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The formula when using the zenith distance and co.-lat., is as follows:

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And substituting the reciprocals of sin. z' and sin. l' we get,

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z Log. cos.

${L. sin. s + L. sin. (s – p) + L. Co-sec. 2 + co-sec. I — 20

2

In the spherical triangle P Z X, fig. 3, given the three sides p, l' and 2 to find angle 2, the azimuth. Using the above formula we get the following result

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N.B.-When using the sides of the triangle the azimuth is always reckoned from the elevated pole

Examples for Practice Example 1.— January 31st at 4h. 29m. Ios. p.m. mean time at ship; it. 37° 8' S., long. 40° 18' E., the observed altitude of the sun's lower mb was 30° 48' 40" bearing by standard compass N. 50° 30' W.; .eight of eye 21 feet. Required the sun's true azimuth and error of the ompass; also, the variation being 28° 30' W., find the deviation for the lirection of the ship's head.

Ans. T. Az. N. 88° 51' W.; Err. of comp. 38° 21' W.; Dev. 9° 51' W.

Example 2.—February 26th at 7h. Iom. 308. a.m. mean time at ship; at. 56° 48' S., long. 136° 7' E., the observed altitude of the sun's lower imb was 15° 5' 10" bearing by standard compass E. 1° 30' S. ; height of eye 26 feet. Required the sun's true azimuth and error of the compass; ilso, the variation by chart being 3° 30' E., find the deviation for the direcion of the ship's head.

Ans. T. Az. N. 82° 49' E. ; Err. of comp. 8° 41' W.; Dev. 12° 11' W.

Example 3.-April 15th at 8h. 26m. a.m. mean time at ship; lat. 40° 59'S., ong. 29° 40' W., the observed altitude of the sun's lower limb was 19° 10' 20" bearing by standard compass S. 87° 30' E.; height of eye 18 feet. Required the sun's true azimuth and the error of the compass; also, the variation by chart being 7° 40' W., find the deviation for the direction of the ship's head.

Ans. T. Az. N. 57° 2' E. ; Err. of comp. 35° 28' W.; Dev. 27° 48' W.

Example 4.-August 23rd a.m. at ship ; lat. 37° 40' N., long. 144° 52' W., when the chronometer (corrected for error and rate) indicated mean time at Greenwich August 23d. 6h. 17m. 335. (Astronomical time), the sun's observed altitude was 37° 15' 40" bearing by compass S. 74° 30' E.; height of eye 20 feet. Required the sun's true azimuth and error of the compass; also, the variation by chart being 15° 20' E., find the deviation for the direction of the ship’s head.

Ans. T. Az. S. 73° 49' E. ; Err. of comp. 0° 40}' E.; Dev. 14° 39' W.

Example 5.-May 5th at 6h. 52m. a.m. apparent time at ship; lat. 39° 40' N., long. 140° 41' W., the observed altitude of the sun's lower limb was 20° 14' 40" bearing by compass E. 1° 30' N.; height of eye 23 feet. Required the sun's true azimuth and the error of the compass, also the variation by chart being 18° 30' E., find the deviation for the direction of the ship's head, which being a N.W. course by compass, give the true course.

Ans. T. Az. N. 85° 17' E. ; Err. of comp. 3° 13' W.; Dev. 21° 43' W.; True course N. 48° 13' W.

Example 6.—November 17th at about 4h. 22m. p.m. at ship ; lat. 6° 15' N., long. by dead reckoning 158° 30' E., the observed altitude of the sun's lower limb was 17° 6' 50" bearing by standard compass S. 64° 40' W.; height of eye 24 feet ; time by chronometer 6h. 2m. 56s. which (allowing for error and rate) was 15m. 23s. fast on mean time at Greenwich. Required

the longitude, together with the sun's true azimuth and the error of the compass; also, the variation by chart being 7° 20' E. at the ship's position, find the deviation and the true course, the ship's compass course at the time being N.W.

Ans. Long. 158° 41' 30" E.; T. Az. S. 67° 51}' W.; Err. of comp. 3o11}' E. ; Dev. 4o8} W.; True course N. 41° 48' W.

Example 7.— January 1st at about 7h. 26m. p.m. at ship ; lat. 49° 42' N., long. by dead reckoning 14° W., the observed altitude of Pollux (B Geminorum) was 26° 4' bearing by standard compass N. 78° 30' E.; height of eye 28 feet; time by chronometer 8h. 3om. 20s. which was (by error and rate) gm. 56s. fast on mean time at Greenwich. Required the longitude, together with the star's true azmuth and the error of the comp ass; also, the variation by chart being 25° 40' W. at the ship's position, find the deviation and the true course, the ship's compass course being E. IN

Ans. Long. 13° 40' 30" W.; T. Az. N. 76° 25' E.; Err. of comp. 2° 5'W.; Dev. on E. I N. = 23° 35' E.; True course E. 4° 54' N.

