and General Differentiation and Integration. 325 

 {.+^Y = /[Ar+ f. A^• + 5- A-V + ^ A-\ ...] 

 = ^:-~^r (G) 



X 



in which the abbreviated operation refers to r as the variable. 

 It should here also be remarked, that the binomial develop- 

 ment terminates at the term immediately before that whose 

 coefficient becomes = ; and if this does not happen, it goes 

 on ad iri/inifum. We shall in more than one instance pre- 

 sently see the value of this apparently trifling observation. 



On General Differentiation and Integration. 

 We will commence with the simple differential 



jr n 

 a- X 



whose numerical coefficient is to be determined, whatever be 

 the values of r, n ; for the value of the exponent is under all 

 circumstances evidently n — r. Let us put n for the coeffi- 

 cient, and when r is a whole positive number the ordinary 

 processes of differentiation give 



(1) 



^.-2« (2) 



terminating at the term immediately before that whose coeffi- 

 cient becomes =0. The same rule of termination also holds 

 good in (1) when r is a whole number and n = r. 



Now when r is a variable integer it is easy to prove that 



r r-l , -1 , »— 2 . -2 o r-3 . -3 o 



n =n —n A, .r +n A, .r —n A, .r ... 



= n —1^ .r (3) 



the operations A~ .r° A~ ./, A~ .r" ...being respec- 

 tively equal to A r, A .r A »•, A .'A .r A r. 

 applied to unity or /". But since 



-1 „ r-l ^ -2 r.r-l . ..r-3 , r...r-2 



A. •'• = r-Y-y A, .r = 5-^ + —3 , 



— 3 o r...)— 5 r...r— 4 , r...r — 3 



^' ■'' = "^1T6" + "^73" '^ "T~ ' 

 it is plain that the coefficient of any term of (3) is composed 

 of factors similar to those of the develo))ment of the binomial, 

 and will conse(|uently admit a «nmilar proof for non-integer 

 values of r. Our (3) is therefore true for all values of r ra- 

 tional, 



