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. 75
CASE II (a): the critical quadratic has equal roots, not a positive integer.
15. It has been proved that, in this case, the original equations possess regular
integrals vanishing with t: and therefore, in order to consider the non-regular integrals
(if any) that vanish with t, we transform the equations as in ~ 6, and we study the
derived system
dt1 = t + 1 (t, t2, t)
t dt2 =icti + et2 + +2 (tl, t2, t)
where <b1 and 02 are regular functions of their arguments, vanish when tl =0, t2 =O,
and contain no terms of dimensions less than 2 in tl, t2, t combined. The integrals t1
and t2 are to be non-regular functions of t, required to vanish with t.
The non-regular integrals are given by the theorem:
When the repeated root e of the critical quadratic has its real part positive, not
itself being a positive integer, there is a double infinitude of non-regular integrals vanishing
with t, these integrals being regular functions of t, t*, te log t.
When the theorem is established, there is an immediate corollary:
If the real part of the repeated root e of the critical quadratic be negative, then
the equations do not possess non-regular integrals vanishing with t; the regular integrals
possessed by the original system of equations are the only integrals that vanish with t.
The forms of the theorem and the corollary are indicated by proceeding nearly to
the limit of the theoremrs for the case of I (a) when the roots of the critical quadratic
are equal to one another. If = + 3, where 8 is infinitesimal, then
tt2 = t (1 + log t +...),
so that a function of t, tt1, t2 becomes a function of t, tEl, t' log t; but further investigation is needed in order to shew that, in passing to the limit, the functions under
consideration continue to exist. Instead of adopting this method of proof, we proceed
independently.
It is convenient to take
= t, -7 = tt log t.
If therefore integrals of the character indicated in the theorem exist, they can be
expressed in the forms
t2 = a1,,E blmn tm t n
10-2