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and Declinations, may be used in the calculation both of eclipses and occultations*.

261. Irradiation and Inflexion. Some astronomers have thought that the apparent diameter of the sun as obtained from observation, and given in the tables, is too great. This has been inferred from a comparison of the observed time of the beginning or end of a solar eclipse at a known meridian, with the time obtained by computation, after making allowances for the errors of the tables in other respects. To account for it, they have supposed that the apparent diameter of the sun, is amplified by the very lively impressions so luminous an object makes on the organ of sight. This amplification is called, irradiation. They have also supposed that the moon has an atmosphere which by its action on the rays of light passing through it, inflects them, and produces an effect such as would be produced by a small diminution in the moon's semidiameter. This is called the inflexion of the moon. Duséjour, an astronomer of note of the last century, concluded that in calculating solar eclipses, the sun's semidiameter as given in the tables, should be diminished 31" for irradiation, and the moon's 2", for inflexion.

The subject of irradiation and inflexion is, however, involved in considerable uncertainty, and several eminent astronomers have doubted the existence of either.

262. Scholium. Eclipses of the sun and occultations, are not only interesting phenomena, but when carefully observed, they are also, practically useful. When observed at places whose latitudes and longitudes are truly known, they furnish means for detecting errors in the tables, used in computing the places, parallaxes, and apparent diameters of the bodies.. For, the difference between the observed and computed times must depend

*For the investigation of formulæ for computing eclipses and occultatious, see the Appendix.

on these errors. When observed at places whose longitudes are not well known, they furnish the means of determining them more accurately. Their application for this latter purpose will be noticed in a subsequent chapter.

CHAPTER XIV.

GENERAL REMARKS ON THE PLANETS.-DEFINITIONS.-ELEMENTS OF THE ORBITS OF THE PLANETS.-CONVERSION OF THE HELIOCENTRIC PLACE OF A PLANET INTO ITS GEOCENTRIC PLACE.-RETROGRADE

MOTIONS OF THE

PLANETS. REAL DISTANCE, ETC., OF THE PLANETS.

263. General Remarks. Each of the planets, like the moon, is found to be during about half its period, on one side of the ecliptic, and during the other half, on the other side. Hence, we may infer that their orbits are all divided by the plane of the ecliptic in nearly equal parts. But the apparent motions of the planets differ essentially in one respect from that of the moon. The apparent motion of the latter is always direct, or from west to east; but the apparent motion of each planet, during a part of its period, is retrograde, or from east to west. When the motion is changing from direct to retrograde, or the contrary, the planet remains stationary, or nearly so, for some days This difference between the motions of the moon and planets, is a consequence of their different centres of motion. As the latter revolve round the sun (17), their apparent motions must depend both on their own motions, and on that of the earth.

DEFINITIONS.

264, Geocentric and Heliocentric Places. The geocentric place of a body is its place as seen from the centre of the

earth; and the heliocentric place, is its place as seen from the centre of the sun.

265. Curtate distance. If a straight line be conceived to be drawn from the centre of a planet, perpendicular to the plane of the ecliptic, the distance from the point in which it meets this plane to the centre of the sun, is called the curtate distance of the planet. The point itself is called the reduced place of the planet. Thus, if P'SN, Fig. 46, be the plane of the ecliptic, S the sun's centre, NP a part of the orbit of a planet, P the place of the planet at 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 commutation, 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 ode. 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.

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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, putr 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 = SN = radius vector of the planet when at the node N, and N VSN the heliocentric longitude of the node. Then, we have, SEN VES — VENS G, and SNE VAN VSN = VEN VSN = G― N. From the triangle SEN, we have,

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or,

or,

sin SNE: sin SEN :: r R,

sin (GN)

sin (SG) ::r R,

=

r. sin (SG) R sin (GN)

In like manner we have,

r'. sin (S'G') = R. sin (G' — N). Therefore, dividing (A) by (B), we have,

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T. sin(S--G) sin(G--N) sin G cos N--cos G sin N sin G-cos G tang. N r'.sinS'--G') ̄ ̄sin G'--N)sinG'cos N--cos G'sin NsinG'--cos G'tang.N' Hence, we easily find,

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r. sin (SG) sin G''. sin (S' — G') sin G r. sin (S — G) cos G' — r'. sin (S' — G') cos G 7. sin (SG)

We, also, have (A), R

=

sin (GN)

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 have 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 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

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