% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
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% The Project Gutenberg EBook of On Riemann's Theory of Algebraic Functions
% and their Integrals, by Felix Klein                                     %
%                                                                         %
% 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: On Riemann's Theory of Algebraic Functions and their Integrals   %
%        A Supplement to the Usual Treatises                              %
%                                                                         %
% Author: Felix Klein                                                     %
%                                                                         %
% Translator: Frances Hardcastle                                          %
%                                                                         %
% Release Date: August 3, 2011 [EBook #36959]                             %
% Most recently updated: June 11, 2021                         %
%                                                                         %
% Language: English                                                       %
%                                                                         %
% Character set encoding: UTF-8                                           %
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The Project Gutenberg EBook of On Riemann's Theory of Algebraic Functions
and their Integrals, by Felix Klein

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: On Riemann's Theory of Algebraic Functions and their Integrals
       A Supplement to the Usual Treatises

Author: Felix Klein

Translator: Frances Hardcastle

Release Date: August 3, 2011 [EBook #36959]
Most recently updated: June 11, 2021

Language: English

Character set encoding: UTF-8     

*** START OF THIS PROJECT GUTENBERG EBOOK ON RIEMANN'S THEORY ***
\end{PGtext}
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\begin{PGtext}
Produced by Andrew D. Hwang
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% [** TN: Previous macro prints:
% ON RIEMANN'S THEORY
% OF
% ALGEBRAIC FUNCTIONS
% AND THEIR
% INTEGRALS.]
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A SUPPLEMENT TO THE USUAL TREATISES.
\vfill\vfill

{\footnotesize BY} \\
FELIX KLEIN.
\vfill\vfill

\footnotesize TRANSLATED FROM THE GERMAN, WITH THE AUTHOR'S
PERMISSION,
\vfill

BY \\
\small FRANCES HARDCASTLE, \\
{\scriptsize GIRTON COLLEGE, CAMBRIDGE.}
\vfill\vfill

{\large\textgoth{Cambridge}:} \\
MACMILLAN AND BOWES. \\
1893
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\textgoth{Cambridge}: \\[4pt]
\scriptsize
PRINTED BY C. J. CLAY, M.A. \&~SONS, \\[4pt]
AT THE UNIVERSITY PRESS.
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\clearpage
\PageSep{v}


\Chapter{Translator's Note.}

\First{The} aim of this translation is to reproduce, as far as
possible, the ideas and style of the original in idiomatic
English, rather than to give a literal rendering of its contents.
Even the verbal deviations, however, are few in number. So
little has been written in English on the subject that a
standard set of technical terms as yet hardly exists. Where
there was any choice between equivalent words, I have followed
the usage of Dr~Forsyth in his recently published work on the
Theory of Functions. A \hyperref[glossary]{Glossary} of the principal technical
terms is appended, giving the original German word together
with the English adopted in the text.

Prof.\ Klein had originally intended to revise the proofs, but
owing to his absence in America he kindly waived his right to
do so, in order not to delay the publication. The proofs have
therefore not been submitted to him, though it was with
considerable reluctance that I determined to publish without
this final revision.

My thanks are due to Miss C.~A. Scott,~D.Sc., Professor of
Mathematics in Bryn Mawr College, for many valuable suggestions
in difficult passages and for her interest in the progress
\PageSep{vi}
of the translation, and also for help in the reading of the
proof-sheets. I must also express my thanks to Mr~James
Harkness,~M.A., Associate Professor of Mathematics in Bryn
Mawr College, for helpful advice given from time to time;
and to Miss P.~G. Fawcett, of Newnham College, Cambridge,
for reading over in manuscript the earlier parts which deal
more especially with the subject of Electricity.

\Signature{FRANCES HARDCASTLE.}
{Bryn Mawr College,}
{Pennsylvania,}
{\textit{June}~1, 1893.}
\PageSep{vii}
\TableofContents
\iffalse

CONTENTS.

PART I.

INTRODUCTORY REMARKS.

SECT.                                          PAGE

1. Steady Streamings in the Plane as an Interpretation of the
Functions of x + iy 1

2. Consideration of the Infinities of w=f(z) .... 5

%[** TN: The phrase "Derivation of the" does not appear in the unit title]
3. Rational Functions and their Integrals. Derivation of the
Infinities of higher Order from those of lower Order . 9

4. Experimental Production of these Streamings . . . 12

5. Transition to the Surface of a Sphere. Streamings on
arbitrary curved Surfaces . . . . . . 15

6. Connection between the foregoing Theory and the Functions
of a complex Argument 19

7. Streamings on the Sphere resumed. Riemann's general
Problem 21


PART II.

RIEMANN'S THEORY.

8. Classification of closed Surfaces according to the Value of
the Integer p 23

9. Preliminary Determination of steady Streamings on arbitrary
Surfaces 26

10. The most general steady Streaming. Proof of the Impossibility
of other Streamings 29

11. Illustration of the Streamings by means of Examples . . 32

12. On the Composition of the most general Function of Position
from single Summands 37
\PageSep{viii}

13. On the Multiformity of the Functions. Special Treatment
of multiform Functions 40

14. The ordinary Riemann's Surfaces over the x+iy Plane . 43

15. The Anchor-ring, p = 1, and the two-sheeted Surface over
the Plane with four Branch-points 46

16. Functions of x+iy which correspond to the Streamings
already investigated 51

17. Scope and Significance of the previous Investigations . . 55

18. Extension of the Theory 56


PART III.

CONCLUSIONS.

19. On the Moduli of Algebraical Equations .... 59

20. Conformal Representation of closed Surfaces upon themselves 64

21. Special Treatment of symmetrical Surfaces .... 66

22. Conformal Representation of different closed Surfaces upon
each other 70

23. Surfaces with Boundaries and unifacial Surfaces ... 72

24. Conclusion 75
\fi
\PageSep{ix}


\Chapter{Preface.}

\First{The} pamphlet which I here lay before the public, has grown
from lectures delivered during the past year,\footnote
  {\textit{Theory of Functions treated geometrically.} Part~\textsc{i}, Winter-semester 1880--81,
  Part~\textsc{ii}, Summer-semester~1881.}
in which,
among other objects, I had in view a presentation of Riemann's
theory of algebraic functions and their integrals.\footnote
  {I denote thus the contents of the investigations with which Riemann was
  concerned in the first part of his \textit{Theory of the Abelian Functions}. The
  theory of the $\Theta$-functions, as developed in the second part of the same treatise,
  is in the first place, as we know, of an essentially different character, and
  is excluded from the following presentation as it was from my course of
  lectures.}
Lectures on
higher mathematics offer peculiar difficulties; with the best will
of the lecturer they ultimately fulfil a very modest purpose.
Being usually intended to give a \emph{systematic} development of the
subject, they are either confined to the elements or are lost
amid details. I thought it well in this case, as previously in
others, to adopt the opposite course. I assumed that the
ordinary presentation, as given in text-books on the elements of
Riemann's theory, was known; moreover, when particular points
required to be more fully dealt with, I referred to the fundamental
monographs. But to compensate for this, I devoted
great care to the presentation of the \emph{true train of thought}, and
endeavoured to obtain a \emph{general view} of the scope and efficiency
of the methods. I believe I have frequently obtained good
results by these means, though, of course, only with a gifted
audience; experience will show whether this pamphlet, based on
the same principles, will prove equally useful.
\PageSep{x}

A presentation of the kind attempted is necessarily very
subjective, and the more so in the case of Riemann's theory,
since but scanty material for the purpose is to be found
explicitly given in Riemann's papers. I am not sure that I
should ever have reached a well-defined conception of the whole
subject, had not Herr Prym, many years ago~(1874), in the course
of an opportune conversation, made me a communication which
has increased in importance to me the longer I have thought
over the matter. He told me that \emph{Riemann's surfaces originally
are not necessarily many-sheeted surfaces over the plane, but that,
on the contrary, complex functions of position can be studied on
arbitrarily given curved surfaces in exactly the same way as on
the surfaces over the plane}. The following presentation will
sufficiently show how valuable this remark has been to me. In
natural combination with this there are certain physical considerations
which have been lately developed, although restricted
to simpler cases, from various points of view.\footnote
  {Cf.\ C.~Neumann, \text{Math.\ Ann.}, t.~\textsc{x}., pp.~569--571. Kirchhoff, \textit{Berl.\
  Monatsber.}, 1875, pp.~487--497. Töpler, \textit{Pogg.\ Ann.}, t.~\textsc{clx}., pp.~375--388.}
I have not
hesitated to take these physical conceptions as the starting-point
of my presentation. Riemann, as we know, used
Dirichlet's Principle in their place in his writings. But I have
no doubt that he started from precisely those physical problems,
and then, in order to give what was physically evident the
support of mathematical reasoning, he afterwards substituted
Dirichlet's Principle. Anyone who clearly understands the
conditions under which Riemann worked in Göttingen, anyone
who has followed Riemann's speculations as they have come
down to us, partly in fragments,\footnote
  {\textit{Ges.\ Werke}, pp.~494~\textit{et~seq.}}
will, I think, share my
opinion.---However that may be, the physical method seemed
the true one for my purpose. For it is well known that
Dirichlet's Principle is not sufficient for the actual foundation
of the theorems to be established; moreover, the heuristic
element, which to me was all-important, is brought out far more
prominently by the physical method. Hence the constant
introduction of intuitive considerations, where a proof by
analysis would not have been difficult and might have been
\PageSep{xi}
simpler, hence also the repeated illustration of general results
by examples and figures.

In this connection I must not omit to mention an important
restriction to which I have adhered in the following pages. We
all know the circuitous and difficult considerations by which, of
late years, part at least of those theorems of Riemann which are
here dealt with have been proved in a reliable manner.\footnote
  {Compare in particular the investigations on this subject by C.~Neumann
  and Schwarz. The general case of \emph{closed} surfaces (which is the most important
  for us in what follows) is indeed, as yet, nowhere explicitly and completely dealt
  with. Herr Schwarz contents himself with a few indications with respect to
  these surfaces (\textit{Berl.\ Monatsber.}, 1870, pp.~767~\textit{et~seq.})\ and Herr C.~Neumann
  only considers those cases in which functions are to be determined by means of
  known values on the boundary.}
These
considerations are entirely neglected in what follows and I thus
forego the use of any except intuitive bases for the theorems to
be enunciated. In fact such proofs must in no way be mixed
up with the sequence of thought I have attempted to preserve;
otherwise the result is a presentation unsatisfactory from all
points of view. But they should assuredly follow after, and I
hope, when opportunity offers, to complete in this sense the
present pamphlet.

For the rest, the scope and limits of my presentation speak
for themselves. The frequent use of my friends' publications
and of my own on kindred subjects had a secondary purpose
important to me for personal reasons: I wished to give my
audience a guide, to help them to find for themselves the
reciprocal connections among these papers, and their position
with respect to the general conception put forth in these pages.
As for the \emph{new} problems which offer themselves in great number,
I have only allowed myself to investigate them as far as seemed
consistent with the general aim of this pamphlet. Nevertheless
I should like to draw attention to the theorems on the conformal
representation of arbitrary surfaces which I have worked
out in the last Part; I followed these out the more readily that
Riemann makes a remarkable statement about this subject at
the end of his Dissertation.

One more remark in conclusion to obviate a misunderstanding
which might otherwise arise from the foregoing words.
\PageSep{xii}
Although I have attempted, in the case of algebraic functions
and their integrals, to follow the original chain of ideas which I
assumed to be Riemann's, I by no means include the whole of
what he intended in the theory of functions. The said functions
were for him an example only, in the treatment of which, it is
true, he was particularly fortunate. Inasmuch as he wished to
include all possible functions of complex variables, he had in
mind far more general methods of determination than those we
employ in the following pages; methods of determination in
which physical analogy, here deemed a sufficient basis, fails us.
Compare, in this connection, §\;19~of his Dissertation, compare
also his work on the hypergeometrical series.---With reference
to this, I must explain that I have no wish to draw aside
from these more general considerations by giving a presentation
of a special part, complete in itself. My innermost
conviction rather is that they are destined to play, in the
developments of the modern Theory of Functions, an important
and prominent part.
\Signature{}{}{Borkum,}{\textit{Oct.}~7, 1881.}
\PageSep{1}
\MainMatter


\Part{I.}
{Introductory Remarks.}

\Section{1.}{Steady Streamings in the Plane as an Interpretation
of the Functions of~$x + iy$.}

The physical interpretation of those functions of~$x + iy$
which are dealt with in the following pages is well known.\footnote
  {In particular, reference should be made to Maxwell's \textit{Treatise on Electricity
  and Magnetism} (Cambridge, 1873). So far as the intuitive treatment of the
  subject is concerned, his point of view is exactly that adopted in the text.}
The principles on which it is based are here indicated, solely
for completeness.

Let $w = u + iv$, $z = x + iy$, $w = f(z)$. Then we have, primarily,
\label{page:1}%[** TN: Sole anchor for page cross-reference]
\[
\frac{\dd u}{\dd x} =  \frac{\dd v}{\dd y},\quad
\frac{\dd u}{\dd y} = -\frac{\dd v}{\dd x},
\Tag{(1)}
\]
and hence
\[
\frac{\dd^{2} u}{\dd x^{2}} + \frac{\dd^{2} u}{\dd y^{2}} = 0,
\Tag{(2)}
\]
and also, for~$v$,
\[
\frac{\dd^{2} v}{\dd x^{2}} + \frac{\dd^{2} v}{\dd y^{2}} = 0.
\Tag{(3)}
\]

In these equations we take $u$~to be the \emph{velocity-potential},
so that $\dfrac{\dd u}{\dd x}$,~$\dfrac{\dd u}{\dd y}$ are the components of the velocity of a fluid
moving parallel to the $xy$~plane. We may either suppose this
fluid to be contained between two planes, parallel to the $xy$~plane,
\PageSep{2}
or we may imagine it to be itself an infinitely thin
homogeneous sheet extending over this plane. Then equation~\Eq{(2)}---and
this is the chief point in the physical interpretation---shows
that the streaming is \Gloss[Steady streaming]{\emph{steady}}. The curves $u = \const$.\
are called the \Gloss[Equipotential curve]{\emph{equipotential curves}}, while the curves $v = \const$.,
which, by~\Eq{(1)}, are orthogonal to the first system, are the \Gloss[Stream-line]{\emph{stream-lines}}.
For the purposes of this interpretation it is of course
indifferent of what nature we may imagine the fluid to be, but
for many reasons it will be convenient to identify it here with
the \emph{electric fluid}; $u$~is then proportional to the electrostatic
potential which gives rise to the streaming, and the apparatus
of experimental physics provide sufficient means for the production
of many interesting systems of streamings.

Moreover, if we increase~$u$ throughout by a constant the
streaming itself remains unchanged, since the differential coefficients
$\dfrac{\dd u}{\dd x}$,~$\dfrac{\dd u}{\dd y}$ alone appear explicitly; this is also true of~$v$.
Hence the function~$u + iv$, whose physical interpretation is in
question, is thus determined only to an additive constant près,
a fact which requires to be carefully observed in what follows.

Further, we may observe that equations \Eq{(1)}--\Eq{(3)} remain
unaltered if we replace $u$~by~$v$, and $v$~by~$-u$. Corresponding to
this we get a second system of streamings in which $v$~is the
velocity-potential and the curves $u = \const$.\ are the stream-lines;
in the sense explained above this represents the function~$v - iu$.
It is often of use to consider this new streaming as
well as the original one in which $u$~was the velocity-potential;
we shall speak of it, for brevity, as the \emph{conjugate} streaming. It
is true that the name is somewhat inaccurate, since $u$~bears the
same relation to~$v$, as $v$~does to~$-u$, but it is sufficiently intelligible
for our purpose.

The differential equations \Eq{(1)}--\Eq{(3)}, and hence also the whole
preceding discussion, apply in the first place solely to that
portion of the plane (otherwise an arbitrary portion) in which
%[** TN: "differential-coefficients" hyphenated at line break in orig; only instance]
$u + iv$ is \Gloss[Uniform]{uniform} and in which neither $u + iv$ nor its differential
coefficients become infinite. In order then that the corresponding
physical conditions maybe clearly comprehended, a
\PageSep{3}
region of this kind must be marked off and then by suitable
appliances on the boundary the steady motion within its limits
must be preserved.

In a bounded region of this description points~$z_{0}$ at which
the differential coefficient~$\dfrac{\dd w}{\dd z}$ vanishes call for special attention
To be perfectly general, I will assume at once that $\dfrac{\dd^{2} w}{\dd z^{2}}$, $\dfrac{\dd^{3} w}{\dd z^{3}}$,~$\dots$\Add{,}
up to~$\dfrac{\dd^{\alpha} w}{\dd z^{\alpha}}$ are all zero as well. To determine the course of the
equipotential curves, or of the stream-lines in the vicinity of
such a point, let $w$~be expanded in a series of ascending powers
of~$z - z_{0}$; in this series, the term immediately after the constant
term is the term in~$(z - z_{0})^{\alpha+1}$. Transforming to polar-coordinates
we obtain the following result: \textit{at the point~$z_{0}$, $\alpha + 1$
curves $u = \const$.\ intersect at equal angles, while the same
number of curves $v = \const$.\ are the bisectors of these angles}.
In consequence of this property I call such a point a \Gloss[Cross-point]{\emph{cross-point}},
and moreover a \emph{cross-point of multiplicity~$\alpha$}.

The following figure (which is of course only diagrammatic)
illustrates this for $\alpha = 2$, and explains, in particular, how a cross-point
\Figure{1}{019}
makes its appearance in the orthogonal system formed by
the curves $u = \const$.\Add{,} $v = \const$.

The stream-lines $v = \const$.\ are the heavy lines in the
figure and the direction of motion in each is indicated by an
\PageSep{4}
arrow; the equipotential curves are given by dotted lines.
We see how the fluid flows in towards the cross-point from
three directions, and flows out again in three other directions,
this being possible because the velocity of the streaming is zero
at the cross-point, or, as we may say, by analogy with known
occurrences, because the fluid is at a standstill, the expression
for the velocity being $\sqrt{\left(\dfrac{\dd u}{\dd x}\right)^{2} + \left(\dfrac{\dd u}{\dd y}\right)^{2}}$.

Further, it is useful to consider a cross-point of multiplicity~$\alpha$
\emph{as the limiting case of $\alpha$~simple cross-points}. The analytical
treatment shows this to be permissible. For at an $\alpha$-ple
cross-point the equation $\dfrac{\dd w}{\dd z} = 0$ has an $\alpha$-ple root and this is
caused, as we know, by the coalescence of $\alpha$~simple roots. The
following figures sufficiently explain this view:
\FiguresH{2}{3}{020}

For simplicity, I have here drawn the stream-lines only.
On the left we have the same cross-point of multiplicity two as
in \Fig{1}; on the right we have a streaming with two simple
cross-points close together. It is at once evident that the one
figure is produced by continuous changes from the other.

Throughout the foregoing discussion it has been tacitly
assumed that the region in question does not extend to infinity.
It is true that no fundamental difficulties present themselves
when we take the point $z = \infty$ into account exactly as we take
\PageSep{5}
any other point $z = z_{0}$; instead of the expansion in ascending
powers of~$z - z_{0}$, we obtain, by known methods, an expansion in
ascending powers of~$\dfrac{1}{z}$; there is an $\alpha$-ple cross-point at $z = \infty$
when the term immediately following the constant term in this
expansion is the term in~$\left(\dfrac{1}{z}\right)^{\alpha+1}$. But we need dwell no further
on the geometrical relations corresponding to a streaming of
this kind, for the separate treatment of $z = \infty$, which here
presents itself, will be obviated once and for all by a method to
be explained shortly, and for this reason the point $z = \infty$ will
be left out of consideration in the following sections (§§\;\SecNum{2}--\SecNum{4}),
although, if a complete treatment were desired, it ought to be
specially mentioned.

