528 PROCEEDINGS OF THE AMERICAX ACADEMY. 



where the third subscript on the functions corresponds to the subscript 

 on the matrix. We have already obtained a solution of an equation 

 of the form 



/^ (•^•) — /" (-i") = T-P (x) 9i,x,m-i+ {x) axj (x) along C 



(compare with (7)) in the form of a definite integral. By forming 

 the sum of /+(.r) and/~(.v) for X = 1,. .,n we obtain for every i and j, 

 elements 2\j,m'^ (x) and Pij_m~ (x) which form the elements of Pm*{x) 

 and Pm~ii^) with the desired property (14). 



Likewise we can break up the second one of the inth pair of matrix 

 equations (14) into n- equations. A solution may here be built up in a 

 similar way (compare (?')• It must be observed that since the de- 

 terminant of A(x) does not vanish along C, the elements of A~^{x) 

 satisfy the conditions imposed on a(.v) at the outset. 



Now if we recall the method of solution of (14), it is clear from 

 (10), (10') that along C the maximum modulus of any element of 

 Pm~ (x) or Qjn^ (x) does not exceed neLj„-i where im-i denotes the 

 maximum modulus of any element of F„t^i~(x) or ^^_i+(.v) along C, 

 and e is arbitrarily small uniformly for all values of m. This relation 

 may be expressed in the simpler form 



(15) Lm ^ tuLm-i. 



The series formed by the elements in any ith. row and jth column of the 

 series of matrices 



Po- (.r) + Pr (x) + . . ., Qo- (x) + Q,- (x) + . . ., 



will therefore converge absolutely and uniformly provided that e is 

 taken so small that 7ie < 1. 



But the sums of m + 1 terms of these two series of matrices are 

 Fm~(x) and 6'^+ (.r) respectively, whose elements therefore converge 

 uniformly to the elements of matrices F~(.v) and 6'+(.r) of regular outer 

 and inner functions respectively. If we recall that Po~{^) = 0, and 

 that the integral form of representation of each element of Pm~ {x) 

 (see (2)) makes each element of this matrix reduce to zero at .r = oc , 

 it is plain that at infinity F~{x) reduces to the matrix 0. 



If we turn now to consider F^"^ {x) and 0^" {x) along C, we see from 

 what precedes and from equations (12), that these matrices also 

 converge uniformly along C, and therefore respectively within C and 

 without C, to matrices F+{x) and G~{x) of regular outer and inner 

 functions. Since Go-{x) = 0, it is clear that G~{x), as well as F-(x). 

 reduces to at .r = x . 



