and General Differentiation and Integration. 331 



numerator and denominator, until the exponents in the deno- 

 minator are reduced to 0, we shall have 



A-"./= I"i + ,^'11'- A./+ i^^-A'.,-.(17) 



employing the same notation for the differential coefficients as 

 before, and ultimately putting t = I. This expression, it is 

 evident, always terminates when u is a whole positive num- 

 ber. It however depends on the properties of the differential 

 numeral coefficients whether the value of the formula be no- 

 thing, finite, or infinite ; but having considered these at some 

 length in the general differentiation it w^ould be tedious to go 

 through a repetition in an application of them in this place. 

 Most of the properties are very similar to and coincide with 

 those for the integration of jt", when m is a non-integer. If n 

 be an integer, then 



— n V x...i — n-\-\ V „ x-)-l...x— 71 + 1 ^ »' , o ■r+2...J — n-)-l 



A"'.x\.. _ _ (18) 



This expression terminates when u is a positive integer 

 whatever be the value of x, and also when x is an integer 

 from <?^ — 1 to -co whatever be the value of d; a property 

 that I have not noticed in any other theorem for integration 

 I have yet met with, and which may obviously be applied to 

 some curious purposes. 



The formula (18), though partially corrected, gives not the 

 complete integral. What we have done is only to bring it 

 from an infinite to a finite value. To have the complete in- 

 tegral difference requires a correction with the inferior na- 

 tural integrals, similar to the correction in the negative diffe- 

 rential. The complete integral is therefore 



when n is an integer. Precisely analogous to (13) comes out 

 likewise tlie complete integral difference when n is a fraction; 

 and I therefore think it unnecessary to give it. 



The transformation in (6) or (7) applies equally well whether 

 we take x", or u^ any function whatever of x. Therefore by 

 (17), (18), and (19) we shall have 

 ^-'.., = J^„ +2 ^^!^A«^ + 3„^-:i^AV-f...(20) 



the natural integral for any value of n ; and 



— n 



A n = 



r...r-n+l , « r+l...x-n-fl 



«- + 2,^£iif-A«,+ ... (21) 

 T t 2 t*»e 



