Thus, if the eye of the upper observer was 68 feet higher than that of the lower, and the two observed altitudes of the sun 20° 0' and 20° 12′, the distance of the land, in sea miles, would be 3.2. For 68 x 0.56 = 38.08, and this, being divided by the difference of the two observed altitudes of the sun, 12', gives 3.2, nearly. Now, if the lower observer be 25 feet above the level of the sea, the dip corresponding to this height and the distance 3.2 miles will be 6', which, being subtracted from 20° 0', leaves 19° 54', the altitude corrected for the dip.

The dip may be calculated, in this kind of observations, to a sufficient degree of accuracy, without using Table 15, in the following manner: Divide the difference of the heights of the two observers, in feet, by the difference of the observed altitude, in minutes, and reserve the quotient. Divide the height of the lower observer, in feet, by this reserved number, and to the quotient add one-quarter of the reserved number, and the sum will be the dip in minutes corresponding to the lower observer. Thus in the above example, = 5.6 is the reserved number, and = 4.4; to this add one-fourth of 5'.6 or 1'.4, and the sum will be the dip 5.8, or nearly 6', corresponding to the lower observer, being the same as was found by the table.

SEMI-DIAMETER.

Art. 255. The semi-diameter of a heavenly body is half the angle subtended by the diameter of the visible disc at the eye of the observer. For the same body the semi-diameter varies with the distance; thus, the difference of the sun's semi-diameter at different times of the year is due to the change of the earth's distance from the sun: and similarly for the moon and the planets.

In the case of the moon the earth's radius bears an appreciable and considerable ratio to the moon's distance from the centre of the earth; hence the moon is appreciably nearer to an observer when she is in or near his zenith than when in or near his horizon, and therefore the semi-diameter besides having a menstrual change has a semidiurnal one also.

The increase of the moon's semi-diameter due to her increase of altitude is called her augmentation. This reduction may be taken from Table 18.

The semi-diameters of the sun, moon, and principal planets are given in their appropriate places in the Nautical Almanac

EXAMPLE. The observed altitude of Sirius was 25° 30' 30'; index corr. I' 30"; height of the eye above the sea-level, 15 feet. Required the true altitude.

Since the stars are at such a distance that the earth's orbit subtends no angle at the star, there is no parallax to be considered.

EXAMPLE. Observed altitude of Venus was 53° 26' 10"; index corr. + 2' 30"; height of eye, 20 feet. Required the true altitude.

EXAMPLE. In Lat. 30° 22′ fower limb was 25 30 30"; Required the true altitude.

N., Long. 8r W., on May 6, 1879, at 8 p. m., the observed altitude of the moon's
Bar., 30.45; Ther., 70; index corr.,

L. m. t., May 6,
Long. 81° W.,

I' 30"; height of the eye, 20 feet.

EXAMPLE. May 6, 1879, the observed altitude of the sun's upper limb was 62° 10′ 40′′; index corr., +3′ 10' : height of the eye, 25 feet.

Required the true altitude.

Observed altitude,

Index corr.,

Dip,

Par.,

Ref.,
S. D.,

True altitude,

CHAPTER VI.

FINDING THE TIME.

RATING THE CHRONOMETER.

Art. 256. Since the chronometer rarely runs for any length of time without change of rate from various causes, it is desirable to ascertain a new rate as often as possible. The length of the interval should depend upon the character of the chronometer and the degree of accuracy required.

In certain ports which are in telegraphic communication with the principal observatory, a time-ball is dropped at some defined instant of time, and by comparing with it the chronometer reading from day to day, and by applying the longitude, the error as well as the rate may be established.

Careful comparisons with a well-regulated sidereal clock in the vicinity will suffice to give the error and rate. If unable to have resort to either of these methods, recourse must be had to astronomical observations, particular care being taken to obtain observations soon after a sea voyage, since the rate of the instrument is liable to change materially at sea from several causes, the chief of which is the disturbance from magnetic influences, the XII hour mark being directed to all points of the horizon.

BY TRANSITS.

Art. 257. The most accurate method of finding the chronometer correction on shore is by means of a transit instrument well adjusted in the meridian, noting the times of transit of a star or the limbs of the sun across the threads of the instrument.

At the instant of a star's passage over the meridian wire note the time by the chronometer. The star's hour angle at the instant = oh, therefore the local sidereal time is equal to the star's right ascension. By converting this sidereal time into the corresponding mean time and applying the longitude, the Greenwich mean time of transit is given. By comparing with this the time shown by chronometer the error is found.

