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evening of the 15th; and the full Moon will take place 56 m. past 9 in the evening of the 23d. The Moon will also be upon the meridian at convenient times for observation on the following days, viz.
The Moon will be in conjunction with a in Libra at 9 in the evening of the 8th. Jupiter and Georgium Sidus will be in conjunction on the 9th, when the latter planet will be 24' south of the former. Saturn will be in quadrature at 7 in the morning of the 22d. The eclipses of Jupiter's satellites are not visible in the vicinity of London this month.
ON THE CALCULATION OF ECLipses.
[Continued from p. 296.]
Method of calculating the Circumstances of Lunar Eclipses.
HAVING endeavoured to explain the nature and extent of the shadows projected by the Earth and the Moon in directions opposite to the Sun, considering them as phenomena of which it is important the student should have a distinct comprehension, we shall now explain the method of calculating the other circumstances of eclipses; and, first, those of the Moon.
The principal circumstances which now demand our attention in this inquiry are, the true time of the opposition, the horary motion of the Sun in longitude, that of the Moon in latitude and longitude, the latitude of the Moon's centre at the instant of opposition; all of which are given in astronomical tables of
the Sun and Moon. In addition to these, it is also requisite to have the distance between the centre of the Moon and that of the Earth's shadow, at any given time either before or after the opposition, and then, with these quantities, the time, magnitude, and duration of lunar eclipses may be readily calculated.
For this purpose, let it be supposed that, at the instant of opposition, the point E (fig. 10) represents the centre of the terrestrial shadow. Let TC be the ecliptic, EP the circle of latitude in which the opposition takes place, and M the centre of the Moon at that time also let T'C be the orbit of the Moon inclined to the ecliptic. Then, in consequence of the Earth's motion in the ecliptic, the centre of the Earth's shadow, which is always diametrically opposite the Sun, moves with it, and always with the same velocity, from west to east, or from C towards T. The centre of the Moon is also in motion at the same time from west to east, or from C towards T'. Now, as these two motions are given in astronomical tables, it is required to determine the instant in which the two circles representing the Moon and the shadow meet, either before or after the true instant of opposition.
This research will be greatly simplified by considering that the apparent distance between the centres of the Moon and the shadow, during the eclipse, which is necessarily very small, may be regarded as rectilinear, and also the difference of latitude and longitude of these centres may be considered as right lines parallel and perpendicular to the ecliptic; so that the move
ments of these two centres may be conceived to be in straight lines, the one taken on the ecliptic CT, and the other on the circle of latitude passing through the centre of the shadow. The duration of a lunar eclipse is always so short, that the motion of the Sun, and consequently that of the shadow also, may be regarded as uniform for that period. The motions of the Moon in both latitude and longitude may likewise be regarded as uniform for the same time; at least in a first approximation. These considerations greatly simplify the problem.
Let E' and M' be two simultaneous positions of the shadow in the ecliptic, and the Moon in her orbit, at any instant either before or after the opposition. Then, since the motions of the Moon and shadow in both latitude and longitude are known, we shall have the values of E'P' and P'M', which represent these relative motions; and as the triangle E'P'M' is right angled at P', the sum of the squares of E'P' and P'M' will give the square of E'M', and consequently the root of this sum is E'M' itself. From the value of this distance, we may always know whether the eclipse has commenced, as we have already shown the value of the distance in other terms beyond which eclipses cannot take place. By forming an analytical expression for any time, it will be easy to determine the precise instant of each phase of the eclipse, by the solution of an equation of the second power.
The results thus obtained will only be strictly correct if the motions of the Sun and Moon were uniform; but if the greatest degree of accuracy be required, it is only necessary to calculate the time for any given phase, then to take that instant for the origin of time, and find the motions of the Sun and Moon by the tables, and recommence the calculations. with these new quantities, and the result would give a correction of the epoch found by the first approxi mation. This final result will possess all desirable accuracy, because the supposition of the uniformity Ee.
of motion involves so short a period. By performing these operations for each phase of the eclipse successively, all the circumstances of the eclipse, with respect to the true motions of the Sun and Moon, will be accurately ascertained.
In the preceding researches, no account has been taken of the diurnal motions of the heavens; as this causes an apparent simultaneous and equal motion of the Sun and Moon, the plane of the ecliptic, and all the celestial circles, without changing the respective positions of these with respect to each. The only effect of this motion is that of successively presenting the eclipse to different parts of the terrestrial globe; its influence affects only the possibility of seeing it,
not its existence.
Now, in order to find the analytical expressions, in which the substitution of the given quantities will furnish the required numerical results, let the horary motion of the Sun, or that of the shadow, in longitude at the instant of the apposition, be denoted by m'; and any time, either before or after the opposition, expressed in hours and fractions of an hour, by t, the time being considered negative before and positive after the moment in which the opposition takes place; then the distance of the centre of the shadow, at any given time from the point E, will be expressed by m't. Also, let n and m denote the horary motions of the Moon in latitude and longitude, commencing at the same epoch. Then the space through which she will have moved, parallel to the ecliptic in the time t, will be expressed by mt, and that on a circle of latitude by nt; so that if the Moon's latitude, at the instant of the opposition, be indicated by l, the two co-ordinates of her centre will be expressed by mt and 1+nt. As the motions of the Sun and Moon are both from west to east, both m' and m will always have the same sign, and are regarded as positive; but that is not the case with respect to n; for it is considered as positive
when the Moon approaches the north pole of the ecliptic, and negative when she removes from it: is also positive for north, but negative for south latitudes. These quantities, with their proper signs, are to be found in astronomical tables.
Let the distance between the centres of the Moon and the shadow, at any instant, be expressed by d, which will evidently be the hypothenuse of a right angled triangle, having for its sides the differences of their movements in latitude and longitude; and 1 + nt, or (m —m')t. We have, therefore,
(m ~ M')2 z2 + (l + nt)2 = d2.
Squaring the second term of this equation, and simplifying it by the substitution of an auxiliary angle a, such that the
mm' will be eliminated, and its solution then gives
t = — (± sin a(d2 — l2 cos2 a)1 — 1 sin2 a)..
Now, the value of d, for any particular phase of the eclipse, being substituted in this expression for t, it will give the epoch corresponding to that phase; and there being always two values of t for every different value of d, the substitution gives the time of the two corresponding phases, the one before the middle of the eclipse, and the other after it; as t is positive in the one case, and negative in the other. The distance of the centres of the Moon and shadow at the beginning and end of the eclipse is known by what precedes; and therefore, by adopting the same notation as in the preceding part of this article, where D and D' denote the apparent diameters of the Sun and Moon, p and p' their horizontal_parallaxes, we have, at these two epochs, d† (D'—D)+p+p'.