the northern limit, and the lower, to one in the southern limit. The contrary has place for a total eclipse. When a series of places at which the eclipse will be central has been found, if a curve line be drawn through their positions on a map, it will represent the line of the central eclipse. If a series of places in the northern, and also in the southern limit of annular or total visibility, be found, and lines be drawn through their positions, they will bound the narrow portion of the earth's surface, within which, the eclipse is annular or total, as in Fig. 66, which applies to the eclipse in May, 1836.* Note. The arcs B and (p' — B) in formulæ (q), may each be taken less than 90°, being marked affirmative or negative according to the sign of the tangent or sine. 79. Occultations. If, instead of the quantities referring to the sun, those referring to a star or planet be taken, the formulæ obtained for computing an eclipse of the sun will also be applicable to the computation of an occultation of the star or planet. For a star, as its diameter and parallax are insensible, we have, r — 0, a=A', d= D', f= o,l=k, hlk=0.2725. Also, as the star's right ascension A', does not sensibly change during the continuance of an occultation, we have, the hourly variation of (u - A') = 15° 2′ 27′′.84 = 54147′′.84. Hence, *For the investigation of formulæ for determining the entire limits of visibility and other circumstances relative to the general eclipse, as represented in Fig. 66, the student may be referred to Woolhouse's Tract on Eclipses, which forms the Appendix to the Nautical Almanac for 1836. This subject has also been very fully investigated by Prof. Hansen in the Astr. Nach. Nos. 339 to 342. The position of the point of contact, in an occultation, is usually denoted, by giving its distance from the north point, or from the vertex of the moon's disc. The expression for this, will evidently be obtained, very nearly, by subtracting 180° from the expression for the position of the point of contact in an eclipse of the sun. We shall thus have, if P' and V now refer to the north point and vertex of the moon's disc, The first expresses the distance to the left of the north point of the moon's disc, and the latter, the distance to the left of the vertex.* 80. Transits of Mercury and Venus. Instead of the quantities which have referred to the moon, using those that refer to the planet, and taking k 0.3766 for Mercury, and 0.9617 for Venus, the formulæ obtained for an eclipse of the sun, will serve to calculate a transit of either of these planets; observing, however, that the values of a, d, and g, must be obtained from the formulæ (D), and not from the approximate formulæ (E). 81. Formulæ for computing an observed eclipse of the sun. Let T' be the observed mean time of beginning or end of the eclipse at a place whose latitude is known, T a mean time at the first meridian, taken to a whole hour near to the time of new moon, Tt the mean time at the first meridian, corresponding to the time T', and d'T (T). Then will d' be the longitude in time of the place at which the eclipse is observed; it being east if affirmative, but west if negative. Let Ρ and be the values of x and y at the time T, and p' and q′ their average hourly variations between the times T and T+t. Then, at the time T + t, we have, x = p + p't, and y = q + qt. Consequently, taking for x", y", and h, their values at this time, the equations of contact Data are given in the Berlin Ephemeris, by which the computations of the principal occultations that occur in the year, is greatly facilitated. These also include data, adapted to formulæ investigated by Prof. Hansen, in the Astr. Nach. No. 360, by means of which the position of the point of contact with reference to the contiguous spots on the moon's disc, is easily computed. m cos (M — N 4) And, consequently, d'T-T+ n cos ↓ To make the computation, the observed time of beginning or end may be reduced to the time T +t, at the first meridian, by using an assumed iongitude of the place. Then, having found the values p', q, x", y", z", and h, for this time, and computed the values M, m, N, and n, from (s), we find d from (u). If d', thus found, does not differ more than a few minutes from the assumed longitude, it may be regarded as the true longitude, as obtained from the observation. But if d' differs considerably from the assumed longitude, the computation should be repeated, taking d' as the assumed longitude. When the beginning and end have both been observed, the computation should be made for each, and the mean of the two results be taken as the longitude of the place. 82. As the solar and lunar tables cannot be regarded as perfectly accurate, the longitude obtained as above is liable to a small error depending on little errors in the elements used in the computation. But when the eclipse has also been observed at Observatories or other places whose positions are accurately known, the means are afforded of correcting the result for the principal errors in the elements. Those liable to the greatest errors, though these are but small, are the right ascension and declination of the moon. Let AA and AD be the corrections which ought to be applied to A and D, the computed right ascension and declination of the moon, so that A + ▲A, and D + ^D, may be the true values. The values of x and y, or their representatives p and q, will require corrections depending on the corrections AA and AD. In obtaining them, we may, without material cos D (A a) error, take, for p and q, the approximate expressions p D-d and q π deduced from the equations (App. 69 F). Substituting, in the first of these, A + AA for A, and, in the second, D + AD for D, Consequently, neglecting the extremely small correction that would be produced by the correction of D, in the factor cos D, the terms AD cos D AA π and are the corrections of p and q. Or, putting ∞ sin for л (App. The correction of may be omitted, as producing scarcely any influence on the value of h, (App. 70 M). Let ẞ and 8 be two unknown quantities whose values depend on those β of ▲A and AD, and assume, Applying the corrections of p and q, thus expressed, to the equations (t), we have, h cos P m sin M + nt sin N + n sin N. B N) — nd N) + nt + nß h cos (P + N) m sin (M h sin (P+N) ⇒m cos (M The squares of these, added together, give, h2 = (' пб 2 m cos (M —- N) + nt + n3)2 + (m sin (M — N) — ns)2 2 Hence, putting h cos & instead of its value, m sin (M — N), we have, m cos (M—N)+nt+n ß)3 — ha — (h cos ↓ — n ♪)3 h3 sin'+2 hn ♪ cos ↓—n3sa Extracting the square root, and observing that, as 8 is a very small quantity, the terms involving its square and higher powers may be omitted, we obtain, n & cos & h sin + п б tang m cos (M-N)+nt+nẞ=h sin 4+ Hence, regarding as always affirmative and less than 180°, and placing the double sign before the terms involving its sine or tangent, as in (App. 70), we have, Let s 3600 number of seconds in an hour, and let the terms con nected with T' T be multiplied by s. Then will the terms be expressed in seconds, and the expression for d' will be, ε When the times of beginning and end of the eclipse have been observed at a place whose longitude d' is accurately known, we shall, by making the computations for these times and this place and taking for d' its known value, have two equations containing the unknown quantities and 5, from which their values may be found. When the eclipse has been well observed at a number of places whose positions are accurately known, it is better to make the computations for several of these, and thus obtain a number of equations containing and 5, from a proper combination of which, their values may be more accurately obtained. ε When the values of and have been found by computation for places. whose positions are known, and substituted in (2), this formula will give the longitude for any other place at which the eclipse has been observed, with the corrections for the errors in the moon's computed right ascension and declination. With the values of ɛ and 5, the values of ▲A and AD may be found from the first two equations of (y). 83. Observed occultation of a star. The formula that have been obtained for computing an observed eclipse of the sun, serve also for the |
