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. 65
where 7rlmn, ti'mn are linear functions of the coefficients in fj and f2, and are integral
functions of the coefficients pl'm'n' and qi'm'n', such that
1'.l, m'm, tn'n, +I'^+m'+n< l+m, +n.
Owing to the forms of f and f,, we have
7qr000 = 0 'oo 000 = 0.
Hence pooo = 0, and qooo is left undetermined; we take
qooo = B,
where B is an arbitrary constant. Moreover, no one of the quantities
n +(l-1) f +m, m, n+li +(m-1),,
vanishes for values of 1, m, n such that n + + m > 2; hence in the equations for
Plmn, qim, no one of the coefficients of plnn, qlmn vanishes when n + m 1. Hence
these equations can be solved for all the coefficients p and q after po00, q000. They
are most conveniently solved in groups for the same value of n + + m, and in
succeeding groups for increasing values of n +1 + m, beginning with 1; the results are
plmn = '7lm,,n qmn = Klmn,
where 7rmn, tclmi are sums of integral functions of the coefficients in fi and f,, each
divided by products of factors of the types
n +(l- l) e + (m + 1)e2, n+ I + mf.
Expressions thus are obtained as formal solutions of the equations: it is necessary to
establish the convergence of the infinite series. As before, we construct dominant
equations for this purpose, as follows.
Let a common region of existence of the functions f, and f2, which are regular
in their arguments, be defined by the ranges
t < r, | zl |< pi, |Z2 < p2, |~l|1 O 2|, 2
and within this region, let N denote the maximum value of Ii and jg', so that N
is a finite quantity. Also let V denote the least among the values of
|n+(l- 1)l+(m+1)el, | n + le +me2\,
for the various combinations of the integers 1, m, n such that I + m + n >l 1; and let
B = B'. Then the dominant equations to be considered are
= vr (q(2 - B')
i 0 N
N Nz, _, Nz2(i)_
(l _ t)( _ ) p2o'1 P2;2 j
VOL. XVIII. 9