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.

82 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
regular integrals vanishing with t. If, however, both a = O and b= O, the equations do
not possess non-regular integrals vanishing with t; the only integrals vanishing with t
are the double infinitude of regular integrals which the equations are known to possess.
This theorem can be established, as in other cases, by the construction of dominant
equations and comparison with their integrals which actually are obtained in explicit
expression.
For this purpose, consider the equations
t - oX + At = S 2 Aijp Xi Y tp
dt
t d Y_ Y + Bt -- ~2 Bijp Xi YJtp
where i +j+ p 2 in the two triple summations. The quantities - and p are real, positive,
and less than unity: ultimately they will be made equal to unity. It follows, from the
theorem of ~ 8, that there is a double infinitude of integrals vanishing with t, these
integrals being regular functions of t, tG, tp.
Let two new variables 0 and q be introduced such that
t = t - (- - 1) 0 + (o- - 1)2 <
tP = t - (p - 1) + (p - 1)2;
we easily find
t j+ t-O=(1-p)(1 —) 0=/3 )!
~~~~~~~dt~
t d + ( = (o + p - + 1) = a I
where a and f3 are constants which, when p = 1, o = 1, are equal to 1 and O respectively.
The regular functions of t, ta, tP are expressible in the form of absolutely converging
power-series; when t' and tP are replaced by their values in terms of 0 and 0, the new
functions are regular functions of t, 0, b. To obtain their expressions in this last form
directly from the differential equations, we substitute
X = X hImn tl S gbn
Y= YCY kImn tl Om on
in the equations which are to be satisfied identically. Now
dX a( dOa d a ~)
tdX (t a + t + t X
dt a t+t dta dt a95
= 25 {(I + m + an ) hlin tl Am pn
- mh.mn tl+l 0m~-1 n - nhm^ t1 0?n+l n-1 + f3mhmn t1 Om-1 ~+l} };