584 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



(-If 



rfx'' /n-n + 1 sin(w-/^)7r 



But /n + 1 = n /n 



In — (« — 1) ln — \ 



ln — fl + 2 = {n — ix + V) /n-fi + l 



/n + 1 = «(«-!) . .. (w— /X + 1) /»-)Ll + l 



^^ sin («-//) TT 



„ , o .. sinwTT 1 1 ( — 1)'' 1 



Now, if « be a fraction, -^^^Z^ = ^^^^ ' and ^^-=1 = 



hence —^— = '» ("-1) • • • ("-M + 1) ••^' " ■ 



But if w be an integer, 



-^-^7 — =17 we obtain by the observations in art. 6. 



am (n - fji) IT •' 



sinwT — ^^.^j^ ^j^^ hinitations imposed on it equivalent to 



'n-IX)w 



sin (: 



sinw TT 



as before ; 



sin W TT cos fX TT COS fX IT 



hence the result is true in all cases. 



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

 If « be a fraction, 



a"**-" _/_ixf^ + i sinwTT /w + 1 /f t-w 



If n be an integer, sm w tt = 0, and _r__ = o . 



1 n 



ax 



3 If fi = n, fi-^— = «. («-l) . . . 2 . 1 . 

 dxi" 



Ex. 3. To find the difiPerential of x" when n is a positive quantity and fx a 



negative integer. 



Here we may adopt our formula (2). 



/« + m + 1 'sin(« + »0'^' 



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



