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.

72 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
Consequently the series
$,Ea'mnOmtn, Z1b'mn fmtn
converge absolutely for a finite region round t= O.
If the original equations
dx
t -= x+ at + i (x, y, t)
t dy= cy + bt+ 02 (x, y, t)
possess integrals vanishing with t in the form of regular functions of t and t log t, these
integrals rnay be assumed to be
= 2fmn Smtn
Y = YS;fn 0M t'
when substituted, they must satisfy the equations identically. Choose fol so that
If1o I = c,
where C is the arbitrary constant in the integrals of the preceding equations.
When the relations that arise from the comparison of the coefficients are solved
so as to give fmn, gmn, it is easy to see that the same results are obtained as would
be given by changing, in a'mn and b',,, A into -a, B into + b, Aij into aop, and
Bijp into bijp, for all values of i, j, p. Bearing in mind that
lal=A, |b|<J|lB, aijp l=Aijp, [b&jp=Bgjp,
it is manifest that the real positive quantities | a',n and b'mn are superior limits for
IfnI and gmn I, that is,
jfrmn I < I a`mnl, I Igmn I< b' mn.
But the series:,a'mn 6tn, -b'mn mtn
converge absolutely: hence also the series
yfmn O tn, ygmn o tn
also converge absolutely, and the equations accordingly possess integrals as stated in the
theorem.
Note. If a is zero, then a',, = O; a'0= O, a'/ = O; and it is immediately obvious
that
a mn = O,
for all values of m > 0 and all values of n. Similarly
b'mn = 0,
for the same combinations of m and n. In this case, 0 disappears entirely from the
expressions
f, fOn' m vt, sygmn Omtn;
so that the integrals become regular functions of t, which are known to be solutions
of the equations when a = O.