266 TRANSITS OF VENUS AND MERCURY. P P 2 = = 87.969 365.256 1 1 1 1 1 1 1 4+ 6+ 1 + 1+ 2+ 1+ 5+ &c. Hence the series of fractions which approximately represent the value of P. is 7 13 33 46 263 or transits of Mercury may take place at the same node after intervals of 7, 13, 33, or 46 years. Transits in fact have been observed at the descending node in the month of May of the years 1661, 1707, 1740, 1753, 1786, 1799, 1832, and 1845; and at the ascending node in the month of November of the years 1677, 1697, 1723, 1736, 1743, 1756, 1769, 1782, 1789, 1802, 1848, and 1861. All these transits (excepting that for 1861) have been discussed by M. Le Verrier in his determination of the orbit of the planet contained in Vol. II. of the Annales de l'Observatoire Impérial de Paris. and transits may be expected to occur at the same node after intervals of 8, 235, or 713 years. The transits of Venus at the ascending node to the end of the present millenary period, commencing with 1631, are those which occurred in the month of December in the years 1631, 1639, and which will occur in 1874 and 1882; and those at the descending node are the celebrated ones of 1761 and 1769, SUN'S PARALLAX FROM TRANSIT OF VENUS. 267 which occurred in the month of June, and that which will occur in the same month of 2004. 7. To find the Sun's horizontal equatorial parallax from the difference of the durations of the same transit of Venus observed at different places on the Earth's surface. The principle on which is founded this method of finding the Sun's parallax has been explained in a preceding chapter (page 199). We now propose to exhibit the mathematical details of the calculations. The first thing to be done is to find the times of the first and last contact as viewed from the centre of the Earth. Let c be the sum or the difference of the angular semidiameters of the Sun and the planet (the sum corresponding to the external contact of the limbs and the difference to the internal contact); L and the geocentric longitudes of the centres of the planet and the Sun at the time of first or last contact; λ and A the geocentric latitudes at the same time. Then we have the following equations: cos c = sin λ sin A+ cos λ cos A cos (L – O), L-O Also, since, at the time of the transit, c, λ, and L-O are very small arcs, we may in general put the arcs for the sines, when we shall have In the Nautical Almanac the values of the Geocentric Longitudes and Latitudes of the planets are not given, but they can * M. Le Verrier at first retains the terms of the fourth order in the expan sions of sin2 с sin2 2' λ-Λ &c., but he finally neglects them as producing no sen sible effect. (Annales de l'Observatoire de Paris, Tome v. page 63.) 268 SUN'S PARALLAX FROM TRANSIT OF VEN US. be calculated from the Heliocentric Longitudes and Latitudes by the formulæ at page 241, or they can be taken immediately from the Spanish Nautical Almanac (Almanique Nautico calculado en el Observatorio de San Fernando). Let these then be computed for three times at equal intervals of four hours, this being, with respect to the duration of the transit, a convenient interval, and let the middle time (T) correspond pretty nearly to the time of conjunction in longitude. We shall then have, for the times T-4a, T, and T+4h, the true longitudes and latitudes of the Sun and Venus. These must then be converted into apparent longitudes and latitudes by the application of the aberration, unless the apparent places are given in the Ephe meris. Then for any time T+t (t being less than four hours, and the hour being taken as the unit of time,) we shall be able to express the latitudes and longitudes in the form a + bt + cť2, and consequently we shall have, for X-A and L-O, expressions of the form and, substituting these values in the equation c2 = (λ — A)2 + (L − 0)2, we shall have an equation of the form which will give the two epochs for which the distance of the centres of the Sun and planet is equal to c. The term of k involving t will, of course, be neglected in the first approximation, but, when an approximate value of t has SUN'S PARALLAX FROM TRANSIT OF VENUS. 269 been found, it can be easily taken into account, though it has an exceedingly small influence on the result. Let now and be the times calculated in the manner explained above, when the first and last contact would be seen at the centre of the Earth; T + ST and 7' + ST' the times at one of the observing stations, ST and ST' being the effects of parallax ; π the Sun's horizontal equatorial parallax, T' the horizontal equatorial parallax of Venus, n' and m' the relative horary motions of the Sun and Venus in latitude and longitude respectively. Then, knowing the hour-angle and declination of the point of contact at the time t + St, we can calculate the coefficients of parallax in R.A. and declination by the formulæ on page 188, and, by the formulæ on page 76, these can be transformed into the coefficients of parallax in longitude and latitude. For the first contact let these coefficients be a and b; so that the whole variation of L-O will be a (π' — π) + m'♪т, and the whole variation of λ - A will be b (π' — π) + n'St. But, differentiating the equation c2 = (λ — A)2 + (L − 0)3, considering c as constant, :. {α (π' — π) + M'dt} (L − O) + {b (π' — π) + N'ST} (λ — A) = 0, m' (L — O) + n' (λ — A) = α (π' — π), suppose. (IF) Similarly for dr' we shall obtain an expression of the form ST' = B (π' — π). Hence, at the station where this observation is made, we shall have: time of first contact = T + α (π' — π), last contact = ' + ß' (π' — πT) ; 270 SUN'S PARALLAX FROM TRANSIT OF VENUS. therefore duration of transit =D=T' -T+ (B — α) (π' — π). If now D' be the duration of transit at another station, and B' and a similarly related quantities, and, as this ratio is known by the theory of elliptic motion, let it be equal to k; This gives the value of the Sun's horizontal equatorial parallax for the day of observation. If then the radius-vector of the Earth's orbit for this day be r, we shall have at the unit of distance, for the constant of solar parallax, πr. By Encke's most careful and laborious discussion of all the observations of the Transit of Venus in 1769, the constant of parallax was found to be 8"-5776, which value however has been proved by modern discussions (page 198) to be considerably too small. |