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Proceedings of Eoyal Society of Edinburgh. [sess. 
S 2 m 
rate of change is nearly constant through a whole page, or is 
extremely small. 
We* may therefore proceed as follows : — Taking the upper sign 
in ( 1 ) we first find q" (or value of q which results from inserting 
the three figure values of 1 U and X in the formula of reduction); 
and subtract this from the known or true value of q (as found 
from the constants) to get Sq. Then 8q = Srrt - (SX)a = (s± l)aSA, 
where s is the number of lines through which the value of the 
3rd figure of A. is unchanged SX = . —7 This form gives 
0 a{s ± 1 ) 
the nearest value of X in four figures. With this and the corre- 
sponding value of nx in four figures we may find the value of X to 
five places by the same form, only that s now represents not the 
number of lines hut the number of horizontal places in the table 
from one change of the 5th figure of X to the next change. 
This must be checked by finding q from Xoy and rrt, and noting 
whether it corresponds with the true value. 
By following this method I have generally been able, in a few 
minutes, to take out the exact values of rrt and Xoy corresponding 
to the known value of q ; and while the solution would be theoreti- 
cally more perfect if we had a supplementary table giving values of 
rrt to the argument q , this is not practically necessary. Under the 
actual conditions, the equation of three terms can be solved to such 
number of places as the tables furnish with very little more trouble 
than is involved in the solution of a quadratic equation. 
I have not given examples of the solution of cubic and quartic 
equations ; but it is evident that the solution by logarithmic 
differences is very much simpler than the methods now in use, 
while the preliminary reduction to three terms is equally necessary 
under either process. 
Note. — Since this paper was written, my attention was called by Dr 
Muir to a short paper in Zeuthen’s Danish Journal of Mathematics , 
1880, p. 135, in which the author makes use of Gaussian Logarithms 
for the solution of the equation x n +px + q. But as that paper is 
less general in its aim and method than mine, and the solution 
given involves some unnecessary transformations, I have thought it 
desirable to publish my paper as read, subject to this explanation. 
