610 Proceedings of the Royal Irish Academy. 
Since {;i-a)-^'=\-^aI)x + ^ I)x{Bx + l)+-^^I)x{I)x + l){Bx + 'l) + . 
= 1 + aDo; + i)2a;2 + -^- i)3jt;3 + . . . . 
we have thus 
(1 + aD^ + y B'^x' + . . .) i^(^) = (1 - «)-^ (r^) ' ^^^^ 
since = 1 + xB, and n'^f{x) =f{nx). 
Ey(6)- 
1 • ^ 
Hence 
eDxD,hx = (1 + Ai)^ + 1! i)2^2 + . . ) .'i^^ (1 _ exp. ( j^^^ . 
This formula (20) is a case of Lagrange's theorem. 
Since Bx-*- = x-^B - rx-*'-^ Bx + r^x-^Bx^'^^, (21) 
and since B-^x = xB-^ - rB-^-^j Bx + r = B^^^xB-^. (22) 
. Erom (21) 
Bx{Bx+\) = Bxx-^Bx^ = B'^x^, 
Bx {Bx^\) [Bx + 2) = 1)2 x^x-'^Bx^ = B^x^, 
Bx {Bx + !)...(/ factors) = B^x^, 
(See Williamson, J)iff. Calc, p. 449.) 
As these factors are commutative, we may reverse the order, which 
will give 
B^x^ -x-^'^{Bx'^Yx-^, (23) 
and (22) gives B^x^ = B-\BHYB-^^^\ (24) 
{Bx^Y ^x^-^B^x-^^^, {B^xY = B^^^x^B'-"^. (25) 
Since, if <^ be any function of its derived function 
B(^-^ = (p-'^B - r<|)-»-V, 
B(f) + r(p' = (p-^-B^*"^^. 
Hence is instantly proved the formula 
B^(f>^ = B({> {B(p + (p') {B(fi + 2(t)') . . . {r factors), 
11. By the method in formula (20) a curious proof may be obtained 
of a theorem in operations found by another process in Fhil. Trans, for 
1869, p. 186 (see also Forsyth, Diff. JSq., p. 353) 
{l-a)-^^''=l+^Bx + ^Bx{Bx+2)+^Bx{Bx + 2){Bx + i) + . . 
