186 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
we have 
Cl f = 1 + (n — n) . 0 + 6 + &c. = I ; 
or if we introduce any subject f (x) ; Cl, <p n being necessarily free, 
[/(*)] ^ f = [/»] ^ =/(*)• 
Hence 71, <p” are mutually inverse operations, the action of the one on the result of 
the other restoring the original subject. Cl may be represented by <p~ n , attaching to 
this symbol the meaning here assigned. 
If n represent any quantity, positive negative, integer, or fractional, understanding 
the conventional notation by the definitions laid down, it follows, that if 
_ 02 03 
© _ l -f- 0 -f- y— ^ 3 +, &c., 
then shall 
n 2 02 n 3 03 
©"= 1 + W ^+rT2+K2T3+’ &C - 
10. Suppose 0 to be simply the operation of multiplying by unity, then 
© = 1 + 1 + YTs ~b TTs "b* ^ c ‘ ’ 
and putting as usual s for the sum of this series, 0 represents the operation of multi- 
plying by s ; put therefore in the formulae of art. 9. 1 for 0, and £ for 0, and then 
7$ 
l — 1 + n + YTq, “b 1,2,3 “b> 
The properties of this series when any way involved are common, as has been seen, 
to those in a series where 0, any linear operation, is put for n, and therefore we may 
write the purely symbolical identity 
6 0 2 03 
£ = l + 0 + 772 ~b 77777 “K &c.. 
where 0 may be an imaginary multiplier, or any species of linear operation. 
Thus if 0 — h d x and denote the operation of changing x into x + h, we have 
Similarly 
J. _ & hd x 
~ X 
4y = i kd y 
4 x 4y 
— c (hd x + kd v ) 
all of which are proved by the formulae of art. 8. 
1 1 . Having seen in the course of the investigations of this section the signification 
of the indices of operations when fractional negative or even purely symbolical of 
linear operations, it is easy to prove by similar steps that in all cases where 0, (? are 
relatively free. 
(0 + 0) n = P + nf~ l ff + 
n (n — l' 
. 0 n 
0' 2 +, &C. ; 
] . 2 
