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Hence, for the time T= 2*, we have the two points,

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321. The curves above computed are all exhibited in the following chart.

[graphic]

For the construction of such charts, on even a much larger scale, the degree of accuracy with which our computations have been made is far greater than is necessary, and many abridgments may be made which will readily occur to the skilful computer.*

* For a graphic method of constructing eclipse charts, aee a paper by Mr. Chauncey Weight, Proceedings of the Am. Association for the Adv. of Science, 8th meeting (1854), p. 65.

Prediction of a Solar EcUpsefor a Given Place.

322. To compute the time of the occurrence of a given phase of a solar eclipse for a given place.—The given phase is expressed by a given value of J, and we are to find the time when this value and the co-ordinates of the given place satisfy the conditions (485). This can only be done by successive approximations.

Let it be proposed to find the time of beginning or ending of the eclipse at the place. The phase is then A = I i£, and we must satisfy the equations (491). Let T0 be an assumed time, and T= T0 + T the required time. Let x, y, x', y', d, I, log i, be taken from the eclipse tables (p. 454) for the time T0. Assuming that x and y vary uniformly, their values at the time T are x + x'r and y + y'z. The co-ordinates of the place at the time T0 are found by (483) or (483*), in which fi is the sidereal time at the place. Putting

= a fij w

in which at is the west longitude of the place and fi^ may be taken from the table (p. 455) for the time T0, the formulas become

A sin B = p sin <p' £ = p cos <p' sin # ^

A cos B p cos <p' cos # ri = A sin {B d) > (550j

C =Acos{B d) )

Let if denote the hourly increments of £ and 37; then, assuming that these increments also are uniform, the values of the co-ordinates at the time T are £ + £'r and rj + ^'f. The values of £' and if are found by the formula? (p. 462)

f — p! p cos <p' cos #

g'=/t'f sin d d'Z

in which and c£' are the hourly changes of // and multiplied by sin 1". The rate of approximation will not be sensibly affected by omitting the small term d'£, and the formulae for £' and if may then be written as follows:

? = p!AcofiB 7]' = p.'? Bind (551)

Put

L = I i:

then, neglecting the variation of this quantity in the first approximation, the conditions (491) become, for the time T,

L sin Q = x — £ + {pi — £') r
£ cos Q = y rj + (y — V) T

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by which r is found. Since the first of these equations does not determine the sign of cos ^, the latter may be taken with either the positive or the negative sign. We thus obtain two value?

of T= T0 + z, the first given by the negative sign of ~^eoa*

n

being the time of beginning, and the second given by the positive sign being the time of ending of the eclipse at the place.

For a'second approximation, let each of the computed times (or two times nearly equal to them) be taken as the assumed time Tg, and compute the equations (550), (551), (552), (553) for beginning and end separately.

The first approximation may be in error several minutes, but the second will always be correct within a few seconds, and, therefore, quite as accurate as can be required; for a perfect prediction cannot be attained in the present state of the Ephemerides.

The formula for r may also be expressed as follows:

To sin (M — Ar — 4.)
n sin 4

which in the second approximation will be more convenient than the former expression; but when sin ^ is very small it will not be so precise.

If we put

t the local mean time of beginning or end,

we have

t=T0+T-W.

323. The prediction for a given place being made for the purpose of preparing to observe the eclipse, it is necessary also to know the point of the sun's limb at which the first contact is to take place, in order to direct the attention to that point. This is given at once by the value of

which is the angular distance of the point of contact reckoned from the north point of the sun's limb towards the east (Art. 295).

The simplest method of distinguishing the point of contact ou the sun's limb is (as Bessel suggested) by a thread in the eye-pieco of the telescope, arranged so that it can be revolved and made tangent to the sun's limb at the point. The observer then, by a slow motion of the instrument, keeps the limb very nearly in contact with the thread until the eclipse begins. The position of the thread is indicated by a small graduated circle on the rim of the eye-piece, as in the common position micrometer.

This method is applicable whatever may be the kind of mounting of the telescope. Nevertheless, if the instrument is arranged with motion in altitude and azimuth, it will be convenient to know the angle of the point of contact from the vertex of the sun's limb, which is that point of the limb which is nearest to the zenith. The distance of the vertex from the north point of the limb is equal to the parallactic angle which being here denoted by y, is found, according to Art. 15, by the formula?

p sin y = cos f sin #

p cos r sin <p cos d — cos <p sin d cos 0

(where we have put p for sin £ and d for the sun's hour angle). As y is not required with very great accuracy, we may here take [see (494)]

p sin y = £ p cos Y = rl

in which £ and yj are the values of the co-ordinates of the place at the instant of contact. But, if c and y denote the values at the time Tw we must take

p sin Y = ? + £ 'r p cos Y = V + Vr (554)

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