Cambridge Philosophical Society. 309 



the permutations corresponding to those elements and to the type, 

 will be 



JtoU * \m'\\ ' jm"!! 



&c. X P. 



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[Q x P], will 

 show how many different permutations can be formed from the given 

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

 computing the number of permutations, by examples. 



3. Let P < " I denote how many different permutations can be 



formed when u elements are taken at a time out of 5 kinds ; and P 

 {m, (TJ denote how many different permutations can be formed when 

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

 If all the elements may be repeated without limit, 



=i>[i«i.[i+..+£i + ....^+. ...]*]• 



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, 



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«l..<->[l+.+ ^+..^] 



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, 7, &c,. 



fi -^'\ 





(XVII.) 



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

 becomes 



