604 Proceedings of the Royal Irish Academy, 
4. It will be useful to state here certain formulas in the Calculus 
of Operations, some of which are not new, before considering one or 
two questions in the Differential Calculus to which we will apply 
them, I would refer to a Paper on the subject in the Quarterly 
Journal of Mathematics , Oct., 1879, and to one in the Froceedings of 
the London Mathematical Society, vol. xii. If a{x) be any function, 
/(i)+a'(^)) = r«W/(i))e«(*), (1) 
f{x+a{D)) = e-^^)f{x)e-^^% (2) 
hence f{x + D) = e-^^'f{D)e ^-^V(^) (3) 
we shall easily find from (1) and (2), 
f{JD + ax) = rl«^V(-^>) = eh^-^I)y{ax) e-i^-'^^\ (4) 
Put here /(-O) = -O**, and let the operand be unity, and we find 
or 
j>.,ia.'^ +l(r_Ll) . .) (5) 
Since Dx = xl)+lj the operators Dx, xD are commutative, either 
being a function of the other, thus 
xD'^x = Dx'^J). 
Also, as we know that (see Art. 10, infra) 
xrD^ = xD{xD-l) . . . {xD-r + 1), 
D^x^ = Dx{Dx+l) . . . {Dx + r -1), 
the operators D^'x'', x'D% xB, Bx, or any functions of them, are com- 
mutative. 
Hence we may prove the theorem 
{DxDy = D^x^D^. (6) 
Por (by interchanging 2)x, a^-'^J)^-^), 
DrxrHr = Dr-^Dxx^^D^-^D = V-iIf-iiJa;!) 
= I)r-H^-^I)r--^{DxI))\ 
and so on. We may also deduce 
{xDxy = xrD^x^. (7) 
