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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;
d, 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 56.91s 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. Any 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.
to such if observed with a chronometer regulated to mean time. When, instead of the two or three chronometers generally taken on board every ship, a number of these instruments, whose rates and errors have been previously carefully ascertained, are conveyed from one meridian to another, the comparison of the mean of the times shown by the chronometers with the local time at each place, affords the means of determining with considerable accuracy the difference of their longitudes; this mode is much practised at present on board surveying vessels *, for measuring the respective meridian distances between a number of maritime towns, ports, and other places on the sea-coast of distant countries. On shore the difference of longitude between two stations can also be determined with precision by the transmission of pocket chronometers between them; provided the errors of the box chronometers or clocks at these stations on sidereal time, and their rates, have been carefully ascertained by transit observations. Where the distance is not very considerable, the operation consists simply in comparing several pocket chronometers with the standard instrument at one of the stations, and then sending them with the greatest care to be compared with the clock or chronometer at the other station, to be returned immediately for another comparison at the starting point; which process of transmission should be repeated several times.
When the time occupied by this operation is considerable, more than four or five days for instance, the accuracy of the result will be increased by stationing a careful assistant at a post midway between the two extreme stations with a box chronometer, with which the transmitted pocket chronometers are to be compared. Mr. Airy recommends commencing from this central position, sending the pocket chronometers (divided into two batches) simultaneously for comparison to the two principal extreme stations, and comparing them again on their return, at nearly the same time, at the intermediate point; by which modification, the time through
* On board H. M. S. Beagle, employed as a surveying vessel principally on the coasts of Australia and Van Diemen's Land, there were at one time as many as twenty-one first-rate chronometers.
This should be done directly after the error of the standard chronometer has been tested by observations with the transit instrument.
which reliance is placed upon the pocket chronometers is diminished one-half, and very little dependence is made to rest upon the steadiness of performance of the box chronometer at the central place of observation.
This method of obtaining the difference of longitudes of two distant places would, it is imagined, seldom be resorted to where the distance was very great, and where an intermediate station was found necessary. On the North American Boundary Survey, the second method was never tried, but the first and more simple process of direct transmission and comparison between the two stations was constantly practised with great success. One example has been selected from Major Robinson's report, calculated according to the directions drawn up by Mr. Airy, each of the three comparisons recorded being the mean of six observations.