490 



Mr. WARBURTON, ON THE PARTITION OF NUMBERS, 



then the number of the permutations, corresponding to every pair of determinate values given to v 

 and u — V, will be 



*P{u, (t] = \v, t] {u -V, o- - t] Piv)P{u - v), 



,w I 1 |U-ii I J * 



And for the line ti of the table, u remaining constant, and v only varying, we have 



1" I ' 

 P{u, ^\=Sl\^P\v, r}P\u-v, cr-r} ^„., _]„_ ]■ 



And the equation of the terras of the table of double entry, in which u is the index of the line, 

 and V of the column, is 



[m, «] = [«, v]y.[u-v, 0]L ^,|, ' (xxviii.), 



I give an example of such a table, applied to the case of the two sets of elements, 

 A, BB, CCC; and DDDD, EEEEE. 



Permutations of ^, B^, C^ ; whose Combinations are I, 3, 5, 6, 5, 3, 1 



and of Z)*, E^ 



1, 2, 3, 4, 5, 5, 4, 3, 2, 1. 



• P {u, (r) means the permutations of the elements contained in the combination {u, <r}. 

 + The truth of Article 9, Section 3, is here exemplified. 



