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. 83
hence, comparing coefficients of t10m O n on the two sides, we have
(I + m + an - a) hl - (m + 1) hil,,. l,- (n + 1) h1,m_,n+l- + (m + 1) fhl, m+l,n- = xa'm,
Similarly
(I + m + an - p) klm, - (m + 1) ki_ m+ n - (n + 1) Ci, m-, n+ + (m + 1) kI, m+,n_ = 3imn
Here amn, /31mn are expressions which are linear in the coefficients A,j Bj respectively,
being an aggregate of terms of the form
Nl Aijphlm,n,... hlim, 'n,'. l,, 'm./n/'
1 l Bijphllnlnl,.. hliminim kllln,'... ki'm'/n/
respectively; the subscripts are subject to the relations
mi +... + mi. m1 +... + mf' = mr
-n, -... + ni + nl: '... + nj' = n
î+ 1+...+ li+ 1,'+...+ Ij/=
and the numerical factor N1 is the number of integer-solutions of these equations.
In particular, we have
hooo = 0, ooo = 0.
When + r + n= 1, the equations for the coefficients in X are
(1 -o-)hoo h0o = — A,
(1 - o) holo- hool = 0,
(a - -) hoo, + holo = 0,
which are satisfied by
h010 = (1 - ) h100oo + A!
hoo = (1 - o) holo J
and hloo is arbitrary. Similarly,
k010 = (1 - p) koo + 8
^001-( L -p) Polo J
and kjoo is arbitrary.
When l+ m + n = 2, the equations for the coefficients in X are
(2 - a) hoo - ho,, = h o 2
(1 + a - o-) h011 + 2h0o2o0 - h002 = a'o01
(2a - o-) hoo2 + /lholl = a'oo0
(2 - r) h10 - 2ho20 - h101 = a'nlo
(1 + a - o) hïoi - ho011 + 3h1o = a'o0.
(2 - o-) h0 oo- h110 = 2'oo
11-2