Ex. 11. 1876, November 27th, A.M. at ship, latitude 39° 20′ S., observed altitude sun's L.L. 34° 37′ 55′′, index corr. + 1′ 15′′, eye 18 feet, time by a chronometer 27d 7h 41m 30s (being P.M. at Greenwich), which was fast 31m 54s for mean noon at Greenwich, October 20th, and on November 9th was 29 40s fast on mean noon at Greenwich.

Ex. 12. 1876, December 24th, A.M. at ship, latitude 9° 59′ S., observed altitude sun's L.L. 10° 38′ 45", index correction 3' 12", eye 18 feet, time by a chronometer 231 17h 36m os (being A.M. at Greenwich), which was slow 34m 191 for mean noon at Greenwich, July 1st, and on July 29th was slow 38m 395 for mean noon at Greenwich.

Ex. 13. 1876, January 1st, P.M. at ship, latitude 38° 28′ S., observed altitude sun's L.L. 39° o', index correction 2′ 25′′, eye 12 feet, time by chronometer 14 11h 58m 29* (being P.M at Greenwich), which was slow 1h 49m 195 for mean noon at Greenwich, on Septerab.r 12th, 1875, and on October 13th was 1h 52m 538 slow for mean noon at Greenwich.

Ex. 14. 1876, February 11th, A.M. at ship, latitude 53° 12′ N., observed altitude sun's L.L. 12° 10, index corr. — 49′, eye 12 feet, time by chronometer 10d 22h 22m 228 (being A.M. at Greenwich), which was fast 34 415 7 for mean noon at Greenwich, October 31st, and on December 1st, 1875, was fast 38m 59s for mean noon at Greenwich.

Ex. 15. 1876, October 26th, A.M. at ship, latitude 28° 10' N., observed altitude sun's U.L. 25° 32′ 20′′, index correction o", eye 17 feet, time by chronometer oh 54m 68 (being P.M. et Greenwich), which was fast 31m 318 on mean time at Greenwich, August 1st, and on Sept. 4th was fast 30m 6s for mean noon at Greenwich.

Ex. 16. 1876, February 6th, P.M. at ship, latitude 6° 58′ N., observed altitude sun's U.L. 21° 43′ 40′′, index corr. o", eye 18 feet, time by a chronometer 11h 40m 26s (being A.M. at Greenwich), which was slow 16m 488 on mean noon at Greenwich, January 2nd, and on January 20th was slow 17m 42% on mean noon at Greenwich.

Ex. 17. 1876, May 1st, P.M. at ship, latitude 21° 8' N., observed altitude sun's L.L. 28°530, index corr. + 2' 50", height of eye 16 feet, time by a chronometer April 30d 18h 50m 2984 (being 6h 50m 29" 4 A.M. at Greenwich), which was 10m 12s slow for mean noon at Greenwich, December 31st, 1875, and on February 17th, 1876, was 7m 33.6 slow for mean noon at Greenwich.

Ex. 18. 1876, April 21st, P.M. at ship, latitude at noon o° 20' N., observed altitude sun's U.L. 32° 21' 10', index correction - 1' 10", eye 12 feet, time by a chronometer 3h 44m 15 (being A.M. at Greenwich) which was slow 9m 78 for mean noon at Greenwich, November 14th, 1875, and on January 11th, 1876, was slow 7m 342 for mean noon at Greenwich, course since noon S. W. by W. (true), distance 36 miles: required the longitude at the time of observation, and also at noon.

Ex. 19. 1876, August 21st, A.M. at ship, latitude at noon o° 20' S., observed altitude sun's L.L. 33° 49′, index correction + 2′ 10", eye 15 feet, time by chronometer 8h 144 0 (being P.M. at Greenwich), which was slow 4m 40% for mean noon at Greenwich, March 13th, and on April 30th was slow 5m 40 for mean noon at Greenwich, course till noon S. W. by W., distance 36 miles: required the longitude at time of sights, and also at noon.

Ex. 20. 1876, March 20th, A.M. at ship, latitude o°, observed altitude sun's L.L. 28° 50' 10", index corr. 1', eye 23 feet, time by chronometer 20d 1h 35m (being P.M. at Greenwich), which was 1m 599 fast for mean noon at Greenwich, February 1st, and on February 28th was fust 2m 8s for mean noon at Greenwich.

