Par. in long. 35′ 46′′ P. L. 0.7017 Par. in lat.35' 221" - P. L. 0.7066 Moon's latitude at beg. 58 41 Parallax in longitude 21 54 4 10 58 0.4771 The apparent time of conjunction at Greenwich, computed by Prob. III. is 3h. 49' 28", and hence the longitude in time is 8' 44". Now the mean of this and 7′ 53′′, the longitude by last example, deduced from a like comparison with the Nautical Almanac, is 8′ 18′′. Since the observations of the above eclipse were, the beginning at one one place and the end at the other, the longitude deduced is, therefore,' affected by the sum of the errors, arising from the error of the Moon's latitude If both observations had been the beginning, or the end, the resulting longitude would be affected by the difference of these errors, and consequently would be nearer the truth. An observer at land, provided with proper instruments, besides observing the beginning and end of the eclipse, takes as many measures as possible of the versed sines of the lucid part of the Sun's disc, or of the distance between the cusps. Hence, by the method of interpolation, he can find the time of the middle of the eclipse, and the nearest approach of the centers of the sun and Moon; from whence the error of the Moon's latitude, and the longitude of the place of observation, are determined. CHAP. VI. The Method of finding the Longitude of a Place BY An Occultation of a fixed Star by the Moon. INTRODUCTION. OF F the method of applying observations of occultations of the fixed stars by the Moon, which is practised by astronomers, Dr. Halley observes, in the Philosophical Transactions, No. 354, as follows: "Of all the methods hitherto proposed for finding the longitude of places for geographical uses, none seems more adapted to the purpose than that by the occultations of the fixed stars by the Moon, observed in distant parts. For those Immersions of the stars, which happen on the dark semicircle of the Moon, and their Emersions from the same, are perfectly momentaneous, without that ambiguity to which the observations of the eclipses of the Moon, and those of Jupiter's satellites, are subject." By observations of the occultation of the planet Mars, 21st August, 1676, Dr. Halley determined the longitude longitude of Oxford and Dantzic;, and the longitudes of many other places have since been determined with great accuracy. The observations necessary for finding the longitude of a place by this method are, the instant when the Moon's eastern limb covers a star called the Immersion, and that of the re appearance of the syar from behind the Moon's west limb, called the Emersion. This method of ascertaining the longitude is given in the Memoires de l'Académie Royale des Sciences, pour l'année 1705, by M. Cassini, page 255, printed at Amsterdam in the year 1707, and has since been employed by astronomers for the same purpose The inintersion of a star is easily observed, as the sight may be directed to the star till that observation happens. In an emersion, the observer should direct his sight to that part of the Moon's limb, from which the star is expected to emerge, some minutes before the supposed time of emersion, and continue looking till the star appears. This time may be found nearly, by adding to the observed time of immersion, the time required by the Moon to pass over a space equal to the estimated chord of the segment cut off by the star. In the calculations, the altitude and longitude of the nonagesimal, and the parallaxes in latitude and longitude, are necessary, as in a solar eclipse. These are, therefore, calculated by Prob. 1. and 11. of last chapter. PROBLEM I. To find the Apparent Time at Greenwich, of the Ecliptic Conjunction of the Moon and a fixed Star. RULE. Reduce the mean longitude of the star, to its apparent longitude by the rule given for that purpose in Vol. II. From the Nautical Almanac, page v. of the month, take the longitudes of the Moon, immediately preceding and following that of the star; find also the mean second difference of the Moon's place. Now to the ar-co. of the log of the change of the Moon's longitude in xi. hours, add the logarithm of the difference between the star's longitude and the preceding longitude of the Moon, and the constant log. 2.557332; the sum will be the log. of the approximate time in minutes to be reckoned from the preceding noon or midnight. When the proportional part is less than 3 hours, the computation may be facilitated by using P. logarithms. In this case, the constant log. is 1.1761; and the degrees and minutes in the change of the Moon's longitude are to be esteemed minutes and seconds. Take VOL. I. 2 K