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. 59
solutions of the equations. Accordingly, regarding t, and t2 as functions of these three
arguments, assume
tl = - m'np zlmZ2ntP'
t2 = Zbnnp zjmZ2LtPJ
where the summation is for all positive (and zero) values of the integers m, n, p, with
the conventions
a000 = O, boo0 = O.
Moreover
d a a a
tdt at azx, a+ 2
Hence the differential equations are
at at, at,
t l + lZ + e2 | = ^tl + Sb0(t0, t2, t)
at _z, az,
at + lZ1at + 2Z2 - 2t2 + b2(tl, t2, t)
Substituting the assumed values of t, and t2, and afterwards equating coefficients of
zlmz2ntP, we have
{(m- 1) i + ne2 + } amnp = =amnp.
{n., + (n - 1) 2 +p} bmînp = 3'mp.mp
where 'mnp is a rational algebraical function of the coefficients in 01, of the coefficients
am',,,,p in ti such that
m' m n, n' < n, p' p, rm' ++ n' + p' < n + n + p,
and of the coefficients brn'np, in t2 with the same restrictions: and likewise for 'mn7p
in relation to b2.
As there is no term in bl (t1, t2, t) of dimension unity in t, t 2, t2, there can be
no term of dimension unity in zl, z2, t after substitution of the values of t1 and t2:
hence
{(m - 1) el + n2 + p} tmnp = O,
when m + n + p = 1. Accordingly
a010 =, acOO = 0;
but there is no limitation upon a100, so that it can be taken arbitrarily: we assume
a100 = A.
For similar reasons
{mel + (n - 1) 2 + p} bn2p = 0,
when m+n+p=l; and we infer that
b100 = O, boo0 = O, bo0o = B,
where B is arbitrary.
8-2