MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
187 
for since 
{0 + 0) n (0 + 0') m = (0 + 0') n + m , 
it follows that the composition of the polynomials 
j> + V+,8tc.|.|r+)«r-V+^^- ) .<r-V+,&c. j 
= 0”+™ + („ + m ) f + —' e + (,i + M) ' > ■ . f + " - g iP, &c. ; 
and since nothing- in the actual verification of this identity depends on their being 
integers, for which case the expansion has been proved, the identity holds generally, 
and therefore if m = n, and we take p , such polynomials, we have 
^(r^n^- l 0'+^Y^-.0 n - 2 0 ,2 + .. y=0 n P+np0 np ~ 1 0' + .0 np ~ 2 0'2 
§^0 n n0 n ~ X 0' + j- ? — 0 nq nq§ iq ~ l 0' &c. 
Put w = y, 
q q j ^ 
{d' + j .o 7 ’ "' + + 
or 
-1-2- X- i ~ *) /X - 2 \ 
(«+« , ) ? =o i ’ d'+ ■-.A p y ^+& c . 
Again, since 
| I d'+^y^<i’ , -V 2 +&c.j.|r’'- K r"-V+^ii)r"-V2-&c.j =i, 
we have 
(0 + 0')-* = n0- n ~ l 0' + &c. 
These formulae applied to quantities or fixed operations, suffice after the usual 
methods for calculating their finite differences, differential coefficients, &c. 
§4. 
12. Suppose 0 to represent any operation which performed on a subject [w] gives 
y as the result, then the inverse operation is denoted by 0~ l , and is such that when 
jj/] is made the subject u becomes the result. 
Thus b denoting the operation of multiplying by a quantity b, we have 
[«] b = c [c] b~ l = a. 
Again, denoting the operation of changing x into x -f- h, we have 
[f (#)] 4 1 = ‘P i x )> where <p (x) = f (x -f- h) 
.’. [<p (a?)] 1 —f{x), where / (a?) = <p (x ■ h) 
MDCCCXXXVII. 2 C 
