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December 1st, at 53 m. after 5 in the morning

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The planets Mercury and Georgium Sidus will be in conjunction on the 5th, at which time the latter planet will be 76' north of the former. The Georgium Sidus will also be in conjunction at 9 in the morning of the 8th; and Mars will be in opposition at past 9 in the evening of the same day. Venus and the star marked ẞ in Scorpio will be in conjunction on the 14th, when the star will be 6' south of the planet. Jupiter will also be in conjunction at 4 in the morning of that day; and the Moon and Mars will be in conjunction at 37 m. after 1 in the morning of the 22d.

The eclipses of Jupiter's satellites are not visible this month, on account of the planet being too near the Sun.

ON THE CALCULATION OF Eclipses.

[Concluded from p. 331.]

Method of Calculating the Circumstances of Solar Eclipses.

THE first thing which presents itself in our inquiries relative to eclipses of the Sun is, whether there will be an eclipse on any part of the Earth's surface; then, what will be its extent and duration, with the epochs of its beginning, middle, and end. By considering eclipses of the Sun as eclipses of the Earth, to an observer situated in the Moon, these circumstances will be easily determined. In that case it is required to ascertain the apparent distance of the

centre of the terrestrial disc from the centre of the lunar shadow for any given instant; and, from the known diameters of the Earth and the shadow, to calculate the times at which they will appear to penetrate each other. This research is, therefore, altogether similar to that we have employed in calculating eclipses of the Moon as seen from the Earth, and the same method will conduct us to similar results; the only difference is in considering the centre of the Moon as the centre of the visual rays.

Let E (fig. 12), therefore, be the centre of the Earth, M that of the Moon, and S of the Sun, at any instant; and consider the triangle ESM as formed by lines joining these three points. If the side SM be produced, it will be the axis of the lunar shadow; and the angle OME, formed by this prolongation and the visual ray EM, drawn from the Moon to the Earth, will evidently be the apparent distance of the centres of the Earth and shadow; which it is required to determine.

M

Fig. 12

Now, in the triangle SME, there are known the two sides SE and ME, which are the respective distances of the Earth from the Sun and Moon, and the angle at E, which is the apparent distance of these two heavenly bodies as seen from the Earth; and which is expressed by the difference of their latitudes and longitudes, in the same manner as for eclipses of the Moon in the preceding article. We shall, therefore, have for any epoch whatever an expression for the distance of the centre of the shadow from that of the terrestrial disc, as seen from the

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Moon; and by putting this expression equal to the different values which correspond to the different phases of the eclipse, and taking the time for the unknown quantity, we may determine the precise epochs of their occurrence.

In the preceding reasoning no regard has been paid to the diurnal revolution of the heavens, or, more properly, the diurnal rotation of the Earth upon its axis. When we consider the Earth in general, this revolution has only the effect already noticed in eclipses of the Moon; but when any determinate physical point on its surface becomes one of the data under consideration, this rotation may not only have an effect upon the possibility of the eclipse being visible, by bringing the observer into the enlightened hemisphere, or removing him from it, but also by altering his distance from the centre of the Moon, and changing her parallax, it may cause a solar eclipse to be visible, which would otherwise not be seen, or occasion one to be seen that would otherwise be invisible.

But the difficulties which the consideration of a given point on the surface introduces into the problem, are easily overcome by reasoning relatively to it in the same manner as for the centre; only employing the given quantities suitable to the new circumstances; that is, the apparent elements of the two bodies, as seen from the assigned point, instead of the true elements answering to the centre of the Earth. And as the two kinds of quantities differ only in the value of the parallaxes of longitude, of latitude, of declination, and right ascension, when the method of calculating the time of the different phases of an eclipse for the centre of the Earth is known, the corrections necessary for bringing it to the surface will be readily found.

If we examine, in a general manner, the motion of the lunar shadow across the terrestrial disc, we shall perceive that its progress, relative to us, ought

to be from west to east, in the proper direction of the Moon's motion: since the angular motion of this body about the Earth is much greater than that which apparently belongs to the Sun, its shadow ought to follow the same direction. Thus, an observer in the Moon, regarding the Earth as eclipsed, would see the eclipse commence on the western part of the disc, and end on the eastern. This is the only part of the phenomena that is constant; for the extent of the part eclipsed, and its position on the solar disc, are subject to great variation.

In the preceding figure, let the angles ESO and EMO be denoted by S and M respectively. The exterior angle EMO of the triangle ESM is evidently the apparent distance of the centre of the Earth from the axis of the lunar shadow; which denote by d, as in the former case for eclipses of the Moon. Then we shall have dE+ S, and consequently Sd E.

From the point E, draw EO perpendicular to MO, the axis of the lunar shadow; then, from the triangle ESO, we have EOES. sin S, or D'. sin S; where D' is the distance of the Sun from the Earth. But in the triangle EMO, we have EO = EM. sin d, or D. sin d, where DEM, the distance of the Moon from the Earth. And by putting these two values of EO equal to each other, we obtain, D'. sin SD sin d, or D'. sin (d— E) = D. sin C.

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of the Sun, and p that of the Moon, and we then have

sin p. sin (d-E) = sin p'. sin d.

This equation is both general and accurate; and when employed at the time of an eclipse, the angle SEM is always, very small as it is, the apparent distance of the centres of the Sun and Moon, when their discs are at least partly interposed before each other; this distance may therefore be considered as the hypothenuse of a right angled triangle, as has been done in treating of eclipses of the Moon. In this case, then, the sides will be the difference of latitude and longitude of the two heavenly bodies at the time of the eclipse; therefore, let the horary motion of the Moon, in latitude and longitude, be denoted by n and m; and that of the Sun in longitude by m'; the latitude of the Moon at the moment of her conjunction by ; and t, the time from the conjunction, expressed in hours and fractions as before; then the difference of latitude at the time t will be equal to l+nt, and of longitude (m—m') t ; and we shall have

E2 = (m — m')2j2 + (l + nt)2.

The value of E, found in the first of these equations, substituted in the second, will enable us to find the value of t in terms of d; and, therefore, if there be given to d its different values corresponding to the different phases of the eclipse, the time t will be the only unknown quantity in the equation, and which may therefore be found by its solution.

The first of these equations would give the value of E in an accurate and elegant manner; but the solution may be simplified by making use of an approximation, that will not introduce an error into the result greater than ths of a second of a degree. Accordingly, in finding the value of E, the small arcs, p, p', d, and d-E, may be considered as proportional to their sines, and then the equation becomes

p (d— E) = p'd, and therefore E=

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