AND ON COMBINATIONS AND PERMUTATIONS. 



475 



From each of the parts, in every one of these partitions, deduct 1. 

 Then we shall have 



L^, P^l = [^ ~P, hi + [N-p, a,] + [N-p, m,] + 



Also, iN + p, p,-\ = [JV, 1,] + [N, 2,] + [N, p,]; 



[A'-jB,]: 



8. I shall proceed to shew the application of these latter formula? to the construction of a 

 iTable of the Partitions of Numbers, and point out tiie leading properties of such a table: and 

 [since all partitions, whatever may be their lower limit, are reducible to partitions whose lower limit 

 lis 1, I shall confine my observations to a table whose lower limit is the Unit. 



To the Equation of Differences, II, we may give the following forms : 



[A^> P.] 



[AT-i, p-i] 



[^-p> P.]; (X)- 



or [N + p, p,] -[N + p-l, p-l] = [N, p,] (X*). 



I 



and these will best serve for tlie construction of the table. 



The annexed table is one of double entry, N being the index of the columns, and p of the 



4incs. [A'', pt] is tlie term in column N, line p. In formula x, the change to A- 1, and p - i, 

 aarks that we are to recede simultaneously one column and one line, that is, diagonally. The 

 [diagonal will cut line at the head of column N — p, and [A'^ - p, /),] is the term on the ju"' line of 

 [that column. Thus the term on the p"' line in the vertical column is the difference between the 

 Iterms on the //'' and (p — 1)"' lines on the diagonal. Suj)])osc that all the terms are known in the 

 [vertical column A'^, an'l that we have determined all the terms on the diagonal, proceeding from the 

 Ihead of that column to the (p — 1)"' line inclusive. Then the term in the diagonal on the ;/'' line. 

 'that is, [N + p, ;;,], is ecjual to [A'^ + p - 1, p - ]~] + [A, p,] ; and in the same way the term 



