ond to
double the percentage of increase in T. Therefore with a seconds
pendulum, in order to make a second's difference in a day, equivalent
to 1/86,400 of the pendulum's rate of vibration, since there are
86,400 seconds in 24 hours, we must have a difference of length
amounting to 2/86,400 = 1/43,200 of the length of the rod. This is
39.138/43,200 = .000906 in. Hence if under the pendulum bob be put a
nut working a screw of 32 threads to the inch and having its head
divided into 30 parts, a turn of this nut through one division will
alter the length of the pendulum by .0009 in. and change the rate of
the clock by about a second a day. To accelerate the clock the nut has
always to be turned to the right, or as you would drive in a corkscrew
and vice versa. But in astronomical and in large turret clocks, it is
desirable to avoid stopping or in any way disturbing the pendulum; and
for the finer adjustments other methods of regulation are adopted. The
best is that of fixing a collar, as shown in fig. 7 at C, about midway
down the rod, capable of having very small weights laid upon it, this
being the place where the addition of any small weight produces the
greatest effect, and where, it may be added, any moving of that weight
up or down on the rod produces the least effect. If M is the weight of
the pendulum and l its length (down to the centre of oscillation), and
m a small weight added at the distance n below the centre of
suspension or above the c.o. (since they are reciprocal), t the time
of vibration, and -dt the acceleration due to adding m; then

-dt m / n n^2 \
--- = --- ( --- - ---- ):
t 2M \ l l^2 /

from which it is evident that if n = l/2, then = dt/t = m/8M. But as
there are 86400 seconds in a day, -dT, the daily acceleration, = 86400
dt, or 10800 m/M, or if m is the 10800th of the weight of the pendulum
it will accelerate the clock a second a day, or 10 grains will do that
on a pendulum of 15 lb weight (7000 gr. being = 1 lb.), or an ounce on
a pendulum of 6 cwt. In like manner if n = l/3 from either top or
bottom, m must = M/7200 to accelerate the clock a second a day. The
higher up the collar the less is the risk of disturbing the pendulum
in putting on or taking off the regulating weights, but the bigger the
weight required to produce the effect. The weights should be made in a
series, and marked 1/4, 1/2,