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. 85
where e is a real positive quantity that can be taken as small as we please. Proceed
therefore to the limit in which o and p acquire the value unity, so that e passes
from small values to zero. The effect is to give to 0 and 0 the values
0=-tlogt, = -t(logt)2;
to change the differential equations to the forms
t -X + At = 2 Ai Ytp
dY
'/;
dt -Y+Bt= Bip XiYyt
and to change the integrals to the forms
X = s h'lmn t (- log t) {t (10g t)} n
whr = 2 C ^ ^nt1 (-t 10g t)m t (log t)2d '
where h'im and k'irn, are the respective values of hlm, and klmn when o-=1, p = 1.
It is necessary to compare the coefficients h'rn, and hmn' and likewise the coefficients k'imn and Imn. Let T be one of the terms in hl,,n as explained above: and
let T' be its value when- = 1, p=l. The effect of the change on the numerator is
to replace (1 - )hlo,+A by A, hooi by 0, (1 -p) kcoo+B by B, k0oo by 0, in every case
a decrease: and therefore, as the numerator is a sum of positive terms, the whole effect
on the numerator is to decrease it, that is,
numerator of T'< numerator of T.
As regards the denominator of T, in the form
P+Q,
the quantity /8 is of the second order of small quantities; Q is an aggregate of a
limited number of products, each containing a limited number of factors; hence Q/8 is
of the second order of small quantities. Let P' be the changed form of P, obtained
from P by changing
l+m+an- - into +m+n-1,
and l+ m+an -p into l+m+n-1.
Now l +m+ n-o — ( +m n- 1)=-(2n -1)e,
a small quantity of the first order unless n= O; so that, unless n=O,
I + m + cn -
I+m+ n-1=