
  = \frac{U(x)}{(x - p) \sqrt{ax^{2} + 2bx + c}},
\]
where $U(x)$~is a polynomial.

Differentiating and clearing of radicals we obtain
\[
\{(x - p) (U' - 1) - U\} (ax^{2} + 2bx + c) = U(x - p)(ax + b).
\]
Suppose that the first term in~$U$ is~$Ax^{m}$. Equating the coefficients of~$x^{m+2}$,
we find at once that $m = 2$. We may therefore take
\[
U = Ax^{2} + 2Bx + C,
\]
so that
\begin{multline*}
\{(x - p) (2Ax + 2B - 1) - Ax^{2} - 2Bx - C\} (ax^{2} + 2bx + c) \\
  = (x - p) (ax + b) (Ax^{2} + 2Bx + C).
\Tag{(1)}
\end{multline*}

From~\Eq{(1)} it follows that
\[
(x - p) (ax + b) (Ax^{2} + 2Bx + C)
\]
is divisible by $ax^{2} + 2bx + c$. But $ax + b$~is not a factor of $ax^{2} + 2bx + c$, as
the latter is not a perfect square. Hence either (i)~$ax^{2} + 2bx + c$ and
$Ax^{2} + 2Bx + C$ differ only by a constant factor or (ii)~the two quadratics have
one and only one factor in common, and $x - p$~is also a factor of $ax^{2} + 2bx + c$.
In the latter case we may write
\[
ax^{2} + 2bx + c = a(x - p) (x - q),\quad
Ax^{2} + 2Bx + C = A(x - q) (x - r),
\]
where $p \neq q$, $p \neq r$. It then follows from~\Eq{(1)} that
\[
a(x - p) (2Ax + 2B - 1) - aA(x - q) (x - r) = A(ax + b) (x - r).
\]
Hence $2Ax + 2B - 1$ is divisible by~$x - r$. Dividing by~$aA(x - r)$ we obtain
\[
2(x - p) - (x - q) = x + \frac{b}{a} = x - \tfrac{1}{2}(p + q),
\]
and so $p = q$, which is untrue.
%% -----File: 054.png---

Hence case~(ii) is impossible, and so $ax^{2} + 2bx + c$ and $Ax^{2} + 2Bx + C$ differ
only by a constant factor. It then follows from~\Eq{(1)} that $x - p$~is a factor
of $ax^{2} + 2bx + c$; and the result becomes
\[
\int \frac{dx}{(x - p) \sqrt{ax^{2} + 2bx + c}}
  = K\frac{\sqrt{ax^{2} + 2bx + c}}{x - p},
\]
where $K$~is a constant. It is easily verified that this equation is actually
true when $ap^{2} + 2bp + c = 0$, and that
\[
K = \frac{1}{\sqrt{b^{2} - ac}}.
\]
The formula is equivalent to
\[
\int \frac{dx}{(x - p) \sqrt{(x - p)(x - q)}}
  = \frac{2}{q - p} \sqrt{\frac{x - q}{x - p}}.
\]

There remains for consideration the case in which $ax^{2} + 2bx + c$ is a
perfect square, say $a(x - q)^{2}$. Then
\[
\int \frac{dx}{(x - p)(x - q)}
\]
must be rational, and so $p = q$.

As a further example, the reader may verify that if
\[
y^{3} - 3y + 2x = 0
\]
then
\[
\int y\,dx = \frac{3}{8} (2xy - y^{2}).\footnotemark
\]
  \footnotetext{Raffy, `Sur les quadratures algébriques et logarithmiques', \textit{Annales de l'École
  Normale}, ser.~3, vol.~2, 1885, pp.~185--206.}}%

\Paragraph{5}{16.} The theorem of §11 enables us to complete the proof of the
two fundamental theorems stated without proof in~\hyperlink{2 para 5.}{\textsc{ii.},~§5}, viz.\

\Item{(\ia)} $e^{x}$ is not an algebraical function of~$x$,

\Item{(\ib)} $\log x$ is not an algebraical function of~$x$.

We shall prove~(\ib) as a special case of a more general theorem, viz.\
`\emph{no sum of the form
\[
A \log(x - \alpha) + B \log(x - \beta) + \dots,
\]
in which the coefficients $A$,~$B$,~\dots\ are not all zero, can be an algebraical
function of~$x$}'. To prove this we have only to observe that the sum
in question is the integral of a rational function of~$x$. If then it is
algebraical it must, by the theorem of~§11, be rational, and this we
have already seen to be impossible~(\hyperlink{4 para 2.}{\textsc{iv.},~2}).

That $e^{x}$~is not algebraical now follows at once from the fact that it
is the inverse function of~$\log x$.

\Paragraph{5}{17.} The general theorem of~§11 gives the first step in the rigid
proof of `Laplace's principle' stated in~\hyperlink{3 para 2.}{\textsc{iii.},~§2}. On account of the
immense importance of this principle we repeat Laplace's words:
%% -----File: 055.png---
`\textit{l'intégrale d'une fonction différentielle ne peut contenir d'autres quantités
\DPtypo{radicaux}{radicales} que celles qui entrent dans cette fonction}'. This general
principle, combined with arguments similar to those used above~(\hyperlink{5 para 15.}{§15}) in
a particular case, enables us to prove without difficulty that a great
many integrals cannot be algebraical, notably the standard elliptic
integrals
\[
\int \frac{dx}{\sqrt{(1 - x^{2}) (1 - k^{2}x^{2})}},\quad
\int \sqrt{\frac{1 - x^{2}}{1 - k^{2}x^{2}}}\, dx,\quad
\int \frac{dx}{\sqrt{4x^{3} - g_{2}x - g_{3}}}
\]
which give rise by inversion to the elliptic functions.

\Paragraph{5}{18.} We must now consider in a very summary manner the more
difficult question of the nature of those integrals of algebraical functions
which are expressible in finite terms by means of the elementary
transcendental functions. In the first place \emph{no integral of any algebraical
function can contain any exponential}. Of this theorem it is, as
we remarked before, easy to become convinced by a little reflection,
as doubtless did Laplace, who certainly possessed no rigorous proof.
The reader will find little difficulty in coming to the conclusion that
exponentials cannot be eliminated from an elementary function by
differentiation. But we would strongly recommend him to study the
exceedingly beautiful and ingenious proof of this proposition given by
Liouville\footnotemark.
  \footnotetext{`Mémoire sur les transcendantes elliptiques considérées comme fonctions de
  leur amplitude', \textit{Journal de l'École Polytechnique}, vol.~14, cahier~23, 1834,
  pp.~37--83. The proof may also be found in Bertrand's \textit{Calcul intégral}, p.~99.}%
We have unfortunately no space to insert it here.

{\small It is instructive to consider particular cases of this theorem. Suppose for
example that $\int y\,dx$, where $y$~is algebraical, were a polynomial in $x$~and~$e^{x}$, say
\[
\sum \sum a_{m, n}x^{m}e^{nx}.
\Tag{(1)}
\]
When this expression is differentiated, $e^{x}$~must disappear from it: otherwise
we should have an algebraical relation between $x$~and~$e^{x}$. Expressing the conditions
that the coefficient of every power of~$e^{x}$ in the differential coefficient
of~\Eq{(1)} vanishes identically, we find that the same must be true of~\Eq{(1)}, so that
after all the integral does not really contain~$e^{x}$. Liouville's proof is in reality
a development of this idea.}

The integral of an algebraical function, if expressible in terms
of elementary functions, can therefore only contain algebraical or
logarithmic functions. The next step is to show that the logarithms
must be simple logarithms of algebraical functions and can only
enter linearly, so that the general integral must be of the type
\[
\int y\,dx = u + A \log v + B \log w + \dots,
\]
%% -----File: 056.png---
where $A$,~$B$,~\dots\ are constants and $u$,~$v$, $w$,~\dots\ algebraical functions.
Only when the logarithms occur in this simple form will differentiation
eliminate them.

Lastly it can be shown by arguments similar to those of \hyperlink{5 para 11.}{§§11--14}
that $u$,~$v$, $w$,~\dots\ are rational functions of $x$~and~$y$. Thus $\int y\,dx$, if
an elementary function, is \emph{the sum of a rational function of~$x$ and~$y$
and of certain constant multiples of logarithms of such functions}.
We can suppose that no two of $A$,~$B$,~\dots\ are commensurable, or indeed,
more generally, that no linear relation
\[
A\alpha + B\beta + \dots = 0,
\]
with rational coefficients, holds between them. For if such a relation
held then we could eliminate~$A$ from the integral, writing it in the
form
\[
\int y\,dx = u + B \log(wv^{-\beta/\alpha}) + \dots\DPtypo{}{.}
\]

{\small It is instructive to verify the truth of this theorem in the special case in
which the curve $f(x,y) = 0$ is unicursal. In this case $x$~and~$y$ are rational
functions $R(t)$,~$S(t)$ of a parameter~$t$, and the integral, being the integral of
a rational function of~$t$, is of the form
\[
u + A \log v + B \log w + \dots,
\]
where $u$,~$v$, $w$,~\dots\ are rational functions of~$t$. But $t$~may be expressed, by
means of elementary algebraical operations, as a rational function of $x$~and~$y$.
Thus $u$,~$v$, $w$,~\dots\ are rational functions of $x$~and~$y$.}

The case of greatest interest is that in which $y$~is a rational function
of $x$~and~$\sqrt{X}$, where $X$~is a polynomial. As we have already seen,
$y$~can in this case be expressed in the form
\[
P + \frac{Q}{\sqrt{X}},
\]
where $P$~and~$Q$ are rational functions of~$x$. We shall suppress the
rational part and suppose that $y = Q/\sqrt{X}$. In this case the general
theorem gives
\[
\int\frac{Q}{\sqrt{X}}\,dx
  = S + \frac{T}{\sqrt{X}}
        + A \log(\alpha + \beta\sqrt{X})
        + B \log(\gamma + \delta\sqrt{X}) + \dots,
\]
where $S$,~$T$, $\alpha$,~$\beta$, $\gamma$,~$\delta$,~\dots\ are rational. If we differentiate this equation
we obtain an algebraical identity in which we can change the sign of~$\sqrt{X}$.
Thus we may change the sign of~$\sqrt{X}$ in the integral equation.
If we do this and subtract, and write $2A$,~\dots\ for $A$,~\dots, we obtain
\[
\int\frac{Q}{\sqrt{X}}\,dx
  = \frac{T}{\sqrt{X}}
    + A \log\frac{\alpha + \beta \sqrt{X}}{\alpha - \beta \sqrt{X}}
    + B \log\frac{\gamma + \delta\sqrt{X}}{\gamma - \delta\sqrt{X}} + \dots,
\]
%% -----File: 057.png---
which is the standard form for such an integral. It is evident that we
may suppose $\alpha$,~$\beta$, $\gamma$,~\dots\ to be polynomials.

{\small \Paragraph{5}{19.} \Item{(i)} By means of this theorem it is possible to prove that a number
of important integrals, and notably the integrals
\[
\int \frac{dx}{\sqrt{(1 - x^{2}) (1 - k^{2}x^{2})}},\quad
\int \sqrt{\frac{1 - x^{2}}{1 - k^{2}x^{2}}}\, dx,\quad
\int \frac{dx}{\sqrt{4x^{3} - g_{2}x - g_{3}}},
\]
are not expressible in terms of elementary functions, and so represent genuinely
new transcendents. The formal proof of this was worked out by Liouville\footnotemark;
  \footnotetext{See Liouville's memoir quoted on \hyperlink{5 para 18.}{p.~45} (pp.~45~\textit{et~seq.}).}
it rests merely on a consideration of the possible forms of the differential
coefficients of expressions of the form
\[
\frac{T}{\sqrt{X}}
  + A\log\frac{\alpha + \beta\sqrt{X}}
              {\alpha - \beta\sqrt{X}} + \dots,
\]
and the arguments used are purely algebraical and of no great theoretical
difficulty. The proof is however too detailed to be inserted here. It is not
difficult to find shorter proofs, but these are of a less elementary character,
being based on ideas drawn from the theory of functions\footnotemark.
  \footnotetext{The proof given by Laurent (\textit{Traité d'analyse}, vol.~4, pp.~153~\textit{et~seq.}) appears at
  first sight to combine the advantages of both methods of proof, but unfortunately
  will not bear a closer examination.}

The general questions of this nature which arise in connection with
integrals of the form
\[
\int \frac{Q}{\sqrt{X}}\, dx,
\]
or, more generally,
\[
\int \frac{Q}{\sqrt[m]{X}}\, dx,
\]
are of extreme interest and difficulty. The case which has received most
attention is that in which $m = 2$ and $X$~is of the third or fourth degree, in
which case the integral is said to be \emph{elliptic}. An integral of this kind is
called \emph{pseudo-elliptic} if it is expressible in terms of algebraical and logarithmic
functions. Two examples were given above (\hyperlink{5 para 10.}{§~10}). General methods have
been given for the construction of such integrals, and it has been shown that
certain interesting forms are pseudo-elliptic. In Goursat's \textit{Cours d'analyse}\footnotemark,
  \footnotetext{Second edition, vol.~1, pp.~267--269.}%
for instance, it is shown that if $f(x)$~is a rational function such that
\[
f(x) + f\left(\frac{1}{k^{2}x}\right) = 0,
\]
then
\[
\int \frac{f(x)\, dx}{\sqrt{x(1 - x) (1 - k^{2}x)}}
\]
is pseudo-elliptic. But no method has been devised as yet by which we can
always determine in a finite number of steps whether a \emph{given} elliptic integral
%% -----File: 058.png---
is pseudo-elliptic, and integrate it if it is, and there is reason to suppose that
no such method can be given. And up to the present it has not, so far as
we know, been proved rigorously and explicitly that (\textit{e.g.})\ the function
\[
u = \int \frac{dx}{\sqrt{(1 - x^{2})(1 - k^{2}x^{2})}}
\]
is not a root of an elementary transcendental equation; all that has been
shown is that it is not \emph{explicitly} expressible in terms of elementary transcendents.
The processes of reasoning employed here, and in the memoirs
to which we have referred, do not therefore suffice to prove that the inverse
function $x = \sn u$ is not an elementary function of~$u$. Such a proof must rest
on the known properties of the function $\sn u$, and would lie altogether outside
the province of this tract.

