4 
44 j® Mr. Herschel on various points of Analysis. 
cisely represented thus,/%/^ . . . ; and this furnishes us 
with a general and very simple notation for the reverst- ope- 
ration of that denoted by /. For since/'”/" [x) ==. 
if we make w = — i, w = + i, we find/ (x) =/° [x) 
= X with the operation/ ?io times performed, = x. Thus/ 
is the characteristic of that operation which must be performed 
on f {x) to reduce it to x\ that is, of the reverse operation. 
This is surely a simpler and more expressive method than that 
of inverting the characteristic,* accentuating it on the left side,/ 
or below or other similar contrivances. For instance, 
=: = 1 + ^ + &c. 
tan~'a: = - _ + - _ &c. 
4. If a combination of operations, as &c. be 71 times re- 
peated in their order, thus; (p^(p^ .... (x), it will, by the 
second and third rules of this article be denoted by ((pipf: x. 
It must be observed, however, that ((pij^)" is not the same as 
ip” J;”, except in some^ particular cases. 
III. 1. If any number of functions of a symbol x be added, 
subtracted, or otherwise combined, the resulting function is 
expressed by the same combination of their characteristics, 
observing the following conditions. 
2. The actual multiplication of two functions (p (x') and ip (x) 
is expressed by inserting a full point between their charac- 
teristics, thus, = (p (x) ./(x). 
3. The actual elevation of a function to any power is thus 
* Laplace. Journal de I’Ecole Polytechnique. No. 15. 
f Monge. Savans Etrangcrs. 1773. ' 
I Knight. Philosophical Transactions, 1811. Parti. 
