202 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
or 
(log v x ) -logP., 
from whence v x or i is determinable. 
§ 8 . 
27. In this section we shall give some examples of the use of the formulae inves- 
tigated in the last section. In general 
d x f = *'(d a -a) 9 
therefore 
d - a = t~ ax di ax . 
X X 
Put for a in this formula the terms of the series 0, h, 2 h, ... . (n — 1 ) . h, and com- 
pound all the binomials which result from these substitutions, hence 
d m {d-h){d-2h) (d-(n~l)h)=d x ^d^\r 2kx dj hx g -C*-U^ g K- D*. 
= d x r hx d x r hx d x r hx d x r hx d x &<■ n - 
d x t~ hx — h d y , putting y = z~ hx . 
Now 
therefore 
d, {d. - h) (d, - 2 h) .... (d, - (n - 1) h) = 4” . 
Example. The expansions of ( a -j- b) n , viz. 
a n + n a n ~ l b -j- n — • a n ~ 2 b 2 +, &c. 
and 
i -f n 1 . . (a -f b) + +’ &c - 
being identical when n is a quantity, ought to remain so when n is a linear operation, 
to verify which, suppose n = d x , now it has been shown that [/ (#)] s hdx = f (x + h) ; 
hence [/(^)] (a + b ) d * =/{a? + 1 . . (a + &)}. 
But 
(a + b) dx — a dx + d x a dx ~ l b + ^ a ^ . a dx 2 b 2 - f- y 7 ^ — — . a dx ~ 3 IP -f-> &c. 
=■-•{' u ■ i +■ ^ ■ (iy + -• w -:^ ’ .o' + m.} 
But by this article 
d x (d x - 1) (d x - 2) . . . (d m - n + 1) = d y n .y n , if y = ? ; 
d y 2 (by' 2 
1.2’ V a > 
and now introducing the subject f (x), we get 
/{* + !••(« + »)} =/ (® + 1. («)) + f + -jr • - | 
which series, if we substitute for x its value 1 . .y, and put f { 1 . .y) — <p (y), becomes 
therefore 
{a + b) d *=a d * {l + d,. b -% + 
+ 1^3 (*)’+> *4 
