University of California • Berkeley '/, f^ If ¥fr ^: ■*^ iy/rC^c*9ruL. "^ Digitized by tlie Internet Arcliive in 2008 witli funding from IVIicrosoft Corporation littp://www.arcliive.org/details/definitiveorbitoOOsearricli Deilive0rioftaell4ITMw| By Frederick H. Scares. \i. 3606-07 1 S99). ^- F Deiinitive Orbit of Comet 1894 IV (E. Swift). By Frederick //. Scarea. 1. Introductory. Comet 1894 IV was discovered by Edward Swift at Echo Mountain, California, on Nov. 20, 1894 at 8'' 30"' p.m. The comet was then very faint and had a short tail. The first observations were obtained by Barnard with the i 2 inch Equatorial of the Lick Observatory on Nov. 21, 22, and 23, and by Javelle at Nice on Nov. 22 and 23. ,A rapidly increasing deviation from the positions pre- dicted by means of the preliminary parabolic elements gave I indication of decided ellipticity which was soon confirmed by later elements. I A great similarity between the elements of the new ' comet and those of l)e Vico 1844 I was soon noted and the I possibility of identity was suggested by Berberich as early as Nov. 23. On Dec. i Tisserand announced that Schulhof had found the two comets to be identical. Owing to the extreme faintness of the object obser- vations were obtained with the greatest difficulty as is suffi- . ciently evident from the notes by the various observers, and in many places bad weather made observing quite impossible. I This was especially the case at Mt. Hamilton where in spite I of constant watchfulness on the part of Barnard the comet I was not seen from Nov. 30, 1894 until Jan. 25, 1895. As a consequence the total number of positions is only 64 for right ascension and 63 for declination. iThe first set of elements approximating a definitive solution were by Chandler from 27 observations grouped into 6 normal places. Basing himself upon this system of elements Chandler undertook a preliminary investigation of the question of identity with De Vico's comet, carrying the perturbations back through the conjunction with Jupiter in 1885. It seemed useless on account of the necessary in- determinateness of the value used for the mean motion to carry the calculation of the perturbations farther , but enough had been done to accomplish a partial adjust- ment of the discrepancies between the two comets and to show that the effect of the conjunctions with Jupiter in 1874 and 1862 would be in the right direction to produce a still better agreement. Notwithstanding the definiteness of the announcement concerning Schulhof's conclusions they were only provisional and were derived from a consideration of the perturbations of De Vico's comet throughout a long series of years. They were substantially the same as those arrived it by Chandler and may be found in k. N. No. 3267. With this encouragement the calculation of the definitive elements of comet Swift was begun, and the results are here pre- sented as a part of an investigation of the question of identity with De Vico's comet which the writer has undertaken. 2. Calculation of Ephemeris. The following elements by Chandler in .A. No. 338 sed as the of the calculation. 7 = .894 Oct. I 2. 1 88 I 7 Gr. M.T. a> = 296° 34' 35:2 I ft = 48 44 37-» 1894.0 i = 2 57 53-9 I log,/ = 0.1436451 e = 0.57189s Period = 2141.6 days With these was calculated an ephemeris of i day inter- vals from 1894 Nov. 10.5 to 1895 Jan. 30.5 Berlin M. T. Ephei is for Berlin Mean Midnigl 1894 a i 6 log J Ab.T. Nov. .9 2 2S4'" 9'94 ' -1 3° 34' 16^04 0.0075 ' 8-27^3 1 -0 17 10.23 13 "3 5629 0108 3> 2 20 9.90 12 53 32-38 0142 . 