550 PROCEEDINGS OF THE AMERICAN' ACADEMY. 



Now write for m. = 1, . . ., A' + 1 



Zm (x) = ^ (x) T (.r) {r'm ^ arg a- ^ r'„,,i). 

 From (41) and (42) there results along arg .i- = t'^+i 



Zrn^l{x) = Zm{x)[I+Cm], | -T | ^ f. 



The functions Z,„ (.r) may accordingly be continued analytically across 

 each ray arg x = T^ and represent matrices analytic for | a* | ^ r, of 

 determinant not zero. The relation (43) leads to the conclusion that 



Zm (.r) ~ S (x) {r'n ^ arg x ^ xm+i). 



where S{x) is of the desired form (35), and thence to the conclusion 

 that this asymptotic representation is valid for r„i ^ arg x < r„j+i. 

 The relation between Zjv+i (.v) and Zi {x) is that stated in (38). 



In § 13 it was shown that, if matrices Zi{x), . . ., Z^'{x) had the above 

 properties and the further properties that they were analytic in the 

 finite plane of determinant not zero for x 7^ 0, and at x = were 

 properly equivalent to a Cauchy matrix, then these matrices were 

 solutions of a canonical system (34) with irregular singular point 

 at a = 00 . The same arguments can be used to establish that the 

 Zi (x),. . .,Zif (x) before us are solutions of a differential system (24) 

 having coefficients rational in character at .r = 00 , with poles of order 

 not more than jJ- But we have proved in § 12 that the matrix solu- 

 tion of such an equation is properly equivalent to that of a canonical 

 linear differential system at x = 00 . Consequently a transformation 

 Z (x) = A(x)Z(x), where A{x) is a matrix of functions analytic in 

 character at a = 00 of determinant not zero there, makes Z{x) the 

 solution of such a canonical system. In particular the matrices 



Zi (x) = A (.!•) Zi (x), , Z^ (x) = A (x) Za- (.r) 



form the solution of oiu* problem, a fact which is apparent if we 

 note that S{x) = A{x) S{x) has the same form as S{x). 



It has therefore been completely established under the stated 

 restrictions that the characteristic constants which occur in the 

 characterization of the matrix solutions of a canonical linear differen- 

 tial system can be chosen at pleasure. 



The restriction that no three of the quantities ai, . . . , a„ shall lie 

 on a straight line is not essential, for if it is not satisfied and if no two 

 of the polynomials pi{x), . . ., p„ (x) arc identical it will be possible to 

 replace the rays ti, . . . , tjv which ha^■e coalesced l\v an equal number 



