Mr . Herschel on various points of Analysis, 44^ 
expressed, |/ (x ) | “ = |/ 1 " ; x, to distinguish it from f (x), 
whose signification has already been explained. 
4 * If F (jc) be developable in any series of the form 
(F : (?) : X = ^ (x) + b , {x) + c , {x) + &c. 
and, (F ) : x = a . ^(p> (x) 1 "* + b , + &c. 
IV. 1. D is used as the sign of derivation. It is, properly 
speaking, the sign of an operation performed, not on quantity, 
but on the characteristic which it immediately precedes ; by 
which the operation denoted by that characteristic is altered. 
For instance : D sin. = Cos. ; D cos. = — sin. But it must 
be observed that D log. — ^ = log. —K 
3. The sign D a Meets only the characteristic next following 
it, thus, D(pf {x) = (D(p) 'f{x). If it be intended to aMect 
a combination of operations, the rule II, 2 must be observed. 
Thus, D ((p/ ): X, D” (ip log. “*) : log. x. 
V. Every functional characteristic is aMected by all the cha- 
racteristics preceding it, in the same manner as if it were a 
symbol of quantity. 
VI. Every characteristic of operation performed on quantity 
affects all which follows it, as if it were one symbol. Thus if 
cc R 
f ( x) = ax bx + &c.; we shall have 
ax^" + bx^ + cx'^ -f &c. 
the following abbreviations are used 
Thus (for example) 
dx 
y- X .r = X + 6-*^ ^ 
} 
