BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 539 



By slightly modifying each element of A(x) along a small segment 

 near either end of the arc of D within Cj for i = 1, . . . , m, it is possible 

 to obtain a matrix A{x) whose elements are analytic save at the end 

 points of these segments, and unlimitedly differentiable along D. 

 Such a matrix A(.v) satisfies all of the conditions necessary for the 

 application of the preliminary theorem in the case r = 1. 



Taking D as the curve and A(x) as the matrix, we can affirm the 

 existence of a certain matrix ^{x), possessing in particular the property 



(28) ."S;.*(.i-) = U"^-*W]^4Gri), 



• the approach to the point .ri of D being from within and without D 

 respectively. Let us choose a = a^. 



If we extend $ (.r) analytically across D between Cj and Cj+i, 

 it becomes ^ix)Ai by (28). If we extend this matrix analytically 

 back across D between Cj+i and Ci+2 [0^+2 = C2], it becomes 

 [<J>(.r).4,:+i-i] Ai-\ or <l>(.r) Tj. That is, the matrix U (x) obtained 

 from ^(x) by analytic extension is analytic outside of Ci, . . .,0^ and 

 undergoes a transformation to U{x) T^ when x makes a positive cir- 

 cuit of «j. Furthermore the determinant of U{x) is not zero outside 

 of Ci,...,C,. 



Denote by Ziix — cv) the Cauchy matrix belonging to the trans- 

 formation Ti, so that Zi {x — tti) undergoes a transformation to 

 Zj (x — (ti) Ti as .r makes a positive circuit of a;. Write 



(29) Y{x) = Z(x)U{x), 



where Y{x) is the solution of the Riemann problem to be constructed. 



The elements of Z(x) must in the first place be single- valued and 

 analytic outside of Ci, . . . , Cm, since U{x) undergoes the same trans- 

 formation as that prescribed for Y(x) about the points cii, . . . ,0^, and 

 I U{x) I ^ 0. 



Furthermore within C^, the elements of Y{x) are to be analytic ex- 

 cept at «i where they may become infinite to finite order. Hence the 

 elements of the matrix Y{x)Zi-''-(x — rtj) must be single-valued and 

 analytic within Ci, by the definition of Zi(x—ai), except for a possi- 

 ble pole at X = cii. Along C^ this matrix may be written 



(30) Zi {x) = Z {x) [Uix) Zr' {x - cu)\ (i = 1, 2, . . . m). 



If we write $ (.r) = Z{x) outside of C'l, . . .,Cn and also $(.r) = Zi{x) 

 within C for i = 1,. . .,?«, the equations (.30) may be written 



