272 
Proceedings of Royal Society of Edinburgh. [sess. 
If instead of log (a + b) we take such a quantity as log ( ax n + /3x p ), ' 
the logarithmic difference may be denoted by the symbol, Xoy%, 
where 
Xoy™ = log ( aX n + /3x p ) - log aX n = log ^1 -f- -X p . ( 1 ). 
In the tables of Logarithms of Addition , the logarithmic differ- 
ence is found from the argument (log a - log b), or in the second 
case from (log ax n - log f3x p ), which I denote by the symbol n\ ® 
or trip, where 
= ( 2 ). 
It seems desirable that quantities which are of general application 
should be denoted by symbols not liable to be mistaken for the 
symbols of other things, which is my reason for proposing this 
notation. 
Similarly, for logarithmic differences of subtraction, we have 
Xoy n p = log ( ax n - I3x p ) - log ax n = log ^1 - x p n ^ . . (3). 
m; as before = log • • • • ( 2 )- 
In the numerical examples given in this paper I have made 
use of Zech’s Tables of Logarithms of Addition and Subtraction 
(Berlin, 1863), in which the argument (in) is given to five signi- 
ficant figures with proportional parts, and the logarithmic difference 
(Xoy) is given to seven significant figures. 
The property of these functions which attracted my attention, and 
led to this new application of them is this : that for any function, 
log (a + &), or log ( ax n + j3x p ), the logarithmic difference does not 
depend on the absolute values of a and 5, or ax n and /3x p , but solely 
on their ratio. 
For if we denote a logarithmic difference (as above) by A.oy B 
have, Xoy A = log (A + B) - log A = log = log what- 
ever may be the absolute values of B and A. 
By means of this property we are enabled to eliminate the powers 
of x from the logarithmic equation, and to obtain equations between 
