MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
189 
which includes the former; since, the form of f{y) being arbitrary, we have g alone, 
amongst the infinite number of values of 
In compound operations, the appendage obtained by the first simple operation be- 
comes a new subject for the succeeding operations, each of which may in like man- 
ner introduce a new appendage. 
14. The operation yp x , taken directly or inversely, is incapable of introducing any 
appendage : for suppose [0] -p ~ l = <p ( x ), then [<p (#)] ^ = 0, or <p (x + h) — 0, which 
identity being general, we may put x instead of x + h, which gives <p (x) = 0 ; from 
which it also follows that [0] -p ~ n = 0. 
To find the appendage introduced by d x x , suppose in the same way [0] d x 1 
— <p (x), hence [<p (#)] d x = 0, therefore [<p (#)] d x 2 = 0 [<p (#)] d x 3 = 0, &c. ; and since 
<t> (x + h) = <p (x) + h + y 
h 2 d 2 <p (x) 
d x 2 
+ &C. 
hence we have <p (pc + Ji) = <p (x), and h being arbitrary, we may put it = — x, there- 
fore <p (x) = p ( 0 ), that is, <p (x) is constant relative to x ; if therefore C be any arbi- 
trary quantity independent of x, we have [0] d~ l = C. 
Again, 
[0] d~ 2 = [0] d~' d- x = [C] d- 1 + [0] d- 1 , 
as above stated ; but since [CJ d x = C, and [0] d v 1 = C', any constant therefore 
[0] d~ 2 = C * + C', and in general 
[0] d~ n = Aj + A 2 + A 3 x re " 3 +.... + A re , 
where A x A 2 . . . . A n denote constant multipliers. 
Lastly, let [0] A^" 1 = <p (x), or [cp (#)] A x = 0, therefore <p (x + h) = cp (x), and 
by Taylor’s theorem (dividing by <p ( x ), 
o = ^A+^a.^ + &c. 
m 1 <5 m 1.2 
where <p' (x) cp" {x) &c. are the differential coefficients of <p ( x ) ; this identity being 
independent of x, the latter quantity must disappear from the series : put therefore 
<P' (•*■) 
n V — l 
<p" (x) = 
9 M 
n V — 1 
h 
, n being independent of x ; hence 
<p' (x) = 
p-<P(x) <p"' 0) 
ri A \/ 
h 3 
A p (x), &c. 
therefore 
\ — V - \ {rc — Y7273 + 1.2. 3.4.5 “ &C - } + { 1 “ 1.2 + 1.2. 3.4“ &C -} ( a ’) 
At present suppose n the least real value which satisfies this equation, then 
<p (x) — C s 
nx V — 1 
h 5 
2 c 2 
