45 



2. If u elements enter at a time into each combination, and the 



hinds are determinate in number, and their number is s, let < > denote 



how many different combinations can then be formed : if the elements 

 are determinate in number, and their number is cr, let the number 

 of the combinations which can then be constructed, be denoted by 

 {u, cr }. If <p (x) be any function of x, let D u <p(x) denote the co- 

 efficient of x w in that function developed according to the powers of*. 



3. The same things as before being assumed, let a given set of 

 elements consist of a elements of the kind A, +/3 elements of the 

 kind B, + &c. Take the product, K, of the s geometrical progres- 

 sions, 



[1+Aa,+A^ + + A"-**], [l+B* + B 2 ^4-....+B^],&c. 



Then K will be of the form, 



1+S[A> + S[A2 + AB>°-+S[A 3 + A*B + ABC>3+ & c<> 

 and D M [K] will be of the form 



S[A^B?0&c], 



the last expression being an aggregate of terms of the form A^B^C . . . , 

 each containing a different combination of u of the given elements, 

 and their sum comprehending all the possible combinations of those 

 elements taken % at a time. Now, if A, B, C, &c. be each made 

 equal to 1, K will become 



A=[l + x + x"+ . , + ar*][l+a; + a? 9 + .. +a^]. &c. ; 

 each of the terms Ap.Bv. O. &c. will become 1, and the number of 

 all the terms of the form A*>B?0 . . . which D M [K] or S [ A*>B?0 . . . ] 

 contains, that is to say, Su, <r\ will be represented by D u [A] ; which 



latter coefficient the author next proceeds to determine. 



Now 



l-x 1-x J I vi.) 



= [i-^ +1 ][i-/ +1 ]...s^»l J 



u Ll M |i J 



p— J|l M l J 



For brevity, write w 1} a l5 j3 u &c. respectively, for u+1, a + l, fi + 1, 

 &c; and also write [1] for [1— a?] -5 ; [2]for[l — ar«i] [1— x~\ ~ s > 

 that is, for [l-* a i].[l]; [3] for [1 -x a ^ [I— aft] [1-*]-*; 

 that is, for [1— a$i].[2], and so on. Then 



D M [2]=D M [1]— D M_ "i[l]; 

 and 



D M [3] =D M [2] -D M ~^i[ 2] ; 



* According to the factorial notation, here used by the author, s"l ±l 

 represents s [s+l][s-\-2]...[s-\-(u— 1)]. 