35-2 23 8.96 12 33 467 01 76 39-3 26 7.40 12 12 33.54 02 10 43-4 29 523 ' ' 5' 59-35 0244 47-5 32 2.44 11 31 22.48 0279 5' 7 T,2 59.00 37 54-93 4329 2 '5 03M 0.0349 8 559 9 0.2 1894 « 6 log J Ab.T. Nov. 28 22 '40"'SO'2 2 -10' 29- i9:'42 0.0384 9"" 4-6 29 43 44.85 .0;' 8 3548 0419 9.1 3° 46 38.82 9'47 5070 0454 13.6 Dec. 1 49 32. <3 9*7 542 0490 18.2 52 24.77 9 6 20.00 0525 22.8 55 "6-73 845 3478 0561 27.4 22 58 8.01 8124 50.10 0597 32.2 23 0 58.61 8' 4 6.28 0634 37.0 23 3 48.52 - 7(43 23-68 0.0670 9 31.8 83 3606 1894-95 « •^ log^ Ab.T. 1895 Jan. 3 « 6 log J Ab.T. 1 Dec. 6 23' 3"'48^52 — 7° 43' 23'.'68 0.0670 9-3 1^8 0'' i8™4i*i9 + .°3i'56:'38 0.17 10 1 2" 1 9^3 7 6 37-75 7 22 42.60 0706 46.7 4 21 13 13 . 50 28.43 1747 25,6 8 9 26.29 7 2 3.35 074. 51-7 5 23 44 58 2 8 53.73 1784 320 9 12 14.14 6 41 26.22 0779 9 56.6 6 26 15 49 2 27 12.17 1822 38.5 10 '5 "^o '• 20 51.53 0815 10 1.7 7 28 45 89 2 45 23.67 1858 45-0 1 1 17 47 77 6 0 19.52 0853 6.8 8 3' «5 81 3 3 28.12 189s 51-5 12 2° 3355 5 39 50-48 0889 I 1 9 9 33 45 2 I 3 21 25.46 '931 12 58.1 13 23 18.64 5 19 24.65 0926 17. 1 10 36 .4 '3 3 39 15-61 1969 13 4-7 14 26 3.05 4 59 2.27 0963 22.4 11 38 42 56 3 56 58.5' 2005 '1 3 '5 28 46.80 4 38 43 53 1 000 278 1 2 41 10 52 4 14 3408 2042 18.0 16 31 29.88 4 18 28.63 '037 33-2 '3 43 38 01 4 32 2.32 2078 24.7 17 34 12.31 3 58 1775 ■075 38-7 '4 46 5 04 4 49 23 19 2115 31 5 18 36 5408 3 38 I ".10 I 1 12 44-2 15 48 3. 63 5 6 36.64 2151 38.3 19 39 35-21 3 18 8.81 ■149 49-7 i6 5° 57 78 5 23 42.66 2188 45-2 20 42 15-73 2 58 .1.05 1186 >° 55 3 '7 53 23 51 5 40 4'-23 2224 52-1 21 44 55-62 2 38 .7.99 1224 II 10 18 55 48 81 5 57 32-33 2260 13 590 22 47 34-89 2 18 29.79 1261 6.7 '9 0 58 -3 71 6 14 15 97 2296 14 6.0 23 50 '3-57 I 58 46.60 1299 12.5 20 I 0 38 22 6 30 52.09 2332 131 24 52 5'-65 I 39 8.57 -336 '83 2 I 3 2 33 6 47 20.65 2367 20.2 25 55 29.14 1 19 35 85 .374 24 2 22 5 26 06 7 3 41-63 2403 27 2 26 23 58 6.05 I 0 8.57 141 I 30.2 23 7 49 42 7 19 55-°3 2438 34.3 27 0 0 42.37 0 40 46.89 1449 36.2 24 10 12 41 7 36 0.81 2474 41.4 28 3 -8.15 0 21 30 98 i486 42.2 25 •2 35 04 7 5' 58.94 2509 48.6 29 5 53-35 -0 2 20.94 1524 48.3 26 14 57 32 8 7 49-39 2544 '4 55-8 3° 8 28.01 + 0 16 43 07 .561 11 54-4 27 17 19 24 8 23 32.14 2579 15 31 3' 11 2. II ° 35 40, Q3 .598 12 0.5 28 19 40 82 8 39 7.16 2614 104 Jan. I 13 35.66 0 54 32.5' ■635 6.7 29 22 2 06 8 54 34-44 2649 17-7 2 0 16 8.68 -+-I 13 1770 0.1673 12 130 30 I 24 22 96 -t-9 9 53.96 0.2683 15 25.0 1 3. Observations and Comparison Stai's. The observations were collected from the usual sources and it is believed that all have been included. The published data of observation were checked wherever possible by in- dependent computation and the parallax factors and reductions to apparent place were recomputed with the constants of the Berlin Jahrbuch. The comparison stars used in the observations are 47 in number. Their positions have been investigated with con- siderable care, perhaps with more than is strictly necessary in view of the large probable errors of the observations. It seemed safer however to reduce the errors in the star places as much as possible in order to make them negli- gible as compared with the errors of observation, especially as the case under consideration is one in which the available material is so scanty as to render difficult the attainment of that degree of accuracy which is to be desired. I am indebted to I'rofessor Leuschner for much of the star catalogue data obtained by him from the libraries of various observatories while abroad in 1895-96, and to Professor Schaeberle as acting director of the Lick Obser- vatory for his courtesy in allowing me to use the cata- logues at Mt. Hamilton. Practically all of the e.xisting catalogues were searched. The catalogue positions wer^ reduced to the beginning of the years 1894 and 1895 b^ Rreutz's tables (A. N. v. 134) which are based