296 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



CoE. The same is true of the formula for the nth. difference of Ux, y : for 



AX, y={l + A^l + Ay-l }X, y 

 d d 



or =(«<*'" ^-1)"m.,_„ 



and the same results are produced as when n is a whole number. 



Prop. 4. F (a) e'--VW=e''-'' F (e-- TTk-l)/(a:) 



Let »x=:e'''^; and «j:=/(«) in Prop. 3. 



=/W(«''-l)" e'''^ + w A/a;(e'--l)'»-i e"+'' + &c. 

 = «'•■'' (e'--l + 6'- A)"/(a;) 



= e''-''(e''r+A-l>/W 

 which being true for aU values of n, shews that the following theorem is also 

 true: 



F(A).e''^/(«) = e''-^F(e'- 1 + A-l) .f{x) 



/ 1 n \ r 108(1 + -) 1 



PkOP. 5. AM,= ^(l + -)"-lj«, = |(1 + A,) -'-l^U, 



Let a;=e^ and let u^ be represented by ug when e" is written for x, let also 

 A/ be the symbol of difference ut+i-ug. Then by (C), when n is an integer, 



;i«^[!iii=D(D-l)(D-2) .... (D-»j + 1)m^ 

 rfa;" 



= ((x4f-,)., 



But A^« = (eD-l)% .-. gl^M^^Cl + A^)?*^ 



or D=log(l + A^). 



Hence (x. i)" = (l. 1)— '=(X..,)'"^C'-') 



and Ae*,= ((l + ^f -l)«,= [(l + A,)'"^^'"^)-!}"* 



log (l + -) 



Cor. «^+„=(i + Atf) ^ ^'m^ 



