TABLE III.-Correction for Mean Horizontal Parallax, to be added to the Lunar Distances on account of the Spheroidal Figure of the Earth, its Ellipticity being 54 58 8 10 15 20 25 30 Lat. O Lat. 0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 46° 48 52 56 12 16 4.7 4.9 4.7 4.6 4.4 4.3 4.1 14 5.5 5.5 5.4 5.3 5.2 5.0 4.8 6.3 6.2 6.2 6.1 5.9 5.7 5.4 62 16.9 16.9 16.7 16.4 15.9 15.4 14.7 17.5 17.4 17.2 16.9 16.4 15.9 15.2 18.0 17.9 17.7 17.4 16.9 16.3 15.6 18.5 18.4 18.2 17.8 17.3 16.7 16.0 18.9 18.8 18.6 18.3 17.8 17.1 16.4 19.3 19.3 19.0 18.7 18.2 17.5 16.7 19.7 19.7 19.4 19.1 18.6 17.917.1 20.1 20.1 19.8 19.5 18.9 18.3 17.5 20.5 20.4 20.219.8 19.3 18.6 17.8 20.820.8 20.5 20.1 19.6 18.9 18.1 21.2 21.1 20.9 20.5 19.9 19.2 18.4 21.5 21.4 21.1 20.7 20.2 19.5 18.6 21.721.6 21.4 21.0 20.4 19.7 18.8 21.9 21.8 21.6 21.220.6 19.9 19.0 22.1 22.1 21.8 21.4 20.8 20.1 19.2 22.3 22.3 22.0 21.6 21.0 20.2 19.3 22.5 22.4 22.1 21.7 21.1 20.4 19.5 22.6 22.5 22.2 21.8 21.2 20.4 19.5 22.7 22.6 22.3 21.9 21.3 20.5 19.6 22.7 22.6 22.4 22.0 21.4 20.6 19.7 22.7 22.6 22.4 22.0 21.4 20.6 19.7 22.8 22.7 22.5 22.1 21.5 20.7 19.8 TABLE IV.-Equations of Second Difference for three Hours. 10 20 30 40 50 60 70 80 90 100 1 2 3 4 5 6 7 8 9 h. m.h. m." M. " " " 7 53 5.80.10.1 0.2 0.2 0.3 0.3 0.4 0.5 0.5 852 6.40.10.1 0.2 0.3 0.3 0.4 0.4 0.5 0.6 951 6.90 10.1 0.2 0.3 0.3 0.4 0.5 0.6 0.6 10 50 7.50.10.1 0.2 0.3 0.4 0.4 0.5 0.60.71149 24 0.8 1.6 2.4 3.24.04.8 5.6 6.4 7.2 8.00.1 0.2 0.2 0.3 0.4 0.5 0.6 0.6 0.7 12 48 210.8 1.7 2.5 3.4 4.25.1 5.9 6.8 7.6 8.5 0.1 0.2 9.3 0.3 0.4 0.5 0.6 0.7 0.81347 18 0.9 1.8 2.7 3.6 4.5 5.4 6.3 7.2 8.1 8.90.1 0.2 0.3 0.4 0.4 0.5 0.6 0.7 0.8 1446 150.9 1.9 2.8 3.8 4.7 5.6 6.6 7.5 8.4 9.40.10.20.30.40.5 0.6 0.7 0.8 0.81545 12 1.0 2.0 2.9 3.9 4.9 5.96.8 7.8 8.8 9.80.10.20.30.4 0.5 0.6 0.7 0.8 0.9 16 44 91.02.03.04.15.16.17.1 8.1 9.1 10.20.10.20.30.4 0.5 0.6 0.7 0.8 0.9 17 43 61.12.13.2 4.25.36.3 7.4 8.4 9.5 10.5 0.1 0.2 0.39.40.5 9.6 0.7 0.8 1.01842 31.12.23.24.35.46.57.6 8.7 9.7 10.8 0.1 0.2 0.3 0.4 0.5 0.6 0.8 0.9 1.0 1941 01.12.23.34.45.6 6.7 7.8 8.9 10.0 11.10.1 0.2 0.3 0.4 0.6 0.7 0.8 0.9 1.0 20 40 57 1.1 2.3 3.4 4.5 5.7 6.8 8.0 9.1 10.2 11.40.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.0 21 39 54 1.2 2.3 3.5 4.65.87.08.1 9.3 10.5 11.6 0.1 0.2 0.3 0.5 0.6 0.7 0.8 0.9 1.022 38 51 1.2 2.43.5 4.7 5.9 7.18.3 9.5 10.6 11.80.1 0.2 0.4 0.5 0.6 0.7 0.8 0.9 1.12337 48 1.2 2.4 3.6 4.86.07.28.4 9.610.8 12.00.1 0.2 0.4 0.5 0.6 0.7 0.8 1.0 1.124 36 45 1.2 2.4 3.6 4.9 6.1 7.38.5 9.7 10.9 12.20.10.20.4 0.5 0.6 0.7 0.9 1.0 1.1 25 35 42 1.2 2.5 3.7 4.96.17.48.6 9.811.1 12.30.10.20.4 0.5 0.6 0.7 0.9 1.0 1.1 26 34 39 1.2 2.5 3.75.06.27.48.7 9.911.112.40.1 0.2 0.4 0.5 0.6 0.7 0.9 1.0 1.1 27 33 36 1.2 2.5 3.75.06.27.58.710.011.212.40.1 0.2 0.4 0.5 0.6 0.7 0.9 1.0 1.1 28 32 33 1.2 2.5 3.7 5.06.27.58.7 10.0 11.212.50.1 0.2 0.4 0.5 0.6 0.70.9 1.0 1.1 29 31 1 302 301.32.5 3.8 5.0 6.3 7.5|8.8|10.0|11.3|12.5 0.1 0.3 0.4 0.5 0.6 0.7 0.9 1.0 1.1 30 30 TABLE V. CORRECTION OF APPARENT TIME FOR EQUATION OF SECOND DIFFERENCE. 1 S. 8. 8. S. S. 8. 