\Section{2.}{Consideration of the Infinities of $w = f(z)$.}

We now further include in this region points~$z_{0}$ at which
$w = f(z)$ becomes infinite. But, since we are about to consider
only a special class of functions, we restrict ourselves in this
direction by the following condition, viz.: \emph{the differential
coefficient $\dfrac{\dd w}{\dd z}$ must have no \Gloss[Essential singularity]{essential singularities}}, or, in other
words, \emph{$w$~is to be infinite only in the same manner as an expression
of the following form}:
\[
%[** TN: "log" italicized in the original]
A \log(z - z_{0})
  + \frac{A_{1}}{z - z_{0}}
  + \frac{A_{2}}{(z - z_{0})^{2}} + \dots
  \Add{+} \frac{A_{\nu}}{(z - z_{0})^{\nu}},
\]
\emph{in which $\nu$~is a determinate finite quantity}.

Corresponding to the various forms which this expression
assumes, we say that at $z = z_{0}$ different discontinuities are
superposed; a \emph{logarithmic} infinity, an \emph{algebraic} infinity of order
one,~etc. For simplicity we here consider each separately, but
it is also a useful exercise to form a clear idea of the result of
the superposition in individual examples.

In the first instance, let $z = z_{0}$ be a \emph{logarithmic} infinity; we
then have:
\[
w = A\log(z - z_{0})
  + C_{0} + C_{1}(z - z_{0}) + C_{2}(z - z_{0})^{2} + \dots.
\]
\PageSep{6}
Here $A$~is that quantity which when multiplied by~$2i\pi$ is
called, in Cauchy's notation, the \emph{residue} of the logarithmic
infinity, a term which will be occasionally employed in what
follows. In the investigation of a streaming in the vicinity of
the discontinuity it is of primary importance to know whether
$A$~is real, imaginary, or complex. The third case can obviously
be regarded as a superposition of the first two and may
therefore be neglected. There are then only two distinct
possibilities to be considered.

(1) If $A$~is real, let $C_{0} = a + ib$. Then, to a first approximation,
we have, writing $w = u + iv$, $z - z_{0} = re^{i\phi}$,
\[
u = A \log r + a,\quad
v = a\phi + b.
\]
Thus the curves $u = \const$.\ are small circles round the infinity,
and the curves $v = \const$.\ radiate from it in all directions
according to the variable values of~$\phi$. \emph{The motion is such that
$z = z_{0}$ is a \Gloss[Source]{source} of a certain positive or negative \Gloss[Strength]{strength}.} To
calculate this strength, multiply the element of arc of a small
circle described about the discontinuity with radius~$r$, by the
proper velocity and integrate this expression round the circle.
Since
\[
\sqrt{\left(\frac{\dd u}{\dd x}\right)^{2}
    + \left(\frac{\dd u}{\dd y}\right)^{2}}
\]
coincides to a first approximation with~$\dfrac{\dd u}{\dd r}$, that is with~$\dfrac{A}{r}$, we
obtain for the strength the expression
\[
\int_{0}^{2\pi} \frac{A}{r}\, r\, d\phi = 2A\pi.
\]
\emph{The strength is therefore equal to the residue, divided by~$i$; it is
positive or negative with~$A$.}

(2) Let $A$~be purely imaginary, equal to~$i\Alpha$. Then, with
the same notation as before, we have to a first approximation,
\[
u = -\Alpha\phi + b,\quad
v =  \Alpha\log r + b.
\]
The parts played by the curves $u = \const$., $v = \const$.\ are thus
exactly interchanged; the equipotential curves now radiate
from $z = z_{0}$, while the stream-lines are small circles round the
infinity. The fluid circulates in these curves round the
\PageSep{7}
point $z = z_{0}$; I call the point a \Gloss[Vortex-point]{\emph{vortex-point}} for this reason.
The sense and intensity of the \Gloss[Circulation]{circulation} are measured by~$\Alpha$.
Since the velocity
\[
\sqrt{\left(\frac{\dd u}{\dd x}\right)^{2}
    + \left(\frac{\dd u}{\dd y}\right)^{2}}
\]
is, to a first approximation, equal to~$\dfrac{\dd u}{\dd \phi}$, \emph{the circulation is
clockwise or counter-clockwise according as $\Alpha$~is positive or
negative}. We may call the intensity of the vortex-point~$2\Alpha\pi$,
it is then equal and opposite to the residue of the infinity in
question.

Further, bearing in mind the definition in the last section
of a conjugate streaming and the ambiguity of sign attached
to it, we may say: \emph{If one of two conjugate streamings has a
source of a certain strength at $z = z_{0}$, the other has, at the same
point, a vortex-point of equal, or equal and opposite, intensity.}

Next, consider \emph{algebraic} discontinuities. The general character
of the streaming is independent of the nature of
the coefficient of the first term of the power-series, be it
real, imaginary or complex. Let
\[
w = \frac{A_{1}}{z - z_{0}} + C_{0} + C_{1}(z - z_{0}) + \dots.
\]
To a first approximation, writing
\begin{gather*}
z - z_{0} = re^{i\phi},\quad
A_{1} = \rho e^{i\psi}, \\
w - C_{0} = \frac{\rho}{r}\bigl\{\cos(\psi - \phi)+ i \sin(\psi - \phi)\bigr\}.
\end{gather*}

Let us first consider the real part on the right. When $r$~is
very small, $\dfrac{\rho}{r}\cos(\psi - \phi)$ may still, by proper choice of~$\phi$ be
made to assume any given arbitrary value; \emph{the function~$u$
therefore assumes every value in the immediate vicinity of the
discontinuity}. For more exact determination, let us, for the
moment, consider $r$~and~$\phi$ as variables and write
\[
\frac{\rho}{r}\cos(\psi - \phi) = \const.\Typo{;}{}
\]
\PageSep{8}
We obtain a pencil of circles, all touching the fixed line
\[
\phi = \psi + \frac{\pi}{2}
\]
and becoming smaller as the \Gloss[Modulus]{modulus} of the constant increases.
\emph{Then, in the vicinity of the discontinuity, the curves $u = \const$.\ are
of a similar description, and, in particular, for very large
positive or negative values of the constant they take the form of
small, closed, simple ovals.}

A similar discussion applies to the imaginary part on the
right and hence to the curves $v = \const$., but the line touched
by all the curves in this case is $\phi = \psi$. The following figure,
in which the equipotential curves are, as before, dotted lines
and the stream-lines heavy lines, will now be intelligible.
\Figure{4}{024a}

An analogous discussion gives the requisite graphic representation
of a $\nu$-ple algebraic discontinuity. It is sufficient
merely to state the result: \emph{Every curve $u = \const$.\ passes $\nu$~times
through the discontinuity and touches $\nu$~fixed tangents, intersecting
at equal angles. Similarly with the curves $v = \const$. For
very great positive or negative values of the constant both systems
\Figure{5}{024b}
\PageSep{9}
of curves are closed in the immediate vicinity of the discontinuity.}
For illustration the figure is given for $\nu = 2$.

These higher singularities, as may be surmised, can be
derived from those of lower order by proceeding to the limit.
I postpone this discussion, however, to the next section, since a
certain class of functions will then easily supply the necessary
examples.

\Section{3.}{Rational Functions and their Integrals. Infinities of
higher Order derived from those of lower Order.}

The foregoing sections have enabled us to picture to ourselves
the whole course of such functions as have no infinities
other than those we have just considered and are with these
exceptions \emph{uniform} over the whole plane. These are, as we
know, \emph{the rational functions and their integrals}. I briefly state,
without figures, the theorems respecting the cross-points and
infinities of these functions, and, for reasons already stated, I
confine myself to the cases in which $z = \infty$ is not a critical
point. This limitation, as was before pointed out, will afterwards
disappear automatically.

(1) The rational function about to be considered presents
itself in the form
\[
w = \frac{\phi(z)}{\psi(z)},
\]
where $\phi$~and~$\psi$ are integral functions of the same order which
may be assumed to have no common factor. If this order is~$n$,
and if every algebraic infinity is counted as often as its
order requires, we obtain, corresponding to the roots of $\psi = 0$,
$n$~algebraic discontinuities. The cross-points are given by
$\psi\phi' - \psi'\phi = 0$, an equation of degree $2n - 2$. \emph{The sum of the
orders of the cross-points is then~$2n - 2$}, where, however, it must
be noticed that every $\nu$-fold root of $\psi = 0$ is a $(\nu - 1)$-fold root
of $\psi' = 0$, and hence that every $\nu$-fold infinity of the function
counts as a $(\nu - 1)$-fold cross-point.

(2) If the integral of a rational function
\[
W = \int \frac{\Phi(z)}{\Psi(z)}\, dz
\]
\PageSep{10}
is to be finite at $z = \infty$, the degree of~$\Phi$ must be less by two
than that of~$\Psi$. It is assumed that $\Phi$~and~$\Psi$ have no
common factor. Then $\Phi = 0$ gives the \emph{free cross-points}, \ie\
those which do not coincide with infinities. The roots of
$\Psi = 0$ give the infinities of the integral; and, moreover, to
a simple root of $\Psi = 0$ corresponds a logarithmic infinity, to a
double root an infinity which is, in general, due to the superposition
of a logarithmic discontinuity and a simple algebraic
discontinuity,~etc. \emph{If then every infinity is counted as often as
the order of the corresponding factor in~$\Psi$ requires, the sum of
the orders of the cross-points is less by two than the sum of the
orders of the infinities.} We must also draw attention to the
known theorem, that the sum of the logarithmic residues of all
the discontinuities is zero.

The foregoing gives two possible methods for the derivation
of discontinuities of higher order from those of lower order.
First---and this is the more important method for our purpose---we
may start from the integrals of rational functions. In
this case an algebraic discontinuity of order~$\nu$ makes its
appearance when $\nu + 1$~factors of~$\Psi$ become equal, that is, \emph{when
$\nu + 1$ logarithmic discontinuities coalesce in the proper manner}.
It is clear that the sum of the residues of the latter must be
zero, if the resulting infinity is to be purely algebraic. The
two following figures, in which only the stream-lines are drawn,
show how to proceed to the limit in the case of the simple
algebraic discontinuity of \Fig{4}.
\Figures{6}{7}{026}

Two different processes are here indicated; in the left-hand
figure two sources are about to coalesce, while in the right-hand
figure these are replaced by vortex-points. \Fig{4} is the
\PageSep{11}
resulting limiting position after either process. The two
following figures bear the corresponding relation to \Fig{5}.
\Figures{8}{9}{027a}

The second possible method is suggested by considering the
rational function $\dfrac{\phi}{\psi}$~itself. Logarithmic discontinuities are
thereby excluded. \emph{The $\nu$-fold algebraic discontinuity now arises
from $\nu$~simple algebraic discontinuities}, for $\nu$~simple linear
factors of~$\psi$ in coalescing form a $\nu$-fold factor. \emph{But at the same
time a number of cross-points coalesce and the sum of their
orders is~$\nu - 1$.} For $\psi\phi' - \phi\psi' = 0$ has, as was pointed out
before, a $(\nu - 1)$-fold factor at the same instant that a $\nu$-fold
factor appears in~$\psi$. The following figure explains the production
by this method of the two-fold algebraic discontinuity
of \Fig{5}.
\Figure{10}{027b}

It is of course easy to include these two methods of proceeding
to the limit in one common and more general method.
If $\nu + \mu + 1$ logarithmic infinities and $\mu$~cross-points coalesce
successively or simultaneously, a $\nu$-fold algebraic discontinuity
will in every case make its appearance. But this is not the
place to enlarge on the idea thus suggested.
\PageSep{12}

\Section{4.}{Experimental Production of these Streamings.}

We now give a different direction to our investigations
and consider how to bring about the physical production of
those states of motion which are associated, as we have just
seen, with rational functions and their integrals. Let it be
assumed that the principle of \emph{superposition} may be freely used,
so that we need only consider the simplest cases. From the
theory of partial fractions it follows that each of the functions
in question can be compounded additively of single parts,
which fall under one of the two following types:
\[
A\log(z - z_{0}),\quad
\frac{A}{(z - z_{0})^{\nu}}.
\]
But since $\log(z - z_{0})$ is discontinuous at $z = \infty$, the first type is
unnecessarily specialised, and may be replaced by the more
general one
\[
A\log\frac{z - z_{0}}{z - z_{1}},
\]
and this again, as in \SecRef{2}, may be divided into two parts---viz.:
writing $A = \Alpha + i\Beta$, we discuss $\Alpha\log\dfrac{z - z_{0}}{z - z_{1}}$ and $i\Beta\log\dfrac{z - z_{0}}{z - z_{1}}$
separately. Hence there are in all three cases to be distinguished.

(1) Corresponding to the type $\Alpha\log\dfrac{z - z_{0}}{z - z_{1}}$ a source of
strength $2\Alpha\pi$ must be produced at~$z_{0}$, and one of strength $-2\Alpha\pi$
at~$z_{1}$. To effect this, conceive the $xy$~plane to be covered with an
infinitely thin, homogeneous conducting film. Then it is clear
that the required state of motion will be produced \emph{by placing
the two poles of a galvanic battery of proper strength at $z_{0}$~and~$z_{1}$}.\footnote
  {See Kirchhoff's fundamental memoir: ``Ueber den Durchgang eines
  elektrischen Stromes durch eine Ebene,'' \textit{Pogg.\ Ann.}\ t.~\textsc{lxiv}.\ (1845).}
The reason that the residue of~$z_{0}$ must be equal and
opposite to that of~$z_{1}$ is now at once evident: the streaming is
to be steady, hence the amount of electricity flowing in at one
point must be equal to that flowing out at the other. There is
obviously an analogous reason for the corresponding theorem
concerning any number of logarithmic infinities, but applying
\PageSep{13}
in the first place only to the purely imaginary parts of the
respective residues (these being associated with sources at the
infinities).

(2) In the second case, where $i\Beta\log\dfrac{z - z_{0}}{z - z_{1}}$ is given, the
experimental construction is rather more difficult. The simplest
arrangement is to join~$z_{0}$ to~$z_{1}$ by a simple arc of a curve
and make this the seat of a constant electromotive force.
A streaming is then set up in the $xy$~plane with vortex-points
at $z_{0}$,~$z_{1}$, but otherwise continuous, and from this, by integration,
we obtain as velocity-potential a function whose value is
increased by a certain modulus of periodicity for every circuit
round $z_{0}$~or~$z_{1}$. We must carefully distinguish between this
velocity-potential and the necessarily one-valued electrostatic
potential. The curve joining~$z_{0}$ to~$z_{1}$ is a curve of discontinuity
for the latter, and this very fact makes the electrostatic potential
one-valued.\footnote
  {The statements in the text are intimately connected, as we know, with the
  theory of ``\textit{Doppelbelegungen}'' for which cf.\ Helmholtz, \textit{Pogg.\ Ann.}\ (1853)
  t.~\textsc{lxxxix}. pp.~224~\textit{et~seq.} (\textit{Ueber einige Gesetze der Vertheilung elektrischer Ströme
  in körperlichen Leitern}), and C. Neumann's treatise \textit{Untersuchungen über das
  Logarithmische und Newton'sche Potential} (Leipzig, Teubner, 1877).}

I cannot say whether there are any experimental means of
producing this simplest arrangement. It would appear that
we must go to work in a more roundabout way. Let us first
think of thermo-electric currents. Let the $xy$~plane be covered,
partly with material~I, partly with material~II, and let the
strength of the films be so arranged that the conductivity shall
be everywhere the same. If we now contrive that the two
parts of the contour separated by $z_{0}$~and~$z_{1}$ may be kept at
constant and different temperatures, an electric streaming of
the kind required will be set up. And the electrostatic potential,
by the principles of the theory of thermo-electricity,
exhibits discontinuities on \emph{both} parts of the said contour. It
would apparently be still more complicated to use electric
currents produced by the ordinary galvanic elements. The
plane must then be divided by at least three curves drawn
from~$z_{0}$ to~$z_{1}$, and two of these parts must be covered by a
\PageSep{14}
metallic film, the other by a conducting liquid film. See
\Fig{12}.
\Figures{11}{12}{030}

In all these constructions it is clear, \textit{ab initio}, that the
vortex-points at $z_{0}$~and~$z_{1}$ must have equal and opposite intensities.
For similar reasons the total intensity of all the vortex-points
must always be zero, and thus the theorem that the
sum of the logarithmic residues must vanish has been placed
on a physically evident basis as regards the real, as well as the
imaginary, parts of these residues.

(3) The states of motion associated with the algebraic
types $\dfrac{A}{(z - z_{0})^{\nu}}$ can, by the results of~\SecRef{3}, be derived from those
just established, by proceeding to the limit. This is, of course,
only possible to a certain degree of approximation. For example,
let $\nu + 1$~wires, connected with the poles of a galvanic
battery, be placed \emph{close together} on the $xy$~plane. Then a
streaming is set up which at a little distance from the ends of
the wires sensibly resembles that associated with an algebraic
discontinuity of multiplicity~$\nu$. At the same time an additional
fact in connection with the above construction is brought
to light. The galvanic battery must be \emph{very strong} if an
electric streaming of even medium strength is to be originated.
This corresponds to the well-known analytical theorem that
the residues of the logarithmic infinities must increase to an
infinite degree in order that the conjunction of logarithmic
\PageSep{15}
discontinuities may lead to an algebraic discontinuity. No
further details need be here given as it is only necessary for
what follows that the general principles should be grasped by
means of Figs.~\FigNum{6}--\FigNum{9}.

