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.

50 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
It follows, as before, that this root of the cubic vanishing with t can be expressed as a
regular function of t in the form of a power-series, which converges absolutely for values
of 1t1 less than the least modulus of a root of the discriminant, that is, for a finite range.
When the power-series for X has been obtained, the power-series for Y is given by
say
X = Pt + P2t2 +... + Pntn +...
Y= Qt t+ Qt2t2+.. + Qtn'+
In the second place it can, as before, be shewn that the analysis, effective for the
determination of pn and qn in connection with the original equations, is effective for the
deterinination of P, and Qn in connection with the dominant equations by merely making
literal changes, and that these literal changes are such as to give
Pn < Pn, I qn < Qn
for all values of n. It therefore follows that the series
plt+p2t + p3t +....
qlt + q2t2 + q3t3 +...,
converge absolutely within a not infinitesimal region round t = 0. Consequently the
equations possess regular integrals vanishing with t: and it is not difficult to prove that
these regular integrals are unique as regular integrals with the assigned conditions.
CASE II (b): when the critical quadratic has equal roots, the repeated
root being a positive integer.
5. The equations are
du
t Ku = m v + + v + (t + (,,t)
tdt =ilu+mv+j3t+ (U, v, t)j
where mn is a positive integer, the functions 0 and b are regular, vanishing with u, v, t,
and containing no terms of dimensions lower than 2.
We transform the equations as in I (b) by successive substitutions, each of which
leads to new equations of a similar form with a diminution by one unit in the
coefficients of u and of v after each operation. We take
u = t (X + U1), v = t (,/ + vI),
choosing X and pU so that u1 and vI vanish with t: then ul and vI are regular functions
of t, if the equations possess regular integrals. To secure this form of transformation,
we must have
(m- 1)X +a=0,
kX + (w - 1) f + /3= 0,