590 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 





+ r 



(n-V)...(n-r-V) 



TT2" (n-l) . . . (n-r-2) •■ ] 

 which gives the above series if r=0 



r{r + l) (n + V)n 



n + 1 1 (ii + l)n 



n-\ 2 (»-l)(w-2) 

 1 ( d-' X 



-yy'dlF^'' '""' ^ ^^^ (n-l)...(n-r)) ' """^ 



-ir 





1 







(»- 



-1)... 



(n 



-r) 



'a; 



/«- 



'-}• 



r = 



:0 



— r 



/» 







^ I /-I /« / 



= -7^, an illusory expression. 



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

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



1 . If w be a positive integer, the series must equal — 1 /^ in order to mak^ 

 the result coincide with the above particular case, and 



IT 



l/-^l = /0=_l = (i)„=0: 



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

 tive fraction, in all which cases / — 1 in the denominator causes the first term to 

 vanish, and the result is 



dx" ^ ' 



fn 



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 



f^ _ (-1)"-' 



7=T ~(-^)(-^-l)---(2) 



rfMog^ __1 (logar-1) (-1)" + ^ 



daf" ~^ '(ITn) (-n-l) . . . (2) /"^i ^ 



or writing —m for n : 



In . P 



rf-™ logic „ loff^-1 , 1. ^ /-»« „„ 



— - — § — =x'^ — , ^ ^, -^ + (-1) " x^ P 



dx-^ mim-V) ... 2 ^ /_1 



— X" 



m 



\ogx- \ x^^ 



(w-l)~.. .2 »«(m-l)...2 



consequently we know that ^ ~ T "*" "2 ^ " ' "*" _ i ^^ ^^^ ^^®^" 



