PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 245 



form under which the /xth differential coefficient of a logarithm can be repre- 

 sented. 



We shall reduce it in the different cases : 



1. When /x is a negative whole number =-m. 



l]JL^P=l-(m+p); and /Z]^=(_jo-l) /-;)-! 



=^i-p-l){-p-2)...{-p-?n)l-(m+p) 



l-P l^-f^+P 



and B^(_i)»^/r^ 



Hence 

 But 



l-g /1-M + ? 



/! + ? 



cMogx^ _-_^.2^ Jl+jp a'-^+^' ^_ .2,^ /1 + g ^ 



]=^ = ,T--=(l-i»A + &c.)whereA=i + ^+ .,.J_ 

 and a;'' = 1 +/) log x &c. 



1 + ? 1 



also 1^ = ,=:^ (1 -? A + &c.) 



Il-fi + q /l-/x' 



a;* = l + y loga7 + &c. 



rf^ log ■^ _^2^ (1 -jp A + &c.) (1 + p log ^ + &c.) 

 d3f jl^' p 



_£Z^ (l-gA + &c.)(l + glogx +&C.) 



= ^ (log a; — A— -loga:+^ h &c. ) 



/i-/x\ P P ) 



(log a;- A), since p and - are both equal to 0. 



/r=7x 



Hence rfa^-" ~/^TT I '°^ '*'" Vi "^2 "^ '^*'' """raj J ^liich is a well known 



c? a;™ log a; 



2. If yu be not a negative whole number, [jx is finite ; and 



B=-/p = -/M(l + B;, + &c.) 

 Pl-P l^-P 



by supposing this function (which is finite) expanded in terms of p,- 



VOL. XVI. PART III. 3 Q 



