any time, and PP' perpendicular to P'SN, then SP' is the curtate distance of the planet at that time, and P' is its reduced place.

266. Elongation, &c. If a plane triangle be formed by joining the reduced place of a planet, the centre of the sun, and centre of the earth, the angle at the earth is called the elongation of the planet, the angle at the sun is called the commutation, and the angle at the reduced place of the planet is called the annual parallax. Thus, SEP' is the elongation, ESP' the computation, and EP'S the annual parallax.

267. Elements of the orbit of a planet. There are seven different quantities necessary to be known in order to compute the place of a planet for a given time. These are called the elements of the orbit. They are, the longitude of the ascending node; the inclination of the plane of the orbit to that of the ecliptic; the periodic time, or the planet's mean motion; the mean distance of the planet from the sun, or, which is the same, the semi-transverse axis of its orbit; the eccentricity of the orbit; the longitude of the perihelion; and the time the planet is at the perihelion, or its mean longitude at a given time or epoch.

ELEMENTS OF THE ORBIT.

268. Longitude of the ascending node. When a planet is at either of its nodes, it is in the plane of the ecliptic, and, consequently, its latitude is then nothing. Let the right ascension and declination of the planet be observed on several consecutive days, at the period it is passing from the south to the north side of the ecliptic, and let its corresponding longitudes and latitudes be computed (119). From these, the time at which the planet's latitude is nothing, and its longitude at that time, may be obtained by proportion or interpolation. This longitude of the planet will evidently be the geocentric longitude of the node. Also, by means of the solar tables, let the longitude of the sun and the radius vector of the earth be found for the time the planet is at the node. By similar observations and computations when the planet returns to the node, let the values of the same quantities be again obtained. From these data, if we assume the node to remain in the same position, its heliocentric longitude may be found.

Let S, Fig. 45, be the sun, PQ a part of the orbit of the planet, N the node, E the place of the earth when the planet was found to be at the node N, from the first set of observations, and E' its place at the time of the planet's return to the node. Also, let EV, E'V, and SV, all parallel to one another, represent the direction of the vernal equinox. Then, assuming the mean radius vector of the earth to be a unit, put r = SE earth's radius vector, S VES= sun's longitude, and G = VEN = geocentric longitude of the node, when the earth is at E; and let r', S' and G' represent the same quantities when the earth is at E'. Also, put R = radius vector of the planet when at the node N, and N the heliocentric longitude of the node. Then, we have, SEN VES VEN SG, and SNE = VAN — VSN

VSN = GN. From the triangle SEN, we have,

or,

or,

sin SNE : sin SEN ::r: R,

sin (GN): sin (S — G) : : r: R,

r. sin (SG)=R sin (GN)

In like manner we have,

?'. sin (S' — G') = R. sin (G' — N)

SN

VSN

VEN

Therefore, dividing (A) by (B), we have,

tang N

r. sin (S — G) sin G' — r'. sin (S'

r'. sin (S' — G′) sin G r. sin (S — G) cos G' — r'. sin (S′ — G′) cos G ̊

We, also, have (A), R =

r. sin (SG)

sin (G-N)

The heliocentric longitude of the descending node may be found in a similar manner.

269. Retrograde motions of the nodes. From observations made at distant periods, it is found that the heliocentric longitudes of the nodes of all the planets are slowly increasing. The greatest increase is about 70' in a century. But, in consequence of the retrograde motion of the vernal equinox, the longitude of a fixed star increases, during a century, nearly 84'. Hence, as the increase in the longitude of each node is less than that of a fixed star, it follows that the nodes of all the planets have slow retrograde motions.

When the motion of the nodes of a planet has been found from observations at distant periods, the slight correction necessary in the longitude of the node, as determined by the last article on the assumption that the node did not move, may be easily made. It it is also obvious, that with the longitude of the node found for any known time and the motion of the node, the longitude may be easily obtained for any other given time.

