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.

84 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
The first three equations, when solved, determine ho2o, ho, hoo2; when the values of h020
and h,0o are substituted in the next two equations, they determine h1lo, h,,,; the last equation
then determines the form of h2oo.
Similarly for the coefficients in Y.
For values of l + m + n 2, the equations can be solved in a similar way. They are
solved in groups for the successively increasing values of + m + n. In each group, say
that for which l+m + n= N (so that the coefficients hi'm'n', kl'm'n', such that
l'+ ' + n'; Ni- 1,
are supposed known), the convenient method is to arrange the equations in sets, determined
by the values of i and in sequence according to increasing values of I beginning with 0:
in each set, the equations are arranged in sequence according to increasing values of n
beginning with 0. In each set, we use the equations in succession to express hIlm in terms
of hl,N-l,o and previously known coefficients and constants; when the first N-I equations
in the set have thus been used, the remaining equation, on substitution of the values of
h1,0,N-, hl,i,Nl-l-î, then determines hv-l,o and so also the values of all the coefficients
hl,,,,, such that m + n= V-1. Likewise for the coefficients kmn.
And then, as the solutions are known to be regular functions of t, 0, f, the series
y,:, h,,.mn t O n~,: k^l, t l,
with the values of hln, kimn which have been obtained, converge absolutely.
As regards the forms of the coefficients hlmn, klmn, they are the aggregates of positive
terms T. The numerator of each term T is the sum of a number of positive quantities:
it is an integral algebraical function of the coefficients Aijp, Bijp: it is also an integral
algebraical function of h 4+m+n, ki+m+n such that + + n = 1. The denominator of the
term T is of the form
P + Q/,
where P is the product of factors of the types
+ m + an- o-, + m + an - p,
and where Q is an aggregate of quantities, each positive and similar to P but containing two factors fewer than P.
As regards the number of factors in P, being a part of a denominator in a term
T in hnn or klmn, it can be proved, by an amplification of Jordan's argument quoted
in ~ 11, that this number
I 31 + 2m + n.
It is known that, so long as a and p are different from unity, the convergence
of the power-series is absolute: hence this will be the case when
=l-, p = 1 -,