COMET 189 Log. /"... 0-1742799 Log. sin. (v "-v") S 9139744 Log. r .., 0-0708694 Log. sin. (v"-v ) 9-0103899 B.~ A B No.. 9-0812593 0-0069950 1-0162368 1-0156182 Log. R ".. 9-9926916 Log. sin. (A "- A") 9 2284039 c...~ 9-2210955 Log. R ... l>-99~3359() Log sin. (A" - A" ). 9-2213297 D.. 9-2146887 .. 0-0064068 p + 0-0006186 s-p- Log. (R . sin. (A" - A ) ) ... + 9-2146887 Log. (q-p)... -7 1384290 Log. TO... -0-4867341 E .. + 6-8398518 1-0148614 1-0156182 q...- 0-0007568 0-0012570 Log. p. ..+67914099 Log. W *}... 9-9932695 Log. (m . sin. (A" - a ) -tan. )... + 97146784 9-8614743 + 6-7846794 No.. . + 0-0006091 Log. p .. F E F , + 9-5761527 + 7-2636991 , 9-9932695 Log. N + 7-25 N.. . + 0-0018070 -C-... + 0-0006091 i-ooooooo H .. 1-0024161 Log. II... 0-0010481 Log. M... 9-6699800 Corrected log. M... 9-6710281 With this corrected value of log. M, we might recalcu late the co-efficients of p and p" 2 in the equations for r " 2 and & 2 , and complete the calculation of the orbit, but as the method of procedure is precisely that already illustrated by an example, it is unnecessary to occupy space here by so doing. In the majority of cases in practice, the first elements of a comet s orbit are calculated from a much shorter interval of observation than has been taken in the pre ceding example, not infrequently from observations on consecutive nights, and in such cases our elements may be open to considerable correction, though the natural desire of the astronomer to learn something of a new comet s position in the system, its track in the heavens, or possible identity with a comet already calculated, induces as speedy a determination of the orbit, however approximate, as practicable. If the observations used in the first computations are near together, or the geocentric motion is slow, it will be preferable to wait for later positions, rather than occupy time in attempting a closer representation of the middle place. When later observations are available, the orbit may be re-calculated, M or -r , being determined by Ol- bers s formulae of correction, employing ? , r ", p , &c., as deduced from the first orbit. But the following general method of correcting approximate elements of a parabolic orbit, which has been widely used, will be found as satis factory, though requiring great care in working. It is generally known as the method of variation of curtate distances. We select three good observations at as wide intervals as practicable. These observations should be corrected for the effects of parallax and aberration by means of distances from the earth (A) calculated from approximate elements. The aberration will be most conveniently taken into account by subtracting 497 8- 8 x A from the time of obser vation, and interpolating the values of A and Log. R from the Nautical Almanac for the time thus reduced. Then, introducing the observed longitude and latitude and the value of p, calculated from the approximate orbil^ find 0, X, and r at the first and third observations from r.cos. A. sin. (6 - a) = R.sin. (A- a) r. cos. A. cos. (6- a) = R. cos. (A- a) + p r . sin. A. = p . tan. From &, X , r , and 6 ", X ", r ", we compute the elements in the same manner as before, and thence the geocentric longitude for the time of the second observation, which call a r (The geocentric latitude may be substituted for the longitude, if it be changing more rapidly.) Also find the time by these elements between the first and third observations, which call ^. Now vary p by a small quan tity, as O Ol or 005 ( = m), and find , X , r again, and with these new values, combined with the previous ones for &", X" , r", compute the elements, and compare again with the second longitude, and call the difference from the longitude first computed r ; also find the interval between the first and third observations, and call the difference in this case p. Next, with the first values of p, 9 , X , r, combine 0" , X" , r" calculated from a similar slightly changed value of p", (p" + n), and, completing the elements, compare again with the longitude at the second observa tion obtained with the unvaried p, p", and also with the corresponding interval between the extreme observations, and call the differences from the longitude and interval with unvaried curtate distances s and q. We have thus in the three calculations, + r , + P , + S The three hypotheses and corrected values of p , p" are then True orbit, p +2! p "+y (t} observed interval. + s [(a) do. longitude. Hyp. I. Hyp. II. Hyp. III. Assumed p .... p p + m p p" .... p " P" p " + n Interval between extreme obser t t +p t +q vations Second longitude 1 cti + r i + s And we have ( t ) t -* i v m n r.x s.y W "i m n n [(0- f] s. TO - [(a) -ajg. m [(t)-t _[(Q-qr.ra-[(a)-aJp.n s.p - r.q r.q-s.p If the resulting corrections of the curtate distances are small, the true, or rather corrected, elements may be obtained by interpolation between the values obtained on the different hypotheses. When x and y are large, it is occasionally necessary to repeat the work, to have a close agreement between the middle longitude calculated from the corrected orbit and the longitude observed. To make our article rather more complete, we may now refer to the calculation of ephemerides of the geocentric places of a comet from the parabolic elements, which are required during its visibility to facilitate observations. If a few places only are required, the right ascensions and declinations may be found in the manner already described ; but if the comet is likely to continue visible any length of time, it is more convenient to work by rectangular equa torial co-ordinates, introducing the X, Y, Z, depending on
the sun s position, which are now given with much detail