round their orbits. For instance, imagine the moon's mass distributed over her orbit in the form of a rigid ring of 480,000 miles diameter, and imagine less of it to exist where the present speed is greater, so that the ring would be thicker at the moon's apogee, and thinner at the perigee. Such a ring round the earth would be similar to Saturn's rings, which have also a precession of nodes, only Saturn's rings are not rigid, else there would be no equilibrium. Now if we leave out of account the earth and imagine this ring to exist by itself, and that its centre simply had a motion round the sun in a year, since it makes an angle of 5½° with the ecliptic it would vibrate into the ecliptic till it made the same angle on the other side and back again. But it revolves once about its centre in twenty-seven solar days, eight hours, and it will no longer swing like a ship in a ground-swell, but will get a motion of precession opposed in direction to its own revolution. As the ring's motion is against the hands of a watch, looking from the north down on the ecliptic, this retrogression of the moon's nodes is in the direction of the hands of a watch. It is exactly the same sort of phenomenon as the precession of the equinoxes, only with a much shorter period of 6798 days instead of 25,866 years.
I told you how, if we knew the moon's mass or the sun's, we could tell the amount of the forces, or
the torque as it is more properly called, with which it tries to tilt the earth. We know the rate at which the earth is spinning, and we have observed the precessional motion. Now when we follow up the method which I have sketched already, we find that the precessional velocity of a spinning body ought to be equal to the torque divided by the spinning velocity and by the moment of inertia[[7]] of the body about the polar axis. Hence the greater the tilting forces, and the less the spin and the less the moment of inertia, the greater is the precessional speed. Given all of these elements except one, it is easy to calculate that unknown element. Usually what we aim at in such a calculation is the determination of the moon's mass, as this phenomenon of precession and the action of the tides are the only two natural phenomena which have as yet enabled the moon's mass to be calculated.
I do not mean to apologize to you for the introduction of such terms as Moment of Inertia, nor do I mean to explain them. In this lecture I have avoided, as much as I could, the introduction of mathematical expressions and the use of technical terms. But I want you to
understand that I am not afraid to introduce technical terms when giving a popular lecture. If there is any offence in such a practice, it must, in my opinion, be greatly aggravated by the addition of explanations of the precise meanings of such terms. The use of a correct technical term serves several useful purposes. First, it gives some satisfaction to the lecturer, as it enables him to state, very concisely, something which satisfies his own weak inclination to have his reasoning complete, but which he luckily has not time to trouble his audience with. Second, it corrects the universal belief of all popular audiences that they know everything now that can be said on the subject. Third, it teaches everybody, including the lecturer, that there is nothing lost and often a great deal gained by the adoption of a casual method of skipping when one is working up a new subject.
Some years ago it was argued that if the earth were a shell filled with liquid, if this liquid were quite frictionless, then the moment of inertia of the shell is all that we should have to take into account in considering precession, and that if it were viscous the precession would very soon disappear altogether. To illustrate the effect of the moment of inertia, I have hung up here a number of glasses—one a filled with sand, another b with treacle, a third c with oil, the fourth d with water,
and the fifth e is empty (Fig. 41). You see that if I twist these suspending wires and release them, a vibratory motion is set up, just like that of the balance of a watch. Observe that the glass with water vibrates quickly, its effective moment of inertia being merely that of the glass itself, and you see that the time of swing is pretty much the same as that of the empty glass; that is, the water does not seem to move with the glass. Observe that the vibration goes on for a fairly long time.