\Section{5.}{Transition to the Surface of a Sphere. Streamings on
arbitrary curved Surfaces.}

To extend the treatment of finite values of~$z$ to infinitely
great values, the use of the surface of a sphere\footnote
  {Following the example of C.~Neumann, \textit{Vorlesungen über Riemann's
  Theorie der Abel'schen Integrale}, Leipzig, 1865.---The introduction of the sphere
  is, so to speak, parallel to the substitution for~$z$ of the ratio~$\dfrac{z_{1}}{z_{2}}$ of \emph{two} variables,
  whereby the treatment of infinitely great values of~$z$ is, as we know, \emph{formally}
  included in that of the finite values.}
derived from
the $xy$~plane by stereographic projection is now adopted in all
text-books. The simple geometrical relations involved in this
representation are known,\footnote
  {If $\xi$, $\eta$, $\zeta$ are rectangular coordinates, let the equation of the sphere be
  $\xi^{2} + \eta^{2} \Typo{+ \zeta^{2}}{} + (\zeta - \frac{1}{2})^{2} = \frac{1}{4}$. Project from the point $\xi = 0$, $\eta = 0$, $\zeta = 1$, let the plane
  of projection be the $xy$~plane, and the opposite tangent-plane the $\xi\eta$~plane.
  Then we have
  \[
  \xi = \frac{x}{x^{2} + y^{2} + 1},\quad
  \eta = \frac{y}{x^{2} + y^{2} + 1},\quad
  \zeta = \frac{1}{x^{2} + y^{2} + 1}.
  \]

  If $ds$~is the element of arc on the plane, $d\sigma$~that corresponding to it on the
  sphere, we have
  \[
  d\sigma = \frac{ds}{x^{2} + y^{2} + 1},
  \]
  a formula of great importance hereafter, inasmuch as it indicates the \Gloss[Conformal representation]{\emph{conformal}}
  character of the representation.}
and we are also perfectly familiar
with the fact that the infinitely distant parts of the plane are
drawn together to one point of the sphere, the point from
which we project, so that it is no longer merely symbolical to
speak of the point $z = \infty$ on the sphere. It appears however
to be a matter of far less general knowledge that by means of
this representation the functions of~$x + iy$ acquire a signification
on the sphere exactly analogous to that they had on the
plane, and hence, that \emph{in the foregoing sections the sphere may
be substituted everywhere for the plane and that thus, from the
outset, there is no question of exceptional conditions for the value
\PageSep{16}
$z = \infty$}.\footnote
  {In connection with this and with the following discussion compare
  Beltrami, ``Delle variabili complesse sopra una superficie qualunque,'' \textit{Ann.\ di
  Mat.}\ ser.~2, t.~\textsc{i}., pp.~329~\Chg{et~seq.}{\textit{et~seq.}}---The particular remark that surface-potentials
  remain such after a conformal transformation is to be found in the treatises
  cited in the preface, by C.~Neumann, Kirchhoff, and Töpler, as well as \eg\ in
  Haton de~la Goupillière, ``Méthodes de transformation en Géométrie et en
  Physique Mathématique,'' \textit{Journ.\ de~l'Éc.\ Poly.}\ t.~\textsc{xxv}. 1867, pp.~169~\textit{et~seq.}}
The propositions of the theory of surfaces from which
this statement follows are now briefly set forth in a form
sufficiently general to serve for certain future purposes.

In the study of fluid motions parallel to the $xy$~plane we
have already had occasion to assume the film of fluid under
investigation to be infinitely thin. The general question of
fluid motion on any surface may obviously be similarly regarded.
An example is afforded by the displacements of fluid-membranes,
freely extended in space, over themselves, as may be
particularly well observed in Plateau's experiments.

We shall attempt to define such states of motion also by a
potential and we shall especially enquire what is the case in
steady motion.

The proper extension of our conception of a potential
presents itself at once. Let $u$ be a function of position on the
surface and let the curves $u = \const$.\ be drawn; moreover let
the direction of fluid-motion on the surface at every point be
\emph{perpendicular} to the curve $u = \const$.\ passing through that
point, and let the velocity be~$\dfrac{\dd u}{\dd n}$, where $\dd n$~is the element of
arc drawn on the surface normal to the curve. Then $u$, as in
the plane, is called the velocity-potential.

This streaming, so defined, is now to be \emph{steady}. To be
definite, let us make use on the surface of a system of curvilinear
coordinates $p$,~$q$, and let the expression for the element
of arc in this system be
\[
\Tag{(1)}
ds^{2} = E\, dp^{2} + 2F\, dp\, dq + G\, dq^{2}.
\]
Then by a few simple steps similar throughout to those usually
employed in the plane, we find that if $u$ is to give rise to a
\PageSep{17}
steady streaming, it must satisfy the following differential
equation of the second order:
\[
\Tag{(2)}
\frac{\ \dd\,\dfrac{F\, \dfrac{\dd u}{\dd q} - G\, \dfrac{\dd u}{\dd p}}
                   {\sqrt{EG - F^{2}}}\ }{\dd p} +
\frac{\ \dd\,\dfrac{F\, \dfrac{\dd u}{\dd p} - E\, \dfrac{\dd u}{\dd q}}
                   {\sqrt{EG - F^{2}}}\ }{\dd q} = 0.
\]

A short discussion in connection with this differential equation
will now bring out the full analogy with the results for
the plane. From the form of~\Eq{(2)} it follows that for every~$u$
which satisfies~\Eq{(2)} another function~$v$ can be found having the
known reciprocal relation to~$u$. For, by~\Eq{(2)}, the following
equations hold simultaneously:
\[
\Tag{(3)}
\left\{
\begin{aligned}
\frac{\dd v}{\dd p}
  &= \frac{F\, \dfrac{\dd u}{\dd p} - E\, \dfrac{\dd u}{\dd q}}
          {\sqrt{EG - F^{2}}}, \\
\frac{\dd v}{\dd q}
  &= \frac{G\, \dfrac{\dd u}{\dd p} - F\, \dfrac{\dd u}{\dd q}}
          {\sqrt{EG - F^{2}}};
\end{aligned}
\right.
\]
and they define~$v$, save as to a necessarily indeterminate constant.
But solving~\Eq{(3)} we have
\[
\Tag{(4)}
\left\{
\begin{aligned}
-\frac{\dd u}{\dd p}
  &= \frac{F\, \dfrac{\dd v}{\dd p} - E\, \dfrac{\dd v}{\dd q}}
          {\sqrt{EG - F^{2}}}, \\
-\frac{\dd u}{\dd q}
  &= \frac{G\, \dfrac{\dd v}{\dd p} - F\, \dfrac{\dd v}{\dd q}}
          {\sqrt{EG - F^{2}}},
\end{aligned}
\right.
\]
and hence,
\[
\Tag{(5)}
\frac{\ \dd\,\dfrac{F\, \dfrac{\dd v}{\dd q} - G\, \dfrac{\dd v}{\dd p}}
                   {\sqrt{EG - F^{2}}}\ }{\dd p} +
\frac{\ \dd\,\dfrac{F\, \dfrac{\dd v}{\dd p} - E\, \dfrac{\dd v}{\dd q}}
                   {\sqrt{EG - F^{2}}}\ }{\dd q} = 0,
\]
so that, on the one hand, $u$~bears to~$v$ the same relation as $v$~to~$-u$,
and on the other hand~$v$, as well as~$u$, satisfies the partial
differential equation~\Eq{(2)}. At the same time the geometrical
meaning of equations \Eq{(3)}~and~\Eq{(4)} respectively shows that the
systems of curves $u = \const$., $v = \const$.\ are in general orthogonal.
\PageSep{18}

As regards the statement at the beginning of this section
with respect to the stereographic projection of the sphere on the
plane, it follows at once from the fact \emph{that the equations \Eq{(2)}--\Eq{(5)}
are homogeneous in $E$,~$F$,~$G$, and of zero dimensions}.\footnote
  {This statement can also be easily verified without the use of formulæ;
  reference may be made to the works of C.~Neumann and of Töpler, already cited.}
If two
surfaces can be mapped conformally upon one another, and if
corresponding curvilinear coordinates are employed, the expression
for the element of arc on the one surface differs from that
on the other only by a factor; but this factor simply disappears
from equations \Eq{(2)}--\Eq{(5)} for the reason just assigned. We have
therefore a general theorem, including, as a special case, the
above statement relating to a sphere and a plane. Forming the
combination $u + iv$ from $u$~and~$v$ and calling this a \emph{complex
function of position on the surface}, this theorem may be stated
as follows:

\emph{If one surface is conformally mapped upon another, every
complex function of position which exists on the first is changed
into a function of the same kind on the second.}

It may perhaps be as well to obviate a misunderstanding
which might arise at this point. To the same function $u + iv$
there corresponds a motion of the fluid on the one surface and
on the other; it might be imagined that the one arose from the
other by the transformation. This is of course true as regards
the position of the equipotential curves and the stream-lines, but
it is in no wise true of the velocity. Where the element of arc
of one surface is greater than the element of arc of the other,
there the velocity is correspondingly \emph{smaller}. This is precisely
the reason that the value $z = \infty$ loses its critical character on the
sphere. At infinity on the plane, the velocity of the streaming,
as we see at once, is infinitely small of the second order, and if
infinity is a singular point, still the velocity there is less by two
degrees than the velocity at a similar point in the finite part of
the plane. Now let us refer to the formula given in the foot-note
at the beginning of this section:
\[
d\sigma = \frac{ds}{x^{2} + y^{2} + 1},
\]
\PageSep{19}
giving the element of arc of the sphere in terms of the element
of arc of the plane. Here $x^{2} + y^{2} + 1$ is a quantity of precisely
the second order and is cancelled in the transition to the sphere.

\Section{6.}{Connection between the foregoing Theory and the Functions
of a complex Argument.}

Since we have now obtained the sphere as basis of operations,
the theorems of §§\;\SecNum{3},~\SecNum{4} respecting rational functions and their
integrals must be restated; we hereby gain in generality, the
previously established theorems holding for infinitely great
values of~$z$ and being thus valid with no exceptions. This
makes it the more interesting to trace the course of any
particular rational function on the sphere and to consider means
for its physical production.\footnote
  {A good example of not too elementary a character is the Icosahedron
  equation (cf.\ \textit{Math.\ Ann.}, t.~\textsc{xii}. pp.~502~\textit{et~seq.}),
  \[
  w = \frac{\bigl(-(z^{20} + 1) + 228 (z^{15} - z^{5}) - 494z^{10}\bigr)^{3}}
           {1728 z^{5} (z^{10} + 11z^{5} - 1)^{5}},
  \]
  which is of the $60$th~degree in~$z$. The infinities of~$w$ are coincident by fives at
  each of $12$~points which form the vertices of an icosahedron inscribed in the
  sphere on which we represent the values of~$z$. Corresponding to the $20$~faces of
  this icosahedron, the sphere is divided into $20$~equilateral spherical triangles.
  The middle points of these triangles are given by $w = 0$ and form cross-points of
  multiplicity two for the function~$w$. Hence of the $2·60 - 2 = 118$ cross-points,
  we already know (including the infinities) $4·12 + 2·20 = 88$.
  \begin{center}
    \Graphic{\DefWidth}{035}
  \end{center}
  The remaining~$30$ are given by the middle points of the $30$~sides of those
  $20$~spherical triangles. The annexed figure is a diagram of one of these $20$~triangles
  with the stream-lines drawn in; the remaining~$19$ are similar.}
But another important question
suggests itself during these investigations:---the different functions
of position on the sphere are at the same time functions
of the \emph{argument}~$x + iy$; whence this connection?
\PageSep{20}

It must first be noticed that $x + iy$ is itself a complex
function of \emph{position} on the sphere, for the quantities $x$~and~$y$
satisfy the differential equations already established in~\SecRef{1} for $u$~and~$v$;
while working in the plane we may imagine that this
function has an essential advantage over all other functions, but
when the scene of operations is transferred to the sphere there
is no longer any inducement to think so. In fact we are at once
led to a generalisation of the remark which gave rise to this
enquiry. If $u + iv$ and $u_{1} + iv_{1}$ are both functions of~$x + iy$,
$u_{1} + iv_{1}$ is also a function of~$u + iv$; hence for plane and sphere
we have the general theorem: \emph{Of two complex functions of
position, with the usual meaning of this expression in the theory
of functions, each is a function of the other.}

But is this a peculiarity of these surfaces alone? It is
certainly transferable to all such surfaces as can be conformally
mapped upon part of a plane or of a sphere; this follows from
the last theorem of the preceding section. But I maintain that
\emph{this peculiarity belongs to all surfaces}, whereby it is implicitly
stated that a part of any \emph{arbitrary} surface can be conformally
mapped upon the plane or the sphere.

The proof follows at once, if we take $x$,~$y$, the real and
imaginary parts of a complex function of position on a surface,
for curvilinear coordinates on that surface. For then the
coefficients $E$,~$F$,~$G$, in the expression for the element of arc,
must be such that equations \Eq[5]{(2)}--\Eq[5]{(5)} of the preceding section
are identically satisfied when $x$~and~$y$ are substituted for $p$~and~$q$
and also for $u$~and~$v$. \emph{This, as we see at a glance, imposes the
conditions $F = 0$, $E = G$.} But then the equations are transformed
into the well-known ones,
\[
\frac{\dd^{2} u}{\dd x^{2}} + \frac{\dd^{2} u}{\dd y^{2}} = 0,\quad
\frac{\dd u}{\dd x} =  \frac{\dd v}{\dd y},\quad
\frac{\dd u}{\dd y} = -\frac{\dd v}{\dd x},\quad\text{etc.},
\]
and these are the equations by which functions of the argument
$x + iy$ are defined; hence $u + iv$ is a function of $x + iy$, as was
to be shown.

At the same time the statement respecting conformal
\PageSep{21}
representation is confirmed. For, from the form of the expression
for the element of arc,
\[
ds^{2} = E\, (dx^{2} + dy^{2}),
\]
it follows at once that the surface can be conformally mapped
upon the $xy$~plane by~$x + iy$. This result may be expressed in
a somewhat more general form, thus:

\emph{If two complex functions of position on two surfaces are
known, and the surfaces are so mapped upon one another that
corresponding points give rise to the same values of the functions,
the surfaces are conformally mapped upon each other.}

This is the converse of the theorem established at the end
of the last section.

These theorems have all, as far as regards arbitrary surfaces,
a definite meaning only when the attention is confined to small
portions of the surface, within which the complex functions of
position have neither infinities nor cross-points. I have therefore
spoken provisionally of \emph{parts} of surfaces only. But it is natural
to enquire concerning the behaviour of these relations when the
\emph{whole} of any closed surface is taken into consideration. This is
a question which is intimately connected with the line of
argument presently to be developed; \Add{§}§\;\SecNum{19}--\SecNum{21} are specially
devoted to it.

\Section{7.}{Streamings on the Sphere resumed. Riemann's general
Problem.}

A point has now been reached from which it is possible to
start afresh and to take up the discussion contained in the
first sections of this introduction in an entirely different
manner; this leads us to a general and most important problem,
in fact to Riemann's problem, the exact statement and solution
of which form the real subject-matter of the present pamphlet.

The most important position in the previous presentation
of the subject has been occupied by the function of~$x + iy$; this
has been interpreted by a steady streaming on the sphere, and
characteristics of the function have been recognized in those of
the streaming. Rational functions in particular, and their
\PageSep{22}
integrals have led to one simple class of streamings---\Gloss[One-valued]{\emph{one-valued}}
streamings---in which \emph{one} streaming only exists at every point
of the sphere. Moreover, subject to the condition that no
discontinuities other than those defined in~\SecRef{2} may present
themselves, these are \emph{the most general} one-valued streamings
possible on a sphere.

Now it seems possible, \textit{ab initio}, to reverse the whole order
of this discussion; \emph{to study the streamings in the first place and
thence to work out the theory of certain analytical functions}.
The question as to the most general admissible streamings can
be answered by physical considerations; the experimental
constructions of~\SecRef{4} and the principle of superposition giving us,
in fact, means of defining each and every such streaming.
The individual streamings define, to a constant of integration
près, a complex function of position whose variations can be
thereby followed throughout their whole range. Every such
function is an analytical function of every other. From the
connection between any two complex functions of position
forms of analytical dependence are found, considered initially
as to their characteristics and only afterwards identified---to
complete the connection---with the usual form of analytical
dependence.

This is all too clear to need a more minute explanation; let
us proceed at once to the proposed generalisation. And even
this, after the previous discussion, is almost self-evident. All
the problems just stated for the sphere may be stated in
exactly the same terms if instead of the sphere \emph{any arbitrary
closed surface is given}. On this surface one-valued streamings
and hence complex functions of position can be defined and their
properties grasped by means of concrete demonstrations. The
simultaneous consideration of various functions of position thus
changes the results obtained into so many theorems of ordinary
analysis. The fulfilment of this design constitutes \emph{Riemann's
Theory}; the chief divisions into which the following exposition
falls have been mentioned incidentally.
\PageSep{23}


\Part{II.}{Riemann's Theory.}

\Section[Classification of closed Surfaces according to the Value of the Integer~$p$.]
{8.}{Classification of closed Surfaces according to the Value
of the Integer~$p$.\footnotemark}
\footnotetext{The presentation of the subject in this section differs occasionally from
  Riemann's, since surfaces with boundaries are not at first taken into account,
  and thus, instead of \Gloss[Cross-cut]{cross-cuts} from one point on the \Gloss[Boundary]{boundary} to another,
  so-called \emph{\Gloss[Loop-cut]{loop-cuts}} are used (cf.\ C.~Neumann, \textit{Vorlesungen über Riemann's Theorie
  der Abel'schen Integrale}, pp.~291~\textit{et~seq.}).}

All closed surfaces which can be conformally represented
upon each other by means of a uniform correspondence, are, of
course, to be regarded as equivalent for our purposes. For
every complex function of position on the one surface will be
changed by this representation into a similar function on the
other surface; hence, the analytical relation which is graphically
expressed by the co-existence of two complex functions on
the one surface is entirely unaffected by the transition to the
other surface. For instance, the ellipsoid may be conformally
represented, by virtue of known investigations, on a sphere, in
such a way that each point of the former corresponds to one
and only one point of the latter; this shows us that the
ellipsoid is as suitable for the representation of rational functions
and their integrals as the sphere.

It is of still greater importance to find an element which is
unchanged, not only by a conformal transformation, but by
\PageSep{24}
any uniform transformation of the surface.\footnote
  {Deformations by means of \emph{continuous} functions only are considered here.
  Moreover in the arbitrary surfaces of the text certain particular occurrences are
  for the present excluded. It is best to imagine them without singular points;
  branch-points and hence the penetration of one sheet by another will be
  considered later on~(\SecRef{13}). The surfaces must not be \emph{unifacial}, \ie\ it must not
  be possible to pass continuously on the surface from one side to the other
  (cf.\ however \SecRef{23}). It is also assumed---as is usual when a surface is \emph{completely}
  given---that it can be separated into simply-connected portions by a \emph{finite}
  number of cuts.}
Such an element
is Riemann's~$p$, the number of loop-cuts which can be drawn
on a surface without resolving it into distinct pieces. The
simplest examples will suffice to impress this idea on our
minds. For the sphere, $p = 0$, since it is divided into two
disconnected regions by any closed curve drawn on its surface.
For the ordinary anchor-ring, $p = 1$; a cut can be made along
one, and only one, closed curve---though this may have a very
arbitrary form---without resolving the surface into distinct
portions.

That it is impossible to represent surfaces having different~$p$'s
upon one another, the correspondence being uniform, seems
evident.\footnote
  {It is not meant, however, that this kind of geometrical certainty needs no
  further investigation; cf.\ the explanations of G.~Cantor (\textit{Crelle}, t.~\textsc{lxxxiv}. pp.~242~\textit{et~seq.}).
  But these investigations are meanwhile excluded from consideration
  in the text, since the principle there insisted upon is to base all reasoning
  ultimately on intuitive relations.}

It is more difficult to prove the converse, that \emph{the equality
of the~$p$'s is a sufficient condition for the possibility of a uniform
correspondence between the two surfaces}. For proof of this
important proposition I must here confine myself to references
in a foot-note.\footnote
  {See C.~Jordan: ``Sur la déformation des surfaces,'' \textit{Liouville's Journal},
  ser.~2, t.~\textsc{xi}.\ (1866). A few points, which seemed to me to call for elucidation,
  are discussed in \textit{Math.\ Ann.}, t.~\textsc{vii}. p.~549, and t.~\textsc{ix}. p.~476.}
In consequence of this, when investigating
closed surfaces, we are justified, so long as purely descriptive
general relations are involved, in adopting the simplest possible
type of surface for each~$p$. We shall speak of these as \emph{\Gloss[Normal surface]{normal surfaces}}.
For the determination of quantitative properties the
\PageSep{25}
normal surfaces are of course insufficient, but even here they
provide a means of orientation.

