542 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The matrix of elements on the right can be written in the form 



{x—amY''{an-\-hn{-r—am) + ...),..., (x~amY'"{ain + &in (.r— Om) + . . . ) 

 (33) 



(.r— am)^"'(««i+fc«i(.v— am) + • ••),-••, (.i'— «m)^'"(«7m+^J««(.r — o^) + . • .) 



Here it is supposed that the exponents of the Cauchy matrix do not 

 differ by integers; but a similar form can be found in all cases. In 

 each column of Y{.v) the highest possible power of x — Ojn is exhibited 

 which leaves the coefficients of this power analytic in character at 



If I ttij \y^ we can write this matrix in the form A{x)I' (x — «^), 

 where I' (x) is the matrix (.r^jSy), and where A (x) is the matrix ob- 

 tained from (33) by striking out the exhibited powers of x — a. From 

 the eciuation 



Y{x) = A(x)r{x-aJC-' 



we see that Y(x) is properly equivalent to a Cauchy matrix belonging 

 to T at X = ttm- 



On the other hand if ] a,j | = we proceed as follows : It is readily 

 verified that if Y(x) be one matrix solution of the Riemann prob- 

 lem satisfying the relation of equivalence given in § 10, then 

 DY (x) [ I D 1 5^ 0] is also a solution. Consequently it is no restric- 

 tion in the consideration of (33) to assume that 



On = Oio. . . = f7,,i = 0, 



since when ] «,; | = this relation may always be made to hold 

 by multiplying on the left by a suitable matrix D. Denote by 

 t/i (.r), . . .,yn (•'*) the elements of the first row after a factor (x — ctmY 

 has been removed, where / is the exponent of the highest power of 

 X — Urn that may be taken out and leave the elements yi (.r), . . . , ^„(.r) 

 of the respective forms 



{x—dnf' (/l + gv (■»■ — a,.J + ...),... , (.r — anf'n ( /„ + gn {X — (Un) + • • . ) 



It follows that /i, . . . , /„ are not all zero. 



We may now add constant multiples of ii\ (.r), . . . , ?/„ (.r) to the succes- 

 sive rows of Y{x) and obtain a new matrix l'(.r). In fact this is equiva- 

 lent to multiplying Y{x) on the left by a matrix 