The reader who desires to pursue the subject further will find references
to the original authorities in \hyperlink{Appendix I}{Appendix~I}.

\Item{(ii)} One particular class of integrals which is of especial interest is
that of the \emph{binomial integrals}
\[
\int x^{m}(ax^{n} + b)^{p}\, dx,
\]
where $m$,~$n$,~$p$ are rational. Putting $ax^{n} = bt$, and neglecting a constant
factor, we obtain an integral of the form
\[
\int t^{q}(1 + t)^{p}\, dt,
\]
where $p$~and~$q$ are rational. If $p$~is an integer, and $q$~a fraction~$r/s$, this
integral can be evaluated at once by putting $t = u^{s}$, a substitution which
rationalises the integrand. If $q$~is an integer, and $p = r/s$, we put $1 + t = u^{s}$.
If $p + q$ is an integer, and $p = r/s$, we put $1 + t = tu^{s}$.

It follows from Tschebyschef's researches (to which references are given
in Appendix~I) that these three cases are the only ones in which the integral
can be evaluated in finite form.}

\Paragraph{5}{20.} In \hyperlink{5 para 7.}{§§7--9} we considered in some detail the integrals connected
with curves whose deficiency is zero. We shall now consider
in a more summary way the case next in simplicity, that in which
the deficiency is unity, so that the number of double points is
\[
\tfrac{1}{2} (n - 1)(n - 2) - 1 = \tfrac{1}{2} n(n - 3).
\]
It has been shown by Clebsch\footnote
  {`Über diejenigen Curven, deren Coordinaten sich als elliptische Functionen
  eines Parameters darstellen lassen', \textit{Journal für Mathematik}, vol.~64, 1865,
  pp.~210--270.}
that in this case the coordinates of
the points of the curve can be expressed as \emph{rational functions of
a parameter~$t$ and of the square root of a polynomial in~$t$ of the third
or fourth degree}.
%% -----File: 059.png---

{\small The fact is that the curves
\begin{align*}
y^{2} &= a + bx + cx^{2} + dx^{3}, \\
y^{2} &= a + bx + cx^{2} + dx^{3} + ex^{4},
\end{align*}
are the simplest curves of deficiency~$1$. The first is the typical cubic
without a double point. The second is a quartic with two double points,
in this case coinciding in a `tacnode' at infinity, as we see by making the
equation homogeneous with~$z$, writing $1$ for~$y$, and then comparing the
resulting equation with the form treated by Salmon on p.~215 of his \textit{Higher
plane curves}. The reader who is familiar with the theory of algebraical plane
curves will remember that the deficiency of a curve is unaltered by any
birational transformation of coordinates, and that any curve can be birationally
transformed into any other curve of the same deficiency, so that any
curve of deficiency~$1$ can be birationally transformed into the cubic whose
equation is written above.}

The argument by which this general theorem is proved is very
much like that by which we proved the corresponding theorem for
unicursal curves. The simplest case is that of the general cubic curve.
We take a point on the curve as origin, so that the equation of the
curve is of the form
\[
ax^{3} + 3bx^{2}y + 3cxy^{2} + dy^{3}
  + ex^{2} + 2fxy + gy^{2}
  + hx + ky = 0.
\]
Let us consider the intersections of this curve with the secant $y = tx$.
Eliminating~$y$, and solving the resulting quadratic in~$x$, we see that the
only irrationality which enters into the expression of~$x$ is
\[
\sqrt {T_{2}^{2} - 4T_{1}T_{3}},
\]
where
\[
T_{1} = h + kt,\quad
T_{2} = e + 2ft + gt^{2},\quad
T_{3} = a + 3bt + 3ct^{2} + dt^{3}.
\]

A more elegant method has been given by Clebsch\footnotemark.
  \footnotetext{See Hermite, \textit{Cours d'analyse}, pp.~422--425.}
If we
write the cubic in the form
\[
LMN = P,
\]
where $L$,~$M$,~$N$, $P$ are linear functions of $x$~and~$y$, so that $L$,~$M$,~$N$ are
the asymptotes, then the hyperbolas $LM = t$ will meet the cubic in
four fixed points at infinity, and therefore in two points only which
depend on~$t$. For these points
\[
LM = t,\quad P = tN.
\]
Eliminating~$y$ from these equations, we obtain an equation of the form
\[
Ax^{2} + 2Bx + C = 0,
\]
where $A$,~$B$,~$C$ are quadratics in~$t$. Hence
\[
x = -\frac{B}{A} ± \frac{\sqrt{B^{2} - AC}}{A} = R(t, \sqrt{T}),
\]
%% -----File: 060.png---
where $T = B^{2} - AC$ is a polynomial in~$t$ of degree not higher than the
fourth.

Thus if the curve is
\[
x^{3} + y^{3} - 3axy + 1 = 0,
\]
so that
\[
L = \omega x + \omega^{2} y + a, \quad
M = \omega^{2} x + \omega y + a, \quad
N = x + y + a, \quad
P = a^{3} - 1,
\]
$\omega$~being an imaginary cube root of unity, then we find that the line
\[
x + y + a = \frac{a^{3} - 1}{t}
\]
meets the curve in the points given by
\[
x = \frac{b - at}{2t} ± \frac{\sqrt{3T}}{6t}, \quad
y = \frac{b - at}{2t} \mp \frac{\sqrt{3T}}{6t},
\]
where $b = a^{3} - 1$ and
\[
T = 4t^{3} - 9a^{2}t^{2} + 6abt - b^{2}.
\]

In particular, for the curve
\[
x^{3} + y^{3} + 1 = 0,
\]
we have
\[
x = \frac{-\sqrt{3} + \sqrt{4t^{3} - 1}}{2t \sqrt{3}}, \quad
y = \frac{-\sqrt{3} - \sqrt{4t^{3} - 1}}{2t \sqrt{3}}.
\]

\Paragraph{5}{21.} It will be plain from what precedes that
\[
\int R\{x, \sqrt[3]{a + bx + cx^{2} + dx^{3}}\} \,dx
\]
can always be reduced to an elliptic integral, the deficiency of the cubic
\[
y^{3} = a + bx + cx^{2} + dx^{3}
\]
being unity.

In general integrals associated with curves whose deficiency is
greater than unity cannot be so reduced. But associated with every
curve of, let us say, deficiency~$2$ there will be an infinity of integrals
\[
\int R(x, y) \,dx
\]
reducible to elliptic integrals or even to elementary functions; and
there are curves of deficiency~$2$ for which \emph{all} such integrals are
reducible.

For example, the integral
\[
\int R\{x, \sqrt{x^{6} + ax^{4} + bx^{2} + c}\} \,dx
\]
%% -----File: 061.png---
may be split up into the sum of the integral of a rational function and
two integrals of the types
\[
\int \frac{R(x^{2})\,dx}{\sqrt{x^{6} + ax^{4} + bx^{2} + c}},\
\int \frac{xR(x^{2})\,dx}{\sqrt{x^{6} + ax^{4} + bx^{2} + c}},
\]
and each of these integrals becomes elliptic on putting $x^{2} = t$. But
the deficiency of
\[
y^{2} = x^{6} + ax^{4} + bx^{2} + c
\]
is~$2$. Another example is given by the integral
\[
\int R\{x,\sqrt[4]{x^{4} + ax^{3} + bx^{2} + cx + d}\}\,dx.\footnotemark
\]
\footnotetext{See Legendre, \textit{Traité des fonctions elliptiques}, vol.~1, chs.~26--27, 32--33;
Bertrand, \textit{Calcul intégral}, pp.~67 \textit{et seq.}; and Enneper, \textit{Elliptische Funktionen},
note~1, where abundant references are given.}

\Paragraph{5}{22.} It would be beside our present purpose to enter into any
details as to the general theory of elliptic integrals, still less of the
integrals (usually called Abelian) associated with curves of deficiency
greater than unity. We have seen that if the deficiency is unity then
the integral can be transformed into the form
\[
\int R(x, \sqrt{X})\,dx
\]
where
\[
X = x^{4} + ax^{3} + bx^{2} + cx + d.\footnotemark
\]
\footnotetext{There is a similar theory for curves of deficiency 2, in which $X$ is of the \emph{sixth}
degree.}%
It can be shown that, by a transformation of the type
\[
x =\frac{\alpha t + \beta}{\gamma t + \delta},
\]
this integral can be transformed into an integral
\[
\int R(t, \sqrt{T})\,dt
\]
where
\[
 T = t^{4} + At^{2} + B.
\]

We can then, as when $T$~is of the second degree (§3), decompose
this integral into two integrals of the forms
\[
\int R(t)\,dt,\ \int \frac{R(t)\,dt}{\sqrt{T}}.
\]
Of these integrals the first is elementary, and the second can be
%% -----File: 062.png---
decomposed\footnote
  {See, \textit{e.g.}, Goursat, \textit{Cours d'analyse}, ed.~2, vol.~1, pp.~257~\textit{et~seq.}}
into the sum of an algebraical term, of certain multiples
of the integrals
\[
\int\frac{dt}{\sqrt{T}},\quad
\int\frac{t^{2}\,dt}{\sqrt{T}},
\]
and of a number of integrals of the type
\[
\int \frac{dt}{(t - \tau)\sqrt{T}}.
\]
These integrals cannot in general be reduced to elementary functions,
and are therefore new transcendents.

We will only add, before leaving this part of our subject, that the
algebraical part of these integrals can be found by means of the
elementary algebraical operations, as was the case with the rational
part of the integral of a rational function, and with the algebraical part
of the simple integrals considered in \hyperlink{5 para 14.}{§§14--15}.

\Section{VI. Transcendental functions}

\Paragraph{6}{1.} The theory of the integration of transcendental functions is
naturally much less complete than that of the integration of rational
or even of algebraical functions. It is obvious from the nature of the
case that this must be so, as there is no general theorem concerning
transcendental functions which in any way corresponds to the theorem
that any algebraical combination of algebraical functions may be
regarded as a simple algebraical function, the root of an equation of
a simple standard type.

It is indeed almost true to say that there is no general theory, or
that the theory reduces to an enumeration of the few cases in which
the integral may be transformed by an appropriate substitution into an
integral of a rational or algebraical function. These few cases are
however of great importance in applications.

\Paragraph{6}{2.} \Item{(i)} The integral
\[
\int F(e^{ax}, e^{bx}, \dots, e^{kx})\,dx
\]
where $F$~is an algebraical function, and $a$,~$b$,~\dots, $k$ commensurable
numbers, can always be reduced to that of an algebraical function.
In particular the integral
\[
\int R(e^{ax}, e^{bx}, \dots, e^{kx})\,dx,
\]
%% -----File: 063.png---
where $R$~is rational, is always an elementary function. In the first
place a substitution of the type $x = \alpha y$ will reduce it to the form
\[
\int R(e^{y})\,dy,
\]
and then the substitution $e^{y} = z$ will reduce this integral to the integral
of a rational function.

{\Loosen In particular, since  $\cosh x$ and $\sinh x$ are rational functions of~$e^{x}$,
and $\cos x$~and~$\sin x$ are rational functions of~$e^{ix}$, the integrals}
\[
\int R(\cosh x, \sinh x)\,dx,\quad
\int R(\cos x, \sin x)\,dx
\]
are always elementary functions. In the second place the substitution
just indicated is imaginary, and it is generally more convenient
to use the substitution
\[
\tan \tfrac{1}{2} x = t,
\]
which reduces the integral to that of a rational function, since
\[
\cos x = \frac{1 - t^{2}}{1 + t^{2}},\quad
\sin x = \frac{2t}{1 + t^{2}},\quad
dx = \frac{2\, dt}{1 + t^{2}}.
\]

\Item{(ii)} The integrals
\begin{gather*}
\int R(\cosh x, \sinh x, \cosh 2x,\dots \sinh mx)\,dx, \\
\int R(\cos x, \sin x, \cos 2x,\dots \sin mx)\,dx,
\end{gather*}
are included in the two standard integrals above.

