[fields] adjoint orders OF COINCIDENCE 253 



be divisible by s — a. It follows that the reduced form of a rational 

 function which is adjoint for the value z = a, must be integral with 

 regard to the element z — a in the case where the equation (1) is in- 

 tegral with regard to this element. In like manner a rational function 

 which is adjoint for the value 2=°° must, in its reduced form, be 

 integral with regard to the element 1/s, if the equation (1) is integral 

 with regard to this element. 



Whether the equation (1) is integral relatively to the element 

 2 — a or not, a rational function of (s, u) in its reduced form will be 

 divisible by the factor 2 — a, if the orders of coincidence of the function 

 with the n branches corresponding to the value z = a are simultane- 

 ously sufficiently great. If namely all w orders of coincidence are 

 indefinitel)^ great and if the function is represented in the form (4), 

 each of the n branches n — P = must have an indefinitely high order 

 of coincidence with some one of the factors u — Q, and with one only, 

 since two of the branches cannot simultaneously have indefinitely 

 high orders of coincidence with the same factor. It immediately 

 follows, that the product on the right-hand side of the identity (4) 

 can be Vi^ritten in the form which appears on the right-hand side of 

 the identity (10), the exponents a^, which appear in this latter form, 

 being at the same time all indefinitely great. Subtracting from the 

 right-hand side of the Litter identity the product {u — Pi) . . . , (u — P„) , 

 and from the left-hand side the function/(2, «), we see that the reduced 

 form H{z, ii) is divisible by an indefinitely high povv^er of z — a. It 

 follows that a rational function whose orders of coincidence for the 

 value z = a are simultaneously sufficiently great, will in its reduced 

 form be divisible by the factor z — a. 



From the identity (10a), we can readily see that the degree 

 of the reduced form H{z, u) of a rational function must be <N—\, 

 if the function is adjoint for the value 2= °° . In this case namely the 

 exponents at, as we have noted, are all > —1. Each factor 



u — Pt — {\/z)°-fSf, on the right-hand side of the identity, consists of the 



sum of two elements u—Pt and — {\/z)°-tS^. The former element is evi- 

 dently of degree > 1 in (2, u), the latter element of degree <1 in 2 and 

 therefore of degree less than that of the former element. On efl^ecting 

 the multiplications on the right-hand side of the identity and sub- 

 tracting the product {11— Pi) . . . {u—P„) from the result, we evi- 

 dently remove the terms of highest degree from the total. The pro- 

 duct just referred to however has for its degree N — the degree of the 

 function J{z, u) — as we see from the identity (2). On subtracting 

 the product {u — Pi) . . . {u — P„) from the right-hand side of the 

 identity (10a), the degree of the remainder must then be < N. 

 This remainder is however identical with the reduced form H{z, ii) 

 which appears on the left-hand side of the identity. The degree of 

 the reduced form H{z, ii) must therefore be<A'^, and consequently 

 <N—1 since the reduced form in question is rational. The reduced 

 form of a rational function, which is adjoint for the value 2=<^, 

 must then be of degreeZ^A^— 1. 



With the aid of the product representation given in (4), we might 

 note that we can readily prove the existence of a rational function of 