upon Struve's constants. Systematic cor- rections derived from the introductions to various catalogues, from vol. VII of the Bonner Beobachtungen and from Auwers' papers in A. N. Nos. 3 195-96, 3413-14, and 3463, were applied to reduce to the system of the Astronomische Ge- sellschaft. When systematic corrections could not be found or when they seemed uncertain the simple catalogue position was used. In a few cases however where the total number of observations is large the positions lacking systematic cor- rections were given zero weight especially if the observations forming the position were old or few in number. The weighting of the catalogues for the formation of the final positions is in accordance with the system published by Davis in his Declinations and Proper Motions of Fiftv- six Stars. This system was derived by a consideration it the probable errors of the catalogues concerned, and ii^ homogeneity has been tested by Dr. Davis in the introduction to his paper. Although established primarily for declinations it has been used for right ascensions as well. A few cata- logues have been used in the present paper which do not appear in the Davis system. Wherever possible these were weighted in accordance with the methods used in forming that system. After the reduction of the catalogue positions to the beginning of the year of observation and the application of systematic corrections and weights, the simple mean by weights was drawn for each comparison star unless there '^5 3606 86 ,vas some indication of proper motion, in which case the final position of the star and its proper motion were obtained )y a least square solution. The following list includes the computed weights of the cotpparison stars, although in the urther comjiutations they were all given equal weight. 1 n a 1 C o 111 I No. Epoch Mag., a Wt. P.M. _A^ w . P.M. , 1894.0 1 8.2 0'' o"'42?i7 4.4 _ — o°2%- 7'.'3 4- — 2 » 9.0 0 2 12.77 1.2 — — 0 24 40 6 I. — 3 92 0 2 2365 19 — — 0 27 32 I i.( — 4 7.2 0 9 10.48 6.6 — 0^0026 -H 0 42 27 7 6.( — S 8.5 0 14 35.86 I.I — -f- 0 34 39 5 I- — 6 7.0 0 54 23.05 14.5 — 0.0003 -H 5 55 0 2 13-! 4-' — 7 1895.0 7.2 0 58 20.13 4-4 — 0.0013 -H 6 12 4 0 — 8 8.2 I "3 59>5 3-6 — 0.0026 -+- 7 5° 34 5 3-6 — 9 8-3 I 17 22.44 2.7 — 0.0045 -f- 8 II 20 0 2-1 — 10 7-9 • >7 3634 2.7 — -H 8 38 22 8 2.7 — I I 95 I 18 5.16 1.0 — -1- 8 3' 45 5 •■1 — 12 9.2 I 20 1.93 I.I — -H 8 45 22 3 I.I — ■3 9.0 1 20 57.93 5-5 -+-0.0081 -H 8 59 3 Q 5-5 — '4 1894.0 8.9 22 21 49.41 1.8 — — 2 48 4' 2 1.8 — ■5 8.5 22 22 30.24 36 — — 2 25 53 2 36 — 16 8.8 22 24 8.06 1 6 — — 2 15 24 5 1.6 — •7 9.1 22 25 34.34 2.6 — — I 39 29 6 2.4 — .8 8.0 22 36 25.29 4.' — — 0 40 47 0 4.2 1 - '9 7.0 22 37 26.63 136 — 0.0017 — 0 39 29 0 I 2.4 1 -(-o'.'oo5 20 7-3 22 39 46.64 I 1.2 — — 0 12 5 0 .0.4 - 21 8.0 22 40 45.68 3-6 — — 0 15 M Q 36 — 0.094 22 9.2 22 43 22.47 1.2 — — 9 29 2g 12 — 23 8.2 22 45 3908 1.7 — 0.0028 — 9 53 >' ■7 — 0.062 24 93 22 47 50.54 I.I — — 9 39 24 I.C — 25 8.5 22 53 27.96 7.2 -0.0037 — 8 46 52 7.^ — 26 9.1 22 S3 47-53 3.> — — 9 7 46 A — 27 7.0 22 54 47.78 I I.I — — 9 26 53 •4 — 0.028 28 6.8 23 26 2.99 20.6 -1-0. 0106 — 4 39 59 19-3 — 0.179 29 9-4 23 26 13.55 I.O — — 5 0 2 i.et — 30 7.2 23 28 I 08 5' — — 4 59 10 5-^ — 31 9.8 23 28 3>.4i 1.0 — — 4 35 16 i.d — 32 9-5 23 28 52.18 1.0 — — 4 34 i> i.