The true declination d of the sun, and D of the moon (these are to be considered as positive if north, and negative if south): the difference between them is D'. Also the true semidiameters of the sun and moon. Find next (art. 315.) the horary angle of the sun and moon, or the apparent time at the place, for the approximate instant T of apparent conjunction in right ascension; this angle must be considered as positive if the moon is on the west of the meridian, and negative if on the east let it be represented by T. Then With the known latitude 7 of the place, and the values of P', D and T', find, from the formula (11) art. 161., the value a of the relative parallax in right ascension at the time T: again, with the same data and the value of a just obtained find, from the formula (v) or (VI) art. 162., the value 8 of the relative parallax in declination at the time T. A-a will express the apparent difference between the right ascensions, and D'-8, the apparent difference between the declinations of the sun and moon at the time T; and it may be observed that the former difference is not zero because T is not exactly the time of apparent conjunction in right ascension. There must now be obtained (art. 358.) the moon's apparent altitude or zenith distance, and with this element there must be computed (art. 163.) the augmentation of the moon's apparent semidiameter, which augmentation being added to the semidiameter of the moon, taken from the Nautical Almanac, the sum will be the moon's apparent semidiameter. The sum of the semidiameters of the sun and moon will be required if the eclipse is partial, and their difference if total or annular. 372. Now (fig. to art. 369.) let sand м be the apparent places of the sun and moon at the time T, or the approximate time of apparent conjunction: let X Y drawn through M be a portion of the moon's apparent orbit, which, for a time equal to the duration of the eclipse, may be considered as a straight line; and, for a partial eclipse, let sx and sy be each equal to the sum of the apparent semidiameters of the sun and moon. Then X and Y, in the orbit, will be the places of the centre of the moon at the commencement and end of the eclipse respectively. Let p be the pole of the equator, and PS a horary circle passing through the sun at the time T. Draw an arc Mp of a great circle perpendicular to PS, and let мq be part of a parallel of declination passing through the moon at M. Draw also the straight line sv perpendicular to XY; then v will be the place of the moon's centre at the middle of the eclipse, or the instant of greatest phase. The angle SPM is equal to a'-a, and the apparent declination of the moon at M is D-8: also the arc мp of the great circle and Mq of the small circle may be considered as equal to one another; therefore (art. 70.) MP (A'-a) cos. (D-8). This value of Mp will be expressed in seconds if a'—a be in seconds. Now (art. 71.) we have pq (in seconds)=(A'-a)' sin. 1" sin. 2 (D-8); therefore sp (=sq+pq)=D' — d + 4 (A′ — a)2 sin. 1′′ sin. 2 (D−8). 373. There must next be found the apparent relative horary motions in right ascension and declination; viz.— there must also be found the value of (d.Ad.ds' dt dt COS. (n-d), the apparent relative horary motion on the parallel Mq of declination. da' Now and are found from the Nautical Almanac, dt dD' dt as stated above; and it has been shown that p' cos. l d.ds' d t 0.2528 cos. T this value will be in seconds of a degree if p' be expressed in seconds, and, in the computation, the values of D and 7 found for the time T must be used. Again, d D, the value of the true relative parallax in declination, being the value of & in the formula (VI) art. 162., may with sufficient correctness for the present purpose be expressed by P' (sin. I cos. D-cos. l cos. 7 sin. D), disregarding a in the term sin. (7+a) and considering cos.a as d.dp' dt equal to unity. Then becomes (7 only being a function of the time, and D-d being put for D) P' cos. 7 sin. (D—8) sin. 1", which will be in seconds if p' be expressed sin. T ατ d t in seconds and thus there may be obtained the value of the apparent relative horary motion in declination. Imagine next, the line mn to be drawn parallel to PS: then these apparent relative horary motions upon the parallel of declination, and in declination, may be represented by мm and mn respectively; therefore the values of these lines are known, and in the plane triangle мmn we have Mm mn: rad. : tan.m Mn; thus the angle m Mn or Psv may be found. Again, cos. m Mn rad. :: Mm: Mn; thus Mn is found, and it will represent the moon's apparent relative horary motion in her orbit. In the triangle мsp, which may be considered as rectilinear, and right angled at p, we have and sp Mp: rad. tan. Msp, cos. Msp rad.:: sp: SM: thus the angle Msp and the distance SM may be found. The steps to be taken for obtaining the times of the commencement and end of the eclipse, and the time of the greatest phase, are similar to those which have been given in the investigation concerning the phenomena of an eclipse of the moon (art. 369.). 