the correct mean time of the moon's transit over the meridian of Greenwich on the given day.-The time of transit in the Nautical Almanac is 10:22 7.-It is much to be regretted that tenths have been made use of in the Ephemeris, on such an important occasion as this. The mean time of the moon's transit, as well as that of each of the bright planets, ought to have been given to seconds, instead of to the nearest tenth of a minute: because, when the practical navigator reduces the tenths to seconds, the result may differ, at times, three or four seconds from the truth; which, of course, will affect the time of transit over the meridian for which he may be calculating; and, in low latitudes, or in places where the moon approaches near the zenith, an error of three or four seconds in the mean time of transit would sensibly affect her meridional altitude. PROBLEM XI. Given the Mean Time of the Moon's Transit over the Royal Observatory at Greenwich, to find the Mean Time of her Transit over the Meridian of any other place. RULE. Take, from page IV. of the month in the Nautical Almanac, the moon's transit over the meridian of Greenwich on the given day, and also on the day following, if the longitude be west; but on the day preceding if east; find the difference, and it will be the retardation of transit, or the excess of the lunar day above the solar day. Now, 24 hours augmented by this excess will be the length of the lunar day :Then say,-As the lunar day, so found, is to the retardation of transit; so is the longitude of the given place, in time, to a correction; which being applied by addition to the mean time of transit over the meridian of Greenwich, if the longitude be west, but by subtraction if east; the sum, or difference, will be the mean time of transit over the given meridian. Note.-This proportion may be readily performed by proportional logarithms, esteeming the hours and minutes in the first and third terms as minutes and seconds; as in the following Example. Required the mean time of the moon's transit, January 1st, 1836, over a meridian 94:30:30" west; the computed mean time of transit at Greenwich being 10 hours, 22 minutes, 42 seconds? D's transit at Greenwich on given day, per Ephemeris on the day following Ditto, Retardation of the moon's transit. the excess of the lunar above the solar day :-hence, Correction 50 Prop. log. ar. comp.. 9.1398 Prop. logarithm 6:18 2: .0.5563 Prop. logarithm . 1.4559 1. 1520 +1241: Prop. logarithm Comp.mean time of transit=10. 22. 42, at Greenwich. Mean time of transit = 10. 35. 23, at the given meridian. 13, page 306, relative to the length of the of excess which the earth must describe beyond a complete revolution on its axis, to bring the moon upon the same meridian that it was on the day before. PROBLEM XII. To Compute the Mean Time of a Planet's Transit over the Meridian of Greenwich. RULE. From the planet's geocentric right ascension at noon of the given day (increased by 24 hours if necessary) subtract the mean sun's right ascension at the same noon; the remainder will be the approximate time of transit.-Take the difference of the mean sun's diurnal motion in right ascension (viz. 3"56:55), and the planet's daily variation in right ascension, if the planet's motion be progressive ;* which difference apply by subtraction to 24 hours, or 86400 seconds, if the planet's diurnal motion be the greatest; but by addition, if it be the least:Then say,-As 24 hours, or 86400 seconds, diminished or augmented by the aforesaid difference, is to 86400 seconds; so is the approximate time of transit, to the mean time of the planet's transit over the meridian of Greenwich. But, if the planet have a retrograde motion,* the sum of its daily variation, and that of the mean sun's (viz. 3′′56:55) is When the planet's motion is progressive, the difference of right ascension between the given day and the day following; but when it is retrograde, the difference of right ascension between the given day and the preceding day, will be its daily variation in right ascension. A A always to be applied by addition to 24 hours :-then 86400 seconds, augmented by that sum, will be the first term in the proportion; with which proceed as above directed. Example 1. Required the mean time of Venus's transit over the meridian of Greenwich on the 1st of January, 1836? in R. A. . . . . . 5:13:42 Mean sun's ditto 3.56.55 -1:16:47 Difference = As 24 hours or 86400: 76:47 86323:53, Log. ar. comp. 5.063871 : 24 hours, or :: approximate time of transit. . 