expressions for the cosine of the arc between S, and S2, and it is as follows: +cos2r cos (R1 - R2) cos (RA) cos (R,' – ▲) + sin2 q cos2 r cos (R1 – +cos2r sin q sin (R1 9 R2) sin (R,' - R2') r cos r sin (R1 - + cos sin - (i) + sin (R2-A)} By assigning suitable values to q and r, this formula can be made to apply to the following astronomical instruments:-the altazimuth, the meridian circle, the prime vertical instrument, the equatorial, and the almucantar. For the meridian circle q and r should be each as near zero as possible, and for the almucantar q is the latitude and r quite arbitrary. The following general proof will show that the complete theory of each of the instruments named must be included in this one formula. From any such instrument we demand no more than that the two readings R and R' obtained by directing the instrument to any particular star shall enable us to calculate the co-ordinates a, d of that star free from all instrumental errors. Let S1, S2, S3 be three standard stars of which the co-ordinates are known, and let each of these stars be observed with the generalised instrument with results R1, R'; R2, R2'; R, R respectively. Substituting for each of the three pairs (S, S2), (S2 S3), (Sg S,) in the typical formula (i), we obtain three independent equations. From these equations, q, r, and ▲ can be found. Nor will there be any indefiniteness in the solution, for in each case we may regard these quantities as approximately known, so that to obtain the accurate values of q, r, and ▲ we shall have to solve only linear equations. We may thus regard (i) as an equation connecting a 8, a 8, R, R1, R2, R2, and known quantities. Let S be the star whose co-ordinates a, & are sought. We write the equation (i) for the pair (S S,), and substitute their numerical values for a1, 81, R1, R. We thus have an equation connecting the co-ordinates a, d of any star with its corresponding R, R' and known numerical quantities. When we substitute for R and R' the values observed for S, the formula reduces to a numerical relation between the a and 8 of the particular star S. From the pair (S S2) we find in like manner another quite independent numerical equation involving a, S. As, however, the equations are not generally sufficient to determine a, & without indefiniteness, we obtain a third equation from (S Sg). This equation is not independent of the others, but if we make x = sin d, y = cos & cos a, z= cos 8 sin a, we shall obtain three linear equations in x, y, z which can be solved, and thus a and 8 are found without any ambiguity whatever. All the ordinary formulæ used in connection with the different instruments named can be deduced as particular cases of the general equation (i). In general, there are no real values of R and R' when the instrument is directed to the pole of circle A. In such a case R would have to be set on one of the imaginary circular points at infinity. 2nd January 1908. The Perturbations of Halley's Comet in the Past. Second Paper. The Apparition of 1222. By P. H. Cowell, M.A., F.R.S., and A. C. D. Crommelin, B.A. In the first paper of this series we identified the comet of October 1301 with Halley's, and found the value 44" 858 for the mean daily motion at that epoch We have now completed (with the aid of Mr. F. R. Cripps) the calculation of the perturbations by Jupiter and Saturn for the preceding revolution. As a first approximation, Hind's date (mid-July 1223) was assumed for the preceding perihelion passage, and on this assumption the results were as follows: Hence mean motion in 1223 = 44" 858-0"015 44′′·843 and calculated period = 1296000" + 1043" = 28924 days. This is This indicated 1222 August 15 as the date of the preceding perihelion passage, or 11 months earlier than Hind's date. too large a discordance to be possible, so Hind's identification of the comet of July 1223 with Halley's is erroneous. There was, however, a much more remarkable comet which appeared at the exact epoch indicated by the calculation; and examination shows that the greater part of the statements made concerning it by contemporary writers are quite consistent with its being Halley's, so that the identity is placed beyond reasonable doubt. The error of the first assumption is so great that it is necessary to recompute the perturbations; this has as yet only been done approximately, the resulting date being 1 day earlier, or August 14; the small difference between this and the preceding result is an illustration of the general proposition that the date of the more remote perihelion passage need only be very roughly known in order to obtain the periodic time correctly. Pingre's description of this comet is as follows; "En Automne, c'est à-dire aux mois