12 27 18.87 Apparent noon Ex. 3.-On the 24th of July, 1822, at 3h 5m 38.7 P. M., and 25th July, at 9h 49m 5.7 A. M. at the same place, the double altitude of the sun's upper limb was 93° 40′; required the apparent time of midnight by the chronometer? Time after noon, July 24th Sum 24 Half sum, or approximate midnight Elapsed time Half elapsed time 3h 5m 38$.7 21 49 59.7 24 55 38.4 12 27 49.2 18 44 21.0 9 22 10.5 Declination at midnight 19° 52′ N., daily variation 12′ 39′′ S. Or increasing the polar distance, and the equation is therefore ne gative for midnight. Latitude 50° 9' cot 9.921503 Proceeding in this manner till a considerable number of observations are made, the error of a chronometer may be determined with great accuracy. If this chronometer be compared with any given number of them, all their errors and rates may be found. The same thing may be done by the stars, though rather less conveniently. The following method of comparing a chronometer with mean time by Dr. Tiarks, communicated by Captain Basil Hall, R. N., will be found very useful. The difference of a chronometer from the mean time at a place being known at three different instants, to find that difference for any intermediate instant, with a proper regard to the change of rate T which may have taken place between the first and second, and between the second and third times. Let the difference at the times o =α t' = a+b t" = = a+b+c So that is the difference between the first and second states of the chronometer, and c the difference between the second and third states of the same chronometer, the state of a chronometer, (namely, its difference from the mean time of a given place), at the moment t will be t (t-t'') St (t + {t"(t—t"') t }b+ -c; or + t (t-t') t" (t"-t') t(-t') t'.t" if t is greater than t', both are positive. EXAMPLE. The difference of a chronometer from the mean time of a certain place was known on the following days: 21.8903 Difference between 1st and 2d = 21.8903 1 and 3 = 26.4007 4.5104 t' = 21.8903 Hence 0 = 0.0 It is now required to find the state of the chronometer for August 17th, at 11h 7m 44 17.4637. Deducting August 9.5243 from August 17.4637 we have the interval t = 7.9394. cor. the time a. 46.90 to be applied to the error of the chronometer at August 9th, 12h 35m chronometer slow for M. T. 51m 57.35 * For Rossel's method of correcting the error in rate of a chronometer, see Biot's Astronomie, vol. III., or Myer's translation of this, page 95, and also this work in the explanation of the tables. Explanation. Table I. contains the decimal fraction of a day of 24h. It is useful for finding what part of a day any number of hours, minutes, and seconds are, and consequently may be conveniently employed in many calculations where daily differences are necessarily involved, such as the daily rate of a clock, the change of which, in any given number of hours, &c., may be thereby readily obtained. It is also very useful in the preceding method of comparing chronometers, and other purposes. Table II. serves the same purpose when an hour is taken for unit, and is useful in several astronomical operations. Table III. is supplementary to the general Table V., which serves to convert time into degrees if less than 6h or 90°. But as 6h answers, to 90°, 12h to 180°, and 18h to 270°, this table will easily be applied to 24h or 360°, the whole circle to every four seconds of time, and by the proportional parts at the bottom to every single second. Whence it is only necessary to convert the decimal part of the time into degrees by this table to complete the whole. III. BY OCCULTATIONS AND ECLIPSES. The moon in her periodical revolution frequently passes between the earth and a fixed star, of which she intercepts the spectator's view, thus producing what is called an occultation. Since the instant of disappearance and reappearance of the star can be ascertained without the use of any instrument liable to error, the longitude may be determined more accurately by an observation of this phenomenon, than by a lunar distance. An observer possessed of an ordinary telescope, a chronometer, and an instrument to determine its error and rate,* can readily make the observations; and the necessary calculations are far from difficult. Several rules have been proposed for this purpose independent of the method of determining the parallaxes by the nonagesimal, and comparatively much more simple. Of these, Dr Inman's (of Portsmouth), which we shall in the mean time adopt with some alterations, appears to us the most convenient.† At the instant of the disappearance or reappearance of the star, the apparent right ascension and declination of the point of the moon's limb in contact with the star is the same as the right ascension and declination of the star, which can be obtained with great facility and accuracy from tables. The apparent right ascension and declination of this point being corrected for parallax, its true right ascension and declination will be determined. Now since the distance of this point from the moon's centre, which is equal to her semidiameter, and the declination of the centre for the estimated time at Greenwich, may be found by the Nautical Almanac, the true right ascension of the moon's centre is easily computed. Should there be an uncertainty in the estimated Greenwich time amounting to about one minute, the operation must be repeated till the estimated and computed Greenwich time be very nearly the same. * If the observations are made at sea, an allowance must be made for the rate of the chronometer between the disappearance and reappearance of the star and the run of the ship, as in lunars. † Almost all of these methods are founded upon a paper by Lagrange. See Connaissance des Tems for 1819. Rule. By applying the estimated longitude in time to the observer's apparent time, the reduced Greenwich time to the nearest minute will be obtained. To this time take from the Nautical Almanac the sun's R. A., the moon's R. A., and their declinations corrected for second differences, together with the variation of declination for 10, for the purpose of repeating the operation when supposed necessary; and the moon's semidiameter, and the horizontal parallax corrected for the spheroidal figure of the earth. Take also the moon's R. A. for 3h after the first estimated time corrected as formerly. Find from the Nautical Almanac, or from other tables, the apparent R. A. and D. of the observed fixed star, and reduce the given latitude for the spheroidal figure of the earth. To the apparent time add the sun's R. A., and from the sum, increased if necessary by 24", subtract the star's R A.; the remainder, if less than 125, will be the hour-angle; if greater than 12", its complement to 24h will be the hour-angle. Now write down the proportional logarithm of the reduced horizontal parallax under the numbers (1), (2), and (3). Under (1) and (2) put the secant of the reduced latitude; under (3) the cosecant of the same; under (1) the cosecant of the hour-angle (a), and take the sum of these. Below the sum of the three logarithms under (1) put the constant logarithm 1.17609, and the cosine of the star's declination; at the same time under (2) put the cosecant, and under (3) the secant of the same; the sum of these three logarithms under (1) will be the proportional logarithm of arc first, or the parallax in R. A. in time, nearly; one half of which (b) is to be subtracted from the hour angle (a), giving (a-b), the corrected hour-angle. Under (2) put the secant of the hour-angle thus corrected. The sum of the logarithms under (3) will be the proportional logarithm of the first part of the parallax in declination, and that under (2) the second. The first part must be applied with such a sign as to diminish the star's distance from the elevated pole: the second must be applied with the same sign as the first, if the hour-angle and polar distance are the one greater, and the other less than 90° or 6h; otherwise with a contrary sign. The result will be the true declination of the observed point of the moon's limb. Take the difference between this true declination of the observed point and the declination of the moon's centre, found from the Nautical Almanac, under which put the moon's horizontal sémidiameter properly corrected for the given time, and take the sum and difference. Add together the proportional logarithm of this sum and difference, and take half the sum, to which add the cosine of the mean of the two declinations just found, the sum will be the proportional logarithm of the moon's semidiameter in R. A. nearly. Under (4) put the constant logarithm 1.17609, the first sum under (1), and the cosine of the declination of the observed point, the sum will be the proportional logarithm of the exact parallax of R. A. in time. This being added to the star's R. A. when west of the meridian, but subtracted if east, will give the true R. A. of the point observed. To the true R. A. thus obtained, add the moon's semidia |