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XIX. On the Algebraic Expression of the number of Partitions of which a given number 
is susceptible. By Sir J. F, W. Herschel^ Bart., K.H., F.R.S. 
Received April 18, — Read May 16, 1850. 
(1.) Before entering on the investigation which forms the object of this commu- 
nication, it will be necessary to recall to recollection some general properties of the 
dilferences of the powers of the natural numbers, or of the numbers comprised in the 
general expression A”* 0”, which I have elsewhere demonstrated, as well as to establish 
certain preliminary theorems by the aid of those properties, which will be useful in 
the progress of the inquiry. I shall employ throughout the separation of the symbols 
of operation from those of quantity, as respects A and 0, in the manner followed 
in my paper “ On the Development of Exponential Functions,” published in the 
Philosophical Transactions, vol. cvi. p. 25 (1816), and further extended in its appli- 
cation in my “ Collection of Examples of the Applications of the Calculus of Finite 
Differences,” appended to the translation of Lacroix’s Differential and Integral Cal- 
culus in 1820*, to which paper and collection the reader is referred for the demon- 
stration of the fundamental properties in question. 
(2.) Denoting by F(a7) any series of powers of x, such as 
F(a?)=:Aa?“-l-Bx*-l-C.r''-l- &c., 
and by f{x) any other as 
f{^x)= Vx^ -b -j- &c . 
the series 
AP.A“0^-l-AQ.A“0^-f-BP.A'0^-b &c., 
continued till the terms vanish, by reason of the peculiar properties of the numbers 
A“0^ &c., will be abbreviatively represented by 
F(A)/(0) ; 
and the following properties of the differenees in question will be either found demon- 
strated in the works above cited, or may very easily be derived from the formulae 
therein given : — 
(1--1-A)*0^=a:^ (1.) 
(1-f A)"F(A)02 '=F(A)(x-1-0)^ (2.) 
(H-A)^F(A)/(0) = F(A)/(x-l-0) (3.) 
(1-f A)"/(0)=/(^) (4.) 
{log(l-l-A)}^F(A)0^=3/.3/- 1 r/— 0^4-1. F(A)0^-* (5.) 
{log(l + A)}T(A)/(0) = F(A)(A)/(0) (6.) 
* A separate edition of this collection (now out of print) is in preparation. 
