bob nearly equal to its own
volume. Hence while the rise of 1 deg. of temperature increases the weight
by .00000012th part, it also decreases the mass by about the same
proportion, and therefore the increase of period due to a rise of
temperature of 1 deg. F. will, instead of being .0104 second a day, be
about .02 second. This must be compensated negatively by lengthening
the pendulum by about .02/1000 in. for each degree of rise of
temperature, which will require a piece of brass about 2 in. long. It
follows, therefore, that with an invar rod having a linear expansion
coefficient of .0000002 per degree F., which requires a piece of brass
about .8 in. long to compensate it, the compensation which is to
regulate both the expansion of the rod and also that of the air must
be .8 in. - 2 in., or -1.2 in.; so that the bob must be hung downwards
from a piece of brass nearly 1-1/5 in. in length. If the coefficient
of expansion of the invar were .00000053 per degree F., then the two
corrections, one for the expansion of the rod and the other for the
expansion of the air, would just neutralize one another, and the
pendulum rod would require no compensator at all. There are a number
of other refinements which might be added, but which are too long for
insertion here. By taking in all the sources of error of higher
orders, it has been possible to calculate a pendulum so accurately
that, when the clock is loaded with the weight sufficient to give the
pendulum the arc of swing for which it is designed, a rate of error
has been produced of only half a minute in a year. These refinements,
however, are only required for clocks of precision; for ordinary
clocks an invar pendulum with a lead bob and brass compensator is
quite sufficient.

Invar pendulum rods are often made of steel with coefficients of
expansion of about .0000012 linear per 1 deg. C.; such a bob as this would
require about 6.7 cm. of brass to compensate it, and, deducting 5 cm.
of brass for the air compensation, this leaves about 1.7 cm. of
positive compensation for the pendulum. But as has been said, the
exact deduction depends on the shape and size of the bob, and the
metal of which it is made. The diameters of the rods are 8 mm. for a
15 lb bob, 5 mm. for a 4 lb bob, and 12 to 15 mm. for a 60 lb bob. The
bob is either a single cylinder or two cylinders with the rod between
them. Lenticular and sph