direction Np. Let E N men be the shadow or projection of the orbit on the plane of the ecliptic, then N Sn is the intersection of these two planes,

for the orbit rises above the plane of the ecliptic towards Np, and sinks below it at NP. The angle p Nm, which these two planes make with one another, is the inclination of the orbit PNPA to the plane of the ecliptic.

NOTE 54, p. 9.

Latitude of a planet. The angle p Sm, fig. 12, or the height of the planet pabove the ecliptic E N m.

In this case the latitude is north. Thus, celestial latitude is the angular distance of a celestial body from the plane of the ecliptic, whereas terrestrial latitude is the angular distance of a place on the surface of the earth from the equator.

NOTE 55, p. 9. Nodes. The two points N and n, fig. 12, in which the orbit N An P of a planet or comet intersects the plane of the ecliptic e N En. The part N An of the orbit lies above the plane of the ecliptic, and the part n PN below it. The ascending node N is the point through which the body passes in rising above the plane of the ecliptic, and the descending node n is the point in which the body sinks below it. The nodes of a satellite's orbit are the points in which it intersects the plane of the orbit of the planet.

Sp in fig. 12. If op be the of the planet p, m Sp is its When these three quantities

NOTE 56, p. 10. Distance from the sun. vernal equinox, then Sp is the longitude latitude, and Sp its distance from the sun. are known, the place of the planet p is determined in space.

NOTE 57, pp. 10, 59. Elements of an orbit. Of these there are seven. Let PN An, fig. 12, be the elliptical orbit of a planet, C its centre, S the sun in one of the foci, o the point of Aries, and EN en the plane of the ecliptic. The elements are-the major axis A P; the excentricity CS; the periodic time, that is, the time of a complete revolution of the body in its orbit; and the fourth is the longitude of the body at any given instantfor example, that at which it passes through the perihelion P, the point of its orbit nearest to the sun. That instant is assumed as the origin of time, whence all preceding and succeeding periods are estimated. These four quantities are sufficient to determine the form of the orbit, and the motion of the body in it. Three other elements are requisite for determining the position of the orbit in space. These are, the angle op SP, the longitude of the perihelion; the angle A Ne, which is the inclination of the orbit to the plane of the ecliptic; and, lastly, the angle op S N, the longitude of N the ascending node.

NOTE 58, p. 10. Whose planes, &c. The planes of the orbits, as PN An, fig. 12, in which the planets move, are inclined or make small angles e N A with the plane of the ecliptic E N e n, and cut it in straight lines, N Sn passing through S, the centre of the sun.

NOTE 59, p. 11. Momentum. Force measured by the weight of a body and its speed, or simple velocity, conjointly. The primitive momentum of the planets is, therefore, the quantity of motion which was impressed upon them when they were first thrown into space.

NOTE 60, p. 11. Unstable equilibrium. A body is said to be in equilibrium when it is so balanced as to remain at rest. But there are two kinds of equilibrium, stable and unstable. If a body balanced in stable equilibrium be slightly disturbed, it will endeavour to return to rest by a number of movements to and fro, which will continually decrease till they cease altogether, and then the body will be restored to its original state of repose. But, if the equilibrium be unstable, these movements to and fro, or oscillations, will become greater and greater till the equilibrium is destroyed.

NOTE 61, p. 14. Retrograde. Going backwards, as from east to west, contrary to the motion of the planets.

NOTE 62, p. 14. Parallel directions. Such as never meet, though prolonged ever so far.

NOTE 63, pp. 14, 16. The whole force, &c. Let S, fig. 13, be the sun, Nm n the plane of the ecliptic, p the disturbed planet moving in its orbit

np N, and d the disturbing planet. Now, d attracts the sun and the planet p with different intensities in the directions d S, dp: 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, pT, and pm. 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 or 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 Sp 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 pm, fig. 13, at right angles to the plane of the orbit Npn, 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 mn, 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 Ppa is greater than Pp 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

Fig. 15.

Fig. 16.

PSA advance to a. In fig. 16, on the contrary, the curvature P pa 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 PSA to recede to a.

Fig. 17.

A

S

P

A"

NOTE 67, pp. 16, 17. Motion of apsides. Let PSA, fig. 17, be the position of the elliptical orbit of a planet, at any time; then, by the action of the disturbing forces, it successively takes the position P'SA', P" SA", &c., till by this direct motion it has accomplished a revolution, and then it begins again; so that the motion is perpetual.

NOTE 68, p. 17. Sidereal revolution. The consecutive return of an object to the same star.

NOTE 69, p. 17. Tropical revolution. object to the same tropic or equinox.

