198 
MR. MURPHY ON THE THEORY OF ANALYTICAL OPERATIONS. 
therefore 
but by § 5. 
therefore 
Hence 
®x d x — d x °X + °x d X> 
*- am d a = d x r"-ar" = {d m -a) i— 
d ' — a = z the extreme required. 
Now if 9 1 = i z, it has been shown that f (6) . i = if («). 
In this case therefore 
f (d x ). ?* = ?*/ (d x - a) 
To find the intermediate operation between d x -\-b and d x -f- c; b and c not containing x. 
Put f (d x ) = d x -f- b in the last identity, and b — a = c we then have 
(d x + b) . z {b ~ c)x = (d x + c) 
f(d x + V) . .<»-'>•* = #->*/ (d. + c). 
22. Suppose the intermediate operation / to denote s p considered as a multiplier, 
'P being a function of x, of which the differential coefficient is P, and 9 to represent 
d x as before, it is required to find z. Since 
therefore 
But 
Hence 
Corollary ; 
j ’p ’p 
d s = g . z 
X 
z =. s p . d . s p . 
X 
-' P d x = d x S -' p + F* d x = d m s” p - P a-' p = (d x - P) a-’ p 
* = d x - P. 
p _ ;p 
= P). 
And if V Q be a function, of which Q is the differential coefficient, we have in like manner 
/w •«'*=« vot- Q)i 
hence 
or 
*' P /K - P) • ^' P = - Q) 
that is, s'^- p ) is the intermediate to the operations d x — P , d x — Q. 
23. Let i now signify the operation of changing y into y -f- (x), 6 being, as be- 
fore, the operation of differentiating relative to x, and the subject being a function of 
x and y, and '<p (x) the quantity, of which <p (.r) is the differential coefficient. 
Here z — r x d. i. 
Now r l is the operation which changes y into y — ( x ), and therefore, as in 5, 
,_l d x — d x r l — <p (#) / -1 d y \ therefore 
z — d x <p (#) . d y . 
