138 Transactions of the Royal Canadian Institute 



for the value s = and the general rational function which is adjoint 

 relatively to this value of the variable z. The number of arbitrary 

 constants involved in the expression of the pr ncipal part of the former 

 function we shall indicate by X. 



A complete basis for the representation of the functions included 

 under {{z,u))a we shall suppose to be constituted by n functions 



11. Z""--^ i^„_i (S, U), s'^~-2i?„_2 (Z, W), . . . , z'^'Oi?o(2, u) 



whose degrees in u are w — 1, w — 2, . . . , respectively. The functions 

 Rn-i , Rn-2 , . . . , Ro are understood to be integral in z as well as in u. 

 Also it is readily seen that we may assume the coefficient of the highest 

 power of u in each of them to be 1. We have Rq(z, u) = 1 and kq has for 

 value the greatest of the integers [/xi], [/X2] , . . . , [i"r]- We may, if we will, 

 assume i?„_i (z, w) to have order numbers, relative to z = 0, none of 

 which is less than the greatest of the numbers m, 1J.2, . . . , Mr- Evidently 



Kn-l =0. 



A complete basis for the representation of the functions included 

 under z~* ((z, ti))o we shall indicate by the notation 



12. Z-^«-lp„_i(2,w\ Z-^-2p„_2{z,ti), . . . , Z-\o{z,u), 



the degrees in u of these n functions being n — l,n — 2,....,0 respect- 

 ively. The funct ons p„_i, p„_2, . . • , po are integral in z as well as in ti. 

 We may if we will suppose them polynomial n z and the same is true 

 of the functions R„-i, Rn-2, • ■ • . ^o- The coefficient of the highest 

 power of u in each of the functions p is taken to be 1. We have po(z, u) = 1, 

 Xo = 0. Also X„_i is equal to the greatest of the integers [/xi] , . . . , [Mrj- 



We may write 

 13. 



M— 1 «— 1 « — 1 



((z, u)),= 2 z's Q,R,, ((z, «))= -2 Q,Rs, z-'((z,«)),= S z"^^ Q, p„ 



S=Q 5=0 5=0 



where Qa, Qi, . . , <2«-i are employed to indicate arbitrary series in 

 powers of z which involve no negative exponents. Evidently z = X„_i. 

 We may take p„_i = i?„_i and in the case where our equation /(z, w) =0 

 is constituted by one cycle only relative to the value z = it is readily 

 seen that we may take ps = Rs, 5 = 0, 1, . . . , w — 1. In this case we may 

 evidently select each function Ps = Rs so that its single order number 

 relative to z = is as great as is compatible with its degree in u, the 

 coefficient of the highest power of u being 1. 



The number of arbitrary constants which present themselves in the 



n-l 



principal part of z~*((z,w)) is X= 2 X^. This is the number of the 



5=0 



conditions which reduce 3~*((z,«))o to the form ((z,w))- Also the 



