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. 77
The terms in t, and t2 of dimension one, obtained as above, are
tl = O, t2 = C+ Bq,
so that, as far as terms of dimension two in ~b and b2 after substitution, we have,b = bt (By + Ct) + k (By + C7)2,
02 = t (By + Ce) + K (Br + CO)2.
Accordingly, for I + m + n = 2, we have
a020 = kB2, a0l = bB, (2 -) ao2 = 0,
a0,, = bC, (1 + ) a110- a020 = 2kBC,:aoo = kC2;
b020 = KB2 + a020, bo11 = B + a011, (2 - ) boo2= Oao2,
bl,, = /3C + Oa1,, (1 +:) b110 - bo02 = Oa110 + 2/BC,
b200 = KCC2 + Oa200;
and therefore the terms in t, and t2, of dimensions two in the arguments, V, t, are
2kBC + B2
J C22 + 1 + B-^ + bB2 + + bBt
in t,: and
(K+ -) - + + +) 2- (+ + Ob) Ct + ( + b) Brt
(K + -- + 2KBC 0 (2kBd + B
l+_ _ + o (1 +: B,
in t2. And so on.
The equations, when solved in groups for the same value of I + m + n beginning
with a zero value of 1, and solved in successive groups for increasing values of
+ m + n, give values of amn, blmn which are sums of integral functions of the literal
coefficients of fj and 0, and of the arbitrary coefficients B and C, each such integral
function being divided by a product of factors of the form n + ( + m - 1). Let the
values thus obtained be
aImn = Clmn, bimn = iP8mn.
As in ~ 9 for the former case, it can be proved that
aoop = O, boop, =
for all positive integer values of p, so that there are no terms in t1 or in t2 involving
t alone; every term involves either ' or r7 or both g and i7.