Mr . Herschel on various points of Analysis, 441 
probability of its coming successfully into play. Such have 
been in part the considerations which determined me to adopt 
it as a vehicle in laying before the Society some results of a 
singular and interesting nature, derived indeed originally from 
other principles, but which, like all the rest, flow with the ut- 
most facility from the first elements of this calculus. 
In the following pages I have uniformly made use of the 
functional or characteristic notation ; together with the method 
of separating (where it could conveniently be done) the sym- 
bols of operation from those of quantity. This method I have, 
perhaps, extended and carried somewhat farther than has 
hitherto been customary ; but, 1 trust, without losing sight of 
its grand and ultimate object, the union of extreme generality 
with conciseness of expression. To avoid the necessity of 
continual explanation, I shall here set down the leading points 
of the svstem. 
1. The signs : x ( ) are used to separate the symbol of ope- 
ration from that of the quantity operated upon, thus : 
/(x), (fi ; log X, U _ 1}" X (f. (j:). 
II. 1. The combination of two operations is represented by 
placing their symbols together in their proper order. Thus, 
(p ) is simply written (pip (a:). For example, if <p (x) = 
1 + j:, and ip (x) = then (a*) = 1 + x% and (x) 
=r= (1 -|- x)\ 
2. When several operations combined are considered as 
one, their cliaracteristics are inclosed in parentheses ( ). rhus 
/log (p (x) = (/log (?) : X. 
3. The repetition of the same operation J being denoted (by 
the first rule of this article) may be more con- 