< — 33 7.8 23 29 47.28 6.5 — — 4 26 27 6.5 — 34 8-5 23 33 22.58 1.2 — - 3 32 47 1.2 — 35 94 23 33 5952 1.0 — — 4 3 49 I.O — 36 8.5 23 38 9.26 38 — — 3 54 > 3 5' — 37 8.8 23 38 5987 — — 3 2 42 I.O — 38 7.0 23 39 6.05 7-5 — — 3 45 47 7.2 — 39 8.9 23 39 56.92 1.2 — — 3 12 23 1.2 — 40 5-4 23 42 29.56 51.6 -1-0.0047 — 3 2' 3 45- -HO. 00 I 4' 9.2 23 47 3' 03 1.0 — — 2 19 5 I.C — 42 7-5 23 49 12.69 5-4 — 0.0031 — 2 32 8 5-4 — 43 8.4 23 53 6.14 > 4 — — 2 16 28 ..4 — 44 7.2 23 54 8.01 2.1 — — 2 26 27 2.1 — 45 8.0 23 57 3«.J9 5-6 — — I 29 0 5.3 — 46 9.0 23 58 49.15 2.2 — — 0 35 52 2.2 — 47 7.0 23 59 37.69 20.1 — — ' 5 30 «7- -0053 Some of the positions of the comet were referred by steps to the comparison stars of the above list. In such oases the intermediate stars are designated by ,/ , /■ . c . etc. and their positions referred to the proper comparison stars IS determined by the various observers are : 8; 3606 88 a - l^ -^-5'"37 -f- 3' 43-.S /'- 21 ^-4 56 -1- 8 110 ,-- 28 -1- 4 29-0 \d- 28 -+-2 49 -H 5 48.0 |,/-28 ^2 48 -+- 5 46.8 '• - 35 + C 2Q -\- 7 40 9 /-36 - 3 40 — 2 8.1 .K - 34 -^3 49 -1- 0 1.7 /'-37 + 3 25 -+- 9 a ! — 42 -4 5 69 — 7 12.0 ./ - 42 ^329 18 — 0 38 5 /■-44 "6 10 43 -+- 2 47 6 /-43 "4 5° 68 -\- I 30 6 Difference in S .li.r.irdant. ■■■) Measured by ISio w — 47 -4™ 10 68 — 10' 37' // — I -HO 12 29 — 10 3 6 P — I -t-O 22 72 — '3 P- 3 — 0 10 93 -+- 52 '/ — 5 -3 '5 10 — 3^ /• — 4 -+-2 28 40 — 46 ,v — 7 -1-0 18 12 -+- 8 / — 8 — 1 4 36 -(- 20 // — 9 — 2 3 04 -(- 17 T — 10 H-O 2 I '9 — '3 26 71' — 1 I — 0 8 '3 — 6 5° ■""- '3 -+- I 40 81 - 2 4> »d by Howe. ^) Me It was not until the Calculation had been finished and the results were being collected for printing that the discord- ance in the two values for the right ascension of r/— 28 was noticed. Doubtless a reference to the original measures would have revealed an error in one or the other and would have made it possible to somewhat improve the third normal place. 4 Comparison of Observations with Eph emeris. No. Date of obs. . Obs. 1 cp. i * Aa Par. 0-C A6 Par. 0- = 1894 1 1 I Nov. 21.66319 B 20.7 '4 -.-.3^4 -Ho'09 — o'o8 - ''45"5 -^6:'5 -+- 4 0 2 22.44399 J 10.5 '5 H-O 24.70 -HO 37 — 0 64 - 8 47-3 -H6.3 - 8 6 3 22.70271 B 12.4 15 -J- I 12.20 -HO 20 -HO 47 - 3 236 -H6.3 - 3 I 4 23.26008 J 5-5 .6 -HI .3.01 — 0 03 — 0 60 — 2 26.7 -+-7 0 — 2 8 5 23.68819 B .0.4 16 -t-2 2994 -HO .6 -HO 25 -4- 6 .8.0 -^6.3 - 6 6 6 2527933 Bi 4-4 (7 -HO 8.39 — 0 01 — 0 40 — 0 29.7 -H7.2 — 0 2 7 25-28594 . * 4.4 a -HO 9.25 -HO 01 — 0 69 — 0 20.2 -H7.2 -H I I 8 25-30237 4-4 a -HO 12.2. -HO 04 — 0 61 - 0^ ^-9 -H7.2 -H 0 I T 28.53.8. 1 S 4.4 ^9^ -+-3 25.84 -HO 08 ' — 0 29 -HIO 27.2 -H6.1 — 2 5 28.53181 » 4-4 18 -H4 26.98 -HO 08 — 0 49 -H.i 459 -h6.i — I 9 II 28.72265 B ■2-4 21 -HO 40.49 -HO 25 -HO 25 - 9 44 5 -H5-7 -H 2 6 12 29.36272 , J 6.6 20 + 3 30-52 -HO 22 — 0 56 -H 0 18.4 -^6.3 — 0 4 13 29.66878 H 10.10 /' -' 3'-55 -HO. 23 — 0 '9 -H ' 37-7 -t-5-9 — 0 9 «4 30-3'758 ; J 6.5 23 -HO 24.50 -HO '3 — 0 43 + 1 '5-3 + 6.3 -H 2 2 'S 30.661 17 ] B 12.4 24 — 2 46.64 -HO -HO 20 - 5 28.5 -*-5-8 - 3 1 16 Dec. 1.32973 J 4.9 22 + 5 36.83 -HO '5 — 0 26 - 5 26.3 -H6.2 -H 2 2 17 1,34110 , K - 27 — 5 46.74 -HO '5 — 0 41 - 3 48.4 -h6.6 -H 2 3 18 2 27433 ^ 4 4 26 -2 5.07 -HO 08 — 0 33 - 3 35-4 -H6.6 -H 0 4 '9 2.32149 J 6.6 26 -' 57-34 -HO '3 — 0 68 - 2 35.1 -h6.2 -H I 5 20 2.36307 i K 20.8 26 -« 4955 -HO 21 -HO 02 - I 43-7 + 6.4 -H I 3 21 336833 5 J 6.6 ^5„ -HI 22.39 -HO 24 _rL° i^ - I 45 8 -h6.o — 2 4 22 14.51562 Br 6.6 29 — 0 11.03 -HO 06 — 0 09 -H I 2.7 ^+673^ ^^6 2 23 •4-53396 4-4 29 — 0 8.15 -HO 12 — 0 16 -H I 25.6 -H49 + 5 3 24 '455057 W 6.6 30 -I 52-75 -HO 04 -HO 02 -t- 0 52.5 + 5-3 -^ 5 2 25 •5 5'355 Br -4 3' ,'■ — — ~ - 3 3' 2 + 4.8 -H 2 0 26 15 52347 » 4-- 3' .'• -HO 16.18 -HO 08 -HO. 16 — — 27 •5-53>82 -4 3' .'■ — ~ - 3 8.3 + 4-8 -H 2 6 28 .5-54189 " -3 32,^ — - 3 59 2 + 48 -+- 3 7 29 '5-55090 iBr&Hu 32.^/ 0 0.00 -HO. .4 -^0.55 — — — 30 15.62069 1 lO.IO 32^ -HO 1.44 -HO. 12 -HO.OI - 2 23.4 -1-4 9 -f- 3 I 3« 16.30764 44 33 + ' 7.97 -HO. .2 — 0.2 I + 3 49' -t-5-' -H 6 3 32 1 •7-5'495 Br 4- 35 -HO 12.29 -HO. 07 -HO. .4 ~ — ~ 33 1 17-52808 » -■5 35 — — — + 5 46.2 -t-4-7 -H 2.6 34 17.54065 » 4.