23d of September, when he enters Libra; and on the 21st of December, when he enters Capricorn.'

This part of the equation of time may be made still more familiar by means of a globe; for if a small black patch be put upon every tenth or fifteenth degree, both of the equator and ecliptic, beginning at the point Aries, and the globe be turned round slowly to the westward, all the patches from Aries to Cancer, and from Libra to Capricorn, will come to the meridian sooner than their corresponding patches on the equator; and all those from Cancer to Libra, and from Capricorn to Aries, will come to the meridian later than their corresponding patches on the equator: while the patches at the beginning of Aries, Cancer, Libra, and Capricorn, being on or even with those of the equator, show that the Sun and star will either meet there, or are even with each other, and for that reason must come to the meridian at the same time.'

The following table exhibits the relation of solar and mean time at certain intervals, as resulting from this cause, and they relate to the period when the sun-dial is faster than the clock; that is, the difference of the arrival of the Sun and the star, as supposed above, at the same meridian, will be on

The other principal cause of the difference between solar and mean time, as already stated, is the unequal motion of the Earth in its orbit, arising from the unequal action of the Sun upon it at different times of the year. The apparent motion of the Sun is slowest in summer, when he is farthest from the Earth, and

swiftest in winter, when he is nearest to it. This is shown by his being longer by about eight days on the northern than in the southern half of the ecliptic; from which circumstance alone the motion of the Sun could not be a true measure of time. This motion sometimes exceeds a degree in 24 h., and at others is less than that quantity; and consequently when it is slowest, any particular meridian will come sooner to the Sun than when his motion is quickest; so that, if there were no other cause of difference, the days cannot be equal to each other.

If two bodies, therefore, were to move in the plane of the ecliptic, so as to go exactly round the Earth in a year; the one describing an equal arc every 24 h., and the other describing sometimes a greater and at others a less arc in the same period, gaining in one part of the year what it lost in another; one of these bodies would obviously come sooner or later to the meridian than the other, according to their respective situations; and when they were both in conjunction, they would come to the meridian at the same instant.

To illustrate this conception more fully, let it be supposed that the Sun and a star commence their annual motions together from that part of the ecliptic in which the Sun is at the greatest distance from the Earth, the former moving with a variable and the latter with a constant velocity, but both completing the whole orbit in exactly the same time, that is, in the space of a year. As the Sun is supposed to be in apogee when the motion commences, his motion is the slowest; and as that of the star is always uniform, it is evident that the star will describe a larger arc in the same time than the Sun will; and consequently, as the Earth turns on its axis from west to east, it will be noon by the Sun sooner than by the star. But as the Sun moves from his apogee where his distance is greatest, to his perigee where it is

least, the velocity of his motion will continually increase till he arrives at his least distance from the Earth, when it will have attained its maximum. But notwithstanding this increase, the star gains so much upon the Sun in the early part of their progress, that the increased velocity of the latter will not be able to compensate for this loss till the star arrives at the perigee, or till each of them has gone just half round the ecliptic; and as they will then be in conjunction, they will be upon the same meridian together; and consequently it will be noon by both of them at the same instant. From this point, where the velocity of the Sun is greatest, it will carry him before the star, in consequence of which, the plane of the same meridian will arrive at the star sooner than it will reach the Sun; and therefore it will be noon by the star before it is noon by the Sun. The motion of the Sun diminishes, in this case, till he arrives again at his greatest distance from the Earth, at which time the two bodies will be once more in conjunction, and it will then be noon by both the Sun and star at the same time. From this it appears that the solar noon is always later than the mean noon, when the Sun is between his perigee and apogee; and before it, during his passage from his apogee to his perigee. So that solar and mean time, as affected by this cause only, would agree on two days in the year; but the difference, which results from the obliquity of the ecliptic only, vanishes four times in the course of the year, as already stated. It is not, however, on any of these days that the equation of time arising from both these causes combined becomes nothing, but at those epochs when the increase from the one just compensates for the diminution from the other.

That part of the equation of time arising from the unequal motion of the Earth in its orbit, during the part of his annual revolution when the dial is slower than the clock, is contained in the following table.

By a comparison of this and the preceding table, it appears that, from the obliquity of the ecliptic, the equation on the 21st of March is nothing, and from the inequality of the apparent solar motion, it is 7 m, 39 s. at the same time; the whole equation is therefore 7 m. 39 s. On the 5th of April, the equation is 4 m. 46 s., as resulting from the first cause, and 7 m. 38 s. from the second; and since the dial is faster than the clock in the one case, and slower in the other, the difference, 7 m. 38 s. —4 m. 46 s. = 2 m. 52 s., is the whole equation resulting from a combination of both causes. By proceeding in this manner throughout the whole, viz. subtracting when the dial is slower than the clock, and adding when it is faster, we shall obtain the equation for any given time of the whole revolution. That which results from a comparison of the two preceding tables is as follows, embracing the whole equation from both these causes.

In the calculation of these numbers, the small effects of nutation have not been taken into the account; but these are so small, as to make very little difference in the results.

The value of the equation of time is also readily

found from the difference of right ascension of the real Sun and of the fictitious one, which we have supposed to proceed along the equator with a uniform motion. The longitude of the mean Sun may be found by means of astronomical tables, and referred to the equator, and then it will be right ascension. And if the nutation of the Earth's axis be neglected, the true and mean poles will coincide, and the true right ascension of the Sun will be measured on the same equator as that of the mean Sun. These two right ascensions will, in general, differ from each other, since the right ascension of the real Sun increases unequally, while the increase in that of the supposed Sun is uniform. This difference reduced into time at the rate of 15 degrees to an hour, will give the time which the real Sun either follows or precedes the supposed one, at the moment for which the calculation is made.

The correctness of these remarks is easily shown, when the effect of nutation is not taken into the account. For, if the right ascension of the real Sun be denoted by A, that of the fictitious Sun, which is supposed to have a uniform and mean motion, by M, and the right ascension of the zenith by a; it is evident

that

α A

аM

will be the apparent time, and 15

15

the mean time. The difference between mean and apparent time will therefore be

since a vanishes by subtraction, and there remains only the difference of the right ascensions of the two Suns. When the effect of nutation is taken into the account, it will introduce a slight modification into this result, but this will never amount to more than a few seconds.

[To be continued.]