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longitude (159, 192), the excess of that of the moon above that of the sun, becomes known. Hence, as this excess: 360° : : 1 day : the moon's mean synodic revolution. Its length is thus found to be about 291 mean solar days.

196. Positions of the moon's nodes. Let E, Fig. 34, be the earth, VNL an arc of the ecliptic, pmag the moon's orbit, and BNH an arc of the great circle in which the plane of the moon's orbit meets the celestial sphere. Then since EN is in the plane of the ecliptic, the point n in which it meets the orbit is the moon's ascending node, and N is the place of the node referred to the celestial sphere.

From a series of the moon's longitudes and latitudes, computed from the observed right ascensions and declinations, we may find the longitude when the latitude is zero. This will evidently be the longitude of one of the nodes. If the latitude is then changing from south to north, it will be the longitude of the ascending node n or N. The longitude of the ascending node increased by 180° must give the longitude of the descending node.

197. Retrograde movement of moon's nodes. From the longitudes of the moon's nodes, repeatedly determined, it is found that they have a retrograde motion along the ecliptic, amounting to about 19° in a year. By this motion, which is not quite uniform, the nodes make a mean tropical revolution in 18 years and 224 days, nearly.

198. Inclination of moon's orbit. From a series of the moon's latitudes, the greatest latitude FF' may be found.

This greatest latitude has place when NF', the excess of the moon's longitude above that of the node, is 90°. It is, therefore, the measure of the angle LNH, which is the inclination of the moon's orbit to the ecliptic. The inclination of the orbit, thus obtained at different times, is found to be subject to some variation. Its least and greatest values are about 5° and 5° 17'.

199. Orbit longitude. When a body moves in an orbit inclined to the ecliptic, the sum of the longitude of the ascending node and the eastwardly angular distance of the body from the node, is called its orbit longitude. Thus, if V, Fig. 34, be the vernal equi

nox, and M the moon's place referred to the celestial sphere, the sum of the angles VEN and NEM is the orbit longitude of the moon when at m. If V' be a point in the orbit when referred to the celestial sphere, corresponding to the vernal equinox, that is, such that the angle V'EN is equal to VEN, then the angle V'EM or arc V'M, is the moon's orbit longitude when she is at m.

When the inclination of the orbit, the longitude of the node, and the longitude of the body are given, the orbit longitude is easily found. Let MD be an arc of a circle of latitude. Then VD is the longitude of the body. Subtracting VN, the longitude of the node, from VD, we have ND. Then in the right angled spherical triangle NDM, we have the base ND and the angle MND, to find NM, the measure of the angle NEM. The angle NEM added to V'EN or VEN, the longitude of the node, gives V'EM, the orbit longitude.

Conversely, the longitude may be found when the orbit longitude is given.

200. Apsides of the moon's orbit. Using the orbit longitudes of the moon, the orbit longitudes of the apsides of her orbit may be found by proceeding in the same manner as for the positions of those of the sun's apparent orbit (155). The orbit longitudes being found, the longitudes may be obtained by the preceding article.

201. Motion of the apsides of the moon's orbit. The longitudes of the apsides, obtained at different periods, are found to increase at the rate of about 41° in a year. They have, therefore, a direct motion, and make a mean tropical revolution in a little less than

9 years.

The motion of the moon's nodes, the variation in the inclination of the orbit, and the motion of the apsides are all effects of the sun's attraction on the moon.*

202. Moon's orbit. From the greatest and least parallaxes of the moon (95), and from her parallax when at any position m, Fig. 34, in her orbit, the least and greatest radius vectors Ep and Ea, and the radius vector Em become known (93). From the

* See Chapter XXII.

orbit longitudes of the perigee and of the moon, when at m, the value of the angle pЕm at that time is known, being equal to their difference.

Assuming the orbit to be an ellipse, of which ap is the transverse axis, E a focus, and C, the middle point of ap, the centre, we have ас 1⁄2 (aE + Ep), EC = 1⁄2 (aE — Ep), and, by the property of the ellipse,

Em

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Now, whatever be the position of m, the value of Em, obtained from this expression, is always found to be nearly equal to its value obtained from the parallax. Hence, the moon's orbit is nearly an ellipse, having the earth in one focus.

It may also be found in the same manner as for the sun (153), that the moon's radius vector describes round the earth nearly equal areas in equal times.