Let us consider some further developments concerning the integral
\[
\int R(\cos x, \sin x)\,dx.\footnotemark
\]
\footnotetext{See Hermite, \textit{Cours d'analyse}, pp.~320~\textit{et~seq}.}%
If we make the substitution $z = e^{ix}$, the subject of integration becomes a
rational function~$H(z)$, which we may suppose split up into

\Item{(\ia)} a constant and certain positive and negative powers of~$z$,

\Item{(\ib)} groups of terms of the type
\[
\frac{A_{0}}{z - a}
  + \frac{A_{1}}{(z - a)^{2}} + \dots
  + \frac{A_{n}}{(z - a)^{n+1}}.
\Tag{(1)}
\]

The terms~(i), when expressed in terms of~$x$, give rise to a term
\[
\sum (c_{k}\cos kx + d_{k}\sin kx).
\]
In the group~\Eq{(1)} we put $z = e ^{ix}$, $a = e^{i\alpha}$ and, using the equation
\[
\frac{1}{z - a}
  = \tfrac{1}{2} e^{-i\alpha} \{-1 - i \cot \tfrac{1}{2} (x - \alpha)\},
\]
%% -----File: 064.png---
we obtain a polynomial of degree~$n + 1$ in $\cot \frac{1}{2} (x - \alpha)$. Since
\[
\cot^{2} x = -1 - \frac{d \cot x}{dx},\quad
\cot^{3} x = -\cot x - \frac{1}{2}\, \frac{d}{dx} (\cot^{2} x),\ \dots,
\]
this polynomial may be transformed into the form
\[
C + C_{0} \cot \tfrac{1}{2} (x - \alpha)
  + C_{1} \frac{d}{dx} \cot \tfrac{1}{2} (x - \alpha) + \dots
  + C_{n} \frac{d^{n}}{dx^{n}} \cot \tfrac{1}{2} (x - \alpha).
\]

The function $R(\cos x, \sin x)$ is now expressed as a sum of a number of
terms each of which is immediately integrable. The integral is a rational
function of $\cos x$~and~$\sin x$ if all the constants~$C_{0}$ vanish; otherwise it includes
a number of terms of the type
\[
2C_{0} \log \sin \tfrac{1}{2} (x - \alpha).
\]

Let us suppose for simplicity that~$H(z)$, when split up into partial fractions,
contains no terms of the types
\[
C,\quad z^{m},\quad z^{-m},\quad (z - a)^{-p}\qquad (p > 1).
\]
Then
\[
R(\cos x, \sin x)
  = C_{0} \cot \tfrac{1}{2} (x - \alpha)
  + D_{0} \cot \tfrac{1}{2} (x - \beta) + \dots,
\]
and the constants $C_{0}$, $D_{0}$,~\dots\ may be determined by multiplying each side of
the equation by $\sin \frac{1}{2} (x - \alpha)$, $\sin \frac{1}{2} (x - \beta)$,~\dots\ and making $x$~tend to $\alpha$,~$\beta$,~\dots.

It is often convenient to use the equation
\[
\cot \tfrac{1}{2} (x - \alpha)
  = \cot(x - \alpha) + \cosec(x - \alpha)
\]
which enables us to decompose the function~$R$ into two parts $U(x)$~and~$V(x)$
such that
\[
U(x + \pi) = U(x),\quad
V(x + \pi) = -V(x).
\]
If $R$~has the period~$\pi$, then $V$~must vanish identically; if it changes sign
when $x$~is increased by~$\pi$, then $U$~must vanish identically. Thus we find
without difficulty that, if $m < n$,
\[
\frac{\sin mx}{\sin nx}
  = \frac{1}{2n} \sum_{0}^{2n-1}\frac{(-1)^{k}\sin m\alpha}{\sin(x - \alpha)}
  = \frac{1}{n} \sum_{0}^{n-1}  \frac{(-1)^{k}\sin m\alpha}{\sin(x - \alpha)},
\]
or
\[
\frac{\sin mx}{\sin nx}
  = \frac{1}{n} \sum_{0}^{n-1} (-1)^{k}\sin m\alpha \cot(x - \alpha),
\]
where $\alpha = k\pi/n$, according as $m + n$~is odd or even.

Similarly
\begin{gather*}
\frac{1}{\sin(x - a)\sin(x - b)\sin(x - c)}
  = \sum\frac{1}{\sin(a - b)\sin(a - c)\sin(x - a)},\\
\frac{\sin(x - d)}{\sin(x - a)\sin(x - b)\sin(x - c)}
  = \sum \frac{\sin(a - d)}{\sin(a - b)\sin(a - c)} \cot(x - a).
\end{gather*}

\Item{(iii)} One of the most important integrals in applications is
\[
\int \frac{dx}{a + b \cos x},
\]
where $a$~and~$b$ are real. This integral may be evaluated in the manner
explained above, or by the transformation $\tan \frac{1}{2} x = t$. A more elegant method
%% -----File: 065.png---
is the following. If $|a| > |b|$, we suppose $a$~positive, and use the transformation
\[
(a + b\cos x)(a - b\cos y) = a^{2} - b^{2},
\]
which leads to
\[
\frac{dx}{a + b\cos x} = \frac{dy}{\sqrt{a^{2} - b^{2}}}.
\]
If $|a| < |b|$, we suppose $b$~positive, and use the transformation
\[
(b\cos x + a)(b\cosh y - a) = b^{2} - a^{2}.
\]

The integral
\[
\int \frac{dx}{a + b\cos x + c\sin x}
\]
may be reduced to this form by the substitution $x + a = y$, where $\cot a = b/c$.
The forms of the integrals
\[
\int \frac{dx}{(a + b\cos x)^{n}},\quad
\int \frac{dx}{(a + b\cos x + c\sin x)^{n}}
\]
may be deduced by the use of formulae of reduction, or by differentiation
with respect to~$a$. The integral
\[
\int \frac{dx}{(A\cos^{2}x + 2B\cos x\sin x + C\sin^{2}x)^{n}}
\]
is really of the same type, since
\[
A\cos^{2}x + 2B\cos x\sin x + C\sin^{2}x
  = \tfrac{1}{2} (A + C) + \tfrac{1}{2} (A - C)\cos 2x + B\sin 2x.
\]
And similar methods may be applied to the corresponding integrals which
contain hyperbolic functions, so that this type includes a large variety of
integrals of common occurrence.

\Item{(iv)} The same substitutions may of course be used when the subject of
integration is an irrational function of $\cos x$~and~$\sin x$, though sometimes
it is better to use the substitutions $\cos x = t$, $\sin x = t$, or $\tan x = t$. Thus
the integral
\[
\int R(\cos x, \sin x, \sqrt{X})\,dx,
\]
where
\[
X = (a, b, c, f, g, h\between \cos x, \sin x, 1)^{2},
\]
is reduced to an elliptic integral by the substitution $\tan \frac{1}{2} x = t$. The most
important integrals of this type are
\[
\int \frac{R(\cos x, \sin x)\,dx}{\sqrt{1 - k^{2}\sin^{2}x}},\quad
\int \frac{R(\cos x, \sin x)\,dx}{\sqrt{\alpha + \beta\cos x + \gamma\sin x}}.
\]

\Paragraph{6}{3.} The integral
\[
\int P(x, e^{ax}, e^{bx}, \dots, e^{kx})\,dx,
\]
where $a$, $b$,~\dots, $k$ are any numbers (commensurable or not), and $P$~is
a polynomial, is always an elementary function. For it is obvious
%% -----File: 066.png---
that the integral can be reduced to the sum of a finite number of
integrals of the type
\[
\int x^{p}e^{Ax}\,dx;
\]
and
\[
\int x^{p}e^{Ax}\,dx
  = \left(\frac{\dd}{\dd A}\right)^{p} \int e^{Ax}\,dx
  = \left(\frac{\dd}{\dd A}\right)^{p} \frac{e^{Ax}}{A}.
\]
This type of integral includes a large variety of integrals, such as
\begin{align*}
\int x^{m}(\cos px)^{\mu}(\sin qx)^{\nu}\,dx,\quad &
\int x^{m}(\cosh px)^{\mu}(\sinh qx)^{\nu}\,dx,\\
%
\int x^{m}e^{-\alpha x}(\cos px)^{\mu}\,dx,\quad &
\int x^{m}e^{-\alpha x}(\sin qx)^{\nu}\,dx,
\end{align*}
($m$,~$\mu$,~$\nu$, being positive integers) for which formulae of reduction are
given in text-books on the integral calculus.

Such integrals as
\[
\int P(x, \log x)\,dx,\quad
\int P(x, \arcsin x)\,dx,\ \dots,
\]
where $P$~is a polynomial, may be reduced to particular cases of the
above general integral by the obvious substitutions
\[
x = e^{y},\quad x = \sin y,\ \dots.
\]

\Paragraph{6}{4.} Except for the two classes of functions considered in the three
preceding paragraphs, there are no really general classes of transcendental
functions which we can \emph{always} integrate in finite terms, although
of course there are innumerable particular forms which may be
integrated by particular devices. There are however many classes
of such integrals for which a systematic reduction theory may be given,
analogous to the reduction theory for elliptic integrals. Such a reduction
theory endeavours in each case

\Item{(i)} to split up any integral of the class under consideration into
the sum of a number of parts of which some are elementary and
the others not;

\Item{(ii)} to reduce the number of the latter terms to the least possible;

\Item{(iii)} to prove that these terms are incapable of further reduction,
and are genuinely new and independent transcendents.

As an example of this process we shall consider the integral
\[
\int e^{x}R(x)\,dx
\]
where $R(x)$~is a rational function of~$x$.\footnote
  {See Hermite, \textit{Cours d'analyse}, pp.~352~\textit{et~seq.}}
The theory of partial
%% -----File: 067.png---
fractions enables us to decompose this integral into the sum of a
number of terms
\[
A \int\frac{e^{x}}{x - a}\,dx,\DPtypo{}{\ \dots,}\quad
A_{m} \int\frac{e^{x}}{(x - a)^{m+1}}\,dx,\DPtypo{\ \dots,}{}\quad
B \int\frac{e^{x}}{x - b}\,dx,\ \dots.
\]

Since
\[
\int\frac{e^{x}}{(x - a)^{m+1}}\,dx
  = -\frac{e^{x}}{m(x - a)^{m}}
   + \frac{1}{m} \int\frac{e^{x}}{(x - a)^{m}}\,dx,
\]
the integral may be further reduced so as to contain only

\Item{(i)} a term
\[
e^{x}S(x)
\]
where $S(x)$~is a rational function;

\Item{(ii)} a number of terms of the type
\[
\alpha \int\frac{e^{x}\,dx}{x - a}.
\]
If all the constants~$\alpha$ vanish, then the integral can be calculated in the
finite form~$e^{x}S(x)$. If they do not we can at any rate assert that the
integral cannot be calculated \emph{in this form}\footnotemark.
  \footnotetext{See the remarks at the end of this paragraph.}%
For no such relation as
\[
\alpha \int\frac{e^{x}\,dx}{x - a}
  + \beta  \int\frac{e^{x}\,dx}{x - b} + \dots
  + \kappa \int\frac{e^{x}\,dx}{x - k}
  = e^{x}T(x),
\]
where $T$~is rational, can hold for all values of~$x$. To see this it is
only necessary to put $x = a + h$ and to expand in ascending powers
of~$h$. Then
\begin{align*}
\alpha \int\frac{e^{x}\,dx}{x - a}
  &= \alpha e^{a} \int\frac{e^{h}}{h}\,dh\\
  &= \alpha e^{a} (\log h + h + \dots),
\end{align*}
and no \emph{logarithm} can occur in any of the other terms\footnotemark.
  \footnotetext{It is not difficult to give a purely algebraical proof on the lines of \hyperlink{4 para 2.}{\textsc{iv}.,~§2}.}

Consider, for example, the integral
\[
\int e^{x}\left(1 - \frac{1}{x}\right)^{3} dx.
\]
This is equal to
\[
e^{x} - 3 \int \frac{e^{x}}{x}\,dx
      + 3 \int \frac{e^{x}}{x^{2}}\,dx
      -   \int \frac{e^{x}}{x^{3}}\,dx,
\]
and since
\[
3 \int \frac{e^{x}}{x^{2}}\,dx
  = -\frac{3e^{x}}{x} + 3 \int \frac{e^{x}}{x}\,dx,
\]
and
\[
-\int \frac{e^{x}}{x^{3}}\,dx
  = \frac{e^{x}}{2x^{2}} - \frac{1}{2} \int \frac{e^{x}}{x^{2}}\,dx
  = \frac{e^{x}}{2x^{2}} + \frac{e^{x}}{2x} - \frac{1}{2} \int \frac{e^{x}}{x}\,dx,
\]
%% -----File: 068.png---
we obtain finally
\[
\int e^{x}\left(1 - \frac{1}{x}\right)^{3} dx
   = e^{x}\left(1 - \frac{7}{2x} + \frac{1}{2x^{2}}\right)
     - \frac{1}{2} \int \frac{e^{x}}{x}\, dx.
\]
Similarly it will be found that
\[
\int e^{x}\left(1 - \frac{2}{x}\right)^{2} dx
  = 2e^{x}\left(\frac{1}{2} - \frac{2}{x}\right),
\]
this integral being an elementary function.

Since
\[
\int \frac{e^{x}}{x - a}\, dx = e^{a} \int \frac{e^{y}}{y}\, dy,
\]
if $x = y + a$, all integrals of this kind may be made to depend on known
functions and on the single transcendent
\[
\int \frac{e^{x}}{x}\, dx,
\]
which is usually denoted by $\Li e^{x}$ and is of great importance in the
theory of numbers. The question of course arises as to whether this
integral is not itself an elementary function.

