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.

78 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
To establish the convergence of the series thus obtained, we proceed in two
stages as in the corresponding question (~~ 10, 11) when the roots of the critical
quadratic are unequal.
Extract from t1 and t2 all the terms which are free from Y]; as each of them
involves ', their aggregate can be taken in the respective forms >p, r-; and the
remaining terms then have qj for a factor, so that we may write
t2 = T + 72.
t2 =T+17e2.
It will be proved, first, that solutions of the form
tl = gp, t2 =?T
exist, where p and r are regular functions of t and f, p vanishing at t= O and r
having an arbitrary value there: so that the functions involve one arbitrary constant,
and there consequently is a simple infinitude of such solutions.
Then substituting
t1= p+ î1, t2=UT+ 2,
it will be proved that functions ~1 and 0, exist, so that they are regular in their
arguments g, r, t, they involve an arbitrary constant C, ~1 vanishes at t = O and 20
acquires the value C there. Thus for an assigned value of B, these will represent another
(and an independent) simple infinitude of integrals.
In each stage, the details of the analysis follow the detailed analysis of the
former case somewhat closely: it therefore will be abbreviated for the present purpose.
16. Substituting ti= p, t = =r in the equations for t1 and t2, we find p and T
determined by
t dt = (P, r, ', t)
td= op++2(p, T,?, t)
dt
where the general character of -I and q2 is as before. If these are satisfied by regular
functions of t and r, their expressions
p = ikq ~t~,
T = Zj~mn mtn,
must, when substituted in the above equations, satisfy them identically. Accordingly,
comparing coefficients of gmtn on the two sides of both equations, we have
(n + mo) Cmn, = K',mn
(n + m~)j.n = J'mn + 6Omn,