374. The moon, by her proper motion, occasionally passes between the earth and some planet or fixed star; in which case the planet or star is suddenly concealed behind her disk, or as suddenly reappears after having been for a time invisible: the disappearance is called an immersion, and the reappearance an emersion; and both phenomena are designated by the general term occultation. The conditions under which an occultation of a planet or star by the moon may be visible at any station are that the difference between the declinations of the moon and star should be less than 1° 30'; that the time of the conjunction should be more than two days before or after the day of new moon; that the sun should be below, or very near the horizon, and that the star should be above it. In the Nautical Almanac there is given a table of the elements for occultations, from which may be obtained the Greenwich mean time of the conjunction of the moon and star in right ascension, as it would appear if a spectator were at the earth's centre; the true right ascensions and declinations of the moon and star at the same time; also the geographical parallels of latitude between which the occultation will take place. And the process of determining, for any given place, the time of immersion and emersion when a star is occulted by the moon may be as follows: 375. Find the right ascension of the midheaven of the station at the instant of true conjunction in right ascension, and subtract from it the star's right ascension; there will remain, in time, the horary angle of the moon and star at the instant of true conjunction in right ascension: let it be represented, in degrees, by From the table of elements for occultations in the Nautical Almanac; that is, for the Greenwich mean time of true conjunction in right ascension, take the difference between the declinations of the moon and star, and the true declination of the star let the latter be represented by d. Take the moon's equatorial parallax; and, if the star be a planet, take its horizontal parallax: in the latter case, the difference between these will be the moon's relative horizontal parallax: then compute the absolute or the relative geocentric horizontal parallax of the moon, according as a fixed star or a planet is used; and let its value be represented by P'. Take out also the moon's horary motion in right ascension when the star is fixed; if a planet, find the difference between the moon's horary motion in right ascension and the planet's horary motion in geocentric right ascension: let this be reda' presented by dt. Compute the parallax in right ascension from formula (1) P' cos. 7 (art. 161.), viz. cos. d sin. 7, which is the value of a or of da'; In these formulæ d, the declination of the star, is used instead of D, the declination of the moon, in order to obtain the parallaxes with respect to that part of the moon's limb which is in contact with the star at the times of immersion and emersion: hence, at those times, the distance of the star from the centre of the moon will be equal to the moon's true semidiameter; and thus the necessity of computing the augmentation of that semidiameter is avoided. In the latter formula, ατ represents the hourly motion of the earth on its axis with dt which multiplied by sin. 1" gives logarithm is 9.41916. The above values being substituted in the formula for t in the investigation relating to eclipses of the sun (art. 371.), the value of t will be obtained, and we shall have 7+ t or T; that is, in Greenwich time, the approximate mean time of apparent conjunction of the moon and star in right ascension at the station. 376. For this time T there must then be found the common horary angle of the moon and star, as in the investigation above mentioned (arts. 371, 375.): let it be represented by and for the same time, compute as above the geocentric horizontal parallax P'; also the true right ascensions and declinations, and the horary motions in right ascension and declination. The difference between the true right ascensions of the moon and star at the time T being called A', subtract from it the parallax in right ascension (the value of a or of dA' above); the remainder A'-a being multiplied by cos. d will give the value of мp or Mq in the figure to art. 369., M being the apparent place of the moon's centre at the time T. The difference between the declinations of the moon and star at the time T being called D', subtract from it the parallax in declination (formula (vi) art. 162.), viz. &= P' (sin. l cos. d- cos. I sin. d cos. T'): the remainder D' - 8 will be the value of sp or sq, s being the apparent place of the star or planet. da dt dD' The values of and or the true hourly motions in right ascension and declination, being found from the Nautical Almanac for the time T; by differentiating the above values of a and 8, we have that of (dr' _d.ds') Thus there may be obtained the values of (A dt cos.d for the apparent hourly motion of the moon on the dD' d.dD' parallel mq of declination; and dt dt for the apparent hourly motion of the moon in declination. These may be represented by Mm and mn; and from thence the moon's horary motion Mn in her orbit may be computed. Now, s being the supposed place of the star or planet, and X and Y the centres of the moon when her limb is in contact with the star, sx and sy will, each, be equal to the true semidiameter of the moon, or to the sum of the semidiameters of the moon and planet: then XM and MY may be calculated as in the investigations concerning the lunar and solar eclipse. Lastly, by means of the horary motion in the orbit the times of describing XM and MY may be found; and the moon being |