590 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



^ -^ [ (n — V) . . . (« — »•) 



(»-l)...(»-r-l) 



1.2 («-l)...(»-r-2) •■■ J 



which gives the above series if )=0 



n + 1 1 (w + 1) w 



Ti^r "^2" (»-i) («-2) 



~V\d=t-^ ^ ' {n-\)...{n-r)) ' 



r Y dx-^ In > 



'• I /-I /n r 



= -Q, an illusory expression. 



The fact is, that the function admits of different values dependent on the 

 value of 71. We shall proceed by a process of comparison to exhibit these values. 



1 . If « be a positive integer, the series must equal - 1 /^ in order to make 

 the result coincide with the above particular case, and 



hence the above numerator is unity. The same is time if n is a positive or nega- 

 tive fraction, in aU which cases /"^ in the denominator causes the first term to 

 vanish, and the result is 



'^"'"g^ = (-D-i /« or- 

 dx" ^ ^ ' 



But if n be a negative whole number (and in no other case), -= is finite ; 



so that in this case the first term does not vanish, and we get 



hi' (-l)'-i 



JTf - i-n) (-«-l) ... (2) 



d-Jogx^l^ _ (lo ga--!) (-1)"^' - 



d!^ a^' (-») (-M-1) . . . (2) /^l a" '^ 



or writing -m for n : 



rf-Jog^^^, log^-1 ^ „ IZZ cC^P 



dx-'" m{m-l) .. .2 /_1 



_ ,„ jog^-1 Jg^P 



""^ ^(m-1) . . .'2 m{m-l) . . . 2 



consequently we know that ^ ~ T "*" "2 ^ ' ' ' "^ m^ '" *^^^ ^^^^' 



