and General Differentiation and Integration. 327 



which admits of an algebraic equality with A^x" when ^=1. 

 From this expression we deduce 



A".o"= n. w — 1. w — 2 . . . (8) 



which terminates at the term immediately before that which 

 becomes = 0, or else proceeds on ad infinitum like the other 

 expressions. Hence we easily get 



a. X ^ . n—r 



ax A . ^ ' 



whatever be the values of n and r.* 



This is the commodious expression I before alluded to ; and 

 perhaps, independent of its importance and its containino- the 

 first attempt that has yet been made towards applying the pro- 

 perties of a". O" to general differentiation, mathematicians will 

 not think my anticipation of it, as an elegant theorem, un- 

 merited. 



We will now slightly glance at two or three of its conse- 

 quences. Suppose r is any positive integer, n being any num- 

 ber, then 



n — r A . 



n — r. n — » — 1. . . to 1 or <» 



= ?/...w-r + l.x"-'" (10) 



and if r be any negative integer, then 



•r" ^JLiS! =.r''+; 7,.n -\...\o\or^ / + '■ 



^""""'.0" '' n-\ r. . .n-n—\ ...toXoTc/i «+l ...n-\-r (^0 



which are known to be the ?th differential and integral of ^". 

 If n be a positive integer and r>n it appears by (10) that 

 the numeral coefficient n =0 



r 



which agrees with the result of ordinary differentiation. And 

 if n and ; be both negative integers, then 



— n.— «— l.-.to'' I 



", — > ~r:""r~r — .r,-_T-r = oo or — 



r-7i.r — »j— 1... tol orco r — n...\ — n 



CV3 holding when r is equal to or greater than n, and the finite 

 value when ;•<?/. This too is perfectly consistent with the 

 common integration ; for our theorem must of course give un- 

 corrected integrals. 'J^hus in the well-known /^ the com- 

 mon value log x is not the natural but a corrected value. 



* This theorem, by chanj^inj^ dirtereiitials into integrals hy a mere change 

 of the sign of the exponent of the order, would induce us to term what 

 we call intcgratior, negative differentiation. 



The . 



