248
RESEARCHES ON THE PARTITION OF NUMBERS.
[140
and the third term is
coefficient x m in
(1 — x 2 ) (1 — xff (1 - af)'
Now the fractions may be reduced to a common denominator
(1 — xf) (1 — # 4 ) (1 — of) (1 — a?)
1 cc®
by multiplying the terms of the second fraction by = (— 1 + x 2 + ¿c 4 ), and the terms
1 oc~
1 0$
of the third fraction by —- 4 (= 1 + as 4 ); performing the operations and adding, the
numerator and denominator of the resulting fraction will each of them contain the
factor 1 — x 2 ; and casting this out, we find
P(0, 1, 2, 3, 4, 5) m \m = coefficient x m in _^)(i->y
I have calculated by this method several other particular cases, which are given
in my “Second Memoir upon Quantics”, [141], the present researches were in fact
made for the sake of their application to that theory.
Received April 20,—Read May 3 and 10, 1855.
Since the preceding portions of the present Memoir were written, Mr Sylvester
has communicated to me a remarkable theorem which has led me to the following
additional investigations 1 .
Let
then if
<J)X
fx
be a rational fraction, and let {x — xf* be a factor of the denominator fx,
denote the portion which is made up of the simple fractions having powers of x — x 1
for their denominators, we have by a known theorem
= coefficient - in
z
1 <f> ( æ i + z )
x — x x — z f(x x + z) '
Now by a theorem of Jacobis and Cauchy’s,
coefficient - in Fz = coefficient j in F (\frt) yfr't ;
Z o
whence, writing x x -\- z — x x e *, we have
= coefficient in
Tj
xj cf) {x x e l )
x x — xe l f{x x e~ l ) '
1 Mr Sylvester’s researches are published in the Quarterly Mathematical Journal, July 1855, [vol. i. pp.
141—152], and he has there given the general formula as well for the circulating as the non-circulating part
of the expression for the number of partitions.—Added 23rd February, 1856.—A. C.