1889 - 90 .] Lord McLaren on Equation of nth Degree. 279 
The nearest tabular values of rq an d Xoy, after applying the 
proportional parts, are 
m 7 4 =0-875680 Xoy? = *054285 
-| XoyJ, 0-040714 
q = 0-834966, which agrees with the value found above (a). 
II. The equation of solution is 
-loga? = J{log3 + XoyJ} 
log /3 = log 3 = 0-47712 
Verification. 
\oyi, 0-05428 
log *, 1-86715 
sum, 0-53140 
log* 4 , 1-46860 
|=-log*, 0-13285 
log 3, 0-47712 
log a?, 1*86715 
log3* 4 , T-94572; 3* 4 = -88250 
:c= 0-736461 
log* 7 , 1-07005; * 7 = -11750 
1 
x 7 + 3x 4 = 1-00000 
Thus the value of the equation resulting from the insertion of the 
value found for x is correct to the last place of decimals. 
Apparently the method of logarithmic differences is incapable of 
extension to equations of more than three terms. 
Note as to Mode of finding Xoy and rq from the Tables by 
Interpolation. 
Supposing values of rq and Xoy to be taken out by inspection 
for the three highest figures or decimal places (which can easily be 
done mentally), we wish to find these quantities to four and eventu- 
ally to five places. 
Let rq", X" be the nearest values of these quantities in three figures, 
and let f be the approximate value of q resulting from the insertion 
of rq", X" in the formula of reduction. Let rq', X', q' be the corre- 
sponding values in four decimal places. The relation between the 
quantities is of the form rq ± aX = q. 
Then since X diminishes as rq increases, we have 
rq'-rq" + a (X - \") = q' - q" ; or Sg = Sm + a(SX) . (1). 
In the table the 3rd figure of the argument rq is changed at each 
line ; but the 3rd figure of X changes much more slowly, and the 
