546 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where | 6'y | 9^ 0. The (juantities ai, . . ., a„ are the roots of the char- 

 acteristic equation alluded to, and the series Sij (.r) are in general di- 

 vergent. We shall assume for the time being that no three of the 

 points ai, . . . , a„ lie on the same straight line in the complex plane. 

 Let now n, . . .,ty denote the iV = n{n — 1) Qj + 1) arguments 

 in order of increasing angular magnitude such that for some j and 

 h (./■ ^ /.•) 

 (36) Ot { (a; - o.k) x^^' \ - 0, arg .r = r„. 



Here "9i" denotes "the real part of". Let us write r^+i = t\ -\- '1-k, 

 and let j„j and Am denote the value of ./' and k corresponding to m, so 

 ordered that the real part (30) changes from positive to negative 

 as arg.r increases through arg.c = r^. There exist then .V matrix- 

 solutions Zi(.t), . . . , Ziv(.r) such that for i = 1, . . . , iV 



(37) Zm {x) ~ S {x), Tm ^ arg .r < Tm+\, 



16 



and such that along arg.r = r^+i, Zm+i(.r) and Zm{.v) differ only in 

 their _/,„th column, the i^th column of Z^+i {x) being obtained from 

 that of Zyn {-f) by the addition of the /."mth column of Z„,(.r), affected 

 with a suitable constant multiplier CJr^, to the j^th column. As a 

 matter of definition we take 



(38) Z.v.i (.r) = Z, {x) r, r = (e^'^^vV^i 5.^). 



The proof of the existence of Zi{x),. . ., Zjv(.r) having these proper- 

 ties can l)e directly based on the existence of a solution Z ix) asymp- 

 totically represented by N (x) along every particular ray.-^'^ 



The properties so far stated are characteristic of the behavior of 

 the matrix solution not alone of a canonical system but of any system 

 with singular point of rank p at x = 00 in the neighborhood 

 of the singular point,^^ and are invariant under a transformation 

 y(.r) = A (x) Y{x) where .4 (x) is a matrix of elements analytic at the 

 singular point in question and of determinant not zero there. 



When Zi (x), . . . , Z^+i (x) are in addition the solution of a canonical 

 system, the matrices Z m{->^) are analytic in the finite plane for x ^ 



16 The relation "z {x)oo s (x), arg x = a," means in the present paper that 

 z (x) is asymptotically represented by s (x) in some sector {however small) that 

 includes arg x = a as an interior ray. This sHght modification of the conven- 

 tional meaning of the symbol " <^^ " and its natural extension to matrices is 

 convenient for the present paper. 



17 I, § 6. 



18 I, § 6. 



