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Ex. 2. Given the true altitude 40° 20′ 13′′, latitude 30° 12′ N., red. decl. 23° 19′ 5′′ N., whence polar dist. is 66° 40′ 55′′: find the hour-angle.

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Required the hour-angle or meridian distance in each of the following examples :

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LONGITUDE BY CHRONOMETER,

FROM AN OBSERVED ALTITUDE OF THE SUN.

175. The Longitude of a place may be determined by various methods, but that of a ship which is beyond sight of land can be ascertained astronomically only by what are called lunar observations, or by obtaining the mean time of the day or night from the observed altitude of a celestial body, and comparing it with the time shown by a chronometer which is supposed to express, at every instant, the mean solar time at Greenwich.

The principle on which the processes of the various methods of determining the longitude of a station are founded is the same in all cases, and consists in a determination of the time at Greenwich and at the place of ship at the same physical instant; the difference between those times being equal to the difference in longitude between Greenwich and the ship. This is obvious, for since apparent time at any place is expressed by the angle (in time) between the meridian of a place and a horary circle passing through the sun at the instant, the difference between the apparent times, and, consequently, between the mean times, at two places at the same instant must be equal to the angle in time between the meridians of the two places. Hence, the problem of finding longitude resolves itself into two distinct parts: 1st. The determination of the mean time at the station of the observer, and, 2nd. The determination of the mean time at the first meridian.

The observation of an altitude of a heavenly body enables us, with the assistance of other elements given in the Nautical Almanac-the declination and necessary corrections, and the known latitude-to compute the hourangle of the body.

The body observed may be (1) the Sun, (2) the Moon, a Star, or a Planet -the moon furnishing the least reliable means of solving the problem.

The error of the chronometer on Greenwich mean time at a given date and its error being known, we thence find the Greenwich mean time corresponding to the instant when the observation is taken for determining the ship mean time; consequently, the mean time at ship having been obtained by an altitude of the sun, star, &c., the mere comparison of that time with the time indicated by chronometer will give the longitude of the ship. This is the most convenient and constant method of finding the longitude at sea. 176. In the following problem two chronometer errors are given, which, by means of the elapsed time, give a daily rate: hence proceed as follows:

1o.

RULE LXXIV.

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Write down the time by chronometer, with the month and day before it, under this time write the second error, adding if it is slow, subtracting if it is fast.

2o. Next, multiply the daily rate by the number of days and decimal parts of a day elapsed between the time at which the rate was last ascertained, and the Greenwich time.

Turn this product into minutes and seconds.

(a) Adding the result to the chronometer time (found in 1° above) when the rate is losing.

(b) Subtracting the result from the chronometer time (found in 1o above) when the rate is gaining.

This will give the Greenwich mean time by the chronometer (see Rule LXXI, page 218).

3°. Take out the sun's declination and the equation of time for the noon of Greenwich date, from page II of the month in the Nautical Almanac, and the corresponding hourly difference for each, from page I, Nautical Almanac: also take out the sun's semidiameter.

4°. Reduce the sun's declination and equation of time to the Greenwich time by correcting them for the mean time at Greenwich (Rules LIX and LXII, pages 183 and 188).

5°. Mark the corrected equation of time "added to " or "subtracted from " apparent time, as directed by the heading of the column of Eq. T. in page I of the month in the Nautical Almanac.

6o. 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 declination.

7°. For the True Altitude.—Correct observed altitude for index error (±), dip (—), (refraction (—), parallax (+), or correction in altitude (—), and semidiameter (+), and thus get the true altitude (Rule LXIII, page 192).

8°. Find the hour-angle or meridian distance (see Rule LXXIII, 2o, 3o).*

9°. When the observation is made in the afternoon, that is, the question gives P.M. at ship, 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, that is, the question gives A.M. at ship, 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.

EXAMPLES.

Ex. 1. January 6th, P.M. at ship; suppose the sun's hour-angle to be 3h 40m 18: what is the apparent time at ship?

Here the time being P.M., we have the ship date app. time, January 6d 3h 40m 18.

Ex. 2. January 6th, A.M. at ship; suppose the sun's hour-angle to be 3h 40m 18: what is the apparent time at ship?

Here the hour-angle is

3b40m 180 24 O O

Ship date app. T., Jan. 5th 20 19 42

• In finding longitude by chronometer the logs. used in finding the hour-angle are required to be taken out for seconds of arc.

GG

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On comparing these examples with paragraph 9°, 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 18s past noon of the 6th day; that is, the ship date (astronomical time) is January 6d 3h 40m 18-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 42′ 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 39o past noon of June 1st, while in Ex. 4, the observation being A.M., the time will be 3h 54m 39° before noon of June 1st, i.e., 20h 5m 21o past noon, May 31st.

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 greater 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 9°) is obviated.

10°. To apparent time at ship apply the reduced equation of time, adding or subtracting as directed in page I, Nautical Almanac, and so get mean time, which must be marked with the proper astronomical day of the month.

11o. 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-degrees (°), minutes ('), and seconds ("); see Rule LVII, page 177, or Table 17, RAPER, or Table XIX, 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.

12°. Call the longitude West when Greenwich time is greater than ship mean time; but East when Greenwich mean time is least.

The following examples illustrate this portion of the Rule:

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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 noon 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 6 (or according to paragraphs 11° and 12°, 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. 1887, 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 118 6h 44m 36′′ (being P.M. at Greenwich), which was 6m 83 fast for mean noon at Greenwich, September 1st, 1886, and on September 30th, 1886, was 8m 42° fast on Greenwich mean time: required the longitude by chronometer.

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