Example 8.— January 11th at about 5h. 15m. p.m. at ship ; lat. 48° 27' N., long. by dead reckoning 28° W., the observed altitude of Vega (a Lyræ) was 28° 1' bearing by standard compass N. 27° 30' W.; height of eye 26 feet; time by chronometer 6h. 53m. 56s. which (allowing for error and rate) was 6m. 56s. slow on mean time at Greenwich. Required the longitude, together with the star's true azimuth and the error of the compass; also, the variation by chart being 32° 30' W. at the ship's position, find the deviation and the true course, the ship's compass course at the time being N.W. by W.

Ans. Long. 28° 12' 45" W.; Star's T. Az. N. 62° 6' W.; Err. of comp. 1 34° 36' W.; Dev. 2° 6' W.; True course S. 89° 9' W.; or about W. 1°S.

LONGITUDE BY CHRONOMETER

(I) BY SUN'S ALTITUDE; OR (2) BY STAR'S ALTITUDE

Longitudes at sca are determined by computing the hour-angle of a heavenly body the altitude of which has been measured by a sextant, and through this hour angle obtaining the local time for comparison with the Greenwich time by chronometer.

Thus longitude becomes the difference between time at place and time at Greenwich at the same instant.

The Greenwich time is ascertained from the chronometer, which has previously been regulated, and its error and rate tabulated; the daily rate being properly applied gives the Greenwich time at any instant.

In determining the local time by means of the object's altitude above the sea-horizon let the time be noted by watch. For greater precision, observe several altitudes in quick succession, noting the time of each, and take the mean of the altitudes as corresponding to the mean of the times. But in taking the mean of several observations in this way it must not be forgotten that we assume that the altitude varies in proportion to the time, which is theoretically true only in the exceptional case where the observer is on the equator and the object's declination is zero. It is, however, practically true for an interval of a few minutes when the heavenly body is not too near the meridian.

Best Position of a Heavenly Body for determining the Time at Place by an Altitude of the object.When the azimuth of the heavenly body is 90°; that is, when it is on the prime vertical, bearing true east or west; the error in time will be the least possible, since, for an object in that position—(1) it rises and falls fastest, allowing its altitude to be observed with the greatest precision ; also (2) the error in the hour angle, corresponding to a small error in the altitude, is least; and (3) the error in the hour angle, corresponding to a small error in the latitude, vanishes.

But no object can reach the prime vertical unless its declination is of the same name as the latitude of the place, and even then observations when the object is nearly east or west must not be carried so far as to include observations at very low altitudes where anomalies in the refraction may produce serious errors.

Tables giving the time when an object is on the prime vertical, that is, when it bears east or west, and its altitude thereon, are given towards the end of Norie's Nautical Tables.

When the latitude and declination are of opposite names the object will not be on the observer's prime vertical, but will be nearest to it when rising or setting; therefore the altitude should, in this case, be taken as soon as it exceeds 10° or 12°.

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BB 2

For the Sun, when you are in the opposite hemisphere, it is not practicable to observe in the most favourable position, hence choose the position as near to it as possible, but not too low; and remember that, generally, for any object, throughout the interval between the best position and the meridian the nearer the object is to the meridian the more unfavourably is it situated for the purpose of computing the time from an altitude. In the tropics, with latitude and declination of same name, proximity of the sun to the meridian has little effect on the hour angle.

The stars and planets are good objects at twilight and dawn, and the moon by day when in a favourable position.

Longitude by Chronometer and Sun's Altitude. RULE-I. Take any odd number of altitudes of the sun, with the corresponding times by chronometer; take the mean of the altitudes, and also of the times, i.e., take the sum of each and divide by the number of observations.

At sea, the first thing to note is—does the time by chronometer require to be increased by 12 hours, in order to express the Greenwich time astronomically ? —Your ship time and longitude will indicate this (see p. 233).

2. For the Greenwich Date, mean time.—To the mean of the times ascertained as above (par. I), and expressed astronomically, apply the original error, by addition if slow, by subtraction if fast.

Then multiply the daily rate by the number of days and parts of a day that have elapsed since the original error on Greenwich was determined ; the product, which is called the accumulated rate, being added to the above sum or remainder, if the chronometer be losing, or subtracted from it if gaining, the result will give the mean time at Greenwich, for which all the elements from the Nautical Almanac must be corrected (see also pp. 124-6).

3. For the Declination.—Take the declination from the Nautical Almanac, p. II. of given month, and correct it (by Var. in Ih. p. I. of Almanac) for the Greenwich date, mean time (see p. 238).

For the Polar Distance. Subtract the corrected declination from 90° when latitude and declination are both N., or both S.; if latitude and declination are one N. and other S., add 90° to the declination. For latitude o, subtract declination from 90°.

4. For the True Altitude, correct the mean of the observed altitudes for dip, semidiameter, refraction, and parallax (see p. 254).

5. For the Equation of Time.—For the given day take the equation from Nautical Almanac, p. II. of month; also take “ Var. in ih." for same day from p. I. of month; multiply the “ Var. in ih." by the hours and tenths of Greenwich time for the correction of equation (see p. 239).

Also, take special notice, on p. I. of Nautical Almanac of the day, when the equation changes; in the column the change is marked by a strong dash and at top thus

add. 6. For the Hour Angle. Write down in succession the true altitude, the latitude, and the polar distance. Take the sum of these quantities, which

add. sub.

or sub.

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