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.

DIFFERENTIAL EQUATIONS. 55
When the roots of the quadratic are unequal, say e and e2, we take new variables
t, = Xx + xy, t2 = 'x + /y,
where
(ai-^+ap ==0l (a,- ~)X' + ~/'= O0
(2l-1) X + a22/ X2 + a,/~,}
/1X + (,2- 1 =) O ' AX + (1 - ) / = oJ '
the equations become
dt,
t = lti + 1 (t,4 t2, t)
Ct t
t-2 =2t2 + 02 (tl, t2, t)
where the regular functions b, and b2 vanish when tl=O, t2 =O, and contain no terms
in ti, t2, t of dimensions lower than 2.
When the roots of the quadratic are equal, the common value being:, the corresponding forms are
t- = et, + 0 (ti, t,,t) )
tdt = t2+b +(ti, t2, t)
dt
with the same characteristic properties of the functions qb2 and 02 as for the former
case; here t2= y and tl= x -+ y, where
(a, - ) x + a2 = 0, flx + (8, - ) = o,
and the constant Ic is given by cX = a2.
We proceed to deal with the various alternative cases, as for the regular integrals:
merely remarking that, for those instances of the original equations which do not possess
regular integrals because the appropriate condition is not satisfied, it will be necessary
to return to those original equations for the discussion of the non-regular integrals.
7. Some indication of the character of the solutions may be derived from the
consideration of two simple examples, one of each form.
A simple example of the case when the roots of the critical quadratic are unequal is
dt\
t dt = Xt2 + att2
cdt
dt2 =,Lt2 + b1tt,
integrals (if they exist) are required which vanish when t= O. The solution of these
equations, which are linear, can be made to depend upon that of a linear equation