PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 59I 



20. We may exhibit the value of P as a definite integral thus : 



p _ n + l 1 (*^^- !> _ 1 (n + 1) (n) 



11-1 2 (ri-i){n-2) 3 (n-2) {n-S) 



= {n + 1) n \ —7 Tr + o 7 Tw oV + • • • i 



^ {n{n-~V) 2 {n — V){n — 2) j 



x" 1 x'^'~^ 



Let therefore, S = ~ — irr +^ j—rv? — ^k + • • • 



then ^ = :«;"-^ + 1 ^"-"^ + 1 «;"-* + . • . 



_ ^1 r 1 1 1 1 1 ^ 



= a:«-Mog(l-l) 

 S = - /'■ /* f/.i; dx X"-' log (1 - - ) 

 and P = — (n + l) • n J f^ dx dx x"-^ log ( 1 | . 



The limits of the integral in this case are not determined. In fact, the in- 

 ferior limit wiU depend on the value of n. If n be negative, this limit is oc . 



21. We shall then, adopt the following value for — . „ » viz. : 



JI^ZE^L {log.-(l + P) 1 ; where' V=''±^,+ _J^±i^^. 1 + &c. 

 /-I . «" ^ ■' **-! (re-l)(»^-2) 2 



Ex. 1. To find the differential to the index ^ of log x. 



dHogx _ {-'^fJ_^ log^ . -^.j Vj (1 + P) 

 6?.r^ /-I V« ^ "^ /-I V.^ 



1 + P 



Now, by what we have seen, -=r- = -1 



^nog-r /^ I , 



^/- 



TT 



Ex.2. To find _fli2£^. 



We have the choice of three methods, which we give as follows : 

 (1) Jllj2«i = (-i)s/J..-i 



= 2 V^^^ «~^ from the formula. 



VOL. XIV. PART II. 5 



