PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 575 



TT 



Now, /»./!-» = -. when » is a fraction less than 1, 



lr—{n — y) /l^^n — fx — r = — 



IT 



sin (r — n + fx) TT 



js — n j\ — s-\-n = 



IT 



sin (s—n)nT 



/-(n-fx) /n + 1 sm(s-n)'7r 



hence — f= — = {—^) 



l-n ^ } • /„_^ + i •sin(y-?i + ^)7r 



ln-\-\ (— 1)— * + ^ . cos sTT sinwTT 



ln-il-v\ ■ (-l)-'' + ^-cosr7rsin(«-/x)'7r 



/w + 1 sinwTT 



ln-\x-^\ ' sin(w-/x)'7r 

 provided that n and «-// are both fractions. 



In this case, then, -j^ = (-1) ^==- • .„-(„— j-^ ^ . 



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



{a) Let n be an integer : 

 then /n-n=(-V)" n (n-1) ... 1 . /^^ 



= (-!)« /w + 1/- 



w 



/r-(n-fx) . /n-fJ.-r + l 

 /jjL-n >— /x + 1 



~(-l)« ■ sm(r-n + fx)v /^_^ + l ' /«-» 



= (-l^'^ + l . "^ /^ + 1 1 



sin(«-/i)7r' /n-n + l ' /n-n 

 =0, '.' /n — n = CO 



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



/fx—n 



(b) Let /»-/z be an integer ; then it follows from the last case that — — =°o , 



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



(c) Let both n and n-fxhe integers ; 



then /n—n=(—l)" /n + 1 /—n 



/« - /x — (% — ju) = ( — I)"—'' /n — fJL+i /fx — n 

 VOL. XIV. PART IL 5 K 



