138 
MR. A. CAYLEY’S RESEARCHES ON THE PARTITION OF NUMBERS. 
which is a function of of the order 36, the coefficients of which are 
1,0, 1, 1,2, 1,3,2, 4,3, 4,4,6, 4, 6, 5, 7, 5, 7, 5, 7, 5, 6, 4, 6, 4, 4, 3, 4, 2, 3, 1,2, 1, 1,0, 1, 
and the first part becomes therefore 
coemcient x m - — wn Jwi bvti s\ 
Tlie multiplier for the second fraction is 
(1— — —x^'^) 
(l—<r‘'^)(l (1 —. 2 ?®) ’ 
which is a function of of the order 14, the coefficients of which are 
1, 1, 2, 1, 3, 2, 3, 1, 3, 2, 3, 1, 2, 1, 1 ; 
and the second term becomes 
. . 2x^ -h 2x‘^ + 3x^ + x^ + x^^ 
-coefficient le" in s 
X^ 
and the third term is coefficient x"^ in 
Now the fractions may be reduced to a common denominator 
T 
by multiplying the terms of the second fraction by = 1 and the terms 
X ““ o(^ • 
of the third fraction by y 3 ;^(= 1 +■3^^) ; performing the operations and adding, the 
numerator and denominator of the resulting fraction will each of them contain the 
factor 1 — 0 ?^; and casting this out, we find 
P(0, 1 , 2, 3, 4, 5)”*-|w= coefficient x”^ in 
I have calculated by this method several other particular cases, which are given in 
my ‘SSecond Memoir upon Quantics the present researches were in fact made for 
tlie 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*. 
<5 <37 
Let^- be a rational fraction, and let (a?— be a factor of the denominator 
then if 
\<^x'\ 
* Mr. Sylvester’s researches are published in the Quarterly Mathematical Journal, July 1855, 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. 
