580 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



Let u, Ti be two functions of x capable of expansion in the form of exponen- 

 tials ; then 



d>^ (uv) _ dt^u dv d/^—^u M(M-1) 



d'^v d^-'^v 

 dx^ dx^-^ 



the order of the Binomial Theorem being observed. 



Let M=2Ae™^, v = 2Be«^ 



MtJ = 2Ae'"^ . 2Be»* 



= 2 ABe'" + "' 

 dx^ ^ ' 



= 2(?w'" + fxnm''-^ + ^^~y^ n^m''-^ + ...) Ae""Be«* 



J. • ^ 



_ c?^^ dv d'^-^u /^(/x-1) rfg^ ; rf^~^M 



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

 the following : to deduce the formulae (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 ^^^". 



''{^'■^) 



dxi' dx^ 



Let, therefore, "=-^' v=x'' -^ and we get from the theorem 



^ ^ ^ jm + u. r- \ / jn + u. — 1 



dxf^ ''^ X 



■^ '^ /m a;'^ + ^i 



1.2 ... / ^ ^ ^ ^ [^ ^m+f.-r 



X^ ( 



= (~1)'" — „ / — 1 /m+a-ar lm + UL—1 + ... 



+ M(M-l)..-(M-'-l)^^^i^}. 



~ -^_f . I (m + /x — 1) .. . (m + fx — r) /m + /Ji—r 



m 

 fxr 



— (m + /;i-2) ... {m + fx-r) /m^ fx-r 



