BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 537 



where Z is a matrix of constants. In fact we have 

 dT{x) _ dl'jx) ^ 



X -. — X ^ ty. 



ax ax 



But the matrix x dl'{x) / dx may be written as 



Kl'ix), K= ikbij). 

 Thus we obtain 



X "^^M = KI' (x) C = KC- C-W {x) C = LT (x). 



§ 9. The Monodromic Group Problem. 



The most elementary existence theorems for ordinary Hnear differ- 

 ential equations show that the linear differential system 



(24) ^^ = R(x)Y{x) 



admits a matrix solution Y(x) whose elements are analytic in the 

 finite plane at every ordinary point where the elements of R(^x) are 

 analytic, and analytic at infinity if the elements of R(x) vanish to at 

 least the second order at the ordinary point infinity; furthermore 

 I Y(x) I is not zero at such points. All other points of the plane are 

 called singular points in contradistinction to the ordinary points. A 

 finite singular point at which the elements of R{x) are analytic or 

 have a pole of the first order, or the point .r = oo if each element of 

 R(x) vanishes to the first but not always to the second order at 

 that point, is termed a regular singidar point. 



Suppose now that the elements of i?(.r) are rational, and that all of 

 the singular points ai, . . . , a^ are regular. These will be taken to lie 

 in the finite plane. It is easy to show that the elements of Y(x) be- 

 come infinite to only a finite order at x = ai,. . . ,Om.-^^ When x makes a 

 positive circuit of one of these points, Y(.v) changes to Y{x) T^, where 

 Ti is a matrix of constants and | Tj- | ?^ ; in fact the most general 

 solution is of the form Y{x) T where T is an arbitrary matrix of con- 

 stants for which | T | 5^ 0, and after the circuit is made in the .r-plane, 

 Y{x) is still a solution of (24). 



If we start from a point x = c and make a circuit of ai, . . ,«„ in 



11 Cf. Trans. Am. Math. Soc, 11, 199-202 (1910). 



