by 206264".8, the seconds in the radius of a circle.* Thus, with the value of E 1° 55'.3 (160), we find for e the same value as above.

162. Secular variations of the equation of the sun's centre and of the eccentricity of his orbit. The greatest equation of the sun's centre, and consequently the eccentricity of his orbit, as determined. at periods distant from one another, are found to be subject to a slow, but continued diminution. The secular diminution of the greatest equation is 18′′.

Investigations in physical astronomy prove that the variation of the eccentricity, from which that of the equation of the centre results, is produced by the actions of the planets on the earth.

163. Kepler's Problem. When the eccentricity of the orbit and the mean anomaly of a body are given, the equation of the centre and true anomaly may be found. This problem, which was proposed and solved by Kepler, is one of some difficulty. Various solutions of it have, however, been since obtained; one of which is given in the appendix.

By formulæ obtained from the solution of the problem, tables have been computed for the sun, moon, and planets, which give the value of the equation of the centre, corresponding to any given mean anomaly.

The equation of the sun's centre for any given time, obtained from its table, and applied to his mean longitude at that time, gives his true longitude from the mean equinox, with the exception of some small corrections to be noticed in the next article.

164. Perturbations. The actions of the moon and planets cause the earth to deviate slightly from an elliptical orbit, and produce small periodic inequalities in its motion. These inequalities are called perturbations. The bodies which produce sensible perturba

* Putting k = the quotient of the value of E, in seconds, divided by 206264′′.8, the following formulæ have been obtained:

tions in the motion of the earth, or apparent motion of the sun, are, the moon, Venus, Mars, Jupiter, and Saturn. The whole amount of these perturbations, when greatest, is about 37".

165. Sun's latitude. As the moon and planets are continually varying their positions with regard to the plane of the ecliptic, the effect of their actions in drawing the earth from this plane or changing its position (137), must also be continually varying. If, therefore, the plane of the ecliptic was regarded as always passing exactly through the centre of the earth, its progressive change of position would be subject to small periodic inequalities. Astronomers, however, find it more convenient to assume the change of position to be regular; and, consequently, to regard the earth's centre as deviating slightly from the plane of the ecliptic, sometimes on one side and sometimes on the other. Hence, as this plane passes through the sun's centre, when the centre of the earth is on one side of it, the centre of the sun must appear to be at an equal distance on the other side, and must have a small latitude. The greatest value of this latitude is only about one second. It may, therefore, be neglected, except in very accurate investigations and computations.*

166. Solar Tables. These are tables for computing the sun's longitude, latitude, radius vector, apparent semidiameter, and the apparent obliquity of the ecliptic at any given time. The best solar tables are those by Carlini, an Italian astronomer, published in the Milan Ephemeris for the year 1833.†

But

* The attractions of the different planets, depending on their masses and distances from the earth, are very different, and some of them extremely small. a very small effect, if produced for a long time in the same direction, so as to accumulate, may at length become sensible. Investigations in physical astronomy show, that the attractions of all planets, except the asteroids, are sensibly operative in producing the progressive change in the position of the ecliptic. But with regard to the periodic inequalities in the effect produced, the case is different. Those inequalities in the attractions, which cause the earth's centre to deviate from the plane of the ecliptic and thus produce the sun's latitude, are only sensible for the moon, Venus, and Jupiter.

A new set of Solar Tables has been computed by MM. Hansen and Olufsen, and just published by the Royal Society of Copenhagen. These tables have the advantage over Carlini's, of an additional twenty years' observations for their basis. They also surpass them in fulness and in convenience of arrangement, and will, doubtless, soon become the standard solar tables.

When the sun's longitude, and the apparent obliquity of the ecliptic, have been computed for a given time, his right ascension and declination may be found by spherical trigonometry (119). The right ascension may also easily be found by a table for the purpose, called a table of reduction of the ecliptic to the equator. A convenient one is given in Carlini's solar tables.

167. Astronomical day. The astronomical day commences at noon of the common day, and the numbering of the hours is continued to 24. The first 12 hours are the same as the afternoon hours of the common day; but the hours from 12 to 24 of the astronomical day correspond to the 12 hours before noon of the next following common day.

