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. 49
introduce an associated set of dominant equations. Let a common region of existence of
fb and P2 be determined by the range
| u\<p, |v |<o-, | |t r;
within this region let the greatest value of i be M, and that of 1 21 be N, so that
within the region
1 + 1 I, l -YlN,
M and N denoting finite magnitudes. Also, let e denote the smallest value of | m- for
values of the integer m; and let c = c. Then we consider the dominant equations
given by
* eX Mt M
eX =-M t + if XiYjtk
eY = cX + - t + i X YJtk,
r pi J rk
where the summations on the right-hand side are for integer values of i, j, k such that
i+j +k2.
The general course of the argument is similar to that in I (a). In the first place, X
and Y can be determined by the simultaneous equations
eX = M -- X-Y
( r) ( p( ) P
(l- E(\ r) (\ p) ) (i
N N N
Y=cX+- - N- -— Y
-- y P
From these we have
NX =M(eY- cX),
so that
Y=X ( e
when this value is substituted for Y in either equation, say in the first, we have
/ M N cM\^(. X\ N X N c
e+- +-+ X - 1 — )1 -- + -}
p -a 6e P - M
--— r-M
t p
r
a cubic equation in X. The term independent of X vanishes when t = 0; and the term
involving the first power of X does not vanish when t = 0, because e is not zero. Hence
the cubic has one (and only one) root vanishing when t=0.
VOL. XVIII. 7