304 Cambridge Philosophical Society. 



and 



D"[4]=D«[3]-D"-5'i[3]; and soon; (VIII.) 



and the developed product of the binomes, 



[l-a;«i],Cl-aA],[i_a;yi],&c.; 

 that is to say, 



1— a?*i+^«i+^i-a'"i+'^i+yi+ &c. 



— &c. +&C. 

 when multiplied into the development of [1 —a;] ~*, 

 manifestly leads to the following formula : 

 D«M=D«[l]-S[D«-«i[l]]+S[D«-«>-^.[l]]l 



- S [D^-^i-'^i-yiCl]] + &c. J 



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

 all positive, no expression of the form 



(«—«,), (m — ai— /3i), (m— ai— /5i— 7,), &c, 

 is to be negative. Then by giving to 



D«[l],D"-«i[l],D«-«i-^i[l],&c. . . (IX.) 

 their respective values, we obtain the series of expressions : 



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

 mula given by Hirsch : 



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

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



J p-iuL _|-„^_^j.-iii J-Lo 



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 m=5. Then 



1 

 {"•''} =1:2:3 



-4.5.6 

 -3.4.5 + 1.2.3 



^•^•S -2.3.4 

 -1.2.3 



= 22. 