- 35 -HO 16.67 -HO.. 2 -HO. 40 - — 35 '7-55356 J : i.- 35 -HO 18.67 + 0..S -1-0.34 - — — 36 '7-5594« -.2 35 — - - - -H 6 25.4 + 4.6 -1- 3 8 3606 90 No, Date of obs. Obs. Cp. -^1 Aa Par. o-c. ^^\ Par. o-c 1S94 — ^ 37 Dec. 1 7.62222 H I.I e./\ o" o!oo -+-o?i3 -o?44 o- IC-o -H4"7 -H 3-3 38 17 65716 » 10.10 e,f^ + 0 533 -HO. 20 -0.70 -H 0 3f 2 -t-47 -H 0.2 39 ■833499 J 6.6 ' "~38^1 — 2 41.62 -HO. 17 -HO.18 ~^ 3 si-s + 5-0 -H 4-4 40 18.64086 H 10.10 .< -HO 1.70 -HO. I 7 -t-o. 34 2 56.6 -H4.6 -H 1.9 4' 1866399 W 8.4 34 -+-3 5560 -HO. 24 -HO. 84 - 2 28.8 -+-5-0 -+- 0-3 42 •935855 J 4-5 40 -3 2029 -HO. 21 -HO. 02 - 0 12.3 -+-4 9 -1- 5-9 43 '958735 St 16.- 39 — 0 10.80 -HO. 2 1 -HO. 06 — — — 44 19.602 18 » -•5 39 — — ~ - 4 6.7 -H4.4 — 14 45 19-65045 H 14.10 40 -2 33-4' -HO. 19 -0.06 -H 5 34-6 -^-45 -H 2.1 46 20.60650 20 to // H-o 3.23 -HO. 10 -0.68 — 2 45-' + 4-5 -t- 2-3 47 21-59743 10.10 / -t-o 0.70 -HO. 08 — 0.39 -t- 2 39-5 -+-4-5 + 3 5 48 21.68884 W 10.6 J — 0 20.05 -HO. 27 -HO. 99 - 3 219 -t-4-7 -H 4-2 49 22.26650 P 55 k -1 301 -HO. 08 — 0.10 -H 0 .8.3 -^-5•l -H 84 50 22.52831 Br 4-- 41 -^0 5-03 -HO. 10 -0.23 — — — 5' 22.54074 -■5 41 — — — -H 1 1.9 -+-4-3 -H 0.5 52 22.54914 4- 4 1 -f-o 8.47 -HO. 14 -0.06 — — — 53 22 60561 H 15.10 / — 0 26.50 -HO. 09 -HO. 37 — 1 41.7 -H4-4 -1- 7-7 54 24.71044 W 6.3 Tf -4 9-44 -HO. 29 -HO 22 — 5 5f4 -H4.6 ( + 28..] 55 25.26631 P 5-5 45 — 2 42.29 -HO.08 -0.39 -H 4 29« -H49 -H 2.0 56 25.60427 H -■5 III — — — — 1 47 3 -H4 2 -H 2.7 57 25.6.772 2- III -+-0 17.74 -HO. 13 -HO. 23 — - — 58 26.34684 J 45 47 -' 58-70 -HO. 19 -HO. 13 -H 2 0.4 -+-4-4 -H 0.9 59 27.30877 c 10.10 46 H-l 19.45 -HO. 16 -0.77 — 9 8.0 -t-4.3 - 7-9 60 27.53061 Br&A -4 11 — — — — ' 5«-4 -H4.0 -H 0.7 61 27.54977 Br 9-- II -0 8.02 + 0,15 -0.52 — — — 62 27 58628 VV 9-5 0 — 0 1 1.01 -Ho.l 2 [+...8] — 4 8.2 -H4-5 -H 1.9 63 28.35724 J 4-4 I -<-2 10.88 -HO. 20 -HO. 29 -H 3 33-2 -+-4-3 + 5-2 64 28.61323 W 8.6 2,/ -1- I 20.21 -HO. 17 -HO. 38 -H 4 43-4 -H4.4 [-12.3] 65 3' 59973 H --5 1 — — -*- 4 8.7 -H38 -H 1.6 66 31.61426 » 4- 0-5 ', 0.8 » 6g, 05 0-5 In this connection the residuals in right ascension of the five observations Nos. i, 3, 5, 11, and 15 made by Barnard with the 1 2 in. equatorial at Mt. Hamilton require special attention. It will be noted that in the first normal place these are the only observations giving rise to positive residuals in a. The same is true of the second normal place with the exception of observation No. 20 which gives a small positive residual. After applying to the residuals of the Barnard observations the ordinates of the normal place curve in the manner above explained the numbers repre- senting the approximate errofs of observation were -i-o'22 -t-o?77 -+-o?55 -Ho'53 +0^48 The prevalence of positive errors of roughly the same order of magnitude would indicate the presence of some systematic difference in these! observations as compared with those of the other observers ; but the other observations entering into these two normal places were made by several different observers, and that a systematic error should exist in all these observations is out of the question. Upon request Professor Barnard kindly communicated the original data for his observations and they were rereduced, but without the discovery of any error in the published values. As to the possibility of his having made settings upon a different point from the other observers Prof. Barnard writes: »The comet was a faint object, and it is perhaps possible to have observed a different point from what others observed. My recollection is that the comet had a faint tail and a faint nucleus, consequently, unless it was well seen — because of its elongated