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.

80 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
and therefore as the series.KmnUqmtn
converges, the series
kmn C +m2tn, C + 2jmnn,t
also converge. The existence of the integrals, connected with the first stage, is therefore
established.
17. Now writing
ti = 'p + 1O7, t2= + 102,
where p and r are the regular functions of t and Ç as just determined, the equations
for,1 and 0, are
dt
t —t = (o~, o2, ~', v, t) )
tde2/ = 0oe + f2(o1, o2, ~ v, t) )
where fi and f, are regular functions of their arguments, vanishing when ~1 = 0 and 02=0;
the coefficients of the first powers of,1 and 0, vanish when t =0; and any term,
involving e1 and 0, in the form OeXe,, contains <x+i-1 as a factor.
The method of proof and the general course of it are the same as before (~ 11).
The regular functions of ', q, t, which are the formal solution of the equations, are
proved to converge, by being compared with the functions which satisfy the dominant
equations
êe`@ =i_ __ IMM M _ _ M
1 | _V -)i- _2) ri —) l - I a
\ r)\ p)\ pa1 \ \ r \ P
M r
o'( )1 )(- PI(1 _ o-~2 /
and are such that, when t=O, =0,?7=0, then <P is zero and 2 = C l. There exists a
single quantity (I, satisfying these equations and vanishing with t, which is expansible as
a regular function of t,,; q in a non-infinitesimal region round t, the power-series which is
its expression being consequently a converging series within that region. And therefore
<>2, being given by
is also expressible as a regular function of t,, r which, when t=O, acquires the
value | CI.