556 PROCEEDINGS OF THE AMERICAN ACADEMY. 



naries is obtained from that on the right-hand side by a complete 

 positive circuit of x = 0, during Avhich T(.v) changes from 



Cx-"^ (pjer^y xl8ij) to (e-'^'' ^^ 'e'-^ ^^ 'Jx"^ {pjC-^Y x'ibij) . 



The /th diagonal element of Ai{x) may now be written 



while the element in the iih. row and _/th coliuun of A (.r) {i 9^ j) may 

 be written 



Bearing (49) in mind we readily perceive that Ai(x) does have the 

 indicated properties along the lower half of the axis of imaginaries. 



According to the preliminary theorem we can then determine a. 

 matrix ^(x) such that 



(50) ,!-_*(.r) = [J-.$(.r)].li(.-), 



w^here Xi is a point of the axis of imaginaries and the approach is from 

 the left-hand and right-hand side of that axis respectively. If we take 

 .r = a = 0, the determinant of $(.r) is not zero in the finite plane except 

 at .r = possibly, and the elements of this matrix are analytic at any 

 point not on the axis. Along the axis as defined from either side these 

 elements have continuous derivatives of all order, and will be analytic 

 at more than certain distance d from the origin. In the vicinity of 

 X = 00 , $(.r) is represented asymptotically by a matrix of series in 1/ .r 

 with determinant of leading coefficients not zero. This matrix is the 

 same on either side of the axis, since A (x) co /. 



Let us denote $(x) by U'^{x) for x in the right half plane and by 

 U~ (x) for X in the left half plane, and write 



Y- (x) = U- (x) T (x), Y- (x) = U- (x) T (.r). 



From equation (50) we see then that 



(51) y-(.r) = YHx)P{x) 



for I .1: I > /• along the axis of imaginaries. From the asymptotic 

 form of U'^ix) and U''(x) at x = ^ we obtain 



r-(.r) cv S{x), r+(.r) - S(x), 



in the left and right half plane, where S(x) is of the same form as S{x) 

 above. The relation (51) shows that Y~{x) is composed of elements, 

 analytic in the right half plane. 



