[harkness] study of ELLIPTIC FUNCTIONS 83 



and m^ crossings of B, all from left bank to right bank, is to increase 

 u by 2mû>+2ni^ co\ 



Since the integral defined by 



-/ 



X, y dx 

 ^0, Jo y 



in which the initial values of x, y are Xg, y^, admits infinitely 

 many values, it is obvious that the one-valuedness of x, y as functions 

 of u is a fact of fundamental importance. It is surprising to find 

 how inadequately the necessity for a rigorous proof of this theorem 

 has been recognized by many of the writers of accepted text-books. 

 The probable reason is that many of the logical possibilities are of a 

 kind that easily escape attention. We propose to examine some of 

 the methods by which the problem of inversion has been successfully 

 attacked, and incidentally to point out the character of the difficulties 

 to be overcome. 



Since we wish to express x, y as one-valued doubly periodic 

 functions of u with the moduli of periodicity of the integral serving 

 as periods of the functions, it is essential that the ratio œ^jœ should 

 not be real. This can be established at once by putting u=v+iw 

 and integrating / vdw over the complete contour of A, B in the 

 sense of the arrows. If 2w=a+ib, 2w^=c-l-id, the contributions of 

 the two banks Aj, Aj. (fig. 1) of A is /"vj duj — /"vj. dUj.= /"(vj — Vj.) du 



-^ A, -^ A, -^ A 



(since duj = duj = a I du = a d ; similarly the contribution of the 



•^ A 

 two banks of B is - be. Thus f vdw = ad-bc ; but the integral 

 can also be put in the form /V' f/ 'u\2 /^^^\^1 dxdy, taken 



■#[i)' - m 



over the surface of T^ and is, therefore, essentially positive. Hence 

 ad-bc ;>o and consequently to^/co has the coefficient of i positive. 



This is, of course, Riemann's proof. It still seems the simplest, 

 though the matter has been examined from many points of view. 



Another preliminary point relates to the expansions in terms of 

 u - u^ for all places (x, y) in the immediate neighborhood of a place 

 (x^, y^), where u^ is the value of u at (x^, y^). For this purpose we 

 introduce a parameter t defined by 



X — x^=t ;x — a; =t";x = 1/t. 



according as (x^, y^) is an ordinary place, a branch-place (aj, o), or a 

 place for which x^ = oo. In these three cases we can express 1/y or 

 l/VR(x) in the forms P (t) ; f^ P (t") ; t" P (t) by ordinary algebraic 

 processes and in each case the power series begins with a constant 



