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diff --git a/.gitattributes b/.gitattributes new file mode 100644 index 0000000..6833f05 --- /dev/null +++ b/.gitattributes @@ -0,0 +1,3 @@ +* text=auto +*.txt text +*.md text diff --git a/36959-pdf.pdf b/36959-pdf.pdf Binary files differnew file mode 100644 index 0000000..3c72210 --- /dev/null +++ b/36959-pdf.pdf diff --git a/36959-pdf.zip b/36959-pdf.zip Binary files differnew file mode 100644 index 0000000..c73ebed --- /dev/null +++ b/36959-pdf.zip diff --git a/36959-t.zip b/36959-t.zip Binary files differnew file mode 100644 index 0000000..7ad6e18 --- /dev/null +++ b/36959-t.zip diff --git a/36959-t/36959-t.tex b/36959-t/36959-t.tex new file mode 100644 index 0000000..d57d22a --- /dev/null +++ b/36959-t/36959-t.tex @@ -0,0 +1,4919 @@ +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % +% % +% 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 *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{36959} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Latin-1 text encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. 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\subsection*{\centering\footnotesize\normalfont\itshape The numbers refer to the pages.} + \small +} + +\newcommand{\Gloss}[2][]{% + \phantomsection% + \ifthenelse{\equal{#1}{}}{% + \label{gtag:#2}% + }{% + \label{gtag:#1}% + }% + \hyperref[glossary]{#2}% +} +\newcommand{\Term}[3]{% + \noindent\hspace*{2\parindent}% + \hyperref[gtag:#1]{#1}, + \textit{#2}, + \pageref{gtag:#1}% +} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} +%% PG BOILERPLATE %% +\PGBoilerPlate +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +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} +\end{minipage} +\end{center} +\newpage +%% Credits and transcriber's note %% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang +\end{PGtext} +\end{minipage} +\vfill +\end{center} + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note.} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\PageSep{i} +\FrontMatter +\ifthenelse{\boolean{ForPrinting}}{% +\null\vfill +\begin{center} +\HalfTitle +% [** TN: Previous macro prints: +% ON RIEMANN'S THEORY +% OF +% ALGEBRAIC FUNCTIONS +% AND THEIR +% INTEGRALS.] +\end{center} +\vfill +\cleardoublepage +}{}% Omit half-title in screen version +\PageSep{ii} +%[Blank page] +\PageSep{iii} +\begin{center} +% [** TN: See above] +\HalfTitle + +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 +\end{center} +\newpage +\PageSep{iv} +\null +\vfill +\begin{center} +\textgoth{Cambridge}: \\[4pt] +\scriptsize +PRINTED BY C. J. CLAY, M.A. \&~SONS, \\[4pt] +AT THE UNIVERSITY PRESS. +\end{center} +\vfill +\normalsize +\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. 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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] % +% % +% Language: English % +% % +% Character set encoding: ISO-8859-1 % +% % +% *** START OF THIS PROJECT GUTENBERG EBOOK ON RIEMANN'S THEORY *** % +% % +% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % + +\def\ebook{36959} +%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% +%% %% +%% Packages and substitutions: %% +%% %% +%% book: Required. %% +%% inputenc: Latin-1 text encoding. Required. %% +%% %% +%% ifthen: Logical conditionals. Required. %% +%% %% +%% amsmath: AMS mathematics enhancements. 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Required. %% +%% %% +%% %% +%% Producer's Comments: %% +%% %% +%% OCR text for this ebook was obtained on July 30, 2011, from %% +%% http://www.archive.org/details/onriemannstheory00kleiuoft. %% +%% %% +%% Minor changes to the original are noted in this file in three %% +%% ways: %% +%% 1. \Typo{}{} for typographical corrections, showing original %% +%% and replacement text side-by-side. %% +%% 2. \Chg{}{} and \Add{}, for inconsistent/missing punctuation,%% +%% italicization, and capitalization. %% +%% 3. [** TN: Note]s for lengthier or stylistic comments. %% +%% %% +%% %% +%% Compilation Flags: %% +%% %% +%% The following behavior may be controlled by boolean flags. %% +%% %% +%% ForPrinting (false by default): %% +%% If false, compile a screen optimized file (one-sided layout, %% +%% blue hyperlinks). 