PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 59I 



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



p_ w + l 1 (m + 1)m 1 (n + 1) (11) 



Let therefore, S = ^^^^^^ +-^ ( .-i)(.-2) "^ " " " 



then S-=^"-^4-^"-^ + ^""^ + -- 



= a:"-Mog (1--) 

 S = - /'■ /"' dx dx a:"-! log (1 -i) 



and P = - (» + 1) • « /\/"' f^^ <i^ a:"^' '°g (1 ~ T ) • 



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

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



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



= { loga;-(l + P) [ ; where ?= — - + . .,. / — -^^ . ;= + &c. 



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



dHogx _ {-'^ff\ \ogx i-^i l_j_ (1 + P) 



dx'' /_1 six ^ ' /_1 S.IX 



1 + p 

 Now, by -vyhat we have seen, -==r ~~^ 



;f^=/T.(-i)i.- 



c?" log 2; _ 



dx^ ~ ' 2 



_ V — TT 



Ex.2. To find .f!i2£^. 



dx'' 



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



(1) ^li2g^ = (-i)W 



^3 ^-i 



dx"- 



= 2 V-'TT a: i from the formula. 



VOL. XIV. PART 11. 5 



