TO FIND THE HOUR-ANGLE. 326. Given the true altitude of an object, its declination, and the latitude of the observer, to find the meridian distance or hour-angle. RULE XCIV. Find the polar distance by Rule LXXXII, page 237. 2o. Add together the true altitude, latitude, and polar distance; take half their sum, and from the half sum subtract the true altitude, which call the remainder. 3°. Add together the secant of latitude, cosecant of polar distance, cosine of half sum, and sine of remainder; the sum of these logs. (rejecting 10 from the index), will be the log. of sun's hour-angle (Table 31, NORIE); or sine square of sun's hour-angle (Table 69, RAPER). When the polar distance exceeds 90°, take out the secant of reduced declination; or subtract the polar distance from 180°, and take the cosecant of the remainder. (See Rules XXXVIII and XXXIX, pages 89—90). (a) When both the latitude and declination are o, take the true altitude from 90°, and so get the zenith distance, which convert into time by Rule LXXVI, page 225, or by Table 19, NORIE, or Table 17, RAPER: the result is the hour-angle. The hour-angle can also be found without a special table, as follows:-Find the sum of the four logs., as above, and divide by 2: the result is the log. sine of half the hour-angle in arc. From the Table of log. sines find the arc corresponding thereto, which multiplied by 2, and converted into time (Rule LXXVI, page 225), is the hour-angle sought. It is thus evident that the complete solution may be obtained by means of the Table of log. sines, &c., alone. LONGITUDE BY CHRONOMETER, FROM AN OBSERVED ALTITUDE OF THE SUN. In the following problem two chronometer errors are given, which, by means of the elapsed time, give a daily rate: hence proceed as follows: RULE XCV. 1o. Write down the time by chronometer, apply its second error, adding if it is slow, subtracting if it is fast; then apply the accumulated rate, adding if losing, subtracting if gaining (see Rule XCII): the result is the Greenwich date at the instant of observation. 2o. Take out of Nautical Almanac, page II, the sun's declination and the equation of time for the noon of Greenwich date, and the corresponding hourly difference for each: also take out the sun's semi-diameter. 3°. Reduce the sun's declination and equation of time to the Greenwich time (Rules LXXIX and LXXXIII). 4° For the Polar Distance.-Subtract the reduced declination from 90°, if latitude and declination of same name; but if of different names add 90° to dealination. 5°. For the True Altitude.-Correct observed altitude for index error, dip, refraction and parallax, or correction in altitude, and semi-diameter, and thus get the true altitude (Rule LXXXIV). 6°. Find the hour-angle or meridian distance by Rule XCII.* 7°. When the observation is made in the afternoon, the hour-angle is apparent time past noon of the given day at ship-before which write the date at ship, but if the observation is made in the morning, take the hour-angle from 24", the remainder is apparent time at ship reckoned from noon of the preceding day, the time at place in both instances being expressed in astronomical time. In finding longitude by chronometer the logs. used in finding the hour-angle are required to be taken out for seconds of arc. On comparing these examples with paragraph 7°, which they are intended to illustrate, the seaman will have no difficulty in understanding that, since the sun's Hour-angle is the Distance (in time) of the object from the meridian, if the observation is made in the afternoon (P.M.), as in Ex. 1, the time will be 3h 40m 18 past noon of the 6th day; that is, the ship date (astronomical time) is January 6d 3h 40m 18s-the astronomical day commencing always at noon; but if the observation be made in the morning (A.M.), the hour-angle will be the time before noon of the 6th day; or, as shown in Ex. 2, 20h 19m 428 past noon of the day before, —that is, January 5d 20h 19m 42o. In Ex. 3, similarly, the observation being P.M., the time will be 3h 54m 39 past noon of June 1st, while in Ex. 4, the observation being A.M., the time before noon of June 1st, i.e., 20h 5m 218 past noon, May 31st. will be 3h 54m 39 Obs. In the new edition of NORIE's Tables the hour-angle is so arranged that when the observation is made P.M. at ship it is read from the top of the page; when A.M., from the bottom, using the next greatest log. to the given one, and it will be the apparent time at ship, reckoning from the day before the ship date; in which case the necessity of deducting from 24h (as explained in paragraph 7°) is obviated. 7°. To apparent time apply the reduced equation of time, adding or subtracting as directed in page I, Nautical Almanac, and so get mean time. 8°. Under ship mean time put Greenwich mean time-not forgetting the day in each case:-subtract the less from the greater; the remainder is longitude in time, which convert into arc '"; see Rule LXXVII, page 226, or Table 17, RAPER, or Table 19, NORIE. In taking the difference of Greenwich mean time and ship mean time, if the days of the month be different, it will be necessary to add 24 to the hours of the more advanced (that is, the one whose days are most), in order to enable the subtraction to be made. 9°. Call the longitude West when Greenwich time is greater than ship mean time; but East when Greenwich mean time is least. NOTE.-When the latitude at noon is given, the latitude in at the time of observation must be found by means of the course steered and distance sailed. The diff. of lat. from noor is to be named North or South, according as the ship at the time of observation is North or South of her latitude at noon. When the longitude is found, as in Exs. 1 to 10 (or according to paragraphs 8° and 9°, above), the diff. of long. between the ship at the time of observation and noon must be applied to find the longitude at noon. The diff. of long, is to be named East or West, according as the ship is East or West of its position at noon. EXAMPLES. Ex. 1. 1882, January 11th, P.M. at ship, latitude 49° 30' N., the observed altitude sun's L.L. was 12° 20′ 30′′, height of eye 18 feet, time by a chronometer January 11d 6h 44m 360 (being P.M. at Greenwich), which was 6m 88.3 fast for mean noon at Greenwich, September 1st, 1881, and on September 30th, 1881, was 8m 42" fast on Greenwich mean time: required the longitude by chronometer. Decl., page II, N.A. 545'9 13 6,0)54,7'3 Acc. rate 9 7:3 By Raper's Tables: dip 4' 10", refr. - 4' 23," par. + 9", semid. + 16' 18", and true alt. 12° 28′ 24". (a) The chronometer having been rated, September 30th, there are no days left in September. Jan. 11th, noon 21° 46' 37" S. deor. Jan. 11th, noon 8m1481 incr. *980 +6·3 6.45 The observation having been made in the afternoon (P.M.) at ship, the hour-angle 2b 17m 42 will be 2h 17m 42 past noon of the 11th day, that is, the apparent time at ship (astronomical time) is January 11d 2h 17m 42a (see Rule XCV, 7o, page 277). Ex. 2. 1882, May 20th, P.M. at ship, latitude 50° 43′ N., obs. alt. sun's L.L. 17° 10′′, index – 1′ 39′′, height of eye 28 feet, time by a chronometer May 20d oh 19m 53s (or oh 19m 53a P.M.), which was 33° fast on mean noon at Greenwich, March 20th, and on April 1st was 234 fast on Greenwich mean time. |