BTEKHOFF. — THE GENERALIZED RIEMANN PROBLEAI. 533 



or become infinite to finite order at x = a. To obtain such a solution 

 $(.r), ^(.r) it is only necessary to divide each element of $(.r), ^(x) 

 by a suitable power of .r — a, so chosen as to make each element of 

 these matrices analytic at x = oo ; and then to apply the normaliza- 

 tion above indicated, letting l/x — a replace x. The curve C can also 

 be taken to be a simply closed analytic curve which passes through 



X= 00 . 



The main part of the conclusion that has been deduced abo"\'e was 

 obtained by Hilbert and Plemelj (loc. cit.) with the aid of the Fredholm 

 theory of linear integral equations. Independently of their work, I 

 treated a special case (see I, § 1) which arose in a different form in 

 connection with, my study of the irregular singular points of ordinary 

 linear differential equations. 



My proof in this special case, suggested to me the al)ove treatment 

 of the general case by the method of successive approximations. 

 The restrictions here placed on the elements of A (x) and on the curve 

 C are not essential to this treatment, and I have very little doubt that 

 these may be replaced by the weaker restrictions of Hilbert and Plemelj. 

 Nevertheless I have been content to use the simplest restrictions 

 consistent with the applications in view. 



A second proof in the special case has recently been given by me. 

 Math. Ann., vol. 74 (1913) pp. 122 (see also Bull. Am. Math. Soc, 

 vol. 18, 1911, p. 64). This second proof, wJiich suggested itself to me 

 at about the same time as the first, is practically the same as that given 

 by Hilbert and Plemelj. To my considerable regret this relation- 

 ship escaped my observation until it was too late for me to make 

 suitable reference. 



§ 7. Tlie Preliminary Theorem. 



The following is an extension of the preceding results which is 

 convenient for the applications: 



Preliminary Theorem. Let Ci,...,Cr he r simply closed analytic 

 curves in the extended complex plane. Let Ai(x),. . ., AX-t-') he matrices 

 of functions defined and unlimitedly differentiahle along Ci, . . . , Cr 

 respectively, analytic save at a finite number of points of these curves and 

 of determinant not zero. If furthermore ai any point of intersection of 

 G^a, Cp, the nmtriceS'AcA:x)yA^{x) are such that the formal derivatives of 

 all orders of the matrix 



(21) AA-v) Affix) -A^{x)A^{.r\ 



