the station being made), the moon's horary angle z Pm was computed: the sun's horary angle ZPS for the same time was also computed, and hence was obtained the difference between the right ascensions of the sun and moon. The right ascension of the moon, in mean time at the station or ship, was thus known: finding then in an ephemeris the time at the first meridian, as Greenwich, when the moon's right ascension was the same; the difference between the times became the required difference of longitude in time.

The quantities employed were, however, too variable to allow sufficient precision in the result; and if, on comparing the moon's declination obtained from the estimated time at the station with that which was obtained from the time determined by the computation, the two were found to disagree, it was necessary to repeat the computation with the last mentioned time. This repetition rendered the operation extremely laborious, and the method has been long since abandoned.

363. A chronometer being set so as to indicate, at a given instant, the mean solar time at Greenwich, may, by its uniform motion, be supposed to continue in every part of the world to indicate at any instant the hour at that place: consequently the mean time at any station or ship being obtained by an altitude of the sun or a star, or by two equal altitudes of either, the mere comparison of that time with the time indicated by the chronometer will give the difference of longitude between the station and Greenwich.

364. In 1763, Mr. Harrison completed a time-keeper which was found to fulfil the required conditions in determining the longitude of a ship, and the improvements since made in horology have led to the construction of machines possessing still greater uniformity of motion. Yet in every machine there may, from insensible defects in the materials, exist causes of derangement which the artist cannot anticipate; therefore the best warrant must not be allowed to supersede the employment of every means which astronomy furnishes for ascertaining the error, in order that due allowance may be made in finding from it the time at Greenwich.

On land, the time indicated by the chronometer at the instant of an observed eclipse of one of Jupiter's satellites being compared with the time set down in the Nautical Almanac, gives at once the error. If a transit telescope be set up, or even a good theodolite, and the successive transits of the same star over the vertical wire of the telescope be observed; or even if the nightly disappearance of a star behind any terrestrial object be observed, the spectator

looking from any fixed point, as through a small perforation in a wall, should the interval between any two transits or disappearances be exactly 24 hours in sidereal time, or 23 ho. 56' 4".09 in solar time, the time-keeper goes correctly: if otherwise, the difference is the error; and thus the daily error, or the daily rate, may be easily found.

In default of such means, the time of mean noon at a place, given by some watch, may be compared with mean noon found by equal altitudes of the sun or a star, and thus the error of such watch may be found and corrected: then, if the longitude of the station be known, on applying to the corrected time of noon, by the watch, the difference of longitude between the station and Greenwich, the result is the time of mean noon at the latter place; and this result ought to agree with the chronometer. Observing the altitude of the sun or a star for the purpose of determining the time at a station, and subsequently comparing that time with the time given by the chronometer, is what is familiarly called taking sights for the chronometer.

The error, and daily rate of a chronometer may be found by means of lunar observation in the following manner. When the contact of the sun and moon has been observed, let the time shown by the chronometer be marked; and then, with the corrected lunar distance, find the Greenwich time from the Nautical Almanac: the difference between that time and the time shown by the chronometer gives the error of the machine. The error found by any of these means, from observations repeated after one day or several days, will serve to determine the required rate.

The error of the chronometer may also be found by what is called the method of cross-bearings. This consists in taking, on board a ship, suppose at s, the bearings from the true or magnetic meridian, of two objects, as A and B, on the shore (the mutual distance of these objects, with their bearing MAB or NBA, frome one another, and both the latitude and longitude of one of them being supposed to be known): then in the triangle ABS, from the parallelism of the meridians, there are data sufficient for the computation, by plane trigonometry, of the distance SA or SB; and letting fall SM or SN perpendicularly on the meridian of A or B, the distance SM or SN may be found in the rightangled triangle ASM or BSN. One of these distances, as SM, being expressed in geographical miles, or equatorial minutes, and divided by the cosine of the latitude of s, the result will be the difference

A

M

N

between the longitudes of the ship and of the station A, whose geographical position is supposed to be known. Thus the longitude of the ship from Greenwich is found: then the time at the ship being obtained by an observation, the time at Greenwich becomes known; and this being compared with the time indicated by the chronometer, the difference is the error of the latter. Meridional distances as they are called, that is the differences of longitude between places, are usually obtained, when chronometers are employed for the purpose, by finding the absolute longitudes of the stations, from comparisons of the computed times with the corresponding chronometer times, and taking the differences.

365. Among the methods which may be put in practice for determining the longitude of a station are those which depend on observed meridional transits of the moon, and on observed transits of the moon combined with those of certain fixed stars, as proposed by the late Mr. Baily in the "Memoirs of the Astronomical Society" (Vol. II.). The last method in particular, by the help of the tables which have been published in the Nautical Almanacs since the year 1834, and with due care in the observations, unites the advantages of affording considerable accuracy in the results, with great facility in the computation.

