202 THE ROYAL SOCIETY OF CANADA 



19. TsT^s = Us —fJ'(T,s — 1 + 1/vs , s = l, . . ., r. 



On designating by H{z, u) the general rational function conditioned 

 by the set of orders of coincidence n, . . . . , r^ , it is easily seen that the 

 conditions imposed on a rational function H{z, u) by the set of orders 

 of coincidence t"! ,. . . ,t^ are obtained on equating to the {n — a)th 

 residue relative to the value s = o in the product H (z,u) H (z,u). 

 The statement holds good for the value 2= co when one replaces the 

 numbers ^5 -M<r,5 -l + l/i's in (19) by the numbers ms -Ma.s +l + l/»'i . 

 Instead of following the course of procedure indicated in what 

 precedes we might have started out arbitrarily with the function 

 F(r{z, u) as defined in (8). Employing, as convenient, either this form 

 or the form given in (17) consider any two rational functions A(z,u) 

 and H(z, u) connected by the relation 



20. A{z,ii)=An-iu''-^+...^Ao =Ffr{z,u)H{z,u). 

 This relation we can write in the form 



21. ^(z,M)=-(^«£r+...+/^) (h„-iU"-(^-^+...+ha+i) 



for (T < w — 1 . For a = n — l the identity (20) becomes A (z, u) = H{z, u) . 

 Both in this case and in the cases included under (20) we see, on 

 identifying coefficients of w""^ on the two sides of the identity, that 

 we have An-\ =fnh(y = hff. The coefficient An-\ of m"~^ in any 

 reduced rational function A (s, w) is then the same as the coefficient 

 of u^ in the reduced form of the quotient A{z, u)/ F(j (z, u). 



The adjoint orders of coincidence corresponding to a value z = a 

 are those which determine that the principal coefficient in a function 

 A{z, u) shall be integral relatively to the element z — a. The adjoint 

 orders of coincidence then determine that the principal coefficient 

 in the product Fcr (2, u)H{z, u) shall be integral with regard to this 

 element. But the principal coefficient in this product, being the 

 coefficient of u^ in the function H{z, u), the orders of coincidence of 

 this function which determine that its coefficient h(j- shall be integral 

 with regard to the element z — a are obtained on subtracting from 

 the adjoint orders of coincidence in question the corresponding 

 orders of coincidence of the function Fff{z,u). The orders of coinci- 

 dence of a rational function H{z, w) for the value z = a which insure 

 that the coefficient hg- of u^^ in the function is integral with regard to 

 the element z — a are therefore those given in (12). 