Now Liouville\footnote
  {`Mémoire sur l'intégration d'une classe de fonctions transcendantes', \textit{Journal
  für Mathematik}, vol.~13, 1835, pp.~93--118. Liouville shows how the integral, when
  of this form, may always be calculated by elementary methods.}
has proved the following theorem: \begin{Result}`if $y$~is any
algebraical function of~$x$, and
\[
\int e^{x}y\,dx
\]
is an elementary function, then
\[
\int e^{x}y\,dx = e^{x}(\alpha + \beta y + \dots + \lambda y^{n-1}),
\]
$\alpha$, $\beta$,~\dots, $\lambda$ being rational functions of~$x$ and $n$~the degree of the
algebraical equation which determines~$y$ as a function of~$x$'.
\end{Result}

Liouville's proof rests on the same general principles as do those of
the corresponding theorems concerning the integral $\int y\,dx$. It will
be observed that no logarithmic terms can occur, and that the theorem
is therefore very similar to that which holds for $\int y\,dx$ in the simple
case in which the integral is \emph{algebraical}. The argument which shows
that no logarithmic terms occur is substantially the same as that which
shows that, when they occur in the integral of an algebraical function,
they must occur linearly. In this case the occurrence of the exponential
factor precludes even this possibility, since differentiation
will not eliminate logarithms when they occur in the form
\[
e^{x} \log f(x).
\]
%% -----File: 069.png---

In particular, if $y$~is a rational function, then the integral must
be of the form
\[
e^{x} R(x)
\]
and this we have already seen to be impossible. Hence the `logarithm-integral'
\[
\Li e^{x} = \int \frac{e^{x}}{x}\, dx = \int^{e^{x}} \frac{dy}{\log y}
\]
is really a new transcendent, which cannot be expressed in finite terms
by means of elementary functions; and the same is true of all integrals
of the type
\[
\int e^{x} R(x)\,dx
\]
which cannot be calculated in finite terms by means of the process of
reduction sketched above.

The integrals
\[
\int \sin x R(x)\,dx,\quad
\int \cos x R(x)\,dx
\]
may be treated in a similar manner. Either the integral is of the form
\[
\cos x R_{1}(x) + \sin x R_{2}(x)
\]
or it consists of a term of this kind together with a number of terms
which involve the transcendents
\[
\int \frac{\cos x}{x}\,dx,\quad
\int \frac{\sin x}{x}\,dx,
\]
which are called the cosine-integral and sine-integral of~$x$, and denoted
by $\Ci x$ and $\Si x$. These transcendents are of course not fundamentally
distinct from the logarithm-integral.

\Paragraph{6}{5.} Liouville has gone further and shown that it is always possible
to determine whether the integral
\[
\int (Pe^{p} + Qe^{q} + \dots + Te^{t})\,dx,
\]
where $P$, $Q$,~\dots, $T$, $p$, $q$,~\dots, $t$ are algebraical functions, is an elementary
function, and to obtain the integral in case it is one\footnotemark.
  \footnotetext{An interesting particular result is that the `error function' $\int e^{-x^{2}}\,dx$ is not an
  elementary function.}%
The most
general theorem which has been proved in this region of mathematics,
and which is also due to Liouville, is the following.
%% -----File: 070.png---

\begin{Result}
`If $y$, $z$,~\dots\ are functions of~$x$ whose differential coefficients are
algebraical functions of $x$,~$y$, $z$,~\dots, and $F$~denotes an algebraical
function, and if
\[
\int F(x, y, z, \dots)\,dx
\]
is an elementary function, then it is of the form
\[
t + A \log u + B \log v + \dots,
\]
where $t$, $u$, $v$,~\dots\ are algebraical functions of $x$,~$y$, $z$,~\dots. If the
differential coefficients are rational in $x$,~$y$, $z$,~\dots, and $F$~is rational,
then $t$,~$u$, $v$,~\dots\ are rational in $x$,~$y$, $z$,~\dots.'
\end{Result}

Thus for example the theorem applies to
\[
F(x, e^{x}, e^{e^{x}}, \log x, \log \log x, \cos x, \sin x),
\]
since, if the various arguments of~$F$ are denoted by $x$,~$y$,~$z$, $\xi$,~$\eta$, $\zeta$,~$\theta$,
we have
\begin{gather*}
\frac{dy}{dx} = y,\qquad
\frac{dz}{dx} = yz,\qquad
\frac{d\xi}{dx} = \frac{1}{x}, \\
%
\frac{d\eta}{dx} = \frac{1}{x\xi},\quad
\frac{d\zeta}{dx} = -\sqrt{1 - \zeta^{2}},\quad
\frac{d\theta}{dx} = \sqrt{1 - \theta^{2}}.
\end{gather*}
The proof of the theorem does not involve ideas different in principle
from those which have been employed continually throughout the
preceding pages.

{\small \Paragraph{6}{6.} As a final example of the manner in which these ideas may be applied,
we shall consider the following question:

`\emph{in what circumstances is
\[
\int R(x) \log x\, dx,
\]
where $R$~is rational, an elementary function?}'

In the first place the integral must be of the form
\[
R_{0}(x, \log x)
  + A_{1}\log R_{1}(x, \log x)
  + A_{2}\log R_{2}(x, \log x) + \dots.
\]
A general consideration of the form of the differential coefficient of this
expression, in which $\log x$ must only occur linearly and multiplied by a
rational function, leads us to anticipate that (i)~$R_{0}(x, \log x)$ must be of the
form
\[
S(x)(\log x)^{2} + T(x)\log x + U(x),
\]
where $S$,~$T$, and~$U$ are rational, and (ii)~$R_{1}$, $R_{2}$,~\dots\ must be rational functions
of $x$~only; so that the integral can be expressed in the form
\[
S(x)(\log x)^{2} + T(x)\log x + U(x) + \sum B_{k}\log(x - a_{k}).
\]
%% -----File: 071.png---

Differentiating, and comparing the result with the subject of integration,
we obtain the equations
\[
S' = 0,\quad
\frac{2S}{x} + T' = R,\quad
\frac{T}{x} + U' + \sum \frac{B_{k}}{x - a_{k}} = 0.
\]
Hence $S$~is a constant, say~$\frac{1}{2} C$, and
\[
T = \int \left(R - \frac{C}{x}\right) dx.
\]

We can always determine by means of elementary operations, as in \hyperlink{4 para 4.}{\textsc{iv.},~§4},
whether this integral is rational for any value of~$C$ or not. If not, then the
given integral is not an elementary function. If $T$~is rational, then we must
calculate its value, and substitute it in the integral
\[
U = -\int \left\{\frac{T}{x} + \sum \frac{B_{k}}{x - a_{k}}\right\} dx
  = -\int \frac{T}{x}\, dx - \sum B_{k}\log(x - a_{k}),
\]
which must be rational for some value of the arbitrary constant implied in~$T$.
We can calculate the rational part of
\[
\int \frac{T}{x}\, dx:
\]
the transcendental part must be cancelled by the logarithmic terms
\[
\sum B_{k}\log(x - a_{k}).
\]

The necessary and sufficient condition that the original integral should be
an elementary function is therefore that $R$~should be of the form
\[
\frac{C}{x} + \frac{d}{dx} \{R_{1}(x)\},
\]
where $C$~is a constant and $R_{1}$~is rational. That the integral is in this case
such a function becomes obvious if we integrate by parts, for
\[
\int \left(\frac{C}{x} + R_{1}'\right) \log x\, dx
  = \tfrac{1}{2} C(\log x)^{2} + R_{1}\log x - \int \frac{R_{1}}{x}\, dx.
\]

In particular
\[
\Item{(i)} \int \frac{\log x}{x - a}\, dx,\qquad\qquad
\Item{(ii)} \int \frac{\log x}{(x - a)(x - b)}\, dx,
\]
are not elementary functions unless in~(i) $a = 0$ and in~(ii) $b = a$. If the
integral is elementary then the integration can always be carried out, with
the same reservation as was necessary in the case of rational functions.

It is evident that the problem considered in this paragraph is but one of
a whole class of similar problems. The reader will find it instructive to
formulate and consider such problems for himself.}
%% -----File: 072.png---

\Paragraph{6}{7.} It will be obvious by now that the number of classes of
transcendental functions whose integrals are always elementary is very
small, and that such integrals as
\begin{alignat*}{2}
& \int f(x, e^{x})\,dx,               && \int f(x, \log x)\,dx,\\
& \int f(x, \cos x, \sin x)\,dx,\quad && \int f(e^{x}, \cos x, \sin x)\,dx,\\
& \multispan{1}{\dotfill},
\end{alignat*}
where $f$~is algebraical, or even rational, are generally new transcendents.
These new transcendents, like the transcendents (such as the elliptic
integrals) which arise from the integration of algebraical functions,
are in many cases of great interest and importance. They may often
be expressed by means of infinite series or definite integrals, or their
properties may be studied by means of the integral expressions which
define them. The very fact that such a function is \emph{not} an elementary
function in so far enhances its importance. And when such functions
have been introduced into analysis new problems of integration arise
in connection with them. We may enquire, for example, under what
circumstances an elliptic integral or elliptic function, or a combination
of such functions with elementary functions, can be integrated in finite
terms by means of elementary and elliptic functions. But before we
can be in a position to restate the fundamental problem of the Integral
Calculus in any such more general form, it is essential that we should
have disposed of the particular problem formulated in \hyperlink{3 para 1.}{Section~III}\@.
%% -----File: 073.png---
\BackMatter
\Appendix{I}{BIBLIOGRAPHY}

{\footnotesize The following is a list of the memoirs by Abel, Liouville and Tschebyschef
which have reference to the subject matter of this tract.

\Author{N.~H. Abel}

\Bibitem[1.] `Über die Integration der Differential-Formel $\dfrac{\rho\,dx}{\sqrt{R}}$, wenn $R$~und~$\rho$ ganze
Funktionen sind', \textit{Journal für Mathematik}, vol.~1, 1826, pp.~185--221
(\OEuvres, vol.~1, pp.~104--144).

\Bibitem[2.] `\DPtypo{Precis}{Précis} d'une théorie des fonctions elliptiques', \textit{Journal für Mathematik},
vol.~4, 1829, pp.~236--277, 309--348 (\OEuvres, vol.~1, pp.~518--617).

\Bibitem[3.] `Théorie des transcendantes elliptiques', \OEuvres, vol.~2, pp.~87--188.

\Author{J. Liouville}

\Bibitem[1.] `Mémoire sur la classification des transcendantes, et sur l'impossibilité
d'exprimer les racines de certaines équations en fonction finie explicite
des coefficients', \textit{Journal de mathématiques}, ser.~1, vol.~2, 1837,
pp.~56--104.

\Bibitem[2.] `{\Loosen Nouvelles recherches sur la détermination des intégrales dont la valeur
est algébrique}', \ibid., vol.~3, 1838, pp.~20--24 (previously published in
the \textit{Comptes Rendus}, 28~Aug.\ 1837).

\Bibitem[3.] `Suite du mémoire sur la classification des transcendantes, et sur
l'impossibilité d'exprimer les racines de certaines équations en fonction
finie explicite des coefficients', \ibid., pp.~523--546\DPtypo{}{.}

\Bibitem[4.] `Note sur les transcendantes elliptiques considérées comme fonctions de
leur module', \textit{ibid.}, vol.~5, 1840, pp.~34--37.

\Bibitem[5.] `Mémoire sur les transcendantes elliptiques considérées comme fonctions
de leur module', \ibid., pp. 441--464.

\Bibitem[6.] `Premier mémoire sur la détermination des intégrales dont la valeur est
algébrique', \textit{Journal de l'École Polytechnique}, vol.~14, cahier~22, 1833,
pp.~124--148 (also published in the \textit{Mémoires présentés par divers
savants à l'Académie des Sciences}, vol.~5, 1838, pp.~76--151).

\Bibitem[7.] `Second mémoire sur la détermination des intégrales dont la valeur est
algébrique', \ibid., pp.~149--193 (also published as above).
%% -----File: 074.png---

\Bibitem[8.] `Mémoire sur les transcendantes elliptiques considérées comme fonctions
de leur amplitude', \ibid., cahier~23, 1834, pp.~37--83.

\Bibitem[9.] `Mémoire sur l'intégration d'une classe de fonctions transcendantes',
\textit{Journal für Mathematik}, vol.~13, 1835, pp.~93--118.

\Author{P. Tschebyschef}

\Bibitem[1.] `Sur l'intégration des différentielles irrationnelles', \textit{Journal de mathématiques},
ser.~1, vol.~18, 1853, pp.~87--111 (\OEuvres, vol.~1, pp.~147--168).

\Bibitem[2.] `Sur l'intégration des différentielles qui contiennent une racine carrée
d'une \DPtypo{polynome}{polynôme} du troisième ou du quatrième degré', \ibid., ser.~2,
vol.~2, 1857, pp.~1--42 (\OEuvres, vol.~1, pp.~171--200; also published
in the \textit{Mémoires de l'Académie Impériale des Sciences de St-Pétersbourg},
ser.~6, vol.~6, 1857, pp.~203--232).

\Bibitem[3.] `Sur l'intégration de la différentielle $\dfrac{x + A}{\sqrt{x^{4} + \alpha x^{3} + \beta x^{2} + \gamma x + \delta}}\,dx$', \ibid.,
ser.~2, vol.~9, 1864, pp.~225--241 (\OEuvres, vol.~1, pp.~517--530;
previously published in the \textit{Bulletin de l'Académie Impériale des
Sciences de St-Pétersbourg}, vol.~3, 1861, pp.~1--12).

