ea 
=) mp 
as 
147 
px ¢D=9D ya-g¢D Wu+ a ¢’D W'x-&e., (10) 
separately, Dr. Hargreave observed that the latter might be 
deduced from the former by changing z into D, and D into 
—w; and on this observed fact he founded the conclusion that, 
in expressions capable of being reduced to the form (9) or 
(10), we are at liberty to effect the above-mentioned inter- 
change of symbols. 
The preceding investigation enables us to account for the 
fact just referred to, and to establish on what seems to be its 
real foundation the validity of the proposed method of de- 
riving formule one from the other. If we take f (x) any func- 
tion of x, we shall have 
D (afx) = x'af + fr, 
or, detaching the subject fx from the operations effected on it, 
we find that 
Daz=2D+1 (11) 
is a symbolical equation which holds good whatever subject be 
operated on by each of its terms. It is, in fact, the fundamen- 
tal equation which defines the law according to which 2 and 
D combine. And as in this equation we may change x into 
D, and D into -2; we may do the same in (9), or in any 
other equation derived from it. 
From this one equation (11) the principal symbolic for- 
mule of the Differential Calculus can be deduced; and we may, 
therefore, regard a great part of it as included in that single 
branch of the Calculus of Operations which refers to the pro- 
perties of symbols connected by the fundamental equation with 
which this Paper commences. 
But there are other changes of symbols which may be made 
in formule deduced from the equation, 
pr=mp+1. 
