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.