552 PROCEEDINGS OF THE AMERICAN ACADEMY. 



This second condition is satisfied if the elementary di\"isors associated 

 with Ti and Ti are the same. 



With the understanding that the two conditions of compatibiUty are 

 satisfied, we can assert that there exist m + 1 canonical systems with 

 the prescribed characteristic constants at .Ti, . . . , .Tm+i and with solu- 

 tions Vi {x),. . ., Vm+i i-c) undergoing a transformation to Vi (x) Ti,..., 

 Vm+\Tm+i as X makes a positive circuit of .Vi, ....Tm+i respectively. 

 Hence there will exist a matrix Y {x) of elements analytic and of de- 

 terminant not zero save at Xi, . . ., Xr^+u properly equivalent to Vi {x) at 

 x = ttj- (^ = 1, . . . ,m), and properly or improperly equivalent to F^+i (.f) 

 at Xm+i = ^ ■ This follows upon application of the results of § 10. 

 The matrix Y (.r) thus obtained is a matrix solution of a differential 

 system of the form required and has the monodromic group and 

 characteristic constants required, if the equivalence at .i-„i+i = ^ be 

 proper. This is an immediate consequence of the results of §§ 13, 14. 



If this is not the case it is easy to show that if one merely increases 

 the characteristic constants ri, . . . ,r„ for the singular point .t„+i = oo 

 by suitable integers this equivalence at infinity becomes proper. For 

 consider the matrix S{x) corresponding to the matrix solution of the 

 equivalent canonical system at .r = co . The corresponding matrix 

 S{x) for Y{x) is of the form A{x) S{x) where the elements of A(x) are 

 rational in character at x = oo . Hence if a proper choice of C be 

 made, the matrix S (x) C may be written (compare (33), § 11). 



e^i(-^/i (an +^ +...],..., c^n(-\r'n Ln + ^^" + . . .) 



e^iWx^i ( an, -f ^ + . . .\ . . ., e^ni^\x-n ( cinn +-+... 



X J \ X 



where vi, . . ., i\ differ from vy, ...,/•„ by integers. By a process of re- 

 duction exactly like that gi^'en in § 11 this matrix may be replaced by 

 a series of matrices of the same form in which the exponents vi, . . ., r^ 

 are increased so long as | Oy- | 5^ 0. We conceive of Y (x) as affected 

 by the same series of operations, which have no effect on the proper- 

 ties above specified. Furthermore the transformed Y{x) and S{x) 

 stand always in the same relation to each other after as before trans- 

 formation. If the process comes to an end so that | a^ \ 9^ 0, the 

 linear differential system with matrix solution r(.v) will have a singu- 

 lar point of rank g-^+i at x = 00 (compare § 13, pp. 547-548) and the 



