88 THE ROYAL SOCIETY OF CANADA 



undetermined save that they are all assumed to be divisible by z. We 

 shall have 



3. f{z, u) = («*+$,_! «*-'+ . . . +go) (zi«-* + <2„-,-i w''-'-^+ . . . +Qo) 



where Qn-k-\, ■ ■ ^ Qo, Sk-i, . . . , ^S",, are power-series in z. Identifying 

 coefficients of w"~\ u*^~^, . . . , u^ on the two sides of (3) we obtain 



n-l 



4. ft=^qk-sQt-k+s; t = n-l,n-2, . . . ,k, 



s=0 



where it is to be understood that g^_, is to be replaced by when the 

 suffix k—s happens to be negative. Since we have qk = l the relations (4) 

 can evidently also be written in the form 



n-t 



5. Qt-k=ft- ^gk-sQt-k+s-, t = n-l,n-2, . . . , k. 



s=l 



Identification of the coefficients of î^*~^, w*~^, . . ., u" on the two sides 

 of (3) gives us 



6. S,=f,- ^q.-sQs-, t = 0,l, ...,k-l. 



We can now readily determine the coefficients in the series g,, > • • ■ , Qk-i 

 and Qo, ' . ■ , Qn-k-i in accord with the relations (5) in such manner that 

 the series St furnished by (6) are divisible by as high a power of z 

 as we will. 



We have assumed that the constant terms in the series Ço, Çi, ■ ■ -, Qk-i 

 are all 0. The identities (5) then evidently determine the constant terms 

 in the series Qo, Qi , ■ ■ • , Ç«_;fe_ito be equal to the constant terms in the 

 series /ft ,/fe+i , . . . ,/„_i respectively. Of importance is the fact that the 

 constant term in Qo, being the same as the constant term in/^, is different 

 from 0. Bearing this in mind and equating to in succession the co- 

 efficients of s in the expressions for S^, Si, . . . , Sk-i given by the identities 

 (6) we successively determine the coefficients of z in g^ Çi, . . . , g^.i in 

 terms of the constant terms in Qo, Qi, . . ., Qk-i- The identities (5) there- 

 after determine the coefficients of s in the series Qo, Qi, . . . , Q„-k-i in 

 terms of the constant terms in these series and of the coefficients of z in 

 the series Qo, qi,- . . . , q^-i. Now equating to in succession the co- 

 efficients of z^ in the expressions for So, Si, . . ., Sk-i given by the identi- 

 ties (6) we successively determine the coefficients of s^ in qo, qi, . . . , qk-\ 

 in terms of the coefficients already determined in these series and in the 

 series Qo, Qu ■ ■ . , Qn-k-i- The identities (5) thereafter determine the 

 coefficients of z- in the series Qo, Q\, . . . , Q„-k-i in terms of the 