Let the normal surface for $p = 0$ be the sphere, for $p = 1$,
the anchor-ring. For greater values of~$p$ we may imagine a
sphere with $p$~appendages (handles) as in the following figure
for $p = 3$.
\FigureH{14}{041a}

There is, of course, a similar normal surface for~$p = 1$; the
surfaces being, by hypothesis, not rigid, but capable of undergoing
arbitrary distortions.

On these normal surfaces there must now be assigned
certain \emph{cross-cuts} which will be needed in the sequel. For the
case $p = 0$ these do not present themselves. For $p = 1$, \ie\ on
the anchor-ring, they may be taken as a meridian~$A$ combined
with a curve of latitude~$B$.
\Figure{15}{041b}

In general $2p$~cross-cuts will be needed. It will, I think,
be intelligible, with reference to the following figure, to speak
\PageSep{26}
of a meridian and a curve of latitude in connection with each
handle of a normal surface.
\Figure{16}{042}

\emph{We choose the $2p$~cross-cuts such that there is a meridian and
a curve of latitude to each handle.} These cross-cuts will be
denoted in order by $A_{1}$,~$A_{2}$,~$\dots$\Add{,}~$A_{p}$, and $B_{1}$,~$B_{2}$,~$\dots$\Add{,}~$B_{p}$.

\Section{9.}{Preliminary Determination of steady Streamings on
arbitrary Surfaces.}

We have now before us the task of defining on arbitrary
(closed) surfaces, the most general, one-valued, steady streamings,
having velocity-potentials, and subject to the condition
that no infinities are admitted other than those named in~\SecRef{2}.\footnote
  {These infinities were first defined for the plane (or the sphere) only. But
  it is clear how to make the definition apply to arbitrary curved surfaces; the
  generalisation must be made in such a manner that the original infinities are
  restored when the surface and the steady streamings on it are mapped by a
  conformal representation upon the plane. This limitation in the nature of the
  infinities implies that only a \emph{finite} number of them is possible in the streamings
  in question, but it must suffice to state this as a fact here. Similarly, as I may
  point out in passing, it follows from our premises that only a finite number of
  cross-points can present themselves in the course of these streamings.}
For this purpose we turn to the normal surfaces of the last
section and once more employ the experimental methods of the
theory of electricity. We imagine the given surface to be
covered with an infinitely thin homogeneous film of a conducting
material, and we then employ those appliances whose use
we learnt in~\SecRef{4}. Thus we may place the two poles of a
galvanic battery at any two points of the surface; a streaming
is then produced having these two points as sources of equal
and opposite strength. Next we may join any two points on
the surface by one or more adjacent but non-intersecting curves
\PageSep{27}
and make these seats of constant electromotive force, bearing
in mind throughout the remarks made in~\SecRef{4} about the
necessary experimental processes for this case. A steady
motion is then obtained, in which the two points are vortex-points
of equal and opposite intensity. Further, we superpose
various forms of motion and finally, when necessary, allow
separate infinities to coalesce in the limit in order to produce
infinities of higher order. Everything proceeds exactly as on
the sphere and we have the following proposition in any case:

\emph{If the infinities are limited to those discussed in~\SecRef{2}, and if
moreover the condition that the sum of all the logarithmic
residues must vanish is satisfied, then there exist on the surface
complex functions of position which become infinite at arbitrarily
assigned points and moreover in an arbitrarily specified manner
and are continuous elsewhere over the whole surface.}

But for $p > 0$ the possibilities are by no means exhausted
by these functions. For there can now be found an experimental
construction which was impossible on the sphere.
There are closed curves on these surfaces along which they
may be cut without being resolved into distinct pieces. There
is nothing to prevent the electricity flowing on the surface from
one side of such a curve to the other. \emph{We have then as much
justification for considering one or more of these consecutive
curves as seats of constant electromotive force as we had in the
case of the curves of~\SecRef{4} which were drawn from one end to the
other.}

The streamings so obtained have no discontinuities; they
may be denoted as \emph{streamings which are finite everywhere} and
the associated complex functions of position as \emph{functions finite
everywhere}. These functions are necessarily infinitely \Gloss[Multiform]{multiform},
for they acquire a real modulus of periodicity, proportional
to the assumed electromotive force, as often as the given
curve is crossed in the same direction.\footnote
  {But this is not to imply that any disposition has herewith been made of the
  periodicity of the imaginary part of the function. For if $u$~is given, $v$~is
  completely determined, to an additive constant près, by the differential equations~\Eq[1]{(1)}
  of \PageRef{1}, and hence the moduli of periodicity which $v$~may possess at the
  cross-cuts $A_{i}$,~$B_{i}$ cannot be arbitrarily assigned.}
\PageSep{28}

We next enquire how many independent streamings there
may be, so defined as finite everywhere. Obviously any two
curves on the surface, seats of equal electromotive forces, are
equivalent for our purpose when by continuous deformation on
the surface one can be brought to coincidence with the other.
If after the process of deformation parts of the curve are
traversed twice in opposite directions, these may be simply
neglected. Consequently it is shown that \emph{every closed curve is
equivalent to an integral combination of the cross-cuts $A_{i}$,~$B_{i}$
defined as in the previous section}.
\Figures{17}{18}{044}

For let us trace the course of any closed curve on a normal
surface;\footnote
  {For another proof see C.~Jordan, ``Des contours tracés sur les surfaces,''
  \textit{Liouville's Journal}, ser.~2, t.~\textsc{xi}.\ (1866).}
for $p = 1$ the correctness of the statement follows
immediately; we need but consider an example as given in the
above figures. The curve drawn on the anchor-ring in \Fig{17}
can be brought to coincidence with that in \Fig{18} by deformation
alone; it is thus equivalent to a triple description of the
meridian~$A$ (cf.\ \Fig{15}) and a single description of the curve of
latitude~$B$.

Further, let $p > 1$. Then whenever a curve passes through
one of the handles a portion can be cut off, consisting of
deformations of an integral combination of the meridians and
corresponding curves of latitude belonging to the handle in
question. When all such portions have been removed there
remains a closed curve, which can either be reduced at once to
\PageSep{29}
a single point on the surface---and then has certainly no effect
on the electric streaming---or it may completely surround one
or more of the handles as in \Fig{19}. \Fig{20} shows how such
a curve can be altered by deformation; by continuation of the
\Figures{19}{20}{045}
process here indicated, it is changed into a curve consisting of
the inner rim of the handle and one of its meridians, but every
portion is traversed twice in opposite directions. Thus this
curve also contributes nothing to the streaming. This conclusion
might indeed have been reached before, from the fact
that this curve, herein resembling a curve which reduces to a
point, resolves the surface into distinct portions.

Nothing \emph{more} is therefore to be gained by the consideration
of arbitrary closed curves than by suitable use of the $2p$~curves
$A_{i}$,~$B_{i}$. The most general streaming we can produce which is
finite everywhere is obtained by making the $2p$~cross-cuts seats
of a constant electromotive force. Or, otherwise expressed:

\emph{The most general function we have to construct, which is
finite everywhere, is the one whose real part has, at the $2p$~cross-cuts, arbitrarily
assigned moduli of periodicity.}

\Section{10.}{The most general steady Streaming. Proof of the
Impossibility of other Streamings.}

If we combine additively the different complex functions of
position constructed in the preceding section, we obtain a
function whose arbitrary character we can take in at a glance.
Without explicitly restating the conditions which we assumed
once and for all respecting the infinities, we may say that \emph{this
\PageSep{30}
function becomes infinite in arbitrarily specified ways at arbitrarily
assigned points, the real part having moreover arbitrarily
assigned moduli of periodicity at the $2p$~cross-cuts}.

I now say, that \emph{this is the most general function to which a
one-valued streaming on the surface corresponds}. For proof we
may reduce this statement to a simpler one. If any complex
function of this kind is given on the surface, we have, by what
precedes, the means of constructing another function, which
becomes infinite in the same manner at the same points and
whose real part has at the cross-cuts $A_{i}$,~$B_{i}$ the same moduli of
periodicity as the real part of the given function. The difference
of these two functions is a new function, nowhere
infinite, whose real part has vanishing moduli of periodicity at
the cross-cuts---this function, of course, again defines a one-valued
streaming. \emph{It is obvious we must prove that such a
function does not exist, or rather, that it reduces to a constant}

The proof is not difficult. As regards the strict demonstration,
I confine myself to the remark that it depends on the
most general statement of Green's Theorem;\footnote
  {For this proposition see Beltrami, \lc, p.~354.}
the following is
intended to make the impossibility of the existence of such a
function immediately obvious. Even if, on account of its indefinite
form, the argument may possibly not be regarded as a
rigorous proof,\footnote
  {I may remind the reader that Green's theorem itself may be proved
  intuitively; cf.\ Tait, ``On Green's and other allied Theorems,'' \textit{Edin.\ Trans.}\
  1869--70, pp.~69~\textit{et~seq.}}
it would still seem profitable to examine, by
this method as well, the principles on which that theorem is
based.

Firstly, then, in the particular case $p = 0$, let us enquire
why a one-valued streaming, finite everywhere, cannot exist on
the sphere. This is most easily shown by tracing the stream-lines.
Since no infinities are to arise, a stream-line cannot
have an abrupt termination, as would be the case at a source
or at an algebraic discontinuity. Moreover it must be remembered
that the flow along adjacent stream-lines is necessarily
in the same direction. It is thus seen that only two kinds of
\PageSep{31}
non-terminating stream-lines are possible; either the curve
winds closer and closer round an asymptotic point---but this
gives rise to an infinity---or the curve is closed. But if \emph{one}
stream-line is closed, so is the next. They thus surround a
smaller and smaller part of the surface of the sphere; consequently
we are unavoidably led to a vortex-point, \ie\ once more
to an infinity, and a streaming finite everywhere is an impossibility.
It is true that we have here not taken into account
the possibilities involved when cross-points present themselves.
But since these points are always finite in number, as was
pointed out above, there can be but a finite number of stream-lines
through them. Let the sphere be divided by these
curves into regions, and in each individual region apply the
foregoing argument, then the same result will be obtained.

Next, if $p > 0$, let us again make use of the normal surfaces
of~\SecRef{8}. By what we have just said, the existence on these
surfaces of one-valued streamings which are finite everywhere,
is due to the presence of the handles. A stream-line cannot be
represented on a normal surface, any more than on a sphere,
by a closed curve which can be reduced to a point. But
further, a curve of the form shown in \Fig{19} is not admissible.
For with this curve there would be associated others of the
form shown in \Fig{20}, so that ultimately a curve would be
obtained with its parts described twice in opposite directions.
A stream-line must therefore necessarily \emph{wind round} one or
other of the handles, that is, it may simply pass once through a
handle or it may wind round it several times along the meridians
and curves of latitude. In all cases then a portion of a
stream-line can be separated from the remainder, equivalent in
the sense of the last section to an integral combination of the
appropriate meridians and curves of latitude. Now the value
of~$u$, the real part of the complex function defined by the
streaming, increases constantly along a stream-line. Further,
the description of two curves, equivalent in the sense of the
last section, necessarily produces the same increment in~$u$.
There exists then a combination of at least one meridian and
one curve of latitude the description of which yields a non-vanishing
increment of~$u$. This is also necessarily true for the
\PageSep{32}
meridian or the curve of latitude alone. But the increment
which $u$~receives by the \emph{description} of the meridian corresponds
to the \emph{crossing} of the curve of latitude and \textit{\Chg{vice~versâ}{vice~versa}}. Hence
at one meridian or curve of latitude, at least, $u$~has a non-vanishing
modulus of periodicity, and a one-valued streaming,
finite everywhere, having all its moduli of periodicity equal to
zero, is impossible.\QED

\Section{11.}{Illustration of the Streamings by means of Examples.}

It would appear advisable to gain, by means of examples, a
clear view of the general course of the streamings thus defined,
in order that our propositions may not be mere abstract statements,
but may be connected with concrete illustrations.\footnote
  {Such a means of orientation, it may be presumed, in also of considerable
  value for the practical physicist.}
This
is comparatively easy in the given cases so long as we confine
ourselves to qualitative relations; exact quantitative determinations
would of course require entirely different appliances.
For simplicity I confine myself to surfaces with a plane of
symmetry coinciding with the plane of the drawing, and on
these I consider only those streamings for which the apparent
boundary of the surface (\ie\ the curve of section of the surface
by the plane of the paper) is either a stream-line or an equipotential
curve. There is a considerable advantage in this, for
the stream-lines need only be drawn for the upper side of
\Figure{21}{048}
\PageSep{33}
the surface, since on the under side they are identically
repeated.\footnote
  {Drawings similar to these were given in my memoir ``Ueber den Verlauf
  der Abel'schen Integrale bei den Curven vierten Grades,'' \textit{Math.\ Ann.}\ t.~\textsc{x}.,
  though indeed a somewhat different meaning is attached there to the Riemann's
  surfaces, so that in connection with them the term fluid-motion can only be
  used in a transferred sense; cf.\ the remarks in~\SecRef{18}.}

Let us begin with streamings, finite everywhere, on the
anchor-ring $p = 1$; let a curve of latitude (or several such
curves) be the seat of electromotive force. Then \Fig{21} is
obtained in which all the stream-lines are meridians and no
cross-points present themselves; the meridians are there shown
as portions of radii; the arrows give the direction of the
streaming on the upper side, on the lower side the direction is
exactly reversed.

In the conjugate streaming, the curves of latitude play the
part of the meridians in the first example; this is shown in the
following drawing:
\FigureH{22}{049}
The direction of motion in this case is the same on the upper
and lower sides.

Let us now deform the anchor-ring, $p = 1$, by causing two
excrescences to the right of the figure, roughly speaking, to
grow from it, which gradually bend towards each other and
finally coalesce. \emph{We then have a surface $p = 2$ and on it
\PageSep{34}
a pair of conjugate streamings as illustrated by Figures \FigNum{23}~and~\FigNum{24}.}

Here, as we may see, two \emph{cross-points} have presented themselves
on the right (of which of course only one is on the upper
\Figures{23}{24}{050a}
side and therefore visible). An analogous result is obtained
when we study streamings which are finite everywhere on a
surface for which $p > 1$. In place of further explanations I give
two more figures with four cross-points in each, relating to the
case $p = 3$.
\Figures{25}{26}{050b}

These arise, if on all ``handles'' of the surface the curves of
latitude or the meridians respectively are seats of electromotive
force. On the two lower handles the directions are the same,
\PageSep{35}
and opposed to that on the upper handle. Of the cross-points,
two are at $a$~and~$b$, the third at~$c$, and the fourth at the corresponding
point on the under side. It is difficult to see the
cross-points at $a$~and~$b$ (\Fig{25}) merely because foreshortening
due to perspective takes place at the boundary of the figure,
and hence both stream-lines which meet at the cross-point
appear to touch the edge. If the streamings on the under side
of the surface (along which the flow is in the opposite direction)
are taken into account, any obscurity of the figure at this point
will disappear.

Let us now return to the anchor-ring, $p = 1$, and let two
logarithmic discontinuities be given on it. The appropriate
figures are obtained if Figs.~\FigNum{23},~\FigNum{24} are subjected to a process of
deformation, which may also be applied, with interesting as well
as profitable results, to more general cases. We draw together
the parts to the left of each figure and stretch out the parts
to the right, so that we obtain, in the first place, the following
figures:
\FiguresH{27}{28}{051}
and then we reduce the handle on the left, which has already
become very narrow, until it is merely a curve, when we reject
it altogether. \emph{Hence, from the streaming, finite everywhere, on
the surface $p = 2$, we have obtained on the surface $p = 1$ a
streaming with two logarithmic discontinuities.} The figures are
now of this form,
\PageSep{36}
\FiguresH{29}{30}{052a}
The two cross-points of Figs.~\FigNum{23},~\FigNum{24} remain, $m$~and~$n$ are the two
logarithmic discontinuities; and these moreover, in \Fig{29}, are
vortex-points of equal and opposite intensity, and, in \Fig{30},
sources of equal and opposite strength. Here, again, it results
from our method of projection that in the second case all the
stream-lines except one seem to touch the boundary at $m$~and~$n$.

If we finally allow $m$~and~$n$ to coalesce, giving rise to a
simple algebraic discontinuity, we obtain the following figures,
in which, as may be perceived, the cross-points retain their
original positions.
\Figures{31}{32}{052b}

There is no occasion to multiply these figures, as it is easy to
construct other examples on the same models. But one more
point must be mentioned. The number of cross-points obviously
increases with the~$p$ of the surface and with the number of
infinities; algebraic infinities of multiplicity~$r$ may be counted
\PageSep{37}
as $r + 1$~logarithmic infinities; then, on the sphere, with $\mu$~logarithmic
infinities, the number of proper cross-points is, in general,
$\mu - 2$. Moreover unit increase in~$p$ is accompanied, in accordance
with our examples, by an increase of two in the number of
cross-points. \emph{Hence it may be surmised that the number of cross-points
is, in every case, $\mu + 2p - 2$.} A strict proof of this
theorem, based on the preceding methods, would present no
especial difficulty;\footnote
  {It would seem above all necessary for such a proof to be perfectly clear
  about the various possibilities connected with the deformation of a given surface
  into the normal surface, cf.~\SecRef{8}.}
but it would lead us too far afield. The
only particular case of the theorem of which use will be
subsequently made, is known to hold by the usual proofs
of analysis situs; it deals~(\SecRef{14}) with streamings presenting
$m$~simple algebraic discontinuities, giving rise therefore to
$2m + 2p - 2$ cross-points.

\Section{12.}{On the Composition of the most general Function of
Position from single Summands.}

The results of~\SecRef{10} enable us to obtain a more concrete
illustration of the most general complex function of position
existing on a surface by adding together single summands of the
simplest types.

Let us first consider functions \emph{finite everywhere}. Let
$u_{1}$,~$u_{2}$,~$\dots$\Add{,}~$u_{\mu}$ be potentials, finite everywhere. These may be
called \emph{linearly dependent} if they satisfy a relation
\[
a_{1}u_{1} + a_{2}u_{2} + \dots \Add{+} a_{\mu}u_{\mu} = A
\]
with constant coefficients. Such a relation leads to corresponding
equations for the $2p$~series of $\mu$~moduli of periodicity possessed
by $u_{1}$,~$u_{2}$,~$\dots$\Add{,}~$u_{\mu}$ at the $2p$~cross-cuts of the surface. Conversely,
by the theorem of~\SecRef{10}, such equations for the moduli of
periodicity would of themselves give rise to a linear relation in
the~$u$'s. It then follows that \emph{$2p$~linearly independent potentials
finite everywhere, $u_{1}$,~$u_{2}$,~$\dots$\Add{,}~$u_{2p}$, can be found in an indefinite
number of ways, but from these every other potential, finite everywhere,
can be linearly constructed}:
\[
u = a_{1}u_{1} + \dots\dots \Add{+} a_{2p}u_{2p} + A.
\]
\PageSep{38}

For $u_{1}$,~$u_{2}$,~$\dots$\Add{,}~$u_{2p}$ can \eg\ be so chosen that each has a
non-vanishing modulus of periodicity at one only of the $2p$~cross-cuts
(where, of course, to each cross-cut, one, and only
one, potential is assigned). And in $\sum\Typo{a_{1}u_{1}}{a_{i}u_{i}}$ the constants~$\Typo{a_{1}}{a_{i}}$ can
be so chosen that this expression has at each cross-cut the same
modulus of periodicity as~$u$. Then $u - \sum\Typo{a_{1}u_{1}}{a_{i}u_{i}}$ is a constant and
we have the formula just given.

