PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 575 



IT 



Now, Ip . l\—p — -. when » is a fraction less than 1. 



'^ -^ &\npir ^ 



fr — {n — fx) ll + n — fx — r 



sin {r — n + fXjTT 



js — n /! — « + « =—7 



sin (*— ») "TT 



hence 



/-{n-^) _ , /w + 1 sin(g-w)'7r 



[^ ' /n~fx + \ ' sin (r-n + /x) TT 



/n + 1 (— l)-^+i . cos«7r sinwTT 



/n—fjL+l (—!)"'■■'■■' -cos J-T sin (w—/x)7r 

 /»» + l sinwTT 



/n-fx + l sin {n-fji)v 

 provided that n and n—fj. are both fractions. 



, ^,. ^, rf'^z" , ,\i. /w + 1 sin WIT ._„ 



In this case, then, = (-1) =r=^ • —-, x" >- . 



dx^ /n-fx + 1 sm{n-fji)'7r 



But if one of the quantities be an integer, we must proceed differently. 

 (a) Let n be an integer : 



then 



hence the formula above will give the result, whether it be correct in form or not. 



(b) Let n-/x be an integer ; then it follows from the last case that ~=^=oi , 



l—n 



so that the formula above gives the correct result in this case also. 



(c) Let both n and n—fihe integers ; 

 then /n-n={-lY /^+T A^ 



/n-fi-{n-ii) = {-iy^i^ /n-fl+1 /fl-n 

 VOL. XIV. PART IL ' 5 K 



