116 



THE ROYAL SOCIETY OF CANADA 



On eliminating the n— 1 functions g , g h ,gn-i from the 



n — 1 equations (3) and the equation 



5. S i4»-<-i B t = S /»-<_! gj + C M _i 



selected from the equations (4), we obtain an expression for C n -\ in 

 the form 



6. C„ 



(-1)"' 



i4„_i 5„_i /„ o o 



2 ^2n-«-3-B< /„_i /„ 0, 



An-tB, ft /> 



2 ^„_/-i 5, /, / s 

 / = i 



./,-! 



In order then that the function B(z, u) maybe adjoint for all finite 

 values of the variable s its coefficients B , Bi, . . . . ,J5„_i must be sub- 

 jected to those conditions which suffice to make the expression on the 

 right-hand side of (6) integral. Now the general rational function 

 A(z,u) of integral algebraic character can be expressed linearly in 



(O) (M-l) 



terms of a basis A (z,u), . . . .,A (z,u) with multipliers which are 

 integral rational functions of z. Furthermore we know that the 

 functions of the basis can be chosen of the forms 



,(»-l) 



and where in 



(k) (Jfe) k (k) k-l (A) 



7. A (z,u)=A k u +A k -\ u + +A ,k=<o,l 



(k) 

 where the coefficients A s are rational functions of s 



(o) (o) 



particular we may take A (z,u)=A = 1. Now in the product 

 (2) the coefficient C n -\ is plainly integral if we have integral co- 

 ca) 

 efficients C n -\ in the several products 



A (z,n) . B (z,u), k = o, 1, ,(«-!). 



