﻿Higher Derivative of a Function. 829 



Therefore 



d n+l y nyirK-l Ag-1\ g ^ K - <t ~ 1 1 1 du Sty 



dai n +l~ tJia^ } \ u / dx n ]{ K -l)\ dm' du K 



-1lJo (_1) u+i) (a+1)u ^ u a_1 J«i dxdl* 



/ tf \ a + 1 __//£ — 1\ 1 



+ 

 But 



therefore 



^» + i y -.n i ,/w\ ^"m"-* ! 1 du d n + l y 



dm^ 1 Lafo ^ ^ U/ " "^J n! <te dt^ 1 



n 1 K ~ 1 a. /kl\ rfn + lyK-a (IK 



We must now show that 



+1 V P^ 1 / -. ^ a f n \ d n u n ~ a -\ 



1 d n+1 r * n «/n+lA a _^+V+^-| 



i_ ^m ^ n+1 z/ r^ 1 



7i ! <i«2? <iw 





(n + l)S o (-l)( a ^-^^^=£ o (-l)( a |„ ^ ra+1 



or if we let n — # = /3, we must show that 



(n+1) ,i ( - 1} Ur *^^ = *o ( - 1} U+i> n "^^r- 



... (6) 



Now ^+1^+1 



d.v n + l 



and (6) reduces to 



^& d S- 





