MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
195 
X 
where it may be remarked that if -j- be an integer, the final terms of the series would 
be . . ./(2 h ) + /( 0), at any of which, if we suppose the series to stop, its 
finite difference would be obviously f(x). 
For the next example suppose the subject to be f(x + y), and that by the opera- 
tion x receives an increment h, and y the same increment by the operation then 
it is obvious that [f(x -}- y)~\ (^ x — — 0, therefore [f (x + y)\ (Ay — AJ = 0 ; 
hence fix -J- y) must be included in the general value of [0] (Ay — A/ -1 . 
Now 
[0] (A, - A .)- 1 = [0] {A y -1 + Ay - 2 . A x + Ay - 3 A/ + Ay - 4 A/ + &C.} 
and also [0] Ay -1 = <p (#) an arbitrary function of x. 
[0] Ay -2 = <p (x) . -jr, omitting the appendage, which being a function of x would 
not vanish with y, and which in the succeeding terms, if included, would only gene- 
rate another series similar to that now formed from <p (x), and consequently in the 
present case not add to its generality. 
Again, 
[o] Ay - 3 = $ (x) • for *)] = (y + h ) y - y (y - h ) = 2 h y- 
Similarly 
[0] Ay^ =>(*) 
y{y—h) iy-zh) 
1 . 2 . 3 . h 3 ’ 
&c. 
Hence 
[0] (A, - A,)- 1 = p (*) + f • Ap (*) • A2 f (*) 
+ 
y(y- h) [y - 2 h ) 
l . 2 . 3 . K 6 
. A 3 <p ( x ) + &C. ; 
and since f(x + y) is included in this general expression, the particular form to be 
assigned to the arbitrary function <p (x) is known by making y — 0 , which gives 
<p (x) = f (x) ; in this formula the ordinary notation has been employed. 
Suppose, for instance,/ (x) = a x and li — 1 , then A”/ (,r) = a x (a — ■ l) ra ; therefore 
■+y — 
= a‘ + y . a’ . (a - 1 ) + a ' (a - l ) 2 + t . 0^3 . a' (« - 1 ) 3 , &c. 
or putting a = 1 + b, and expunging a x from both sides, 
(1 + b)’= 1 + y . b + ■ & 2 + - (y 7 .'a [ 3 ~ g) • &c - 
which is the binomial theorem, whatever may be the value of y. 
For the next example let us take the same subject, f(x + y), and let d x , d y denote 
the differential coefficients relative to x and y • then since 
L /(* +.v) >y- A x = 0 
h 
2 D 
MDCCCXXXVII. 
