48 



versd, the author shows that hi, <r} will he identical with {<r — u, sr}, 

 provided |v, r} is identical with {t — v, t"\, and provided also iu—v-> 

 <r—rj is identical with {<?— 1~ (u — v), <r — r]. 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 fv, rl and every term of the series 

 hi — v, <r—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,<r} = [<r—u,o-} (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—u. 



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 m + m' + m"+ &c. ; 

 and u, the number of elements, is mv + m'v' + m"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 5 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" ,v" , 

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

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

 sented by 



and those of the latter desci-iption by 



P'i 



1to|1 # \m'\l^ 1?«"|1 



and their joint effect by the product 



gZ\-\ i-U 



PU X lm',1. \m'\\ % 1»."|I # ' • • • (XIII.) 



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 