8. S. 03.06.09.0 12.0 15.0 18.0 21.0 24.0 27.030.00.30.6 0.9 1.2 1.5 1.82.1 2.4 2.7 22.9 5.8 8.7 11.6 14.5 17.4 20.323.226,129.00.3 0.6 0.9 1.21.51.72.02.32.6 42.8 5.6 8.4 11.2 14.1 16.9 19.7 22.5 25.3 28.10.3 0.6 0.8 1.11.41.72.0 2.2 2.5 62.7 5.5 8.2 10.9 13.6 16.4 19.1 21.8 24.5 27.30.30.6 0.8 1.11.4 1.6 1.9 2.2 2.5 82.7 5.3 7.9 10.6 13.2 15.9 18.5 21.2 23.8 26.50.30.50.81.11.31.6 1.9 2.1 2.4 102.65.17.7 10.3 12.9 15.4 18.0 20.6 23.1 25.70.30.50.8 1.0 1.31.5 1.82.1 2.3|| 122.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 25.00.30.50.8 1.0 1.3 1.5 1.8 2.0 2.3 142.4 4.9 7.3 9.7 12.2 14.6 17.0 19.5 21.9 24.30.20.50.7 1.0 1.2 1.5 1.72.02.2 162.4 4.7 7.1 9.5 11.8 14.2 16.6 18.9 21.323.70.20.50.71.01.21.4 1.7 1.9 2.1 182.34.6 6.9 9.2 11.5 13.8 16.218.5 20.8 23.10.20.50.7 0.9 1.2 1.41.61.92.1 20 2.24.5 6.7 9.0 11.2 13.5 15.7 18.0 20.222.50.20.50.70.91.11.4 1.6 1.8 2.0 1 222.24.46.6 8.8 11.0 13.2 15.4 17.6 19.821.90.20.40.70.9 1.11.3 1.5 1.8 2.0 242.14.36.4 8.6 10.7 12.9 15.0 17.1 19.321.40.20.40.60.91.1 1.3 1.5 1.7 1.9 262.1 4.2 6.3 8.4 10.5 12.6 14.7 16.7 18.820.90.20.40.60.81.11.31.5 1.7 1.9 282.04.16.1 8.2 10.2 12.3 14.3 16.4 18.420.50.20.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 302.04.06.0 8.0 10.0 12.0 14.0 16.0 18.0 20.0 0.2 0.40.60.81.0 1.2 1.4 1.6 1.8 322.03.9 5.9 7.8 9.8 11.7 13.7 15.6 17.6 19.50.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 34 1.9 3.8 5.7 7.7 9.6 11.5 13.4 15.3 17.219.10.20.40.60.81.01.2 1.3 1.51.7 36 1.9 3.7 5.6 7.5 9.4 11.2 13.1 15.0 16.9 18.70.2 0.40.60.8 0.9 1.1 1.3 1.5 1.7 381.83.75.5 7.3 9.2 11.0 12.914.7 16.5 18.40.2 0.40.60.70.91.1 1.3 1.5 1.7 40 1.8 3.6 5.4 7.2 9.0 10.8 12.6 14.4 16.218.00.2 0.40.50.70.91.1 1.3 1.4 1.6 8.8 10.6 12.4 14.1 15.9 17.60.20.40.50.70.91.1 1.2 1.4 1.6 8.7 10.4 12.1 13.8 15.6 17.30.20.40.50.70.9 1.0 1.2 1.4 1.6 8.5 10.2 11.9 13.6 15.317.00.20.30.50.70.91.01.21.4 1.5 8.3 10.0 11.7 13.3 15.0 16.70.20.30.50.70,81.01.21.31.5 8.2 9.8 11.5 13.1 14.716.40.20.30.50.70.8 1.0 1.2 1.3 1.5 9.6 11.2 12.9 14.5 16.10.20.30.50.6 0.8 1.0 1.1 1.3 1.4 9.5 11.1 12.6 14.215.80.20.30.50.60.8 1.0 1.1 1.3 1.4 9.3 11.0 12.4 14.0 15.50.20.30.50.60.80.9 1.1 1.2 1.4 9.2 10.7 12.213.715.20.20.30.50.60.8 0.9 1.1 1.2 1.4 9.0 10.5 12.0 13.5 15.00-10.30.40.60.70.9 1.0 1.2 1.3 1 421.83.55.3 7.1 In the practice of lunars four persons are frequently employed in making the observations, the first to take the distance, the second to take the altitude of the sun or star, the third to take the altitude of the moon, and the fourth to write down the observations. One person, however, may make the whole himself, according to the following method, which was obligingly communicated by that distinguished practical navigator, Captain Basil Hall. Speaking of his own practice, he says," I always take all my altitudes and distances with the same instrument. First the altitudes of the sun, then those of the moon, then several distances; next the altitudes of the moon, then those of the sun, and interpolating by proportional logarithms for the altitudes at the mean time of the distances.