responding to the Greenwich mean noon of the given day. The above decimal changes on March 22d, 1845, but a like process must be used during the next equinoctial year; and

so on.

If it were required to find the equinoctial time corresponding to July 10th, 1844, at 5 P. M., mean time at Greenwich; since from March 22d, to July 10th, there are 110 days, and 5 hours are equal to, or 0.208333 days, that time would be 1843 years 110.208333+0.082875 days, or 1843 years 110.291207 110.291207 days; or again, 1843 =1843.30196) years. Should the equinoctial time be required for a given instant expressed in mean solar time for any place distant, in longitude, from Greenwich; that mean time may be reduced. to Greenwich mean time by adding the difference of longitude in time, if the place be west of Greenwich, or by subtracting it if eastward, and then proceeding as before.

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312. The right ascension of the meridian or midheaven, at any given instant, is expressed by the time which a sidereal clock indicates at that instant (art. 73.); and therefore, the apparent or mean solar time being given, that right ascension may be obtained by finding the corresponding sidereal time, as explained at the end of art. 308.; such sidereal time is the right ascension required.

SI

S

If a watch or clock regulated according to solar time indicate correctly mean time at a station, the moment at which the sun will be on the meridian will be ascertained on applying the equation of time to the hour (12) of mean noon, by addition or subtraction, as directed in the Nautical Almanac (page II. of each month). But the time at which any fixed star will be on the meridian of a station is to be found by the following process: let Qss' represent a trace of the equator in the heavens; E the earth, Pm a projection of the plane of the meridian, and the position of the equinoctial point; let also Qs' represent the right ascension of the star, which is to be taken from the Nautical Almanac. Then, from the same work let the right ascension of the sun at apparent, or at mean noon, at Greenwich be taken, and let there be added to it, if the station be westward of Greenwich, or subtracted from it, if eastward, the variation of the sun's right ascension for a time corresponding to the distance of the place in longitude from Greenwich; the result (Qs), which is the sun's right ascension at apparent or mean noon at the

R

B

E

A

P

station, being subtracted from the star's right ascension, gives ss', the approximate time at which the star culminates, or the time in which the meridian Pm would revolve by the diurnal rotation from PS to PS'. Again, let the variation ss of the sun's right ascension during that approximate time be found and subtracted from the approximate time; the remainder will be the angular distance SPS' (in time) of the sun from the meridian when the plane of the latter passes through the star; that is, very nearly, the apparent, or mean, solar time at which the star will culminate.

If the sun's right ascension should exceed that of the star, it would be merely necessary to add twenty-four hours to the latter before the sun's right ascension is subtracted from it; the remainder will still be the time which is to elapse before the star culminates.

313. It is of importance to have a knowledge of the instant at which a planet will culminate, or be on the meridian of any particular station whose longitude from Greenwich is known exactly or nearly, for the purpose of determining, by means of an observed altitude of the planet, the hour of the observation, or in order to be prepared for observing the meridian altitude of the planet and subsequently of determining the latitude of the station. This knowledge may be obtained in the following manner. If ss', as above, represent the difference between the right ascension of the sun and the geocentric right ascension of a planet, at mean noon at the station, and consequently the approximate mean time at which the planet will culminate; let there be next found, for that approximate time, the variation ss of the sun's right ascension and the variation s's' of the planet's geocentric right ascension (the latter being set out in the direction s's' or s's", according as the right ascension is increasing or decreasing). Then the arc ss' or ss" (ss' + s's' ss, or ss's's" - ss) may be considered as a very near approximation to the mean time at which the planet will culminate at the station.

The time at which the moon will culminate at any station may be found in like manner, s's' representing the variation of the moon's right ascension in the first approximate time; and this variation may be obtained from the hourly right ascensions of the moon in the Nautical Almanac.

The culmination of a planet at any station may also be determined from the times of the daily transits or passages of the planets over the meridian of Greenwich which are given in the pages of the Nautical Almanac containing the geocentric places. For the day of the month being given, on

subtracting from one another the times of transit on that day and the next, there will be found the excess or deficiency of the interval between the times, with respect to a mean solar day: then, supposing that the excess or deficiency takes place uniformly, there may be had by a simple proportion the value of it for an interval of time equal to the given difference of longitude.

For example, let it be required to find the time that Venus culminated on March 1st, 1843, at a station whose longitude from Greenwich, in time, is three hours eastward. From the Nautical Almanac it is found that the difference between the times of the meridian passages on the first and second days of March is 0.4 (an increase); therefore

24 ho. 3 ho. :: 0'.4 : 0.05,

which being subtracted from 21 ho. 5'.6 (the time of the transit at Greenwich on the first day of March) gives 21ho. 5'.55, or 21 ho. 5′ 33′′ for the time of the transit, in mean time, at the station.

The excess is subtracted because the station is eastward of Greenwich, and the transit takes place there earlier than at the latter place: it should have been added if the station had been westward of Greenwich. It must be observed also that, if the times of transit should diminish from one day to the next, the difference corresponding to the difference of longitude must be added when the station is east of Greenwich, and subtracted when it is westward.

