The approximate mean zenith distance of the two stars is then set off on the vertical circle, and the level bubble brought to the center of the tube. When the star appears in the field of the telescope the horizontal cross wire is brought to bisect the star and such bisection retained until the star crosses the vertical cross wire of the telescope. The micrometer head is then read. The telescope is then revolved through 180° about its vertical axis and brought to the same inclination with the horizontal by moving the telescope itself about its horizontal axis until the level bubble is at the center of the tube. In like manner the second star is bisected by the horizontal cross wire and the micrometer head again read. The difference between the two micrometer readings gives the difference of zenith distances of the two stars in terms of divisions of the micrometer, which when multiplied by the known angular value of one division of the micrometer gives the angular difference of the zenith distances of the two stars. CHAPTER XIII. LONGITUDE. 336. The longitude of a position on the earth's surface is measured by the are of the equator intercepted between the prime meridian and the meridian passing through the place, or by the angle at the pole between those two meridians. Meridians are great circles of the terrestrial sphere passing through the poles. The prime meridian is that one assumed as the origin, passing through the location of some principal observatory, such as Greenwich, Paris, or Washington. That of Greenwich is the prime meridian not only for English and American navigators, but for those of many other nations. Secondary meridians are those connected with the primary meridian, directly or indirectly, by exchange of telegraphic time signals. Tertiary meridians are those connected with secondaries by carrying time in the most careful manner with all possible corrections. Longitude is found by taking the difference between the hour angle of a celestial body from the prime meridian and its hour angle, at the same instant, from the local meridian. In determinations ashore the hour angle from the prime meridian may be found either from chronometers or from telegraphic signals; the local hour angle may be found by transit instrument or by sextant. In determinations at sea the chronometer and sextant give the only means available. DETERMINATION ON SHORE. 337. TELEGRAPHIC DETERMINATION OF SECONDARY MERIDIANS.-In order to locate with accuracy the positions of prominent points on the coasts, it is necessary to refer them, by chronometric measurements, to secondary meridians of longitude which have been determined with the utmost degree of care. Before the establishment of telegraphic cables, this was attempted principally through the observation of moon culminations, which seemed always to carry with them unavoidable errors, or by transporting to and fro a large number of chronometers between the principal observatory and the position to be located; and in this method it can be conceived that errors would be involved, no matter how thorough the theoretical compensation for error of the instruments. By the aid of telegraph and radio, differences of longitude are determined with great accuracy, and an ever-increasing number of secondary meridional positions are thus established over the world; these afford the necessary bases in carrying on the surveys to map correctly the various coast lines, and render possible the publication of reliable and accurate navigators' charts. 338. To determine telegraphically the difference of longitude between two points, a small observatory containing a transit instrument, chronograph, break-circuit sidereal chronometer, and a set of telegraph instruments is established at each of the two points, and, being connected by a temporary wire with the cable or land line at each place, the two observatories are placed in telegraphic communication with each other. By means of transit observations of stars, the error of the chronometer at each place on its own local sidereal time is well determined, and the chronometers are then accurately compared by signals sent first one way and then the other, the times of sending and receiving being very exactly noted at the respective stations. The error of each chronometer on local sidereal time being applied to its reading, the difference between the local times of the two places may be found, and consequently the difference of longitude. The time of transmission over the telegraph line is eliminated by sending signals both ways. By the employment of chronometers keeping sidereal time, the computation is simplified, though mean-time chronometers may be used. 339. ESTABLISHMENT OF TERTIARY MERIDIANS.