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.

42 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
CASE I (b): the critical quadratic has unequal roots, one of which is a positive integer
and the other of which is not a positive integer.
2. The equations may be taken in the form
dt = mu + at + (u, v, t))
dv
td = v + 't + y (, V, t)
where m is a positive integer, e is not a positive integer, 0 and 0 are regular functions
of their arguments, vanishing with u, v, t, and contain no terms of dimensions lower
than 2.
If regular solutions exist, which vanish with t, we can take
u = t (\ + '11), v = t (p~ + vi),
choosing the constants X and ui so that ul and v, vanish with t. Then t2 is a factor of
0 and b after this substitution is made, say
0 (u, v, t) = t2'1O (u,, v1, t), b (u, v, t) = t21 (u,, yI, t);
but 0,' and b,' do not necessarily vanish when t, u,, v, vanish. The equations for the
new variables are
t l = (m - 1)X+a + (n- l) u, + t'(ul,, v, t)
dvl
dt tc = ( - 1)f++t (- 1)v,+tV,' (t,,,1, t)
Now as u,, v, are regular functions of t, the expressions on the left-hand side vanish
when t=O; hence
(m- )X+a= 0, (-1)~ +/ = 0.
If 01'(0, O, 0)=a, 0,'(0, 0, 0)=,, the equations are
t dt' = (m-1) 4 + at ^ + toi (u, yv,, t)
dv
t d= ( - 1), + 3,t + to,(l,, v,, t)
where 0, and (, are regular functions of their arguments and vanish when u =O, v =O,
t = 0. The equations are, in form, the same as before, except that the coefficients of
the first power of the dependent variables on the right-hand side have been reduced by
unity; and the relation between the two sets of dependent variables is
t6=t - 1+ ), v=t -.l+V).V
mz - 1 ~:e-1
It is manifest that the equations in ul and vy can be subjected to a similar transformation with a corresponding result; and that, as m is a positive integer while t is