522 PROCEEDINGS OF THE AMERICAN ACADEMY. 



classic form was first solved by Hilbert.* His treatment and Plemelj's 

 elegant completion thereof ^ reposed alike upon a certain theorem 

 whose proof was made by means of the Fredholm theory. Owing 

 to the deep-seated analogy between linear differential and difference 

 and g-difference equations, I have been able to apply a convenient 

 extension of the same theorem in all cases; my proof is based on a 

 method of successive approximations. 



Inasmuch as I have been able to simplify Hilbert's and Plemelj's 

 treatment of the classic Riemann problem, I have ventured to include 

 my treatment of it also. 



Part I. The Preliminary Theorem. 

 § 1. Some Dcfinifions. 



Let C be a simply closed analytic curve in the complex a:-plane. 

 If the arc length along this curve from a fixed to a variable point is 

 measured by s, and if / be the length of C, it is clear that x is a single- 

 valued analytic function of s with period / for s real, and that dx/ds 

 is not zero. Consequently if we introduce a new variable r defined 



by 



2t ^|—\s 



a one-to-one analytic correspondence is set up between the points 

 of the unit circle | r | = 1 in the r-plane and the points of C It will 

 therefore be possible to choose p > 1 so that the circular ring in the 

 T-plane, 



-<\t\^P, 

 P ~ 



is transformed in a one-to-one and conformal manner into a ring in 

 the x-plane bounded by simply closed analytic curves Ci and C2, 

 within and without C respectively, while at the same time the circle 

 I r I = 1 is transformed into C. Let t = t (x) be the function wliich 

 effects this transformation. 



Also let a (x) be any function continuous together with its deriva- 

 tives of all orders along C,^ and analytic save at a finite number of 



4 Gott. Nachr. (1905), pp. 307-338. 



5 Monatsh. f. Math. u. Phys., 19, 205-246 (1908). 



6 By definition we take df\x)/dx along a curve L as follows: 



dfp_ ^ urn fix')- fix) ^ ^^ 



dx ^ -^ X — X 



