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.

88 PROF. FORSYTH, ON THE INTEGRALS OF SYSTEMS OF
But the formal expression is more general than the actual value. The equations
determining the coefficients are
(1 + rn + n - 1) Hmn - (2n + 1) H1,, n - (n + ) H, m-i, n+1 = Eljnn
(I + m + n - 1) Kmn - (m + 1) i, m+i, n - (n + 1) Kl, n-i, n+li = F~ n
with
1lHoo = CI, Holo = - a, H0oo = O,
Kl0oo = C Koo = - Koo1 = 0.
It is clear that, when + m + n=2,
Emn=O, Fin=0, if n=1, 2;
hence Hmn, Kimn both vanish for + m + n = 2 if n = 1, 2.
Thus for l+m+n=3,
Eimn=O, Fin=0, if n=l, 2, 3;
hence also Hlm, Kimn, both vanish for I + n+n=3 if n=1, 2, 3. And so on: all the
coefficients Himn, Klm vanish if
n >0;
that is, the quantity ~b does not actually occur in the expressions for x and y which
accordingly are regular functions of t and tlogt.
The theorem is therefore established.
Note 1. Any term in x and y is of the form
Kttm (t log t)'n,
that is, Ktm+n (log t)n; and therefore the index of log t is never greater than the index
of t.
If, however, the equations were
t d- = x + at + ct log t + aijpq xiyjt (t log t)
dy
t = y + bt + c't log t + lSlzbijpq xyJtP (t log t)q
where i+j+p+ q>2 for the summations, then the values of x and y satisfying the
equations are
x = - ct (log t)2 + at log t + Clt + SEZH,n tO
y =- lc't (log t)2+ bt log t+ Ct + ZsKimn tlOîBnJ)
where t, O, q have the same values as above, and the summations are for values of