-Let it be supposed that the meridional distance between A and B is to be measured, of which A is a secondary meridional position accurately determined, and B a tertiary meridional position to be determined. If possible, two sets of observations should be taken at A to ascertain the errors and rates of the chronometers. The run is then made to B, and observations made to determine local time, and hence the difference of longitude; and on the same spot altitudes of the sun, or of a number of pairs of stars, or both, should be taken to determine the latitude. Now, if chronometer rates could be relied on to be uniform, this measurement would suffice, but since variations may always arise, the run back to A should be made, or to another secondary meridional position, C, and new rates there obtained. Finally, the errors of the chronometers on the day when the observations were made at the tertiary position should be corrected for the loss or gain in rate, and for the difference of the errors as thus determined. When opportunity does not permit obtaining a rate at the secondary meridional station or stations, both before and after the observations at B, the navigator may obtain the errors only, and assume that the rate has been uniform between those errors. A modification of the foregoing method which may sometimes prove convenient is to make the first and third sets of observations at the position of the tertiary meridian, and the intermediate one at the secondary meridian; in this case the error will be obtained at the secondary station and the rate at the tertiary. EXAMPLE: A vessel at a station A, of known longitude, obtained chronometer errors as follows: May 27, noon, chro. slow, 7m 18.9, June 3, noon, chro. slow, 7 12.7; then proceeding to a station B a series of observations for longitude was taken on June 17; after which, returning to A, the following errors were obtained: Therefore, assuming that these rates were correct at the middle of the periods for which they were determined, we have, May 30, Midnight, Rate, +0.89 340. SINGLE ALTITUDES.-The determination of longitudes on shore by single altitudes of a celestial body is identical in principle with the determination at sea by that method, which will be explained hereafter (art. 341). It may be remarked, however, that by taking observations on opposite sides of the meridian, at altitudes as nearly equal as posssible, a means is afforded, which is not available at sea, of elimi DETERMINATION AT SEA. 341. THE TIME SIGHT.-A method of determining longitude at sea is that of the time sight, sometimes called the chronometer method. The altitude of the body above the sea horizon is measured with a sextant and the chronometer time noted; the hour angle of the body is then found by the process described in article 316, Chapter XI. "If the sun is observed, the hour angle is equal to the local apparent time; the Greenwich apparent time may be determined by applying the equation of time to the Greenwich mean time as shown by the chronometer; the longitude is then equal to the difference between the local and the Greenwich apparent times, being east when the local time is the later and west when it is the earlier of the two. If any other celestial body is employed, the hour angle from the local meridian, found from the sight, is compared with the hour angle from the Greenwich meridian to obtain the longitude; the Greenwich hour angle is found by converting the Greenwich mean time into Greenwich sidereal time in the usual manner, and then taking the difference between the latter and the right ascension of the body, the remainder being marked east or west, according as the Greenwich sidereal time is the lesser or greater of the two quantities; and as the local hour angle may be marked east or west according to the side of the meridian upon which it was observed, the name of the longitude will be indicated in combining the quantities. 342. As has been stated, the most favorable position of the celestial body for finding the hour angle from its altitude is when nearest the prime vertical, provided the altitude is not so small as to be seriously affected by refraction. 343. In determining the longitude at sea by this method, it is necessary to employ the latitude by account. This is seldom exactly correct, and a chance of error is therefore introduced in the resulting hour angle; the magnitude of such an error depends upon the position of the body relative to the observer. The employment of the Sumner line, which is to be explained in a later chapter, insures the navigator against being misled by this cause, and its importance is to be estimated accordingly. EXAMPLE: At sea, May 18, 1916, a. m.; Lat. 41° 33′ N.; Long. 33° 37′ W., by D. R., the following altitudes of the sun's lower limb were observed, and times noted by a watch compared with the Greenwich chronometer. Chro. corr., +4 59.2; I. C., -30; height of the eye, 23 feet; C-W, 2h 17 06. Required the true longitude. 347291w sin.t, 9.74176 EXAMPLE: At sea, April 16, 1916, p. m., in Lat. 11° 47' S., Long. 0° 20′ E., by D. R., observed an altitude of the star Aldebaran, west of the meridian, 23° 13' 20"; chronometer time, 6h 58m 29s, chronometer fast of G. M. T., 2m 27; I. C.,-2' 00"; height of eye, 26 feet. What was the longitude? Long.,00 36" 45") EXAMPLE: At sea, July 26, 1916, a. m., in Lat. 25° 12′ S., Long. 75° 30′ W., by D. R., observed an altitude of the planet Jupiter, east of the meridian, 32° 46′ 10′′; watch time, 2h 48m 02s; C-W, 5h 05m 42s; C. C.,+2m 18; I. C.,+1' 30"; height of eye, 18 feet. Required the longitude. W. T., C-W, C. C., + 2 18 11 26.8 Red. (Tab. 9), + 3 16.5 Dec., |