Now if the iron tube and the lead cylinder be each made of the same length as the zinc, and arranged as we have described, the compensation will be perfect. To prove this, find, by means of the expansions given in Table I., the actual expansion of each of the substances employed in the pendulum, and we shall have the following results:-- The expansion of 12·66 inches of zinc expanding upwards is ·0002186 Deduct that of 12·66 inches of iron expanding downwards ·0000869 -------- Remaining effect of expansion upwards, referred to the lower extremity of the iron tube ·0001317 Now, for the lead.--On the principle of the mercurial compensation, subtract one tenth part of the length of the cylinder, and take half the remainder, and we shall have six inches of lead, the expansion of which upwards is ·0000955 -------- Total expansion of the compensation upwards ·0002272 -------- To find the expansion of the rod, we have the expansion of 43 inches of glass ·0002059 Of two inches of steel ·0000127 -------- Total expansion of the pendulum rod ·0002186 Agreeing near enough with that of the compensation before found. As we conceive we have been sufficiently explicit in our description of this pendulum, in the construction of which no difficulty presents itself, we think an engraved representation of it would be superfluous. We have hitherto treated only of compensations for temperature; but there is another kind of error, which has been sometimes insisted upon, arising from a variation in the density of the atmosphere. If the density of the atmosphere be increased, the pendulum will experience a greater resistance, the arc of vibration will in consequence be diminished, and the pendulum will vibrate faster. This, however, is in some measure counteracted by the increased buoyancy of the atmosphere, which, acting in opposition to gravity, occasions the pendulum to vibrate slower. If the one effect exactly equalled the other, it is evident no error would arise; and in a paper by Mr. Davies Gilbert, President of the Royal Society of London, published in the Quarterly Journal for 1826, he has proved that, by a happy chance, the arc in which pendulums of clocks are usually made to vibrate is the arc at which this compensation of error takes place. This arc, for a pendulum having a brass bob, is 1° 56′ 30″ on each side of the perpendicular; and for a mercurial pendulum, 1° 31′ 44″, or about one degree and a half.