196 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
put h — 0 and ^ becomes d x , ^ being similarly d y ; therefore 
U( xJ ry)~\ (4 “ d x) = 
hence f(x y) must be included in the general value of [0] (d y — d x )~ ] m 
But 
[0] (dy — d x )~ l = [0] (dy- 1 + dy~ 2 ^ "f ^y^ "f dy" ^ d +, &C.) 
Now 
[ 0 ] dy- 1 -(p (x) 
an arbitrary function of x ; therefore 
[0] d y - 2 ~<p(x).y 
omitting the appendage for the reason abovementioned ; also 
[Q] d y * = <p(x).r2-g 
since [j/ 2 ] d y = 2 y ; and 
substituting we have 
[0] dy =<p (x) . YToTs 
[o] (4 - 4)' 1 = p (.) +y ^ &C. 
dx 2 ""^1.2.3’ dx 3 
employing the common notation ; the form of <p ( x ) is determined by making y = 0, 
which gives <p (x) =f (x) ; therefore 
1 .2 
/(* + y) =/0) + ( x ) + fa •/" (*) +> &c. 
where /' (#) /" (.r), &c. are the successive differential coefficients of f (x ) ; this is 
Taylor’s expansion. 
If we put f (x) = a and for the limit of 
a n — 1 
write log (a), we get from this 
a J = 1 + y 1 . • (a) + • ( !• • a Y +> &c ° 
These examples suffice to show the mode and use of the inversion of binomial ope- 
rations. 
20. To return to the general theory, suppose 0, /, x to represent three operations 
connected by the equation 0 1 = t x, where the subject is omitted, the identity being 
supposed general ; the symbol / represents an operation which may be said to be in- 
termediate to those designed by 0, x. 
If either of the extreme operations 0, x be given, and the intermediate i be also 
given, the other extreme may be readily found for 
0 = / nr 1 and x = i~ x 0 /. 
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