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.

60 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
Suppose now that no one of the quantities
(m-l)el+re+,+p, ml+ (n- 1) 2+p,
for positive integer values of m, n, p such that
mn + n +p > 2,
vanishes. Then when the equations
(m - 1) el + ne2 + p} acmnp = mnp
{me, + (n - 1) e2 + p} bmnp = / mnp J
are solved in groups for the same value of m+n -+p, and in successive groups for increasing values of z +n+p beginning with 2, they lead to results of the form
amnp = ~mnp, brnnp = 3mnp>
where annp, l3mnp are rational integral functions of the coefficients that occur in jb and 02,
these functions being divided by a product of factors of the forms
(m - )8 ++ n9+p, mi +(n - 1) 2 + p, for m n + p >2.
It has been seen that aol,=0, bool= 0: we easily see that aoop=0, boop= for all values
of p. For every term in b1 (t1, t2, t) and every term in 2 (t2, t2, t) involve ti, or t2, or
both: and the equations for aoop, boop are
(p - e1) aoop = Aop, (p - 2)) boop = Bop,
where Aoo, Boo are integral functions of the coefficients in (l and 2,, and of coefficients
a,,,, b,,, such that p' <p, these integral functions being divided by factors of the form
p'- e, p'- 2. No term occurs either in Aoop, Boop independent of aop,, boop, because there
is no term in b1 or in 02 independent of ti and t2. Hence if all the coefficients aop,,
boop, vanish when p'< p, then aoop, boop also vanish. But a,0 = 0, boo0 = 0: hence a,02 = 0,
b002=0: and so on with the whole series.
Consequently in the expressions for t1 and t2, there occur no terms that involve t
alone without either z1, or z2, or z1 and z,: which is therefore one general characteristic of
the non-regular integrals if they exist.
From t, and t,, let all the terms which do not involve z, be gathered together.
By what has just been proved, there are no terms which involve t alone: hence the
aggregates of the selected terms contain zl as a factor, and the aggregates of the remainders
contain z, as a factor, so that we can write
t, = z1p + Z201,
t2 = Z17 + z2092,
where p and T are regular functions of t and z1, which will be proved to be such that
p=A, 7=0, when t=O, A being an arbitrary constant: and (), 02 are regular functions