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.

58 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
CASE I (a): the critical quadratic has unequal roots, neither of them being
a positive integer.
8. It has been proved that the original equations in this case possess regular
integrals vanishing with t: and therefore, in order to consider the non-regular integrals
(if any) that vanish with t, we transform the equations as in ~ 6, and we study the
derived system
t - = ti + l(t, t2, t)
dt2J
t = 2t2 + 2 (tl, t2, t)
where 01 and ^0 are regular functions of their arguments, vanishing when t =O, t2 = O,
and containing no terms of dimensions less than 2 in t1, t2, t. The integrals t1 and t2
are to be non-regular functions of t, required to vanish with t.
The main theorem is as follows:When the roots of the critical quadratic e and e2 have their real parts positive,
and are such that no one of the quantities
(X-1)l +/z2+ V, +x+(g - 1) 2+ V,
vanishes for positive integer values of X, i, v such that X + ~+ v 2, then the equations
possess a double infinitude of non-regular integrals vanishing with t, these integrals being
regular functions of t, t6e, t$2.
Immediate corollaries, when once this theorem is established, are as follows:If the real part of e be positive and that of e be negative, there is only a single
infinitude of non-regular integrals vanishing with t: they are regular functions of t
and tal.
Likewise, if the real part of, be positive and that of e be negative, there is only
a single infinitude of non-regular integrals vanishing with t: they are regular functions
of t and te2.
If the real part both of el and of e2 be negative, there are no non-regular integrals
of the equations that vanish with t.
These results (the last of which is due to Goursat in the first instance) will be
found sufficiently obvious to dispense with any proof subsequent to the establishment
of the main theorem.
9. In discussing the equations, it will be convenient to replace te' and tt2 by new
variables, say
t t= bz, t2= Z2,
so that, by the general theorem, regular functions of Z1, Z2, t are to be established as