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AN INTRODUCTION TO THE THEORY OF MULTIPLY PERIODIC FUNCTIONS BY H. F. BAKER, Sc. D., F. R. S., FELLOW OF ST JOHNS COLLEGE AND LECTURER IN MATHEMATICS IN THE UNIVERSITY OF CAMBRIDGE PROPERTY OF mum msrmiTE Of CAMBRIDGE at the University Press 1907 Sie erinnern Sich aber auch vielleicht zu gleieher Zeifc meiner Klagen, liber einen Satz, dor thoils schon an aich sehr interessdnt 1st, theils einem sehr betrachtlichen Theile jener Untersuchungen als Grundlage oder als Schlussstein dient, den ich damals schon liber 2 Jahr kannte, und der alle meine Bemiihungon, einen gcnligendon Bowels zu finden, vereitelt hatte, diesor Satz ist schon in meiner Theorie der Zahlcn angocloutct, und bctrifft die Bcstimmung eines Wurzelzeichens, sie hat rnich immer gequalt. Dieser Mangel hat rair allos Uebrige, was ich fand, verleidet und seit 4 Jahren wird selten eine Woche hingegangen seiu, wo ich nicht einen oder den anderon vergeblichen Versuch, diesen Knoten zu losen, gemacht hatte besonders lebhaft nun auch wieder in der letzten Zcit. Aber alles Bruten, allos Suchcn ist umsonst gowesen, traurig habe ich jedesmal die Feder wieder niederlegen mlissen. Endlich vor ein Paar Tagen ists gelungen GAUSS an OLBERS, September 1805 Sobering, Festrodo. PREFACE. present volume consists of two parts the first of these deals with the theory of hyperelliptic functions of two variables, the second with the reduction of the theory of general multiply-periodic functions to the theory of algebraic functions taken together they furnish what is intended to be an elementary and self-contained introduction to many of the leading ideas of the theory of multiply-periodic functions, with the incidental aim of aiding the comprehensionof the importance of this theory in analytical geometry. The first part is centred round some remarkable differential equations satisfied by the functions, which appear to be equally illuminative both of the analytical and geometrical aspects of the theory it was in fact to explain this that the book was originally entered upon. The account has no pretensions to completeness being anxious to explain the properties of the functions from the beginning, I have been debarred from following Humberts brilliant monograph, which assumes from the first Poincares theorem as to the number of zeros common to two theta functions this theorem is reached in this volume, certainly in a generalised form, only in the last chapter of Part n. being anxious to render the geometrical portions of the volume quite elementary, I have not been able to utilise the theory of quadratic complexes, which vi Preface. has proved so powerful in this connexion in the hands of Kummer and Klein and, for both these reasons, the account given here, and that given in the remarkable book from the pen of R. W. H. T. Hudson, will, I believe, only be regarded by readers as comple mentary. The theory of Kummers surface, and of the theta functions, has been much studied since the year 1847 or before in which Gopel first obtained the biquadratic relation connecting four theta functions and Wirtinger has shewn, in his Unter suchungen iiber Thetafunctionen, which has helped me in several ways in the second part of this volume, that the theory is capable of generalisation, in many of its results, to space of 2 1 dimensions but even in the case of two variables there is a certain inducement, not to come to too close quarters with thedetails, in the fact of the existence of sixteen theta functions connected together by many relations, at least in the minds of beginners. I hope therefore that the treatment here followed, which reduces the theory, in a very practical way, to that of one theta function and three periodic functions connected by an algebraic equation, may recommend itself to others, and, in a humble way, serve the purpose of the earlier books on elliptic functions, of encouraging a wider use of the functions in other branches of mathematics...