character — one might not observe the precise center of the head, but from the fad J_ that it was very small he could not be far out in his settings. « 93 3606 94 Althougli no source could be found for the systematic difference the residuals were arbitrarily corrected by 0^5 1 which is the mean of the five quantities given above, and ;he resulting residuals were given the weight 0.5. With the application of these corrections and the new system of weights, the residuals were again combined to form normal place residuals with the following results : I II III IV V VI VII Nov. 23.70 30.68 Dec. 16.21 20.62 27.70 Jan. ig.oi 27.68 0-5 ' 6 ■0-374 -0.036 -0.022 0.00 I ■0.197 ■0.210 Mean Date Nov. 23.70 30.68 Dec. 16.04 20.49 27.92 Jan. 1901 27.68 -2:2 6 -I-0-37 -+-3.68 -t-3-73 -(-2. 1 I ^6,54 — o 48 Red. — o'.'o8 — 0.06 -(-0.02 -HO.O I 4-0.05 0.00 -l-O.O I The columns headed Red. give the values for reducing the Aa and /Id from the mean date of observation to •.he nearest Berlin mean midnight. The final for the normal places are I II III IV V VI VII Nov. 23 5 30 5 Dec. 16.5 20.5 27-5 Jan. 19.5 27-5 — 0:520 -0.378 -0.031 -1-0.02 I ; p h e m e r 1 No. Pos. 8 •3 '3 — 2"34 -HO. 31 -t-3-70 -t-3-74 -H2.16 -H6.54 ■0 47 5 9 10.8 10.7 9.1 5-7 No. Pos. 8 ■3 The weights for the normal [ilaces are the sums of the veights of the individual observations in the normal place jroups. 6. Computation of Perturbations. The effect of Jupiter, .Saturn, Mars, and the Earth ipon the positions of the comet during the period of vi- ibility were computed by the method of variation of con- tants. The masses used were : Jupiter The Earth 2680337 calculation was based upon Chandler's elements .\. J. No. 338, referred to 1900.0. They are: Epoch Dec. i.o 1894. Equino.x 1900.0. .)/ = 8° 2 2- r.2 •''I = 345 24 '38 a = 48 48 529 '■ = 2 57 55 5 ff == 34 52 56.9 « = 60571520 Dec. 1 0.0 1894 was choosen as the epoch of oscu- lation, and the perturbations were computed for 20 day intervals beginning with 1894 Nov. 11. The resulting per- turbations in the elements were : Date zJ/ //a Jt Jjt zi/. 4« 894 Nov. I 1 -HO'.'0 2 4 -o'.'25 7 - 0-776 - 4'-'7 88 -3"4i6 -Ho'.'oio7 Dec, I -HO 013 — O.I 16 - 0-525 - 1.804 — 1. 100 -HO. 0062 2 I — 0.02 1 -HO. 127 -H 0.809 -H 2.082 + 0.987 — 0.0091 89s Jan. 10 -0.086 + 0-335 -H 3.291 -+- 7264 -H2.371 — 0.0361 30 -0..96 -HO. 302 + 6.933 -HI4.078 -H2.512 -0.0753 Febr. 19 -0.371 — 0.262 -HI 1.708 -H22.765 -Ho.8.4 -0.1268 From these the values for the dates of the normal )laces were found by interpolation — the calculated values )eing checked by a graphical interpolation from the curves v by means of ormed by plotting the perturbations in the elements. The (uantities desired, however, are the efiects of the pertur- lations in « and 6 and these were derived from the per- urbations in llie elements by means of the differential orniulae given in the following section for determining the lefinitive corrections to the elements. These formulae are he ones given by Schonfeld in A. N. No. 2692-93 and since interpolation gives the following hey involve the three elements x, X and r in place of the for the dates of the normal p usual three /, hi and m, the perturbations in the latter ele- ments must be transformed into perturbations m x , X and 6x = dfo -H coa / dj J d.i = sin CO d/ — cos "o35 -t-o'.'oogo — o'.'74i -0-013 -i-o:'oi7 3° 5 — 1.888 -+- o-73> -1-0.0064 -0 544 — 0.009 H-o. 01 I Dec. 16. 5 -H 1. 115 - 0-558 — 0.0049 -HO. 415 -1-0 008 — 0.008 20. 5 + 1.970 — 1 028 — 0.0086 -1-0. 761 -HO. 015 — 0.014 27. 5 -4- 3.600 — 2.057 — 0 0165 -(-1.484 -HO. 029 — 0.026 1895 J'ln iq. 5 -+- 10.267 — 7-624 -0.0531 -(-4-874 -(-0.109 — 0.076 27. 