488 



Mr. WARBURTON, ON THE PARTITION OF NUMBERS, 



This theorem, I fear, is not likely to facilitate much the practical computation of the permutations of 

 plural elements, though perhaps it may lead to curious Algebraic results. Professor De Morgan, since 

 I made known to him this Theorem, has done much to remove the difficulties which beset the com- 

 puting of permutations by means of it. But I doubt whether the method will be rendered more simple 

 than that derived from a direct consideration of the problem of permutations, given in Article 2 of 

 the present Section. Thoroughly examined, the two methods must in the end prove identical. 

 I had some expectation that by giving to the Polynomes the form 



(e' - a) (e' - b) (e' - c), &c. = e'' - ae""' + abe''-^ - &c. 



- b + ac 



— c + be, 



some facilities might be afforded to the computing of permutations in certain cases ; but I do not 

 at present believe that any such results are to be anticipated. 



7. The theorem, xxvi, has led me to the determination, in one particular case, of an 

 explicit function of u, for expressing the number of the permutations formed by « kinds of elements 

 taken u at a time: the case is that where, in all the kinds, the elements are dual. If we develope 



« 

 l_Ao + AiW + Aiio^ + &c.]', 



by Arbogast's method, we obtain for the coefficient of x", 



where D* (^1)""'' is the coefficient of x'' in 



and a second developement leads to a double series, in which, if A„ and Ai are made equal to 1, 

 and A., to 1, and all the other terms A^, A^, &c., are made equal to 0, we obtain terms 



expressing the coefficient of a;" in j 1 + ,r? h ] ; and that series multiplied by l"|' gives the 



number of the permutations in the case stated*. 



I here transcribe the coefficient of j:", in [1 + x +.-l.^x' + A^x'+u^]' obtained by Arbogast's method, slightly moditied. 



