4G 



and 



D"[4]=D a [3]— D M "*i[3]; and soon; (VIII.) 



and the developed product of the binomes, 



[l-Vi],[l—a^], []-**],&&.: 

 that is to say, 



l_ a? a i-|_ a .ai+/3i_ ir «i4-/3i+yi-j- & c . 



— a^i+a^i+tt— &c. 

 -xVi+afii+n 



— &C. + &C. 



when multiplied into the development of [1 —x"] ~~ s , 

 manifestly leads to the following formula : 



D"C»]=D»[1]-S [D M - a i[l]]+S [D«— i-ft[l]]l yim) 



-S [D w -*i-&-^[l]] + &c. 



where, since the powers of a,', in (VI.) or (VII.) developed, are to be 

 all positive, no expression of the form 



(u — a,), (w— a l — /3 t ), (m— oc 1 — f3 l — y x ), &c. 

 is to be negative. Then by giving to 



D«[l],D M -"i[l],D w -*i-fc[l],&c. . . (IX.) 

 their respective values, we obtain the series of expressions : 



" LiJ— pZTfx^ 1 1«H \_s J 



where in all the kinds the elements are plural without limit ; a for- 

 mula given by Hirsch : 



D«[2]=^r« 1 -Mi-[« i -a 1 ]^i!^ = Q] 



where the elements A are limited in number to a, but those of the 

 other (s— 1) kinds are plural without limit : 



D<<m= — r^- 1 ' 1 -[^-aj^-^+^-ai-A?- 111 "] _ r*n 



-i ls -ui|_ — n«i— A3* -1 ' 1 -I L<J 



where, moreove-r, the elements B are limited in number to (3, but 

 those of the other (s— 2) kinds are plural without limit : and so for 

 the rest. The law of the terms being evident, they need not be 

 continued further. 



Example of (IX.). Given one element of 1 kind, two elements of 

 a 2nd kind, three of a 3rd, and four of a 4th ; and let ti=5. Then 



{ u >'l~TM 



r -4.5.6 ~) 



Rti Q -3.4.54-1.2.3 _ 99 



6 ' 7 - 8 -2.3.4 



1.2.3 



J 



