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.

38 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
It is convenient to transform the variables. When the roots of the critical quadratic
(- a0) (f-/3)- c1 = 0
are unequal, say e and e2, we introduce new variables u and v, such that
u=XU+pUV, v=X'U+p'V,
where (al - ()X + c/ar = 0 (a, - 2) X' + a21 =
/1x + (a3- l) =o' ~ lx' + (2 - )2) '= O
the ratios X: /L and X': L' are unequal, and consequently the new variables u, v are
distinct. The equations become
dv
t - = 2V + 02 (u, v, t)
dt = t2 +, (u t)
where <^, çf are regular functions of their arguments, vanishing with them; except for
a term in t, they have all their terms of the second or higher orders in u, v, t combined.
When the roots of the critical quadratic are equal, having a value s, say, we
introduce a new variable u such that
u = xU+ V,
where (al- ) X + a2u/ =, /1X + (2- ) = 0 -Then we have
tdt eu-t+01(U' v, t),
dt U
udV- a2 t
t = fu + + 0 (u, y, t),
=,u~+ fV+p.,(u, V, ),
say.
It therefore appears that the equations corresponding to the cases I(a), I (b), I(c),
are
t- = u+ i(, t)
dv
= e.V + b2 (', V, t))
where el and 2 are unequal to one another: and that the equations corresponding to
the cases II (a), II(b), are
dv,,
dt = fu++ cb2u v 4t)i