203. Greatest equation of the centre and eccentricity of moon's orbit. These may be obtained in the same manner as for the sun (160, 161). Or, the eccentricity may be found from the values of aC and EC, deduced from the greatest and least parallaxes, as in the preceding article. The eccentricity being known, the greatest equation may be computed by the formulæ in the note to article (161).

The greatest equation is found to exceed 6°, being more than three times that of the sun. Consequently, the eccentricity of the lunar orbit must also be more than three times that of the apparent orbit of the sun or orbit of the earth.

204. Other equations of the moon's motion. The moon's motion is subject to numerous inequalities besides the equation of the centre. The three principal ones are called, respectively, Evection, Variation, and Annual Equation. The Evection was discovered by Ptolemy. It depends on the angular distances of the moon from the sun and the perigee. When greatest it amounts to about 11o. The Variation was discovered by Tycho Brahe. It disappears when the moon is in the syzygies and quadratures, and is greatest when she is in octants. It then amounts to 35'.7. The

Annual Equation depends on the sun's mean anomaly, and, when greatest, amounts to 11'.2.

Investigations, in physical astronomy, by Laplace and others, have made known the causes of these inequalities, and have discovered various smaller ones with which the moon's motion is af fected. By means of these investigations, and long continued accurate observations, the moon's motion is now known, and her place at any given time may be computed, with a very near approach to precision.

205. Lunar Tables. There are two sets of lunar tables, of nearly equal accuracy: one by Burkhardt, and the other by Damoiseau. The former, in which 36 equations are employed in finding the longitude, is used in computing the Nautical Almanac, Connaissance des Tems, and Berlin Jahrbuch.

206. Moon's phases. The different forms which the moon's visible disc presents, during a synodic revolution, are called phases. The moon's phases are completely accounted for by assuming her to be an opaque globular body, rendered visible by reflecting light, received from the sun. Let E, Fig. 35, be the earth, and ABCD the orbit of the moon; the sun being supposed to be at a great distance in the direction ES. When the moon is in conjunction at A, the enlightened half* is turned directly from the earth, and she must then be invisible. It is then said to be new

moon.

About 7 days after new moon, when she is in quadrature at B, one half of her illuminated surface is turned towards the earth, and her enlightened disc then appears as a semi-circle. She is then said to be at her first quarter.

About 15 days after new moon, when she is in opposition at C, the whole of her illuminated surface is turned towards the earth, and she appears as a full circle. It is then said to be full moon.

About 7 days after this, when she is again in quadrature, at D, one half of her illuminated surface being towards the earth, she again appears as a semi-circle. She is then said to be at her last quarter.

* As the sun is far greater than the moon, he enlightens rather more than half her surface. But this slight excess need not be here considered.

From new moon to first quarter, and from last quarter to new moon, her enlightened disc is called a crescent. This phase is represented near a and d. The two extremities of the crescent are called cusps or horns. From first quarter to full moon, and from full moon to last quarter, the form of her enlightened disc is said to be gibbous. This phase is represented near 6 and c.

207. Lunation or Lunar Month. The interval from new moon to new moon again, is called a lunation or lunar month. It is evidently the same as a synodic revolution of the moon.

208. Mean New or Full Moon. The time at which it would be new moon or full moon, according to the mean motions of the sun and moon, is called mean new moon or mean full moon.

When the moon is first disc is quite perceptible,

209. Obscure part of the moon's disc. visible after new moon, the whole of her the part not fully illuminated appearing with a faint light. As the moon's age, that is, the time from new moon, increases, the obscure part becomes more and more faint; and it entirely disappears before full moon. This phenomenon depends on light reflected from the earth to the moon, and from the moon back to the earth.

When the moon is near to a, she evidently receives light from nearly the whole of the earth's illuminated surface; and this light being in part reflected back, renders visible that portion of the disc that is not directly illuminated by the sun. As the moon advances towards opposition at C, the quantity of light she receives from the illuminated surface of the earth must evidently decrease; and its effect in rendering the obscure part visible, is still further diminished by the increased size and, consequently, increased light of the directly illuminated part, which finally prevents the faint light of the former from making any impression.

210. The earth as seen from the moon. It is obvious, from the explanation in the preceding article, that to an observer at the moon, the earth must appear as a splendid moon, assuming all the phases of the latter body as seen from the earth, and having more than three times the apparent diameter (100).

211. Moon's surface. When the moon is viewed with a telescope, the line separating the enlightened part of the disc from the ob

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