Ex. 21. 1876, June 14th, A.M. at ship, latitude 29° 10' S., observed altitude sun's L.L. 30° 40', eye 25 feet, time by chronometer, June 13d 22h 59m 208 (being P.M. on 14th at Greenwich), which was fast 4m 359 for mean noon at Greenwich, March 20th, and on May 3rd was slow 1m 17s for mean noon at Greenwich.

VARIATION BY AN AZIMUTH.

In this problem the Error of the Compass is required by computing the true bearing of the sun, and taking the difference between the true bearing and the bearing by an Azimuth Compass.

300. The Azimuth of a heavenly body is the arc of the horizon intercepted between the cardinal point adjacent to the elevated pole, and the circle of altitude passing through the body, or it is the angle at the Zenith contained between the vertical circle passing through the elevated pole (the meridian) and the vertical circle passing through the object. Azimuth is

usually reckoned from the north or south point, eastward and westward from 0° to 180°.

301. True Azimuth is the bearing of an object from the true north or south point, and is the azimuth found by calculation from the observed altitude or hour-angle of the body. It is in general simply called The “Azimuth” but it is thus qualified as the True Azimuth to distinguish it from the Magnetic Azimuth, which is the bearing of the object from the compass North or South point, and which is found by direct observation with an instrument carrying a magnetic needle. The difference between the true and magnetiç azimuth gives the entire correction of the compass-variation and deviation combined.

301. Given the latitude, altitude, and declination of an object, to find the true azimuth.

RULE XCVIII.

1. Add together the polar distance, the latitude, and the altitude, take half the sum, and take the difference between the half sum and the polar dist.

NOTE. When the latitude is o, suppose it to be of contrary name to the declination when finding the polar distance.

2°. Add together the log. sec. of latitude, the log. sec. of altitude, (rejecting tens), the log. cosines of the half sum and remainder; the sum (rejecting tens) is log, sine square of true azimuth (Table 69 Raper). Or half the sum of the four logs. is the log. sine of half of the true azimuth, which take out of the table (Table 24 Norie), and double it; the result is the true azimuth.

3°. Reckon the true azimuth from S., when the latitude is N., but from the N. when the latitude is S.; towards E. when it is A.M., or when the altitude is increasing, but towards W. when it is P.M., or when the altitude is decreasing.

(a) When latitude is o°, if declination is N. reckon the azimuth from the South; if declination is S., reckon the azimuth from the North.

(b) When both latitude and declination are o°, the object moves on the prime vertical, or is E. while the altitude is increasing and W. while the altitude is decreasing.

NOTE. The logs. are taken out in these examples to the nearest second.

The learner will observe that in this formulæ the pol. dist., lat., and alt. occur in the reverse order of that in Rule XCVI (finding hour angle) in which the initials form the word alp. In finding the azimuth the initials form pla. The 2nd and 3rd terms, take secants; the last two, cosines. By this arrangement the term which has to be taken for the half sum is always on the top.

EXAMPLES FOR PRACTICE.

In each of the following examples it is required to find the true azimuth : (The sign

means P.M.)

means A.M., and the sign

301. Given the true bearing and compass bearing, to find the error of the compass.

RULE XCIX.

1o. To find the amount of the Error of the Compass.-Reckon the True and Magnetic Azimuths from the same point of the compass-North or South.

(a) If one of the azimuths be expressed from the North and the other from the South, take either of them from 180°, and it will then be reckoned from the same point as the other.

(b) If the bearing by compass be reckoned from East or West, towards North or South, take it from 90°, and reverse the position of the letters; or, add 90°, and it will then be expressed from the opposite point to that from which it is reckoned when taken from 90°.

EXAMPLE.

Ex. Suppose magnetic azimuth to be W. 78° 30' N; then subtract the magnetic azimuth from 90° thus:—

90° 00' W. 78 30 N.

The azimuth is thus reckoned N. 11 30 W. from the north pole.

(c) When the magnetic azimuth is either East or West, it is to be reckoned as 90° from North or South, according as the true azimuth is North or South.

2°. Take the difference of the true and magnetic azimuths when measured towards the same point of the compass, East or West; but when measured towards different points, i.e., when one is reckoned towards East and the other towards West, take the sum; the result is the error of the compass or correction.

3°. To name the Correction of Compass.-Let the observer look at the two azimuths (or bearings) from the centre of the compass-then if the true azimuth is to the right of the magnetic azimuth, the correction is East; but if the true azimuth is to the left of the magnetic azimuth, the error is West.