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.

46 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
so that (i - Kc) q vanishes for I = fc, we must have the coefficient of tK in k (u', v', t) zero.
If this coefficient be zero, the value of qK is undetermined; let a value f/, provisionally
arbitrary, be assigned to it. For the remaining values of l, the equations determine
formal expressions for the remaining coefficients, involving a and /: and no further
formal conditions need to be imposed. When the values of _p,..., p,_, q,..., q,- are
inserted in k (u', v', t), the coefficient of tK in that quantity may be an identical zero;
in that case, the functions u', v' involve two arbitrary constants a and f3 so that, if the
functions actually exist, there is a double infinitude of regular solutions vanishing with t.
Or the coefficient may be zero only if some relation among the constants of the original
equations be satisfied; if the relation is not satisfied, there are no regular integrals of
the original equations vanishing with t: if the relation is satisfied, there is a double
infinitude of regular integrals. Or the coefficient may be zero only if some relation
among the constants of the original equations and a be satisfied; this relation is then
to be regarded as determining a, and then for each value of a so determined, there is a
single infinitude of regular solutions vanishing with t.
These results are stated on the assumption that the power-series, as obtained with
the coefficients p and q, converge: the assumption can be justified as follows. Let
AK._ = p1t + p2t2 +... + p,-lt —1,
BK_ = qt + q2t2 +...+ q-_lt-l,
the coefficients p and q being known; then if functions u' and v' exist of the specified
form, we can assume
u; = A_-1 + t-1 U',
v' = BK-1 + t<-1V',
where U' and V' are regular functions of t that vanish with t. Thus, assuming a=0,
we have
t dt- + tK dt + (K - 1)tK-1U' = A,, + tK-1U' + h (A _- + t-1,U BK- + tK- V t)
Now the quantity
dA _
t dt 1-AK-1
dt
is equal to the aggregate of the terms involving t, t2,..., tK-l in
h (AK1, BK-, t).
Also in h (u', v', t) there are no terms of dimensions lower than 2 so that, in
h (AK_1 + tK-1 U, BK-_ + tK- V', t) -h (AK1, BK-1, t),
the coefficients of tK- U', tK- V' are of dimension at least unity, and therefore this
expression may be taken as equal to
tK- Hi (U', V', t),