306. Parallax is always additive; combined parallax and refraction additive in the case of the moon, but subtractive for the sun.

As the correction for parallax of the moon is so large, it is essential that it be taken from the table with considerable accuracy; the corrections for index correction, semidiameter, and dip should therefore be applied first, and the "approximate altitude" thus obtained should be used as an argument in entering Table 24 for parallax and refraction.

SEMIDIAMETER.

307. The semidiameter of a heavenly body is half the angle subtended by the diameter of the visible disk at the eye of the observer. For the same body the semidiameter varies with the distance; thus, the difference of the sun's semidiameter 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 center of the earth; hence the moon is materially nearer to an observer when in or near his zenith than when in or near his horizon, and therefore the semidiameter, besides having a menstrual change, has a semidiurnal one also.

The increase of the moon's semidiameter due to increase of altitude is called its augmentation. This reduction may be taken from Table 18.

The semidiameters of the sun, moon, and planets are given in their appropriate places in the Nautical Almanac.

The semidiameter is to be added to the observed altitude in case the lower limb of the body is brought into contact with the horizon, and to be subtracted in the case of the upper limb. When the artificial horizon is used, the limb of the reflected image is that which determines the sign of this correction, it being additive for the lower and subtractive for the upper.

EXAMPLE: May 6, 1916, the observed altitude of the sun's upper limb was 62° 10′ 40′′; I. C., + 3′ 10′′; height of the eye, 25 feet. Required the true altitude.

EXAMPLE: The altitude of Sirius as observed with an artificial horizon was 50° 59′ 30′′; I. C., — 1′30′′. Required the true altitude.

EXAMPLE: April 16, 1916, observed altitude of Venus 53° 26′ 10′′; I. C., + 2′ 30′′; height of eye, 20 feet. Required the true altitude.

EXAMPLE: May 6, 1916, at 13h 24 G. M. T., the observed altitude of the moon's lower limb was 25° 30′ 30′′; I. C.,-1' 30''; height of eye, 20 feet. Required the true altitude.

308. The corrections for dip, parallax, refraction, and semidiameter, which must be applied to the observed altitude of a star or of the sun's lower limb in order to obtain the true altitude, have been combined in Table 46, and for the moon's upper and lower limb in Table 49, and will henceforth be used in all subsequent problems. This is done in order to save the time and labor involved in referring to separate tables of these corrections.

The tabulated correction for an observed altitude of a star combines the mean refraction and the dip; and that for the observed altitude of the sun's lower limb, the mean refraction, the dip, the parallax, and the mean semidiameter, which is taken as 16'. A supplementary table, taking account of the variation of the sun's semidiameter in the different months of the year, is given in connection with the main table.

Thus, in the first example under article 324, we may, when variations from the mean state of the atmosphere (barometer 30 inches, Fahr. thermometer 50°) are left out of consideration, proceed as follows:

Correction from Table 46, height of eye 20 feet. +10' 35'
Supplementary table for June 21

True altitude.......

0 14

40 07 00
10 21

40 17 21

CHAPTER XI.

THE CHRONOMETER ERROR

309. It has already been explained (art. 261, Chap. VIII) that the error of a chronometer is the difference between the time indicated by it and the correct standard time to which it is referred; and that the daily rate is the amount that it gains or loses each day. In practice, chronometer errors are usually stated with reference to Greenwich mean time. It is not required that either the error or the rate shall be zero, but in order to be enabled to determine the correct time it is essential that both rate and error be known and that the rate shall have been uniform since its last determination.

310. DETERMINING THE RATE.-Since all chronometers are subject to some variation in rate under the changeable conditions existing on shipboard, it is desirable to ascertain a new rate as often as possible. The process of obtaining a rate involves the determination of the error on two different occasions separated by an interval of time of such length as may be convenient; the change of error during this interval, divided by the number of days, gives the daily rate.

EXAMPLE: On March 10, at noon, found chronometer No. 576 to be 0m 32.5 fast of G. M. T.; on March 20, at noon, the same chronometer was 0m 48.0 fast of G. M. T. What was the rate?

The chronometer is therefore gaining 18.55 per day.

311. DETERMINING ERROR FROM RATE. The error on any given day being known, together with the daily rate, to find the error on any other day it is only necessary to multiply the rate by the number of days that may have elapsed and to apply the product with proper sign to the given error.

