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cury at the time of observation. This planet is found to vary its greatest elongation from the sun very considerably, the angle syv varying from 28°48' to 16° 12′; whence we conclude that its orbit is very elliptical. The planet Venus, on the contrary, shews but a slight deviation from a circular path; her angle of greatest elongation ranges between 47° 48′ and 44° 57'.

The distance of a superior planet from the sun may be found by measuring the arc through which it appears to retrograde, when in opposition to the sun. Let x in fig. 16 be the place of Mars in opposition, and y that of the earth; y y the arc described by the earth in a short period of time-say one day; x ẞ the arc described by Mars in the same time; the periodic times being known, these arcs may be found simply by dividing 360° by the number of days occupied in one revolution. * We have before stated that the nearer to the sun a planet is, the more rapid is its motion round him: thus, the Earth will pass by Mars, which planet will appear to retrograde through the arc Y d, whereas in fact he has been moving through Y 0; that is to say, at x the geocentric and heliocentric longitudes of Mars will be the same; the arc 08 will be the difference between the geocentric and heliocentric longitudes of Mars when at the former will be found from observation, the latter from tables of the planet's motion. Now, in the triangle sy ẞ, the angle sẞy=0 ẞd, the difference between the heliocentric and geocentric places of Mars; the angle ẞs y, which is the difference between the angular advance of the Earth and Mars between the two

*This will only give the mean motion: a correction must be introduced, which is here neglected, to prevent confusion.

observations, and the side s y are given; whence may be found by trigonometry s C, which is the distance of Mars from the sun.

Or, the distance of one planet being known, we may find the distance of another, by the application of the third law of Kepler, as shewn in § 203.

SECTION VII.

ON TIME.

REGULARITY OF THE EARTH'S MOTION-SIDEREAL DAY-SOLAR DAY -APPARENT AND MEAN TIME-METHOD OF DETERMINING MEAN SOLAR TIME FROM SIDEREAL, AND THE CONTRARY-TO FIND THE MEAN TIME OF TRANSIT OF A FIXED STAR-TROPICAL, SIDEREAL, AND ANOMALISTIC YEAR.

270. ONE of the most important elements in astronomical observation is an exact measurement of time. To obtain this, some standard, not depending on mere sensation, is evidently requisite. Happily, in nature there are certain motions which are ever invariable; and the time taken to accomplish these may be so divided and subdivided as to assist the astronomer in his observations, as well as to answer all the purposes of common life.

Now, it has been found by repeated and continual observation, that the exact time which elapses between two successive arrivals of a star at the meridian of any place, which space of time constitutes a sidereal day, is ever and unchangeably the same; and that not only is the duration of a sidereal day, that is, the time the earth

takes to turn on its axis, constant, but that, during every part of the earth's rotation, its motion is equain other words, the earth revolves with a uniform velocity.

ble;

271. The difference between the solar and sidereal day has already been alluded to (§ 233). Suppose the sun and a star to culminate at the same instant on a particular day; on the following day, the sun, from the earth's advance in her orbit, will have passed to the eastward of the star; and therefore the earth must make an additional portion of a daily revolution before the meridian which has arrived opposite the star will be opposite the sun; which portion of a diurnal arc will always be equal to the number of degrees and minutes the sun is in advance of the star. Now the sun describes a complete circle of 360° in the year; in that time, then, the earth will have to perform a diurnal arc of 360°, which will occupy 24 hours of sidereal time, before the meridian of any place, having left the star, will overtake the sun; but this complete revolution will bring the sun and the star on the meridian at the same instant. We see, then, that the 365 solar days, which constitute the year, will contain 366 sidereal days. The length of a sidereal day is uniformly 23 hrs. 56 min. 4.092 sec. of solar reckoning; and the difference between it and a mean solar day of 24 hrs. is 3 min. 55.908 sec., or 3 min. 56 sec. nearly.

272. We say a MEAN SOLAR DAY, because the days reckoned by two successive appulses of the sun to the meridian will be found to be of unequal duration; indeed, on comparing them with a well-regulated clock, it is found that no two days in the year are of the same

very nearly agree.

length, but that the corresponding days of every year This variation arises from two causes, the first of which is, the unequable motion of the earth in its orbit.

It was shewn in § 179, that the orbit of the earth is not circular, but elliptical; and in § 184 that motion in an elliptical orbit is not uniform, but performed more slowly about the aphelion, increasing in velocity till the earth arrives at its perihelion. Now, if the orbit of the earth were a circle, whose plane coincided with that of the equator, the earth's motion would be equable, and then the difference between a solar and sidereal day would ever be the same; it would in fact be equal in arc to 360° divided by the number of days in the year, or 0° 59' 8"-33, and in time to 24 hours divided by the same number of days. But near its aphelion the earth's daily arc is no more than 57' 12"; the meridian of a place will then turn forward through less than a mean arc after leaving a star (as we have supposed in § 271) before it arrives opposite to the sun; or, in other words, before the sun culminates: apparent noon will therefore occur before mean noon. Near her perihelion the earth will describe a daily arc of 1° 1' 9". Here, then, apparent noon will be behind the mean; for the earth must describe an arc greater than the mean diurnal arc before the sun will be on the meridian. When the earth is exactly in her aphelion or perihelion, and when, also, she is at two other points between these, mean and apparent time will be the same, as far as this cause is concerned.

273. The other cause of the want of uniformity in the length of the solar days arises from the circumstance

of the plane of the equator not coinciding with the plane of the ecliptic, so that the sun's apparent motion in longitude is not equal to his motion in right ascension. The celestial globe will assist us in rendering this cause apparent. Put spots of colour at every 30° on the ecliptic, and others at every 30° on the equinoctial, beginning at the first of Aries. Those on the ecliptic will represent twelve positions of the true sun during the year; those on the equator the corresponding positions of a fictitious sun, whose orbit we will suppose to coincide with that circle. Turn the globe from east to west, and you will observe that from Aries to Cancer the real sun will arrive at the meridian before the fictitious: at Cancer the two will be on the meridian together; from Cancer to Libra the equatorial sun will anticipate the real, i. e. mean time will be before the apparent. From Libra to Capricorn the apparent time will be before, and from Capricorn to Aries after the mean time; the two coinciding, however, at the first point of each of the signs mentioned, i. e. four times a year.

The solar days then, it will appear, are subject to an inequality from two causes-the time between two successive apparent noons being sometimes greater and sometimes less than a day of mean duration. Now it is found that if the deficiency and excess all the year round be registered and compared, they will neutralize each other, and the result will be a mean solar day of 24 hours' duration. To this day clocks and watches are set; and, supposing their rate to be constant, we should find that they agree with apparent time at only four periods in the year; namely, on or about April 15, June 15, September 1, and December 24.

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