188 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
and in general if 
[w] 0 = y then [y] 0~ 1 = u, and therefore [ij] 6~ 1 0 = y, 
the compound operation 0~ 1 0 or 0° being equivalent to no operation. 
To invert a compound operation we must invert the order as well as the nature of 
the component operations, which rule must be strictly observed when the latter are 
relatively fixed. For let 
[#] 0 = y [y] & — z and therefore [#] 0 & = z. 
Then [z] 0' - 1 = y \_y] 0“ 1 = x and [z] 1 0“ 1 = x, 
which proof is applicable, whatever be the number of the component operations. 
If all the component operations be alike, we have then but to change the sign of 
the index to obtain the inverse, thus if 
[w] 0 n ~ y, then [?/] &~ n — u, 
as in the last section. 
13. The consideration of inverse operations leads to the introduction of the ap- 
pendage, which when the operations are linear must be annexed to the result to give 
it the most general value of which it is susceptible, for the inverses of such operations 
are themselves linear ; thus if 0 be a linear operation, 
[X + |] 0 = [X] 0 + [?] 6. 
Put [X] 0 = X l5 [|] 6 = gj, 
hence [Xj + gj 6~ 1 = X -f- g 
= [X!] r 1 + KJ 6 - \ 
which shows the linearity of & h 
Now suppose the nature of 6 to be such that [P] 0 = 0, the subject P being thus 
in some way connected with the nature of the operation 0, then if we suppose 
[X] 0 = y, we have also [X + P] 0 = y, hence [y] 0~ 1 = X + P, 
this being the same as writing y + 0 for y, since [0] 0" 1 = P. 
The appendage therefore in a linear operation is the result of its action on zero ; 
P will express a form, but its magnitude must be susceptible of an infinity of values, 
that is, it contains arbitrary constants which enter as multipliers, for if a be such a 
constant, we have in general [X] a 0 = [X] 0 a ; and supposing X = 0, we have 
[0 . a] 0 = [0] 0 a : therefore whatever particular value may be assigned to [0] 0, a 
more comprehensive value is attained by its arbitrary multiplication by a. A multi- 
plier is the most general form in which the operation represented by a can enter 
when X is a function of but one variable ; but it admits of other forms more extended 
in cases of several variables, as may easily be perceived : thus \_fy~] representing a 
function independent of x, then [fy\ = f{y)> 4* x denoting the operation of chan- 
ging x into x + h : hence [/’(«/)] = 0. Then if X be any function of x, and | be 
any particular value of [X] A” 1 , we shall have more generally [X] A~ l = i + f(y ), 
