Again, on the 11th of May, 1824, the altitude of the sun's lower limb taken with the same instruments as before, the index error being constant, was 19° 9' 50", when the chronometer showed 18h 57m 56. This gives the mean time at Falmouth 18h 30m 23.5, and the error of the chronometer for the meridian of the place 27TM 32.5. Whence, on May 1st, the error was 11th The loss in ten day is Or in one day it is Hence the daily rate is 28m 44.5 27 32.5 1 12 7.2 -7.2 It is to be observed, that the altitudes should be taken nearly at the same time of the day, otherwise an allowance must be made for the rate during the interval. 1. On the 22d of May, 1824, in latitude 32° 36′ N., and longitude by account 16° 40′ W., the altitude of the sun's lower limb at sea was 37° 24', when the chronometer showed 5h 12m 24.5, the height of the eye being 20 feet; required the longitude? For the usual computations at sea it is unnecessary to push the calculations farther than the nearest minute of a degree. 2. On the 11th of October, 1824, at noon, on the meridian of Greenwich, a chronometer was 11m 19.4 fast, and the daily rate was +4.1. On the 21st of October, at 6h 42m 10° A. M. by the same chronometer, the observed altitude of the sun's lower limb was 42° 17' 20", and the height of the eye 20 feet; required the longitude? Ans.-33° 25′ E. 3. On the 16th August, 1828, in latitude 38° 20' S., the mean of several altitudes of Antares west of the meridian was 14° 29′, the height of the eye being 12 feet, and the mean of the times per watch 11h 41m 38 P. M., which had been compared with mean time at the Cape of Good Hope on the 22d of June, and was found to be 1h 10m 28 too slow, and gaining 3.54 a day; required the longitude of the ship? Ans.-17° 36' E. EQUATION TO EQUAL ALTITUDES. The equation of equal altitudes is a correction for the change of declination of a celestial body during the interval of observation, to be applied to the middle time between the instants shown by a chronometer, at which, on a given day, that body has equal altitudes; to find the true time by the chronometer when the object was upon the meridian. ~ In ordinary cases the error and rate of a chronometer may be determined by single altitudes; but when great accuracy is required equal altitudes are very superior, especially when a transit instrument cannot be obtained. On this account various tables have been computed to facilitate this operation, though it is believed few of them afford great advantage in practice. By reason of the inconvenience of taking proportional parts, it is often better to give an easy practical rule, requiring the use of the ordinary tables, where neither double entries, different signs, nor proportional parts are necessary. Rule.* To the log cosine of half the interval between the times of observation add the cotangent of the latitude, the sum, rejecting 10 in the index, will be the tangent of arc first, the difference between which and the polar distance will be arc second. (44727) Constan Now to the constant logarithm 5.364517 add the cotangent of half the elapsed time, the cosecant of arc first, the cosecant of the polar distance, the sine of arc second, the logarithm of the elapsed time in minutes, the logarithm of the daily variation of the declination in seconds,† the sum will be the logarithm of the equation of equal altitudes in seconds of time, which, when applied to NOON, is additive if the polar distance is increasing, and subtractive if it is decreasing. If the equation is applied to MIDNIGHT, it is additive if the polar distance is decreasing, and subtractive if the polar distance is increasing.t Ex. 1.-On the 23d of March, 1809, at Pisa, in latitude 43° 43′ 11′′ N. equal altitudes of the planet Venus were taken before and after transit, the elapsed time between which was 8h 50m; required the equation of equal altitudes when her declination was 20° 42′ 40′′ N., and her daily variation +20′ 5′′ or +1205′′ increasing, and consequently the polar distance decreasing? Eq. E. Alts. 12$.99 1.113585 Daily var. dec. 20′ 5′′ = 1205′′ log. Or subtractive, because the polar distance is decreasing, and is to be applied to noon. Ex. 2.-On the afternoon of the 17th of September, 1810, altitudes of the sun were observed at Marseilles, in latitude 43° 17′ 50′′ N., * See Dr Mackay's or Mr Riddle's Navigation for a similar rule, analogous in principle, though perhaps in the detail somewhat less simple. + Half the sum of the variations for the given and preceding days should properly be employed, if the equation of equal altitudes be required for the noon of the given day; but the variation for the given day simply, if for midnight, the longitude not differing much from Greenwich. By polar distance in the computation is meant the distance of the object from the elevated pole, which may be either referred to the north or south pole, according to the name of the latitude. (See page 96.) and equal altitudes were taken on the forenoon of the 18th, after an interval of 21h 50m, the sun's declination for the 17th at midnight being 2° 14′ 23′′ N., and daily variation of declination == 1394"; required the equation of equal altitudes? 23' 14" Or Ans. Equation of equal altitudes 136 70.-2 16.7. subtractive, for the polar distance is increasing, and is to be applied to midnight. Ex. 3.-At Florence, in latitude 43° 46′ 40′′ N., on the 8th of April, 1809, equal altitudes of the planet Mars were taken at an interval of 8h 20m when his declination was 5° 9' 40" S., decreasing at the rate of 6′ 38′′ daily; required the correction for the planet's superior passage? Ans.-Equation of equal altitudes — 5o.196. Or subtractive, because the polar distance is decreasing, and is to be applied to the superior transit. TO FIND THE ERROR OF A CHRONOMETER BY EQUAL ALTITUDES.. By the Sun. The sun is in general the most convenient object for determining the error of a chronometer by equal altitudes, and the forenoon and afternoon of the same civil day are often preferred, though the evening and succeeding morning may sometimes be employed with advantage. In the morning, when the sun is more than two hours distant from the meridian, in mean latitudes, let a set of observations be taken with the corresponding times by a chronometer. In the afternoon observe the instants when the sun comes to the same alitude, writing each time down opposite its corresponding altitude. Now half the sum of any two times, answering to the same altitude, will be the approximate time of noon. Find the mean of all the times of noon in this manner from each corresponding pair of observations; to which the equation of equal altitudes being applied, the result will be the time of apparent noon, or the instant that the sun's centre is on the meridian by the chronometer. The difference between this and noon is the error of the chronometer, which will be fast or slow according as the time of noon thereby is greater or less than twelve hours. Ex. 1.-On the 29th of January, 1826, in latitude 57° 9′ N., longitude 2o 8′ W., the following equal altitudes of the sun were observed; required the error of the chronometer? 12 1 37.3 Time of mean noon by chronometer Hence the chronometer was 15m 5 fast for apparent noon, and 1′′ 37.1 fast for mean noon. Ex. 2.-On the 24th of July, 1822, at Pendennis castle, near Falmouth, in latitude 50° 8′ 48′′ N., Dr Tiarks, with a sextant of ten inches radius by Mr Troughton, and an artificial horizon, together with a chronometer by Morice, found the double alitude of the sun's upper limb to be 69° 47′ 20′′, at 8h 29m 13 A. M., and 4h 25m 5.3 P. M.; required the time of apparent noon by the chronometer?* The declination of the sun, at noon 24th, is 19° 58′ nearly. See a Report on Chronometrical Observations to ascertain the Longitude of the Island of Madeira, by J. L. Tiarks, 1822. Indeed the variation of the declination of the sun 12h preceding and 12h following the time, whether noon or midnight, ought to be taken; but, unless very great accuracy be required, the sun's variation for the given day as it stands in the Nautical Almanac will be sufficiently correct. |
