AND ON COMBINATIONS AND PERMUTATIONS. 491 



9. The following corollary results from the preceding Article. 



j<r| 1 



The formula ^„ p ^^ ,, ^^ , , ^^ expresses the number of the permutations, not only of the 



a + /3 + 7 + &c. elements, when they are all permuted together, but also of [a + /3 + 7 + &c.] - 1 

 elements, when permuted together : of which the following is the demonstration. 



Since, by xxviii, P{<t-1, a} =P{t-1, t\ P\a-T, a - t] — p, !"' ^ ' 1, , 



•^ ^ ' ' )-|T— ll-j(7 — TJl" 



+ P[t, t\P\g-t-\, a-T\ ,f 'I',,, ; 



assume, for a moment, that 



P{t, r} = P{t - 1, t}; and let each =7, 

 and that P\<j - t, a - t] = P\a - t - \, a - t\ ; and let each = K. 



The 



p{<r-i, c\ =j'.iir.r-')'^-^^-^ 



1 



,ir-T 



= J .K 



T I 1 i<T~T I 1 



^ , 1 



But, P\<T, a] =P{t, T}P\a-T, a-T} 



= J.K 



'1 



1 ,(r-i I 1 



r 



1 itr-r 1.1 ' 



Hence the law enunciated will be true of the two sets of elements conjoined, if it be ever true 

 of each of the two sets separately. But it is true of two separate sets, when each consists of elements 

 of only one kind ; for then, whatever may be the number of the elements permuted at a time, the 

 number of the permutations is constantly one. Consequently, the law holds true when there are two 

 kinds of elements conjoined ; consequently, when there are three kinds ; and therefore universally. 



Hence, in the product of the Polynomes, 



the penultimate coefficient = o- x ultimate coefficient. 



HENRY WARBURTON. 



Mai/, 1847. 



ADDENDUM. 



Since this Paper was corrected for publication, a member of the Society, distinguished for his mathematical 

 erudition, has caused the Author's attention to be drawn to the work of Bczout* on Elimination, as containing 

 a formula similar in structure to the Author's formula xiv. 



In the Author's researches in Combinations, his concern has been exclusively with such of the terms of a 

 polynome function of the s quantities, //, B, C, &c., as were of some one, say, the «"', dimension. liy such 

 modeB of investigation as occurred to him, he obtained an expression representing the number of such terms. 



• Thiorie GhUrale ilea Equations Algibriques, par M. B(!zoul. 4to, 471 pp. Purls, 1779. 



