Crofton — Applications of the Method of Operative Symbols. 603 
Hence, by reducing, we find 
_1 + 2A + — + — + — + .... 
Such results are not easy to verify by common algebra. 
3. The reciprocal or inverse of the operator 
where 2>, as always, stands foJ^ ^ • 
This may be seen in different ways; I will use the following. 
Let us put 
BH^xf 
Vi 
L-^a'DF{x) = F{w), 
where tf = r» - »//(«;). 
I^'ow F[z)=F{x-ir ^x) = I)-^CiDF{x) 
operate on both sides with B-^cl'D, 
I)-^Cl'I)F{x + ^x) = D-^CL'D.I)F {x) ; 
.'. F{w + ^w) = F{x) = D-^a'aDF{x) ; 
. .'. D-^n'nB = 1 ; n'n = 1. 
As an instance, let cf3{x) = x- x^, and we have the theorem : — The 
two operators 
l+Bix-x"^) + ~ I)\x - xY + 
l-.I)(,xi-x)-^~L\x^-xf~kG. 
1.2 
are reciprocals. 
then = 1 - + - &c. Hence (2) 