\Bibitem[4.] `Sur l'intégration des différentielles irrationnelles', \ibid., pp.~242--246
(\OEuvres, vol.~1, pp.~511--514; previously published in the \textit{Comptes
Rendus}, 9~July 1860).

\Bibitem[5.] `Sur l'intégration des différentielles qui contiennent une racine cubique'
(\OEuvres, vol.~1, pp.~563--608; previously published only in Russian).

Other memoirs which may be consulted are:

\Author{A. Clebsch}

\Bibitem[] `Über diejenigen Curven, deren Coordinaten sich als elliptische Functionen
eines Parameters darstellen lassen', \textit{Journal für Mathematik}, vol.~64,
1865, pp.~210--270.

\Author{J. Dolbnia}

\Bibitem[] `Sur les intégrales pseudo-elliptiques d'Abel', \textit{Journal de mathématiques},
ser.~4, vol.~6, 1890, pp.~293--311.

\Author{Sir A. G. Greenhill}

\Bibitem[] `Pseudo-elliptic integrals and their dynamical applications', \textit{Proc.\ London
Math.\ Soc.}, ser.~1, vol.~25, 1894, pp.~195--304.

\Author{G. H. Hardy}

\Bibitem[] `Properties of logarithmico-exponential functions', \textit{Proc.\ London Math.\
Soc.}, ser.~2, vol.~10, 1910, pp.~54--90.

\Author{L. Königsberger}

\Bibitem[] `Bemerkungen zu Liouville's Classificirung der Transcendenten', \textit{Mathematische
Annalen}, vol.~28, 1886, pp.~483--492.
%% -----File: 075.png---

\Author{L. Raffy}

\Bibitem[] `Sur les quadratures algébriques et logarithmiques', \textit{Annales de l'École
Normale}, ser.~3, vol.~2, 1885, pp.~185--206.

\Author{K. Weierstrass}

\Bibitem[] `Über die Integration algebraischer Differentiale vermittelst Logarithmen',
\textit{Monatsberichte der Akademie der Wissenschaften zu Berlin}, 1857,
pp.~148--157 (\textit{Werke}, vol.~1, pp.~227--232).

\Author{G. Zolotareff}

\Bibitem[] `Sur la méthode d'intégration de M.~Tschebyschef', \textit{Journal de mathématiques},
ser.~2, vol.~19, 1874, pp.~161--188.

Further information concerning pseudo-elliptic integrals, and degenerate
cases of Abelian integrals generally, will be found in a number of short notes
by Dolbnia, Kapteyn and Ptaszycki in the \textit{Bulletin des sciences mathématiques},
and by Goursat, Gunther, Picard, Poincaré, and Raffy in the \textit{Bulletin de la
Société Mathématique de France}, in Legendre's \textit{Traité des functions elliptiques}
(vol.~1, ch.~26), in Halphen's \textit{Traité des fonctions elliptiques} (vol.~2, ch.~14),
and in Enneper's \textit{Elliptische Funktionen}. The literature concerning the
general theory of algebraical functions and their integrals is too extensive to
be summarised here: the reader may be referred to Appell and Goursat's
\textit{Théorie des fonctions algébriques}, and Wirtinger's article \textit{Algebraische Funktionen
und ihre Integrale} in the \textit{Encyclopädie der Mathematischen Wissenschaften},
\textsc{ii}~B~2.}
%% -----File: 076.png---

\Appendix{II}{ON ABEL'S PROOF OF THE THEOREM OF V., §11}

{\small Abel's proof (\OEuvres, vol.~1, p.~545) is as follows\footnotemark:
  \footnotetext{The theorem with which Abel is engaged is a very much more general
theorem.}

We have
\[
\psi(x, u) = 0,
\Tag{(1)}
\]
where $\psi$~is an irreducible polynomial of degree~$m$ in~$u$. If we make use of the
equation $f(x, y) = 0$, we can introduce~$y$ into this equation, and write it in the
form
\[
\phi(x, y, u) = 0,
\Tag{(2)}
\]
where $\phi$~is a polynomial in the three variables $x$,~$y$, and~$u$\footnotemark;
  \footnotetext{`Or, au lieu de supposer ces coefficiens rationnels en~$x$, nous les supposerons
  rationnels en $x$,~$y$; \emph{car cette supposition permise simplifiera beaucoup le
  raisonnement}'.}
and we can
suppose~$\phi$, like~$\psi$, of degree~$m$ in~$u$ and irreducible, that is to say not
divisible by any polynomial of the same form  which is not a constant
multiple of~$\phi$ or itself a constant.

From $f = 0$, $\phi = 0$ we deduce
\[
\frac{\dd f}{\dd x}
  + \frac{\dd f}{\dd y}\, \frac{dy}{dx} = 0,\quad
\frac{\dd\phi}{\dd x}
  + \frac{\dd\phi}{\dd y}\, \frac{dy}{dx}
  + \frac{\dd\phi}{\dd u}\, \frac{du}{dx} = 0;
\]
and, eliminating $\dfrac{dy}{dx}$, we obtain an equation of the form
\[
\frac{du}{dx} = \frac{\lambda(x, y, u)}{\mu(x, y, u)},
\]
where $\lambda$~and~$\mu$ are polynomials in $x$,~$y$, and~$u$. And in order that $u$~should
be an integral of~$y$ it is necessary and sufficient that
\[
\lambda - y\mu = 0.
\Tag{(3)}
\]

Abel now applies Lemma~(2) of \hyperlink{5 para 11.}{§11}, or rather its analogue for polynomials
in~$u$ whose coefficients are polynomials in $x$~and~$y$, to the two polynomials $\phi$
and $\lambda - y\mu$, and infers that \emph{all} the roots $u$,~$u'$,~\dots\ of $\phi = 0$ satisfy~\Eq{(3)}. From
this he deduces that $u$,~$u'$,~\dots\ are all integrals of~$y$, and so that
\[
\frac{u + u' + \dots}{m + 1}
\Tag{(4)}
\]
%% -----File: 077.png---
is an integral of~$y$. As \Eq{(4)}~is a symmetric function of the roots of~\Eq{(2)}, it is a
rational function of $x$~and~$y$, whence his conclusion follows\footnotemark.
  \footnotetext{Bertrand (\textit{Calcul intégral}, ch.~5) replaces the last step in Abel's argument by
  the observation that if $u$~and~$u'$ are both integrals of~$y$ then $u - u'$~is constant (cf.\
  p.~39, bottom). It follows that the degree of the equation which defines~$u$ can be
  decreased, which contradicts the hypothesis that it is irreducible.}

It will be observed that the hypothesis that \Eq{(2)}~does actually involve~$y$
is essential, if we are to avoid the absurd conclusion that $u$~is necessarily
\emph{a rational function of $x$~only}. On the other hand it is not obvious how
the presence of~$y$ in~$\phi$ affects the other steps in the argument.

The crucial inference is that which asserts that because the equations
$\phi = 0$ and $\lambda - y\mu = 0$, considered as equations in~$u$, have a root in common,
and $\phi$~is irreducible, therefore $\lambda - y\mu$ is divisible by~$\phi$. This inference is
invalid.

We could only apply the lemma in this way if the equation~\Eq{(3)} were
satisfied by one of the roots of~\Eq{(2)} \emph{identically}, that is to say for all values of
$x$~and~$y$. But this is not the case. The equations are satisfied by the same
value of~$u$ \emph{only when $x$~and~$y$ are connected by the equation~\Eq{(1)}}.

Suppose, for example, that
\[
y = \frac{1}{\sqrt{1 + x}},\quad
u = 2\sqrt{1 + x}.
\]
Then we may take
\begin{align*}
f &= (1 + x)y^{2} - 1, \\
\psi &= u^{2} - 4(1 + x),
\intertext{and}
\phi &= uy - 2.
\end{align*}
Differentiating the equations $f = 0$ and $\phi = 0$, and eliminating~$\dfrac{dy}{dx}$, we find
\[
\frac{du}{dx} = \frac{u}{2(1 + x)} = \frac{\lambda}{\mu}.
\]
Thus
\[
\phi = uy - 2,\quad
\lambda - y\mu = u - 2y(1 + x);
\]
and these polynomials have a common factor only in virtue of the equation
$f = 0$.}
%% -----File: 078.png---
%[Blank Page]
%% -----File: 079.png---
%[Blank Page]
%% -----File: 080.png---
\newpage
\medskip

\begin{center}
PUBLISHED BY

THE SYNDICS OF THE CAMBRIDGE UNIVERSITY PRESS

\smallskip

Bentley House, 200 Euston Road, London, N.W.1 \\
American Branch: 32 East 57th Street, New York 10022, N.Y.

\vfill

PRINTED IN GREAT BRITAIN
\end{center}

\medskip
\bigskip

\PGLicense
\begin{PGtext}
End of the Project Gutenberg EBook of The Integration of Functions of a
Single Variable, by G. H. Hardy

*** END OF THIS PROJECT GUTENBERG EBOOK INTEGRATION OF FUNCTIONS OF ONE VARIABLE ***

***** This file should be named 38993-t.tex or 38993-t.zip *****
This and all associated files of various formats will be found in:
        http://www.gutenberg.org/3/8/9/9/38993/

Produced by Brenda Lewis, Anna Hall and the Online
Distributed Proofreading Team at http://www.pgdp.net (This
file was produced from images generously made available
by The Internet Archive/American Libraries.)


Updated editions will replace the previous one--the old editions
will be renamed.

Creating the works from public domain print editions means that no
one owns a United States copyright in these works, so the Foundation
(and you!) can copy and distribute it in the United States without
permission and without paying copyright royalties.  Special rules,
set forth in the General Terms of Use part of this license, apply to
copying and distributing Project Gutenberg-tm electronic works to
protect the PROJECT GUTENBERG-tm concept and trademark.  Project
Gutenberg is a registered trademark, and may not be used if you
charge for the eBooks, unless you receive specific permission.  If you
do not charge anything for copies of this eBook, complying with the
rules is very easy.  You may use this eBook for nearly any purpose
such as creation of derivative works, reports, performances and
research.  They may be modified and printed and given away--you may do
practically ANYTHING with public domain eBooks.  Redistribution is
subject to the trademark license, especially commercial
redistribution.



*** START: FULL LICENSE ***

THE FULL PROJECT GUTENBERG LICENSE
PLEASE READ THIS BEFORE YOU DISTRIBUTE OR USE THIS WORK

To protect the Project Gutenberg-tm mission of promoting the free
distribution of electronic works, by using or distributing this work
(or any other work associated in any way with the phrase "Project
Gutenberg"), you agree to comply with all the terms of the Full Project
Gutenberg-tm License available with this file or online at
  www.gutenberg.org/license.


Section 1.  General Terms of Use and Redistributing Project Gutenberg-tm
electronic works

1.A.  By reading or using any part of this Project Gutenberg-tm
electronic work, you indicate that you have read, understand, agree to
and accept all the terms of this license and intellectual property
(trademark/copyright) agreement.  If you do not agree to abide by all
the terms of this agreement, you must cease using and return or destroy
all copies of Project Gutenberg-tm electronic works in your possession.
If you paid a fee for obtaining a copy of or access to a Project
Gutenberg-tm electronic work and you do not agree to be bound by the
terms of this agreement, you may obtain a refund from the person or
entity to whom you paid the fee as set forth in paragraph 1.E.8.

1.B.  "Project Gutenberg" is a registered trademark.  It may only be
used on or associated in any way with an electronic work by people who
agree to be bound by the terms of this agreement.  There are a few
things that you can do with most Project Gutenberg-tm electronic works
even without complying with the full terms of this agreement.  See
paragraph 1.C below.  There are a lot of things you can do with Project
Gutenberg-tm electronic works if you follow the terms of this agreement
and help preserve free future access to Project Gutenberg-tm electronic
works.  See paragraph 1.E below.

1.C.  The Project Gutenberg Literary Archive Foundation ("the Foundation"
or PGLAF), owns a compilation copyright in the collection of Project
Gutenberg-tm electronic works.  Nearly all the individual works in the
collection are in the public domain in the United States.  If an
individual work is in the public domain in the United States and you are
located in the United States, we do not claim a right to prevent you from
copying, distributing, performing, displaying or creating derivative
works based on the work as long as all references to Project Gutenberg
are removed.  Of course, we hope that you will support the Project
Gutenberg-tm mission of promoting free access to electronic works by
freely sharing Project Gutenberg-tm works in compliance with the terms of
this agreement for keeping the Project Gutenberg-tm name associated with
the work.  You can easily comply with the terms of this agreement by
keeping this work in the same format with its attached full Project
Gutenberg-tm License when you share it without charge with others.

1.D.  The copyright laws of the place where you are located also govern
what you can do with this work.  Copyright laws in most countries are in
a constant state of change.  If you are outside the United States, check
the laws of your country in addition to the terms of this agreement
before downloading, copying, displaying, performing, distributing or
creating derivative works based on this work or any other Project
Gutenberg-tm work.  The Foundation makes no representations concerning
the copyright status of any work in any country outside the United
States.

1.E.  Unless you have removed all references to Project Gutenberg:

1.E.1.  The following sentence, with active links to, or other immediate
access to, the full Project Gutenberg-tm License must appear prominently
whenever any copy of a Project Gutenberg-tm work (any work on which the
phrase "Project Gutenberg" appears, or with which the phrase "Project
Gutenberg" is associated) is accessed, displayed, performed, viewed,
copied or distributed:

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.org

1.E.2.  If an individual Project Gutenberg-tm electronic work is derived
from the public domain (does not contain a notice indicating that it is
posted with permission of the copyright holder), the work can be copied
and distributed to anyone in the United States without paying any fees
or charges.  If you are redistributing or providing access to a work
with the phrase "Project Gutenberg" associated with or appearing on the
work, you must comply either with the requirements of paragraphs 1.E.1
through 1.E.7 or obtain permission for the use of the work and the
Project Gutenberg-tm trademark as set forth in paragraphs 1.E.8 or
1.E.9.

