90 THE ROYAL SOCIETY OF CANADA 



modular factorisation of F (s, v) after the manner of that given in (8) 

 for/(2, u) in the special case which we there had in view. Retransforming 

 F{z,v) to f{z,u) on writing v = u — p the modular factorisation of F{z,v) 

 conserves itself and the modular factorisation of f{z, u) in (8) may then 

 evidently be taken to hold good in all cases where the roots of /(o, u)=0 

 are not all equal, the constant terms in Qg, qi, . . . , q^-i however being 

 no longer assumed to have the value 0. 



Let us now consider the case in which the roots of the equation 



f{o,ii)=Oare all equal. On writing u=v— — f„_i we transform 



n 



f{z, u) = into an equation 



where the coefficient of v"~^ is 0. The roots of the equation g{o,v) =0 

 will evidently all be equal. Also the common value of these roots is 

 since their sum is 0. In the equation (9) then the coefficients ^„_2, ■ ■ • , g© 

 must all have the value for 2 = and must therefore each be divisible 

 by some positive power of z. 



Putting v = z'^y in (9) and dividing through by s"" we arrive at an 

 equation 



10. G(s,>.)=/+g„_2S-'''/-'+. . .+goZ-'"' = 0. 



Indicating the lowest exponents in the series g„-2, ■ ■ • , go^Y "■n-2, ■ ■ ■ ,<^o 

 respectively let us choose a so that none of the numbers 



11. a„_^ — ra;r = 2,S,...,n 



is negative and so that one of them at least is 0. The coefficients of the 

 powers of y in G{z,y) then involve no negative exponents. Also the 

 roots of the equation G {o, y) = do not all have the value and are 

 evidently therefore not all equal to each other, since their sum is 0. By 

 what precedes then there will be a factorisation of G{z, y) analogous to 



that of /(s, w) in (8) . Writing 3' = s~"i' = s~"(w+ — /„_i) and retrans- 



n 



forming the factored G{z, y) to terms of u we shall evidently have, in the 



case here in question also, a factorisation of/(2, u) of the type given in (8). 



In all cases then we may suppose the modular factorisation of /(s, u) 



given in (8) to hold good the constant terms in q^, Çi, . . . , qk-i however, 



as already indicated, being no longer assumed to have the value 0. 



To the factors in (8) we can apply the factoring process employed 

 in what precedes. By repeated applications of this process we can 

 replace the right hand side of (8) by a product of n factors linear in u 