* At night I never take an altitude, unless it be about twilight, when it can be done with accuracy and ease.” "The method which I use to connect lunars and chronometers is This is similar to the method given in Norie's Navigation. . not very general, but infinitely the best, and ought to be universally adopted, as it renders all allowance for the distance run in the interval of little or no consequence." "The use of lunars at sea I conceive is, in a great degree, to check the chronometers: the method by lunars being infallible, though not very nice; that by chronometers being fallible, but as nice as possible. So that a number of lunars are necessary to check a chronometer, and the object is to bring the whole of such lunars to bear rigorously on the chronometer without making use of the logboard." "This will be best illustrated by an example. At noon, or any other hour during the day most convenient for taking a lunar, I observe a set or half dozen sets of lunars with the sun, carefully noting what the chronometer shows, but without taking any account of the actual time. At any other hour when the sun is near the prime vertical, or most suitable for determining the time, I take altitudes expressly with this view, from which I discover the error of the same chronometer used for the lunars. Again, during the night I take lunar distances with the stars, on both sides of the moon if possible, at the moments most favourable, but never mind the exact time, only carefully recording what the chronometer shows. Now by the sights for absolute time I ascertain what was the error of the chronometer on apparent time at that meridian, and this same error, corrected for rate during the interval, I apply to each of the different times by the chronometer when the lunars were taken. By this means I get the apparent times due to the meridian, on which the absolute time sights were taken, with as much accuracy as if the whole, lunars and all, had been taken at that fixed meridian. The distances give the several times at Greenwich, and thus they all concur in settling the difference of time, between the first meridian and that chosen for taking the time, with a view of seeing what longitude the chronometer gives. Hence, if there had been an unseen current of some miles an hour of which no account could possibly be taken, still the result would not be vitiated thereby, but all the lunars would be found to contribute to the same end, thus making, according to Dr Wollaston's simile, the moon serve the purpose of a great Greenwich clock in the heavens. After having determined the true longitude and error of the chronometers when within a few days' sail of the land, I run the remainder of the voyage, in a great degree, by the chronometers alone." These remarks may be illustrated by the following observations, communicated by Lieutenant-general Sir Thomas Macdougal Brisbane, K. C. B., taken on his passage from New South Wales to Britain. With Troughton's Circle. Clear. 12th May, 1825. Chr. + A.. 3h 1m 19s (1.) Chr. Time. A. Dist. Sun from Moon. Moon's App. Alt. Sun's App. Alt. 5h 13m 53s 5 09 62° 57′ 1′′ Moon's hor. par. 54° 25' 2" 55' 46" 55° 47' 50" T. C. 5h 13m 53s Long. in time 2 51 33 42° 53′ 15′′ W. Long. by 1st series 42° 52′ 30" W. C+G. M. T. 13m 44s 2d 3d Mean long. 42 52 0 42 53 15 13 46 13 41 Mean+ 13 44 Error chr. 42 52 35 W. Mean long. by lunars 42° 52′ 35′′ W. by chro. 42 35 0 By proceeding in the same manner in the evening, and taking observations from objects on both sides of the moon, a very great degree of precision may be obtained. On finding the Longitude. II. BY CHRONOMETERS. The foregoing method of finding the longitude by lunars is very valuable at sea, on account of the