506 
[154 
154 
SUPPLEMENTARY RESEARCHES ON THE PARTITION OF 
NUMBERS. 
[From the Philosophical Transactions of the Royal Society of London, vol. cxlviii. for 
the year 1858, pp. 47—52. Received March 19,—Read June 18, 1857.] 
The general formula given at the conclusion of my memoir, “Researches on the 
Partition of Numbers’’(A is somewhat different from the corresponding formula of 
Professor Sylvester 1 2 , and leads more directly to the actual expression for the number of 
partitions, in the form made use of in my memoir ; to complete my former researches, 
I propose to explain the mode of obtaining from the formula the expression for the 
number of partitions. 
nator of which is made up of factors (the same or different) of the form 1 — x m , and 
if a is a divisor of one or more of the indices m, and k is the number of indices of 
which it is a divisor, then 
where 
1 Philosophical Transactions, tom. cxlvi. (1856) p. 127, [140]. 
2 Professor Sylvester’s researches are published in the Quarterly Mathematical Journal, tom. i. [1857, 
pp. 141—152]; there are some numerical errors in his value of P (1, 2, 3, 4, 5, 6) q.