580 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Let M, T be two functions of x capable of expansion in the form of exponen- 

 tials ; then 



dx" ~^\lx'' '^' dx' ci^i--i 1.2 



d^v di'-'^v 



the order of the Binomial Theorem being observed. 

 Let u = l.ke^' ,v-1'&e''' 



=2ABe'» + "=' 



d'^{uv) „ . -n . , -.u. m + nx 

 ^ — =^ A. a (in + 11)'^ e 



dx'^ 



= 2 (»« + «)'' Af"". Be" 



= 2 {m'^ + fin »«''-' + ~^^~- n^ »re''~^ + . . . ) A e"^ B e"- 



+ ... 



di^u dv^ di"-' u /J^iix-l) d^v rf'' 



dxi^ ^^dx- dx>'-^ ^ 1-2" ■ ''^ rf:*:''-^ 



12. The application which we pm-pose to make of this theorem at present is 

 the following : to deduce the formulse (2), (3), and (4), from the fundamental 

 formula. 



Let r be an integer such that n=r—m, where m is a fraction less than unity ; 



then ''"''" - 



"'('4) 



dxi' dxi' 



Let, therefore, «= — - ''=^' ; and we get from the theorem 



1 



1.2 ... >• ^ ' ^ ' [^ 



_ (-1)'^'~^ I ^^ + ^_i) ... („j + ^_y) Im + fx-r 



fxr 



■I-{m + ix-2) ... (m + fJL-r) /m-i-fi-r 



