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.

70 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
so that
a10=(1 - ) a + A, (1 - c)bol =+B, bo= 0;
thus ao, is undetermined and therefore can be taken arbitrarily, say = C, where C is positive.
Thus ao0, ao, bol are positive.
For m+n=2, so that m=2, n=0: m=1, n=l: and m=0, n=2; then
oa20 = A 200 a2o 1
(2- - K) b20 = B200a210
a 11- 2a20 = 2A200 o,,, a, + A-o11a10 bo1 A101 ao,
(1 + - ) b - 2b26o = 2B2,oaioao + B1l0al0bol + BlOI al0
(2 - o) a02 - ai = Aoa201 + o ao b + oao o 1 b2oo + Ao2o a2o + Alol + bol + Aoo2 2
(2 - c) b02 - b6n = B2oo a2o + B,, ao bo B2b2 + Ba, B Boo 2
And so on, taking in succession the groups of terms for increasing values of m + n, and
taking, in each group, the equations for increasing values of n beginning with zero.
The result is to give
amn = 0mn, bmn = mmn,
where m,, and omn are sums of a number of terms; each term is a quotient, the
numerator being a positive integral function of the coefficients of 01 and 0. and
containing a,0m as a factor, and the denominator being a product of quantities of the
form
n +- -m -o, n + cm- K.
It can be proved, by an argument precisely similar to that in Jordan's Cours d'Analyse,
t. iii, ~ 97, that the number of quantities entering into the denominator product for
each of the terms in 09m and fmn is
< m + 2n-1.
On account of the theorem of ~ 10, establishing the existence of the integrals as regular
functions of t and ta, it follows that the series
2amn O6tnn, Ybmn, Omtn
converge absolutely.
Now proceed to the limit in which o- increases to, and ultimately acquires, the value
unity; then 0 becomes -tlogt, the differential equations become
dtd X + At = o(X, Y, t)
dY
-dt- /Y + Bt = 2(X, Y, t)
and the integrals change to
ae mn mtn, a b/ mn om tn,
where amn and b/n are the values of amn and bmn when o is replaced by 1.