457 



II. If we take the differential coefficient of x", and mul- 

 tiply it by X, the result will be x" n ; that is to say, 



d 

 x" n.z=:x-r x" ; 

 ax 



and as a consequence of this equation, we shall likewise have, 



X- y {n) = F [x^^ x\ (2) 



In the right-hand member of this equation, let us put 

 1 + re — 1 in place of x; and then expand by the binomial 

 theorem ; the result will be 



x''F{n) — y[x-r\x^^ ; M + -,-1^ n(n- I) 



\ dxj 1 1.2 



+ &e. (3) 



The coefficient of h {n — 1) . . . (« — »i-^\) in this deve- 

 lopment will be 



\ ( TYliw. 1^ 1 



v-jr — {x"'s{m)~mx"'-^-p(m — \)-\ — \-——x"'-H(m—'2) — kc. \ 

 1.2. .m( ^ '1.2 J 



and, if we now suppose x ■=. I, we shall have the development 

 of F (n) in the desired form ; the coefficient of the factorial 

 n(n—\) (n—m-\- 1) being 



y^^-^JF (/«) - mv{m-\) -f- Y2 '^ (>«-2)-&c.| 



Comparing the two expressions (I) and (3) we find, as we 

 ought to do, 



7H ( iH — ' 1 ) 



A™ F (o) — F (/«) — m F (m — l) -\ ~x — - f(?«— 2)- &c., 



a formula which might be obtained directly by making xz=.0 

 in the fundamental equation of the calculus of finite differences, 



, m(m—\) J 



A'" Ma- = Uj-+m — lllif.i+"t-l i j-^ M.r+m-2 " «C. 