The consecutive return of an

NOTE 70, p. 17. The orbit only bulges, &c. In fig. 18 the effect of the variation in the excentricity is shown where Pp A is the elliptical orbit at any given instant; after a time it will take the form Pp' A, in consequence of the decrease in the excentricity CS; then the forms Pp" A, P p"" A, &c., consecutively from the same cause; P and, as the major axis P A always retains the same length, the orbit approaches more and more nearly to the circular form. But, after this has gone on for some thousands of years, the orbit contracts again, and becomes more and more elliptical.

NOTE 71, pp. 18, 19. the heavens. See note 46.

Fig. 18.
p""

p"

A

The ecliptic is the apparent path of the sun in

NOTE 72, p. 18. This force tends to pull, &c. The force in question, acting in the direction pm, fig. 13, pulls the planet p towards the plane Nm n, or pushes it farther above it, giving the planet a tendency to move in an orbit above or below its undisturbed orbit N pn, which alters the angle p N m, and makes the node N and the line of nodes N n change their positions.

NOTE 73, p. 18. Motion of the nodes. Let S, fig. 19, be the sun; SNn the plane of the ecliptic; P the disturbing body; and p a planet moving in its orbit pn, of which pn is so small a part that it is represented as a straight line. The plane Snp of this orbit cuts the plane of the ecliptic in the straight line Sn. Suppose the disturbing force begins to act on p, so as to draw the planet into the arc pp'; then, instead of moving in the orbit pn, it will tend to move in the orbit pp'n', whose plane cuts the ecliptic in the straight line S n'. If the disturbing force acts again upon

the body when at p', so as to draw it into the arc p'p", the planet will now

tend to move in the orbit p' p" n", whose plane cuts the ecliptic in the straight line Sn". The action of the disturbing force on the planet when at p" will bring the node to n"", and so on. In this manner the node goes backwards through the successive points n, n', n", n"", &c., and the line of nodes Sn has a perpetual retrograde motion about S, the centre of the sun. The disturbing force has been represented as acting at intervals for the sake of illustration: in nature it is continuous, so that the motion of the node is continuous also; though it is sometimes rapid and sometimes slow, now retrograde and now direct; but, on the whole, the motion is slowly retrograde.

NOTE 74, p. 18. When the disturbing planet is anywhere in the line SN, fig. 19, or in its prolongation, it is in the same plane with the disturbed planet; and, however much it may affect its motions in that plane, it can have no tendency to draw it out of it. But when the disturbing planet is in P, at right angles to the line S N, and not in the plane of the orbit, it has a powerful effect on the motion of the nodes: between these two positions there is great variety of action.

NOTE 75, p. 19. The changes in the inclination are extremely minute when compared with the motion of the node, as evidently appears from fig. 19, where the angles np n', n'p' n", &c., are much smaller than the corresponding angles n S n', Sn", &c.

NOTE 76, p. 20. Sines and cosines. Figure 4 is a circle; np is the sine, and Cp is the cosine of an arc mn. Suppose the radius Cm to begin to revolve at m, in the direction mna; then at the point m the sine is zero, and the cosine is equal to the radius C m. As the line C m revolves and takes the successive positions Cn, Ca, Cb, &c., the sines np, aq, br, &c., of the arcs mn, ma, mh, &c., increase, while the corresponding cosines Cp, Cq, Cr, &c., decrease; and when the revolving radius takes the position Cd, at right angles to the diameter gm, the sine becomes equal to the radius Cd, and the cosine is zero. After passing the point d, the contrary happens; for the sines e K, IV, &c., diminish, and the cosines CK, CV, &c., go on increasing, till at g the sine is zero, and the cosine is equal to the radius Cg. The same alternation takes place through the remaining parts gh, hm, of the circle, so that a sine or cosine never can exceed the radius. As the rotation of the earth is invariable, each point of its surface passes through a complete circle, or 360 degrees, in twenty-four hours, at a rate of 15 degrees in an hour. Time, therefore, becomes a measure of angular motion, and vice versâ, the arcs of a circle a measure of time, since these two quantities vary simultaneously and equably; and, as the sines and cosines of the arcs are expressed in terms of the time, they vary with it. Therefore, however long the time may be, and how often soever the radius may revolve round the circle, the sines and cosines never can exceed the radius; and, as the radius is assumed to be equal to unity, their values oscillate between unity and zero.

NOTE 77, p. 20. The small excentricities and inclinations of the planetary orbits, and the revolutions of all the bodies in the same direction, were proved by Euler, La Grange, and La Place, to be conditions necessary for the stability of the solar system. Subsequently, however, the periodicity of the terms of the series expressing the perturbations was