Orofton — Applications of the Method of Operative Symbols. 611 
{Dx + 4) {Dx + 2)Dx = D\B-^ xf, &c. ; 
also (l-a)-^^i^(^) = (l-«)-^i^^(-^), 
hence 
{l^^B\B-^x) + ^B^[D-^xY + &c.)F{x) = {\-a)-'^F(-^\. (26) 
Let F{x)^e^^''', then = ; 
put a = lch^ and we have the theorem in question, viz. : — 
\1 — AA/ 
In like manner, it can be shown that 
(1 _ =1 + 1 BHB-^x) + ^ B^B-'-xy + &c. 
3 3.6 
If, then, we can find a function y such that D~^xy = y : that is, if 
y-F{x) be the transcendent derived from the differential equation 
then e^"^' F{x)={l^a)-^F(-^\. 
\{l-a)^J 
By formula (22) we may derive, as above, 
Bx {Bx +2) {Bx + 4)....{r factors) = B'^*-{B-^xY, 
and by (21) ={Bx-^yx^^; 
.-. i)2r(jr)-la;)r = (i);^l)r^2r (27), 
for any operand. 