frequent opportunities which occur for observation. About the time of new moon, and in unsteady weather, the necessary observations for the practice of this method cannot be obtained, and the dead reckoning is not to be depended on for any length of time, therefore recourse must be had to other methods. On account of the very high degree of perfection to which chronometers have been brought, the longitude determined by a mean of three or four of these delicate machines merits great confidence. If the rate of a chronometer be determined on shore, or rather perhaps on board in the situation it is intended to occupy during the voyage, where the various causes which act upon it, and are likely to alter its rate, are in operation, it is likely this rate will remain pretty uniform for some time, and the amount of the gain or loss, being allowed for on the time indicated by it at any future period, the true time may be obtained at the meridian of the place where its rate and original error was determined, with as much accuracy as if it had been adjusted to go accurately to mean solar time on that me ridian. Hence, it is obvious, that if the original error, and the gain or loss in 24 hours, called the daily rate, of a chronometer, be known, on any meridian, such, for example, as that of Greenwich; by making proper allowance for these, the mean time at Greenwich may be readily known to such a degree of accuracy as the going of the chronometer will warrant. It is now only necessary to find the apparent time at ship, by an altitude of any celestial body properly situated by some of the methods already given; to which the equation of time being taken from the Nautical Almanac and properly applied, the result will be the mean time to be compared with that at the given meridian to show the longitude of the ship. The rate of a chronometer is readily obtained by observing daily, if possible, the altitude of one or more celestial objects near the prime vertical, from which the mean time may be accurately determined, and, being compared with that shown by the chronometer, its gain or loss in 24 hours, and also its error on the day of the last observation, called the original error, will become known.* Ex. 1.-May 1, 1824, near Falmouth, in latitude 50° 8' 48" N., and longitude 20m 10 W., at about 18h 47m 20, the following altitudes of the sun's lower limb were taken, with an artificial horizon, in order to ascertain the daily rate of a chronometer previously set to Greenwich time. The observations were made with a sextant of which the index error was +1'30", the barometer 29.6 inches, and the thermometer 56° Fahrenheit. * These would be more accurately performed on shore by using an artificial horizon and the method of equal altitudes. In this case a pocket-chronometer should be employed, to be compared with those on board, which ought to be as numerous as possible. I am also of opinion, that the rate of a chronometer to be used at sea should not be ascertained in an observatory. At least three of the chronometers should be properly secured in their respective situations on board, and by a good pocket-chronometer a comparison of these should be made daily with an observatory clock, or by direct observation, in order to determine accurately the sea-rate. |