[fields] 



AN ALGEBRAIC CURVE 



In order that the function B(z, it) may be adjoint for all finite values 



of z therefore it suffices to impose on its coefficients B , B u B n -i 



the conditions requisite to give integral character to the expressions 



A k B„-i fn o 



(*) 



(k) k 



C-i-(-l) 



2 An+k-t-2 B t fn-1 fn 



,(«-!) 



2 A, 

 / = « - k 



l Bi f n - fe+2 fn 



S ,4„-(-l -B< /«-fc4 1 fn-k+1 /«-I 



These expressions are obtained from the expression given for 

 C„-i in (6) by indexing the functions A , A ly ,i»-i and bearing 



in mind that we have A s = o for s>k. If the function B(z,u) 

 conditioned as above be further subjected to the conditions requisite 

 for adjointness relative to the value z = ~ it becomes the general 

 adjoint function. If furthermore we require it to be adjoint to the 

 order 2 relative to the value z= ~ it will become the function 0(z,«) 

 in the numerator of the general Abelian integrand of the first kind 

 4>(z,u)/f»{z,u). In the case where the equation (1) is an integral 

 algebraic equation we know* that a rational function, which is adjoint 

 for all finite values of the variable z, must be an integral function of 



(z, u). In this case then in imposing on the functions B ,B X , ,B„- 1 



the conditions requisite to secure integral character to the 



functions C«_i in (9.) we may in advance take for granted that 

 tbese functions are integral polynomials in z. . If the equation _ (1) 

 is of degree n in (z, u) and if the terms of this degree split up into 

 n distinct linear factors, it is readily seen that adjointness relative 

 to the value z= « on the part of the polynomial B(z,u) simply re- 

 quires that it be of degree < w-1 in (z, u). • Adjointness to theorder 2 

 relative to the value z=°° would require its degree to be < n — 3. 

 In passing from adjointness relative to the value z= °° to adjointness 

 to the order 2 relative to this value of the variable we impose** on 

 the constant coefficients in the function B{z, u), already assumed to be 

 adjoint for finite values of z, just 2« -p conditions, where p is the num- 

 ber of irreducible equations involved in the equation (1). This 



* On the foundations, etc., p. 349. 

 ** Theory of the algebraic functions of a complex variable, p. 155. Mayer & 

 Miiller, Berlin, 1906. 



