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. 89
l, m, n such that l + m + n 2: and the coefficients Hifn,, /Kimn are determinable as
before. Any term in x is
Htl+'m+n (log t)m+2-,
that is, the index of log t is not greater than twice the index of t.
Note 2. If a vanishes but not b, the solutions are still non-regular functions of
t; likewise if b vanishes but not a. In these cases, it is known that no regular integrals
vanishing with t are possessed by the equation.
If a = 0, b = O, then Hl,= 0, KiM =0, if m > 1: that is, t log t disappears from the
expressions for x and y, which then become regular functions and are the double
infinitude of regular integrals that vanish with t. In this case, the regular integrals
are the only integrals vanishing with t that are possessed by the equation.
20. Second sub-case: Kc not zero.
The theorem is:
The equations possess in general a double infinitude of non-regular integrals vanishing
with t which are regular fuicctions of t, t log t, t (log t); and it is known that there
are no regular integrals which vanish with t. If however a= 0, then the integrals can
be arranged in two sets; one is a simple infinitude of non-regular integrals vanishing
with t which are regular functions of t and t log t; the other is the simple infinitude of
regular integrals vanishing with t which the equation is known to possess. (It is necessary
that the constant K be different from zero: otherwise some of the coefficients in
the second set are infinite unless b also is zero, in which form we revert to the first
sub-case already considered.)
The method of establishment is similar to those which precede: it need therefore
not be repeated after the many instances of it which already have been given.
The initial terms in the integrals of the equations as taken in ~ 15 are
ti = aO + At+...,
t, = Kca + (KA + b) 0 + Bt +...,
the unexpressed terms being of higher order in t, 0, he: here A and B are arbitrary,
0O=tlogt, and = t (log t)2. Any term in the expansion of t, or t2 which involves <
contains K in its coefficient; the disappearance of the terms in () from the integrals
in the first sub-case is thus explained, for Kc then is zero.
Concluding Note.
21. Some sub-cases still remain over from Case I(a), when the roots el and:2 of
the critical quadratic do not satisfy the conditions that (~ 8) prevent some one (or
more) of the quantities
(X- 1)VO+L. +I, X1+(~-l)2,+ 2,
VOL. XVIII. 12