540 PROCEEDINGS OF THE AMERICAN ACADEMY. 



(31) ,!!^.+ $ (.t) = [,!!^- <!> (x)] Ai (.1-0, Ai (x) = U (x) Zr' (x-m). 



(i = 1, . . ., m). 



This suggests another appHcation of the prehminary theorem, since 

 the curves Ci,...,Cm and the matrices Ai{x),..., Ajn{x) of known 

 functions satisfy the necessary restrictions. 



Let ^(x) be the solution given by the theorem for a = dm, and let 

 Zi(;x),. . .,Zm(x), Z(x), be defined as equal to <J>(.r) within Ci,..., Cm 

 and outside of these curves respectively. These functions will then 

 satisfy (30). Let Y{x) be defined by (29). This matrix is clearly 

 composed of elements analytic without and along Ci, . . . , C^, as are 

 those of U(x); within C^, Y{x) continues analytically into 

 Zi{x) Zi{x — ttj) by (30) for i — 1,. . ., m, and consequently its ele- 

 ments are analytic throughout the plane except possibly at ai, . . .,a^; 

 its elements become infinite only to a finite order at % since the 

 elements of Zj(.i'— a^) become infinite only to finite order at Oj. 

 Furthermore by (29) Y{x) undergoes a linear transformation to 

 Y {x) Ti as X makes a positive circuit of Oj. Thus the Riemann prob- 

 lem has been solved. 



It is worthy of note that | Y (.r) | does not vanish for .r ^ o, 

 (i = 1, . . .,m). This is an immediate consequence of the fact that 

 I Zi{x) |, . . ., I Zm(.r) |, I Z{x) I do not vanish in their regions of defini- 

 tions save at these points, and of the fact that the Cauchy matrix 

 Zj(.r — Oj) has a determinant which does not vanish save possibly at 

 X = Oj and x = go . 



§ 10. yl Generalization. Equivalence. 



x\ more general result can be deduced exactly as the results of § 9 

 were obtained. Let us say that two matrices of functions 1 1 (.r) and 

 Yo{x) whose elements are analytic in the vicinity of x = a, but not 

 in general single-^'alued or analytic at x = a, are properly equivalent 

 at .V = a if we have 



Y,{.v) = A{.v)Y,ix). 



where A{.x) is composed of elements single-valued and analytic at x=a, 

 of determinant not zero there; if this condition is not satisfied, 

 but if the elements of A{.v) have a pole or are analytic at x = a, let 

 us say that Fi(.r) and Yoix) are improperly equivalent at x = a. 



This definition is convenient for the statement of the following 

 result : Let ai, . . ., a„ be m given points; let Ti, . . ., T^ be matrices of con- 

 stants such that Tm T^-i- . . Ti = I; let Zi {x), . . ., Z^ {x) be matrices of 



