found that the number of such revolutions, or days, in the mean tropical year is (art. 174.) 365.242217. The obliquity of the terrestrial equator to the plane of the ecliptic, considered as fixed, is here supposed to suffer no variation, and the line of the equinoxes to continue parallel to itself except the uniform angular deviation from parallelism which is due to the effect of general precession (=50′′.2 yearly). The arrival of the sun at the plane of a terrestrial meridian constitutes noon on that meridian, and the shadow of the gnomon indicates on a sun-dial the hour of the day: but the lengths of these solar days are unequal, both because the motion of the earth in its orbit is variable, and because the plane of the orbit is not perpendicular to the axis of the diurnal rotation. 360° 306. Now (= 59'8".33) expresses the 365.242217 da. mean daily motion of the earth in the ecliptic: and if it be conceived that the centre of the earth is at rest while a fictitious sun revolves about it in a circular orbit in the plane of the equator with this uniform daily motion, the earth at the same time revolving on its axis as before, each interval between the instants at which successively the meridian of a station on the earth passes through the fictitious sun would constitute a mean solar day. Thus, let QSN be part of the circumference of a circle in the celestial sphere in the plane of the equator BD, of the earth; s and s', in that circumference, the centres of the fictitious sun in such positions N that the angle SPS' is equal to 59'8".33. Also let PA be a projected meridian passing through any point A, which represents a station on the earth: then the time in which that point would, in consequence of the diurnal rotation, describe about P (the projected pole D S/S/S A a E B P of the equator) the arc AEa is the length of the mean solar day, while the time of describing the circumference AEA expresses, very nearly, the length of the sidereal day. Thus the length of the mean solar day exceeds that of the sidereal day by the time in which PA would, by the diurnal rotation, move to the position Pa; and this time is expressed by 59'8".33 or 3'56".55 in sidereal time. 15 , The length of the mean solar day may be divided into twenty-four equal parts, or hours, or the half of it into twelve hours; and a clock whose dial plate is divided into twelve hours being regulated so that the hour hand may make a complete revolution on the plate in the time of half the mean solar day, as above described, is a perfect indicator of mean solar time. 307. The times at which two consecutive transits of the moon's centre take place at the meridian of any station being observed by a clock regulated according to mean solar time, the interval between those times is designated a lunar day, and is the solar time in which the moon appears to deviate 360 degrees, or twenty-four hours of terrestrial longitude, from the meridian of the station: the length of such lunar day is evidently variable on account of the inequalities of the moon's motion in right ascension; but the period of a synodical revolution being 29 da. 12 ho. 44 min. (art. 215.) its mean excess above the length of a mean solar day is found, on dividing 24 hours by the length of that period, to be 48'46"-8. In like manner, if the times of two consecutive transits of a planet over the meridian of any station be observed by a clock regulated according to mean solar time, the interval between those times may be considered as the length, in solar time, of a planetary day. This length, besides being different for the different planets, is variable for the same planet; and it exceeds or falls short of the length of a solar day: its variations depending on the order of the planet's movement (direct or retrograde) in its orbit, and upon the inequalities of the motion in right ascension. 308. If it be required to find the hour, minute, &c. of mean solar time corresponding to any given sidereal time; let QN (fig. to art. 306.) be a portion of the trace of the equator in the celestial sphere, and Q be the true place of the vernal equinox: now there must be a time when the meridian PA of a station lies in the direction PQ, when the sidereal clock at the station A should indicate O ho., and if the terrestrial meridian be, at a subsequent instant, in any other position as PE, the angle QPE (in time) will express the sidereal time, at the same instant, on the same meridian; also if QPS express the sidereal time, at the mean noon, on subtracting it from QPE, the remainder SPE diminished by SPS", which may represent the increase of the fictitious sun's mean longitude (in time), while the meridian is revolving from PS to PE, will be the angular distance of the mean sun s" from the meridian; that is, mean solar time, at the same instant. In the Nautical Almanac there is given, for every day, the sidereal time at the instant of mean noon (when a clock regulated by mean solar time indicates O ho.) at Greenwich. And if it were required to find the mean solar time at that place, corresponding to any given sidereal time, the following process may be used. From the given sidereal time subtract the sidereal time at mean noon; the remainder will be the number of sidereal hours which have elapsed since mean noon, and this may be considered as the approximate solar time: but the excess of a mean solar hour above a sidereal hour is equal 3'56'.55 to or to 9".8565 (sid. ti.); therefore the number of 24 sidereal hours in a given interval of time is greater than the number of solar hours in the same interval; and in order to obtain the required number of solar hours it is necessary to subtract the product of 9".8565 by the above remainder, from that remainder. The quantity 9".8565t (when t expresses any given number of sidereal hours) is called the acceleration of sidereal above mean solar time for that number of hours. A converse operation is performed in the reduction of a given solar time to the corresponding sidereal time. Thus, having found the acceleration for the given solar time, add together the given time, the acceleration and the sidereal time at mean noon; the