86400, Logarithm. .4.936514 5821.41, Logarithm . .3.765028 Mean time of transit in seconds 5827: Logarithm. .3.765413 Ditto raised to hours, &c. correct mean time of the planet's transit over the meridian of Greenwich on the given day ;-in the Nautical Almanac it is 1:37:1, or 1:37 6: 1:37 7; which, therefore, is the Note. Had the planet's diurnal motion been less than that of the mean sun, the difference of the variations in right ascension, viz. 76'47 would have been additive to 86400 seconds; because, in this case, the planet would come to the meridian in less than 24 hours after its preceding time of transit. Example 2. Required the mean time of Venus's transit over the meridian of Greenwich on the 30th day of July, 1836? As 24 hours, or 86400: +384:81-86784:81, Log. ar. comp. 5.061556 ́ : 24 hours, or 86400, Logarithm. .4.936514 :: approximate time of transit 84666. 28, Logarithm Mean time of transit in seconds. 84291, Logarithm .4.927710 4.925780 Ditto raised to hours, &c. 23 2451; which, therefore, is the correct mean time of the planet's transit over the meridian of Greenwich on the given day :--The time of transit in the Ephemeris is 23 248, or 23:24" 48; which is 3 seconds less than the above: this is owing to the time in that work being only given to the nearest tenth of a minute. Note.-When the planet is stationary, the mean time of transit over the meridian of Greenwich is found at once, by simply subtracting the mean sun's right ascension from the geocentric right ascension of the planet: diminishing the remainder by the corresponding equation in Table XLVI. PROBLEM XIII. Given the Mean Time of a Planet's Transit over the Meridian of Greenwich; to find the Mean Time of Transit over any other Meridian. RULE. Find, in the Nautical Almanac, the difference between two consecutive transits, agreeably to the following precepts, viz. :-If the longitude be west, and the times of transit increasing, take the difference between the transits on the given day, and the day following; but if decreasing, between the transits on the given day and the day preceding. Again-If the longitude be east, and the times of transit increasing, take the difference between the transits on the given and the preceding days; but if decreasing, between the transits on the given and following days.-Find, also, the interval between the two transits: this, when the times are increasing, will be expressed by 24+ the difference of transit; but when decreasing, by 24 hours-the difference of transit. Then, to the proportional Logarithm Ar. Comp. of this expression, add the proportional logarithm of the difference of transit, and the proportional logarithm of the longitude in time; the sum, abating 10 in the index, will be the proportional logarithm of a correction: which, in west longitude, is to be applied, by addition, to the "meridian passage" at Greenwich on the given day, when the transits are increasing, or, by subtraction, if decreasing.-But, if the longitude be east, the correction is to be applied conversely, viz., by subtraction when the transits are increasing, and by addition when decreasing :-In either case the result will be the correct mean time of transit over the given meridian. Example 1. Required the mean time, on the 2nd January, 1836, that the planet Venus will pass the meridian of a place, which is 175:30 east of the Royal Observatory at Greenwich? Venus's meridian passage, January 1st. Ditto ditto 2nd. = 1:37 6: per Ephemeris. = 1.38.24 ditto. Difference of transit 0' 1:18:: Then, 24+118! = 24118, is the interval of time between the two transits. Note.-The difference is taken between the transits on the given day and the day preceding, because the longitude is east, and the time increasing. Now, 1.18 Interval between the transits=24: 118: Prop. log. ar. comp. 9. 1253 Correction = 0:38: Prop. logarithm Mn. time of tran. on given day 1.38.24 at Greenwich. 2.1413 . 1.1871 . 2.4537 Mn. time of tran. on given day=1:37" 46, at the given meridian. Remark.-The correction is subtractive, because the longitude is east, and the times of transit increasing; had they been decreasing, the correction would be additive to the meridian passage at Greenwich on the given day. Example 2. Required the mean time, on the 23rd of July, 1836, that the planet Venus will pass the meridian of a place, which is 140:45 west of Greenwich? 24-6:30:23:53:30:, is the interval between the two transits. |