MATHEMATICS: F. R. MOULTON 351 
Therefore under these Kmitations not only do the terms of each degree 
in the right member of equations (1) converge, but the whole right 
members converge. 
The particular form of (^"2) was chosen so as to secure by one analysis 
as wide a range as possible of permissible values of For 
example, if infinitely many of the a are bounded from zero the x» must 
tend to zero for i = 0° . On the other hand, if Sc» converges all the 
conditions so far imposed can be satisfied by which are bounded from 
zero. In the latter case 2/i does not converge. 
If an analytic solution of (1) satisfying the initial conditions (Hi) 
exists, it will have the form 
xi = Ai^'H + Ai^'H' + ... = 1, 2, . . .). (6) 
On substituting these series in (1) and equating coefiicients of corre- 
sponding powers of /, it is found that 
=ai (i=l,2, ' ' •), 
2af>=!i^+tg.f, (7) 
where Pi^^) is a polynomial in aj^^\ . . ., ay^""^^ whose coefiicients are 
Hnear functions of the coefficients of /< with positive numerical mul- 
tipKers. Hence the formal analytic solution of (1) is unique. 
In order to prove the convergence of the series (6) for values of / 
whose moduli are sufiiciently small, consider the solution of 
^i = Au^a+j^^ a = 1,2, 
'^ = Ar,{a+^} (i = l,2, •••), (8) 
where 
a = Cot + Ci^i + C2^2 + • • • . (9) 
The right members of (8) dominate the respective right members of (1). 
The formal analytic solution of (8) is 
i, = a?>t + a?>t'+ ■ ■ ■ (i= 1, 2, • • •)• (10) 
The coefficients of these series can be obtained by equations analogous 
to (7). They are therefore real and positive, and it follows from the 
fact that the right members of (8) dominate the right members of (1) 
that 
(^*,y = i, 2, . . 0. (11) 
