Conditions Imposed by a Set of Order Numbers 139 



number of the conditions which reduce ((z, «)) to the form ((z,u))a 



n-l r 



is K= S Ks. We have X + '<= 2 (yu^ — 1 + 1 (f 5)1;^ since this, as we know, 



s=Q s=l 



is the number of the conditions which reduce z~\{z,u))q to the 

 adjoint form ((s, u))a- 

 The products 



14. z''"-'-^'-'^Rn-sPs-i ; ^=1. 2,. ..n 



are all adjoint relatively to 2 = 0. Also the products R„^sPs-i are all of 

 degree n—l'mu with 1 for principal coefihcient. The order numbers of 

 these latter products for z = cannot therefore be simultaneously greater 

 than /ii, . . . , Hr- It follows that none of th^ differences /c„_j — X,_i can 

 be negative. As a consequence we have /c > X. 



We impose on the coefficients of the function ((z, u)) the conditions 

 requisite to adjointness on equating to the principal residue relative to 

 2 = in the product 



z~\(z,u))o {{z,u)). 

 Since X is the numbei of the arbitrary constants in the principal part 

 of z~'((2, m))o the number of the conditions requisite to adjointness on 

 the part of ((z, 11)) is < X. But the number of these conditions is k. 

 We therefore have k<'X. Since it has been ptoved that k>X we 

 derive /c==X. Now the number of the conditions which reduce the 



r 



form z~'((z,«))o to the form ((z, w))a is X + /c= S {tx^ — I -\- l/v,)Vf. 



s=l 



We consequently have 



r 



15. X = K= 1 S (^^_i + i/j,J„^. 



s=l 



§3. From the equality /c = X and the inequalities k:„_^ > X^-i , obtained 

 in what precedes, we derive k„_5 =X,_i ; s = l, 2, . . . , n. It would 

 suggest itself to arrive at our formula by proving these equalities directly. 

 We shall not here attempt the detail involved in the consideration of 

 the general case but shall for the moment restrict ourselves to the special 

 case where /(z, u) =0 presents only one cycle corresponding to the value 

 z = 0. The order number of /m(z,m) we shall designate by fj,. A poly- 

 nomial in u with coefficients of rational character relative to the value 

 2 = 0, must, if it is adjoint for this value of the variable z, be of integral 

 character relative to 2. Also it must be divisible by z if its order number 

 is greater than //. 



In the products (14) for the case here under consideration we may, 

 as has already been pointed out, assume p, = R,; s = 0, 1, . . . ,n — l. As 

 has already been noted, we may also suppose these functions to have 