d'Août et de Septembre, on vit une étoile de première grandeur, fort rouge, et accompagnée d'une grande queue qu'elle étendait vers le haut du ciel, en forme d'un cone extrêmement aigu: elle paraissait fort près de la Terre: on l'observa (d'abord) vers le lieu où le soleil se couche au mois de Décembre. Le 15 Août, jour de la première apparition de cette comète (peut-être à Milan) la Lune fut comme morte; elle n'avait plus d'éclat, et elle joignit la comète. On vit ensuite cette comète à l'occident, et même vers le nord, avant la fin du mois d'Août. En Chine on l'observa le 10 Septembre, entre la constellation Kang (les pieds de la Vierge,=, K, λ, 0 Virginis), Arcturus et la chevelure de Bérénice: elle disparut le 8 Octobre. Le Père de Mailla dit que les Chinois virent une comète à l'ouest en 1222, à la première lune: c'est sans doute une erreur du copiste, il faut lire à la huitième lune." The words in parentheses are not part of the original documents, and Pingre's interpolation "d'abord appears to be erroneous; it is the place where the comet was last seen, not first seen. When the comet extended its tail towards the zenith it must have been nearly vertically above the Sun. The following is the description in Williams' Chinese Observations of Comets:-"In the reign of Ning Tsung, the 15th year of the epoch Kea Ting, the 8th moon, day Kea Woo (= 1222 September 15), a comet appeared in Yew She Te (n, T, v Boötis). Its luminous envelope was 30 cubits long. Its body was small, like the planet Jupiter. It was seen for two months. It passed through Te (a, ẞ, y, Libræ), Fang (B, 8, π Scorpii), and Sing (Antares, etc.), and then disappeared." (Williams, in this and other places, erroneously gives v Boötis instead of v as one of the components of Yew She Te.) Most of the above statements are satisfied by the following orbit, in which the longitude of perihelion is that actually derived from the perturbations, and the date of perihelion passage is 8 days later than that indicated above; a perihelion distance somewhat greater than the present value has been used. Hind found that this was also indicated in the return of 1066. We have had to assume that the date when the comet was first seen in China (in the region 7, т, v Boötis) should be one lunar month earlier than that given, or August 17 instead of September 15. Our reasons are: (1) it is distinctly stated that the comet was seen for two months in China, the final date being October 8, the position then being in Scorpio; (2) we cannot make this a month later, for in November Scorpio was invisible; (3) further, we are distinctly told that the comet was a splendid object in Europe in mid-August, so that it is most improbable that the Chinese should have missed it till a month later; (4) further, its geocentric motion was at that time so rapid that had it been in Boötes in mid-September it would have been a morning-star on August 15, which it apparently was not. Another difficulty that we have to surmount is the phrase "La Lune joignit la comète." It was doubtless this phrase that led Pingré to the conclusion that the motion was from south to north, which is in opposition to the Chinese account. Considering the vagueness of all European cometary observations at that epoch, we may look on the phrase as sufficiently satisfied by the fact that the Moon (at her first quarter) and comet were symmetrically placed above the western horizon an hour after sunset, though separated by some 70°. There is one more point to notice. The comet is said to have been near Antares when last seen (October 8). Our chart shows that it was then in Libra, but the head was too near the Sun to be seen; the tail would point to Antares, and, considering its length, it may well have reached that star. It will be seen that we have given the European observations most weight for dates, the Chinese most weight for the track among the stars. This conclusion is in agreement with that reached in the case of other comets, including that of 1301. We have thus shown that the observations are satisfied with no further alteration than they would in any case require to render them self-consistent, and the exact coincidence of date leaves scarcely any room for doubt as to the identity. Pingré was prevented from making this identification by not knowing the wide range through which the period of Halley can alter, and doubtless his inference that the comet moved at first from south to north put Hind off the track. It is of interest to point out that the revolution 1222-1301 is the longest on record, being 79 years 2 months. It just exceeds that from 1066 to 1145, which was hitherto looked on as the longest. The round now being accomplished is the shortest on |
