THEORY OF THE PARTITION OF NUMBERS. 
673 
Hence 
( 1 )( 2 ). 
. (m) (1 — x m2 + m ) Jc m ( x) — 
1 
F - 
(2) (3) . . . (m + 1) (2) (3) . . . (m + 1) 
Therefore 
km (3?) — 
(1) (2 f (3 f . . . (m - l) 2 (my (m + 1) 
Hence, by induction, it has been established that k m ( x ) has this expression for 
all values of m. 
Therefore the result 
( 2 > m > 00 ) = (1 - *) (1 - x*y (1 - . . . (1 - x" l f (T“- x m+r ) 
agreeing with that obtained in a totally different manner by Forsyth. 
I hope to continue the theory, adumbrated in this paper, in a future communication 
to the Royal Society. 
4 R 
mdcccxcyi.—A. 