Thus, let it be required to find the time that Jupiter culminated on August 1st, 1843, at a station whose longitude from Greenwich, in time, is 3 hours eastward. From the Nautical Almanac it is found that the difference between the times of the meridian passages on the first and second days of August is 4'.4 (a diminution); therefore

24 ho. 3 ho. :: 4'4 0'.55,

which being added to 13 ho. 8'.2 (the time of the transit at Greenwich on the first day of August) gives 13 ho. 8'.75 for the mean time of the transit at the station. If the station had been westward of Greenwich the diminution must have been subtracted.

The time of the moon's transit over the meridian of any station whose longitude from Greenwich is known may also be determined with sufficient precision for the purpose of preparing to observe the meridian altitude of that luminary. For example; on the first day of September, 1843, the moon passed the meridian of Greenwich at 6 ho. 15'.2, and on the second day at 7 ho. 12.5; the difference between them is

57'.3 (the excess of the time between two consecutive transits above a solar day): therefore, if the station be distant from Greenwich in longitude 45 degrees, or 3 hours, eastward,

24 ho. 3 ho. :: 57.3 7.16;

:

and this excess subtracted from 6 ho. 15'.2 leaves 6 ho. 8'.04, or 6 ho. 8′ 2′′.4, for the mean solar time of the transit at the station. The excess obtained from the proportion must have been added if the station had been westward of Greenwich: in the first case the transit takes place at the station before it takes place at Greenwich; and in the other, it takes place later.

314. If a celestial body having no proper motion were observed at two different instants, the angle at the pole between the hour circles passing through the apparent places of the body at those instants would be expressed in degrees by 15 T, T representing the number of sidereal hours in the interval of time during which a meridian of the earth would, by the diurnal rotation, pass from one place of the celestial body to the other. Also, when the celestial body is the sun, and the interval between the observations is expressed as usual in mean solar time, or that which is given by a clock or watch going correctly, the same interval multiplied by 15 will give the angle between the hour circles in degrees.

For, let Qss' (fig. to art. 312.) represent the equator, P its pole, Q the equinoctial point; and let the plane of the meridian of a station pass through PA and PB at the two times of observation: also let qs be the right ascension of the sun at the first observation; and, by the change in the sun's right ascension in the interval, let Qs be the right ascension at the second observation. Then, the arc As in degrees (the angular distance of the meridian from the hour circle passing through the sun at the first observation) being represented by a, the arc AB by b, and ss by c; the angular distance BS of the meridian from the hour circle passing through the sun at the second observation, is equal to ab+c, and the angular distance between the hour circles passing through the two visible places of the sun is equal to as Bs or bC. In sidereal time that angle is expressed by (b−c): but c is the acceleration corresponding to the sidereal time 15 b; and (bc) expresses also the solar time corresponding to 13 b, or 15 (b-c) is the given interval. Thus the angle in degrees between the hour circles is equal to the product of 15 by the given interval between the observations, in solar time. In strictness, the angle between the two hour circles being

-

affected by the inequality of the sun's movement in right ascension during the interval, that interval, in mean time, should be reduced to the corresponding interval in apparent solar time, by applying, according to its sign, the variation in the equation of time during the interval before it is multiplied by 15. The variation of the equation does not, however, exceed 30" in 24 hours, and is generally much less.

If the interval between the observations on a fixed star at two different instants, were expressed in solar time, it would evidently be necessary, in order to have in degrees the angle at the pole between the hour circles passing through the two visible places of the star, that the interval so expressed should be converted into sidereal time, either by a table of “ Time equivalents" (Naut. Alm., pages 554-557.), or by adding the acceleration, before the multiplication by 15 is made. When the interval between two observations on the moon is expressed in solar time, it must in like manner be converted into sidereal time: this last may be represented by AB in the figure, that is, by the angular distance (in time) between the places of the meridian at the times of the two observations; therefore, subtracting from it the increase of the moon's right ascension during the interval (represented by ss), and multiplying the remainder by 15, the result will be the required angle at the pole between the hour circles passing through the apparent places of the moon at the observations. In like manner, the interval between two times of observation on a planet being reduced to sidereal time, if from the result there be subtracted the increase, or added the decrease of the planet's geocentric right ascension for the interval; there will be obtained (in time) the required angle at the pole.

315. It follows from what has been said, that when the hour angle of a celestial body (the angle at the pole between the meridian of a station, and à horary circle passing through the celestial body) is obtained, in degrees, from an observed altitude of the body, the reduction of that angle to solar time, and the converse operation, will be different for the sun, the moon, and for a star. The angle at the pole between the meridian of a station, and a hour circle passing through the sun, being divided by 15, or multiplied by according to the practice of mariners, expresses the solar time of the observation, reckoning from the instant of apparent noon. For a fixed star, the like angle at the pole, in degrees, on being divided by 15, is expressed in sidereal time; and the acceleration for the number of hours must be subtracted from the quotient, in order that it may be expressed in solar time:

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