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.

48 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
shall vanish. Take pm =a; and then from the original equations determine p,+i,., n-n,
qm,..., qn-. The second condition is that the coefficient of tn in
1n-1 n-1 \
4, -plt\, Yqi t\t)
1=1 1=1
shall vanish. It is not difficult to verify these statements.
Summarising the results, it appears that, unless one condition be satisfied, the equations
possess no regular integrals vanishing with t. When the condition is satisfied, another relation
must be satisfied. If this relation determines a parameter, the equations possess a single
infinitude of regular integrals; if it involves only the constants in the differential equations,
then, when it is not satisfied, there are no regular integrals vanishing with t: and, when it is
satisfied, there is a double infinitude of such integrals.
CASE II (a): the critical quadratic has equal roots, not a positive integer.
4. The equations are
du
t d = eu + 01 (u, V, t)
dv
t dv= KU+ ev+ 2, (u, v, t)
where e is not a positive integer; the functions <0 and 02 are regular and (with the
possible exception of a termin t) contain no terms of order lower than 2. If they possess
regular integrals vanishing with t, they must have the forms
U =, pntn, v = 2 q.tn.
n=l n=l
Substituting these expressions and equating coefficients, we find
(n-_)pn=fn
(n - ) qn = gn + KPn
where f, and gn are the coefficients of t" in <0 and <k2 respectively, when the series for u
and v are substituted. It is clear that fn and gn are linear in the coefficients of ~< and
(p, that they are integral algebraical combinations of pi, p,,..., q 2, q,..., and that they
contain no coefficient p or q in the succession later than pn- and qn-. As e is not an
integer, the foregoing equations, taken for successive values of n, determine formal expressions for the whole set of coefficients p and q; in particular, p, and q,, are obtained as
sums of quotients, the numerators of which are integral functions of the coefficients in q,
and b2,, and the denominators of which are products of powers of the quantities
1-r, 2-,..., n-.
To discuss the convergence of the power-series for u and v with these coefficients, we