Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/838

on this assumption. It is clear that whatever the sun s daily retardation may be, he loses one complete circuit of the heavens in a year of 365| solar days. In other words, while the sun has been carried round 365^ times by the diurnal rotation, the star-sphere has been carried round

36 6^ times. Therefore, on our assumption

365 J solar days = 366^ sidereal days and a sidereal day = - f solar day af sim s motion. = 23 h. 56 HI. 4 s., approximately.

This, in fact, indicates roughly the manner in which the mean solar day is connected with the sidereal day. It is only necessary in the above process to substitute the true length of the year for the value 3G5J days, meaning by the year, the year of seasons, measured by the successive returns of the sun to the equator as he crosses that circle with northward motion. But it will not be until we con sider the actual nature of the motion by which the sun s annual apparent motion is explained, that we need inquire into the exact relation between the mean solar day and the sidereal day.

As the sun moves at a varying rate, it is manifest that the actual solar day measured by the successive returns of the sun to the meridian could not be constant in value, even if the sun moved round the equator. For the excess of a solar day over the sidereal day is caused by the motion of the sun on the star-sphere, and will be therefore greater or less according as the sun s motion on the star-sphere is greater or less. The actual solar day, therefore, exceeding the constant sidereal day by a variable quantity, must necessarily be itself variable. It is greater than the mean in December and January, when the sun is moving at a rate greater than his mean rate, and less in June and July when he moves at a less rate. And it is clear that if at the end of December the moment of the real sun passing the meridian were taken as the beginning of the mean solar day of twenty-four hours, then the next passage of the meridian by the actual sun would occur after the twenty-four hours of mean solar time had elapsed. Day after day the sun would come to the meridian at a later and later hour of mean solar time, until towards the end of March, when, the sun s rate having acquired its mean value, the actual sun would not lag any farther behind. From this time lie would gain, until towards the end of June he Avould come to the meridian at noon of mean solar time. In the remaining half year he would be in advance, that is, he would cross the meridian before noon of mean solar time. Towards the end of September he would have made his greatest advance compared with mean time, and in the remaining quarter of the year he would gradually lose more and more of that gain, until at the end of December he would again cross the meridian at noon of mean solar time.

But besides this cause of variation in the length of the true solar day, there is another depending on the inclination of the sun s apparent path on the heavens to the celestial equator. To conceive the effect of this cause, it is neces sary to have regard to the motion of the sun with reference to the equator. The sun describes every day a small arc of the ecliptic. Through the extremities of this arc sup pose two meridians to pass ; the arc of the equator, which they intercept, is the sun s motion for that day referred to the equator, and the time which that arc takes to pass the meridian is equal to the excess of the astronomical day over the sidereal. But it is obvious that at the equinoxes the arc of the equator is smaller than the corresponding arc^of the ecliptic, in the proportion of the cosine of the obliquity of the ecliptic ; at the solstices, on the contrary, it is greater in the proportion of the secant of the same obliquity. The astronomical day is diminished in the first case, and lengthened in the second.

To have a mean astronomical day independent of these Mca causes of inequality, astronomers have supposed a second sun to move uniformly on the ecliptic, and to pass over the extremities of the axis of the sun s orbit at the same in stant as the real sun. This removes the inequality arising from the inequality of the sun s motion. To remove the inequality arising from the obliquity of the ecliptic, con ceive a third sun to pass through the equinoxes at the same instant with the second sun, and to move along the equator in such a manner that the angular distances of the two suns at the vernal equinox shall be always equal. The interval between two consecutive returns of this third sun to the meridian forms the mean astronomical day. Mean time is measured by the number of the returns of this third sun to the meridian ; and true time is measured by the number of returns of the real sun to the meridian. The arc of the equator, intercepted between two meridian Equ circles drawn through the centres of the true sun and the of ^ imaginary third sun, when reduced to time, is what is called the Equation of Time. This will be rendered plainer by the following diagram.

Fig. 13.—Motions of real and mean Sun.

Let Z T z (fig. 13) be the star-sphere; Zz its axis; abcde, &c., the equator ; ABODE, tc., the northern half of the ecliptic from T to , on the side of the globe next the eye ; and MNOP, &c., the southern half on the opposite side from toT. Let the points at A, B, C, D, E, F, &c., mark off equal portions of the ecliptic gone through in equal times by the real sun, and those at a, b, c, d, e, f, &c., equal portions of the equator described in equal times by the fictitious sun ; and let Z T z be the meridian.

As the real sun moves obliquely in the ecliptic, and the fictitious sun directly in the equator, any point between _HP and F on the ecliptic must be nearer the meridian Z T z, than the corresponding point on the equator from T to /, that is to say, than the point whose distance from T is expressed by the same number of degree* ; and the more so, as the obliquity is greater; and therefore the true sun comes sooner to the meridian every day whilst he is in the quadrant TF, than the fictitious sun does in the quadrant T/; for which reason the solar noon precedes noon by the clock, until the real sun comes to F, and the fictitious to /: which two points being equidistant from the meri dian, both suns will come to it precisely at noon by the clock.

Whilst the real sun describes the second quadrant of the

ecliptic FGHIKL from Cancer to , he comes later to the meridian every day than the fictitious sun moving through

the second quadrant of the equator from / to ; for tha