EXAMPLE. 1879, May 9 (Ast. day), in Long. 44° 39′ E., observed the transit of Arcturus over the middle wire of the telescope; the time noted by a chronometer regulated to Greenwich mean time being 8h 5m 338.5. R. A. Arcturus, May 9 = 14o 10m 11a.71 = local sid. t. of transit.

EXAMPLE. 1879, May 10 (Ast. day), in Long. 81° 15′ W., observed the transit of Spica over the five wires of the telescope; the times given below having been noted by a chronometer regulated to Greenwich mean time. Find the error of the chronometer.

Art. 258. By observing the transit of one limb of the sun over the meridian wire of the telescope, and noting the time by the chronometer, its error may be determined. The instant of transit of the sun's centre is apparent noon. On page I of the Calendar of the Nautical Almanac is given the "Sidereal time of the Semi-diameter passing the Meridian, which, added to the time of the transit of the first limb, or subtracted from the time of transit of the second limb, will give the chronometer time of transit of the centre.

By noting the times of transit of both limbs the mean will give the chronometer time of transit of the centre. EXAMPLE. 1879, May 10, in Long. 30 W., observed the transit of the first, or western, limb of the sun, noting the time by a chronometer regulated to Greenwich mean time 1h 58m 78. Find the chronometer error.

EXAMPLE. 1879, June 25, in Long. 60 E., observed the transit of both limbs of the sun over the meridian wire of the telescope, noting the times by a chronometer regulated to Greenwich mean time. Find the chronom

EXAMPLE. 1879, July 23, in Long. 74° W., observed the transits of the sun's limbs over the five wires of the telescope, noting the times by a chronometer regulated to Greenwich mean time.

Art. 259. In this method of rating chronometers, as well as that of equal altitudes, stars are to be preferred, partly because many can be chosen during the same night, and the instrument is not exposed to the sun's rays. Observations of the moon to determine time cannot be relied upon for accuracy.

Having obtained an error for the chronometer, the observation should be repeated at a proper interval of time for a new error; then from the two a rate can be determined.

Should the transit instrument not be in perfect adjustment, or the instrumental corrections uncertain, a sufficiently reliable rate can be determined by observations of the same star, or set of stars, at proper intervals, provided the instrument has been left undisturbed.

An approximate rate may be found by observing the disappearance of a star behind a well-defined vertical edge of a terrestrial object, regard being had that the position of the eye is the same at all the observations. Stars should be selected whose right ascensions and declinations remain constant during the interval.

BY A SINGLE ALTITUDE.

Art. 260. This problem is simply how to find the mean time at any place; and, by comparing this time with that indicated by the chronometer, the error on local mean time is given. If the chronometer is regulated to Greenwich mean time, apply the longitude to the local mean time obtained by observation, and compare with this the time noted by chronometer.

Art. 261. It should be borne in mind that the most favorable position of the heavenly body for time observations is when near the prime vertical. When exactly in the prime vertical a small error in the latitude produces no appreciable effect. Therefore if the latitude is uncertain, good results may be obtained by selecting for observation stars near the prime vertical.

Art. 262. Observe several altitudes of a heavenly body in quick succession, noting the times by chronometer, or by a watch compared with the chronometer, and find the mean of the altitudes corresponding to the mean of the times. Condensing the observation into a brief interval of time eliminates the error caused by assuming that the altitude varies in proportion to the time.

Correct the observed altitude for instrumental errors, and, if a double altitude observed by the artificial horizon, reduce the apparent altitude to the true altitude.

If the sun, the moon, or a planet is observed, the declination is to be taken from the Nautical Almanac for the time of the observation. If the time is not very accurately known, the first hour-angle found will be an approx

imate one; the declination found by this new value of the time will produce a more exact value of the hour-angle; and thus proceed until a sufficiently precise value is determined.

Art. 263. Then knowing, in Fig. 65,

S

PD, the polar distance

PZ Co-L, the co-latitude

To find the angle

which may be found from the formula

- d. L.

=

t, the hour-angle of the body O;

But if it is preferred to use the altitude instead of the zenith distance, the formula becomes by letting S = 1⁄2 (h+L+PD)

PD, the polar distance, is always reckoned from the elevated pole; the latitude is regarded positive; hence, no attention need be paid to the signs of the quantities in the second member of the equation.

If a star is the body observed, the hour-angle, with the R. A. of the star, gives sidereal time, from which the mean time can be found. If the sun is the body observed, the hour-angle gives the apparent time, from which the mean time can be deduced.

EXAMPLE. At a place in Lat. 30° 25′ 22′′ N., Long, 5h 25m 42" W., April 20th, 1879, the following double altitudes of the sun west of the meridian were observed with a sextant and artificial horizon, the times being noted by a chronometer regulated to Greenwich mean time: