between them cannot be seen from both, a chain consisting of three or more signal stations may be formed between the two places whose difference of longitude is required; and in this case, the observer at each of the two extremities of the chain registers the time indicated by his chronometer or sidereal clock at the instant of the explosion at the signal station nearest to him, while the person at each of the intermediate stations registers the time indicated by his chronometer at the instant of the explosion which takes place on each side of him.

In the year 1825 the difference in longitude between the meridians of Greenwich and Paris was determined by means of rockets which, at intermediate stations, were fired at regulated times; and the method then pursued may be conveniently employed on all similar occasions.

Fairlight Down in Sussex, and La Canche on the opposite coast of Normandy, were stations separated from each other by about 50 miles of sea, and Paris was connected with the latter by two intermediate stations, Mont Javoult and Lignières: on the English side, Fairlight, by an intermediate station at Wrotham, was connected with Greenwich; the several stations, in the order here mentioned, from Paris to Greenwich, lying successively westward of one another. The azimuths, or bearings, of the several stations from one another were previously determined by computation, and, at the appointed times, by means of azimuth circles, telescopes were placed in the proper directions, in order that the explosions of the rockets, at the instants of their greatest altitudes, might be observed in the fields of view. The observers being prepared, on a certain night a rocket was fired at Mont Javoult and observed at Paris to explode at 18 ho. 39' 52".5, sidereal time (let this time be represented by A), and the same explosion was observed at Lignières at 10 ho. 49′ 41′′, mean solar time for that station, by a chronometer; let this time be represented by B. About 5 minutes afterwards a rocket fired at La Canche was observed at Lignières at 10 ho. 54′ 53′′.2 (call this time c), and at Fairlight station at 10 ho. 46′ 37′′.5, mean solar times at the two places; represent the latter time by D. Finally, about 5 minutes later, a rocket was fired at Wrotham, which was observed at Fairlight at 10 ho. 51′ 59′′.4 mean solar time (call it E), and at Greenwich at 18 ho. 41' 7".11 sidereal time; let this last be represented by F. The intervals between the discharges of the successive rockets were purposely small, in order that the errors in the rates of the chronometers might be inconsiderable.

Now C B 5'12".2 and E-D=5′21′′.9; therefore their

Y

sum, 10′34′′.1 in solar time, or 10′35′′.83 in sidereal time, was the interval between the observation of the first rocket at Paris and of the third rocket at Greenwich. This interval being added to the time A gives 18 ho. 50′28′′.33, the sidereal time which would be reckoned at Paris at the instant that the sidereal time at Greenwich was expressed by F: the difference between these times, 9′21′′.22, is therefore the distance between the two meridians, in time. In the above example it has been supposed that the mean time chronometers at the stations between Paris and Greenwich were subject to no gain or loss on sidereal time; but this will seldom be the case, and the rates of the chronometers with respect to sidereal time were determined in the following

manner.

Rockets being fired on two successive days at Mont Javoult, for example, were observed

ho.

ho. "1 at Paris, the first day, 18 32 21.88; the second, 18 19 41.83 sid. times; at Lignières

the differences were

10 46 13.6;

7 46 8.28 and

10 28 33.93 mean times by chronom.

7 50 7.9

The excess of the latter difference above the former (=3' 59".62.) was the gain of the chronometer on sidereal time during the interval(23 ho. 47' 19".95.) between the sidereal times of the two observations; this interval being found by adding 24 hours to the time on the second day, and subtracting from the sum the time on the first day. But

:

23 ho. 47'19".95: 3′59′′.62 :: 24 ho. 4'1".74, and this last term was the gain of the chronometer in one sidereal day now the solar day exceeds the sidereal day by 3'55".91; therefore the difference (5".83) was the gain of the chronometer on one solar day. Hence

24 ho. 5.83: c-B(=5′12′′.2): 0.02,

and the last term subtracted from 5'12".2 gives the correct value of C― B, In like manner might be found the correction of E-D, which may also be considered as equal to 0.02: thus the interval between the observations of the rockets at Paris and Greenwich becomes 10'34".06 or, in sidereal time, 10′35′′.79, and the difference of the longitudes becomes 9'21".18.

