BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 535 



of the curves or at those points where an element of A^ (x) fails to be 

 analytic; 



, ,. x^2f.+ U{.r) = x^f.- U(x) {i = 1, . . ., k), 



^ ^ .^x" ,-. U (x) $, (.r) = [xJ^^- U (x) $, (.r)] Ak,^ {xk,{). 



The condition (3') necessitates that U (x) is analytic at any point 

 of Ci (i = 1, . . ., A') and throughout the extended plane save at a and 

 at points of C^+i. The condition (3') gives us in addition 



x-'ir^,-. U{x) = [Ji^_^^- U{x)]^kixk^i)Ak^d-rk^i)^k~'{-rk^i). 



Conversely if U{x) satisfies these conditions (1'), {-'), (3') it is ap- 

 parent that $4+1 (x), given by (22), will satisfy the conditions prescribed 

 for <l>(.r) in the theorem. 



But in view of the above simplification of (3')> the conditions (!'), 

 (2'), (3') on U(x) are precisely those of the theorem on $(.r) if we take 

 r = 1, C = Ck+i, Ai(x) = ^kix) Ak^i{x)^k~K-f^)- To complete a 

 proof we need only show that this matrix fulfils the conditions pre- 

 scribed in the theorem along C^+i; for then a matrix U{x) will exist 

 which satisfies these conditions. 



The elements of the matrix 



(23) ^kix)Ak^i(x)^k-H'^) 



are analytic at points of Ck+i which are not points of intersection or 

 contact with Ci, . . ., C^, or points at which an element of ^^+1(0*) fails 

 to be analytic; at these latter points the elements of ^^(.r) are analytic 

 so that the above matrix is unlimitedly differentiable in the neighbor- 

 hood of such a point. The determinant of this matrix is equal to 

 I Ak+i(x) I and nowhere vanishes along Ck+\- 



It remains only to examine the elements of the matrix near points 

 of intersection or contact of C^+i with one of the curves Cy, . . .,Ck- 

 Suppose first that C^+i intersects a single curve Cf at such a point. As 

 X moves along the curve Cj+i and passes from the positive to the 

 negative side of Cj, the matrix $4 (.r) changes to <l>jt (.r) Ai (.r), so that 

 the matrix (23) changes to 



^k (x) Ai (x) Ak+i {x) Ar' (.r) ^k' {x). 



Bearing in mind the condition of permutability imposed on 

 Ai{x),. . ., A^ (x) at such a point of intersection (see (21)), it becomes 

 apparent that the elements of the matrix (23) are continuous at this 

 point, and have equal backward and forward derivatives of all orders 



