In general the sidereal day is shorter than the solar day. The reason of this is obvious. It arises from the circumstance of the annual motion of the sun being contrary in direction to the diurnal motion of the stars. Let us suppose the sun and a star to be upon the meridian at the same time. After the lapse of a sidereal day the star will again have returned to the meridian; but the sun, in virtue of its annual motion, will have travelled somewhat to the eastward during the intermediate period, and an additional short interval of time must elapse before it is again transported to the meridian by the diurnal motion. If we suppose the mean solar day to consist of twenty-four hours, the sidereal day will be found to number 23 hours, 56 minutes, 4 seconds of mean solar time. The sidereal day is therefore shorter than the mean solar day by 3 minutes 56 seconds.

It has been already stated that the standard of time by which ordinary clocks and watches are regulated is the mean solar day. If we suppose a fictitious sun to travel eastward in the celestial equator with a uniform motion, and to perform a complete revolution in a solar year, the interval between two successive arrivals of such a sun on the meridian will be a mean solar day, and its presence on the meridian will determine the instant of mean noon, as shown by an accurately going clock or watch. Time-keepers which are regulated by the annual motion of this fictitious or mean sun are said to indicate mean solar time.

Apparent noon is determined by the instant when the real sun appears on the meridian. Since the interval between two successive returns of the real sun to the meridian is perpetually varying throughout the year, it is obvious that the instant of apparent noon will generally differ throughout the year from the instant of mean noon, as determined by the interval between two successive arrivals of the fictitious or mean sun on the meridian. This difference between the instant of apparent noon and the instant of mean noon is termed by astronomers the equation of time. Its value for every day in the year is given in the 'Nautical Almanac.' It manifestly enables us to pass from apparent to mean noon, and vice versâ.

Apparent time coincides with mean time on four days in the year-viz., April 15, June 14, August 31, December 24. On those days the sun arrives on the meridian at mean noon -in other words, at twelve o'clock, as shown by an ordinary clock or watch. It is also obvious that a sun-dial, which usually indicates apparent time, will at once give the true time on any of the four days just mentioned. On any other day of the year it will be necessary to apply the equation of time to the indications of the dial, so as to pass from apparent to mean solar time.

It has been stated that on four days of the year apparent time coincides with mean time; in other words, the sun is on the meridian at the instant when the clock shows twelve. On every other day of the year the clock is either before the sun or behind the sun; or, which amounts to the same thing, when the clock shows the instant of noon, the sun either has not come to the meridian or has already passed the meridian. The following illustration will make this more clear.

On January 1, 1869, the clock is 3 minutes 58 seconds before the sun. This indicates that when the sun is actually on the meridian, the clock ought to show oh. 3m. 58s. The sun continues to fall behind the clock till February 11, when the difference between them amounts to 14 minutes 29 seconds. Henceforward the sun gains daily upon the clock until April 15, when they coincide, and the equation of time vanishes. From this day the clock is behind the sun, the difference attaining its maximum on May 14, when it amounts to 3 minutes 53 secondsthat is to say, when the sun is on the meridian the clock ought to show 11h. 56m. 7s. The sun now begins to fall back relatively to the clock, and the difference between them continues to diminish until June 14, when they finally coincide again. From this day the clock is again before the sun, the difference attaining its maximum on July 26, when it amounts

to 6 minutes 14 seconds. From July 26 the sun continues to gain upon the clock until August 31, when the difference between them vanishes, and apparent time coincides again with mean time. From August 31 the sun gains upon the clock, or the clock is behind the sun, the difference attaining its maximum on November 3, when it amounts to 16 minutes 18 seconds. Henceforward the sun begins to fall behind, and this goes on until December 24, when apparent time again coincides with true time. From December 24 the clock begins to be before the sun, the difference on December 31 amounting to 3 minutes 23 seconds. On January 1, 1870, the same cycle of changes commences, and is repeated in the ensuing year.

The difference which generally exists between apparent and true time arises from two distinct causes the variable motion of the sun in its orbit, and the inclination of its path to the equator. If only one of those causes acted, or if both causes were synchronous in their action, there would be only one period in the year during which the clock was before the sun, and only one period during which it was behind that body. But the times of maximum and minimum action of both causes being different, there hence originate two periods of each kind in the course of the year.

