MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
191 
M i X = X X 6 x + h 
[«KA,= M (A^. +4 + 'o 
M i*d.= MCii + O- 
Again, let the subject be f (x, y) ; and suppose now 0 x to be the operation of changing 
y into y + © (x), then 0 x \p or 4-[-« is the operation of changing y into y -{■ p (x h), 
6 x A x or s Q x is the operation which changes f (x. y) into 
f{x,y -f p (x + h)} —f{x,y + p (as)} =/{x, y -f p (x)} 
df{*,y + p (a-)} 
+ 
dy 
p' (■»)•* + &c- —f{x,y-\-p (x)}, 
- Oj f \ X If “|“ (b oo\ • • 
and therefore 0 x d x will give - v d -~ ! and is equivalent to 0 X d y p' (x) ; in this 
case we should have 
[/ 0 : 3/)] K d x = C ffa y)h ( ( L K + K d v ¥ {?)), 
which result maybe also deduced by putting for 0 x its equivalent symbol s rfylp X> 
which gives 
K d x = W • d y p' (x) = 0 X d y p' (x), 
. . . n dp (x) 
where ©' («r) is written for — 
This example shows how operations may themselves be the subjects of other ope- 
rations. 
16. We now proceed to consider the transformed values of 0 X 4> n x , A A n x , 0 v d n x , 
when n is any positive integer. First, 
K X = X ^ X = 5 
suppose therefore 
0 d ^ — (7) d d © d • 
a: t i ~x * x x > x t x 
Now p x $ x regards p x solely as the subject of the operation 4*) and 
$ d d zn d $ d d 
X • X I X *X XIX • X 
by the first formula ; therefore 
A'x X = ^ 2 * ^ 4V 
and in general if we suppose 
then 
but 
K = ’L* -1 f.” -1 = t*> 
0 ’ll/ n — © d = d © d : 
X Ti ra; Yx Yx T x Yx 5 
0 d W_1 d =J/ 
1 “i tj: "a; a: < a: 
by the first formula ; therefore 
0 d re = d "- 1 d 0 d K = d ” 4 d \ 
a: i u t# a? ' a: t # # t or 
