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.

68 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
where c, = -m+l, is not a positive integer, the functions f, and f2 are regular
functions of their arguments of the same type as 0 and b above, and the integrals
t, and t2 are to vanish with t.
It has been proved that there are no regular integrals of the equation vanishing
with t unless a is zero: and that, if a= O, 'there exists a simple infinitude of regular
integrals satisfying the equations. We proceed, not in the first place to the complete
theorem but only to a partial theorem, by shewing that when a is not zero, there
exists a simple infinitude of non-regular integrals vanishing with t, these integrals being
regular functions of t and t log t: and when a is zero, these non-regular integrals do
not exist.
To establish this result, we proceed from equations
dx
t d = ox + at + 01 (x, y, t)
dt
t d- = cy + bt + 02 (x, y, t)
where a is taken to be a real positive quantity, a little less than 1 initially and
equal to 1 ultimately: and, as the explicit forms of 0, and 02 are required, we
suppose
0(x, y, t)= =22aijxiyJtP,
(i +j + p > 2).
0O (x, y, t) = Sbj xyJttP,
With these equations, we associate a set of dominant equations. Let
laijpl = Aijp, bijp l= Bijp, a = A;
then the dominant equations are
dX
t — -j X + At = (X, Y, t)|
dt
t - Y + Bt = 2 (X, Y, t)
where
)1(X, Y, t)=SY;AijjXjy Yjtp
(2 (X, Y, t)-= SX BijpXiyjtpJ
If K be real, not being a positive integer, we choose that sign for the term + Bt,
which makes
B
K-i
a positive quantity; if Kc be complex, we choose a term + Bt, such that
B
K-l
is a real positive quantity and \B bj.