Fig. 13.
Fig. 14.
Note 63, pp. [14], [16]. The whole force, &c. Let S, fig. 13, be the sun, N m n the plane of the ecliptic, p the disturbed planet moving in its orbit n p N, and d the disturbing planet. Now, d attracts the sun and the planet p with different intensities in the directions d S, d p: the difference only of these forces disturbs the motion of p; it is therefore called the disturbing force. But this whole disturbing force may be regarded as equivalent to three forces, acting in the directions p S, p T, and p m. The force acting in the radius vector p S, joining the centres of the sun and planet, is called the radial force. It sometimes draws the disturbed planet p from the sun, and sometimes brings it nearer to him. The force which acts in the direction of the tangent p T is called the tangential force. It disturbs the motion of p in longitude, that is, it accelerates its motion in some parts of its orbit and retards it in others, so that the radius vector S p does not move over equal areas in equal times. (See [note 26].) For example, in the position of the bodies in fig. 14, it is evident that, in consequence of the attraction of d, the planet p will have its motion accelerated from Q to C, retarded from C to D, again accelerated from D to O, and lastly retarded from O to Q. The disturbing body is here supposed to be at rest, and the orbit circular; but, as both bodies are perpetually moving with different velocities in ellipses, the perturbations or changes in the motions of p are very numerous. Lastly, that part of the disturbing force which acts in the direction of a line p m, fig. 13, at right angles to the plane of the orbit N p n, may be called the perpendicular force. It sometimes causes the body to approach nearer, and sometimes to recede farther from, the plane of the ecliptic N m n, than it would otherwise do. The action of the disturbing forces is admirably explained in a work on gravitation, by Mr. Airy, the Astronomer Royal.
Note 64, pp. [16], [74]. Perihelion. Fig. 10, P, the point of an orbit nearest the sun.
Note 65, [p. 16]. Aphelion. Fig. 10, A, the point of an orbit farthest from the sun.
Note 66, pp. [16], [17]. In fig. 15 the central force is greater than the exact law of gravity; therefore the curvature P p a is greater than P p A the real ellipse; hence the planet p comes to the point a, called the aphelion, sooner than if it moved in the orbit P p A, which makes the line P S A advance to a. In fig. 16, on the contrary, the curvature P p a is less than in the true ellipse, so that the planet p must move through more than the arc P p A, or 180°, before it comes to the aphelion a, which causes the greater axis P S A to recede to a.
Fig. 15.