Passing now from the potentials~$u$ to the functions~$u + iv$,
finite everywhere, suppose, for simplicity, that coordinates $x$,~$y$,
employed on the surface~(\SecRef{6}), are such that $u$~and~$v$ are connected
by the equations
\[
\frac{\dd u}{\dd x} =  \frac{\dd v}{\dd y},\quad
\frac{\dd u}{\dd y} = -\frac{\dd v}{\dd x}.
\]
Now let $u_{1}$~be an arbitrary potential, finite everywhere. Construct
the corresponding~$v_{1}$; then \emph{$u_{1}$~and~$v_{1}$ are linearly independent}.
For if between $u_{1}$~and~$v_{1}$ there were an equation
\[
a_{1}u_{1} + b_{1}v_{1} = \const.
\]
with constant coefficients, this would entail the following
equations:
\[
a_{1}\, \frac{\dd u_{1}}{\dd x} + b_{1}\, \frac{\dd v_{1}}{\dd x} = 0,\quad
a_{1}\, \frac{\dd u_{1}}{\dd y} + b_{1}\, \frac{\dd v_{1}}{\dd y} = 0,
\]
whence, by means of the given relations, the following contradictory
result would be obtained:
\[
\frac{\dd u_{1}}{\dd x} = 0,\quad
\frac{\dd u_{1}}{\dd y} = 0.
\]

Further, let $u_{2}$~be linearly independent of $u_{1}$,~$v_{1}$. Then we
may take the corresponding~$v_{2}$ and obtain the more general
theorem: \emph{The four functions $u_{1}$,~$u_{2}$, $v_{1}$,~$v_{2}$, are likewise linearly
independent.} For from any linear relation
\[
a_{1}u_{1} + a_{2}u_{2} + b_{1}v_{1} + b_{2}v_{2} = \const.,
\]
by means of the relations among the~$u$'s and the~$v$'s, we should
obtain the following equations:
%[** TN: a_{2}(d/dx) + b_{2}(d/dy) gives the first; reverse for the second]
\begin{alignat*}{3}
(a_{1}a_{2} + b_{1}b_{2})\, \frac{\dd u_{1}}{\dd x}
  &- (a_{1}b_{2} - a_{2}b_{1})\, \frac{\dd v_{1}}{\dd x}
  &&+ (a_{2}^{2} + b_{2}^{2})\, \frac{\dd u_{2}}{\dd x} &&= 0, \\
%
(a_{1}a_{2} + b_{1}b_{2})\, \frac{\dd u_{1}}{\dd y}
  &- (a_{1}b_{2} - a_{2}b_{1})\, \frac{\dd v_{1}}{\dd \Typo{x}{y}}
  &&+ (a_{2}^{2} + b_{2}^{2})\, \frac{\dd u_{2}}{\dd \Typo{x}{y}} &&= 0,
\end{alignat*}
\PageSep{39}
from which by integration a linear relation among $u_{1}$,~$v_{1}$,~$\Typo{v_{2}}{u_{2}}$
would follow.

Proceeding thus we obtain finally $2p$~linearly independent
potentials,
\[
u_{1},\ v_{1}\Chg{;}{,}\quad
u_{2},\ v_{2}\Chg{;\ \dots\dots\ }{,\quad\dots\dots,\quad}
u_{p},\ v_{p},
\]
where each~$v$ is associated with the~$u$ having the same suffix.
Writing $u_{\alpha} + iv_{\alpha} = w_{\alpha}$ and calling the functions $w_{1}$,~$w_{2}$,~$\dots$\Add{,}~$w_{\mu}$,
which are finite everywhere, linearly independent if no relation
\[
c_{1}w_{1} + c_{2}w_{2} + \dots\dots \Add{+} c_{\mu}w_{\mu} = C
\]
exists among them, where $c_{1}$,~$\dots$\Add{,}~$c_{\mu}$,~$C$ are arbitrary \emph{complex}
constants, we have at once: \emph{The $p$~functions $w_{1}$\Add{,}~$\dots$\Add{,}~$w_{p}$\Add{,} finite everywhere, are linearly independent.} For if there were a linear
relation we could separate the real and imaginary parts and
thus obtain linear relations among the $u$'s~and~$v$'s.

But, further, it follows \emph{that every arbitrary function, finite
everywhere, can be made up from $w_{1}$,~$w_{2}$,~$\dots$\Add{,}~$w_{p}$ in the following
form}:
\[
w = c_{1}w_{1} + c_{2}w_{2} + \dots \Add{+} c_{p}w_{p} + C.
\]
For by proper choice of the complex constants $c_{1}$,~$c_{2}$,~$\dots$\Add{,}~$c_{p}$, since
$u_{1}$,~$\dots$\Add{,}~$u_{p}$, $v_{1}$,~$\dots$\Add{,}~$v_{p}$ are linearly independent, we can assign to the
real part of the function~$w$ defined by this formula, arbitrary
moduli of periodicity at the $2p$~cross-cuts.

This is the theorem we were to prove in the present section,
in so far as it relates to the construction of functions finite
everywhere. The transition to \emph{functions with infinities} is now
easily effected.

Let $\xi_{1}$,~$\xi_{2}$,~$\dots$\Add{,}~$\xi_{\mu}$ be the points at which the function is to
become infinite in any specified manner. Introduce an auxiliary
point~$\eta$ and construct a series of single functions
\[
F_{1},\ F_{2},\ \dots\Add{,}\ F_{\mu},
\]
each of which becomes infinite, and that in the specified
manner, at one only of the points~$\xi$, and in addition has, at~$\eta$, a
logarithmic discontinuity whose residue is equal and opposite
to the logarithmic residue of the $\xi$~in question. The sum
\[
F_{1} + F_{2} + \dots \Add{+} F_{\mu}
\]
\PageSep{40}
is then continuous at~$\eta$, for the sum of all the residues of the
discontinuities~$\xi$ is known to be zero. Moreover, this sum
only becomes infinite at the~$\xi$'s, and there in the specified
manner. It therefore differs from the required function only
by a function which is finite everywhere. \emph{The required function
may thus be written in the form}
\[
F_{1} + F_{2} + \dots \Add{+} F_{\mu}
  + c_{1}w_{1} + c_{2}w_{2} + \dots \Add{+} c_{p}w_{p} + C,
\]
whereby the theorem in question has been established for the
general case.

This result obviously corresponds to the dismemberment of
complex functions on a sphere considered in~\SecRef{4}, and there
deduced in the usual way from the reduction of rational
functions to partial fractions.

\Section{13.}{On the Multiformity of the Functions. Special Treatment
of uniform Functions.}

The functions $u + iv$, under investigation on the surfaces
in question, are in general infinitely multiform, for on the one
hand a modulus of periodicity is associated with every logarithmic
infinity, and on the other hand we have the moduli of
periodicity at the $2p$~cross-cuts $A_{i}$,~$B_{i}$, whose real parts may be
arbitrarily chosen. I assert that \emph{in no other manner can $u + iv$
become multiform}. To prove this we must go back to the
conception of the equivalence of two curves on a given surface
which was brought forward in~\SecRef{9}, primarily for other purposes.
Since the differential coefficients of $u$~and~$v$ (or, what is the
same thing, the components of the velocity of the corresponding
streaming) are one-valued at every point of the surface, two
equivalent closed curves not separated by a logarithmic discontinuity
yield the same increment in~$u$, and also in~$v$. But we
found that every closed curve was equivalent to an integral
combination of the cross-cuts $A_{i}$,~$B_{i}$. We further remarked
(\SecRef{10}) that the description of~$A_{i}$ produced the same modulus of
periodicity as the crossing of~$B_{i}$ it and \textit{vice~versa}. And from this
the above theorem follows by known methods.

It will now be of special interest to consider \emph{uniform}
functions of position; from the foregoing all such functions
\PageSep{41}
can be obtained by admitting only purely \emph{algebraical} infinities
and by causing all the $2p$~moduli of periodicity at the cross-cuts
$A_{i}$,~$B_{i}$ to vanish. To simplify the discussion, \emph{simple} algebraic
discontinuities alone need be considered. For we know from
\SecRef{3} that the $\nu$-fold algebraic discontinuity can be derived from
the coalescence of $\nu$~simple ones, in which case, it should be
borne in mind, cross-points are absorbed whose total multiplicity
is $\nu - 1$. Let $m$~points then be given as the simple
algebraic infinities of the required function. We first construct
any $m$~functions of position $Z_{1}$,~$\dots$\Add{,}~$Z_{m}$ each of which has a simple
algebraic infinity at one only of the given points but is otherwise
arbitrarily multiform. From these~$Z$'s the most general
complex function of position with simple algebraic infinities at
the given points can be compounded by the last section in the
form
\[
a_{1}Z_{1} + a_{2}Z_{2} + \dots \Add{+} a_{m}Z_{m}
  + c_{1}w_{1} + c_{2}w_{2} + \dots \Add{+} c_{p}w_{p} + C,
\]
where $a_{1}$\Add{,}~$\dots$\Add{,}~$a_{m}$ are arbitrary constant coefficients. To make
this function \emph{uniform} the modulus of periodicity for each of
the $2p$~cross-cuts must be equated to zero; but these moduli of
periodicity are linearly compounded, by means of the~$a$'s and~$c$'s,
of the moduli of periodicity of the $z$'s~and~$w$'s; \emph{there are
thus $2p$~linear homogeneous equations for the $m + p$ constants $a$~and~$c$}.
Assume that these equations are linearly independent,\footnote
  {If they are not so, the consequence will be that the number of uniform
  functions which are infinite at the $m$~given points will be \emph{greater} than that given
  in the text. The investigations of this possibility, especially Roch's (\Chg{Crelle}{\textit{Crelle}},
  t.~\textsc{lxiv}.), are well known; cf.\ also for the algebraical formulation, Brill and
  Nöther: ``Ueber die algebraischen Functionen und ihre Verwendung in der
  Geometrie,'' \textit{Math.\ Ann.}\ t.~\textsc{vii}. I cannot pursue these investigations in the text,
  although they are easily connected with Abel's Theorem as given by Riemann
  in No.~14 of the Abelian Functions, and will merely point out with reference
  to later developments in the text (cf.~\SecRef{19}) that \emph{the $2p$~equations are certainly
  not linearly independent if $m$~surpasses the limit~$2p - 2$}.}
this important proposition follows:

\emph{Subject to this condition, uniform functions of position with
$m$~arbitrarily assigned simple algebraic discontinuities exist
only if $m \geqq p + 1$; and these functions contain $m - p + 1$ arbitrary
constants which enter linearly.}

Now let the $m$~infinities be moveable, then $m$~new degrees
\PageSep{42}
of freedom are introduced. Moreover it is clear that $m$~arbitrary
points on the surface can be changed by continuous
displacement into $m$~others equally arbitrary. It may therefore
be stated---bearing in mind, however, under what conditions---that
\emph{ the totality of uniform functions with $m$~simple algebraic
discontinuities existing on a given surface forms a continuum of
$2m - p + 1$ dimensions}.

Having now proved the existence and ascertained the
degrees of freedom of the uniform functions, we will, as simply
and directly as possible, enunciate and prove another important
property that they possess. The number of their infinities~$m$
is of far greater import than has yet appeared, for I now state
that \emph{the function~$u + iv$ assumes any arbitrarily assigned value
$u_{0} + iv_{0}$ at precisely $m$~points}.

To prove this, follow the course of the curves $u = u_{0}$, $v = v_{0}$
on the surface. It is clear from~\SecRef{2} that each of these curves
passes once through every one of the $m$~infinities. On the
other hand it follows by the reasoning of~\SecRef{10} that every
\Gloss[Circuit]{circuit} of each of these curves must have at least one infinity
on it. Hence the statement is at once proved for very great
values of $u_{0}$,~$v_{0}$; for it was shewn in~\SecRef{2} that the corresponding
curves $u = u_{0}$, $v = v_{0}$ assume in the vicinity of each infinity
the form of small circles through these points, which necessarily
intersect in \emph{one} point other than the discontinuity (which last
is hereafter to be left out of account).
\Figure{33}{058}

But from this the theorem follows universally, \emph{since, by
continuous variation of $u_{0}$,~$v_{0}$, an intersection of the curves $u = u_{0}$,
$v = v_{0}$ can never be lost}; for, from the foregoing, this could only
\PageSep{43}
occur if several points of intersection were to coalesce, separating
afterwards in diminished numbers. Now the systems of
curves $u$,~$v$ are orthogonal; real points of intersection can then
only coalesce at cross-points (at which points coalescence does
actually take place); but these cross-points are finite in number
and therefore cannot divide the surface into different regions.
Thus the possibility of a coalescence need not be considered
and the statement is proved.

It is valuable in what follows to have a clear conception of
the distribution of the values of~$u + iv$ near a cross-point. A
careful study of \Fig{1} will suffice for this purpose. For instance,
it will be observed that of the $m$~moveable points of intersection
of the curves $u = u_{0}$, $v = v_{0}$, $\nu + 1$~coalesce at the $\nu$-fold
cross-point.

Considerations similar to those here applied to uniform
functions apply also to multiform functions; I do not enlarge
on them, simply because the limitations of the subject-matter
render them unnecessary; moreover it is only in the very
simplest case that a comprehensible result can be obtained.
Suffice it to refer in passing to the fact that a complex function
with more than two incommensurable moduli of periodicity can
be made to approach infinitely near every arbitrary value at
every point.

\Section{14.}{The ordinary Riemann's Surfaces over the $x + iy$
Plane.}

Instead of considering the distribution of the values of the
function $u + iv$ over the original surface, the process may, so to
speak, be reversed. We may represent the values of the
function---which for this reason is now denoted by~$x + iy$---in
the usual way on the plane (or on the sphere)\footnote
  {I speak throughout the following discussion of the plane rather than of the
  sphere in order to adhere as far as possible to the usual point of view.}
and we may
study the \emph{conformal representation} of the original surface
which (by~\SecRef{5}) is thus obtained. For simplicity, we again
confine our attention to uniform functions, although the consideration
\PageSep{44}
of conformal representation by means of multiform
functions is of particular interest.\footnote
  {Cf.\ Riemann's remarks on representation by means of functions which are
  finite everywhere, in No.~12 of his Abelian Functions.}

A moment's thought shows that we \emph{are thus led to the
very surface, many-sheeted, connected by \Gloss[Branch-point]{branch-points}, extending
over the $xy$~plane, which is commonly known as a Riemann's
surface}.

For let $m$ be the number of simple infinities of $x + iy$ on
the original surface; then $x + iy$, as we have seen, takes \emph{every}
value $m$~times on the given surface. \emph{Hence the conformal
representation of the original surface on the $x + iy$ plane covers
that plane, in general, with $m$~sheets.} The only exceptional
positions are taken by those values of~$x + iy$ for which some of
the $m$~associated points on the original surface coalesce,
positions therefore which correspond to \emph{cross-points}. To be
perfectly clear let us once more make use of \Fig{1}. It follows
from this figure that the vicinity of a $\nu$-fold cross-point can be
divided into $\nu + 1$~sectors in such a way that $x + iy$ assumes
the same system of values in each sector. \emph{Hence, above the
corresponding point of the $x + iy$~plane, $\nu + 1$~sheets of the
conformal representation are connected in such a way that in
describing a circuit round the point the variable passes from one
sheet to the next, from this to a third and so on, a $(\nu + 1)$-fold
circuit being required to bring it back to the starting-point.} But
this is exactly what is usually called a \emph{branch-point}.\footnote
  {In \SecRef{11} the number of cross-points of~$x + iy$ was stated without proof to be
  $2m + 2p - 2$. We now see that this statement was a simple inversion of the
  known relation among the number of branch-points (or rather their total
  multiplicity), the number of sheets~$m$, and the~$p$ of a many-sheeted surface (where
  $p$~is the maximum number of loop-cuts which can be drawn on this many-sheeted
  surface without resolving it into distinct portions).}
The
representation at this point is of course not conformal; it is
easily shown that the angle between any two curves which
meet at the cross-point on the original surface is multiplied by
precisely $\nu + 1$ on the Riemann's surface over the $x + iy$~plane.

\emph{But at the same time we recognize the importance of this
many-sheeted surface for the present purpose.} All surfaces
\PageSep{45}
which can be derived from one another by a conformal representation
with a uniform correspondence of points are equivalent
for our purposes~(\SecRef{8}). We may therefore adopt the $m$-sheeted
surface over the plane as the basis of our operations instead of
the surface hitherto employed, which was supposed without
singularities, anywhere in space. And the difficulty which
might be feared owing to the introduction of branch-points is
avoided from the first; for we consider on the $m$-sheeted surface
only those streamings whose behaviour near a branch-point
is such that when they are traced on the original surface
by a reversal of the process, the only singularities produced
are those included in the foregoing discussion. To this end
it is not even necessary to know of a corresponding surface
in space; for we are only concerned with ratios in the
immediate vicinity of the branch-points, \ie\ with differential
relations to be satisfied by the streamings.\footnote
  {For the explicit statement of these relations cf.\ the usual text-books, also
  in particular C.~Neumann: \textit{Das Dirichlet'sche Princip in seiner Anwendung auf
  die Riemann'schen Flächen}. Leipzig, 1865.}
And there
is no longer any reason, in speaking of arbitrarily curved
surfaces, for postulating them as free from singularities; \emph{they
may even consist of several sheets connected by branch-points
and along \Gloss[Branch-line]{branch-lines}}. But whichever of the unlimited number
of equivalent surfaces may be selected as basis, we must
distinguish between \emph{essential} properties common to all equivalent
surfaces, and \emph{non-essential} associated with particular
individuals. To the former belongs the integer~$p$; and the
``moduli,'' which are discussed more fully in~\SecRef{18}, also belong
to them;---to the latter belong the kind and position of the
branch-points of many-sheeted surfaces. If we take an ideal
surface possessing only the essential properties, then the
branch-points of a many-sheeted surface correspond on this
simply to ordinary points which, generally speaking, are not
distinguished from the other points and which are only noticeable
from the fact that, in the conformal representation leading
from the ideal to the particular surface, they give rise to
cross-points.
\PageSep{46}

We have then as a final result that \emph{a greater freedom of
choice has been obtained among the surfaces on which it is
possible to operate and the accidental properties involved by the
consideration of any particular surface can be at once recognized}.
Consequently, many-sheeted surfaces over the $x + iy$~plane are
henceforward employed whenever convenient, but this in no
measure detracts from the generality of the results.\footnote
  {The interesting question here arises whether it is always possible to transform
  many-sheeted surfaces, with arbitrary branch-points, by a conformal process
  into surfaces with no singular points. This question transcends the limits of
  the subject under discussion in the text, but nevertheless I wish to bring it
  forward. Even if this transformation is impossible in individual cases, still the
  preceding discussion in the text is of importance, in that it led to general ideas
  by means of the simplest examples and thus rendered the treatment of more
  complicated occurrences possible.}

\Section{15.}{The Anchor-ring, $p = 1$, and the two-sheeted Surface
over the Plane with four Branch-points.}

It was possible in the preceding section to make our explanation
comparatively brief as a knowledge of the ordinary
Riemann's surface over the plane with its branch-points could
be assumed. But it may nevertheless be useful to illustrate
these results by means of an example. Consider an anchor-ring,
$p = 1$; on it there exist, by~\SecRef{13}, $\infty^{4}$~uniform functions
with two infinities only; each of these, by the general formula
of~\SecRef{11}, has four cross-points. The anchor-ring can therefore be
mapped in an indefinite number of ways upon a two-sheeted
plane surface with four branch-points. With a view to those
readers who are not very familiar with purely intuitive
operations, I give explicit formulæ for the special case
of this representation which I am about to consider, even
though, in so doing, I partly anticipate the work of the next
section.