1.E.3.  If an individual Project Gutenberg-tm electronic work is posted
with the permission of the copyright holder, your use and distribution
must comply with both paragraphs 1.E.1 through 1.E.7 and any additional
terms imposed by the copyright holder.  Additional terms will be linked
to the Project Gutenberg-tm License for all works posted with the
permission of the copyright holder found at the beginning of this work.

1.E.4.  Do not unlink or detach or remove the full Project Gutenberg-tm
License terms from this work, or any files containing a part of this
work or any other work associated with Project Gutenberg-tm.

1.E.5.  Do not copy, display, perform, distribute or redistribute this
electronic work, or any part of this electronic work, without
prominently displaying the sentence set forth in paragraph 1.E.1 with
active links or immediate access to the full terms of the Project
Gutenberg-tm License.

1.E.6.  You may convert to and distribute this work in any binary,
compressed, marked up, nonproprietary or proprietary form, including any
word processing or hypertext form.  However, if you provide access to or
distribute copies of a Project Gutenberg-tm work in a format other than
"Plain Vanilla ASCII" or other format used in the official version
posted on the official Project Gutenberg-tm web site (www.gutenberg.org),
you must, at no additional cost, fee or expense to the user, provide a
copy, a means of exporting a copy, or a means of obtaining a copy upon
request, of the work in its original "Plain Vanilla ASCII" or other
form.  Any alternate format must include the full Project Gutenberg-tm
License as specified in paragraph 1.E.1.

1.E.7.  Do not charge a fee for access to, viewing, displaying,
performing, copying or distributing any Project Gutenberg-tm works
unless you comply with paragraph 1.E.8 or 1.E.9.

1.E.8.  You may charge a reasonable fee for copies of or providing
access to or distributing Project Gutenberg-tm electronic works provided
that

- You pay a royalty fee of 20% of the gross profits you derive from
     the use of Project Gutenberg-tm works calculated using the method
     you already use to calculate your applicable taxes.  The fee is
     owed to the owner of the Project Gutenberg-tm trademark, but he
     has agreed to donate royalties under this paragraph to the
     Project Gutenberg Literary Archive Foundation.  Royalty payments
     must be paid within 60 days following each date on which you
     prepare (or are legally required to prepare) your periodic tax
     returns.  Royalty payments should be clearly marked as such and
     sent to the Project Gutenberg Literary Archive Foundation at the
     address specified in Section 4, "Information about donations to
     the Project Gutenberg Literary Archive Foundation."

- You provide a full refund of any money paid by a user who notifies
     you in writing (or by e-mail) within 30 days of receipt that s/he
     does not agree to the terms of the full Project Gutenberg-tm
     License.  You must require such a user to return or
     destroy all copies of the works possessed in a physical medium
     and discontinue all use of and all access to other copies of
     Project Gutenberg-tm works.

- You provide, in accordance with paragraph 1.F.3, a full refund of any
     money paid for a work or a replacement copy, if a defect in the
     electronic work is discovered and reported to you within 90 days
     of receipt of the work.

- You comply with all other terms of this agreement for free
     distribution of Project Gutenberg-tm works.

1.E.9.  If you wish to charge a fee or distribute a Project Gutenberg-tm
electronic work or group of works on different terms than are set
forth in this agreement, you must obtain permission in writing from
both the Project Gutenberg Literary Archive Foundation and Michael
Hart, the owner of the Project Gutenberg-tm trademark.  Contact the
Foundation as set forth in Section 3 below.

1.F.

1.F.1.  Project Gutenberg volunteers and employees expend considerable
effort to identify, do copyright research on, transcribe and proofread
public domain works in creating the Project Gutenberg-tm
collection.  Despite these efforts, Project Gutenberg-tm electronic
works, and the medium on which they may be stored, may contain
"Defects," such as, but not limited to, incomplete, inaccurate or
corrupt data, transcription errors, a copyright or other intellectual
property infringement, a defective or damaged disk or other medium, a
computer virus, or computer codes that damage or cannot be read by
your equipment.

1.F.2.  LIMITED WARRANTY, DISCLAIMER OF DAMAGES - Except for the "Right
of Replacement or Refund" described in paragraph 1.F.3, the Project
Gutenberg Literary Archive Foundation, the owner of the Project
Gutenberg-tm trademark, and any other party distributing a Project
Gutenberg-tm electronic work under this agreement, disclaim all
liability to you for damages, costs and expenses, including legal
fees.  YOU AGREE THAT YOU HAVE NO REMEDIES FOR NEGLIGENCE, STRICT
LIABILITY, BREACH OF WARRANTY OR BREACH OF CONTRACT EXCEPT THOSE
PROVIDED IN PARAGRAPH 1.F.3.  YOU AGREE THAT THE FOUNDATION, THE
TRADEMARK OWNER, AND ANY DISTRIBUTOR UNDER THIS AGREEMENT WILL NOT BE
LIABLE TO YOU FOR ACTUAL, DIRECT, INDIRECT, CONSEQUENTIAL, PUNITIVE OR
INCIDENTAL DAMAGES EVEN IF YOU GIVE NOTICE OF THE POSSIBILITY OF SUCH
DAMAGE.

1.F.3.  LIMITED RIGHT OF REPLACEMENT OR REFUND - If you discover a
defect in this electronic work within 90 days of receiving it, you can
receive a refund of the money (if any) you paid for it by sending a
written explanation to the person you received the work from.  If you
received the work on a physical medium, you must return the medium with
your written explanation.  The person or entity that provided you with
the defective work may elect to provide a replacement copy in lieu of a
refund.  If you received the work electronically, the person or entity
providing it to you may choose to give you a second opportunity to
receive the work electronically in lieu of a refund.  If the second copy
is also defective, you may demand a refund in writing without further
opportunities to fix the problem.

1.F.4.  Except for the limited right of replacement or refund set forth
in paragraph 1.F.3, this work is provided to you 'AS-IS', WITH NO OTHER
WARRANTIES OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO
WARRANTIES OF MERCHANTABILITY OR FITNESS FOR ANY PURPOSE.

1.F.5.  Some states do not allow disclaimers of certain implied
warranties or the exclusion or limitation of certain types of damages.
If any disclaimer or limitation set forth in this agreement violates the
law of the state applicable to this agreement, the agreement shall be
interpreted to make the maximum disclaimer or limitation permitted by
the applicable state law.  The invalidity or unenforceability of any
provision of this agreement shall not void the remaining provisions.

1.F.6.  INDEMNITY - You agree to indemnify and hold the Foundation, the
trademark owner, any agent or employee of the Foundation, anyone
providing copies of Project Gutenberg-tm electronic works in accordance
with this agreement, and any volunteers associated with the production,
promotion and distribution of Project Gutenberg-tm electronic works,
harmless from all liability, costs and expenses, including legal fees,
that arise directly or indirectly from any of the following which you do
or cause to occur: (a) distribution of this or any Project Gutenberg-tm
work, (b) alteration, modification, or additions or deletions to any
Project Gutenberg-tm work, and (c) any Defect you cause.


Section  2.  Information about the Mission of Project Gutenberg-tm

Project Gutenberg-tm is synonymous with the free distribution of
electronic works in formats readable by the widest variety of computers
including obsolete, old, middle-aged and new computers.  It exists
because of the efforts of hundreds of volunteers and donations from
people in all walks of life.

Volunteers and financial support to provide volunteers with the
assistance they need are critical to reaching Project Gutenberg-tm's
goals and ensuring that the Project Gutenberg-tm collection will
remain freely available for generations to come.  In 2001, the Project
Gutenberg Literary Archive Foundation was created to provide a secure
and permanent future for Project Gutenberg-tm and future generations.
To learn more about the Project Gutenberg Literary Archive Foundation
and how your efforts and donations can help, see Sections 3 and 4
and the Foundation information page at www.gutenberg.org


Section 3.  Information about the Project Gutenberg Literary Archive
Foundation

The Project Gutenberg Literary Archive Foundation is a non profit
501(c)(3) educational corporation organized under the laws of the
state of Mississippi and granted tax exempt status by the Internal
Revenue Service.  The Foundation's EIN or federal tax identification
number is 64-6221541.  Contributions to the Project Gutenberg
Literary Archive Foundation are tax deductible to the full extent
permitted by U.S. federal laws and your state's laws.

The Foundation's principal office is located at 4557 Melan Dr. S.
Fairbanks, AK, 99712., but its volunteers and employees are scattered
throughout numerous locations.  Its business office is located at 809
North 1500 West, Salt Lake City, UT 84116, (801) 596-1887.  Email
contact links and up to date contact information can be found at the
Foundation's web site and official page at www.gutenberg.org/contact

For additional contact information:
     Dr. Gregory B. Newby
     Chief Executive and Director
     gbnewby@pglaf.org

Section 4.  Information about Donations to the Project Gutenberg
Literary Archive Foundation

Project Gutenberg-tm depends upon and cannot survive without wide
spread public support and donations to carry out its mission of
increasing the number of public domain and licensed works that can be
freely distributed in machine readable form accessible by the widest
array of equipment including outdated equipment.  Many small donations
($1 to $5,000) are particularly important to maintaining tax exempt
status with the IRS.

The Foundation is committed to complying with the laws regulating
charities and charitable donations in all 50 states of the United
States.  Compliance requirements are not uniform and it takes a
considerable effort, much paperwork and many fees to meet and keep up
with these requirements.  We do not solicit donations in locations
where we have not received written confirmation of compliance.  To
SEND DONATIONS or determine the status of compliance for any
particular state visit www.gutenberg.org/donate

While we cannot and do not solicit contributions from states where we
have not met the solicitation requirements, we know of no prohibition
against accepting unsolicited donations from donors in such states who
approach us with offers to donate.

International donations are gratefully accepted, but we cannot make
any statements concerning tax treatment of donations received from
outside the United States.  U.S. laws alone swamp our small staff.

Please check the Project Gutenberg Web pages for current donation
methods and addresses.  Donations are accepted in a number of other
ways including checks, online payments and credit card donations.
To donate, please visit:  www.gutenberg.org/donate


Section 5.  General Information About Project Gutenberg-tm electronic
works.

Professor Michael S. Hart was the originator of the Project Gutenberg-tm
concept of a library of electronic works that could be freely shared
with anyone.  For forty years, he produced and distributed Project
Gutenberg-tm eBooks with only a loose network of volunteer support.

Project Gutenberg-tm eBooks are often created from several printed
editions, all of which are confirmed as Public Domain in the U.S.
unless a copyright notice is included.  Thus, we do not necessarily
keep eBooks in compliance with any particular paper edition.

Most people start at our Web site which has the main PG search facility:

     www.gutenberg.org

This Web site includes information about Project Gutenberg-tm,
including how to make donations to the Project Gutenberg Literary
Archive Foundation, how to help produce our new eBooks, and how to
subscribe to our email newsletter to hear about new eBooks.
\end{PGtext}

% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% End of the Project Gutenberg EBook of The Integration of Functions of a %
% Single Variable, by G. H. Hardy                                         %
%                                                                         %
% *** END OF THIS PROJECT GUTENBERG EBOOK INTEGRATION OF FUNCTIONS OF ONE VARIABLE ***
%                                                                         %
% ***** This file should be named 38993-t.tex or 38993-t.zip *****        %
% This and all associated files of various formats will be found in:      %
%         http://www.gutenberg.org/3/8/9/9/38993/                         %
%                                                                         %
% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %

