526 PROCEEDINGS OF THE AMERICAN ACADEMY. 



and (6) likewise by 



(6') /- ix)+r-n-v) g%v)a{:c) = — ^ f r'^ (t) g- (t) "^^)~"^-''^ dt 



27r V — 1 Jc t — X 



27rV— lJc2 t — x 



From these two equations there results at once 



(7') r (x) - 1 (x) = T-^ (x) g- (x) a (x) along C. 



In order to develop an inequality for the modulus of /"^(.r) in this 

 case, we note that the contour C in (4') may be modified to Do. The 

 modification yields 



(9') |r(.r)l^f^i f\r-^{t)dt\+ f\"^dt \ 



where L is the maximum of | g'(x) \ along C. But | t~^{x) \ tends to 

 zero for x outside of C as p becomes infinite, since for such an x we 

 have I t(.t) I > 1. We conclude therefore that for any positive e 

 however small, the integer p may be taken so large that for every 

 regular outer function g~(x) we have 



(10') maximum of |/^ (x) | ^ e { maximum of g~ (x) | } along C. 



A further property of the function /~(.r), which is apparent from its 

 definition, is that this function vanishes at .r = cc . 



§ 3. Solution of a Pair of Matrix Equations. 



Throughout the present paper we shall be concerned with linear 

 equations in n unknown functions, whose complete solution may be 

 expressed in terms of n particular solutions. On this account we 

 shall employ the matrix notation. 



We consider first a pair of matrix equations 



,, ,, ( f* (-t) - F- (x) = r' (x) G* (x) A (x), 



(11' \a- (,) - G* (X) = r-' (x) F- (X) -4-1 (x) - I, "'""8 ^- 



Here r{x) is the function defined in § 1; the matrix A{x) is a given 

 matrix {aij{x)) (i, j = 1,. . .,n) of which each element is defined along 

 C and has the properties specified in § 1 for the function a{x) (namely, 

 it is continuous together with its derivatives of all orders along C, 

 and analytic save at a finite number of points); furthermore the de- 

 terminant I ^(x) I is not to be zero along C. The symbol / stands 



