200 
ME. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
hence 
/( \) (1 + bf = (1 + »)*/(■ \ - b). 
Now by the nature of an identity both sides in this expression would be alike, if 
expanded according to the powers of b, the coefficients of like powers must therefore 
be equal, and in consequence the identity must remain if b instead of being a mul- 
tiplier represent any operation which is free, relative to A ffl , thus if b = & ys 1 -p b = 
and therefore 
/( A J • ^y h = ~ A y) 
As a particular example of this, suppose the subject to be <p (y) independent of x } and 
/(AJ merely to stand for A^, then 
0 ( y)~i ( A J = o, 
the general identity therefore becomes 
L<p (3/)] V ( A * - Ay) = 
or 
IXs' + t)] (^-a») = o, 
A- being the increment of 3/ ; now 
[p (# + x)] = ^ ~ r~) = p (y + A + x) = (y + x;k 
which verifies the result deduced from the general identity. 
26 . Thus when the intermediate operation and one extreme are given the other 
extreme may generally be found, but it seems more difficult to discover the interme- 
diate operation when both extremes are known ; here follow some examples of the 
latter process. 
Let 6 represent any linear operation, d x that of taking the differential coefficient 
of the subject relative to x, and 1 the required intermediate. 
Then 
0 1 = 1 d x 
— d x i 1 d x 
perform on both sides the inverse operation < - l . 
Hence 
6 = d x + id x r 1 
or 
1 d x • = 9 - d x- 
, = ✓(»-« =!+/(•- dj + } 2 &c. 
Suppose 
