51 



the permutations corresponding to those elements and to the type, 

 will be 



fm\ — 1 ifm'\ — l "pn"\ — \ 



&C. X P. 



ym\\ ' jm'U " \m"\\ ' ~ 



Every different partition of u that may be formed within the limits 

 pointed out in art. 10, Section II., will give rise to a similar product, 

 QxP ; and the sum of all these particular products, S[QxP], will 

 show how many different permutations can be formed from the given 

 elements, taken u at a time. The author illustrates this method of 

 computing the number of permutations, by examples.. 



3. Let P < U > denote how many different permutations can be 



formed when u elements are taken at a time out of s kinds ; and P 

 [u, <rj denote how many different permutations can be formed when 

 u elements are taken at a time out of cr, a finite number of elements. 

 If all the elements may be repeated without limit, 



l u \ =D M [l w l 1 .£ s *]=s M 



=D M [i^[i+, + £l+....^+....] s ]. 



Hence the author infers that, if the elements A are limited in num- 

 ber to a, while those of the other (s— 1) kinds are plural without 

 limit, 



p{;}=D-[i« ii ^ i >ii+»+3+..+^-]]i 



that if, moreover, the elements B are limited in number to /3, while 

 the other (s — 2) kinds are plural without limit, 



P {«}=D»[l.i. e <.-*[l+,+ i5l + ..^] 



and so on, until finally, if all the elements are finite in number, and 

 the elements A, B, C, &c. are respectively limited, in point of num- 

 ber, to a, /3, y, &c, 



P { »,, } =D»[ 1 «|.[l+,+ ..^ r ][l+,+ ..^ T ] 



(XVII.) 



4. Hence, if in all the s kinds the elements are dual, (XVII.) 

 becomes 



