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.

52 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
so that
dU' dV'
dt= H (U',, t), = K (U', V', t),
where H and K are regular functions of their arguments. To these equations, Cauchy's
general existence-theorem can be applied; it shews that they possess integrals which
are regular functions of t and assume assigned (arbitrary) values when t= O. Accordingly, the equations in u' and v', in the case when the conditions a=O, b=O are satisfied
and when the constant K is zero, possess a double infinitude of regular integrals which
vanish when t = O.
Secondly, let K be different from zero. If the equations possess regular integrals,
they are expressible in the form
u'= aLt + a2t2 +..., v' = b+t + b2t2 +...;
substituting these, and taking account of the first power of t on the two sides of both
equations, we have
aL = a, + a, b1 = ca, + b + b.
Hence we must have a =; then b1 is undetermined, and
b
al = - -
a finite quantity because K is not zero.
Assuming that the condition a= O is satisfied, and assigning an arbitrary value
A to b1, let
(=t ( —+ ), v'=t(A +2),
/C
so that?,7 and r2 are to be regular functions of t vanishing with t; the equations
for?, and 2 are
-td = h -t- + t,, tA+tq, t)
= t2H (i, 2, t),
t2 2= Ktvl+k(- t-+tgl, tA + t2, t)
dat \ /C tdc~ /
= tctr + t2K(7q1, 2, t),
that is, they are
t = tH(l1, 2, t)
dt
t d= 71q + tK(7l, 12, t)
where H, K are regular functions of their arguments and involve the arbitrary constant A.