it will thus denote the solar time of the observation reckoned from the instant at which the star culminates. If a like angle at the pole be obtained in degrees from an observed altitude of the moon, the number of degrees divided by 15, give the sidereal time corresponding to the interval: if to this be added the increase of the moon's right ascension for the same interval the sum will be (nearly) the hour angle in sidereal time, which may be reduced to solar time by subtracting the acceleration. The result will be (nearly) the time of the observation, reckoning from the instant at which the moon culminates. A process, precisely similar, must be employed when the hour angle of a planet is obtained from an observation, the variation of the planet's right ascension being added to, or subtracted from the sidereal interval according as that right ascension is increasing or decreasing. The variations in the right ascension of the sun, the moon, or a planet, for a given interval of time, may be obtained from the Nautical Almanac, by methods which will be presently noticed (art. 316.). Conversely, if the time of the day be given, the sun's hour angle is immediately found on multiplying by 15 the apparent time elapsed since the preceding noon. When the time of night is given, the most simple way of finding the horary angle of a fixed star, a planet, or of the moon, is to take the difference between the right ascension of the midheaven, and of the celestial body (both of them being found for the given time); for the result, when multiplied by 15, gives the required angle in degrees. The following example will serve to illustrate the methods of finding the hour angle of a fixed star for any given time and station. Let it be required to find the hour angle of a Polaris for Sandhurst at 10 ho. 46 m. 21 sec. mean time, or 11 ho. 1 m. 13 sec. apparent time, November 17. 1843. Sidereal time at mean noon, Greenwich - 15 43 34·6 (Naut. Alm.) .5 15 43 35.1 10 48 7. 2 31 42.1 1 4 0.8 1 27 41.3 Note.--If there be a sidereal clock at the station, the right ascension of the midheaven is given by it for the instant; therefore, by taking the difference between such right ascension and that of the star, the hour angle is immediately found. CHAP. XV. INTERPOLATIONS PRECISION OF OBSERVATIONS. 316. THE longitudes and latitudes of the sun, moon, and planets, and the right ascensions and declinations of the sun and planets, with the semidiameter of the former, and the logarithms of the radii vectores, are given in the Nautical Almanac for the noon of every day at Greenwich, that is, for intervals of time equal to twenty-four hours; the semidiameter and horizontal parallax of the moon for intervals of twelve hours, and the right ascensions and declinations of the moon for every hour and certain corrections only are necessary, in order to obtain the values of the elements for any given time at the place of observation. : For the ordinary purposes of practical astronomy, it will suffice to find, by a simple proportion, the variation of the element in the interval between the given instant, and the hour at Greenwich for which the value of the element is given in the almanac. The longitude of the station, from Greenwich, being known by estimation or otherwise, the time at Greenwich corresponding to the given instant can be found by subtracting the difference of longitude, in time, from the given time at the place, if the place be eastward of Greenwich, or adding the two together if westward; for the given time being reckoned from the preceding noon, the remainder or the sum is the required time at Greenwich, reckoned also from the preceding noon. Now, in the Nautical Almanac, adjoining the column of the sun's right ascension, and also of his declination, there is given a column containing the hourly variation of the element; and this variation being multiplied by the number of hours in a given interval, as that between the Greenwich noon and the time found as above, will be the required correction of the element, which being applied to the value for Greenwich noon, the sum or difference will be the value for the given instant. In like manner, may the correction of the moon's right ascension and declination for a given interval be obtained from the Nautical Almanac; the former, by means of the difference between the hourly right ascensions, and the latter by means of the column of variations for ten minutes. Thus, for example, on the 7th of April, 1843, at 4 ho. 34' 30" P. M. mean time at a station whose longitude west of Greenwich is 1 ho. 10', the corresponding mean time at Greenwich is 5 ho. 44' 30"; and, if it were required to have the correct declination for that instant, the variation for 10 min. at 5 P. M. being - 63".45, the following proportion may be made: 282".35, or 10' : 63"-45 :: 44'•5: (4′ 42′′-32), which subtracted from 21° 58' 1"-9, the declination at 5 P. M., leaves 21° 53′ 19′′55 for the corrected declination. Corrections thus found are said to be for first differences. The variations of a planet's geocentric right ascension and declination in any given interval of time, may be obtained from the columns containing the planet's right ascension and declination for the given month in the Nautical Almanac, by taking proportional parts of the daily differences. 317. But when the variations of the elements are considerable, and when the values are required with great accuracy, the corrections for second, and occasionally for third, as well as for first differences become necessary: these may be obtained by means of the usual formula for interpolation, which is investigated in the following manner : Let y, the quantity to be interpolated, be considered as a function of a variable quantity m, and assume y = A + Bm + cm2 + Dm3 + &c. Now, let m have successively values represented by o, n, 2n, 3 n, &c., then the corresponding values of y will be Subtracting successively (1) from (2), (2) from (3), &c., and representing the remainders by P', Q', R', &c. we have Again, subtracting successively (5) from (6), (6) from (7), &c., and representing the remainders by p", Q", &c., we get Next, subtracting (8) from (9), &c., and representing the remainders by p'", &c., we have posing, 3n Substituting this value of D in the equation (8), and trans but p" = , n and putting ▲" for Q'— P', we get Again, substituting these values of C and D in (5), and transposing, B= P' A" A"" A" + or putting A' for P' and simplifying, Δ' B= n A" A"" Finally, substituting these values of B, C, and D in the assumed equation, we obtain |