by finding its true anomaly. The problem which has for its object the determination of the true anomaly from the mean, was first resolved by Kepler, and is called Kepler's Problem. The solution of it may be found in the Appendix. Another and more convenient method of obtaining the true anomaly, is to compute the equation of the centre from the mean anomaly, and add it to the mean anomaly, or subtract it from it, according to the position of the body in its orbit (Art. 185).

Heliocentric Place of a Planet.

245. The place of a planet in the plane of its orbit is designated by its orbit longitude, and radius vector. To find the orbit longitude we have the equation,

long. long. of perihelion + true anomaly.

=

The orbit longitude may also be deduced from the mean longitude, by adding or subtracting the equation of the centre. The radius vector results from the polar equation of the elliptic orbit (Art. 227), viz:

in which a denotes the true anomaly, e the eccentricity, and a the semi-major axis.

246. Now to find the heliocentric longitude and latitude, which ascertain the position of the planet with respect to the ecliptic, the triangle N P p (Fig. 41) gives,

sin P p = sin N P sin P Np;

or, sin lat. = sin (orbit long. — long. of node) sin (inclin.).. (62);

and

cos PN p = tang Np cot N P, or tang N p = tang NP cos PNp,

or,

tang (long. — long. of node) = tang (orbit long. — long. of node) Xcos (inclination). . . (63).

Geocentric Place of a Planet.

247. From the heliocentric longitude and latitude and the radius vector of a planet, to find the geocentric longitude and latitude. Let S (Fig, 41) be the sun, E the earth, P the planet,

its reduced place, and V the vernal equinox. Denote the heliocentric longitude V S by L, the heliocentric latitude PS by l, and the radius vector S P by v; and denote the geocentric

longitude by G, and the geocentric latitude by λ. Also let E = SE the elongation; C = ES the commutation; A =S&E the annual parallax; and r = SE the radius vector of the earth.

Now,

or,

VET SET+VES,

GE+long. of sun.

This equation will make known the geocentric longitude, when the value of E is found. In the triangle S E, the side S =SP cos PS = v cos l, and is therefore known, the side E S is given by the elliptical theory (Art. 245), and the angle C may be derived from the following equation: C=VSE — V S « = long. of earth long. of planet and to find E we have, by Trigonometry,

ES+S: ES-S :: tan or, rv cos l: r-v cos

whence, tang (A — E) :

(ES+SE): tan (ES—SET), :: tang (A+E): tang (A - E); } †

=

=

Then,

=

tang (A + E);

or, tang (A - E) = tang (45° -8) tang

(A+ E)... (64).

But, A+E 180°C, and E = } (A + E) — } (A — E).

=

Next, to find the geocentric laitude.

Stangl = P = E « tang λ,

248. When a planet is in conjunction or opposition, the sines of the angles of elongation and commutation are each nothing. In these cases, then, the geocentric latitude cannot be found by the preceding formula, it may however be easily determined in a different manner. Suppose the planet to be in conjunction at P (Fig. 42); then,

249. To find the distance of the planet from the earth, represent the distance by D; then, from the triangles S P and EP (Fig. 41), we have,

250. The distance of a planet being known, its horizontal parallax may be computed from the equation

251. The place of the sun, as seen from the earth, may be easily deduced from the heliocentric place of the earth; for, the longitude of the sun is equal to the heliocentric longitude of the earth plus 180°, and the radius vector of the earth's orbit is the same as the distance of the sun from the earth. But it is more convenient to regard the sun as describing an orbit around the earth, and to compute its true anomaly, (Art. 244), and thence the longitude and radius vector by the equation

long. = true anomaly + long. of perigee,

and the polar equation of the orbit.

252. The orbit longitude and the radius vector of the moon

* For opposition and inferior conjunction, the sign of cos I must be changed.

are found by the same process as the longitude and radius vector of the sun. The orbit longitude being known, the ecliptic longitude and the latitude may be determined by a process precisely similar to that by which the heliocentric longitude and latitude of a planet are found (Art. 245).

Verification of Kepler's Laws.

253. If Kepler's first two laws be true, then the geocentric places of the planets, computed by the process that we have described (Art. 246), which is founded upon them, ought to agree with the true geocentric places as obtained for the same. time by direct observation: or, the heliocentric places computed from the observed geocentric places (Art. 223), ought to agree with the same as computed by the elliptic theory (Arts. 245, 246). Now, a great number of comparisons have been made between the observed and computed places, and in every instance a close agreement between the two has been found to subsist. We infer, therefore, that the motions of the planets must be very nearly in conformity with these laws.

The truth of the third law has been established by a direct comparison of the mean distances of the different planets with their periodic times.

254. Kepler's laws have been verified for the sun and moon, in a similar manner.

255. The relative distances of the sun, or moon, at different times, result from observations upon the apparent diameter, upon the principle that any two distances are inversely proportional to the corresponding apparent diameters. Let = semidiameter corresponding to the mean distance, and ♪ = semidiameter, corresponding to any distance D: then

an equation which, when ▲ has been found, will make known the distance corresponding to any observed semi-diameter 8, in terms of the mean distance as a unit.

Now, to find a, denote the greatest and least semi-diameters respectively by d', ', and the corresponding distances by D' and D", and we have,

and thence,

whence,

↓ (D' + D") = ( ~~+~) or, 1 = 금(+0);

256. The distance of the sun or moon in terms of the mean distance as a unit, may be found in a similar manner; but it may be had more accurately by means of a principle which has been discovered from observation, namely, that the distance is inversely proportional to the square root of the daily angular motion.

CHAPTER X.

OF THE INEQUALITIES OF THE MOTIONS OF THE PLANETS AND OF THE MOON; AND OF THE CONSTRUCTION OF TABLES FOR FINDING THE PLACES OF THESE BODIES.

257. It is a general law of nature, discovered by Sir Isaac Newton, that bodies tend, or gravitate towards each other, with a force directly proportional to their masses and inversely proportional to the square of their distance. The force which causes one body to gravitate towards another, is supposed to arise from a mutual attraction existing between the particles of the two bodies, and is hence called the Attraction of Gravitation. This force of attraction, common to all the bodies of the Solar System, is the general physical cause of their motions. The sun's attraction retains the planets in their orbits, and the planets by their mutual attractions slightly alter each other's motions. The reasoning by which Newton's Theory of Universal Gravitation is established, appertains to Physical Astronomy, and will be presented in another part of the work.

258. If a planet were acted on by no other force than the attraction of the sun, it is proved that its orbit would be accu