•242 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



'-—^=(-1)'* — a;""'', whatever be n and u. This form can be proved to be 



til* ^ l-n 



the correct one in every interpretable case, and can be deduced from the gene- 



raUzation of when n is negative.* We shall at present assume it as the 



d>^ x" 

 defininy property or definition of . 



When, from this definition, we can deduce the differential coefBcients of e' 

 and of log x, that is, of the ascending and descending index-function, we are in 

 possession of the three fundamental forms from which aU others may be de- 

 rived. The following mode of arriving at those differential coefficients is differ- 

 ent from that which has hitherto been given, and appears to leave nothing to be 

 desired. 



1. To find ^-^. 



,, ^ c^ x' c'^ x^ . c" of 



<^^^-, _^^^■J=l±ili <^'^' 



■={-vrv 



" d^ l-r jTTl 



= (-cf^ [z-'^ + ~z-'^+ ^2-'' + &c.], where2=c^; 

 /O I az dz ' ) 



* See Part I., and the excellent Memoir of M. Liouvilie, referred to in that Treatise. 

 Another formula has been proposed, viz. 



dot jl+n-^ 



I have lately received from Mr W. Center, of Langside, some judicious remarks on these formulae, 

 contrasting the results arrived at by them respectively. He shews that (without continual introduction 

 of an infinite arbitrary constant) the latter formula is inapplicable in many of the most simple cases : 



for example, in d'' of ^ expanded positively, it gives, when applied, infinity on one side and not on 



the other, and when expanded negatively, infinity on both sides ; and again, it gives for or 



the value ~^;::^ a"'', which is a function of x when /* is a positive proper fraction. 



