492 Mr. WARBURTON, ON THE PARTITION OF NUMBERS, ETC. 



with the special object of applying it to denote the number of the combinations which can be formed with 

 plural elements. 



Bezout's object, at least the sole use to which he applies his formulae throughout his work, is elimination. 

 His concern is with all the terms, in the aggregate, of how many dimensions soever, belonging to such a 

 polynome function as is above described. By a mode of investigation, entirely different from that of the 

 Author, he obtains a formula expressing the number of the terms which, in a polynome, complete in all its 

 terms of every dimension from o to ii, are not divisible by any of the factors, A'^,B^, C-, &c.* He finds that, 



iu the complete polynome, the number of all the terms is represented by ~ — TTli — > ^"'^ '•'^^ number of 

 the terms not divisible by any of the said factors, by 



/[!( + 1]' I ' - [k + 1 - a]' I ' + [« + 1 - a - /jj I ' - &c.\ 



-[« + l-/3]-|'+&c. I (1.) 



-&c. / 



and this is the formula to which the Author's attention has been directed. It agrees in its general structure 

 with the Author's formula xiv : the points in which it differs will presently come under notice. 



In his 4th problem, Bezout considers a particular case of an incomplete polynome, meaning thereby a 

 polynome in which the highest dimension of one of the s quantities, A, is a, of another B, is /3, and so on; 

 a, ft, &'c., being less than ?/, the highest dimension of the polynome itself: and he here makes the observation, 

 that there are as many terms in such a polynome as there would be of terms not divisible by any of the factors 

 ^ " + ', B ^ * ', &c., in the polynome, supposing it to be complete ; but he gives no formula coextensive with 

 the generality of that observation. By following out that observation, we may, by two steps, deduce the 

 Author's formula xiv. from that of Bezout, 



The first step is the following. The terms which in the polynome, if complete, would be non-divisible 

 by any of the factors ^ » + ', S ^ + ', &c., amount in point of number to 



.[„ + 1J|'_|^„+1_(„ + 1)]<| ' + [« + ! -(a +l)-(/3+])]-|'-&c.\ 



1/ _|:„+]_(^ + l)J|.&c. 1 (2.) 



V -&c. . / 



and such, therefore, is the number of the terms in the incomplete polynome function of s quantities, where 

 a, 13, 7, &c., are the limits of the dimensions of A, B, C, &c., respectively ; the highest dimension of the 

 polynome itself being ;/. 



The second step is the following. If from a polynome whose highest dimension is «, all the terms of the 

 dimensions not exceeding (u- I) be deducted, the remainder will be the terms which the polynome contains 

 of the k"" dimension. Hence the number of the terms of the «"" dimension in the incomplete polynome 

 will be obtained, if in (2) we substitute n for (« + 1), and deduct the result from (2). That is to say, the 

 required number of terms will be A (2), meaning, by A (2), [[1 - £"'] (2) ; i. e., 



/[« + 1]-' I ' - [m + 1 - (a + 1)]'-' 1 1 + [!( + 1 - (n + 1) - (/3 + 1)]-' 1 1 - &c. \ 

 l^J -[« + l-(/3+l)]-|' + &c. ) (3.) 



which agrees with the Author's formula xiv. 



Considering that Bezout's work has now been published nearly seventy years, it will no doubt excite the 

 surprise of many members of the Society, that a deduction from Bezout's formula so easy as the foregoing, 

 should not have been made long ago, and applied to the solution of the problem of the combinations of 

 plural elements. 



" The complexity of Bezout's notation rendered it inexpedient to retain it in its original form. To facilitate comparison, the letters 

 have been assimilated to those used by the Author. 



