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.

74 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
CASE I (c): the roots of the critical quadratic are unequal, and both
are positive integers.
14. Denoting the roots by m and n, of which m may be taken as the smaller
integer, the equations can be transformed so as to become
du
t -=u + at + 0 (u, v, t)
dt
dv
t = nv +/3t + 0(u, v, t)
They can be modified by substitutions sirnilar to those adopted in the preceding case;
such substitutions can be applied m - 1 times in succession, leading to the forms
dt1
t = ti +at +f, (tL, t2
~~~~~dt2~~
t-t = Kt2bt+f(t t t)t
where K, =n-m+1, is a positive integer greater than 1, the integrals t, and t2 are to
vanish with t, and the functions fi, f2 are regular functions which vanish with their
arguments and contain no terms of dimensions lower than 2 in ti, t2, t combined.
It has already been proved (~ 3) that the equations possess no regular integrals
vanishing with t, unless two relations among the constants be satisfied; one of them
is represented by a =, the other by (say) C= 0, where C is a definite combination
of a, b, and the constant coefficients in f1 and f2. The theorem as regards the nonregular integrals is:
The equations in general possess a double infinitude of non-regular integrals which
vanish with t; they are regular fjinctions of t, and t log t. If both of the conditions
represented by a = 0, C= 0 are satisfied, the equations possess no non-regular solutions
vanishing with t: they are known to possess a double infinitude of regular integrals
which vanish with t.
The method of establishing this theorem is similar to that for the case when K
is unity so that the critical quadratic has a repeated root. As that case will be
discussed later in full detail, we shall not here reproduce the analysis and the argument,
which follow closely the corresponding analysis and argument in that later discussion.
It may be added that the conditions for the equations
t d = ~mu + at + 0 (u, v, t)
dv
teesented for the modified forms by a (,, have already ( 3) been given.
represented for the modified forms by a = O, C=O, have already (~ 3) been given.