544 PROCEEDINGS OF THE AMERICAN ACADEMY. 



is improperly etiuivalent to ii ( •aucliy matrix which at .v = undergoes 

 a transformation inverse to that which 1' (;r) undergoes at x = oo . 

 Here we take m = 2, oi = 0, »> = oo . The condition T2T1 = I is 

 satisfied. 



By means of a modification precisely like that of § 11 we can make 

 Z(x) properly equivalent to a suitable Cauehy matrix at x = 0, and 

 yet preserve the other properties listed in § 10. 



Now consider 



i?,(.r) = ^^Z-Ha-). 



Since. | Z \ (x) \ ^ for x f^ 0, QO, and since Z(x) and dZ{x)/dx 

 undergo the same substitution about x = 0, the elements of Ri{x) are 

 single-valued, and analytic for x ?^ 0, 00 . Since Z(x) is properly 

 equivalent to a Cauehy matrix at x = 0, the elements of Ri{x) have 

 poles of at most the first order at x = 0.^* Moreover since Z(x) is 

 properly equivalent to Y(x) at x = 00 we have 



Z{x) ^ A{x)Y{x), 



where the elements of ^(.r) are analytic at .r = 00 and also | A{x) \ 7^ 

 at X = CO . Therefore we obtain 



^^("•)=^(.r)^ + ^7(.r) 



dx dx dx 



and 



A{x)R\x)^'^ 



Y{x), 



.(.,).,.) + '^- 



A-' (.r). 



Hence the elements of i?i(.r) are analytic or have a pole of order not 

 greater than p at .r = co . 



From this analysis it follows that xRi{x) is a matrix of polynomials 

 of degree at most 2> ~\~ 1, so that Z{x) is itself the solution of a linear 

 differential system 



(34) »f=^'W^. 



where P{x) is a matrix of polynomials of degree at most p -]- 1. This 

 is the canoniccd form of equation with an irregular singular point of 

 rank j* + 1 at .r = 00 . 



At a finite singular point x = a, the canonical system is of a type 

 obtained from (34) by a transformation x' — a = l/x. 



14 The detailed proof is entirely similar to that given herewith to determine 

 the nature of the elements of R{x) at x = go . Cf. Sclilesinger, loc. cit., 

 pp. 143-144. 



