u133We have said, that comets are a sort of planets revolved in very eccentric orbits about the sun; and as, in the planets which are without tails, those are commonly less which are revolved in lesser orbits, and nearer to the sun, so in comets it is probable that those which in their perihelion approach nearer to the sun are generally of less magnitude, that they may not agitate the sun too much by their attractions. But as to the transverse diameters of their orbits, and the periodic times of their revolutions, I leave them to be determined by comparing comets together which after long intervals of time return again in the same orbit. In the mean time, the following Proposition may give some light in that inquiry.

PROPOSITION XLII. PROBLEM XXII.

To correct a comet's trajectory found as above.

OPERATION 1. Assume that position of the plane of the trajectory which was determined according to the preceding proposition; and select three places of the comet, deduced from very accurate observations, and at great distances one from the other. Then suppose A to represent the time between the first observation and the second, and B the time between the second and the third; but it will be convenient that in one of those times the comet be in its perigeon, or at least not far from it. From those apparent places find, by trigonometric operations, the three true places of the comet in that assumed plane of the trajectory; then through the places found, and about the centre of the sun as the focus, describe a conic section by arithmetical operations, according to Prop. XXI., Book I. Let the areas of this figure which are terminated by radii drawn from the sun to the places found be D and E; to wit, D the area between the first observation and the second, and E the area between the second and third; and let T represent the whole time in which the whole area D + E should be described with the velocity of the comet found by Prop. XVI., Book I.

OPER. 2. Retaining the inclination of the plane of the trajectory to the plane of the ecliptic, let the longitude of the nodes of the plane of the trajectory be increased by the addition of 20 or 30 minutes, which call P. Then from the aforesaid three observed places of the comet let the three true places be found (as before) in this new plane; as also the orbit passing through those places, and the two areas of the same described between the two observations, which call d and e; and let t be the whole time in which the whole area d + e should be described.

OPER. 3. Retaining the longitude of the nodes in the first operation, let the inclination of the plane of the trajectory to the plane of the ecliptic be increased by adding thereto 20' or 30', which call Q. Then from the[Pg 496] aforesaid three observed apparent places of the comet let the three true places be found in this new plane, as well as the orbit passing through them, and the two areas of the same described between the observation, which call and ; and let be the whole time in which the whole area should be described.

Then taking C to 1 as A to B; and G to 1 as D to E; and g to 1 as d to e; and to 1 as to ; let S be the true time between the first observation and the third; and, observing well the signs + and -, let such numbers m and n be found out as will make ; and . And if, in the first operation, I represents the inclination of the plane of the trajectory to the plane of the ecliptic, and K the longitude of either node, then I + nQ will be the true inclination of the plane of the trajectory to the plane of the ecliptic, and K + mP the true longitude of the node. And, lastly, if in the first, second, and third operations, the quantities R, r, and , represent the parameters of the trajectory, and the quantities , , , the transverse diameters of the same, then will be the true parameter, and will be the true transverse diameter of the trajectory which the comet describes; and from the transverse diameter given the periodic time of the comet is also given. Q.E.I. But the periodic times of the revolutions of comets, and the transverse diameters of their orbits, cannot be accurately enough determined but by comparing comets together which appear at different times. If, after equal intervals of time, several comets are found to have described the same orbit, we may thence conclude that they are all but one and the same comet revolved in the same orbit; and then from the times of their revolutions the transverse diameters of their orbits will be given, and from those diameters the elliptic orbits themselves will be determined.

To this purpose the trajectories of many comets ought to be computed, supposing those trajectories to be parabolic; for such trajectories will always nearly agree with the phænomena, as appears not only from the parabolic trajectory of the comet of the year 1680, which I compared above with the observations, but likewise from that of the notable comet which appeared in the year 1664 and 1665, and was observed by Hevelius, who, from his own observations, calculated the longitudes and latitudes thereof, though with little accuracy. But from the same observations Dr. Halley did again compute its places; and from those new places determined its trajectory, finding its ascending node in ♊ 21° 13' 55''; the inclination of the orbit to the plane of the ecliptic 21° 18' 40''; the distance of its perihelion from the node, estimated in the comets orbit, 49° 27' 30'', its perihelion in ♌ 8° 40' 30'', with heliocentric latitude south[Pg 497] 16° 01' 45''; the comet to have been in its perihelion November 24d. 11h. 52' P.M. equal time at London, or 13h. 8' at Dantzick, O. S.; and that the latus rectum of the parabola was 410286 such parts as the sun's mean distance from the earth is supposed to contain 100000. And how nearly the places of the comet computed in this orbit agree with the observations, will appear from the annexed table, calculated by Dr. Halley.

In February, the beginning of the year 1665, the first star of Aries, which I shall hereafter call , was in ♈ 28° 30' 15'', with 7° 8' 58'' north[Pg 498] lat.; the second star of Aries was in ♈ 29° 17' 18'', with 8° 28' 16'' north lat.; and another star of the seventh magnitude, which I call A, was in ♈ 28° 24' 45'', with 8° 28' 33'' north lat. The comet Feb. 7d. 7h. 30' at Paris (that is, Feb. 7d. 8h. 37' at Dantzick) O. S. made a triangle with those stars and A, which was right-angled in ; and the distance of the comet from the star was equal to the distance of the stars and A, that is, 1° 19' 46'' of a great circle; and therefore in the parallel of the latitude of the star it was 1° 20' 26''. Therefore if from the longitude of the star there be subducted the longitude 1° 20' 26'', there will remain the longitude of the comet ♈ 27° 9' 49''. M. Auzout, from this observation of his, placed the comet in ♈ 27° 0', nearly; and, by the scheme in which Dr. Hooke delineated its motion, it was then in ♈ 26° 59' 24''. I place it in ♈ 27° 4' 46'', taking the middle between the two extremes.

From the same observations, M. Auzout made the latitude of the comet at that time 7° and 4' or 5' to the north; but he had done better to have made it 7° 3' 29'', the difference of the latitudes of the comet and the star being equal to the difference of the longitude of the stars and A.

February 22d. 7h. 30' at London, that is, February 22d. 8h. 46' at Dantzick, the distance of the comet from the star A, according to Dr. Hooke's observation, as was delineated by himself in a scheme, and also by the observations of M. Auzout, delineated in like manner by M. Petit, was a fifth part of the distance between the star A and the first star of Aries, or 15' 57''; and the distance of the comet from a right line joining the star A and the first of Aries was a fourth part of the same fifth part, that is, 4'; and therefore the comet was in ♈ 28° 29' 46'', with 8° 12' 36'' north lat.

March 1, 7h. 0' at London, that is, March 1, 8h. 16' at Dantzick, the comet was observed near the second star in Aries, the distance between them being to the distance between the first and second stars in Aries, that is, to 1° 33', as 4 to 45 according to Dr. Hooke, or as 2 to 23 according to M. Gottignies. And, therefore, the distance of the comet from the second star in Aries was 8' 16'' according to Dr. Hooke, or 8' 5'' according to M. Gottignies; or, taking a mean between both, 8' 10''. But, according to M. Gottignies, the comet had gone beyond the second star of Aries about a fourth or a fifth part of the space that it commonly went over in a day, to wit, about 1' 35'' (in which he agrees very well with M. Auzout); or, according to Dr. Hooke, not quite so much, as perhaps only 1'. Wherefore if to the longitude of the first star in Aries we add 1', and 8' 10'' to its latitude, we shall have the longitude of the comet ♈ 29° 18', with 8° 36' 26'' north lat.