If it were required to find the longitude of a station by an observed transit of the moon over the meridian of the station, the following process must be used. Take from the Nautical Almanac, in the table of "moon-culminating stars," the right ascension of the moon's centre; and having observed the transit of that centre over the meridian of the station, which, for example, may be westward of Greenwich, let E be a point in the meridian of the station

when the moon's centre at C culminates at Greenwich. Now while the western meridian is

A

E

revolving to the moon, the centre of the latter will, by the proper movement of the luminary, have advanced to some point D, where it will culminate on that meridian; consequently the difference between the sidereal times of the transits, or culminations, which is the difference between the right ascension of the moon's centre in the Nautical Almanac and the observed sidereal time of the transit at the station, or is the change in the moon's right ascension while the meridian of the station is revolving from E to D, may be represented by CD. But the variation of the moon's right ascension corresponding to one hour of longitude on the earth, that is in the time that a meridian of the earth is revolving through an

angle of 15 degrees, is given in the Nautical Almanac; therefore, v representing the variation of the moon's right ascension (in time) corresponding to one hour,

v: 1 hour :: CD: ED (in time),

and ED CD EC, the difference of longitude required. It has been supposed, for simplicity, that the transit of the moon's centre was observed at the station; but, in fact, the transit of the moon's enlightened limb is that which ought to be observed, and the time of such transit should be reduced to the transit of the centre by adding or subtracting the time in which the moon's semidiameter passes over the meridian wire of the telescope. This reduction is not necessary when the station is near the meridian of Greenwich, because the variation in the moon's proper movement would then be insensible between the times of the transit at Greenwich and at the station: but if the latter be so far distant in longitude that the difference between the times of passing the wire becomes sensible, the reduction must be made.

366. In order to determine the longitude of a station by the culminations of the moon and certain stars having nearly the same right ascension and declination, the following process is employed.

Let t represent the sidereal time at Greenwich, when, by the revolution of the earth on its axis, the meridian of that place passes through a star, supposed to be at A; and imagine that when the same meridian, in revolving, arrives at the enlightened edge of the moon at C, the time is also shown by the sidereal clock. To this last time apply, by addition or subtraction, according as the first or second limb of the moon is on the meridian, the time in which the moon's semidiameter passes the meridian, and let the sum or difference be represented by T; then, the moon being supposed, for example, to culminate first, t-T will express the difference (represented by AC) between the right ascensions of the moon's centre and of the star when the former culminates at Greenwich.) (These right ascensions may be taken from the Nautical Almanac, from the table of "moon-culminating stars.") Let the meridian of the station pass through some point E when the moon culminates at Greenwich; then while that meridian is revolving to the moon the latter will have moved to some point D. Now let the transits of the star at A, and of the moon's limb be observed, and adding to the latter the time in which the semidiameter passes the meridian, let the first term be represented by t' and the last by T'; then t' T' (represented by AD) will be the difference between the right ascen

sions of the moon's centre and of the star when the former culminates at the station. The difference CD, between AD and AC, is therefore known; and consequently, by the proportion above mentioned, we may have ED.

This value of ED expresses, in time, the angle through which the meridian of the station revolves while the moon moves from c to D by her proper motion. It is evidently equal to EC + CD; and EC is, in time, the required difference in longitude between Greenwich and the station. In the above description it has been supposed that Greenwich is eastward of the observer's station, but the process would have been the same in the contrary case.

Ex. 1. At Sandhurst, October 8. 1840, there was observed by the sidereal clock, the transit of the moon's first limb on the meridian.

Error of the clock (too slow)

Correct time of the transit

Right ascension of the moon's first limb (from the table of moon-culminating stars in the Naut. Alm.)

Difference (CD)

Then, the variation of the moon's right ascension for one hour being 122".59,

122.59 1 hour :: 6".36: 3′6′′.8 (=ED)
6.36 (=CD)

Difference of longitude, in time, 3 0.44 (EC)

Ex. 2. At Sandhurst, Oct. 8. 1840, there were observed the transits

Then by proportion, as above, we get ED=3′ 4′′; and subtracting the present value of CD, there remain, for the difference of longitude, in time, 2′ 57′′.73 (= EC.)

367. The difference between the longitudes of two stations may be determined by means of signals, which may be visible at both at the same instant, and such a signal is the flash of fired gunpowder, the extinction of a lamp, or the explosion of a rocket. Let, for example, a rocket be fired from some place between the stations, and let the instant of explosion be observed by persons at those stations: then, the times of the explosion being given by watches previously adjusted so as to indicate the mean times proper to the stations, the difference between them will be, in time, the required difference of longitude. If the stations be so remote that the rocket fired