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already been described. In this case, the three sides, viz., the colatitude, the zenith, and polar distances, are given to find the hour angle P, which, when the sun is the object observed, will (as was explained in page 193) be the apparent time from apparent noon at the place of observation; and it is converted into mean time by applying to it the equation of time with its proper sign. In the case of a star, it will denote its distance in time from the meridian, which being added to its right ascension, if the observation be made to the westward of the meridian, or subtracted from the right ascension (increase by 24 hours if necessary) if to the eastward, will give the sidereal time, to be converted into mean solar time, if required.
A simple formula for finding the angle of a spherical triangle whose three sides are given is sin. 2x= sin (Sc) (sin & S—b)
sin c. sin b
where S denotes the sum of the three sides a, b, and c; of which a is assumed as the one opposite the required angle. In the present case a represents the co-altitude or zenith distance; b the co-declination, or polar distance; and c the co-latitude.
Observed altitude of the 1838, was 14° 44′ 58′′ at -5
upper limb of the sun on May 4, 47m 15s by chronometer; latitude
51° 23′ 40′′; longitude 2m 21.5o east; index error of sextant 28′′.
* The most favourable time for observing single, or absolute, altitudes of the sun or a star, to obtain the local time, is when they are on or near the prime vertical, since their motion in altitude is then most rapid, and a slight error in the assumed latitude is not of so much consequence. The corrections for the refraction, however, are then often considerable. The same observation will of course give the azimuth Z, and also the variation of the needle, if the magnetic bearing of the star, or of either limb of the sun, is taken by another observer at the same moment as the altitude. This will be further explained.
Method 2nd. From equal altitudes of a star or the sun, and the interval of time between the observations.
If a star is the object observed, it is evident that half the interval of time elapsed between its returning to any observed altitude, after its culmination, will give the moment of its passing the meridian without any correction, from whence the error of the clock or chronometer is at once found. But with regard to the sun, there is a correction to be applied to this half interval, on account of his constant change of declination. From midwinter to midsummer the sun gradually approaches the North Pole, and therefore a longer period will intervene after, than before noon,—between the sun's descent to the same altitude in the evening as at the morning observation: and the reverse takes place from midsummer to midwinter. The amount of this correction depends partly upon the change of declination, proportioned to the interval of time on the day of observation; and partly upon the latitude of the place.The difference of the sun's horary angles at the morning and afternoon observations is easily calculated by the following formula of Mr. Baily's:
Atan L + B♪ tan D, where
T = the interval of time expressed in hours;
L, the latitude of the place, minus when south;
D, the declination at noon, also minus when south;
the double daily variation in declination in seconds, deduced from the noon of the preceding day to that of the following, minus when the sun is proceeding to the south; and
x= the required correction in seconds, A* being minus when the time of noon is required.
The result is of course apparent noon, to which must be applied the equation of time, in order to compare a chronometer with
If the rate only of a chronometer is required, it can be obtained by observing the transits of a star on successive days, or by equal altitudes of the same star, on the same side of the meridian, on different evenings; as a star attains the same altitude after each
The logs. of A and B will be found in table 14.
interval of a sidereal day, which is 3m 56.91 less than a mean solar day; but if the refraction is not alike on the days of observation, a correction will be required.
By reading the azimuths, when the sun or a star has equal altitudes, we obtain the true meridian line, which will be again alluded to. Very frequently the afternoon altitude cannot be observed on account of intervening clouds, but the time can still be calculated from the observed single altitude, as in the last problem.
TO DETERMINE THE LONGITUDE.
The usual method of finding the longitude at sea is by comparing the local time, found by observation, with that shown by a chronometer whose error and rate for Greenwich mean time are known. The accuracy of the result depends of course upon the chronometer maintaining a strictly equal rate under all circumstances, which cannot always be relied upon*, and various methods have been resorted to, to render the solution of this most important problem independent of such uncertain data, or at all events to afford frequent and certain checks upon its correctness. celestial phenomenon which should be visible at the same predicted instant of time in different parts of the globe, would of course furnish the necessary standard of comparison; and all the methods in use for determining the longitude are based upon this foundation; but they are not generally practicable at sea, with the exception of that derived from the observed angular distances between the moon and the sun, or certain stars, which are calculated for every three hours of Greenwich time, and which lunar distance is measured with a sextant, or other reflecting instrument.Artificial signals have been resorted to as a means of ascertaining the difference of longitude, with considerable success, between places not separated from each other by any very considerable distance.
In the Philosophical Transactions for 1826 is an account drawn up by Sir J. Herschel, of a series of observations made in the
*It is usual to have several chronometers on board, and to take the mean of those most to be depended upon. If one varies considerably from the others it is rejected.
summer of 1825, for the purpose of connecting the royal observatories of Greenwich and Paris, undertaken by the Board of Longitude, in conjunction with the French Minister of War. The signals were made by the explosion of small portions of gunpowder fired at a great elevation by means of rockets, from three stations, two on the French, and one on the English side of the Channel; and were observed at Greenwich and Paris, as well as at two intermediate places, Legnieres, and Fairlight-Downs, near Hastings. The difference of longitude thus obtained, 9'21.6", is supposed by Sir J. Herschel to be correct within one tenth of a second, and the observations were taken with such care, that those of the French and English observers at the intermediate stations only differed one-hundreth part of a second.
At page 198 also, of Francœur's "Géodesie," is a description of similar operations for the purpose of ascertaining the difference of longitude between Paris and Strasburg. In operations of this nature, it is only necessary that the rates of the chronometers used should be uniform for the short period of time occupied by the transmission of the signals.
Suppose A and B are two places, whose difference of longitude is required, and that they are too far distant to allow of one signal being seen from each
C and D are taken as intermediate stations, and the first signal, made at S, is observed from A and C, and the times noted; the second signal at S', is observed from C and D, some fixed number of minutes after; and then that at S" from D and B. Suppose these two intervals to have been five minutes each, then the difference of longitude is equal to the difference between the local time at A+ ten minutes, and that observed at B at the moment of the last signal.
Everything in this operation depends upon the correct observation of the times, which should be kept in sidereal intervals, or reduced
*Flashes of gunpowder upon a metal plate are visible at night for a very considerable distance, upwards of 40 miles,—this method is far superior to firing rockets,—the quantity may be from 4 to 16 drachms or more for moderate distances, and a quarter of a pound for long ones.