584 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



«?'' a;« _ ^ /n + 1 sin n tt 



(-1) 



dxf^ /n-fx + l sin(«-/i)7r 



But /n+1 = n /n 



X 



.n — ^ 



/n = (« — 1) /n — 1 



/n — fX + 2 = (n — fx + V) /n — fJL + l 



/n + 1 = n(n — l) ... (n — fx + l) /n- /jl + I 

 d^ x" , tsu, , -,N / . -.N sin w 77 



and ::^^={-lYn(n-l)... (n-fx + l)-^ 



ji—f^ 



ds^ sin («-//) TT 



Now, if w be a fraction, -r—, -r— = ; and — — - = 1 : 



' sin {n — fj.j'K cos fxir cos fxir 



d^ x^ 



hence = » («-l) • • • ("-^ + 1) a-" •'■ . 



dx'^ 



But if n be an integer, 



sinwTT ix • 1 ji , 



imT^T^nuTTT^'O ^^ obtain by the observations in art. 6. 

 -r—, r — with the limitations imposed on it equivalent to 



sin (n— 1^)11 ^ '■ 



sin » TT 1 , „ 



as before ; 



sin W TT COS fX TT COS fX TT 



hence the result is true in all cases. 



2. If M be greater than n, we may use formula (3). 

 If n be a fraction. 



d'^af' _/_-,x^i + l sinnTT /n + l/fx-n 



dx^ TT ^—n 



ifi n 



If n be an integer, sin w tt = 0, and ^ = o 



dx" 



df^ t"' 

 3. If IJi = n, ^^L^- = n(n-l) ...2.1. 



dx^ 



Ex. 3. To find the differential of a?" when ?z is a positive quantity and [i a 

 negative integer. 



Here we may adopt our formula (2). 



dx-"^ jn-\m + l sin (w + »?) TT ■ 



and by the restrictions, if /^ be an integer, or actually if it be not 



