Cambridge Philosophical Society. SOI 



record my own recollections as to the progress and introduc- 

 tion of this invention : and though they relate to transactions 

 which took place from thirty to forty years ago, I believe they 

 are in the main correct, and can be confirmed by documentary 

 evidence. 



XLIX. Proceedings of Learned Societies, 



CAMBRIDGE PHILOSOPHICAL SOCIETY. 

 [Continued from p. 143.] 



ON the Partitions of Numbers, on Combinations, and on Permu- 

 tations. By Henry Warburton, M.P., F.R.S., F.G.S., Mem- 

 ber of the Senate of the University of London ; formerly of Trinity 

 College, A.M. 



The use made by Waring of the Partitions of numbers in develo- 

 ping the power of a polynome, induced the author to seek for some 

 general and ready method of determining in how many different 

 ways a given number can be resolved into a given number of parts. 

 On his communicating the method described in article 5 of Section 

 L of this abstract, to Professor De Morgan, in the autumn of 1846, 

 that gentleman intimated a wish that the author would turn his 

 attention also to Combinations ; and such was the origin of the re- 

 searches which form the subject of the 2nd and 3rd sections. 



I. Gn the Partitions of Numbers. 



1. Let [N, p»j] denote how many different ways there are of re- 

 solving the integer N into ^J integral parts, none less than ij. Then 



[N.p„] = [N±;,9,p„+^] (I.) 



2. Such of the p-partitions of N as contain ij as a part, and no 

 part less than t], are obtained by resolving N— ij into p — 1 parts not 

 less than tj, and by adding ij, as a pXh part, to every such (p— 1)- 

 partition. That is, 



[N,2.«]-[N,p]=[N-o,,i.-l]. . . . (IL) 



3. In (IL), substitute ij + l, ij + 2, &c. successively for ij. The 

 sum of the results is 



[N,^,]-[N,2?,+^+i]=Sj[N-,j-ri?,p-l]. . (III.) 



Z tl 



Inthisexpression, when 9=1* (—)—ij, the term [N,^„^^^.l] 



vanishes, and the formula then becomes analogous to one published 

 anonymously by Professor De Morgan in a paper printed in the 

 fourth volume, p. 87, of the Cambridge Mathematical Journal. 



4. In(IL), for [N,p„^i] substitute [N— pij.p,], and transpose 



« I 



f — jis employed to avoid the long phrase, "the integer nearest to 



N 

 and not exceeding — ." 



P 



