142 Transactions of the Royal Canadian Institute 



In the case of any polynomial H{z, u) in u with coefficients of rational 

 character in z we can write 



r 



18. B.{z,u)^ S Gs{z,u)fi(z,u) .../,- 1(2, u)fs+i{z,u) . . .fr{z,ti), 



s=l 



mod f{z,u), 



where Gi(z,u), . . . , Gr(z,u) are polynomials in u of degrees 

 I'l — 1, . . . , v^ — l respectively. The values of the function H(z,ti) are, 

 as we know, given by the summation on the right-hand side of the 

 congruence when u satisfies the equation /(s, zO =0- 



In the case of the general function {{z,u)) we can, in analogous 

 manner, evidently write 



t 



19. ({z,u))= S gs(z,u)Mz,u) . . .f,^iiz,u)fs+i(z,ti) . . .ft(z,u), 



s = l 



modfi(z,u) . . .ft{z,u), 



where gi{z.u) , . . . , gt{z, u) are polynomials in « of the respective degrees 

 vi—l,...,vt — l with coefficients which are of rational character rela- 

 tively to the value 2 = 0. 



Let us now assume that we are imposing on the coefficients of the 

 general function ((2, u)) the conditions requisite to adjointness relative 

 to the equation /(2, m) =0, not simultaneously, but successively for the 

 branches of the several cyclical equations /i (2, w) =0, /2(2,w) =0, . . ., 

 fr{z,u)=0. Suppose we have imposed these adjoint conditions 

 successively for the branches of the cyclical equations fi{z, u)=0, . . . , 

 /t_i (2,^0=0- Let us consider the congruence (19). By virtue of the 

 orders of coincidence in question it is evident that among other restric- 

 tions on the /-I polynomials gi{z,u), . . . , g<_i(2,z/) they will have 

 coefficients which are of integral character for the value 2 = 0. Because 

 of the integral character of ((2, u)) it will then follow from the congruence 

 that the coefficients of the polynomial gi{z, u) must also be of integral 

 character relative to 2 = 0. It is also evident that the orders of coincidence 

 in question impose no further conditions on the polynomial gt{z, u). On 

 considering the /th element in the summation on the right-hand side of 

 the congruence we see that with the imposition on the function ((2, u)) 

 of the adjoint orders of coincidence corresponding to the branches of the 

 ^-1 cyclical equations /i (2, ?i) = 0, . . . , ft-\{z,u)=0 we have at the 

 same time imposed on the function an order of coincidence with the 

 branches of the cyclical equation /<(2, «)=0 which is equal to the order 

 of coincidence of the product /i (2, m) . . .Jt-i{z,u) with one of these 

 branches. 



