and General Differentiation and Integration. 329 



It is also obvious from n,- in (9) that 



d''-''^'-=d-'dW' = dWd-'=d'd-''d'' 



a well-known property. 



I have before observed that our formula (9) gives only the 

 uncorrected integral. If the correct and complete integral be 

 sought when r is an integer it will be 



d.~''.x' + aod^~\a:° + a,d^~\x°...a^_^d°.x° (12) 



in which a^, «,, a.,. . . are the arbitrary constants. And if r be 

 a rational fraction = ^, the complete integral will be 



^ m—l m—2 1 



d ' .x^+aod" ' .x° +a,d^~'^.x°...a^_^d^~T^:c°{l3) 

 the arbitrary constants being such, if the problem requires it, 

 as to neutralize the fractional differentiation on the integer 

 power 0. We must therefore in considering (13) regard the 

 several orders of d as referring to the powers of .r only, and 

 the numeral coefficients as comprehended in the arbitrary con- 

 stants. This remark is of considerable importance towards a 

 right understanding of the import of (13). It would other- 

 wise appear that a successive differentiation of the order J_ to 



— , which successively nulls the right hand terms and ulti- 

 mately leaves as it should = x", is different from a sino-le dif- 

 ferentiation of the order -^ ; whereas it ought to be and in- 

 deed is the same. For the — differential of any term the 



a _ d * . x° is actually 



"^ . _"»— y i_!?r^ J!L 



^'-^y-\<^ ' '■^° = "„^\(^ 'd'>.x° 



which, since by our preceding d* .x° = and ad ' . x° 

 is finite, is null. 



Hence it appears that the complete ?th integral of any sim- 

 ple quantity x" consists of as many terms as there are units 

 plus one in the numerator of the order r, the order being in 

 its lowest terms. Therefore when the order is irrational or 

 imaginary the number of terms is infinite. 



We may now easily deduce some interesting properties of 

 the series (3) and (5), and show how to approximate to the 



n n 



value of the ratio - ^^^ ^^_^ , when the factors run on to in- 

 \'<i]. (if). Sn. fi'2.->. M,7i/ ]H2r,. Tl /illily 



