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mean equinox, is the place at which the equinox would be if there was no nutation. The same applies to the mean position of the equator, called the mean equator; and to the mean obliquity of the ecliptic. The mean right ascension of a body is the right ascension of the mean place of the body, reckoned from the mean equinox along the mean equator; and similarly for mean declination, longitude, or latitude.

The actual place of the vernal equinox at any time is called the apparent or true equinox; the actual obliquity of the ecliptic is called the apparent obliquity; and the equator in its actual position is called the apparent or true equator. The apparent or true right ascension of a body, is the right ascension of the apparent or true place of a body, reckoned from the apparent equinox along the apparent equator; and similarly for apparent or true declination, longitude, or latitude.

The point in which the arc of a declination circle, passing through the mean equinox, meets the apparent equator, is the reduced place of the mean equinox. The small distance between this point and the apparent equinox, is evidently the nutation of the equinox in right ascension, or the equation of the equinoxes in right ascension.

141. Tables of reduction. The exact apparent positions of the fixed stars, are so continually wanted by the practical astronomer in adjusting and examining the adjustments of his instruments, and as points of reference, that much attention has been devoted, to obtain concise and accurate methods of deducing these from the mean places given in catalogues. In the catalogue of 8377 principal fixed stars, published a few years since under the direction of the British Association for the Advancement of Science, besides the mean places, annual precessions, and secular variations, there are given certain constant logarithms for each star, by means of which, with others given, in the Nautical Almanac, for each day in the year, the apparent places of these stars may be found for any given time, with great facility.

Professor Bessel, in his Tabula Regiomontanæ, has given general formulæ and tables for reducing the mean places of the stars to apparent places.

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142. Sidereal Time. The sidereal day, as now used by astronomers, commences at the instant the apparent vernal equinox is on the meridian, and is reckoned through 24 hours to the return of the equinox to the meridian.* Consequently the sidereal time at any instant expresses the apparent right ascension of the meridian at that instant, or of any body that is then on the meridian. Thus if ε, Fig. 1, be the position of the apparent vernal equinox at any instant, the arc Q, which expresses the sidereal time at the place A at that instant, expresses also the apparent right ascension of any body that is then on the meridian PZP'.


The sidereal clock is adjusted, or its error and rate determined, by observations of the passages over the meridian of certain fixed stars, whose apparent right ascensions are known, or may be computed with great precision. The apparent right ascension of any other body, when on the meridian, may then be found by observing the time of its passage as shown by the clock, and correcting this time for the error of the clock.

143. Solar Time. The interval between two consecutive returns of the sun's centre to the meridian, is called a solar day; and time reckoned by solar days is called solar time. The length of the solar day is found to be somewhat different at different seasons of the year; its mean or average length is called a mean solar day.

If on any day the sun is on the meridian of a place, at the same


* The equinox, having a precession in right ascension or westwardly motion of 46′′ in a year (127) or of a second in a day, must return to the meridian sooner than a fixed star by To of a second in time. The sidereal day, as here defined, is therefore shorter than as defined in Art. 53, by д of a second. In consequence of the nutation of the equinoxes, it is not strictly uniform; but the deviation is extremely small.


instant with some fixed star, he will, in consequence of his apparent eastwardly motion (7), be to the east when the star returns to the meridian next day, and will not arrive at it till some minutes later than the star. Consequently, the solar day is longer than the sidereal day. The mean solar day is found to be equal to 24h. 3m. 56.555sec. of sidereal time.

144. Tropical Year. The interval between two consecutive returns of the sun to the vernal equinox is called a tropical year.

145. Length of the tropical year. From the sun's declination, and the sidereal time when he is on the meridian, obtained for a number of consecutive days about the 21st of March, the time when the declination is nothing: that is, the time when he is at the equinox, may be accurately determined. If this be done in successive years, the length of the year, in sidereal time, becomes known.

The length of the year, determined at different periods, is found to be subject to a slight variation. Its mean length at the present period, expressed in mean solar time, is 365d. 5h. 48m. 48sec.

146. Sidereal year. The time during which the sun, by his apparent motion, makes an entire revolution in the ecliptic, is called a sidereal year.

147. Length of the sidereal year. In consequence of the retrograde motion of the equinoxes (124), the arc of the ecliptic which the sun passes through during a tropical year, is less than 360° by 50".2. Hence, as 360°-50".2: 360°:: length of the tropical year: length of the sidereal year. The length of the sidereal year is thus found to be 365d. 6h. 9m. 10sec. It is therefore 20m. 22sec. longer than the tropical year.

148. Sun's apparent orbit. The path described by the sun's centre in the plane of the ecliptic, during an apparent revolution round the earth, is called the sun's apparent orbit or the solar orbit.

149. The Solar Orbit is an ellipse, having the earth in one focus. Let PSAB, Fig. 24, represent the sun's apparent orbit, E, the place of the earth, P, the sun's place in his orbit when his apparent diameter is greatest, A, his place when it is least, and S, his place at some intermediate time.

The obliquity of the ecliptic being known, the sun's longitude, at any time, becomes known from his observed right ascension (119). Hence, from a series of observations of the sun's right ascension and corresponding apparent diameter, continued throughout the year, we may obtain a series of corresponding longitudes and apparent diameters. From these, the two longitudes, corresponding to the greatest and least apparent diameters at P and A, may be easily found. These longitudes, thus determined, are found to differ 180°. Hence EA and EP are in the same straight line. It is also found, from a general examination of the corresponding longitudes and apparent diameters, that the apparent diameter at any place S, is equal to half the sum of the greatest and least diameters, less the product of half their difference by the cosine of the angle AES, which is the difference of the sun's longitudes at A and S.

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Then, from the known relation between the angle AES and the apparent diameters, we have

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Substituting the values of m, n, and ♪" in (A), and dividing by

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or, ES (AC – EC cos AES) = AE.EP = AE.EP = (AC + EC) (AC-EC)

or, ES





This is the polar equation of an ellipse, of which AP is the transverse axis, C the centre, and E a focus.

150. Earth's orbit. Let S, Fig. 25, be the sun, and PEAE' the earth's orbit, or path described by its centre in the plane of the ecliptic, during a revolution round the sun. Then substituting E for S, and the contrary, the demonstration in the last article proves that the earth's orbit is an ellipse, having the sun in one focus.

The discovery that the sun's apparent orbit, or the earth's real orbit, is an ellipse, was made in the early part of the 17th century by Kepler, a celebrated German astronomer. He first ascertained that the orbit of the planet Mars was an ellipse, and, pursuing his investigations, he found that the orbits of the earth and other planets were also ellipses. This, being one of several important discoveries made by him relative to the planetary motions, is called Kepler's first law.

151. Definitions. A Radius Vector is a straight line joining the centres of the sun and a planet, or the centres of a planet and satellite.

Perihelion, &c. In the orbit of the earth, or a planet, the point nearest the sun is called the perihelion, and that which is most distant, the aphelion. In the moon's orbit, or sun's apparent orbit, the point nearest the earth is called the perigee, and the most distant point, the apogee. These points have also the general appellation of apsides: the nearest point being called the lower apsis, and the most distant, the higher apsis. The transverse axis of the ellipse, or the line joining the apsides, is called the line of the apsides.

The Eccentricity of an elliptical orbit, as the term is generally used in astronomy, is the distance between the centre and a focus, expressed in terms of the semi-transverse axis regarded as a unit; or, which amounts to the same, it is the quotient of the distance between the centre and focus, divided by the semi-transverse axis.

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