result will be the required sidereal time. 309. If the station of the observer be at any distance, in longitude, from Greenwich, the sidereal time at noon, at the station, must be employed instead of the sidereal time at noon at Greenwich. Now the difference between any two consecutive times in the column of sidereal time at noon in the Nautical Almanac is 3'56".55, which is, in fact, the acceleration of sidereal time for twenty-four hours; or the variation of sidereal time in the column is 9".8565 for one hour of time: but the instant of mean noon at any station distant from Greenwich, in longitude, 15 degrees westward, is later by one hour than the instant of mean noon at Greenwich, and at a station 15 degrees eastward, is earlier by one hour. Therefore, if t, in hours, be the longitude of the station from Greenwich, 9".8565t must be added to the sidereal time at Greenwich mean noon, if the station be west of Greenwich, or the same must be subtracted if the station be east of Greenwich, in order to have the required sidereal time at mean noon. 310. The difference between apparent and mean solar time at any instant is called the equation of time, and in the Nautical Almanac its value is given for the instants both of apparent and mean noon at Greenwich on every day in the year. In order to understand the manner of finding this value, imagine a point to revolve about the earth in the plane of the equator with a uniform angular motion equal to 59′ 8′′.33 in a mean solar day; and let ST, TV, &c. subtend each, at P, the pole of the earth's equator, an angle equal to that quantity: then, since this motion is performed in the time that the hour hand of a clock regulated according to mean solar time revolves twice round upon S P given instant, suppose that of mean noon; then Q is the place of the mean equinox. Make QQ' equal to the equation of the equinoxes in right ascension, in arc or in minutes, &c. of a degree; then Q' will be the place of the true equinox. Let Q's' represent part of the trace of the ecliptic in the heavens; make that arc equal to the sun's true longitude at the same instant, and let fall s't perpendicularly on Qs; then s' may represent the place of the true sun, and q't the sun's true right ascension. Now the true sun revolving, apparently, about the earth in the plane of the ecliptic with its proper variable motion; the time at which the plane of a terrestrial meridian passes, by the diurnal rotation, through s', is the instant of apparent noon, or that at which the shadow of the gnomon on a sun-dial indicates 0 hrs.: but the terrestrial meridians being perpendicular to the equator, that which passes through s' also passes through t; and the angle PS through which a meridian revolves between the instant of apparent noon and the instant of mean noon is evidently equal to the difference between the sun's mean longitude and his true right ascension. Therefore since the places of s and s' are supposed to be computed for the instant of mean noon, the difference so found is the equation of time for mean noon. If the places of s and s' had been computed for the instant of apparent noon, the difference would have been the equation for that instant. If L, for any given instant, represent the mean longitude of the sun, reckoned from the mean vernal equinox, as it would be found from astronomical tables; and if ±n represent the equation of the equinoctial points in right ascension, in seconds of a degree; then q's L+n. Now the true right ascension of the sun in degrees might be obtained from the Nautical Almanac for the same instant; but when found from astronomical tables, it consists of the following terms: the mean longitude (L) of the sun from the mean equinox; the equation (+E) of the centre; the equation of the equinoxes (N) in longitude; the effects (+P) of planetary perturbation; and lastly, a term (+R) which being applied according to its sign, reduces the whole to its projection on the equator: the sum of all these terms is represented by q't. Therefore ts, the equation of time at any instant, is expressed by the formula 311. The instant of the occurrence of a celestial phenomenon is usually expressed in mean, or apparent solar time for the station at which the phenomenon is observed; and it is consequently necessary to mention also the place of observation in order that the time so expressed may be reduced to the corresponding times for other stations. For the purpose, however, of rendering unnecessary a reference to the particular station, a practice has been of late introduced of giving the time of a phenomenon in mean solar time reckoned from the instant when a mean sun was in the mean equinoctial point in the year 0 of the Christian era. Time so reckoned is called equinoctial time, and being independent of any station on the earth's surface it serves in some measure to supply the want of a fixed meridian in expressing the time of a phenomenon. The mean sun alluded to is one which is supposed to revolve about the earth with a uniform motion equal to the mean velocity with which the latter revolves about the sun; and the mean equinoctial point is a place in the heavens which the intersection of the ecliptic and equator would occupy if its movement were that only which constitutes the general precession. The time in which the mean sun would make one revolution from the mean equinoctial point to the same again, or the length of the equinoctial year, is, from the investigations of M. Bessel, equal to 365.242217 mean solar days, and at the instant of mean noon at Greenwich on March 22d, 1844, the equinoctial time was 1843 years, 0.082875 days, or an equinoctial year terminated at 0.082875 days (or 1 hr. 59′ 20′′.4) before the instant of mean noon at Greenwich. Now in the Nautical Almanac there is given, page XXII. of each month, the number of mean solar days which have elapsed since the mean noon of the day in the preceding March on which the sun was in the mean equinoctial point; therefore (till March 22d, 1845) the decimal 0.082875 days must be added to the complete number of days elapsed since March 22d, 1844, in order to have the equinoctialˇtime cor |