%[** TN: Manual insetting of tall diagram]
\smallskip\noindent\setlength{\TmpLen}{\parindent}%
\begin{minipage}[b]{\textwidth-1.25in}
\setlength{\parindent}{\TmpLen}%
Imagine the anchor-ring as an ordinary tore generated by
the rotation of a circle about a non-intersecting axis in its
plane. Let $\rho$ be the radius of this circle, $R$~the distance of the
centre from the axis, $\alpha$~the polar-angle.
\PageSep{47}

Take the axis of rotation for axis of~$Z$, the point~$O$ in the
figure as origin for a system of rectangular coordinates,
and distinguish the planes through~$OZ$
by means of the angle~$\phi$ which they
make with the positive direction of the axis
of~$X$. Then, for any point on the anchor-ring,
we have,
\end{minipage}
\Graphic{1.25in}{063a} \\
\[
%[** TN: Added brace]
\Tag{(1)}
\left\{
\begin{aligned}
X &= (R - \rho\cos\alpha) \cos\phi, \\
Y &= (R - \rho\cos\alpha) \sin\phi, \\
Z &= \rho\sin\alpha.
\end{aligned}
\right.
\]

Hence the element of arc is
\begin{align*}
\Tag{(2)}
ds &= \sqrt{dX^{2} + dY^{2} + dZ^{2}} \\
   &= \sqrt{(R - \rho\cos\alpha)^{2}\, d\phi^{2} + \rho^{2}\, d\alpha^{2}},
\intertext{or,}
\Tag{(3)}
ds &= (R - \rho\cos\alpha)\sqrt{d\xi^{2} + d\eta^{2}},
\end{align*}
where $\xi$,~$\eta$ are written for $\phi$, $\displaystyle\int_{0}^{\alpha} \frac{\rho\, d\alpha}{R - \rho\cos\alpha}$.

By~\Eq{(3)} we have a conformal representation of the surface
of the anchor-ring on the $\xi\eta$~plane. The whole surface is
obviously covered once when $\phi$~and~$\alpha$ $\bigl(\text{in~\Eq{(1)}}\bigr)$ each range from
$-\pi$~to~$+\pi$. \emph{The conformal representation of the surface of the
anchor-ring therefore covers a rectangle of the plane, as in the
following figure,}
\FigureH{35}{063b}
where $p$~stands for
\[
\int_{0}^{\pi} \frac{\rho\, d\alpha}{R - \rho\cos\alpha}.
\]
\PageSep{48}

To make the relation between the rectangle and the anchor-ring
intuitively clear, imagine the former made of some material
which is capable of being stretched and let the opposite edges
of the rectangle be brought together without twisting. Or
the anchor-ring may be made of a similar material, and after
cutting along a curve of latitude and a meridian it can be
stretched out over the $\xi\eta$~plane. Instead of further explanation
I subjoin in a figure the projection of the anchor-ring from the
positive end of the axis of~$Z$ upon the $xy$~plane, and in this
figure I have marked the relation to the $\xi\eta$~plane.
\FigureH{36}{064a}

The upper surface of the anchor-ring is, of course, alone
visible, the quadrants 3~and~4 on the under side are covered by
2~and~1 respectively.

Again, let a two-sheeted surface with four branch-points
$z = ±1$,~$±\dfrac{1}{\kappa}$ be given, where $\kappa$~is real and~$< 1$, and
\Figure{37}{064b}
\PageSep{49}
imagine the two positive half-sheets of the plane to be shaded
as in the figure. Let the branch-lines coincide with the straight
lines between $+1$~and~$\dfrac{1}{\kappa}$, and between $-1$~and~$-\dfrac{1}{\kappa}$ respectively.
This two-sheeted surface is known to represent the branching
of $w = \sqrt{\Chg{1 - z^{2}·1 - \kappa^{2}z^{2}}{(1 - z^{2})·(1 - \kappa^{2}z^{2})}}$ and by proper choice of branch-lines we
can arrange that the real part of~$w$ shall be positive throughout
the upper sheet. Now consider the integral
\[
W = \int_{0}^{z} \frac{dz}{w}.
\]

This also, as is well-known, gives a representation of the
two-sheeted surface upon a rectangle, the relation between the
two being given in detail in the following figure, where the
shading and other divisions of \Fig{37} are reproduced. To the
\Figure{38}{065}
upper sheet of \Fig{37} corresponds the left side of this figure.
The representation near the branch-points of the two-sheeted
surface should be specially noticed.

It would perhaps be simplest to proceed first from \Fig{37}
by stereographic projection to a doubly-covered sphere with
four branch-points on a meridian---then to cut this surface
along the meridian into four hemispheres, which by proper
bending and stretching in the vicinity of the branch-points
are then to be changed into plane rectangles---and lastly to
place these four rectangles, in accordance with the relation
among the four hemispheres, side by side as in \Fig{38}. Moreover
it is thus made evident that in \Fig{38} to one and the
\PageSep{50}
same point on the original surface correspond exactly \emph{two}
(associated) points on the edge. And now to arrive at the
required relation between the anchor-ring and the two-sheeted
surface we have only to ensure by proper choice of~$\kappa$ that the
rectangle of \Fig{38} shall be \emph{similar} to that of \Fig{35}. A
proportional magnification of the one rectangle (which again is
effected by a conformal deformation) will then make it exactly
cover the other and the result is a uniform conformal representation
of the two-sheeted surface upon the anchor-ring or
\textit{vice~versa}. Here again it is sufficient to give a figure corresponding
exactly to \Fig{36}. The shading in this figure is
%[** TN: Next three diagrams manually set narrower to improve page breaks]
\Figure[4in]{39}{066}
confined to the upper part of the anchor-ring; on the remainder,
the lower half should be shaded while the upper half is
blank.

The required conformal representation has thus been actually
effected. Now, conversely, we will determine on the surface of
the anchor-ring the streamings by means of which (according
to~\SecRef{14}) the representation is brought about. There are cross-points
at $±1$,~$±\dfrac{1}{\kappa}$, and algebraic infinities of unit multiplicity
at the two points at~$\infty$. The equipotential curves and the
stream-lines are most easily found by using the rectangle as an
intermediate figure. The curves $x = \const$., $y = \const$.\ of the
$z$-plane, \Fig{37}, obviously correspond on the rectangle of
\Fig{38} to those shown in \Fig{40} and \Fig{41}. The arrows are
\PageSep{51}
confined to the curves $y = \const$.\ to distinguish them as stream-lines.
\Figures[4in]{40}{41}{067a}

We have now only to treat these figures in the manner
described for \Fig{35} and we obtain an anchor-ring and the
required system of curves on its surface. The result is the
following.
\FiguresH[4in]{42}{43}{067b}

In \Fig{42}, by reason of the method of projection, the four
cross-points of the streaming appear as points of contact of the
equipotential curves with the apparent rim of the anchor-ring.

\Section{16.}{Functions of~$x + iy$ which correspond to the Streamings
already investigated.}

Let $x + iy$, as in~\SecRef{14}, be a uniform complex function of
position on the surface, with $m$~simple algebraic infinities; let
us transform the surface by the methods there given into an
\PageSep{52}
$m$-sheeted surface over the $x + iy$~plane\footnote
  {This geometrical transformation is of course not essential; it merely
  preserves the connection with the usual presentations of the subject.}
and let us then ask
\emph{into what functions of the argument $x + iy$ the complex functions
of position we have hitherto investigated have been changed}?
The results of~\SecRef{6} should here be borne in mind.

First, let $w$~be a complex function of position which, like
$x + iy$, is \emph{uniform} on the surface. From the assumptions
respecting the infinities of the functions, and particularly those
of uniform functions, it follows at once that~$w$, as a function of~$x + iy$,
has no \emph{essential} singularity. Again,~$w$, on the $m$-sheeted
surface as on the original surface, is uniform. Hence it follows
by known propositions that $w$~is an \emph{algebraic function} of~$z$.

We have here not excluded the possibility of the $m$~values
of~$w$ which correspond to the same~$z$ coinciding everywhere $\nu$~at
a time (where $\nu$~must of course be a divisor of~$m$). But it
must be possible to choose functions~$w$ such that this may not
be the case. We have already~(\SecRef{13}) determined uniform
functions with arbitrarily assigned infinities; thus, to avoid the
above contingency, we need only choose the infinities of~$w$ in
such a way that no~$\nu$~of them lead to the same~$z$. Then we
have:

\emph{The irreducible equation between $w$~and~$z$
\[
f(w, z) = 0
\]
is of the $m$th~degree in~$w$.}

Similarly, it will be of the $n$th~degree in~$z$, if $n$~is the sum
of the orders of the infinities of~$w$.

But the connection between the equation $f = 0$ and the
surface is still closer than is shown by the mere agreement of
the degree with the number of the sheets. To every point of
the surface there belongs only \emph{one} pair of values $w$,~$z$, which
satisfy the equation; and conversely, to every such pair of
values there belongs, in general,\footnote
  {In special cases this may not be so. If we regard $w$,~$z$, as coordinates and
  interpret the equation between them by a curve, the double-points of this curve,
  as we know, correspond to these exceptional cases.}
only one point of the surface.
\PageSep{53}
\emph{Equation and surface are, so to speak, connected by a uniform
relation.}

Now let $w_{1}$~be another uniform function on the surface; it
is therefore certainly an algebraic function of~$z$. Then, when
once the equation $f(w, z) = 0$ has been formed, with the above
assumption, the character of this algebraic function can be
expressed in half a dozen words. \emph{For it can be shown that $w_{1}$~is
a rational function of $w$~and~$z$, and, conversely, that every
rational function of $w$~and~$z$ is a function with the characteristics
of~$w_{1}$.} This last is self-evident. For a rational function
of $w$~and~$z$ is uniform on the surface; moreover, as an analytical
function of~$z$, it is a complex function of position on the
surface. The first part is easily proved. Let the $m$~values of~$w$
belonging to a special value of~$z$ be $w^{(1)}$,~$w^{(2)}$,~$\dots$\Add{,}~$w^{(m)}$ (in
general,~$w^{(\alpha)}$) and the corresponding values of~$w_{1}$ (which are
not all necessarily distinct) $w_{1}^{(1)}$,~$w_{1}^{(2)}$,~$\dots$\Add{,}~$w_{1}^{(m)}$. Then the sum,
\[
w_{1}^{(1)}{w^{(1)}}^{\nu} +
w_{1}^{(2)}{w^{(2)}}^{\nu} + \dots \Add{+}
w_{1}^{(m)}{w^{(m)}}^{\nu}
\]
(where $\nu$~is an arbitrary integer, positive or negative), being a
symmetric function of the various values~$w_{1}^{(\alpha)}{w^{(\alpha)}}^{\nu}$, is a uniform
function of~$z$, and therefore, being an algebraic function, is a
\emph{rational} function of~$z$. From any $m$~of such equations
\[
w_{1}^{(1)},\ w_{1}^{(2)},\ \dots\Add{,}\ w_{1}^{(m)},
\]
being linearly involved, can be found, and it can easily be
shown that each~$w_{1}^{(\alpha)}$ is, as it should be, a rational function of
the corresponding~$w^{(\alpha)}$ and of~$z$.

With the help of this proposition we can at once determine
the character of those functions of~$z$ which arise from the
\emph{multiform} functions of position of which we have been treating.
Let $W$ be such a function. Then $W$~must certainly be an
analytical function of~$z$; we may therefore speak of a \emph{differential
coefficient}~$\dfrac{dW}{dz}$, and this again is a complex function
of position on the surface. Quà function of position it is
necessarily uniform; for the multiformity of~$W$ is confined
to constant moduli of periodicity, any multiples of which may
be additively associated with the initial value. Hence $\dfrac{dW}{dz}$~is,
\PageSep{54}
by what has just been proved, a rational function of $w$~and~$z$,
and \emph{$W$~is therefore the integral of such a function, viz.}:
\[
W = {\textstyle\int} R(w, z)\, dz.
\]

The converse proposition, that every such integral gives
rise to a complex function of position on the surface belonging
to the class of functions hitherto discussed, is self-evident on
the grounds of a known argument which considers, on the one
hand, the infinities of the integrals, on the other, the changes
in the values of the integrals caused by alterations in the path
of integration. It is not necessary to discuss this here at
greater length.

We have now arrived at a well-defined result. \emph{Having
once determined the algebraical equation which defines the relation
between $z$~and~$w$, where $w$~is highly arbitrary, all other
functions of position are given in kind; they are co-extensive in
their totality with the rational functions of $w$~and~$z$ and the
integrals of such functions.}

A convenient example is the repeatedly considered case of
the anchor-ring, $p = 1$, with, for $z$~and~$w$, the functions discussed
in the last section, the function~$z$ being the one illustrated by
Figs.~\FigNum{42},~\FigNum{43}. The equation between these being simply
\[
w^{2} = \Chg{1 - z^{2}·1 - \kappa^{2}z^{2}}{(1 - z^{2})·(1 - \kappa^{2}z^{2})},
\]
the integrals $\int R(w, z)\, dz$ are those generally known as \emph{elliptic
integrals}. Among them, by~\SecRef{12}, there is one single integral,
``finite everywhere.'' From the representation given in \Fig{38}
it follows that this is no other than $\displaystyle\int\frac{dz}{w}$ there considered, the
so-called \emph{integral of the first kind}. The equipotential curves
and stream-lines are shown in Figs.~\FigNum{21},~\FigNum{22}. But the functions
corresponding to Figs.~\FigNum{29},~\FigNum{30} and to Figs.~\FigNum{\Typo{30}{31}},~\FigNum{\Typo{31}{32}} are also
familiar in ordinary analysis. In one case we have a function
with two logarithmic discontinuities, in the other case one
with one algebraic discontinuity. Regarded as functions of~$z$
these are the elliptic integrals usually called \emph{integrals of the
third kind}, and \emph{integrals of the second kind} respectively.
\PageSep{55}

\Section{17.}{Scope and Significance of the previous Investigations.}

The last section has actually accomplished the solution of
the general problem indicated in~\SecRef{7}. The most general of
the complex functions of position here treated of have been
determined on an arbitrary surface, and the analytical relations
among these have been defined by observation of the fact that
all are dependent, in the sense of ordinary analysis, on a single,
uniform, but otherwise arbitrarily chosen function of position.
To complete the discussion, therefore, a synoptic review of the
subject alone is wanting, to ascertain the total result of the
investigation. We have obtained, though not the whole content,
yet at least the principles of Riemann's theory, and for further
deductions Riemann's original work as well as other presentations
of the theory may be referred to.

First, to establish that \emph{these investigations do actually
comprehend the totality of algebraic functions and their integrals}.
For if any algebraical equation $f(w, z) = 0$ is given, we can
construct, as usual, the proper many-sheeted surface over the
$z$-plane, and on this we can then study the one-valued streamings
and complex functions of position (cf.~\SecRef{15}).

We then enquire, is the knowledge of these functions
really furthered by these investigations? In this connection
we must remember that it was chiefly the multiplicity of value
of the integrals which for so long hindered any advance in their
theory. That integrals acquire a multiplicity of value when
logarithmic discontinuities make their appearance had been
already observed by Cauchy. But it was only through
Riemann's surfaces that the other kind of periodicity was
clearly brought to light,---that, namely, which has its origin in
the \emph{connectivity} of the surface, and is measured along the
cross-cuts of that surface. Another point is this:---transformation
by substitutions had long been employed in the
examination of integrals, but without much more result than
their mere empirical evaluation. In Riemann's theory an
extensive class of substitutions presents itself automatically,
and is to be critically examined in operation. The variables
$w$,~$z$, are merely any two independent, uniform functions of
\PageSep{56}
position; any other two, $w_{1}$,~$z_{1}$, can be equally well assumed as
fundamental, whereby $w_{1}$,~$z_{1}$ prove to be any rational, but
otherwise arbitrary functions of $w$,~$z$, and these in their turn to
be rational functions of $w_{1}$,~$z_{1}$. The Riemann's surface is not
necessarily affected by this change. Hence among the numerous
\emph{accidental} properties of the functions, we distinguish certain
\emph{essential} ones which are unaltered by uniform transformations.
And in the number~$p$ especially such an invariantive element
presents itself from the outset. Thus Riemann's theory,
avoiding these two difficulties which had hampered former
investigations, proceeds at once to determine in what way the
functions in question are arbitrary. This was accomplished in~\SecRef{10}
by the proposition: \emph{the infinities of the functions \(with the
restrictions we have assumed throughout\) and the moduli of
periodicity of its real part at the cross-cuts, are arbitrary and
sufficient data for the determination of the function}.

This fairly represents the advantage gained by this treatment
if, with most mathematicians, we place the interests of
the theory of functions foremost. But it must be borne in
mind that the opposite point of view is as fundamentally
justifiable. The knowledge of one-valued streamings on given
surfaces may with good reason be regarded as an end in itself,
since in numerous \emph{physical} problems it leads directly to a
solution. Among the infinite possible varieties of these
streamings Riemann's theory is a valuable guide for it indicates
the connection between the streamings and the algebraic
functions of analysis.

Finally, we may bring forward the geometrical side of the
subject and consider Riemann's theory as a means of making
the theory of the conformal representation of one closed
surface upon another accessible to analytical treatment. The
third part of this pamphlet is devoted to this view of the
subject; it is unnecessary to dwell on it at present at greater
length.

\Section{18.}{Extension of the Theory.}

In Riemann's own train of thought, as I have here attempted
\PageSep{57}
to show, the Riemann's surface not only provides an intuitive
illustration of the functions in question, but it actually \emph{defines}
them. It seems possible to separate these two parts, to take
the definition of the function from elsewhere and to retain the
surface only as a means of intuitive illustration. This is, in
fact, what has been done by most mathematicians, the more
readily that Riemann's definition of a function involves considerable
difficulties\footnote
  {Cf.\ the remarks on this subject in the Preface.}
when subjected to more exact scrutiny. They
therefore usually begin with the algebraical equation and the
definition of the integral and then construct the appropriate
Riemann's surface.

But this method produces \textit{ipso facto} a considerable generalisation
of the original conception. Hitherto, two surfaces were
only held to be equivalent when one could be derived from the
other by a conformal representation with a uniform correspondence
of points. Now there is no longer any reason for
retaining the conformal character of the representation. \emph{Every
surface which by a continuous uniform transformation can be
changed into the given surface, in fact any geometrical configuration
whose elements can be projected upon the original surface
by a continuous uniform projection, serves equally well to give a
graphic representation of the functions in question.} I have, in
former papers, followed out this idea in two different ways, to
which I should like to refer.

