20 
Proceedings of the Royal Society of Edinburgh. [Sess. 
This style of solution, we shall see, can be extended to equations of 
every degree. 
§ IT, Harmonising of this Calculus with the Fluxional One. 
To bring this calculus into harmony with that of Newton and Leibnitz, 
we begin by regarding the statement 
d_ 
dx 
x 1l = nx n 1 
as universally true — true, that is, for all values of n, including zero or 
infinitesimally small quantities. 
This will cause no difficulty, but the converse, 
x n ~ l — — 
n 
must also be regarded as holding without exception. In this view it is 
a mere corollary to the former. 
We remark next that there exists no formal proof for the statement 
- = log x. 
x 
This is accepted in virtue of the converse 
d 1 1 
— log x— - 
dx 
x 
a statement which admits of proof. 
Consider, however, the operation D -1 - generally 
00 
tASj 
According to the rule just laid down, this must be evaluated as -g or 
where li is an infinitesimal, and the indeterminateness thus introduced is, 
in this new calculus, counteracted in the manner shown, and we reach the 
determinate value —1. In the Fluxional Calculus, again, the same inde- 
terminateness is counteracted by the addition of a constant — after the 
manner of this calculus — the particular constant (if we can so call it) 
rjd 1 2 . 
added being — oo in the form — j, so that we have -y-, whose limit, 
for h = 0, is log x. And our justification for choosing this particular 
constant is, of course, that we thereby reach that evaluation which is in 
harmony with the established fact, 
d , 1 
— log x = ~. 
dx ° x 