Take the equation of second difference from Table xxxvII. answering to the approximate time, and the mean second difference, with which enter Table xxxvIII. at the top, and the change of the Moon's longitude in twelve hours in the side column, and take out the corresponding equation; which being added to the approximate time, if the Moon's motion in twelve hours is increasing, or subtracted if decreasing, will give the apparent time at Greenwich of the ecliptic conjunction of the Moon and star. EXAMPLE. Required the apparent time of the ecliptic conjunction of the Moon and II, November 26, 1787? Mean long at giv. t. 3° 0° 28′ 45′′ Aberration 18 + 18 37 43 20 Diff. 7 29 41 ar-co. P.L. 8.6195 The mean 2d diff. is l′ 57′′; hence, in Tab. XXXVII. the equat. of 2d diff. is 1".9,towhich, and 7°30 in T. xxXVIII. the correction is Apparent time of conjunction PROBLEM II. To find the Longitude of a Place by an Occultation of a fixed Star by the Moon, when both the Immersion and Emersion are observed, RULE. Reduce the apparent times of observation to the meridian of Greenwich, by applying thereto the estimated longitude. Find the Moon's horizontal parallax and semidiameter, and, hence, the reduced parallax and semidiameter as before. Find the apparent longitude of the star; from thence, and the Moon's longitude, compute the apparent time at Greenwich of the ecliptic conjunction of the Moon and star, by the last problem. Diminish the latitude of the place of observation by the quantity from Table XXXVI. With With the reduced latitude, and the right ascension of the meridian, compute the altitude and longitude of the nonagesimal at the instants of observation; and find the difference between the longitudes of the nonagesimal and observed star, which will, therefore, be the apparent differences of longitude between the nonagesimal and the Moon's observed limb. Compute the parallaxes of the Moon in latitude and longitude by Prob. 11. page 236, using the apparent latitude of the star in the computation. Find the Moon's motion in latitude and longitude during the observed interval, and from the change of longitude subtract the difference of the parallaxes in longitude, if the Moon is either east or west of the nonagesimal at both observations; but if east of the nonagesimal at the time of immersion, and west at that of emersion, the sum of the parallaxes in longitude is to be subtracted from the change of the Moon's longitude in the above interval. Hence, the apparent difference of longitude will be obtained. If the Moon is approaching the elevated pole of the ecliptic, find the difference between the parallax in latitude at the emersion, and the sum of the change of the Moon's latitude in the observed interval and the parallax at the immersion. But if the Moon is receding from the elevated pole, let the difference between the parallax in latitude at the immersion, and the sum of the change of the Moon s latitude and the parallax in latitude at the emersion, be found Hence, the apparent change of the Moon's latitude in the observed interval will be known. From the P. Log. of the apparent difference of longitude, its index being increased by 10, subtract the P. Log. of the apparent difference of latitude; the remainder will be the log. tangent of the apparent inclination; to the log co-sine of which, the P. Log. of the apparent difference of longitude being added, the sum will be the P. Log. of the apparent motion of the Moon in its relative orbit. From the P. Log. of the Moon's apparent motion in its proper orbit, subtract the P.Log. of the Moon's augmented diameter; the remainder will be the log. secant of the central angle. Now, if the Moon be approaching the elevated pole of the ecliptic, the sum of the apparent inclination, and the central angle, will be arch first, and their difference arch second; the star being between the Moon and the elevated pole of the ecliptic; but, when the Moon is between the star and the elevated pole of the ecliptic, the difference of these angles is arch first; and their sum arch second. If the Moon is receding from the elevated pole of the ecliptic, the sum of the apparent inclination and central angle is arch first, and their difference arch second; the Moon being between the star and elevated pole of the ecliptic; otherwise, their difference is arch first, and their sum arch second. Now, to the log. secant of arch first, add the P. Log. of the Moon's In strictness, this angle ought to be computed both at the immersion and emersion, using the Moon's apparent diameter at these times. This degree of precision is, however, almost unnecessary. 2 K2 semi |