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\subsection*{\centering\footnotesize\normalfont\itshape The numbers refer to the pages.} + \small +} + +\newcommand{\Gloss}[2][]{% + \phantomsection% + \ifthenelse{\equal{#1}{}}{% + \label{gtag:#2}% + }{% + \label{gtag:#1}% + }% + \hyperref[glossary]{#2}% +} +\newcommand{\Term}[3]{% + \noindent\hspace*{2\parindent}% + \hyperref[gtag:#1]{#1}, + \textit{#2}, + \pageref{gtag:#1}% +} + +%%%%%%%%%%%%%%%%%%%%%%%% START OF DOCUMENT %%%%%%%%%%%%%%%%%%%%%%%%%% +\begin{document} +%% PG BOILERPLATE %% +\PGBoilerPlate +\begin{center} +\begin{minipage}{\textwidth} +\small +\begin{PGtext} +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] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK ON RIEMANN'S THEORY *** +\end{PGtext} +\end{minipage} +\end{center} +\newpage +%% Credits and transcriber's note %% +\begin{center} +\begin{minipage}{\textwidth} +\begin{PGtext} +Produced by Andrew D. Hwang +\end{PGtext} +\end{minipage} +\vfill +\end{center} + +\begin{minipage}{0.85\textwidth} +\small +\BookMark{0}{Transcriber's Note.} +\subsection*{\centering\normalfont\scshape% +\normalsize\MakeLowercase{\TransNote}}% + +\raggedright +\TransNoteText +\end{minipage} +%%%%%%%%%%%%%%%%%%%%%%%%%%% FRONT MATTER %%%%%%%%%%%%%%%%%%%%%%%%%% +\PageSep{i} +\FrontMatter +\ifthenelse{\boolean{ForPrinting}}{% +\null\vfill +\begin{center} +\HalfTitle +% [** TN: Previous macro prints: +% ON RIEMANN'S THEORY +% OF +% ALGEBRAIC FUNCTIONS +% AND THEIR +% INTEGRALS.] +\end{center} +\vfill +\cleardoublepage +}{}% Omit half-title in screen version +\PageSep{ii} +%[Blank page] +\PageSep{iii} +\begin{center} +% [** TN: See above] +\HalfTitle + +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 +\end{center} +\newpage +\PageSep{iv} +\null +\vfill +\begin{center} +\textgoth{Cambridge}: \\[4pt] +\scriptsize +PRINTED BY C. J. CLAY, M.A. \&~SONS, \\[4pt] +AT THE UNIVERSITY PRESS. +\end{center} +\vfill +\normalsize +\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 +%%%%%%%%%%%%%%%%%%%%%%%%% GUTENBERG LICENSE %%%%%%%%%%%%%%%%%%%%%%%%%% +\PGLicense +\begin{PGtext} +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-pdf.pdf or 36959-pdf.zip ***** +This and all associated files of various formats will be found in: + http://www.gutenberg.org/3/6/9/5/36959/ + +Produced by Andrew D. Hwang + +Updated editions will replace the previous one--the old editions +will be renamed. + +Creating the works from public domain print editions means that no +one owns a United States copyright in these works, so the Foundation +(and you!) can copy and distribute it in the United States without +permission and without paying copyright royalties. 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(max. 131072) + 271 words of extra memory for PDF output out of 10000 (max. 10000000) + diff --git a/36959-t/old/36959-t.zip b/36959-t/old/36959-t.zip Binary files differnew file mode 100644 index 0000000..6d35042 --- /dev/null +++ b/36959-t/old/36959-t.zip diff --git a/LICENSE.txt b/LICENSE.txt new file mode 100644 index 0000000..6312041 --- /dev/null +++ b/LICENSE.txt @@ -0,0 +1,11 @@ +This eBook, including all associated images, markup, improvements, +metadata, and any other content or labor, has been confirmed to be +in the PUBLIC DOMAIN IN THE UNITED STATES. + +Procedures for determining public domain status are described in +the "Copyright How-To" at https://www.gutenberg.org. + +No investigation has been made concerning possible copyrights in +jurisdictions other than the United States. Anyone seeking to utilize +this eBook outside of the United States should confirm copyright +status under the laws that apply to them. diff --git a/README.md b/README.md new file mode 100644 index 0000000..596ebc4 --- /dev/null +++ b/README.md @@ -0,0 +1,2 @@ +Project Gutenberg (https://www.gutenberg.org) public repository for +eBook #36959 (https://www.gutenberg.org/ebooks/36959) |
