133 8 The longitude of the ascending node. f The inclination of the comet s orbit to the ecliptic. $... The perihelion distance, expressed, like other distances in astronomical calculations, in parts of the earth s mean distance from the sun. v The comet s true anomaly. We suppose that the observations furnish three complete positions of the comet referred as usual to the equator, or expressed in right ascension and declination, with the mean times of observation at the respective places. The first step will be to convert the observed right ascensions and declinations into longitudes (a) and latitudes (13), thus : Put tan. N= tan S . Then, tan. o = ( sin. R.A. . tan R.A. And tan. /3 = tan. (N e). sin. a e. cos. the obliquity of the eclip tic at date, from the Nautical Almanac. Thus we find a , a", a ", and /3 , /3", /? ", where the quantities with one accent apply to the first observation, those with two accents to the second place, and with three accents, to the last observation. This is to be understood throughout our formulce for the calculation of the orbit. Now reduce the times of observation to the meridian of Greenwich by applying the longitude of the place of ob servation with its proper sign, and convert the times so reduced into decimals of a day ; thus we have t , t", t ". For each of these times interpolate from monthly page iii., in the Nautical Almanac, the sun s longitude (A) and the logarithm of the earth s radius-vector (R) ; the sun s longitude in the Almanac being apparent, the amount of aberration (20" 42 -r-R), which is given in another part of the ephemeris, must be added to the apparent longitude, to obtain the true longitude required in the calculation. We have then A , A", A " and the logarithms of R , R", R ", and are ready to proceed with the application of Olbers s method. We commence by calculating M or ^- , the ratio of the P comet s curtate distances from the earth at the first and third observations from jj _ tan, ft", sin, (a A") tan, ft , sin, (a" A") tan. ft ", sin. (a" A") tan. ft", sin. ( " A") or rather more conveniently, by putting m = from , r _ wi.sin. (a -A") -tan. ft t " -t" (I.) t! t tan, ft" in. (a" - A") .,, . tan. ft " - m . sin. (a" - A") t" - t The following equations must then be formed (k is the chord of the comet-orbit between the extreme observations) : 2 = R 2 - 2R . cos. (a - A ) . p + sec. 2 . // 2 " 2 = R " 2 -2R ". M.cos. (a "- A ")p + sec. ft " 2 . M 2 . /> = (r 2 + r" 2 ) - 2R . R ". cos. (A " - A ) + 2R ". cos. (a - A ")p + 211 . M . cos. (a " - A ) . p - 2M . cos. (a " - a )p 2 - 2M . tan. ft , tan. ft ", p* ill.) If (t " -t ) be the interval of time between the first and third observations we have, by Lambert s theorem, t "-t = 3m . V2 With an assumed value for p we calculate r , r" , and k, and then t " for comparison with the observed interval be tween the first and third observations, and vary p in successive trials until the observed and calculated values agree. In this solution of the above equations by the method of trial and error, a first approximate value of p may be inferred as follows : Writing the equation for & - 2H assume tan. l/= -^- then p =tan. "JS (IV.) The amount and direction of the error of interval be tween the extreme times of observation, resulting from this first value of p, will, after a little experience, guide the computer to another value nearer to the true one; and the error of the second assumption, compared with that of the first, again leads to a much closer value for the third approximation, and so on till the assumed value of p pro duces an agreement between the calculated and observed intervals. In practice we have not found any great ad vantage on adopting one or other of the devices suggested for obtaining successive values of p by use of tables or otherwise the simple method of continued approximation, by deducing a new value of the curtate distance propor tional to the errors in the two preceding assumptions, will be found in the great majority of cases sufficiently exp editioua and as little troublesome as any other. In working Lambert s equation, proceed as follows : Put r + r 1 -Tc D = H (V.) log. z^log. B + ^log. B + l. 4378117 log. 2^ = log. p + ilog. D + 1.43781 17 ^ "= the time of describing the chord, expressed in days and decimals. The approximations to p may be continued until z - z" agrees with (t " - 1 ), within 2 or 3 in the fifth place of decimals, though if the computer has only rough observa tions at command, a larger error may be tolerated. The comet s curtate distance from the earth at the third observation is given by p "=Mp With the final values of r , r" , p and p ", the direct calculation of the elements of the orbit commences. The heliocentric longitudes , ", and latitudes X , " f are obtained from and r . cos. A . sin. (8 -A }-p . sin. ( -A ) r . cos. A . cos. (tf -A )-p . cos. (a -A )-R i sin. A =o . tan. ft r ". cos. A ", sin. (0" - A" )-p" . sin. (a " -A ") r " cos. A ", cos. (0 "- A ") = p". cos. (a " - A ") - R " r ". sin. A " o". tan. ft " (VI.) in which equations the right-hand quantities are known. The values of r and r ", resulting from these equations, should agree with the preceding ones if the calculations have been correctly performed. This agreement forms the first verification of the work. If &" is in advance of , the motion in the orbit is direct ; if the contrary be the case, the motion is retrograde. Then, if the motion be direct, the longitude of the ascend ing node (Si) and inclination of the orbit to the ecliptic (?) will be found from tan. i. sin. (0 - ) = tan. A . tan. A " -tan. A , cos. (8 " - 6 ) [ (VII.); and if tlie motion be retrograde from tan. i. sin. (Q, -0 ) = tan. A tan. i. sin. ( ft - ) = tan - K> " ~ tan " . x / . S (& ~~ 6 " The distances of the comet from the ascending node reckoned upon the orbit, at the first and third observa tions (u } u" ), are given in the case of direct motion by ^ , r= tan.Jtf^-fl ) 1 (VIJL) COS. I ) COS. I or, if the motion be retrograde, by
tan. u -toMa-O tan.^