BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 541 



functions analytic of deterininant not zero in the vicinity of Oj- and under- 

 going a transformation to Z\{x) Ti, . . ., Z^ix) Tm as x makes a positive 

 circuit of oi, . . ., a^, respectively. There exists then a matrix Y{x) of 

 functions of determinant not zero for x 7^ a\,. . ., a^ and analytic 

 save at these points, which undergoes a transformation to Y (.r) T^ as x 

 makes a positive circuit of a^ {i = 1,. . .m); furthermore Y{x) is prop- 

 erly equivalent to Z\ (x),. . .jZ^-iCi") at Oi,. . ., a„j_i and properly or im- 

 properly equivalent to Z^ (x) at am- 



The result above stated is obtained when we replace the Cauchy 

 matrices Zi( x — ai), . . ., Z^ix — o„i) of § 9 by matrices Zi{x), . . ., Z^ix) 

 having the properties specified. The line of attack is identical with 

 that given in § 9. The facts concerning equivalence follow at once 

 from the relations analogous to (30) : 



(32) Y(x) = Z,{x)Zi(x) (i= l,...,m). 



Here Zj(.i') is composed of elements analytic at .r = a^ (i = 1, . . . ,m — 1), 

 and of determinant not zero there; also Z^ix) is composed of elements 

 analytic at .i- = a^ or with a pole at that point. 



§ 11. Final Form of Solution. 



There is a certain lack of symmetry between the role of ai, . . ., r;^ 

 in the solution of the Riemann problem obtained in § 9, provided the 

 given Cauchy matrices Zi(.r — oj),. . ., Z„^{x — a„j) were taken in the 

 most general possible form. We shall now proceed to show that if 

 ^m (•^' — ^m) he a properly chosen Cauchy matrix associated toith the 

 transformation Tm, the equivalence of Y {x) and Z^ (x) can be made 

 proper at x = a^ also. 



A form of reduction that is well-known suffices to establish this 

 fact.^^ Nevertheless, inasmuch as a similar type of reduction is neces- 

 sary later in the present paper, I give this reduction herewith. 



Let us begin with the matrix Y{x), obtained in § 9, which we may 

 assume to be improperly equivalent at «,„ to the Cauchy matrix 

 Zm {x — am). Now we have (§ S) 



so that from (30) 



Zm {x — am) = r (x — a J C 

 Y(x)C-' = Zm(x)r{x-am] 



12 Plemelj, loc. cit., pp. 237-240. 