We have supposed, in the foregoing explanation, that true or mean time is obtained solely by observations of the sun. First, we determine the instant of apparent noon by observing the sun when it is on the meridian, and then, by means of the equation of time, we pass from apparent to mean noon. It may be well to state, however, that true time is generally ascertained at astronomical observatories, not by the sun, but by the stars. There are several important reasons in favour of this course of procedure. First, the sun appears on the meridian only once in twenty-four hours, and at the instant of arrival its disc may be obscured by clouds. On the other hand, the stars are dotted over the whole heavens, and the astronomer consequently, by availing himself of them, is more independent of temporary casualties of the weather. He is also enabled, by observing the transits of several stars over the meridian on the same night, to arrive at a more accurate result as regards the determination of the true time than if he depended upon the observation of only one object. Again, the instant of a star's passage across the meridian may be observed with much greater precision than the instant of the sun's passage. It is true that the observation of a star supplies us only with sidereal time, but the results in the 'Nautical Almanac,' founded upon the theory of the sun's apparent motion, enables us to pass with facility from sidereal to mean solar time.

CHAPTER V.

On the Corrections to the Apparent Position of a Celestial Object-Refraction— Aberration-Precession-Nutation—Parallax.

The basis of every theory relating to the movements of the heavenly bodies, consists in accurate determinations of their apparent positions made at well known times with instruments constructed for the purpose. But there are certain disturbing causes which exercise an influence on the apparent position of an object in the celestial sphere, the effects of which must be carefully eliminated before the results thus derived from observation can be available as materials for any ulterior researches relating to the advancement of astronomy. We may enumerate these disturbing influences thus,-Refraction, Aberration, Precession, Nutation, Parallax. They

are usually termed corrections to the observed position of a celestial object, because, in all instances wherein they are of sensible magnitude, it is necessary to correct the observed position of the celestial object by eliminating from it the effects which they severally produce. We proceed to make a few remarks on each of these corrections.

Refraction.—When a ray of light passes from a rarer into a denser medium-as, for instance, from air into water-it invariably undergoes a change of direction. Thus, let R O represent

a ray of light emanating from an object at R, and encountering at O a medium of greater density than that through which it previously passed. Let A B be the surface which separates the rarer from the denser medium, and let P p be a perpendicular to this surface. If the ray, after passing O, had continued to traverse a medium of the same density, as in the first instance, it would have proceeded to S in a line coincident with its original direction. In consequence, however, of its passage from a rarer into a denser medium, its course is bent at O, and the result is that it now takes up a new course, O T, nearer to P p than OS is. This phenomenon is termed Refraction.

The angle ROP, which the incident ray makes with a perpendicular to the surface bounding the two media, is termed the angle of incidence; the angle T O p, which the refracted ray makes with the same line P p, is termed the angle of refraction. It is obvious that in all cases of the passage of a ray of light from a rarer into a denser medium, the angle of refraction is less than the angle of incidence.

When a ray of light passes from a denser into a rarer medium, the effect produced upon

its course is the reverse of what occurs in the former instance. Thus, if we suppose the ray RO to pass from a denser into a rarer medium at O, its subsequent course O T

will be farther removed from the perpendicular P p than O S, the line which marks the direction of its original course. In this case the angle of refraction is manifestly greater than the angle of incidence.

The law of refraction, whether it relates to the passage of a ray of light from a denser into a rarer medium or vice versâ, is usually enunciated in the following terms: The sine of the angle of refraction is proportional to the sine of the angle of incidence.

The rays of light from a celestial object necessarily undergo refraction in passing through the atmospheric fluid which encompasses the earth. In this case, however, the medium through which the rays pass being composed of concentric strata of different densities, the course of the refracted ray will not be a straight line, but a curve. This will be understood better by referring to fig. 2, Plate 1. Until the ray of light enters the atmosphere its course is a straight line. Henceforward, however, it is bent more and more as it passes through each successive stratum of air; and the result is, that its course through the atmosphere assumes the form of a curve, which is concave towards the earth's surface. In the figure referred to, the reader will perceive a tangent drawn to the course of the ray at the point where it reaches the earth's surface. This represents the direction in which the celestial object is seen by an observer at the earth's surface, and is manifestly different from the line joining the object and the same point, which indicates the true direction of the object relatively to the observer. The effect, then, of refraction, is obviously to elevate a celestial object apparently above its true place.

The refraction of a celestial object gradually diminishes from the horizon, where it is greatest, to the zenith, where it vanishes altogether. For objects situate in the horizon, the refraction amounts to rather more than 34'; at 5° of altitude above the horizon, its mean value is 10'; at 20°, it is 2' 30"; at 45°, it is 57′′—and so on, as illustrated in fig. 2, Plate 1.

The refraction of a celestial object does not depend solely on the altitude of the object above the horizon. It is also to a certain extent affected by the temperature and pressure of the atmosphere, and by the quantity of aqueous vapour contained therein; and as these elements are continually fluctuating within small limits, the amount of refraction is also incessantly undergoing corresponding small variations.

