MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
201 
then by § 3, 
= , -/«-<•> = i -f(e - d,) - &c., 
and 
' d x = •*'“*> ./(0 - d x 
also 
(8 - d.) < = (8 - <*.) 
N 0 w 0 — d x is free relative to its own functions, hence 
therefore 
f(0-d x ) = (9-d x ).d~\ 
from whence i is known. 
For a second example, suppose ip the operation of changing x into x + h, A that of 
taking the finite difference on the same hypothesis, 0 any linear operation, then / is 
required to be such that 
(4 - 0) i = t A 0. 
By art. 15, 
i A = A i \p + / A 
— ip i t i A 
— ('P' — t . i , *P~ 1 ) * "4 '• 
Substituting we obtain 
( \p — 0) i — (\p — t . i , pj~ ■*) i \p 0. 
To satisfy this identity, suppose 
i . i ^~ l — 9 ; 
now i 4 1 is free relative to i, hence 
I ■pi~ 1 s — 0 , 
prefix to each side the operation / \p, which only alters the subject, which is perfectly 
general ; therefore 
I = i \p . 0 ; 
hence the preceding supposition fully satisfies ; therefore we have this theorem, if t 
be determined such that 
= 0 , 
then shall 
(4 - 9) i = i A 0, 
and 
-»), = ./( A 8). 
As a particular case, suppose 0 to be a multiplier P^, then i will be another multi- 
plier v x , such that 
