608 Proceedings of the Royal Irish Academy. 
Pirst, let us state a principle (referred to in the Paper, London 
Math. Soc. Proc, vol. xii.) that in any operator 
U==f{x,D) 
we may substitute for -Z), + understanding that I) solely affects 
the X which is contained in Z7, and that D is constant or inoperative 
as regards this and only operates on the x which enters the operand 
which follows. 
This may be expressed thus : — 
ir=f{x,D)=f{x,i)-\-D), (17) 
where = ^> and x is regarded as a quantity independent of x, which 
is to be replaced by x at the end of the operations. 
Por instance, we have 
(xD)^ = [x{D + i))]2 = x{b + I))x{3 + D) 
= x{I)+ D) xD, since i) . 1 = 0 ; 
.-. {xBf = xW^ + xD = x^D'^ + xB. 
It is easy to show also that 
{xDY = x^D^ + SxW^ + xD, 
(a;2i)2)2 = x^Di + ixW^ + 2xW^. 
ITow let us consider 
D^f(x») : if y = ic" 4- = ^^'^ 4" ; 
^ ' ^ dx dy 
now if we put (13) C== n-^x-^^^^, C = — » ^^^^ have 
i>"-/(^) = + (18) 
but as ^ is constant as to x, we have by formula (1) above 
I)rf{x^) = ^"^/(a;"), as in (15). 