EXAMPLE: On December 17 a chronometer is 3m 27.5 slow of G. M. T. and losing 0.47 daily. What is the error on December 26?

The chronometer is therefore slow of G. M. T. on December 26, 3m 31.7. 312. It is necessary to distinguish between the signs of the chronometer correction and of the chronometer error. A chronometer fast of the standard time is considered as having a positive error, since its readings are positive to (greater than) those of an instrument showing correct time; but the same chronometer has negative correction, as the amount must be subtracted to reduce chronometer readings to correct readings.

313. Numerous methods are available for determining the error of a chronometer in port. The principal of these will be given.

BY TIME SIGNALS.

314. In nearly all of the important ports of the world a time signal is made each day at some defined instant. In many cases this consists in the dropping of a time ball-the correct instant being given telegraphically from an observatory. In a number of places where there is no time ball a signal may be received on the instruments at the telegraph offices, whereby mariners may ascertain the errors of their chronometers. Such signals are to be had in almost every port of the United States, and similar signals are being sent out from Government radio stations, so that it is now possible to find the error of the chronometer on board ships fitted with

receiving instruments when lying in port and also when underway within radio distance of these stations.

The time signal may be given by a gunfire or other sound, in which case allowance must be made by the observer for the length of time necessary for the sound to travel from the point of origin to his position. Sound travels 1,090 feet per second at 32° F., and its velocity increases at the rate of 1.15 feet per second with each degree increase of temperature. If V be the velocity of sound in feet per second at the existing temperature, and D the distance in feet to be traversed, is the number of seconds V to be subtracted from the chronometer reading at the instant of hearing the signal to ascertain the reading at the instant the signal was made.

This method of obtaining the chronometer error consists in taking the difference between the standard time and chronometer time at the time of observation and marking the result with appropriate sign.

EXAMPLE: A time ball drops at 5h 0m 0, G. M. T., and the reading of a chronometer at the same moment is 4h 57m 52.5. What is the chronometer error?

That is, chronometer is slow 2m 07.5; chronometer correction additive.

BY TRANSITS.

315. The most accurate method of finding the chronometer correction 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 the body's passage over the meridian wire, mark the time by the chronometer. The hour angle at the instant is 0h; therefore the local sidereal time is equal to the right ascension of the body in the case of a star, or the local apparent time is 0h in the case of the sun's center. By converting this sidereal or apparent 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: 1916, 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 05m 33.5. Required the error.

EXAMPLE: June 25, 1916, 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. Find the error of the chronometer on G. M. T.

N

BY A SINGLE ALTITUDE (TIME SIGHT).

316. The problem involved in this solution, by reason of its frequent application in determining the longitude at sea, is one of the most important ones in Nautical Astronomy. It consists in finding the hour angle from given values of the altitude, latitude, and polar distance. The hour angle thus obtained is converted by means of the longitude and equation of time in the case of the sun, or longitude and right ascension in the case of other celestial bodies, into Greenwich mean time; and this, compared with the chronometer time, gives the error.

317. It should be borne in mind that the most favorable position of the heavenly body for time observations is when near to 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 observing the sun or other body when bearing east or west. If observations are made at the same or nearly the same altitude on each side of the meridian and the mean of the results is taken, various errors are eliminated of which it is otherwise impossible to take account, and a very accurate determination is thus afforded.

318. With a sextant and artificial horizon or good sea horizon, several altitudes of a body should be observed in quick succession, noting in each case the time as shown by a hack chronometer or comparing watch whose error upon the standard chronometer is known. Condensing the observation into a brief interval justifies the assumption that the altitude varies uniformly with the time. A very satisfactory method is to set the sextant in advance at definite intervals of altitude and note the time as contact is observed.

319. Correct the observed altitude for instrumental and other errors, reducing the apparent 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 chronometer correction is not approximately known and it is therefore impossible to determine the Greenwich

mean time of observation with a fair degree of accuracy, the first hour angle found will be an approximate one; the declination corrected by this new value of the time will produce a more exact value of the hour angle, and the operation may be repeated until a sufficiently precise value is determined.

320. In figures 48 and 49 are given:

AM=h, the altitude of the body M;

DMd, the declination; and

Q'Z-L, the latitude of the place.

In the astronomical triangle PMZ there may be found from the foregoing:

S