BIRKHOFF. — THE GENERALIZED RIEMANN PROBLEM. 545 



In order not to introduce artificial difficulties we shall consider the 

 regular singular point to be of rank zero. To this case the above 

 argument applies also, and the canonical system is that satisfied by a 

 Cauchy matrix. 



If Y(x) is the matrix solution of a differential system (24) in which 

 the elements of R{.v) are rational, the above argument shows that at 

 each of its singular points ai, . . ., a,n, Y(x) is properly equivalent to 

 the matrix solutions Zi(.v), . . ., Z,„(.v — a„i) of canonical differential 

 systems. 



Conversely the results of § 10 lead to the conclusion that given 

 Ti,...,Tm such that T^Tm-i- . . Ti = I, and canonical differential sys- 

 tems belonging to the singular points Oi, . . ., a^ with matrix solutions 

 Zi (.r), . . .,Zm i-c) undergoing a transformation to Zi (x) Ti, . . ., Z^ix) T^ 

 at these points respectively, there will exist a matrix solution Y(x) 

 of a rational differential system (24) which undergoes a transformation 

 to Y(x) Ti as X makes a positive circuit of Oj and which is properly 

 equivalent to Zj(.r) at this point, for i = 1, . . ., in — 1, and properly or 

 improperly equi^'alent to Z^ (.r) at x = a^. 



It is therefore essential to obtain a characterization of the matrix 

 solution of a canonical system (34), and further to solve the associated 

 inverse problem, before solving the general problem of characteriza- 

 tion for a system (24) with irregular singular points. 



§ 13. The Problem of the Irregular Singular Point. 



In my paper referred to ^^ I characterized the nature of the matrix 

 solution of a canonical linear differential system (34), at least in the 

 case that the roots of a certain characteristic equation were distinct; 

 the case of equal roots introduces complications of an algebraical 

 nature, and is put to one side in the present paper. 



The results which I obtained may be recapitulated as follows: 

 If the singular point is taken at .v = oo , there exists a formal matrix 

 solution of (34) 



S(x) = {e^j<~'\v^JSijix)), 



j.p+i -J.P 



(35) p, (x) = aj — — + /3, ^+ . . . + \-.v, (i = 1, . . . , n), 



p + I p 



Sij (x) = Sij + 5i/') -+..., {l,j=l,...,7l), 



15 I, §§ 6, 7. 



