Ex. 2. 1865, March 20th, P.M. at ship, an observation was made when the time by chronometer was 20d oh 7m 55%, which was 50m 51s fast on Greenwich mean time November 21st, and losing 68.8 daily.

Ex. 3. 1865, September 8th, P.M. at ship, an observation was made when a chronometer showed 7d 23h 16m 28s, which was 57m 47s slow on Greenwich mean time June 30th, and losing 4o5; find the Greenwich date by chronometer.

Ex. 4. 1865, June 28th, A.M. at ship, an observation was made when a chronometer showed 28d 8h 5m 40s, which was 15m 22s fast on Greenwich mean time May 4th, and losing 5.8 daily.

Ex. 5. 1865, February 28th, A.M. at ship, an observation was made when chronometer showed 28d 1h 12m 40, which was fast 9m 55s on Greenwich mean time January 27th, and gaining 389 daily.

EXAMPLES FOR PRACTICE.

Ex. 1. 1865, February 16th, A.M. at ship, an observation was taken when the corresponding time by a chronometer was 16d 8h 59m 25s, which was 1h 14m 238 fast on Greenwich mean time January 3rd, and losing 10s 9 daily.

Ex. 2. 1865, April 19th, P.M. at ship, an observation was taken when a chronometer showed 19d 5h om os, which was fast 33m 30s on Greenwich mean time March 19th, and losing 5o daily.

Ex. 3. 1865, May 7th, A.м. at ship, an observation was taken when a chronometer showed 7d 6h 9m 48s, which was fast 11m 954 on Greenwich mean time February 16th, and losing 55.5 daily.

Ex. 4. 1865, June 26th, P.M. at ship, an observation was taken when the chronometer showed 25d 21h 29m 533, which was 30m 12s fast on Greenwich mean time March 31st, and gaining 28-2 daily.

Ex. 5. 1865, October 25th, P.M. at ship, time by chronometer 25d 8h 31m 103, which was 12m 10s slow on Greenwich mean time July 20th, and gaining 4o7 daily.

Ex. 6. 1865, January 20th, P.M. at ship, time by chronometer 19d 13h 21m 25%, which was 53m 47s fast on mean time at Greenwich October 31st, 1864, and losing 4s daily.

Ex. 7. Time by chronometer, November 8d 16h 2m 35, which was 33m os slow on mean time at Greenwich July 31st, and gaining 43·8 daily.

Ex. 8. Time by chronometer, August 1d 1h 3m os, which was 1h 6m 4s fast on mean time at Greenwich May 31st, and losing 87 daily.

Ex.

9. Time by chronometer, May 1d 13h 23m 10s, chronometer slow 3m 24 on mean time at Greenwich February 2nd, and losing oo 97. daily.

TO FIND THE HOUR-ANGLE.

Given the true altitude of an object, its declination, and the latitude of the observer, to find the meridian distance or hour-angle.

RULE LV.

1°. Find the polar distance by Rule XLV, page 123.

2°. 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 af 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.

(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 XL, 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 XL, page 114) is the hourangle sought. It is thus evident that the complete solution may be obtained by means of the Table of log. sines, &c., alone.

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

1o. To the time by chronometer apply its original error and accumulated rate, as directed in Rule LIV; the result is the Greenwich date at the instant of observation.

2°. Take out of Nautical Almanac, page II, the sun's declination and the equation of time for the noon of Greenwich date and that following it; also take out the sun's semi-diameter.

3°. Reduce the sun's declination and equation of time to the Greenwich time (Rules XLIII and XLVI); also find the polar distance (Rule XLV).

4. Correct observed altitude for index error, dip, correction in altitude, and semi-diameter, and thus get the true altitude.

5°. Find the hour-angle by Rule LV..

6°. When the observation is made in the afternoon, the hour-angle is apparent time past noon of the given day at ship; but if the observation

is made in the morning, take the hour-angle from 24", the remainder is apparent time past noon of the day before that at ship.

7°. To apparent time apply the reduced equation of time, and so get mean time.

8°. Under ship mean time put Greenwich mean time, and take their difference; the remainder is longitude in time, which convert into arc (Table 17, Raper; or Table 19, Norie).

9°. Call the longitude West, when Greenwich time is greater than ship mean time; but East, when Greenwich mean time is least.

In taking the difference of Greenwich mean time and ship mean time, if the days of the month be different, put the more advanced one day back, and add 24 to the hours.

EXAMPLES.

Ex. I. 1865, 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 11d 8h 24m 363, (being P.M. at Greenwich) which was 1h 48m 42o fast for mean noon at Greenwich, September 30th, 1864, and gaining 55.3 daily; required the longitude by chronometer.