BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 551 



of cul•^•eci rays so chosen that the real part of only one of the differences 

 Pi (.r) — pj {x) changes sign along the ray, and thence to apply the 

 preliminary theorem in much the same fashion as before. If two or 

 more polynomials pi (a*),. . ., Pn {x) are identical it is not necessary to 

 modify the nature of the rays. 



Furthermore it would be possible to construct an analogous existence 

 proof when S{x) has for its elements so-called anormal series. 



In other words, the results here obtained are of an entirely general 

 nature. 



§ 15. The Generalized Riemann Problem. 



The problem which I proposed in my paper on singular points (I) 

 was the following: " To construct a system of linear differential equations 

 of the first order with prescribed singular points 



Xi, Xo,. . .Xrr,, .V^+i = 00 



of respective ranks 



q\,- ■■, qm+u 



and with a gircn monodromic group, the characteristic constants being 

 assigned for each singular point." This problem is virtually solved by 

 what precedes. 



It is of course understood that certain obvious conditions of com- 

 patibility are satisfied, the first being the one already noted, 



i m+l i m • • • 1 1 ^ 1. 



However a second necessary condition must also be imposed. Take 

 any assigned point x = Xi. The solution Zi (.v) appertaining to this 

 point (see § 14) is transformed successively into 



Zo (.!■) (1 + Ci)-i, Z3 (.v) (7 + C2)-' (/ + Ci)-', 



...,Z^^i{x)(I+CMy... (/+Ci)-S 



as X passes over the rays ti, . . . tj^ respectively. Hence after a com- 

 plete circuit of O;, Zj (.v) alters to 



Zi i.v) r (7 + Cn)-^ . . . (7 + Ci), 



where the matrix T^ of transformation is explicitly determined in 

 terms of the characteristic constants. But in order for this set of 

 characteristic constants to be possible, some solution Z (x) = Zi (.r) C 

 must undergo precisely the transformation by Ti, i. e., 



fi= CTiC-' ii= 1,. . ., m). 



