MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
199 
Corollary ; 
f{d x ) t = *f(d x -<p (x) d y ) ; 
and as in the last article, if / be the operation of changing y into y + {cp (x) — F (x ) } . 
Then 
f(d x - F (x) dy) / = if{d x - cp (a?) dy). 
The example last given is capable of being extended to any number of variables in 
the subject of the operations ; if that subject contain x, y , z, u. See., and i denote the 
operation of changing y into y cp (x), z into z & (x), u into u + Cl (x), &c., where 
<p ( x ) 7s (x) Cl ( x ), &c. represent any functions of x, then will t be the intermediate 
operation, of which d x and d x — <p' (x) . d y — w' (#) d z — Cl' ( x ) .d u —, &c. form the 
extremes, the accents being used to represent the differential coefficients of the func- 
tions over which they are placed. 
24. Let one of the extremes, 9, be now supposed to represent the operation A x of 
finite differences, on the supposition that h is the increment of x. 
Let i represent a multiplier P x , constant or variable with x . Then 
z = P ~ 1 A P . 
X XX 
But by § 5, 
thence 
V 1 A. = A. . P-| 4 + P, 
- 1 
and then 
Thus, if 
we obtain 
z — A„ . 
+ .P - 1 A 
• X X 
X ’ P 
X + k 
f(\) • P* = P j{\ ■ P 2 *- + P«- A, . pj 
x -\- h 
-l 
P x = a h .*.P s+h = a .a \ 
K = (a - 1) .«*, 
f(A x ).a h - « 
Let \p x denote the operation of changing x into x + h, then \ + 1, and the 
general formula of this article becomes 
x-\- h 
Again, if the subject be a function of x and y, let < represent the operation of 
changing y into y + cp (x) and / that which would change y into?/ — A x {cp (x)}, 
where A x cp ( x ) according to the common notation stands for cp (x + h) — cp (x), then 
by a similar procedure we shall obtain the identity 
25. In the identity 
/(A x ).«“a = a ~ h fa(A x + put - _ b 
l 
1 + h 
