THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 
349 
where x (2, u) is the reduced form of the product on the left-hand side of this identity. 
The factors d) (s, u) and (2, u) of this product are also supposed to be expressed in 
their reduced forms, so that the degree in u of the product is ^ 2 n —2 and the degree 
of 3 -( 2 , in u as a consequence is 2 . If d> (2,represents the most general 
function of (2, w) of rational character for a given value z — a (or 2 = co ) conditioned 
by a given set of orders of coincidence tj, t,, for this value of the variable 2, the 
vanishing of the corresponding residue in the principal coefficient of x gives, in 
the case of a finite value 2 = «, the necessary and sufficient condition that the orders 
of coincidence of the function (2, ii) for this value of 2 should be complementary 
adjoint to the orders of coincidence while, in the case of the value z= 
the vanishing of the corresponding residue in the principal coefficient of x {z, a) gives 
the necessary and sufficient condition that the orders of coincidence of the function 
d'(2, for the value 2 = co sliould be complementary adjoint to tlie order 2 to the 
orders of coincidence ..., t,.. 
§ 3 . We shall now assume the equation (l) to bean integral algeliraic equation. 
ITie series representing the luanclies of the equation for any finite value z = a will 
then involve no negative exponents. In the representation of a rational function 
H {z,u) in the form ( 8 ) corresponding to the value 2 = a, the functions Q. (2, u) will 
therefore evidently be integral with regard to the element z — a. If the function 
H (2, u) be adjoint for the value z = a it is readily seen that it must be integral with 
regard to the element z — a. For in this case the lowest exponent in each of the 
series 8 ^ in ( 8 ) is > —1 and the same is therefore true of the lowest exponent in each 
of the coefficients of the several products (2, u). It follows that, in the coefficients 
of the rational function H (z,u) represented by the sum on the right-hand side of ( 8 ), 
the lowest exponent is ^ 0 . A rational function H (2, m), which is adjoint for the 
value 2 = a, must then be integral with regard to the element z — a. Furthermore, a 
rational function of (2, u), which is adjoint for all finite values of the variable 2, must 
evidently be an integral rational function of (2, u). 
While a rational function H (2, u) must be integral with regard to the element 
2 —a if its orders of coincidence are to be adjoint for the value 2 = a, divisibility* by 
z—a is required from it by a set of orders of coincidence tj, ..., corresponding to the 
value 2 = a when these orders of coincidence severally exceed the corresponding 
numbers jUi, ft-r —but not otherwise. If, namely, the orders of coincidence of H (2, u') 
severally exceed the corresponding numbers /xj, ..., /x,., the quotient of the function by 
z — a will be adjoint for the value 2 = «, and must, therefore, be integral with regard to 
the element z—a. If, however, a single one, r^, of the orders of coincidence which 
condition the rational function H (2, u) is not greater than the corresponding number 
then in the representation of the general function in the form ( 8 ) the corre- 
* We here find it convenient to say of a rational function of (?, \C) that it is divisible by the element 
2 - a if the function can be represented as the product of 2 - a and a function of (2, u) in which the 
coefficients of the powers of u are power-series in 2 — a not involving negative exponents. 
