GB-OFTOi^i— Applications of the Method of Operative Symbols, 605 
Again, ix'^Dy=,^*\Drxr-^^ (8) 
for {x^Df = x^Bx^B = X {xJDx) {xBx) x-\ 
{fI)f:=x{xI)xYx-^; 
,'. {x^ny = X {xBxyx-'^ = x^^'^b^x^-'^ by (7). 
5. Let us now apply (8) to the question of finding the differential 
coefficients of f{x-^), f being any function whose successive derived 
functions 
fh fit /s, . • . . 
are known. Let x~^ = 't/; then 
dx- ' 
now ^Lj±L^.y.L^ 
dx dx dy dy 
Brf{x-^) = (- \y {y^ ^ y{y) = (- lyyr*^ Vviy)- 
(^) V-y(y)=2/-yry+r(r-i)2/-y,.iy+!ife^^ . . . 
Put for and we find D'f^x-') = 
(- iyx-^r{f^^^i)Mr{r- l)xfr.iix-^) + ''-^^^{r- 1) (r-2) x^fr.2{x-') 
^ r(r-l)(r-2) ^^_^^ (r-2)(r-3):^/.-3M+ . . 
^ Some curious results in Algebra follow from this formula. Thus if 
f{x) = x^, f{x-^) = x-»; 
B^f{x-^) = B^x-»='{-iyn{n + l),,.{n + r-l)x-»-r. 
New fr{x-'^) = «(« - 1) . . . (« - r + 1) x-"^'', &c. Hence the above formula gives, 
on dividing by a;-"-*", 
n{n+l){n + 2)..{n + r-~l) r-1 r{r-l) {r-l){r-2) 
n(n-l){n-2) . . {n-r+1) ^ n-r+l 2 * («_r+ l)(«-r + 2) * 
If we take /(a;) = f{x-'^) = x^, or change the sign of 
n(n-I)(^-2)..(n-r+l) _ r-1 r(r-l) (r-l)(r--2) 
n(w+l)(w + 2). . + 2 {» + r- l)(« + r-2) 
which thus is the reciprocal of the former series. 