5 ■+- 13 128 -10.546 — 0 0706 -(-6.415 -HO. 152 -0.095 96 These quantities were then substituted into the differ- ential formulae whose coefficients are given below and the corresponding perturbation.s in a and 6 were found to be: Obse r vat ion - - Un disturbed Pos tion. Date Att cos 6 Ad I 894 Nov. 235 — o:'347 -o:'i4 7 II 30-5 — 0017 — 0.008 III Dec. ,6.5 — 0.266 — 0.128 IV 20,5 - 0583 — 0.281 V 275 - '439 — 0.701 VI 895 Jan. ■Q 5 - 7.009 -3340 vn 275 — 10.000 — 4.668 Applying these perturbations with the reversed sign to the normal place residuals, after the right ascensions of the latter have been multiplied by the cosines of the decli- nations we derive the residuals Undisturbed Position minus Ephemeris. These are the absolute terms of the equations of condition used in determining the definitive osculating elements. Undisturbed Position — Ejihemeris. 1 II in IV V VI VII 1895 Ja Date Nov. 23.5 305 Dec. 16.5 20.5 275 iQ-5 275 I II III IV V VI VII Aa cos 0 -7r28 -5-57 — 0.20 -(-o.go -1-1.44 -1-4 03 -(-6. go Date 1894 Nov. 2 Ad The residual in 6 for the normal place of Jan. 19.5 appears to be discordant when compared with those of the other normal places. That this is actually the case becomes more certain when it is noted that all of the normal pla except this one depend upon from 5 to 13 observation' while this is based upon only 2, Nos. 68 and 69, and the latter of these depends upon an assumed coincidence betweer comet and comparison star. It was suspected that it woulc be impossible to pass through the normals an orbit which would give a good representation for the declination of this date, and a preliminary solution proved this to be the case Although the errors of the positions forming this norma are not larger than those occurring in a number of othej observations they are of the same sign, thus preventing com pensation. A consideration of all the data led me to be lieve that the retention of these observations as a separate normal place would add nothing to the accuracy of the results. Nor did it seem advisable to combine them will the normals of Dec. 27.5 or Jan. 27.5 on account of the magnitude of the intervening intervals. The declination.' were therefore e.xcluded from the calculation while the righ ascensions, not presenting any special discordance, wen retained and given a small weight. 7. Differential Formulae and Least Square Solution for Definitive Elements. Transforming the ephemeris positions of the come for the dates of the normals to the equinox of 1900.0, whicl has been choosen for the calculation, they become : 1895 Jan. 235 30.5 .6.5 20.5 275 •9-5 27-5 336° 35' 53"78 341 43 40.94 352 56 19.98 355 37 46.84 o 14 25.27 14 37 14.90 19 23 38.89 -12° 10' 5 5r96 — 9 46 1 1 . 10 — 4 '6 47-57 — 2 56 3025 — o 39 7-03 -(- 6 15 48.73 -+- 8 25 1.24 logz/ 0.02 I o I 5 0.045424 o. 103769 0.118683 0,144884 0.229572 0.257886 These coordinates togpther with Chandler's elements referred to the equinox of 1900.0 form the basis for the calculation of the differential j formulae, which, as has already been stated, was carried oijt according to the method of Schonfeld. The computation )f these coefficients was checked 9.9794 dx 9-9645 9.9342 9-9275 9.0159 0.6 0.6; by assigning arbitrary variations to the elements and deter mining the resulting changes in « and 6 both by the differ ential formulae and by the ordinary ephemeris formulae. The eciualions of condition thus derived are : 65 dil/o -I- 2.3668,1 d/^ -I- 9 48 -I- 2.2762,1 -+- 9 91 -H 2.01 I9„ -(-9 1.9257,1 -(- o 1-7397.1 -t- o 5U2 5:74 log V / .0488 A(p -+- 9.6572,1 dX -(- 9.4308,, di' = 0.86 19,, 0-3854 .6080 -(-9.6362,, -H 9.4873,1 = 0.7459,, 0 5167 -9723 -+- 9-5599.1 -1- 9-5660,, = 9.2967,, 0.4286 .0184 -H 9.5354,, -1-9-5773.. == 99538 0.4722 .0788 H- 9 