MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
181 
But 
[x n ] a p = [a /] \|/ = « ( x h) n 
[x n ] p a = [{x + hy\ a = a (x + A)". 
in the latter case the order of the operations is indifferent, because the operation p 
does not act on the multiplier a , and for the contrary reason the order of x p is fixed. 
in the first case. 
Operations are therefore relatively fixed or free ; in the first case a change in the 
order in which they are to be performed would affect the result, in the second case it 
would not do so. 
In a compound operation any part of the symbols may be taken conjointly with 
the subject in the square brackets, their result being the subject for the compound 
operation of the remaining symbols. Thus 
o 2 ] a p A d x = [ax 2 ] ^ A d x . 
§2. 
4. Linear operations in analysis are those of which the action on any subject is 
made up by the several actions on the parts, connected by the sign + or — , of 
which the subject is composed. 
Let p denote the operation of multiplying by a quantity p , then 
[a + b] p = [a] p + [ b ] p ; 
this operation is therefore linear. 
Let p denote the operation of changing x into x + h, then if / (a?), p (x) be any 
functions of x, we have 
[/» + <P (*)] P —f(* + h) + p (x + h) = l f{x )] P + [p (*)] P, 
which shows that p is also a linear operation. 
Let X + \ represent the subject acted on, and 6, & any linear operations, then 
[X + |] {6 + 0) = [X + 5]H[X + i] 0 
= [X] 6 + [I] 6 + [X] 0 4- [f] 0 
= [X] (tf + o + [I] (tf + O; 
hence polynomial operations of which the parts are linear possess themselves the 
same character. 
Thus A the operation of Finite Differences is linear, because A = p — 1, the ope- 
ration p of changing x into x + h and the multiplying by unity being both linear. 
Also 
[X + %]60 = [X6 + £'0]0 
= [X] 6 0 + [|] 6 0, 
which shows that the compounds of linear operations are also linear. 
The operation of taking the differential coefficient is therefore linear, for the opera- 
2 b 2 