270. The plane of a planet's orbit. When the heliocentric longitudes of the two nodes of the same orbit are obtained for the same instant of time, they are found to differ 180°. Hence, it follows that the line of the nodes, and, consequently, the plane of the orbit, pass through the centre of the sun.

271. Inclination of the orbit. To determine the inclination of the orbit, let the time at which the sun's longitude is the same as the heliocentric longitude of the node be found, by means of the solar tables; and let the longitudes and latitudes of the planet be found from its observed right ascension and declination, for several consecutive days, contiguous to this time. From these, its geocentric longitude and latitude at that time become known.

Let E, Fig. 46, be the earth, S the sun, and N the node, when the longitude of the sun and node are the same, and let P be the place of the planet in its orbit at that time. Let PP' be perpendicular to the plane of the ecliptic, meeting it in P' and P'D perpendicular to SN. Then will the angle PDP' be the inclination of the plane of the orbit. Put E

centric longitude of the planet

SEP' = VEP' VES geosun's longitude, a PEP' = PDP'inclination of

geocentric latitude of the planet, and I the orbit. Then,

The orbits of the planets, with one exception, have small inclinations. Those of Venus, Mars, Jupiter, Saturn, Uranus, and Neptune, are all less than 4°; that of Mercury is about 7°; that

of Pallas is nearly 35°; and those of all the other asteroids are between 3° and 15o.

The inclinations are found to have small secular variations. Some of them are increasing and others decreasing.

272. Periodic Time and Mean Motion. The period which elapses from the time that the planet is at one of its nodes till its return to the same, allowance being made for the motion of the node, is the sidereal revolution of the planet.

Another method of finding the sidereal revolution, is to deduce it from the synodic revolution. The latter is obtained by observing the times of two consecutive conjunctions of the same kind, or two consecutive oppositions. The intervening period is the synodic revolution, from which the sidereal is easily obtained. Let s = the synodic revolution, p = the periodic time or sidereal revolution, and p' the sidereal revolution of the earth, all expressed in mean solar days. Also, let m = mean daily motion of the planet, and m' = mean daily motion of the earth. Then we evidently have, m'm: 360° : 1 day : s.

Substituting these, and dividing the first and second terms of the proportion by 360, we have,

In which, the upper sign is to be used for a superior planet, and the lower for an inferior.

When P is found, m, the mean daily motion, becomes known (C).

Cor. From the proportion (D), we have, for the synodic revolution in terms of the periodic revolutions of the planet and earth, pp' p p

273. Heliocentric Longitude and Latitude. When the place

of the ascending node, and the inclination of the orbit of a planet are known, and the geocentric longitude and latitude of the planet at any time have been found, from its observed right ascension and declination, its heliocentric longitude and latitude, and also, its radius vector, at that time, may be obtained by computation.

Let the sun's longitude and radius vector, at the time, be calculated; and referring to Fig. 47, put,

NVSN

L'= VES = sun's longitude,

heliocentric longitude of ascending node,

180°, or D = L' + 180° N, and E =

VEP - VES G

L', are known. We have also, p= SpE VSpVAp = VSp

+

- VEp L G, S = NSE NSP Dx, and L = NSpNx. Now, by trigonometry, we have, Dp sin x, and Sp tang 7 = Pp Dp tang I = tang l sin x tang I...........

or,

or,

Also, Ep tang λ= Pp Sp tang 1,

Sp sin x tang I,

VSN

Sp

sin D

tang tang :: Sp: Ep: sin E: sin S........ (F) But, since SD-x, we have, sin S = sin (D x) − COS X cos D sin x.

Substituting in the proportion (F), the

values of sin S and tang 7, it becomes,

tanga sin x tang I :: sin E: sin D cos x

cos D sin x,

or, sin a tang I sin E

tang a sin D cos x

tang a cos D sin x,

tang a sin D

tang a cos D tang x.

or, tang x tang I sin E

Hence, tang x =

tang a sin D

tang I sin E + tang a cos D

Consequently, LN+x, becomes known.

As SDx, is also known when x is known, we may obtain 7 from either (E) or (F), the latter of which gives,