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.

62 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
Since no one of the quantities
(m - l) +n +p, m l+(n- l) +p
vanishes for integer values of rn, n, p such that m + n + p ~ 2, it follows that no one of the
quantities
n+ m,, n + (m +1)- 2
vanishes for integer values of m and n such that m + n > 1. Hence when the coefficientequations for k and 1 are solved in groups for the same value of m + n, and in
successive groups for increasing values of m + n beginning with 1, they lead to results
of the form
kmn == rnnn, mn == Xmn
where y and X are integral functions of the coefficients that occur in fi and f,, each
divided by a product of factors of the forms
n + mi,, n + (m + 1)-.
Moreover each of the coefficients k and 1, thus determined, contains A as a factor.
It now is necessary to prove that the series for p and r, the formal expressions
of which have been deduced, are converging series. For this purpose, we construct
dominant equations as follows.
Let a region of common existence of the functions Jr and p2 be defined by the
ranges tl r, Iz l ri, p l a, 1 r /3: so that "1 and f2 are regular functions of their
arguments within these ranges. In this region, let M1 be the greatest value of [ i1
and M2 the greatest value of 1 1: let M denote the greater of the two quantities
M1 and M2. Further, since the quantities n + mnl, n + (m + 1) f - 2 do not vanish for
integer values of m and n such that m + n>1, there must be a least value for the
moduli of the quantities for the various combinations of m and n; let this value be
v, so that
n 4 men!??, | n+ (m + 1) ti- 2 1,
in all instances. Also let A =A'. Then the dominant equations are chosen to be
(P-A') = ST
1M M______ zP ZPMZT,
Z, - i ^V - ti rio rl >
(l r,(1 o r
Clearly P-A'=T': their common values are given as the roots of the cubic
equation
(+ -/+ - T+ 1 - 1 _ T (1 _ 1A _ T
(V rl le r^S- r+ o r r )T ri aa rla
1 r \r r,3 raa
7'