\end{document}

This is pdfTeX, Version 3.1415926-1.40.10 (TeX Live 2009/Debian) (format=pdflatex 2012.9.24)  27 SEP 2012 19:13
entering extended mode
 %&-line parsing enabled.
**38993-t.tex
(./38993-t.tex
LaTeX2e <2009/09/24>
Babel <v3.8l> and hyphenation patterns for english, usenglishmax, dumylang, noh
yphenation, farsi, arabic, croatian, bulgarian, ukrainian, russian, czech, slov
ak, danish, dutch, finnish, french, basque, ngerman, german, german-x-2009-06-1
9, ngerman-x-2009-06-19, ibycus, monogreek, greek, ancientgreek, hungarian, san
skrit, italian, latin, latvian, lithuanian, mongolian2a, mongolian, bokmal, nyn
orsk, romanian, irish, coptic, serbian, turkish, welsh, esperanto, uppersorbian
, estonian, indonesian, interlingua, icelandic, kurmanji, slovenian, polish, po
rtuguese, spanish, galician, catalan, swedish, ukenglish, pinyin, loaded.
(/usr/share/texmf-texlive/tex/latex/base/book.cls
Document Class: book 2007/10/19 v1.4h Standard LaTeX document class
(/usr/share/texmf-texlive/tex/latex/base/bk12.clo
File: bk12.clo 2007/10/19 v1.4h Standard LaTeX file (size option)
)
\c@part=\count79
\c@chapter=\count80
\c@section=\count81
\c@subsection=\count82
\c@subsubsection=\count83
\c@paragraph=\count84
\c@subparagraph=\count85
\c@figure=\count86
\c@table=\count87
\abovecaptionskip=\skip41
\belowcaptionskip=\skip42
\bibindent=\dimen102
) (/usr/share/texmf-texlive/tex/latex/base/inputenc.sty
Package: inputenc 2008/03/30 v1.1d Input encoding file
\inpenc@prehook=\toks14
\inpenc@posthook=\toks15
(/usr/share/texmf-texlive/tex/latex/base/latin1.def
File: latin1.def 2008/03/30 v1.1d Input encoding file
)) (/usr/share/texmf-texlive/tex/latex/base/ifthen.sty
Package: ifthen 2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
) (/usr/share/texmf-texlive/tex/latex/amsmath/amsmath.sty
Package: amsmath 2000/07/18 v2.13 AMS math features
\@mathmargin=\skip43
For additional information on amsmath, use the `?' option.
(/usr/share/texmf-texlive/tex/latex/amsmath/amstext.sty
Package: amstext 2000/06/29 v2.01
(/usr/share/texmf-texlive/tex/latex/amsmath/amsgen.sty
File: amsgen.sty 1999/11/30 v2.0
\@emptytoks=\toks16
\ex@=\dimen103
)) (/usr/share/texmf-texlive/tex/latex/amsmath/amsbsy.sty
Package: amsbsy 1999/11/29 v1.2d
\pmbraise@=\dimen104
) (/usr/share/texmf-texlive/tex/latex/amsmath/amsopn.sty
Package: amsopn 1999/12/14 v2.01 operator names
)
\inf@bad=\count88
LaTeX Info: Redefining \frac on input line 211.
\uproot@=\count89
\leftroot@=\count90
LaTeX Info: Redefining \overline on input line 307.
\classnum@=\count91
\DOTSCASE@=\count92
LaTeX Info: Redefining \ldots on input line 379.
LaTeX Info: Redefining \dots on input line 382.
LaTeX Info: Redefining \cdots on input line 467.
\Mathstrutbox@=\box26
\strutbox@=\box27
\big@size=\dimen105
LaTeX Font Info:    Redeclaring font encoding OML on input line 567.
LaTeX Font Info:    Redeclaring font encoding OMS on input line 568.
\macc@depth=\count93
\c@MaxMatrixCols=\count94
\dotsspace@=\muskip10
\c@parentequation=\count95
\dspbrk@lvl=\count96
\tag@help=\toks17
\row@=\count97
\column@=\count98
\maxfields@=\count99
\andhelp@=\toks18
\eqnshift@=\dimen106
\alignsep@=\dimen107
\tagshift@=\dimen108
\tagwidth@=\dimen109
\totwidth@=\dimen110
\lineht@=\dimen111
\@envbody=\toks19
\multlinegap=\skip44
\multlinetaggap=\skip45
\mathdisplay@stack=\toks20
LaTeX Info: Redefining \[ on input line 2666.
LaTeX Info: Redefining \] on input line 2667.
) (/usr/share/texmf-texlive/tex/latex/amsfonts/amssymb.sty
Package: amssymb 2009/06/22 v3.00
(/usr/share/texmf-texlive/tex/latex/amsfonts/amsfonts.sty
Package: amsfonts 2009/06/22 v3.00 Basic AMSFonts support
\symAMSa=\mathgroup4
\symAMSb=\mathgroup5
LaTeX Font Info:    Overwriting math alphabet `\mathfrak' in version `bold'
(Font)                  U/euf/m/n --> U/euf/b/n on input line 96.
)) (/usr/share/texmf-texlive/tex/latex/graphics/graphicx.sty
Package: graphicx 1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
(/usr/share/texmf-texlive/tex/latex/graphics/keyval.sty
Package: keyval 1999/03/16 v1.13 key=value parser (DPC)
\KV@toks@=\toks21
) (/usr/share/texmf-texlive/tex/latex/graphics/graphics.sty
Package: graphics 2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR)
(/usr/share/texmf-texlive/tex/latex/graphics/trig.sty
Package: trig 1999/03/16 v1.09 sin cos tan (DPC)
) (/etc/texmf/tex/latex/config/graphics.cfg
File: graphics.cfg 2009/08/28 v1.8 graphics configuration of TeX Live
)
Package graphics Info: Driver file: pdftex.def on input line 91.
(/usr/share/texmf-texlive/tex/latex/pdftex-def/pdftex.def
File: pdftex.def 2009/08/25 v0.04m Graphics/color for pdfTeX
\Gread@gobject=\count100
))
\Gin@req@height=\dimen112
\Gin@req@width=\dimen113
) (/usr/share/texmf-texlive/tex/latex/base/alltt.sty
Package: alltt 1997/06/16 v2.0g defines alltt environment
) (/usr/share/texmf-texlive/tex/latex/yfonts/yfonts.sty
Package: yfonts 2003/01/08 v1.3 (WaS)
) (/usr/share/texmf-texlive/tex/latex/footmisc/footmisc.sty
Package: footmisc 2009/09/15 v5.5a a miscellany of footnote facilities
\FN@temptoken=\toks22
\footnotemargin=\dimen114
\c@pp@next@reset=\count101
\c@@fnserial=\count102
Package footmisc Info: Declaring symbol style bringhurst on input line 855.
Package footmisc Info: Declaring symbol style chicago on input line 863.
Package footmisc Info: Declaring symbol style wiley on input line 872.
Package footmisc Info: Declaring symbol style lamport-robust on input line 883.

Package footmisc Info: Declaring symbol style lamport* on input line 903.
Package footmisc Info: Declaring symbol style lamport*-robust on input line 924
.
)

LaTeX Warning: You have requested, on input line 104, version
               `2011/06/06' of package footmisc,
               but only version
               `2009/09/15 v5.5a a miscellany of footnote facilities'
               is available.

(/usr/share/texmf-texlive/tex/latex/fancyhdr/fancyhdr.sty
\fancy@headwidth=\skip46
\f@ncyO@elh=\skip47
\f@ncyO@erh=\skip48
\f@ncyO@olh=\skip49
\f@ncyO@orh=\skip50
\f@ncyO@elf=\skip51
\f@ncyO@erf=\skip52
\f@ncyO@olf=\skip53
\f@ncyO@orf=\skip54
) (/usr/share/texmf-texlive/tex/latex/geometry/geometry.sty
Package: geometry 2008/12/21 v4.2 Page Geometry
(/usr/share/texmf-texlive/tex/generic/oberdiek/ifpdf.sty
Package: ifpdf 2009/04/10 v2.0 Provides the ifpdf switch (HO)
Package ifpdf Info: pdfTeX in pdf mode detected.
) (/usr/share/texmf-texlive/tex/generic/oberdiek/ifvtex.sty
Package: ifvtex 2008/11/04 v1.4 Switches for detecting VTeX and its modes (HO)
Package ifvtex Info: VTeX not detected.
)
\Gm@cnth=\count103
\Gm@cntv=\count104
\c@Gm@tempcnt=\count105
\Gm@bindingoffset=\dimen115
\Gm@wd@mp=\dimen116
\Gm@odd@mp=\dimen117
\Gm@even@mp=\dimen118
\Gm@dimlist=\toks23
(/usr/share/texmf-texlive/tex/xelatex/xetexconfig/geometry.cfg))

LaTeX Warning: You have requested, on input line 163, version
               `2010/09/12' of package geometry,
               but only version
               `2008/12/21 v4.2 Page Geometry'
               is available.

(/usr/share/texmf-texlive/tex/latex/hyperref/hyperref.sty
Package: hyperref 2009/10/09 v6.79a Hypertext links for LaTeX
(/usr/share/texmf-texlive/tex/generic/ifxetex/ifxetex.sty
Package: ifxetex 2009/01/23 v0.5 Provides ifxetex conditional
) (/usr/share/texmf-texlive/tex/latex/oberdiek/hycolor.sty
Package: hycolor 2009/10/02 v1.5 Code for color options of hyperref/bookmark (H
O)
(/usr/share/texmf-texlive/tex/latex/oberdiek/xcolor-patch.sty
Package: xcolor-patch 2009/10/02 xcolor patch
))
\@linkdim=\dimen119
\Hy@linkcounter=\count106
\Hy@pagecounter=\count107
(/usr/share/texmf-texlive/tex/latex/hyperref/pd1enc.def
File: pd1enc.def 2009/10/09 v6.79a Hyperref: PDFDocEncoding definition (HO)
) (/usr/share/texmf-texlive/tex/generic/oberdiek/etexcmds.sty
Package: etexcmds 2007/12/12 v1.2 Prefix for e-TeX command names (HO)
(/usr/share/texmf-texlive/tex/generic/oberdiek/infwarerr.sty
Package: infwarerr 2007/09/09 v1.2 Providing info/warning/message (HO)
)
Package etexcmds Info: Could not find \expanded.
(etexcmds)             That can mean that you are not using pdfTeX 1.50 or
(etexcmds)             that some package has redefined \expanded.
(etexcmds)             In the latter case, load this package earlier.
) (/etc/texmf/tex/latex/config/hyperref.cfg
File: hyperref.cfg 2002/06/06 v1.2 hyperref configuration of TeXLive
) (/usr/share/texmf-texlive/tex/latex/oberdiek/kvoptions.sty
Package: kvoptions 2009/08/13 v3.4 Keyval support for LaTeX options (HO)
(/usr/share/texmf-texlive/tex/generic/oberdiek/kvsetkeys.sty
Package: kvsetkeys 2009/07/30 v1.5 Key value parser with default handler suppor
t (HO)
))
Package hyperref Info: Option `hyperfootnotes' set `false' on input line 2864.
Package hyperref Info: Option `bookmarks' set `true' on input line 2864.
Package hyperref Info: Option `linktocpage' set `false' on input line 2864.
Package hyperref Info: Option `pdfdisplaydoctitle' set `true' on input line 286
4.
Package hyperref Info: Option `pdfpagelabels' set `true' on input line 2864.
Package hyperref Info: Option `bookmarksopen' set `true' on input line 2864.
Package hyperref Info: Option `colorlinks' set `true' on input line 2864.
Package hyperref Info: Hyper figures OFF on input line 2975.
Package hyperref Info: Link nesting OFF on input line 2980.
Package hyperref Info: Hyper index ON on input line 2983.
Package hyperref Info: Plain pages OFF on input line 2990.
Package hyperref Info: Backreferencing OFF on input line 2995.
Implicit mode ON; LaTeX internals redefined
Package hyperref Info: Bookmarks ON on input line 3191.
(/usr/share/texmf-texlive/tex/latex/ltxmisc/url.sty
\Urlmuskip=\muskip11
Package: url 2006/04/12  ver 3.3  Verb mode for urls, etc.
)
LaTeX Info: Redefining \url on input line 3428.
(/usr/share/texmf-texlive/tex/generic/oberdiek/bitset.sty
Package: bitset 2007/09/28 v1.0 Data type bit set (HO)
(/usr/share/texmf-texlive/tex/generic/oberdiek/intcalc.sty
Package: intcalc 2007/09/27 v1.1 Expandable integer calculations (HO)
) (/usr/share/texmf-texlive/tex/generic/oberdiek/bigintcalc.sty
Package: bigintcalc 2007/11/11 v1.1 Expandable big integer calculations (HO)
(/usr/share/texmf-texlive/tex/generic/oberdiek/pdftexcmds.sty
Package: pdftexcmds 2009/09/23 v0.6 LuaTeX support for pdfTeX utility functions
 (HO)
(/usr/share/texmf-texlive/tex/generic/oberdiek/ifluatex.sty
Package: ifluatex 2009/04/17 v1.2 Provides the ifluatex switch (HO)
Package ifluatex Info: LuaTeX not detected.
) (/usr/share/texmf-texlive/tex/generic/oberdiek/ltxcmds.sty
Package: ltxcmds 2009/08/05 v1.0 Some LaTeX kernel commands for general use (HO
)
)
Package pdftexcmds Info: LuaTeX not detected.
Package pdftexcmds Info: \pdf@primitive is available.
Package pdftexcmds Info: \pdf@ifprimitive is available.
)))
\Fld@menulength=\count108
\Field@Width=\dimen120
\Fld@charsize=\dimen121
\Field@toks=\toks24
Package hyperref Info: Hyper figures OFF on input line 4377.
Package hyperref Info: Link nesting OFF on input line 4382.
Package hyperref Info: Hyper index ON on input line 4385.
Package hyperref Info: backreferencing OFF on input line 4392.
Package hyperref Info: Link coloring ON on input line 4395.
Package hyperref Info: Link coloring with OCG OFF on input line 4402.
Package hyperref Info: PDF/A mode OFF on input line 4407.
(/usr/share/texmf-texlive/tex/generic/oberdiek/atbegshi.sty
Package: atbegshi 2008/07/31 v1.9 At begin shipout hook (HO)
)
\Hy@abspage=\count109
\c@Item=\count110
)
*hyperref using driver hpdftex*
(/usr/share/texmf-texlive/tex/latex/hyperref/hpdftex.def
File: hpdftex.def 2009/10/09 v6.79a Hyperref driver for pdfTeX
\Fld@listcount=\count111
) (./38993-t.aux)
\openout1 = `38993-t.aux'.

