Cambridge Philosophical Society. 303 



kinds are determinate in number, and their number is 5, let < > denote 



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

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

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

 {m, (tJ. If (p (a:) be any function oi x, let D" (p(a;) denote the co- 

 efficient of a?" in that function developed according to the powers oix. 

 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 5 geometrical progres- 

 sions, 



•[H-Aa + A2a;2+ + A"a?"], [l+B^ + B2a;'+....+B'^a^],&c. 



Then K will be of the form, 



l+S[A]a? + S[A2 + AB]a;2+S[A3 + A«B + ABC]a:3+&c., 



and D«[K] will be of the form 



S[A/'B^/C'-&c.], 



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

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

 and their sum comprehending all the possible combinations of those 

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

 equal to 1, K will become 



each of the terms A^.B*?. C''. &c. will become 1, and the number of 

 all the terms of the form A^B'i'C'" .. . which Dw[K] or S [A^B^C''...] 

 contains, that is to say, |m, (t} will be represented by D" [A] ; which 



latter coefficient the author next proceeds to determine. 

 Now 



« ^ . — 1— — -. &c. =[1— .-r ^ iri— ar^ ]..[1 — a;] 



\—X \—X ^ -"- J L J 



=[i---+']ci-»^+']...sTr^*-»i 



u Ll"|i J 



=[i-^"+'][i— ^+']...p^,s^[[«+ir-''V](vii.) 



For brevity, write u^, a,^, /3i, &c. respectively, forM-l-l,a+l,/3 + l, 

 &o.; andalsowrite [l]for[l-^x]-^ [2]for[l-a;«i][l-a.]-*; 

 that is, for [1-.t"i] . [1]; [3] for [l-a;«i] [l-.r^i] [l-ar]"*; 

 that is, for [1— a;^i].C2], and so on. Then 



D«[2]=D«[l]-D«-«i[l]; 

 and 



D«[3] =D«[2] -D«-^i[ 2] ; 



* According to the factorial notation, here used by the author, «wl±i 

 represents s [s-4 1][5+2J. ..[*+_(«— 1)]. 



^VI.) 



