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. 81
A comparison of the coefficients of l4vmtn in,1 and 20 with those of the same
combination of the variables in (I and V2 is easily seen to lead to the inference that
the moduli of the former are less than the modulus of the latter; consequently the
former series converge and therefore integrals of the equations, defined by the specified
conditions, are proved to exist. Their explicit expressions, as power-series, are obtained as
in ~ 11.
CASE II (b): the critical quadratic has a repeated root which is a positive integer.
18. Denoting the repeated root by m, the equations are
du
t =u + + 0 (u, v, t)
t, = KU + mv + ft + (u,, t, )
where the functions 0, 4 are regular, vanish with u, v, t, and contain no terms of
dimensions lower than 2 in their arguments.
The equations can be transformed as before (~ 5) by the appropriate substitutions;
and this transformation can be effected - 1 times, leading to new equations of the
form
tdt + = ti + at + 01(t1, t2, t)
t t2 = ct+ t++ 0+ bt+ (t,, t2, t)
where t, and t2 are to vanish with t; and 01, 02 are of the same type and properties
as 0, > in the first form.
There are two sub-cases according as K is zero, or K is not zero.
19. First sub-case: Kc=0. The equations can be taken in the form
t d= x + at + 01 (x, y, t)
tdt =y+bt +0(x, y, t) j
the integrals are to vanish with t; and the functions 01, 0, are regular functions of their
arguments, which vanish when x = 0, y = 0, t = O and contain no terns of order lower
than 2 in x, y, t combined.
The integrals vanishing with t are defined by the theorein:
The equations possess, in general, a double infinitude of non-regular integrals vanishing
with t, which are regular functions of t a.ud t log t; and it is known that there are no
VOL. XVIII. 11