252 THE ROYAL SOCIETY OF CANADA 



e element I/2 respectively. Bearing in mind the identity (2), we 

 then derive from (10) and (10a) the identities 



and 



(12) II{z,ii) = {z-a)°-G{z-a,u) 

 (12a) H{z,u) = {\/z)°-G{\/z,ii). 



The lowest exponent in the coefficient of «"~^ in the reduced form 

 H{z, ii) must then be > —1, if the orders of coincidence of the func- 

 tion for the value z = a (or s= <^) are severally greater than the corre- 

 sponding numbers /xi — 1, . . . , /û„ — 1. The coefficient in question 

 however being rational, its lowest exponent cannot be fractional and 

 must therefore be>0. In the reduced form of a rational function 

 which is adjoint for the value z = a (or 2=°°), it follows that the 

 coefficient of u"~'^ must be integral with regard to the element z — a 

 (or I/2). 



If the equation (1) is integral relatively to the element z — a, the 

 series P\, . . . , P„ corresponding to the value z = a are also integral 

 relatively to this element. In this case, representing in the form (4) 

 a rational function of (2, w), which is not only adjoint for the value 

 z = a, but whose orders of coincidence with the n branches u — Pi = 0, 

 . . . , u — P„=0 are greater than the numbers mi, . . . , AT,, respectively, 

 we successively deform the product on the right-hand side of (4) into 

 the products (8) and (9) precisely as before. As before, the order of 

 coincidence of the product (9) with the branch u — Pk =0 will be at 



least as great as the order of coincidence of the product (4) with this 

 branch, and must therefore in this case be > ai,^ . It follows that the 



order of coincidence of the factor u — Qt with the branch in question 

 must be>0. As before however, the order of coincidence of the 

 factor w — (2/ with the branch u—Pt =0 is at least as great as its order 

 of coincidence with the branch u — P^ =0, and must therefore be 



>0. We can then write Qt =Pt -\-{z — a)°'tSt whereas is >0 and where 

 St is a series in powers of 2 — a none of whose exponents is negative. 

 The identity (4) will then take the form (10) the exponents at in the 

 case here in question being all>0. 



Effecting the multiplications indicated on the right-hand side of 

 the identity (10), it is clear that all terms in the result which are not 

 divisible by a positive power of 2 — a are included under the product 

 {u — P\) . . . {u — P„). On subtracting this product from the right- 

 hand side of the identity and f{z, u) from the left-hand side, we see 

 that the reduced form H{z, ti) on the left-hand side is identical with 

 an aggregate of terms on the right-hand side, each one of which is 

 divisible by a positive power of z — a. The reduced form H{z, u) of a 

 rational functi(^n is then divisible by a positive power of z — a, and 

 therefore by the factor z — a, if the orders of coincidence of the func- 

 tion with the n branches ïy— Pi = 0, . . . , u — P„ =0 are greater than 

 the numbers /ïi, • • • , iu« respectively. The product of 2 — a and the 

 reduced form of a rational function which is adjoint for the value 

 z = a, will have orders of coincidence with the n branches which are 

 greater than the numbers 'jli, . . . ,]!„ respectively, and must therefore 



