XXXV. On the Partition of Numbers, and on Combinations and Permutations. 

 By Henry Warburton, M.A., M. P., F.R.S., Y.G.S., Jormerly of 

 Trinity College. 



[Read March 1, 1847-] 

 Introduction. 



The researches of which an account is here presented, had their origin in tlie following 

 manner. In the Autumn of 1846, having communicated a Theorem (which will be found in the 

 sequel) on the Partition of Numbers to Professor A. De Morgan, I received from him an oblio-infj 

 reply, wherein he intimated a wish that I would turn my attention to Combination.s, as a depart- 

 ment in Mathematics, which, he thought, much needed cultivation. I acted upon this suo-o-estion 

 and shortly afterwards sent to Mr. De M. results, and subsequently from time to time further 

 results, which he wished me to render public. These I placed at his disposal ; and, with my 

 concurrence, he drew up an account of my Researches, in a Paper which was read before the 

 Society on the 1st of March, 1S47. 



After the reading of this Paper, further suggestions presented themselves to me, of which I 

 drew up an account, and this was laid before the Society by way of Supplement to the former Paper 

 of Professor De Morgan. Still further in)provements again occurred to me ; and it then seemed to 

 me desirable that both Papers should be withdrawn, to give me an opportunity of revising my own 

 researches, and of incorporating the revision in one Paper to be communicated to the Society. 



Many important original observations on the same heads of inquiry, proceeding from Professor 

 De Morgan himself, were contained in the Paper which he drew up ; and I should much regret 

 if, in consequence of the course which I have suggested of withdrawing that communication, those 

 observations were to be lost to the Society and the public. 



It was as impossible for me, as for any other person, to hold communication with tliat gentle- 

 man on Mathematical questions, and avoid deriving great advantage from his sagacity and erudition 

 in Mathematics. I have not, I trust, abused those advantages by appropriating to myself anything 

 which belongs to him ; but I have endeavoured, while possessing those advantages, to carry on my 

 researches with originality and independence. 



SECTION I. 

 On the Partitions of Numbers. 



1. Fkom a recollection of the inipcjrtant application made by Waring of tlie Partitions of 

 Numbers to the developcment of tlie power of a Polynonip, I was leil to investigate their ])ro[)erties, 

 in the hope of discovering some ready method of determining in how many different ways u given 

 Number can l)e resolved into a given number of parts. 



Assuming the Unit to be the lower limit of the magnitude of tlie ])arts, 1 I'omikI that it the 

 Number to be partitioned, A'^, were expressid in terms of a certain Modulus, m, so that N was 

 Vol.. Viii. Paut IV. ,T P 



