only approximate times, and may differ some minutes from the true times. Let, therefore, the moon's apparent longitude and latitude be computed for the approximate time of beginning. We then, easily, obtain the average values of the moon's apparent hourly motions in longitude and latitude, between this time and the time of the middle of the eclipse. The latter hourly motion. will be the average value of q', and the difference between the former and the sun's hourly motion, will be the average value of p'. Also, let the moon's augmented semi-diameter (97) be found. for the approximate time of beginning,* and take h= the sum of s' and this augmented semi-diameter. Then, using the values of p', q' and h, let the time of beginning be again computed. This time will be very nearly the true time. If still greater accuracy is desired, it may be obtained by another repetition of the computation. In a similar manner, the true time of the end of the eclipse may be found. OCCULTATIONS. 256. Definition. When the moon passes between the earth and a star or planet, she must, during the passage, render the body invisible to some parts of the earth. This phenomenon is called an occultation of the star or planet. 257. Extent of visibility of an occultation of a fixed star. The breadth of the portion of the earth's surface, in which an occultation of a star is visible, is much less than that for an eclipse of the sun. It is about 2150 miles. Let E, Fig. 44, be the earth's centre, and Es the direction of the star; and, supposing the moon to pass directly between the star and centre of the earth, let M be the place of her centre when in that position. As the star has no sensible parallax, lines, as Aa and Bb, drawn from it and tangents to the moon, will be parallel to sE. Hence, in the portion of the surface whose breadth ab is limited by these parallel lines, the occultation must be visible. * A formula for this purpose is always given with the formula for computing the parallaxes. M Now, as ha is parallel to ME, the angle aME Mah 8, the moon's semi-diameter, nearly. We, also, have (93 E), ME Hence, from the triangle MEa, we have, Ea sin π Ea Ea: sin s But (99), sin sin Therefore, the angle MaE= 164° 9'. Whence, taking aME ac = 15° 35'; and for the length :: sin & : sin MaE sin & sin .2730. = 8 = 16′, we obtain MEa of ab, 2150 miles, nearly. As the moon moves in her orbit from u towards n, the occultation will be visible in succession to different portions of the earth, lying in a direction, nearly, from west to east; the common breadth of the whole not differing greatly from that obtained above, except when the moon passes considerably north or south of the star. The parallaxes and apparent diameters of all the planets are small. The extent of visibility of an occultation of any one of them will not, therefore, differ much from that of an occultation of a fixed star. 258. Distance of moon's centre from the star, at the beginning or end of an occultation for the earth in general. This distance is equal to the sum of the moon's horizontal parallax and semidiameter. Let CD, a tangent to the earth, be parallel to sE. Then the occultation must commence for the earth, in general, when the moon's edge comes to this line. Hence, the distance sEu sEg + gEu EgD+gEu = x + d. The greatest value of +8 is 78' 20".5. Hence, when the moon's least distance from the star exceeds this quantity, there cannot be an occultation at any place on the earth. From the greatest and least values of +8, and by taking into view the inequalities in the moon's motion, it has been found, that, when at the time of the moon's mean conjunction with a star, the difference between the mean latitude of the moon and that of the star is 1° 37', there cannot be an occultation; but when the difference is less than 51', there must be one. Between these limits there is a doubt, which can only be removed by computing the true place of the moon. 259. Stars that may suffer an occultation. As the sum of the moon's greatest latitude (198), and the greatest distance of the moon from a star, when an occultation can take place, is about 6° 36', it follows that no star whose latitude is greater than this, can suffer an occultation, and that all those whose latitudes are a little less may be occulted. 260. Computation of an occultation for a given place. The computation of an occultation for a given place, either of a star or planet, differs but little from that of a solar eclipse. The star or planet takes the place of the sun. In the case of a star, it is to be observed that the star has no sensible parallax, apparent semidiameter, or hourly motion. In the case of a planet, the moon's apparent relative hourly motion in latitude depends on the hourly motions in latitude of both the moon and planet. In making the computation, the difference between the latitude of the moon and star or planet, at the time of apparent conjunction, is used instead of the moon's latitude. Consequently, the arc AB, Fig. 43, which, in an eclipse of the sun, represents an arc of the ecliptic, in the case of an occultation, represents an arc passing through the star or planet, and parallel to the ecliptic. As the distance on this arc between two circles of latitude is less than on the ecliptic, the apparent distance of the moon in longitude from the star or planet, and the moon's apparent relative motion in longitude require small reductions. These are made by multiplying each, by the cosine of the latitude of the star or planet. Instead of Longitudes and Latitudes, Right Ascensions 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 For the investigation of formulæ for computing eclipses and occultations, see the Appendix. 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. Du Séjour, an astronomer of note of the last century, concluded, that in calculating solar eclipses, the sun's semi-diameter, 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 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. DEFINITIONS. 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. 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 |