Example 2. May 1st, 1825, in latitude 30:15 S., the mean of several altitudes of the sun's lower limb was 11:17:14", and that of the corresponding times 132310, by a chronometer, the error and rate of which were established at noon, February 1st, when it was found 325: slow for mean time at Greenwich, and losing 0.97 daily; the error of the sextant was 2:30? subtractive, and the height of the eye 23 feet; required the longitude? Obs. alt. of the sun's 1. limb=11:17:14"; hence, its true cent. alt. is 11:24:5" Sun's horary dist. from the merid. or noon=426" 40: Log.ris.=5.78097.9 Apparent time at the place of observation=19:33:20: Apparent time of observation at Greenw.=13.31. 10 Long. of the place of observ., in time = 6 210:90:32:30% east. PROBLEM IV. Given the Latitude of a Place, and the observed Altitude of a known fixed Star; to find the Longitude of the Place of Observation, by a Chronometer or Time-Keeper. RULE. Let several altitudes of the star be observed, at a proper distance from the meridian,* and the corresponding times, per chronometer, noted down; of these, take the means respectively. Let the mean altitude of the star be reduced to the true altitude, by Problem XVII., page 327. To the mean of the times of observation apply the original error of the chronometer, by addition or subtraction, according as it was slow or fast for mean time at the meridian of Greenwich when its rate was established; to which let its accumulated rate be applied affirmatively or negatively, according as the machine may be losing or gaining, and the result will be the mean time of observation at Greenwich; which is to be converted into apparent time, by Problem II., page 416. To the apparent time at Greenwich let the sun's right ascension be reduced, by Problem V., page 298; and let the star's right ascension and declination, as given in Table XLIV., be reduced to the period of observation. Then, with the star's true altitude, its declination, and the latitude of the place, compute its horary distance from the meridian, by any of the methods given in Problem III., page 383. Now, if the star be observed in the western hemisphere, its horary distance from the meridian, thus found, is to be added to its reduced right ascension; but if in the eastern hemisphere, subtracted from it: the sum, or remainder, will be the right ascension of the meridian; from which, (increased by 24 hours, if necessary,) subtract the sun's reduced right ascension, and the remainder will be the apparent time at the place of observation; the difference between which and the apparent time at Greenwich will be the longitude of the place of observation in time :-east, if the computed apparent time be the greatest; if otherwise, west. Example 1. January 29th, 1825, in latitude 40:30' N.. the mean of several altitudes of the star Aldebaran, west of the meridian, was 24:57:0%, and that of the See Note, page 417. corresponding times 16:56:3, by a chronometer, the error and rate of which were determined at noon, January 1st, when it was found 729: fast for mean time at Greenwich, and losing 1'.53 daily; the error of the sextant was 3'10" additive, and the height of the eye above the level of the horizon 22 feet; required the longitude? Mean time of observation at Greenwich = Mean time at Greenwich = 16:56" 3: 7.29 + 0.44 16:49 18: 13.38 Apparent time of observation at Greenwich = 16:35:40: == Sun's right ascension at noon, January 29th, 20:47:19: + 2.50 20:50" 9: 4*25*54: 16: 8:57"N. 73:51 3 139:14:41 69:37:20 Log. co-sine Star's horary distance, west of the Right ascension of the meridian 9.541836 44.43.42 Log. sine. 9.847417 mer.=4:4310: Log.rising=5.82672.1 Apparent time at the place of observ. 12:18:55: Apparent time of observ. at Greenwich=16. 35.40 Longitude of the place of obs., in time=41645:=64:11:15% west. The right ascension of the meridian is to be considered as being increased by 24 hours, because it is less than the sun's reduced right ascension. Example 2. January 29th, 1825, in latitude 39:15 S., the mean of several altitudes of the star Regulus, east of the meridian, was 10:28:48", and that of the corresponding times 33646, by a chronometer, the error and rate of which had been established at noon, December 1st, 1824, when it was found 437 slow for mean time at Greenwich, and gaining 1'.17 daily; the error of the sextant was 1.34′′ subtractive, and the height of the eye above the level of the sea 21 feet; required the longitude of the place of observation? Sun's right ascension at noon, January 29th, 20:47:19: Star's horary distance, east of the merid.=420" 0:Log.rising=5.76145.8 Star's horary distance, east of the merid.=420" 0: Star's reduced right ascension = 9.59. 3 Right ascension of the meridian = 5:39 3: 20.47.54 8:51 9: Apparent time at the place of observ. Long. of the place of observ., in time 5:2428: 81:7:0" east. PROBLEM V. Given the Latitude of a Place, and the observed Altitude of a Planet; to find the Longitude of the Place of Observation, by a Chronometer or Time-Keeper. RULE. Let several altitudes of the planet be observed, at a proper distance from the meridian,* and the corresponding times, per chronometer, noted down; of these take the means respectively. Let the mean altitude of the planet be reduced to its true central altitude, by Problem XVI., page 325. To the mean of the times of observation apply the original error and the accumulated rate of the chronometer, as directed in the last Problem: hence the mean time of observation at Greenwich will be obtained; which is to be converted into apparent time, by Problem II., page 416. To the apparent time of observation at Greenwich let the sun's right ascension be reduced, by Problem V., page 298; and let the planet's right ascension and declination be reduced to the same time, by Problem VII., page 307. Then, with the latitude of the place, the planet's reduced declination, and its true central altitude, compute its horary distance from the meridian, and, hence, the apparent time at the place of observation, by Problem V., page 397. Now, the difference between the computed apparent time of observation and the apparent time at Greenwich will be the longitude of the place of observation in time;-east, if the former exceed the latter; otherwise, west. See Note, page 417. |