There is always some degree of uncertainty with respect to the refraction of celestial objects, the apparent positions of which are very near the horizon. This arises from our not being sufficiently acquainted with the physical changes which are perpetually occurring in the lower regions of the atmosphere, and which especially affect the refraction of rays traversing horizontally the lower strata of air. It is consequently the practice of astronomers to confine their observations to objects the apparent positions of which are elevated to some extent above the horizon.

It is owing to refraction that we see the sun and moon on the horizon, when they are in reality depressed beneath it. In point of fact, the sun is visible in the morning about four minutes before it has ascended above the horizon, and is visible in the evening about four minutes after it has descended beneath it. The explanation of this is very simple. The refraction at the horizon amounts to at least 33', or rather more than the apparent diameter of either the sun or moon. Hence those bodies may be wholly visible above the horizon, from the effect of refraction, when they are in reality depressed beneath it; and as they occupy about four minutes in ascending above the horizon, and the same time in sinking beneath it, the duration of sunlight or moonlight is manifestly prolonged by refraction to the extent of about eight minutes.

is

Another remarkable phenomenon explicable by refraction is this: when the sun or moon upon the horizon, the disc exhibits an oval shape, the vertical diameter being sensibly smaller

than the horizontal diameter. This arises from the unequal effect of refraction upon the upper and lower extremities of the vertical diameter, combined with the fact of the horizontal diameter being altogether unaffected by the same cause. Let us consider the case of the sun. The mean value of the sun's apparent diameter may be said to amount to 32'. Again, the variation of refraction at the horizon amounts to about 4' for an elevation above the horizon to the extent of 32'. Hence it is clear that when the sun is upon the horizon the effect of refraction upon the lower extremity of the vertical diameter will exceed the effect upon the upper extremity to the extent of 4', and the result consequently is, that the vertical diameter will be reduced from 32' to 28'. On the other hand, the horizontal diameter being wholly unaffected by refraction, will retain its normal value of 32'. Hence originates the oval shape exhibited by the sun's disc when it is on the horizon. The same explanation is obviously applicable to the case of the

moon.

Aberration. The principle of the aberration of light may be thus illustrated: Suppose a shower of rain is falling in heavy drops during a dead calm. To a person standing still the drops will be seen descending in their true direction, which is perpendicular to the horizon. Let us, however, suppose the person to walk forward. In this case the drops will meet his face, and the direction in which they seem to descend will no longer be strictly vertical, but somewhat inclined to that direction, as represented in fig. 4, Plate 1. This apparent displacement of the falling direction of the drops of rain is an accurate illustration of the aberration of light. Another illustration, due to Mr Airy, the Astronomer-Royal, is this: Let us suppose a shot to be fired from a land-battery upon a vessel sailing quickly on the sea or on a river, and to pass through the sides of the vessel. A comparison of the place where the shot entered the vessel with the place where it quitted the vessel on the opposite side will not represent the true direction of the shot, but will rather indicate it as having come from a quarter somewhat ahead, the amount of displacement depending on the relative velocities of the shot and the vessel. If we suppose in this case the shot to represent a ray of light, and the vessel to represent the earth travelling in its annual orbit round the sun, the displacement referred to will constitute an exact representation of the apparent displacement produced in the position of a celestial object by the aberration of light (see fig. 5, Plate 1).

The aberration of a celestial object for any assigned instant of time depends on the corresponding position of the earth in its orbit; it is therefore perpetually varying throughout the year. Its general effect is to cause every star to describe apparently a small ellipse in the heavens round its true place. The major axis of this ellipse is invariable; it amounts to 40".8. The magnitude of the minor axis, on the other hand, depends on the angular distance of the star from the ecliptic. It is greatest for a star situate in the pole of the ecliptic, where it is equal to the major axis, and where, consequently, the ellipse becomes a circle. From this point it gradually diminishes towards the ecliptic, where it finally vanishes altogether, and the ellipse of aberration becomes a straight line, whose length is equal to the major axis of the ellipse, or, in other words, equal to 40".8.

Precession. This consists in a slow motion of the equinoctial points of the celestial sphere in a direction opposite to the apparent motion of the sun and planets. The result is a uniform increase in the longitudes of the heavenly bodies when observed from year to year, the latitudes remaining unchanged. The precession in longitude amounts to 50".1 annually, and is the same for all the celestial bodies. If the place of an object be represented by its right ascension and declination, the effect of precession will be to cause a slow but continual variation of both these elements, the amount in either case depending on the position of the object in the celestial sphere.