MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
183 
Again, = A + 1, therefore 
r = (A + I)’ = 1 + n A + . A 2 + ” - r^ ( |— 2) . V + A’..i 
or, introducing the subject f(x), 
f(x + nh ) =f(x) + n A f{x) + ” . A ?f{x) + 1 [” ■ - . A 3 /(*) + (H.) 
Again, 1 = 4 — A, therefore 
1 = (+- A)" = (— 1)“A", 
or 
/(*) =/(<* + n h) -n Af{x + (w — 1) . h) + ~ f; ^ - 1 ) . A 2 ] 
f{x + (n — 2) .h) - n ^ n - — ~ A?f{x-\- (n- 3) A},&c. j 
> . 
(HI.) 
In the expansion (II.) put n h = k, or n — : therefore 
/M-*)=/« +*£/(*) + • (ilVw + ^ - ^r 1 • (i)/WAc. 
Now suppose n to increase infinitely, h remaining constant, the quantity h, which 
a 
is the increment of x, diminishes infinitely, and the operation in the limiting state 
when h vanishes, becomes d x . Hence 
7,2 7.3 
/(* + *) =/ (*) + * d J (*) + 172 d ?f (*) + 17773 ^ 3 / (*) + 7 &e. • (IV.) 
The expansions (II.) (IV.) are Taylor’s theorems for the development of functions 
by means of their finite differences and their differential coefficients respectively. 
Again, if h be written for h in the expansion (IV.), and the subject be omitted, it 
becomes 
;,2 (] 2 7,3 (l 3 
4 = 1 +hd x + +, &c (V.) 
and 
Tfi d ® d ^ 
§3. 
7- The expansion given for the operation \p, of changing x into x + h, possesses 
remarkable properties, which we propose to develop in the present section, from the 
importance of the theorem of Taylor, which it expresses. 
Representing, as usual, by 4 the operation of changing x into x -j- h, and by \p' 
that of changing x into x-\-h ' , the quantities h, h' being independent of x, and, lastly, 
denoting by 4/ the operation which changes x into x + h + h', we have obviously the 
identity 
4 4' = v, ; 
