308 Cambridge Philosophical Society. 



Consequently the required sum is 



,_j-, [„-i].-,i- 



If in (XVI.) ^ varies from to m— 1, 



J [^ pii ^ 2,_i|, J j„u' 



this summation being a particular case of formula (XL). The result 

 agrees with D"[l] formula (IX.), art. 3. 



10. When the given elements are all finite in number, we may 

 determine •[«, crj, by taking the sum of all the particular determina- 

 tions that may be obtained pursuant to art. 9, by giving to z the 

 successive values 0, 1,2, 3, &c. If w <: s, the upper limit of z is m, 

 and the number of types to be formed is [2m, m,]; which becomes 

 [2s, Sj], if M=s. If M > s, the upper limit of 2 is .9; and the number 

 of tj^pes to be formed is [w +5, sj . (See articles 4 and 5, Section I.) 

 But, if the repetition is finite, some of these partitions may fail to 

 yield combinations. 



11. If the elements A, B, C, &c. represent different prime num- 

 bers, all the methods and theorems contained in this section will 

 apply, mutatis mutandis, to the composite numbers of whicli those 

 primes, or the powers of those primes, are divisors. 



III. On Permutations. 



1 . Let the given elements be of 5 different kinds. We can de- 

 termine in two known cases, by an explicit function of u, when the 

 elements are taken m at a time, in how many different ways they can 

 be permuted. The number of the permutations is denoted, when 

 there is but one element of a kind, by s'«l-i ; and when in all the 

 kinds the elements are plural without limit, by s". When the plu- 

 rality is finite, it is only in the particular case of all the elements 

 being permuted at a time, that there is a known formula to express 

 the number of their permutations. 



2. Every combination constructed on a given type, u=.mv-\-m'v' 

 + m"v"+ &c., will generate the same number of permutations, 



]^ ^p 



Mciinwi rii)'ii-im'n""in»»" . &c. 



Therefore, if the number of the different combinations which can be 

 constructed out of the given elements in conformity with that type, 

 is represented by Q, Q X P will be the number of the permutations 

 corresponding to the type and to those elements. If the plurality 

 be without limit, 



xP 



JWlll^ 1»»'|1. l»i"|l^ gjg 



will be the number of the permutations. If the given elements be 

 finite in number, as in formulas (XIV.) and (XV.), the number of 