On one occasion I used the conception of a normal surface
(cf.~\SecRef{8}) which, although representative, was open to various
modifications, and on this I attempted to illustrate the course
of the functions in question by various graphical means.\footnote
  {Cf.\ my papers on Elliptic Modular-functions in \textit{Math.\ Ann.}, t.~\textsc{xiv}., \textsc{xv}.,~\textsc{xvii}.}
The
nets of polygons which I have repeatedly used\footnote
  {Cf.\ especially the diagrams in \textit{Math.\ Ann.}, t.~\textsc{xiv}. (``Zur Transformation
  siebenter Ordnung der elliptischen Functionen''), and Dyck's paper, to be cited
  presently, ib., t.~\textsc{xvii}.}
fall also under
this head; these I constructed by means of an appropriate dissection
of the Riemann's surface afterwards spread out over the
plane. It need not here be discussed whether these figures,
\PageSep{58}
which in the first place are susceptible of continuous deformation,
may not hereafter, for the sake of further investigations in
the theory of functions, be restricted by a law of form whereby
it may be possible to \emph{define} the functions graphically represented
by each figure.

On another occasion\footnote
  {``Ueber eine neue Art Riemann'scher Flächen,'' \textit{Math.\ Ann.}\Add{,} t.~\textsc{vii}.,~\textsc{x}.}
I undertook to bring out as intuitively
as possible the connection between the conceptions of the
theory of functions and those of ordinary analytical geometry,
in which last an equation in two variables means a \emph{curve}.
Starting from the proposition that every imaginary straight
line on the plane, and therefore also every imaginary tangent
to a curve, has one and only one real point, I obtained a
Riemann's surface depending essentially on the course of the
curve at every point. These surfaces I have hitherto employed,
following my original purpose, only to illustrate intuitively the
behaviour of certain simple integrals.\footnote
  {See Harnack (``Ueber die Verwerthung der elliptischen Functionen für die
  Geometrie der Curven dritten Grades''), \textit{Math.\ Ann.}, t.~\textsc{ix}.; and my paper referred
  to above, ``Ueber den Verlauf der Abel'schen Integrale bei den Curven vierten
  Grades,'' \textit{Math.\ Ann.}, t.~\textsc{x}.}
But a remark similar
to that on the nets of polygons may here be made. In so far
as the surface is subjected to a law of form, it must be possible
to use it as a \emph{definition} of the functions which exist on it. And
it is actually possible to form a partial differential equation for
these functions somewhat analogous to the differential equation
of the second order considered in §§\;\SecNum{1}~and~\SecNum{5}; except that the
differential expression on which this equation depends cannot
be directly interpreted by the element of arc.

These few remarks must suffice to indicate developments
which appear to me worthy of consideration.
\PageSep{59}


\Part{III.}{Conclusions.}

\Section{19.}{On the Moduli of Algebraical Equations.}

In one important point, Riemann's theory of algebraic
functions surpasses in results as well as in methods the usual
presentations of this theory. It tells us that, \emph{given graphically
a many-sheeted surface over the $z$~plane, it is possible to construct
associated algebraic functions}, where it must be observed that
these functions if they exist at all are of a highly arbitrary
character, $R(w, z)$~having in general the same branchings as~$w$.
This theorem is the more remarkable, in that it implies a
statement about an interesting equation of higher order. For
if the branch-points of an $m$-sheeted surface are given, there is
a finite number of essentially different possible ways of arranging
these among the sheets; this number can be found by
considerations belonging entirely to pure analysis situs.\footnote
  {This number has been determined by Herr Kasten, for instance, in his
  Inaugural Dissertation: \textit{Zur Theorie der dreiblättrigen Riemann'schen Fläche.}
  Bremen, 1876.}
But,
by the above proposition this number has its algebraical
meaning. Let us with Riemann speak of all algebraic functions
of~$z$ as belonging to the same class when by means of~$z$ they can
be rationally expressed in terms of one another. \emph{Then the
number in question\footnote
  {If I may be allowed to refer once more to my own writings, let me do so
  with respect to a passage in \textit{Math.\ Ann.}\Add{,} t.~\textsc{xii}. (p.~173), which establishes the
  result that certain rational functions are fully determined by the number of
  their branchings, and again to ib., t.~\textsc{xv}., p.~533, where a detailed discussion
  shows that there are ten rational functions of the eleventh degree with certain
  branch-points.}
is the number of different classes of
\PageSep{60}
algebraic functions which, with respect to~$z$, have the given
branch-values.}

In the present and following sections various consequences
are drawn from this preliminary proposition and among these
we may consider in the first place the question of the \emph{moduli}
of the algebraic functions, \ie\ of those constants which play the
part of the invariants in a uniform transformation of the
equation $f(w, z) = 0$.

For this purpose let $\rho$ be a number initially unknown,
expressing the number of degrees of freedom in any one-one
transformation of a surface into itself, \ie\ in a conformal
representation of the surface upon itself. Then let us recall
the number of available constants in uniform functions on given
surfaces~(\SecRef{13}). We found that there were in general $\infty^{2m-p+1}$
uniform functions with $m$~infinities and that this, as we stated
without proof, is the exact number when $m > 2p - 2$. Now
each of these functions maps the given surface by a uniform
transformation upon an $m$-sheeted surface over the plane.
\emph{Hence the totality of the $m$-sheeted surfaces upon which a given
surface can be conformally mapped by a uniform transformation,
and therefore also the number of $m$-sheeted surfaces with which
an equation $f(w, z) = 0$ can be associated, is~$\infty^{2m-p+1-\rho}$}; for $\infty^{\rho}$~representations
give the same $m$-sheeted surface, by hypothesis.

But there are in all $\infty^{w}$ $m$-sheeted surfaces, where $w$~is the
number of branch-points, \ie~$2m + 2p - 2$. For, as we observed
above, the surface is given by the branch-points to within a
finite number of degrees of freedom, and branch-points of
higher multiplicity arise from coalescence of simple branch-points
as we have already explained in connection with the
corresponding cross-points in~\SecRef{1} (cf.\ Figs.~\FigNum{2},~\FigNum{3}). With each of
these surfaces there are, as we know, algebraic functions
associated. \emph{The number of moduli is therefore}
\[
w - (2m + 1 - p - \rho) = 3p - 3 + \rho.
\]

It should be noticed here that the totality of $m$-sheeted
surfaces with $w$~branch-points form a \emph{continuum},\footnote
  {This follows \eg\ from the theorems of Lüroth and of Clebsch, \textit{Math.\
  Ann.}, t.~\textsc{iv}.,~\textsc{v}.}
corresponding
\PageSep{61}
to the same fact, pointed out in~\SecRef{13} with respect to uniform
functions with $m$~infinities on a given surface. Hence we
conclude \emph{that all algebraical equations with a given~$p$ form a
single continuous manifoldness}, in which all equations derivable
from one another by a uniform transformation constitute an
individual element. Thus, for the first time, a precise meaning
attaches itself to the number of the moduli; \emph{it determines the
dimensions of this continuous manifoldness}.

The number~$\rho$ has still to be determined and this is done
by means of the following propositions.

1. \emph{Every equation for which $p = 0$ can by means of a one-one
relation be transformed into itself $\infty^{3}$~times.} For on the
corresponding Riemann's surface uniform functions with one
infinity only are triply infinite in number~(\SecRef{13}), and in order
that the transformation of the surface into itself may be uniform,
it is sufficient to make any two of these correspond to each
other. Or the proof may be more fully given as follows. If
one function is called~$z$, all the rest are (by~\SecRef{16}) algebraic and
uniform, \Chg{i.e.}{\ie}\ rational functions of~$z$, and since the relation must
be reciprocal, \emph{linear} functions of~$z$. Conversely every linear
function of~$z$ is a uniform function of position on the surface
having one infinity only. Hence the most general uniform
transformation of the equation into itself is obtained by transforming
every point of the Riemann's surface by means of the
formula
\[
z_{1} = \frac{\alpha z + \beta}{\gamma z + \delta},
\]
$\alpha : \beta : \gamma : \delta$ being arbitrary.

2. \emph{Every equation for which $p = 1$ can be transformed
into itself in a singly infinite number of ways.} For proof
consider the integral~$W$ finite over the whole surface, and in
particular the representation upon the $W$-plane of the Riemann's
surface when properly dissected. This has already been done
in a particular case (\SecRef{15}, \Fig{38}) and a minute investigation
of the general case is hardly necessary as the considerations
involved are usually fully worked out in the theory of elliptic
functions. The result is that to every value of~$W$ belongs one
\PageSep{62}
and only one point of the Riemann's surface, while the infinitely
many values of~$W$ corresponding to the same point of the
Riemann's surface can be constructed from one of these values
in the form $W + m_{1}\omega_{1} + m_{2}\omega_{2}$, where $m_{1}$,~$m_{2}$ are any integers and
$\omega_{1}$,~$\omega_{2}$ are the periods of the integral. For a uniform deformation
a point~$W_{1}$ must be associated with each point~$W$ in such
a way that every increase of~$W$ by a period gives rise to a
similar increase of~$W_{1}$ and \textit{vice~versa}. This is certainly
possible, but in general only by writing $W_{1} = ±W + C$; in
special cases (when the ratio of the periods~$\dfrac{\omega_{1}}{\omega_{2}}$ possesses certain
properties belonging to the theory of numbers) $W_{1}$~may also
$= ±iW + C$ or $±\rho W + C$ ($\rho$~being a third root of unity).\footnote
  {This result, which is well known from the theory of elliptic functions,
  is stated in the text without proof.}
However that may be we have in each case in the formulæ of
transformation only one arbitrary constant and hence corresponding
to its different values we have a singly infinite
number of transformations, as stated above.

3. \emph{Equations for which $p > 1$ cannot be changed into
themselves in an infinite number of ways.}\footnote
  {This theorem refers to a \emph{continuous} group of transformations, those with
  arbitrarily variable parameters. It is not discussed in the text whether, under
  certain circumstances, a surface for which $p > 1$ may not be transformed into
  itself by an infinite number of \emph{discrete} transformations; though when $p$~is
  finite in value this also seems to be impossible.}
For the analytical
proof of this statement I refer to Schwarz (\textit{Crelle}, t.~\textsc{lxxxvii}.)
and to Hettner (\textit{Gött.\ Nachr.}, 1880, p.~386). By intuitive
methods the correctness of the statement may be shown as
follows. If there were an infinite number of uniform transformations
of the equation into itself, it would be possible to
displace the Riemann's surface continuously over itself in such
a way that every smallest part should remain similar to itself.
The curves of displacement must plainly cover the surface
completely and at the same time simply; there can be no
\emph{cross-point} in this system, for such a point would have to be
regarded as a stationary point in order to avoid multiformity in
the transformation and the rate of displacement would there
\PageSep{63}
necessarily be zero. But then an infinitesimal element of
surface approaching the cross-point in the course of the displacement
would necessarily be compressed in the direction of
motion and perpendicular to that direction it would be stretched;
it could therefore not remain similar to itself, contrary to the
conception of conformal representation. But on the other
hand all systems of curves covering a surface for which $p > 1$
completely and simply must have cross-points; this is the
proposition proved in somewhat less general form in~\SecRef{11}. The
continuous displacement of the surface over itself is thus
impossible, as was to be proved.

By these propositions, $\rho = 3$ for $p = 0$, $\rho = 1$ for $p = 1$, and
for all greater values of~$p$, $\rho = 0$. \emph{The number of moduli is
therefore, for $p = 0$ zero, for $p = 1$ one, and for $p > 1$
$3p - 3$.}

It may be worth while to add the following remarks. To
determine a point in a space of $3p - 3$ dimensions we do not
generally confine ourselves to $3p - 3$ coordinates; more are
employed connected by algebraical, or transcendental relations.
But moreover it is occasionally convenient to introduce parameters,
of which different series denote the same point of the
manifoldness. The relations which then hold among the $3p - 3$
moduli necessarily existing for $p > 1$ have been but little
investigated. On the other hand the theory of elliptic functions
has given us an exact knowledge of the subject for the case
$p = 1$. I mention the results for this case in order to be able
to express myself precisely and yet briefly in what follows.
Above all let me point out that for $p = 1$ the algebraical
element (to use the expression employed above) is actually
distinguished by one and only one quantity: \emph{the absolute
invariant}~$J = \dfrac{g_{2}^{2}}{\Delta}$.\footnote
  {Cf.\ \textit{Math.\ Ann.}, t.~\textsc{xiv}., pp.~112~\Chg{et~seq.}{\textit{et~seq.}}}
Whenever, in what follows, it is said that
in order to transform two equations for which $p = 1$ into each
other it is not only sufficient but also necessary that the
moduli should be equal, the invariant~$J$ is always meant.
\PageSep{64}
In its place, as we know, it is usual to put Legendre's~$\kappa^{2}$, which,
given~$J$, is six-valued, so that by its use a certain clumsiness in
the formulation of general propositions is inevitable. And it is
even worse if the ratio of the periods~$\dfrac{\omega_{1}}{\omega_{2}}$ of the elliptic integral
of the first kind is taken for the modulus, though this is
convenient in other ways; for an infinite number of values of
the modulus then denote the same algebraical element.

\Section{20.}{Conformed Representation of closed Surfaces upon
themselves.}

In accordance with our original plan we now develop the
geometrical side of the subject, in order to obtain at least the
foundations of the theory of conformal representation of surfaces
upon each other,\footnote
  {The theorems to be established in the text are, for the most part, not
  explicitly given in the literature of the subject. For the surfaces for which
  $p = 0$, compare Schwarz's memoir (\textit{Berl.\ Monatsber.}, 1870), already cited.
  And, further, a paper by Schottky: \textit{Ueber die conforme Abbildung mehrfach
  zusammenhängender Flächen}, which appeared in~1875 as a Berlin Inaugural
  Dissertation and was reprinted in a modified form in \textit{Crelle}, t.~\textsc{lxxxiii}. It
  treats of those plane surfaces of connectivity~$p$ which have $p + 1$~boundaries.}
so following up the indications which, as we
have already remarked in the Preface, were given by Riemann
at the close of his Dissertation. For the cases $p = 0$, $p = 1$, I
shall for the most part, to avoid diffuseness, confine myself to
mere statements of results or indications of proofs. And first,
in treating of the conformal representations of a closed surface
upon itself, a distinction which has been hitherto ignored must
be introduced: \emph{the representation may be accomplished without
or with reversal of angles}. We have an example of the first
case when a sphere is made to coincide with itself by rotation
about its centre; of the second case when it is reflected across
a diametral plane with the same result. The analytical treatment
hitherto employed corresponds to representations of the
first kind only. If $u + iv$ and $u_{1} + iv_{1}$ are two complex functions
of position on the same surface, $u = u_{1}$, $v = v_{1}$ gives the most
general representation of the first kind (cf.~\SecRef{6}). But it is
easy to see how to extend the formula in order to include
\PageSep{65}
representations of the second kind as well. \emph{We have simply
to write $u = u_{1}$, $v = -v_{1}$ in order to obtain a representation of the
second kind.}

Let us first take from the theorems of the last section those
parts which refer to representations of the first kind; in the
most geometrical language possible we have then the following
theorems:

\emph{It is always possible to transform into themselves in an
infinite number of ways by a representation of the first kind
surfaces for which $p = 0$, $p = 1$, but never surfaces for which $p > 1$.}

\emph{For the surfaces for which $p = 0$ the only representation of
the first kind is determined if three arbitrary points of the surface
are associated with three other arbitrary points of the same.}

\emph{If $p = 1$, to any arbitrary point of the surface a second
point may be arbitrarily assigned, and there is then in general
a two-fold possibility of determination of the representation of
the first kind, though in special cases there may be a four-fold or
six-fold possibility.}

These propositions of course do not exclude the possibility
that special surfaces for which $p > 1$ may be transformed into
themselves by \emph{discontinuous} transformations of the first kind.
If this occurs it constitutes an invariantive property for any
conformal deformation of the surface and by its existence and
modality specially interesting classes of surfaces may be distinguished
from the remainder.\footnote
  {Algebraical equations with a group of uniform transformations into themselves
  correspond to these surfaces. The observations in the text thus refer to
  investigations such as those lately undertaken by Herr Dyck (cf.\ \textit{Math.\ Ann.},
  t.~\textsc{xvii}., ``Aufstellung und Untersuchung von Gruppe und Irrationalität regulärer
  Riemann'scher Flächen'').}
This point of view, however,
need not be discussed more fully here.

With respect to the transformations of the second kind
we may first say that \emph{every such transformation, combined with
one of the first kind, produces a new transformation of the
second kind}. Now by the above theorems we have complete
knowledge of the transformations of the first kind for surfaces
for which $p = 0$, $p = 1$; in these cases therefore it suffices to
\PageSep{66}
enquire whether \emph{one} transformation of the second kind exists.
\emph{For the surfaces for which $p = 0$ this is at once answered in the
affirmative.} For it is sufficient to take any one of the uniform
functions of position with only one infinity, $x + iy$, and then
to write $x_{1} = x$, $y_{1} = -y$. For the surfaces for which $p = 1$ the
case is different. \emph{We find that in general no transformation of
the second kind exists.} The easiest way to prove this is to
consider the values which the integral~$W$, finite over the
whole surface, assumes on the anchor-ring, $p = 1$. Let the points
$W = m_{1}\omega_{1} + m_{2}\omega_{2}$ be marked on the $W$~plane, $m_{1}$,~$m_{2}$ being as
before arbitrary positive or negative integers. It is then easily
shown that a transformation of the second kind can change the
surface for which $p = 1$ into itself only if this system of points
has an axis of symmetry. This case occurs when the invariant~$J$,
defined above, is \emph{real}; according as $J$~is~$< 1$ or~$> 1$, these
points in the $W$~plane are corners of a rhomboidal or rectangular
system.

Now let $p > 1$. If one transformation of the second kind
exists for this surface, there will in general be no other of the
same kind.\footnote
  {There are, of course, surfaces capable of a certain number of transformations
  of the first kind, together with an equal number of transformations
  of the second kind; these correspond to the \emph{regular symmetrical} surfaces of
  Dyck's work.}
For otherwise the repetition or combination of
these transformations would produce a transformation of the
first kind distinct from the identical transformation. The
transformation must then necessarily be \emph{symmetrical}, \ie\ it
must connect the points of the surface in \emph{pairs}. The surface
itself will for this reason be called \emph{symmetrical}. Moreover
under this name I shall in future include all those surfaces
for which there exists a transformation of the second kind
leading, when repeated, to identity. To this class belong
evidently all surfaces for which $p = 0$, and such surfaces for
which $p = 1$ as have real invariants.

\Section{21.}{Special Treatment of symmetrical Surfaces.}

Among the symmetrical surfaces now to be considered,
divisions at once present themselves according to the number
\PageSep{67}
and kind of the \Gloss[Curve of transition]{``\emph{curves of transition}''} on the surfaces; \Chg{i.e.}{\ie}\ of
those curves whose points remain unchanged during the symmetrical
transformation in question.

\emph{The number of these curves can in no case exceed~$p + 1$.}
For if a surface is cut along all its curves of transition with
the exception of one, it will still remain an undivided whole, the
symmetrical halves hanging together along the one remaining
curve of transition. Thus if there were more than $p + 1$ of
these, more than $p$~loop-cuts in the surface could be effected
without resolving it into distinct portions, thus contradicting
the definition of~$p$.

\emph{On the other hand there may be any number of curves of
transition below this limit.} It will be sufficient here to discuss
the cases $p = 0$, $p = 1$; for the higher~$p$'s examples will present
themselves naturally.

