, THE ROYAL SOCIETY OF CANADA 
in terms of the A’s with lower suffixes and of the coefficients hy in 
the power-series /y (z). Evidently the expression for A, is polynomial 
in Ay, À....À,-1 and linear in terms of the coefficients hy, Here 
we have A,;=/,,. and it is furthermore plain that we can now 
successively express À, A3,....As ....in terms of the coefficients hy¢ 
alone by aid of additions and multiplications only. The question is 
whether the series (35.), with the coefficients A, so determined, is 
convergent. 
Since the series h,(z) on the right-hand side of (34.) all converge 
for the value s=1 we can name a finite positive number A which is 
numerically greater than any of the coefficients in these series. If 
now we replace the equation (34.) by another of like form in which 
the coefficients of the series are all positive and each one numerically 
greater than the corresponding coefficient in (34.) and if this new 
equation is satisfied by a convergent power-series 
30. 7 = Zt po g2+ Botbrere His esse 
it will follow that the series in (35.) with the coefficients A; is also 
convergent, for evidently y, >|A,| for all values of s, since p, is 
built up by additions and multiplications out of positive numbers, by 
the same law by which A, is built up out of the coefficients hy, of the 
series h»(z) in (34), each number involved in the structure of ps 
however being numerically greater than the corresponding number 
Nog in Xs. If in particular we should replace each of the series h (2) 
in (34) to the series A(1+2+2?+ = A(1 —2) 1 the coefficients ps in 
the series (36), assumed to formally satisfy equation (34), would be 
numerically greater than A,. If at the same time we should regard 
the sum in parentheses in (34.) as containing terms of higher power 
than 7” each with the coefficient o and if we should replace each of 
these coefficients also by the series A(1+2+....) the effect would be 
to increase some of the coefficients u, in the series (36.), assumed to 
satisfy the resulting equatlon, and would decrease none of them. If 
then the series (36.) is assumed to satisfy the equation 
87. q=2{A(L+e+... d+ Qasr... JF } 
the required coefficients ws will be ah numerically greater than the 
corresponding coefficient A; in the series (35.). In order to prove 
the convergence of the series (35.) it will therefore suffice to establish 
the existence of a convergent series of the form (36.) which satisfies 
the equation (37.). This equation however we can evidently write 
in the form 
BAS 1 re 
38. 7 7) 
whence 
er” Boas Az cE 
TANT ASH 
and one of the solutions of this quadratic equation is the function 
39. H=4-2{1-(8441)2}) {14(A-Dz}%, 
which function can evidently be expanded in a power-series of the 