LaTeX Font Info:    Checking defaults for OML/cmm/m/it on input line 332.
LaTeX Font Info:    ... okay on input line 332.
LaTeX Font Info:    Checking defaults for T1/cmr/m/n on input line 332.
LaTeX Font Info:    ... okay on input line 332.
LaTeX Font Info:    Checking defaults for OT1/cmr/m/n on input line 332.
LaTeX Font Info:    ... okay on input line 332.
LaTeX Font Info:    Checking defaults for OMS/cmsy/m/n on input line 332.
LaTeX Font Info:    ... okay on input line 332.
LaTeX Font Info:    Checking defaults for OMX/cmex/m/n on input line 332.
LaTeX Font Info:    ... okay on input line 332.
LaTeX Font Info:    Checking defaults for U/cmr/m/n on input line 332.
LaTeX Font Info:    ... okay on input line 332.
LaTeX Font Info:    Checking defaults for LY/yfrak/m/n on input line 332.
LaTeX Font Info:    ... okay on input line 332.
LaTeX Font Info:    Checking defaults for LYG/ygoth/m/n on input line 332.
LaTeX Font Info:    ... okay on input line 332.
LaTeX Font Info:    Checking defaults for PD1/pdf/m/n on input line 332.
LaTeX Font Info:    ... okay on input line 332.
(/usr/share/texmf/tex/context/base/supp-pdf.mkii
[Loading MPS to PDF converter (version 2006.09.02).]
\scratchcounter=\count112
\scratchdimen=\dimen122
\scratchbox=\box28
\nofMPsegments=\count113
\nofMParguments=\count114
\everyMPshowfont=\toks25
\MPscratchCnt=\count115
\MPscratchDim=\dimen123
\MPnumerator=\count116
\everyMPtoPDFconversion=\toks26
)
*geometry auto-detecting driver*
*geometry detected driver: pdftex*
-------------------- Geometry parameters
paper: class default
landscape: --
twocolumn: --
twoside: true
asymmetric: --
h-parts: 117.43875pt, 379.4175pt, 117.43875pt
v-parts: 75.16095pt, 607.06755pt, 112.74149pt
hmarginratio: 1:1
vmarginratio: 2:3
lines: --
heightrounded: --
bindingoffset: 0.0pt
truedimen: --
includehead: --
includefoot: --
includemp: --
driver: pdftex
-------------------- Page layout dimensions and switches
\paperwidth  614.295pt
\paperheight 794.96999pt
\textwidth  379.4175pt
\textheight 607.06755pt
\oddsidemargin  45.16876pt
\evensidemargin 45.16876pt
\topmargin  -28.98283pt
\headheight 12.0pt
\headsep    19.8738pt
\footskip   30.0pt
\marginparwidth 98.0pt
\marginparsep   7.0pt
\columnsep  10.0pt
\skip\footins  10.8pt plus 4.0pt minus 2.0pt
\hoffset 0.0pt
\voffset 0.0pt
\mag 1000
\@twosidetrue \@mparswitchtrue 
(1in=72.27pt, 1cm=28.45pt)
-----------------------
(/usr/share/texmf-texlive/tex/latex/graphics/color.sty
Package: color 2005/11/14 v1.0j Standard LaTeX Color (DPC)
(/etc/texmf/tex/latex/config/color.cfg
File: color.cfg 2007/01/18 v1.5 color configuration of teTeX/TeXLive
)
Package color Info: Driver file: pdftex.def on input line 130.
)
Package hyperref Info: Link coloring ON on input line 332.
(/usr/share/texmf-texlive/tex/latex/hyperref/nameref.sty
Package: nameref 2007/05/29 v2.31 Cross-referencing by name of section
(/usr/share/texmf-texlive/tex/latex/oberdiek/refcount.sty
Package: refcount 2008/08/11 v3.1 Data extraction from references (HO)
)
\c@section@level=\count117
)
LaTeX Info: Redefining \ref on input line 332.
LaTeX Info: Redefining \pageref on input line 332.
(./38993-t.out) (./38993-t.out)
\@outlinefile=\write3
\openout3 = `38993-t.out'.

\AtBeginShipoutBox=\box29
[1

{/var/lib/texmf/fonts/map/pdftex/updmap/pdftex.map}]
LaTeX Font Info:    Try loading font information for U+msa on input line 373.
(/usr/share/texmf-texlive/tex/latex/amsfonts/umsa.fd
File: umsa.fd 2009/06/22 v3.00 AMS symbols A
)
LaTeX Font Info:    Try loading font information for U+msb on input line 373.
(/usr/share/texmf-texlive/tex/latex/amsfonts/umsb.fd
File: umsb.fd 2009/06/22 v3.00 AMS symbols B
) [2

] <./images/device.pdf, id=31, 469.755pt x 561.09625pt>
File: ./images/device.pdf Graphic file (type pdf)
<use ./images/device.pdf> [1



 <./images/device.pdf>] [2] [3] [4] [5] [6]
LaTeX Font Info:    Try loading font information for OMS+cmr on input line 562.

(/usr/share/texmf-texlive/tex/latex/base/omscmr.fd
File: omscmr.fd 1999/05/25 v2.5h Standard LaTeX font definitions
)
LaTeX Font Info:    Font shape `OMS/cmr/m/n' in size <12> not available
(Font)              Font shape `OMS/cmsy/m/n' tried instead on input line 562.
[7] [1


]
LaTeX Font Info:    Font shape `OMS/cmr/m/n' in size <8> not available
(Font)              Font shape `OMS/cmsy/m/n' tried instead on input line 638.
LaTeX Font Info:    Font shape `OMS/cmr/m/n' in size <7> not available
(Font)              Font shape `OMS/cmsy/m/n' tried instead on input line 644.
[2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
LaTeX Font Info:    Font shape `OMS/cmr/m/n' in size <10.95> not available
(Font)              Font shape `OMS/cmsy/m/n' tried instead on input line 1350.

[15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
[31] [32] [33]
LaTeX Font Info:    Font shape `OMS/cmr/m/n' in size <10> not available
(Font)              Font shape `OMS/cmsy/m/n' tried instead on input line 2402.

[34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49]
[50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65]
[66


] [67] [68]
LaTeX Font Info:    Font shape `OMS/cmr/bx/n' in size <10.95> not available
(Font)              Font shape `OMS/cmsy/b/n' tried instead on input line 4381.

[69

] [70] [1]
Underfull \vbox (badness 10000) has occurred while \output is active []

[2

]
Underfull \vbox (badness 10000) has occurred while \output is active []

[3]
Underfull \vbox (badness 10000) has occurred while \output is active []

[4]
Underfull \vbox (badness 10000) has occurred while \output is active []

[5]
Underfull \vbox (badness 10000) has occurred while \output is active []

[6] [7] (./38993-t.aux)

 *File List*
    book.cls    2007/10/19 v1.4h Standard LaTeX document class
    bk12.clo    2007/10/19 v1.4h Standard LaTeX file (size option)
inputenc.sty    2008/03/30 v1.1d Input encoding file
  latin1.def    2008/03/30 v1.1d Input encoding file
  ifthen.sty    2001/05/26 v1.1c Standard LaTeX ifthen package (DPC)
 amsmath.sty    2000/07/18 v2.13 AMS math features
 amstext.sty    2000/06/29 v2.01
  amsgen.sty    1999/11/30 v2.0
  amsbsy.sty    1999/11/29 v1.2d
  amsopn.sty    1999/12/14 v2.01 operator names
 amssymb.sty    2009/06/22 v3.00
amsfonts.sty    2009/06/22 v3.00 Basic AMSFonts support
graphicx.sty    1999/02/16 v1.0f Enhanced LaTeX Graphics (DPC,SPQR)
  keyval.sty    1999/03/16 v1.13 key=value parser (DPC)
graphics.sty    2009/02/05 v1.0o Standard LaTeX Graphics (DPC,SPQR)
    trig.sty    1999/03/16 v1.09 sin cos tan (DPC)
graphics.cfg    2009/08/28 v1.8 graphics configuration of TeX Live
  pdftex.def    2009/08/25 v0.04m Graphics/color for pdfTeX
   alltt.sty    1997/06/16 v2.0g defines alltt environment
  yfonts.sty    2003/01/08 v1.3 (WaS)
footmisc.sty    2009/09/15 v5.5a a miscellany of footnote facilities
fancyhdr.sty    
geometry.sty    2008/12/21 v4.2 Page Geometry
   ifpdf.sty    2009/04/10 v2.0 Provides the ifpdf switch (HO)
  ifvtex.sty    2008/11/04 v1.4 Switches for detecting VTeX and its modes (HO)
geometry.cfg
hyperref.sty    2009/10/09 v6.79a Hypertext links for LaTeX
 ifxetex.sty    2009/01/23 v0.5 Provides ifxetex conditional
 hycolor.sty    2009/10/02 v1.5 Code for color options of hyperref/bookmark (HO
)
xcolor-patch.sty    2009/10/02 xcolor patch
  pd1enc.def    2009/10/09 v6.79a Hyperref: PDFDocEncoding definition (HO)
etexcmds.sty    2007/12/12 v1.2 Prefix for e-TeX command names (HO)
infwarerr.sty    2007/09/09 v1.2 Providing info/warning/message (HO)
hyperref.cfg    2002/06/06 v1.2 hyperref configuration of TeXLive
kvoptions.sty    2009/08/13 v3.4 Keyval support for LaTeX options (HO)
kvsetkeys.sty    2009/07/30 v1.5 Key value parser with default handler support 
(HO)
     url.sty    2006/04/12  ver 3.3  Verb mode for urls, etc.
  bitset.sty    2007/09/28 v1.0 Data type bit set (HO)
 intcalc.sty    2007/09/27 v1.1 Expandable integer calculations (HO)
bigintcalc.sty    2007/11/11 v1.1 Expandable big integer calculations (HO)
pdftexcmds.sty    2009/09/23 v0.6 LuaTeX support for pdfTeX utility functions (
HO)
ifluatex.sty    2009/04/17 v1.2 Provides the ifluatex switch (HO)
 ltxcmds.sty    2009/08/05 v1.0 Some LaTeX kernel commands for general use (HO)

atbegshi.sty    2008/07/31 v1.9 At begin shipout hook (HO)
 hpdftex.def    2009/10/09 v6.79a Hyperref driver for pdfTeX
supp-pdf.mkii
   color.sty    2005/11/14 v1.0j Standard LaTeX Color (DPC)
   color.cfg    2007/01/18 v1.5 color configuration of teTeX/TeXLive
 nameref.sty    2007/05/29 v2.31 Cross-referencing by name of section
refcount.sty    2008/08/11 v3.1 Data extraction from references (HO)
 38993-t.out
 38993-t.out
    umsa.fd    2009/06/22 v3.00 AMS symbols A
    umsb.fd    2009/06/22 v3.00 AMS symbols B
./images/device.pdf
  omscmr.fd    1999/05/25 v2.5h Standard LaTeX font definitions
 ***********


LaTeX Font Warning: Size substitutions with differences
(Font)              up to 4.0pt have occurred.

 )
(\end occurred inside a group at level 2)

### simple group (level 2) entered at line 2326 ({)
### simple group (level 1) entered at line 1667 ({)
### bottom level 
Here is how much of TeX's memory you used:
 6368 strings out of 493848
 87257 string characters out of 1152823
 171395 words of memory out of 3000000
 9219 multiletter control sequences out of 15000+50000
 19342 words of font info for 72 fonts, out of 3000000 for 9000
 714 hyphenation exceptions out of 8191
 37i,12n,43p,298b,664s stack positions out of 5000i,500n,10000p,200000b,50000s
</usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmbsy1
0.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmbx10.pfb></usr
/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmbx12.pfb></usr/share/texm
f-texlive/fonts/type1/public/amsfonts/cm/cmcsc10.pfb></usr/share/texmf-texlive/
fonts/type1/public/amsfonts/cm/cmex10.pfb></usr/share/texmf-texlive/fonts/type1
/public/amsfonts/cm/cmmi10.pfb></usr/share/texmf-texlive/fonts/type1/public/ams
fonts/cm/cmmi12.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cm
mi6.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmmi7.pfb></us
r/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmmi8.pfb></usr/share/texm
f-texlive/fonts/type1/public/amsfonts/cm/cmr10.pfb></usr/share/texmf-texlive/fo
nts/type1/public/amsfonts/cm/cmr12.pfb></usr/share/texmf-texlive/fonts/type1/pu
blic/amsfonts/cm/cmr17.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfont
s/cm/cmr5.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmr6.pfb
></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmr7.pfb></usr/share/
texmf-texlive/fonts/type1/public/amsfonts/cm/cmr8.pfb></usr/share/texmf-texlive
/fonts/type1/public/amsfonts/cm/cmsy10.pfb></usr/share/texmf-texlive/fonts/type
1/public/amsfonts/cm/cmsy6.pfb></usr/share/texmf-texlive/fonts/type1/public/ams
fonts/cm/cmsy7.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cms
y8.pfb></usr/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmti10.pfb></us
r/share/texmf-texlive/fonts/type1/public/amsfonts/cm/cmti12.pfb></usr/share/tex
mf-texlive/fonts/type1/public/amsfonts/cm/cmtt8.pfb></usr/share/texmf-texlive/f
onts/type1/public/amsfonts/symbols/msam10.pfb></usr/share/texmf-texlive/fonts/t
ype1/public/gothic/ygoth.pfb>
Output written on 38993-t.pdf (86 pages, 682798 bytes).
PDF statistics:
 850 PDF objects out of 1000 (max. 8388607)
 308 named destinations out of 1000 (max. 500000)
 30 words of extra memory for PDF output out of 10000 (max. 10000000)