(1) When a sphere is made to coincide with itself by
reflection in a diametral plane, the great circle by which the
diametral plane cuts it, is the \emph{one} curve of transition. An
example of the other kind is obtained by making every point
of the sphere correspond to the point at the opposite end of
its diameter. Both examples can be easily generalised; the
analysis is as follows. If one curve of transition exists, there
are uniform functions of position with only one infinity, which
assume real values at all points of the curve of transition. If
one of these functions is~$x + iy$ the transformation, already
given as an example above, is $x_{1} = x$, $y_{1} = -y$. For the second
case, a function~$x + iy$ can be so chosen that $\infty$~and~$0$, and
$+1$~and~$-1$, are corresponding points. Then
\[
x_{1} - iy_{1} = \frac{-1}{x + iy}
\]
is the analytical formula for the corresponding transformation.

(2) In the case $p = 1$, the invariant~$J$ must in the first
place, as we know, be assumed to be real. First, let it be~$> 1$.
Then the integral~$W$, which is finite over the whole surface,
can be reduced to a normal form by the introduction of an
appropriate constant factor in such a manner that one period
\PageSep{68}
becomes \emph{real}${} = a$ and the other \emph{purely imaginary}${} = ib$. If we
then write
\[
U_{1} = U,\qquad V_{1} = V,\quad\text{in}\quad W = U + iV,
\]
we obtain a symmetrical transformation of the surface for
which $p = 1$, with the \emph{two} curves of transition,
\[
V = 0,\qquad V = \frac{b}{2},
\]
but if we write
\[
U_{1} = U + \frac{a}{2},\qquad V_{1} = -V,
\]
which again is a symmetrical transformation of the original
surface, we have the case in which there is \emph{no} curve of
transition. The case with only \emph{one} curve of transition occurs
when $J < 1$. $W$~can then be so chosen that its two periods are
conjugately complex. We write, as before,
\[
U_{1} = U,\qquad V_{1} = -V,
\]
and obtain a symmetrical transformation with the \emph{one} curve of
transition, $V = 0$.

Besides this first division of symmetrical surfaces according
to the \emph{number} of the curves of transition there is yet a second.
The cases of no curves of transition and of $p + 1$~curves of
transition are to be excluded for one moment. Then a two-fold
possibility presents itself: \emph{Dissection of the \Typo{surfaces}{surface} along
all the curves of transition may or may not resolve it into
distinct portions.} Let $\pi$~be the number of curves of transition.
It is easily shown that $p - \pi$~must be uneven if the surface
is resolved into distinct portions; that there is no further
limitation may be shown by examples. We shall therefore
distinguish between symmetrical surfaces of one kind or of the
other and count the surfaces with $p + 1$~curves of transition
among the first kind---those that are resolved into distinct
portions---and the surfaces with no curves of transition among
the second kind.

These propositions have a certain analogy with the results
obtained in analytical geometry by investigating the forms of
curves with a given~$p$.\footnote
  {Cf.\ Harnack, ``Ueber die Vieltheiligkeit der ebenen algebraischen Curven,''
  \textit{Math.\ Ann.}, t.~\textsc{x}., pp.~189~\Chg{et~seq.}{\textit{et~seq.}}; cf.\ also pp.~415,~416, ib.\ where I have given
  the two divisions of those curves. It is perhaps as well in these investigations
  to start from the symmetrical surfaces and Riemann's Theory as presented in
  the text.}
And in fact we see that this analogy
\PageSep{69}
is justified. Analytical geometry is (primarily) concerned only
with equations, $f(w, z) = 0$, with real coefficients. Let us first
observe that every such equation determines a symmetrical
Riemann's surface over the $z$-plane, inasmuch as the equation,
and therefore the surface, remains unchanged if $w$~and~$z$ are
simultaneously replaced by their conjugate values, and that the
curves of transition on this surface correspond to the \emph{real} series
of values of $w$,~$z$, which satisfy $f = 0$, \ie\ to the various circuits
of the curve $f = 0$, in the sense of analytical geometry.

But the converse is also easily obtained. Let a symmetrical
surface, and on it any arbitrary complex function of position,
$u + iv$, be given. The symmetrical deformation causes a reversal
of angles on the surface. If then to every point of the surface
values $u_{1}$,~$v_{1}$, are ascribed equal to those $u$,~$v$, given by the
symmetrical point,~$u_{1} - iv_{1}$ will be a new complex function of
position. Now construct
\[
U + iV = (u + u_{1}) + i(v - v_{1}),
\]
so obtaining an expression which in general does not vanish
identically; to ensure this, it is sufficient to assume that the
infinities of~$u + iv$ are unsymmetrically placed. \emph{We have then
a complex function of position with equal real parts, but equal
and opposite imaginary parts at symmetrically placed points.}
Of such functions,~$U + iV$, let any two, $W$,~$Z$, be taken, these
being moreover \emph{uniform} functions of position. The algebraical
equation existing between these two has then the characteristic
of remaining unaltered if $W$,~$Z$ are simultaneously replaced
by their conjugate values. \emph{It is therefore an equation with real
coefficients} and the required proof has been obtained.

I supplement this discussion with a few remarks on the \emph{real}
uniform transformations of \emph{real} equations $f(w, z) = 0$ into
themselves, or, what amounts to the same thing, on conformal
representations, of the first kind, of symmetrical surfaces upon
themselves, in which symmetrical points pass over into other
symmetrical points. Such transformations, by the general
\PageSep{70}
proposition of~\SecRef{19}, can occur in infinite number only for
$p = 0$, $p = 1$; we therefore confine ourselves to these cases.
Let us first take $p = 1$. Then we see at once that among the
transformations already established, we need now only consider
the one
\[
W_{1} = ±W + C,
\]
\emph{where $C$~is a real constant}. Similarly when $p = 0$, for the first
case. The relations $x_{1} = x$, $y_{1} = -y$ remain unaltered if
\[
x + iy = z\quad\text{and}\quad x_{1} + iy_{1} = z_{1}
\]
are simultaneously transformed by the substitution
\[
z' = \frac{\alpha z + \beta}{\gamma z + \delta}\;,
\]
\emph{where the ratios $\alpha : \beta : \gamma : \delta$ are real}. When $p = 0$, for the
second case, the matter is rather more complicated. \emph{Similar
transformations with three real parameters are again possible};
but these assume the following form, $z$~being the same as above,
\[
z' = \frac{(a + ib)z + (c + id)}{-(c - id)z + (a - ib)}\;,
\]
where $a : b : c : d$ are the three real parameters. This result
is implicitly contained in the investigations referring to the
analytical representation of the rotations of the $x + iy$~sphere
about its centre.\footnote
  {Cf.\ Cayley, ``On the correspondence between homographies and rotations,''
  \textit{Math.\ Ann.}, t.~\textsc{xv}., pp.~238--240.}

\Section{22.}{Conformal Representation of different closed Surfaces
upon each other.}

If we now wish to map different closed surfaces upon each
other, the foregoing investigation of the conformal representation
of closed surfaces upon themselves will give us the means
of determining how often such a representation can occur, if it
is once possible. Surfaces which can be conformally represented
upon each other certainly possess (as has been already pointed
out) transformations into themselves, consistent with these.
Thus all representations of the one surface upon the other are
obtained by combining one arbitrary representation with all
those which change \emph{one} of the given surfaces into itself. To
this I need not return.
\PageSep{71}

Let us first consider general, \ie\ non-symmetrical surfaces.
Then the enumerations of the moduli of algebraical equations
given in~\SecRef{19} are at once applicable.

We have first: \emph{Surfaces for which $p = 0$ can always be conformally
represented upon each other}, and we find besides that
surfaces for which $p = 1$ have one modulus, surfaces for which
$p > 1$, $3p - 3$~moduli, unaltered by conformal representation.
Every such modulus is in general a \emph{complex} constant. Since in
the case of symmetrical surfaces real parameters alone must be
considered, we shall suppose the modulus to be separated into
its real and imaginary parts. Then we have: \emph{If two surfaces
for which $p > 0$ can be represented upon each other there must
exist equations among the real constants of the surface, $2$~for
$p = 1$, and $6p - 6$ for~$p > 1$.}

Turning now to the \emph{symmetrical} surfaces, we must make
one preliminary remark. It is evident that two such surfaces
can be ``symmetrically'' projected upon one another only if they
have, as well as the same~$p$, the same number~$\pi$ of curves of
transition, and moreover if they both belong either to the first
or to the second kind. The enumeration in~\SecRef{13} of the number
of constants in uniform functions is now to be made over again,
with the special condition required for symmetrical surfaces
that those functions only are to be considered whose values at
symmetrical places are conjugately imaginary. And then, as in~\SecRef{19},
we must combine with this the number of those many-sheeted
surfaces which can be spread over the $z$-plane and are
symmetrical with respect to the axis of real quantities. To
avoid an infinite number of transformations into themselves, I
will here assume $p > 1$. The work is then so simple that I do
not need to reproduce it for this special case. The only
difference is that those constants which were before perfectly
free from conditions must now be \emph{either every one real} or else
\emph{conjugately complex in pairs}. Hence all the arbitrary quantities
are reduced to half the number. This may be stated as follows:
\emph{In order that it may be possible to represent two symmetrical
surfaces for which $p > 1$ upon one another, it is necessary that,
over and above the agreement of attributes, $3p - 3$~equations
should subsist among the real constants of the surface.}
\PageSep{72}

The cases $p = 0$, $p = 1$, which were here excluded, are
implicitly considered in the preceding section. Of course two
symmetrical surfaces for which $p = 1$ which are to be represented
upon one another must have the same invariant~$J$,
giving \emph{one} condition for the constants of the surface, inasmuch
as $J$~is certainly real. But besides this we find at once that the
representation is always possible, so long as the symmetrical
surfaces agree in the \emph{number of curves of transition}, a condition
which is obviously always necessary.

\Section{23.}{Surfaces with Boundaries and unifacial Surfaces.}

By means of the results just obtained an apparently
important generalisation may be made in the investigation of
the representations of \emph{closed} surfaces, and it was for the sake of
this generalisation that symmetrical surfaces were discussed in
so much detail. For surfaces \emph{with boundaries} and \Gloss[Unifacial surface]{\emph{unifacial}
surfaces} (which may or may not be bounded) may now be
taken into account and the problems referring to them all
solved at once. With reference to the introduction of boundaries
here, a certain limitation hitherto implicitly accepted must be
removed. The surfaces employed have been all assumed to be
of continuous curvature or at least to have discontinuities at
isolated points only (the branch-points). But there is now no
reason against the admission of other discontinuities. For
instance, we may suppose that the surface is made up of a
finite number of different pieces (in general, of various curvatures)
which meet at finite angles after the manner of a
polyhedron; for there is nothing to prevent the conception of
electric currents on these surfaces as well as on those of
continuous curvature. Now surfaces with boundaries are included
among such surfaces.\footnote
  {I owe this idea to an opportune conversation with Herr Schwarz (Easter,
  1881). Compare Schottky's paper, already cited, \textit{Crelle}, t.~\textsc{lxxxiii}., and
  Schwarz's original investigations in the representations of closed polyhedral
  surfaces upon the sphere. (\textit{Berl.\ Monatsber.}, 1865, pp.~150~\Chg{et~seq.}{\textit{et~seq.}} \textit{Crelle}, t.~\textsc{lxx}.,
  pp.~121--136, t.~\textsc{lxxv}., p.~330.)}
\emph{For let the two sides of the
bounded surface be conceived to be two faces of a polyhedron
\PageSep{73}
meeting along a boundary \(and therefore everywhere at an angle
of~$360°$\), and employ the \Gloss[Total surface]{total surface} composed of these two
faces instead of the original bounded surface.}\footnote
  {I express myself in the text, for brevity, as if the original surface were
  bifacial, but the case of unifacial surfaces is not to be excluded.}

This total surface is then in fact a closed surface; but it is
moreover symmetrical, for if the points which lie one above the
other are interchanged, the total surface undergoes a conformal
transformation into itself, the angles being reversed; the
boundaries are here the curves of transition. \emph{But at the same
time the division of symmetrical surfaces into two kinds obtains
an important significance.} The usual bounded surfaces, in
which the two sides are distinguishable, evidently correspond
to the first kind; but unifacial surfaces, in which it is possible
to pass continuously from one side to the other on the
surface itself, belong to the second kind. The case, above
mentioned, in which the unifacial surface has no boundary has
also to be considered. \emph{It is a symmetrical surface without a
curve of transition.}

Let us now consider in order the various cases to be
distinguished.

(1) \emph{First, let a simply-connected surface with one boundary
be given.} This surface now appears as a closed surface for
which $p = 0$, which, since there is a curve of transition, can be
symmetrically represented upon itself. \emph{We find therefore that
two such surfaces can always be conformally represented upon
one another by transformations of either kind, and that there are
always three real disposable constants.} These can be employed
to make an arbitrary interior point on the one surface correspond
to an arbitrary interior point on the other surface and
also an arbitrary point on the boundary of one to an arbitrary
point on the boundary of the other. This method of determination
corresponds to the well-known proposition concerning the
conformal representation of a simply-connected \emph{plane} surface
with one boundary upon the surface of a circle, given by
Riemann, and explained at length in No.~21 of his Dissertation
\PageSep{74}
as an example of the application of his theory to problems of
conformal representation.

(2) \emph{Further we consider unifacial surfaces for which $p = 0$,
with no boundaries.} From §§\;\SecNum{21},~\SecNum{22} it follows at once that two
such surfaces can always be conformally represented upon one
another and that there still remain (by the formulæ at the end
of~\SecRef{21}) three real disposable constants.

(3) \emph{The different cases arising from a total surface
for which $p = 1$, may be considered together.} These include,
first, the \emph{doubly-connected surfaces with two boundaries}, that
is, surfaces which in the simplest form may be thought of
as closed ribbons; and, next, the well-known \emph{unifacial surfaces
with only one boundary}, obtained by bringing together the
two ends of a rectangular strip of paper after twisting it
through an angle of~$180°$. Finally, certain \emph{unifacial surfaces
with no boundaries} belong to this class. An idea of these
may be formed by turning one end of a piece of india-rubber
tubing inside out and then making it pass through
itself so that the outer surface of one end meets the inner
surface of the other. With reference to all these surfaces it
has been established by former propositions that the representation
of one surface upon another of the same kind is possible if
\emph{one}, but only one, equation exists among the real constants of
the surface; and that the representation, if possible at all, is
possible in an infinite number of ways, since a double sign and
a real constant remain at our disposal.

(4) \emph{We now take the general case of a \Gloss[Bifacial]{bifacial} surface.}
The surface has $\pi$~boundaries and admits moreover of $p'$~loop-cuts
which do not resolve it into distinct portions, where either
$p'$~must be~$> 0$, or $\pi > 2$. Then the total surface composed of the
upper and under sides admits of $2p' + \pi - 1$~loop-cuts which leave
it still connected; for first the $p'$~possible loop-cuts can be effected
twice over (on the upper, as well as on the under side), and then
cuts may be made along $\pi - 1$~of the boundaries, and the total
surface is still simply-connected. We will therefore write
$p = 2p' + \pi - 1$ in the theorems of the foregoing section and we
have the following theorem: \emph{Two surfaces of the kind in question
\PageSep{75}
can be represented upon each other, if at all, only in a finite
number of ways. The transformation depends on $6p' + 3\pi - 6$
equations among the real constants of the surface.}

(5) \emph{We have, finally, the general case of unifacial surfaces}
with $\pi$~boundaries and $P$~other possible loop-cuts when the
surface is considered as a bifacial total surface. Leaving aside
the three cases given in (1),~(2), and~(3) ($P = 0$, $\pi = 0$~or~$1$, and
$P = 1$, $\pi = 0$) we have the same proposition as in~(4) only that
for $2p' + \pi - 1$ we must write~$P + \pi$, where $p$~may be odd or
even. \emph{In particular, the number of real constants of a unifacial
surface which are unchanged by conformal transformation is}
\[
3P + 3\pi - 3.
\]

The general theorems and discussions given by Herr Schottky
in the paper we have repeatedly cited, are all included in these
results as special cases.

\Section{24.}{Conclusion.}

The discussion in this last section now drawing to its
conclusion is, as we have repeatedly mentioned, intended to
correspond to the indications given by Riemann at the close of
his Dissertation. It is true we have here confined ourselves to
uniform correspondence between two surfaces by means of
conformal representation, whereas Riemann, as he explicitly
states, was also thinking of multiform correspondence. For
this case it would be necessary to imagine each of the surfaces
covered by several sheets and to find then a conformal relation
establishing uniform correspondence between the many-sheeted
surfaces so obtained. For every branch-point which these
surfaces might possess a new complex constant would be at our
disposal.

It may here be remarked that we have already considered
in detail at least \emph{one} case of such a relation. When an arbitrary
surface is spread over the plane in several sheets~(\SecRef{15}), there
is established between the surface and plane a correspondence
which is multiform on one side. Further we may point out
that by means of this special case two arbitrary surfaces are in
\PageSep{76}
fact connected by a relation establishing a multiform correspondence.
For if the two surfaces are each represented on
the plane, then, by means of the plane, there is a relation
between them. The subject of multiform correspondence is of
course by no means exhausted by these remarks. But we have
laid a foundation for its treatment by showing its connection
with Riemann's other speculations in the Theory of Functions,
to an account of which these pages have been devoted.


\BackMatter
%[** TN: No page break in the original]
\Glossary
% ** TN: Macro prints the following text:
% GLOSSARY OF TECHNICAL TERMS.
% The numbers refer to the pages.

\Term{Bifacial}{zweiseitig}{73}

\Term{Boundary}{Randcurve}{23}

\Term{Branch-line}{Verzweigungsschnitt}{45}

\Term{Branch-point}{Verzweigungspunct}{44}

\Term{Circuit}{Ast, Zug}{42}

\Term{Circulation}{Wirbel}{7}

\Term{Conformal representation}{conforme Abbildung}{15}

\Term{Cross-cut}{Querschnitt}{23}

\Term{Cross-point}{Kreuzungspunct}{3}

\Term{Curve of transition}{Uebergangscurve}{67}

\Term{Equipotential curve}{Niveaucurve}{2}

\Term{Essential singularity}{wesentlich singuläre Stelle}{5}

\Term{Loop-cut}{Rückkehrschnitt}{23}

\Term{Modulus}{absoluter Betrag}{8}

\Term{Multiform}{vieldeutig}{27}

\Term{Normal surface}{Normalfläche}{24}

\Term{One-valued}{einförmig}{22}

\Term{Source}{Quelle}{6}

\Term{Steady streaming}{stationäre Strömung}{1}

\Term{Stream-line}{Strömungscurve}{2}

\Term{Strength}{Ergiebigkeit}{6}

\Term{Total surface}{Gesammtfläche}{73}

\Term{Unifacial surface}{Doppelfläche}{72}

\Term{Uniform}{eindeutig}{2}

\Term{Vortex-point}{Wirbelpunct}{7}
\vfill
\enlargethispage{16pt}
\noindent\hrule
\smallskip

\noindent{\tiny\centering CAMBRIDGE: PRINTED BY C. J. CLAY, M.A. AND SONS. AT THE UNIVERSITY PRESS.\\}
\normalsize
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% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %
%                                                                         %
% End of the Project Gutenberg EBook of On Riemann's Theory of Algebraic  %
% Functions and their Integrals, by Felix Klein                           %
%                                                                         %
% *** END OF THIS PROJECT GUTENBERG EBOOK ON RIEMANN'S THEORY ***         %
%                                                                         %
% ***** This file should be named 36959-t.tex or 36959-t.zip *****        %
% This and all associated files of various formats will be found in:      %
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 36959-t.out
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