Page:Popular Astronomy - Airy - 1881.djvu/277

Rh (from experiment), that when a rod carrying balls is suspended in this manner by a wire, the space through which the balls will be pulled sideways is exactly in proportion to the force which pulls them sideways. In this respect the law of forces acting on the suspended rod, is exactly similar to the law of forces acting sideways on a pendulum vibrating in a moderately small arc, for the motion of a pendulum is thus produced. If the pressure caused by the weight of the pendulum-bob, which acts vertically, is resolved into two parts, of which one part is in the direction of the pendulum rod, and the other acts sideways upon the pendulum, the former does not affect the movement of the pendulum at all, and the latter, which produces the movement, is proportional to the distance of the pendulum from its place of rest, and therefore is similar in its law to the law of the force of twist of the suspending wire by which a rod with balls is supported (which force of twist is the same thing as the force which pulls the balls aside, because it exactly resists that force). Moreover, the force which acts sideways on the pendulum-bob, is in the same proportion to the whole weight of the bob, as the displacement sideways is to the length of the pendulum. Now the length of a pendulum which vibrates in a second, is 39·139 inches; and for such a pendulum, if it is pulled one inch sideways, the dead pull sideways (as I have just explained) will be $1⁄39·139$ part of its weight: and thus we know that, for any balls or other things which vibrate in one second, the dead pull sideways corresponding to an inch of displacement is $1⁄39·139$ part of their weight.

Then it is known as a general theorem regarding vibrations, that to make the vibrations twice as slow,