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. 67
effect, upon the initial expression for jIrlj, of each of the modifications is to appreciate the value; hence taking the cumulative effect, we have
7rmn1 < hImne
Similarly I lmn I < rIlmn.
Now the series for <P2, when Pmn is replaced by Hlmn, converges for a finite
region round the origin; hence the series
e,1 -= E rlmnZl'22mtn)
2, = B + C5Klmn. Zllz2mtn
also converge for that region. Consequently the modified equations have integrals of the
character
T1 = Z201, Tg = z2O2
and therefore the original equations have integrals
ti = pzi + z21, t2 = TzI + Z2~2,
where p and r are regular functions of t and zl: and e1, e) are regular functions of
t, Z3, Z2.
This completes the proof of the main theorem with the specified conditions.
CASE I (b): one root of the critical quadratic is a positive integer, the other is not
a positive integer.
12. Let the integer root be denoted by m, the non-integer root by e; the equations can be taken in the form
du
t d = mu + at + O (u, v, t)
tdt = v t+ (t + v, t)0
where 0 and Q are regular functions of their arguments, vanish with u, v, t and
contain no terms of dimensions lower than 2. The sanie transformations as were used
in ~ 2, viz.
u=t(-m-1 +u) v=t( IV
can be applied m - 1 times in succession: and ultimately we have equations
dti
t d = tl + at +fi (t9, t2, t)
9 2