The signals are made by previous agreement on several different nights, in order that a mean of the results may be taken, and because the observations often fail. During the seasons in which the operations here alluded to were carried

on the rockets sometimes exploded without ascending, at times they exploded twice; and one rocket passed through the field of view without exploding.

The details of all the operations for connecting, by means of signals, the meridians of Paris and Greenwich may be seen in Mr. (Sir John) Herschel's paper in the "Philosophical Transactions" for 1826, part 2,

CHAP. XVII.

ECLIPSES.

OUTLINES OF THE METHODS OF COMPUTING THE OCCURRENCE OF THE PRINCIPAL PHENOMENA RELATING TO AN ECLIPSE OF THE MOON, AN ECLIPSE OF THE SUN FOR A PARTICULAR PLACE, THE OCCULTATION OF A STAR OR PLANET BY THE MOON, AND THE TRANSITS OF MERCURY AND VENUS OVER THE SUN'S DISK.— THE LONGITUDES CF STATIONS FOUND BY ECLIPSES AND OC

CULTATIONS.

368. AN Appendix to the Nautical Almanac for 1836, by Mr. Woolhouse, contains investigations of all the formulæ necessary for computing the phenomena of eclipses of the moon and sun, the occultations of stars by the moon, and the transits of Mercury and Venus over the sun's disk: the formulæ relating to eclipses of the sun consisting of such as serve to determine the phenomena with respect to the earth generally, the terrestrial limits of the phases, the phenomena for particular places, and of such as serve to reduce the phenomena for one station to those which correspond to them for another. It is intended in the present chapter to follow, nearly, the methods employed in that essay, but outlines only will be given of the processes for determining the commencement and end of an eclipse of the moon; and, for particular stations, those of eclipses of the sun, occultations of stars and the transits of planets: to these will be added the rules for computing terrestrial longitudes from observed phenomena of eclipses and occultations.

There exists, beyond the earth with respect to the sun, a space within which the rays proceeding from the sun's disk and touching the surface of the earth do not enter; and this space, which is of a conical form, has its vertex beyond the region of the moon. There exists, likewise beyond the earth, and on the exterior of that umbra or shadow, a space within which the sun's rays only partially enter: this, which is called the penumbra, is bounded by rays which coming from every part of the circumference of the sun's disk cross one another at a point between the sun and earth, and diverging from thence touch the surface of the latter; and in determining the phenomena of eclipses of the sun and moon, it is neces

sary first to find the semidiameters of the shadow and penumbra in a plane passing through the moon's centre perpendicularly to the common axis of the cones, or the line joining the centres of the sun, earth and moon, it being supposed here that those centres are in one right line. Therefore, let OR be a diameter of the sun, whose centre is s; ET

a semidiameter of the earth, and мm a semidiameter of the moon, supposed to be in direct opposition, so that S, E and M are in a straight line: also let a line touching the surface of the sun at o and of the earth at T meet the line SM produced in c; then the triangle TCE will represent half a longitudinal section through the cone of shadow beyond the earth, c being the vertex of the cone, and MN will be a semidiameter of a transverse section in the region of the moon. If a line touching the surfaces of the sun at R and of the earth at T meet MN produced in Q; then MQ will be a semidiameter of a tranverse section of the penumbra in the same region. Imagine the other lines in the figure to be drawn.

In the Nautical Almanac there will be found, at page XII. of each month, the day and hour (in mean time) of full moon, or of the opposition of the sun and moon in longitude. Therefore, for any convenient hour of that day, suppose that which is nearest to the mean time of opposition, take from thence the equatorial horizontal parallaxes P and p of the moon and sun, and the semidiameters of the luminaries, and find the horizontal parallax of the moon for the station of the observer, its latitude 7 being given. This latter parallax is expressed (art. 157.) by p (1-8in.27) putting P, in seconds,

305

for sin. P: let this expression be represented by P'.

The angle TCS is equal to OTS-TSE, or to the difference between the horizontal parallax and the angle subtended at T, or at E nearly, by the semidiameter of the sun; therefore if s denote the angular semidiameter of the sun (in seconds) the angle TCS = s p. Now the angle NEM is that which is subtended at E by the semidiameter of the earth's shadow, and it is equal to TNE-TCS: but TNE may be considered as equal to TME, the moon's horizontal parallax; therefore NEM = P' − s + p nearly.