244 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Now C is altogether independent of n : if, therefore, we take n positive and 

 greater than (-/"), which can always be done, we shall have proved generally, 

 that 



It wiU be observed that the properties on which the truth of equation (2) is 

 based, are these, — 



dxi^ /~" J. whatever be M. 



1. 



2. jn + l = nn 



3. ,,=/ e~°''^a"~ da, when n is positive. 



2. To find -J^- 



In my previous Memoir, Art. 19, I obtained an expression for ^SJL^ by as- 



Pd X 

 suming that / — =:loga;; an assumption which owes its correctness to the admit- 

 ted possibility of the introduction of an arbitrary constant of integration. Con- 

 sequently, the conclusions at which I arrived can only be correct generally, by the 

 aid of an arbitrary function of differentiation. Noav, it is our object to avoid the 

 use of such functions, and to obtain expressions for the general differential coeffi- 

 cient of all functions which shall be complete in themselves, so far as relates to 

 the satisfaction of every law of combination to which they may be subjected. It 



— = log X, and to substitute 



in its place some other function of x. The following process appears to be per- 

 fectly satisfactory. 



The value of ^-^=^' is l+i> log^+&e-l-y log ■^-&c- 

 p P 



= loga'— — logx + A^ + &c. 

 If, therefore, y be of a higher order than p, such as p^, it is manifest that 

 — will be a simple representation of log x, provided p=0 and ^=0. 



By adopting this mode of representation we obtain, 



dialog x ^ ly^^ ~ -(-rf^^Zl — - 



rf.t* ^ ' p^-pxf'-r ^ > p-qoif—i' 

 This expression comprehends every case, and appears to be the most simple 



