532 PROCEEDINGS OF THE AMERICAN ACADEMY. 



X = c by one unit, without the introduction of further zeros of | $(.r) j 

 or I ^(.r) 1 in the finite plane. 



An entirely similar process eliminates a zero of | ^(.r) | without C, 

 or a zero of ] $(.r) | and | ^(x) \ along C. In consequence of the 

 results of § 5, if either of these functions vanishes along C, the other 

 does also, both at least to the first order. 



This process may be continued so long as there remain zeros of 

 $(.r) or ^(.r). It must however finally come to an end. If this were 

 not the fact it would follow at once that both | $(.r) | and | ^(x) \ have 

 a zero of infinite multiplicity at some one point of C where an element 

 of A(x) fails to be analytic. But this cannot be the case, for let $(.r) 

 and ^{x) be a solution of the following matrix equation 



(20) ^(x) = A-'ix)^{x) along C, 



where the elements of $(.r) and "^(.r) are restricted hke $(.r) and ^(x) 

 were found to be in § 4. The existence of such a solution becomes 

 manifest by a mere interchange of the role of rows and columns in 

 what precedes. Now from (17) and (20) we conclude that 



i $ (.r) H $ (.r) i = i ^ (.r) I • I ^ (x) \ along C. 



The function represented by either side of this equality is not identi- 

 cally zero; and it appears from the two representations that it is 

 analytic in the finite plane, and analytic or with a pole at infinity. 

 Hence this function is a polynomial, and the multiplicity of the zeros 

 of either | $(.r) ] or | ^(x) \ at any point of C is finite. 



When the process comes to an end the following result has been 

 obtained: if the elements of A{x) are unlimitedly differentiable along 

 C, analytic save at a finite number of points of C, and if | A (x) \ is 

 not zero along C, there exists a solution $(.v), ^(.r) of the equation (17) 



$ (x) = ^ (x) A (x) along C, 



in which the elements of $(.r) are analytic within C, unlimitedly differ- 

 entiable along C, and $(.r) is of determinant not zero within or on C; 

 and in which the elements of ^(x) are analytic without C in the ex- 

 tended plane save for a possible pole at x = oo , unlimitedly differen- 

 tiable along C, and "^(x) is of determinant not zero without C. 



Here the point x = oo appears as an exceptional point. It is evi- 

 dent that an arbitrary point x = a not on C may be used to take the 

 role of the point at infinity. In fact a may also be taken to l)e a point 

 of C. When this is the case, the elements of <l>(.r) and ^f.r) are finite, 



