EXAMPLE: August 6, 1879, Lat. 59° S., Long. 175° 27′ E., during evening twilight, observed an altitude of Achernar, near lower transit, 26° 52'; watch time, 4h 31m 12; C-W, 0h 18m 07; chro. fast of G. M. T., 12" 42"; I. C., +1' 20"; height of eye, 24 ft. Find hour angle by both methods; thence the latitude. R. A. 12h L. S. T. lower trans.f 339. This observation should be limited to conditions where the body is within three hours of meridian passage and where it is not more than 45° from the meridian in azimuth. On the prime vertical the solution by this method is inexact, and when the hour angle is 6h it is impracticable. The problem is: Given the hour angle, declination, and altitude, to find the latitude. The solution is accomplished by letting fall, in the usual astronomical triangle, a perpendicular from the body to the meridian, and considering separately the distances on the meridian, from the pole and zenith, respectively, to the point of intersection of the perpendicular; the sum or difference of these distances is the co-latitude. Following the usual designation of terms and introducing the auxiliaries q' and q", the formulæ are as follows: tan "tan d sec t; cos q' = sin h sin p" cosec d; The terms and o" will have different directions of application according to the position of the body relatively to the observer. From a knowledge of the approximate latitude, the method of combining them will usually be apparent; it is better, however, to have a definite plan for so doing, and this may be based upon the following rule: Mark " north or south, according to the name of the declination; mark o' north or south, according to the name of the zenith distance, it being north if the body bears south and east or south and west, and south if the body bears north and east or north and west. Then combine " and p' according to their names; the result will be the latitude, except in the case of bodies near lower transit, when 180°-q" must be substituted for q" to obtain the latitude. It may readily be noted that if we substitute " for declination and q' for zenith distance, the problem takes the form of a meridian altitude; indeed, the method resolves itself into the finding of the zenith distance and declination of that point on the meridian at which the latter is intersected by a perpendicular let fall from the observed body. The time should be noted at the instant of observation, from which is found the local time, and 100 If the sun is observed, the hour angle is the L. A. T. in the case of a p. m. sight, or 12h - L. A. T. for an a. m. sight. If any other body, the hour angle may be found as hitherto explained. EXAMPLE: June 7, 1879, in Lat. 30° 25′ N., Long. 81° 25′ 30′′ W., by account; chro. time, 6h 22m 528; obs. 75° 13', bearing south and east; I. C.-3' 00"; height of the eye, 25 feet; chro. corr. Find the latitude. 2m 36. Chro. t., 6h 22m 528 + Obs. alt. Q, 75° 13′ 00′′ Dec., 7 40 EXAMPLE: August 6, 1879, p. m., in Lat. 52° 47′ S. by D. R., Long. 146° 32′ E., observed altitude of Achernar, near lower transit, 24° 01′ 20′′ bearing south and west; watch time, 6h 48m 22; C-W, 9h 46m 27; chro. corr. on G. M. T., +1 57%; height of eye, 18 feet; I. C. 1' 00". Find the latitude. Watch time, 6h 48m 29s Corr., 5 19 t, 9 47 02 W. 2h 12m 58s 340. This method, confined to northern latitudes, is available when the star Polaris and the horizon are distinctly visible, the time of the observation being noted at the moment the altitude is measured. Two methods will be given. The first is sufficiently precise for nautical purposes, involving the computation of the formula: h = true altitude, deduced from the observed altitude; p = polar distance = 90° - d, the apparent declination being taken from the Nautical Almanac for the date; t = star's hour angle. Find the right ascension and declination of Polaris from the Nautical Almanac; then find the hour angle in the usual way. To the log cosine of the hour angle add the logarithm of the polar distance in minutes; the number corresponding to the resulting logarithm will be a correction in minutes to be subtracted from the star's true altitude to find the latitude. Attention must be paid to the sign of the correction p cos t. If t is more than 6h and less than 18", the sign of cost is -; hence the formula becomes arithmetically: L=h+p cost. EXAMPLE: June 11, 1879, from an observed altitude of Polaris the true altitude was found to be 29° 5' 55". The time noted by a Greenwich chronometer was 13h 41m 26; chro. corr.-2m 225; Long. 5 25 42 W. 341. The second method is more rigorous, and should be employed when greater accuracy is sought. It is embodied in Table 28. Reduce the observed altitude of the star to the true altitude. Find from the Nautical Almanac the apparent right ascension and declination of the star at the time of observation. Find the hour angle in the usual manner. With the hour angle take out the first correction, A, from Table 28, giving to it the sign — when the hour angle is numerically less than 6h; the sign + when the hour angle is greater than 65. With the hour angle and altitude take out the second correction, B, from Table 28. The sign of this correction is always. (If the altitude is greater than 60°, this correction may be found by taking that for 45° and multiplying it by the tangent of the altitude; adding, if desirable, the second term in the expression for B, viz: +0.0076 sin' t tan3 h.) With B and the declination take out the third correction, C, from Table 28, giving it the sign + when the declination is less than 88° 48'; when the declination is greater than 88° 48'. With A and the declination take out the fourth correction, D, from Table 28, giving it the same sign as that of A when the declination is less than 88° 48′; the opposite sign when the declination is greater than 88° 48′. Combine these corrections with the true altitude according to their signs; the result is the latitude of the place of observation. If, when several sights are taken, great precision is required, or the intervals are great, it will be mean of the times may be used. The means of these two corrections may always be used for finding the third and fourth corrections; and these four quantities may be combined with the mean of the altitudes. If the nearest 10" suffices for each, the corrections may be taken out for the nearest arguments without interpolation, and all but the first may thus be taken out when a precision of 3" is required. If a precision of 1' is sufficient for each correction, as is ordinarily the case at sea, an hour angle within 3m will suffice for A; C and D may be neglected, and B used only when the altitude exceeds 47°. EXAMPLE: January 1, 1903, about 9 p. m., Longitude 79° 54′ 07′′ W., observed double altitude of Polaris with artificial horizon, 81° 57′ 20′′; chro. time 1h 55m 12; chro. corr. on G. M. T. +1 078; I. C.-0′ 50′′. (The necessary quantities, taken from the Nautical Almanac for 1903, are given below.) Required the latitude. CHAPTER XIII. LONGITUDE. 342. The longitude of a position on the earth's surface is measured by the arc 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 but also for American navigators, and 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 instruments or by sextant. In determinations at sea the chronometer and sextant give the only means available. DETERMINATION ASHORE. 343. 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 the electric telegraph, 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. 344. 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. 345. 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. nometers. If possible, two sets of observations should be taken at A to ascertain the errors and rates of the chroThe 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 that 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 second |