BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 567 



According to the theorem there exists a matrix ^(x) such that 



(71) ,!i-.<i>(.r) = [;i-_$(.r)]^li(.vi), 



where Xi is an arbitrary point of d and the + and — signs denote 

 approach to xi from within and without Ci respectively; the matrix 

 <J>(.r) possesses certain other properties: it is composed of elements 

 analytic for x not on C'l, save at a point a which we shall take to be at 

 infinity; furthermore its determinant does not vanish in the finite 

 plane. 



Let us denote by Uq (x) the matrix $ (x) within Ci, and its 

 analytic extension across Ci; similarly by U^ (x) let us denote the 

 matrix $(.r) without Ci and its analytic extension across Ci. Consider 

 then the matrices 



IV (.r) = Uo (x) T, (x), y, (x) = U^ (.r) T^ (.r). 



From (71) we obtain at once 



(72) Fo- (.!•)= Y.,(x)P(x), 



and prove without difficulty that ro""(.r) and Y^-^(x) as thus defined 

 have the characteristics demanded save possibly that 1^ M ^^^Y ^^ot 

 be precisely of the form {bS) at x = co , as it would be if $(.i-) were com- 

 posed of elements analytic at x = oo and if also | <J>(.r) | were different 

 from zero at .r = oc . Nevertheless one can always write Y^ (x) in 

 the form 



qi (t-—i) 



an + ^-^ + . . .\ . . . x--^n (a,,. + ^'"^ + 



r / V X 



(Inl + " + ...),... .l-^« ( Unn + — + 



X J \ X 



where (Ti, . . ., cr„ cUffer from ai, . . ., a^ by integers. By a process of 

 reduction precisely like that employed earlier (§ 11) one may further 

 modify Yq(x) and simultaneously Y^ (x) so as to preserve all of the 

 properties already not^d and to finally obtain | Oy | 5^ 0. It is to be 

 recalled that | Yq (x) | is not identically zero and can at most vanish 

 to a finite order at .r = 00 ; for it is this fact that enables us to conclude 

 that the process of reduction terminates. 



The argument of the preceding paragraph shows that Yq(x) and 

 Y^ (x) will be matrix solutions of a system (54) with coefficients poly- 

 nomials in .r of degree /x at most. 



