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. 39
In both forms, the functions b, and 02 are regular functions of their arguments
and vanish with them; and the only term of the first order in qb and 52 is possibly
a term in t. For both forms, the initial conditions are that u= O, v = O, when t =0.
For brevity, integrals, which are regular functions of t, will be called regular
integrals: and integrals, which are not regular functions of t but are regular functions
of quantities that themselves are not regular in t, will be called non-reqular integrals.
The results are obtained for the transformed equations in u and v; since U and V
are linear homogeneous combinations of u and v, the results apply to the original equations.
REGULAR INTEGRALS.
CASE I (a): the critical quadratic has unequTal roots, neither being a positive integer.
1. If the equations
du dv A
t - = e6 + 01 (u, v, t), tdt =- v + 02 (u, v, t),
possess regular integrals vanishing with t, these integrals must have the form
00 00
U = Y ant, V = V bnt.
n=l n=l
That they may have significance, the power-series must converge; that they may be
solutions, they must satisfy the equations identically.
Accordingly, substituting the expressions and comparing coefficients of tn, we have
(n - e1) an =fn, (n - 2) bn = gn,
where f, and g, are the coefficients of tn in 01 and 02 respectively after the expressions for ut and v are substituted. From the forms of <j and b2, it is clear that f,
and g, are linear combinations of the coefficients in 0j and n,, that they are rational
integral combinations of the coefficients a,, a.,..., b,, b2,..., and that they contain no
coefficient a after a,,_ and no coefficient b after bn,_ in the respective sets. Since
neither e nor:2 is a positive integer, the equations can be solved in succession for
increasing values of n, so as to determine formal expressions for all the coefficients.
In particular, a,, and b, are obtained each of them as sums of quotients; the numerators
of these quotients are integral algebraical functions of the coefficients in b, and 2,,
and the denominators are products of powers of the quantities