Hence, when a time in common reckoning is before noon, we must, in order to convert it into astronomical time, subtract a unit from the number of the day and add 12 to the number of the hour. Thus, May 12th, at 5 h. A. M., common reckoning, is the same as May 11th, at 17 h., in astronomical reckoning.

168. Apparent time.

Time reckoned by the position of the sun's centre with reference to the meridian of a place, is called apparent time. And the instant at which the centre is on the meridian is apparent noon.

The length of the apparent day is variable, and from two causes. In the first place, the sun's motion in the ecliptic is not uniform, being affected by the equation of the centre and the planetary perturbations. In the second, as equal arcs of the equator pass the meridian in equal times, the arcs of the ecliptic which pass it in equal times must, in consequence of its obliquity to the equator, be unequal; so that, if the sun's motion in the ecliptic were uniform, the intervals between his consecutive returns to the meridian would not be equal.

As the length of the apparent day is not uniform, clocks or chronometers could not be made to exhibit apparent time without continually repeated adjustments.

169. Mean time. If we assume an imaginary sun to be at the reduced place of the mean vernal equinox (140), at the same instant that the real sun is at the apparent equinox, and to move from thence in the equator with a uniform motion, equal to the real

H

sun's mean motion in longitude, the time measured by the position of this imaginary sun, with reference to the meridian, is called mean time. And the instant at which the centre is at the meridian is mean noon.

The mean day, according to this definition, is of uniform length, and is evidently the same as that defined in a preceding article (143). Mean time, therefore, flowing uniformly, is that to which clocks are adjusted for the common purposes of society, and also for many astronomical purposes. Observatories are usually furnished with at least two clocks; one of which is adjusted to sidereal time, and the other to mean solar time.

170. Equation of time. The difference, at any instant, between apparent and mean time, is called the equation of time. It depends on the unequal motion of the sun in the ecliptic and the obliquity of the ecliptic to the equator (168).

171. The equation of time is equal to the difference between the sun's true right ascension and the sum of his mean longitude and the equation of the equinoxes in right ascension, converted into time.

Let EQ, Fig. 27, be an arc of the equator, P its pole, PZM an arc of the meridian of a place, of which Z is the zenith, EC an arc of the ecliptic, E the apparent or true equinox, E' the mean equinox, S the true place of the sun in the ecliptic, and PE'G and PSH arcs of declination circles. Then G is the reduced place of the mean equinox, EG the equation of the equinoxes in right ascension, EH the sun's true right ascension, and the angle HPM, of which HM is the measure, is the hour angle for apparent time.

Let GS' be equal to the sun's mean longitude. Then S' is the place of the imaginary sun, assumed to move uniformly in the equator (169), ES' = GS' + EG, is the sum of the sun's mean longitude and the equation of the equinoxes in right ascension, and the angle S'PM, of which S'M is the measure, is the hour angle for mean time.

Put T = the apparent time and T' the mean time. Then since each 15° of the hour angle corresponds to an hour, we have in hours or parts of an hour,

Consequently, by subtraction, we have for the equation of time,

From these we have, also, for the expression of either time in terms of the other and the equation of time,

The equation of time is given in the Nautical Almanac for every day in the year, with instructions whether to add or subtract in changing one time to the other.

172. Times at which the equation of time is nothing. When the effects of the two causes on which the equation of time depends (170), are opposed to each other and are equal, the equation of time must be nothing; apparent and mean time must then be the This occurs four times in the year; about the 15th of April, 15th of June, 1st of September, and 24th of December.

same.

1

173. Sidereal time. The arc EM of the equator converted into time, expresses the sidereal time (142). But EM = EH + HM. Hence, the sidereal time is obtained by adding the apparent time to the sun's true right ascension, expressed in time.

Again, EM = GS' + EG + S'M. Consequently, the sidereal time is also obtained, by adding the mean time to the sum of the sun's mean longitude and the equation of the equinoxes in right ascension, expressed in time.

As EM is the right ascension of the zenith Z, it follows that the sidereal time expresses the right ascension of the zenith in time.

174. Solar Spots. When the sun is viewed with a telescope furnished with a coloured glass to protect the eye, a number of dark spots are usually seen on his surface. Each spot generally consists of a central part of irregular form, which is black, surrounded by a margin or border, called the penumbra, of much lighter colour, as represented in Fig. 28. The spots differ greatly