Demonstration of Mr. Cayley. 201 



identical) is for some of the combinations as great as, but for 

 none of the combinations greater than (m), as was to be proved. 

 It will of course be seen that, for the purposes of the demonstra- 

 tion above given, it would have been sufficient to have been 

 able to assume that the number of partitions, when the greatest 

 part is not allowed to exceed (m), is not greater than the number 

 of partitions when the number of parts in any one partitionment 

 does not exceed (m). The equality of these two numbers would 

 then evince itself in the course of the demonstration as a conse- 

 quence of this assumption. 



A word now as to Euler's beautiful law upon which the 

 above demonstration is based. 



A corollary from it, obtained by subtracting the equation 

 which it gives when the limiting number is taken (m — 1) from 

 the equation which it gives when the limiting number is {m), 

 will be the following proposition. The number of modes of par- 

 titioning {n) into (m) parts is equal to the number of modes of 

 partitioning {n) into parts, one of which is always m, and the 

 others (m) or less than [m). This proposition was mentioned to 

 me by Mr. N. M. Ferrers*, whose demonstration of it (probably 

 not different from that of Euler^s for the other proposition, of 

 which it may be viewed as a corollary) is so simple and instruct- 

 ive, that I am sure every logician will be delighted to meet with 

 it here or elsewhere. It affords a most admirable example of 

 that rather uncommon kind of reasoning whereby two abstract 

 integers are proved to be equal indirectly, by showing that neither 

 can be greater than the other. 



If there be a group of A^s and a group of B*s, and every (A) 

 can be shown to produce a (B), and every (B) can be shown to 

 produce an (A), no matter whether the (A) producing a Bis the 

 same as, or different from, the A produced by that B, it is obvious 

 that the number of A^s cannot exceed that of the B's, nor of the 

 B^s that of the A^s, and the two numbers will therefore be equal. 



Take any such grouping as 3, 3, 2, I, say A. This may be 

 written as 



1, 1, 1 



I, 1, 1 

 1, 1 



and by reading off the columns as lines, may be transformed into 



* I learn from Mr. Ferrers that this theorem was brought under his 

 cognizance through a Cambridge examination paper set by Mr. Adams of 

 Neptune notability. 



Phil Mag. S. 4. Vol. 5. No. 31. March 1853. P 



