Page:Encyclopædia Britannica, Ninth Edition, v. 6.djvu/218

190 COMET in the Nautical Almanac. Before proceeding further, however, it will be desirable to explain the use of Barker s Table, to which reference has already been made, in the calculation of the true anomalies, as it dispenses with the longer computation introduced above, with the view to render our example independent of any other publication. The table had appeared from time to time, in one form or another, in various astronomical works; but in 1847 it was re-computed with extreme precision by Dr Luther, and printed in Encke s edition of Olbers s Abhandluncj iiber die leichteste und bequemste Mcthode die Bahn eines Cometen zu berechnen. It is much too extensive to be reproduced here. The true anomaly in the parabola is related to the time from perihelion by the equation 75&(<-T) = 75 tan. * v + 25 tan. 1 * v <fl V 2 where Ic is the Gaussian constant [log. = 8-2355814], and q as before the perihelion distance. In the table M = 75 tan. v + 25. tan. 3 v 75..(<-T) or M = s 7^=~ q% & an equation, which, when q is known, allows either of (t - T) being found from M, aud consequently from v, or when (t - T) is known, gives M, and then, by means of the table, the corresponding v. Put C = 7= ; C is therefore a constant a*id log. C = 9 9601277. v 2 If, then, there be calculated for any comet the quantity C ve shall Lave -T) = 75 tan. . tan. 3 To afford the reader a clearer idea of the great assist ance which a table of this kind renders in cometary calcu lations, we will apply Luther s table in the two cases where we have used direct formulae in our example, (1.) To obtain (t -T) from v = 98 59 43" 0. Log. q 9-6960002 i Log. q 9-8480001 Log. ?! 9-5440003 Log. C 9-9601277 Log. in 0-4161274 The table, of which the argument is the true anomaly (v), with interval 100", furnishes these values of log. M, near the above value of v v Log. M. Diff. for 1". // 98 58 20 2-1066640 , , Q 99 2-1070109 Wherefore, by simple interpolation, we find 98 59 43 corresponds to log. M...2 10G9519 ; from this value of log. M subtract log. m, as found above, and we have T6908245 for the logarithm of the time (in days and decimals) from perihelion, corresponding to 49 d -07096, as before. (2.) In the reverse process, the determination of v, when (t - T) is given, we have in our example, as referring to the second observation with which the elements were compared (t"-T) +58-48950 Log. (if -T) Log. m 1-7670779 0-4161274 Log. M 2-1832053 Near this value of M the table gives us - v Log. M. 104 104 40 2-1830332 20 2-1831081 Dif. for 1" 37-50 Whence, again by simple proportion, we find log. M = 2-1832053, which corresponds to 104 52 25 9", differing only 0" l from the value found in the example. The student should procure the last edition of the werk above-named for the sake of this table, and for the exten sive catalogue of orbits of cornets, coming down to 18G4, and by far the most complete and reliable yet published. . It has been remarked that, where an ephemeris of geo centric positions (right ascension and declination) for any length of time is required, it is convenient to calculate with rectangular equatorial co-ordinates, instead of by the process we have followed in comparing the orbit with the middle observation. For this purpose we compute what may be termed co ordinate constants, from the elements TT, Q, and i, and obliquity of ecliptic (t), by the following formulae : P = cos. & P = siu. a cos. w P" = sin. Q sin. w tan. i Q=-sm. a . cos. i tan.**- - Q = s.siu. i Q =tan. A cos. & sin. i P" A = A+ (TT- Q- = tan. B C = C + (7r- P sin. B sin. b - a) As a partial check upon the calculation, we have sin, fc.sin. e.sin. (C -B ) tan - * = sin. a. cos. A The angle ^ is to be taken in the first quadrant with its proper sign, and when the comet s motion is retrograde, i must be used with a negative sign. The heliocentric co-ordinates of the comet (x, y, z) will then be obtained from x = r.sin. .sin. (A + v) y = r.sin. b . sin. (B + i>) z = r.sin. c . sin. (C + v), x being measured in the direction of the first point of Aries, from the sun as origin of co-ordinates, y towards 90 of Right Ascension, and z from the plane of the equa tor, positive to the north. Similar co-ordinates of the sun, X, Y, Z, with the earth as origin, are found in the Nautical Almanac for Green wich noon and midnight. The Right Ascension and Declination are given by Y + v Z + z T, . tan R.A. = tan 5= - - . cos E.A. X + a; X + x The true distance of the comet from the earth (A) = sin. 8 As an example of this calculation we will find the co ordinate constants applying to the elements of Borrelly s comet in this article. Here = 282 12 48" l n - & =344 36 45" 4. i = 80 56 27" 5 (the motion being retrograde). The mean obliquity of ecliptic, 1875 = 23 27 19" 9. P = Log. cos. a + 9-3254186 -Log. sin. a +9-9900575 Los. cos. i + 9-1971480 Log. tan. i -0 7974009 Log. cos. 8, + 9 "325 ,113 Log. tan. V -1-4719823 Q.., -log. tan. A.., A... Log. sin. A.. P .+9-1872055 . + 0-1382131 , 53 58 1" 2 . + 9-9077758 ^....- 4 5 -3 28 27 19" 9 Log. sin. i -9-9945489 Log. sin. 4/ -9-9997531 Loir, s +9-9947958 , . =log.sin.a 9-4176428 Log. sin. A

A = A+(7r- a) 3S 34 46" 6