Mr. Jarrett on Algebraic Notation. 67 



■'• ^2m-l = fll • "r •(£'2r-l)- 



These two examples will be found of great use in the investigation of the 

 general term of many series. 



6. Prop. If b is independent of m, then shall 



t 



For, P"^.{a„,b) = Uib.a.b.a^b . . .a„b 



— !f .Oi 02 Ug. . . On 



= b" . P",„ («,„). • 



7. Prop. Whatever is the form of «„„ P°,„ . («„,) = !. 



For, P',„. («„,) = P„r^^'.(a,„) = P„r'.{a,„) . P\, («,„+„_,), (Art. 5.) 

 . p ii-i- /„ \ III \^'i') 



•* 711 V ^m + n — r} 



Put now r = n, and we get P",,,. (a,,,) = -p^~^\ = 1. 



8. The most common form in which factorials occur, is that 



of an arithmetical series. A factorial of this kind consisting of 



m factors, of which n is the first, and of which the common 



difference is ± r, may be denoted by \n ; the particular case 



in which the common difference is - 1, we may represent by 

 |n, and if, in this case, m = n, the index subscript may be 



m 



omitted : 



