182 Proceedings of Royal Society of Edinburgh. [sess. 
time-keeping, and we may, therefore, first in a general way, consider 
what may be the relative values of the coefficients P and Q, and 
also the signs which these may take. 
In my mean-time clock, the bob is a cylinder of lead cased in 
brass, having its diameter 3 inches, its length 5, its weight, by 
estimate, being 101 200 grains; and therefore for each inch of 
deflection at the mean distance the redressing tendency is 2586 
grains. 
The air’s resistance on a square inch of surface in grains for a 
velocity v measured in inches per second, varies, as deduced from 
£7 2 /^2 
several formulae in use, from — — to — ; now the bob offers a 
’ 1200 1600 
v 2 
surface of 15 square inches, and thus we may assume in grains 
as the air’s resistance in our case. 
Now the oscillation extends to 1*3 on each side, wherefore the 
greatest incitement amounts to PX = 3362 grains; the maximum 
velocity is V = 1*3 x tt = 4*08 inches per second, and causes a 
maximum retardation of ’166, or the sixth part of a grain at the 
lowest point ; and thus it appears that the maximum resistance by 
the air is only the twenty-thousandth part of the greatest redressing 
tendency. 
Because of the complexity of the parts, it is not so easy to form 
an estimate in the case of the chronometer balance. In my two-day 
chronometer the compensating weights have a diameter of *3, or 
just the tenth part of the pendulum bob ; and they oscillate to a 
distance, measured along the arc of 1*2 or 1*3, almost the same as 
that of the pendulum. Hence, if the times of oscillation had been 
alike, the ratio of disparity would have been 2000 instead of 
20000. But the balance makes two complete oscillations per 
second, while the pendulum makes only one-half. Now, in order 
to quadruple the number of oscillations per second, we must 
augment P sixteen times, Q being unchanged ; however, since the 
velocity is now quadrupled, the product QV 2 is made sixteen times 
greater, and thus the ratio of PX to QV 2 remains 2000 to 1. The 
rim of the balance meets with direct resistance only on its ends ; its 
motion is impeded by what may be called friction, on the sides, and 
in all likelihood the ratio of disparity is higher than for the weights; 
