306 Cambridge Philosophical Society, 



versd, the author shows that [u,or^ will be identical with /«■— m, tr}, 

 provided {«.''} is identical with {t—v, r|, and provided also {u—v, 

 cr—t'^ is identical with /a— r— (m— jj), tr— rl. But this identity 

 actually exists when T consists of elements of one kind only, and 

 when T' also consists of elements of one kind only. For, in that 

 case, every term of the series |f, t'I and every term of the series 

 |m--u, cr— r} is equal to 1. Let the elements of the single kind 

 which T contains, be different from those of the single kind which 

 T' contains. Then the identity in question will exist, when S con- 

 sists of elements, finite in number, of two different kinds : conse- 

 quently, it exists also when T consists of elements, finite in number, 

 of two different kinds, and T' consists of elements, finite in number, 

 of one or two other kind or kinds ; that is, when S consists of ele- 

 ments, finite in number, of three or four different kinds. And there- 

 fore universally, in the case as well of finitely plural, as of singular 

 elements, the following law obtains : 



{u,v) = [<T-u,(r) (XII.) 



Hence it follows that in applying formulas (IX.) and (X.) to parti- 

 cular cases, the labour of computation will be shortened by substi- 

 tuting for the variable the lesser of the two numbers u and <r— m. 



8. The author next considers how many different combinations 

 can be formed from a given set of elements, when every combination 

 is to be constructed in conformity with a given type ; in which type 

 there are m different kinds containing v elements each, m' other dif- 

 ferent kinds containing v' elements each, m" other different kinds 

 containing v" elements each, and so on ; and where, consequently, 

 in each combination, z, the number of kinds, is wj-{-Jw'-j-/w"+ &C' ; 

 and u, the number of elements, is mv + m'v' -^tri'v" + &c. The type 

 remaining constant, any combination conformable thereto may be 

 altered, either by changing the particular z kinds which are selected 

 out of the s given kinds ; or, the kinds remaining the same, by alter- 

 ing the distribution of the parts v,v,v, . . . (m)v', v', v', . . . {m*)v'' ,v" yv" , 

 . . . {m") &c., among those kinds. When all the elements are plural 

 without limit, the changes of the former description will be repre- 

 sented by 



1*11 ' 

 and those of the latter description by 



\m\\^ \m,'\\^ \iii."\\ , , . * 



and their joint effect by the product 



~pii~ ^ lm|l^ im'li, l»n"|l, * • • •* (XIU.) 



But when the elements of all the given kinds are finite in number, 

 class these kinds, so that each kind in class 1 contains not fewer 



