Memoirs presented to the Cambridge philosophical society on the occasion of the jubilee of Sir George Gabriel Stokes, bart., Hon. LL. D., Hon. SC. D., Lucasian professor.

DIFFERENTIAL EQUATIONS. 69
By the theorem of ~ 10, we know that solutions exist, which vanish with t and
are expressible as regular functions of t and t'. Let a new variable 0 be introduced,
defined by the equation
t- t = (1 - -) 0;
and, in the solutions indicated, replace t' by t + (1 - ) 0; they then become regular
functions of t and 0, expressed as converging power-series. To obtain their coefficients
in this form directly, let
X = E a,m Omtn,
Y = - bmn Omtn,
where ao,=O, boo= O; then since
dO
t = a-O-t,
we have
dX
dt = 22 a~, {nOmtn + m-1 t (a- - t)}
= 2$ {(n + a-m) Omtn - mn-l1tn+1} amn,
and
tdY
dt = 2- {(n + o-m) omtn - mo-1tn+1} b.
Substituting in the differential equations and comparing coefficients, we have
(n + o-m - o-) a,, - (m + 1) a,+i, n-i = Hm, n
(n + -rnm - c) bmn - (m + 1) bm+i, n-l = Km, n
where H., and K, n are sums of terms of the form
Imn = NAYip,v jamn... amn,, bmi'n/'..* bmnj',
and similarly for Km, n, such that
i-+j+p>2
m n+... +n mi + m+... + m' = m
p + ni +... + ni +... + nj = n
N being a numerical quantity, representing the number of integer solutions of the last
two equations.
As regards the initial coefficients, we have the following expressions.
For m + n =, so that m = 0, n = 0; then
aoo = 0, boo = 0.
For m+n=1, so that m=1, n=0: and m=O, n=l; then
O.a = O, (o —t)b0 = O;
(1 - or) cxoi - aio = - A, (1 - K~) 601 - 610 = î 5;