96 THE ROYAL SOCIETY OF CANADA 



in regard to the necessity for tlie use of more than one variable when 

 the periods are more than two in number. "I shall like to make just 

 one remark on the objection which this distinguished scholar [Jacobi] 

 advances against the consideration of functions of a single variable 

 on account of the absurdity of a triple periodicity. The assertion, 

 namely, that every triply periodic function admits a period less than 

 every given quantity and that this is absurd, is one with which I be- 

 lieve we must unreservedly agree as regards the first part but we do 

 not see clearly (with all due respect to so eminent a scholar) wherein 

 the absurdity lies. Where, in fine, are we to assume that the cause of this 

 absurdity lies? Is it in the elements, or in the concept of the integral, 

 or of the function?" As Jacobi prepared Gopel's manuscript for the 

 press and added the brief foot-note ! ? to the sentence immediately follow- 

 ing those quoted, it would seem that in 1847 Jacobi had still grave doubts 

 as to the possibility of inverting a hyperelliptic integral of genus 2. 



Ten 3^ears later Riemann cleared away all difficulties as regards 

 the inversion of an Abelian integral of the first kind. In section 12 of his 

 classical memoir on the Abelian Functions, he takes an Abelian Integral 

 of the first kind on the simply connected Riemann surface T^, which 

 arises from the Riemann Surface associated with a basis-equation of 

 genus p by the drawing of 2p cross cuts Aj, Bj (i = l, . ., p), and maps 

 T^ conformally, by means of this integral, on a finite region bounded 

 by p parallelograms. These parallelograms are superposed in p sheets, 

 are arbitrarily placed and are connected by p — 1 branch-cuts which 

 join the 2p — 2 branch-points in pairs in such a way as to make of the 

 complete figure a fundamental region. The opposite sides of any 

 parallelogram correspond in the ordinary way. To take the simplest 

 case, suppose that there is one parallelogram whose sides correspond 

 in pairs. By uniting corresponding sides, the figure can be deformed 

 continuously into a tore (p = l). Just as the net-work of parallelo- 

 grams in the case of elliptic functions is obtained from the fundamental 

 parallelogram by translations equal to the moduli of periodicity of 

 the elliptic integral of the first kind, so also here by successive trans- 

 lations, equivalent to periods, of the p superposed parallelograms we 

 are led to a surface that covers the plane infinitely many times, yet about 

 any place on any sheet the function x (u) is analytic. 



Note. March, 1914. 



Professor Fricke in his Encyclopaedia article on Elliptic Functions, 

 which has just been published, has given a very careful original discussion 

 of the Inversion Problem which satisfies all requirements as regards sim- 

 plicity and rigour. Reference should also be made to his earlier treatment 

 of the same subject in his published lectures on the Theory of Functions. 



