472 Mr. WARBURTON, ON THE PARTITION OF NUMBERS, 



= mt + r, the number of the ways of resolving N into p parts could be expressed in the form of a 

 rational and integral function of the factor, t. Thus, in the case of Bi-partition, 2 being the 

 Modulus, and the Number being 21, or 2<+ I, < is the number of the partitions. In the case of 

 Tri-partition, 6 being the Modulus, 



For N = 6t, the number of the partitions is Sf, 



= 6< ± 1 <[3i!±l], 



= 6<±2 <[3i!±2], 



^ 6t + 3 3f + 3f + I, 



and in the case of Quadri-partition, when the Modulus is 12, and t becomes of 3 dimensions, I also 

 ascertained the formula;. But perceiving that, since the modulus and the dimensions would increase 

 with the number of the parts, the functions obtained would be so many, and of such complexity as 

 to be of little or no practical utility, I abandoned that method, and sought for some other. Having 

 at last discovered the method here proposed, (Arts. 7 and 8, Sect. 1,) I communicated the same to 

 Professor A. De Morgan, trusting to his known Mathematical erudition for obtaining the information 

 I required — whether the method was novel. By his reply, I was made aware that the Partitions 

 of Numbers had received a share of his attention, and that he had written a paper on the subject, 

 which was publislied anonymously in the -ith Volume of the Cambridge Mathematical Journal. He 

 further stated that, after the date of that publication, he had also discovered the Theorem which I 

 communicated to him ; though he had not announced it ; and since I iiave no doubt of the entire 

 accuracy of that statement, he must participate fully in any credit that may attach to the discovery 

 of the formula in question. 



In this Section of my present Paper, I have limited myself, as regards these partitions, to 

 what I considered necessary for the proof and illustration of the Theorem in question. Other 

 matters bearing on the question of Partitions occur in the Section on Combinations. 



2. The number of the different ways in which a Number, N, can be resolved into p parts, 

 when no number is admitted as a part, but such as is either equal to, or greater than, the arbitrary 

 number, rj, may be denoted by \_N, p^']. We may term j; the lower limit of the parts, or parti- 

 tion, or, for brevity, the lower limit. By a, p — partition of N, I mean any set of p numbers, 

 having N for their sum. 



A partition included among those, the number of which is denoted by [_N, p ], may consist of 

 parts exclusively equal to, or exclusively greater than >; ; or it may contain some parts equal to, 

 and some parts greater than tj. 



\_N, p^'\ includes the whole of \_N, jo, + i] ; 



\_N, p,+i] includes the whole of [A^, p,+s] ; 



and, generally, the partitions which have t) for their lower limit, include all those partitions in 

 which the lower limit is greater than »;. 



3. If to or from eacli part in every partition of N whose lower limit is »;, a given number 

 be added or subtracted, N will be increased or diminished by the amount p9; but the number of 

 the partitions, and the number of the parts in every partition, will remain unchanged: i.e. 



\_N, pj = [_N^pe, p,^,] (I). 



This involves the conclusion, that we recognize 0, and negative numbers also, among the 

 admissible parts; unless we expressly assume that they are to be excluded. It also involves the 

 recognition of negative numbers, as the subjects of partition, unless their exclusion be expressly 

 stipulated. 



