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. 47
where H1 is a regular function of its arguments, which vanishes with them and contains
no terms of dimension lower than 2. Also let the terms in h (A,_1, B,_,, t) of order higher
than K - 1 be
cKtK + cK+1 t+ +...;
then
dU'
t - + (c - 2) t-U = cKt" + c,,+,t"+ +... + t-1 H ( U', V' t),
and therefore
au'
dt = (2 - ) + + H(UI, V', t),
on absorbing the other powers of t into H1 and denoting by H the new function which
has the same character as H1. Similarly
d V'
t += V'+ bKt+K( U, V',t),
where the terms in k (AK_1, BK_i, t) of order higher than K - 1 are
bKtK +...,
and K is a function of the same character as H.
As K is a positive integer > 1, 2 - c is not a positive integer 1. Thus the coefficient
of U' is not a positive integer, while the coefficient of V' is unity; and thus the two
equations for U' and V' are a particular instance of the general form discussed in I(b).
There is no regular integral vanishing with t unless b" = O; the significance of this
condition, either as an identity, or as a relation among the constants of the original
equations, or as an equation determining a, has already been discussed. Assuming the
condition bK = O satisfied, it is known from the preceding result that the equations in U'
and V' possess regular integrals, which vanish with t and involve an arbitrary constant
that does not appear in the differential equations. The inferences stated earlier are
therefore established.
It appears from the investigation that two conditions must be satisfied in order to
the possession of regular integrals: one of them is a relation among the constants of the
equation represented by a = 0: the other of them is the relation represented by b, = O.
To obtain them directly from the original forms, we can proceed as follows. Let
n = pIt, v= qltl,
1=1 1=1
be substituted in the original equations: and determine p,..., pi-_, q1,..., q_. The first
condition is that the coefficient of tm in
/ (-1 -1 \t