4874.. -(-9-5907n = 0.1581 0.3702 97 3607 6) 9 8882 d;( + 0.4481 d.I/, + I 3.76CI/ + 0.1897 ( kp + 9.2873nd X + 95927 7) 9 8808 + 0.4.68 + I 6076 + 0.2102 + 9.2012,, + 9 5816 8) 9 6025 + 0.2756 + I 9662,, + 8.9129 + 0.0463 + 9-8199 9) 9 6141 + 0.2782 + I 8881,, -+- 9-3537 + 0 0029 + 9.8540 lo) 9 6219 + 0.2566 + I 6>57n + 9.7051 -+- 9 8937 + 9.8998 ■0 9 6205 + 0.2462 + I 5M9n + 9.7506 + 9.8643 + 9.9062 12) 9 6155 + 0.2244 + I 27 '5m + 9.8100 + 9.8108 + 9.9141 ■3) 9 5623 + 0.0837 -+- ' 40 1 1 + 9.9030 -^- 9-539° + 9 9194 l> The coefficients are logarithmic and the last jf the equations of condition. Applying these weights X = [0.4900] dx y = [ 1. 1600] d.l/| i = [2.8000] (!// ind further choosing 1.2700 as the logarithm of tli quations of condition (logarithmic coefficients) : = 0.6050 = 0.8387 = 0.3404 = 9-505" = 0.5832 = 0.6042 1= 0.4564 = 0.6232 column contains the logarithm of tht square root of the weight and introducing new unknowns defined by the relations ■ ■■ I I 0.3010 0.3854 0.5167 0.5147 o 4795 0.3779 0.3495 / = [0.5200J d^ 1/ = [0.5200] dX lii = [0.4200] d/' (A) unit of error, there resulted the following weighted homogeneous 0 2) 3) 4) 5) 6) 7) 8) 9) .0) >•) 12) >3) 9.8748 9.9912 9.8728 9.9097 9.7961 9.4862 9.6918 9-4979 9.6408 9.6466 9.6100 9-5034 9.4218 8819 99>5 8477 8764 7476 3761 5578 5010 6349 6.13 5657 4423 2732 + 9.9522,, -*- 9-9929.. + 9.6405,, -+- 9-5979n -I- 9-3°99.. + 8.6056 + 9.1086 -+- 9-55'6., + 9.6048,, -t- 9-3304.. -I- 9-'944., + 8.8494,, + 8.9506 + 8. 9 142/ + 9.52260 ' + 9.6047 + 9.6329,, + 9.8809 + 9.4685,, + 9.9706 + 9.4876,, + 9.9290 + 9. 3376,, -I- 9-7577 -I- 8.8553,, + 9.99 I 2 + 8.98220 8-7783 9-3504 9.6998 9.7101 99117 9.9996 9.8884 9.8238 9.6679 + 9.6687 97325 -+-9-3685 -+- 9 3962n + 9.58400 -+- 9-57460 -+- 9.62950 + 9.5409,, -1- 9.26070 + 9.46260 -I- 9-7853 -+- 99507 -+- 9 9945 -+- 99657 + 9.8720 + 9.8489 99260 4553n 1560 2583 4230 8697 4558,. 75'8 8279 8137 5643 7027 The usual least square method gave as equations (numerical coefficients) : + 4.2893.V+ 4 + 4.0257 +3 — 2.7810 — 2 + 3-8095 + 3 + 0.1734 +0 + 0.2153 + o 0257.1 8040 7452 4000 '372 I 669 27452 2-5497 1. 4801 0.0256 0.0375 + 3.8095/+ o + 3.4000 + o — 1. 4801 + o -t- 47055 -t- o + 0.3146 +3 + 0.3909 + 3 i734« 1372 0256 3«46 5>63 8867 + 0,2153 + 0.1669 + 00375 + 0.3909 + 3.8867 + 4.8210 + 0.2722 + 0.5226 - 16937 - 1.7799 - 1.6398 - 2.0085 The similarity of the coefficients of the first and second ind of the fifth and sixth equations indicated that one or nore of the unknowns would be affected with considerable racertainty, and a preliminary solution showed .v and j to be indeterminate. Rewriting the normals so that these un. knowns appeared last in the solution the following elimination equations were found Jogarithmic coefficients) : 0.40649 : + 0.170300/ + 8.40909 , 0 2) 0.58504 3) 4) By successive substitution 95«7« 0-542; + 8.57461 70 + 9.61563 -*- 0.58557 9.71950 -+- 0.44420n . -+- 0.34145 + 8.I27IO + 7-77815 43858„> + 0.228840 25682 00432 389«7 + 0.441400 + 0.141790 + 9.194930 and r through (log. coefficients) : / and ; were expressed as functions of 70 = 8.058650 .V + 7.66967n_i- + 9.47543 > 1 «= 7.94448 +7. 3541 1 +8.82905 j 1 ^, t^ 9.756040 +9.671500 +9.83286 I ^ s = 9.88069 + 9.90540 + 0.02295 ! ind substituting these into the original homogeneous weighted equations of condition the following series was found for he determination of .v and r (logarithmic coefficients) : 99 0 2) 3) 4) 5) 6) 7) 8) 9) lo) ii) 12) ■3) Check 3607 3414 55f'3 2455n 2900,1 0755.1 0792 5'q8 G043 3424 97 7 7n 9395n 7-5119. 71139 8.5587.1 8.8817 -+- 7-477IU + 8.6702,, + 7-$563n + 8.8621U 7-3802,, -+- 7-3222 + 7-7559 -+- 7-»i39 -h 6.6990 7-7993 = 90328 = 9 2750 7.7672 7.)6i4n -H 9 3228 7.176.,, 6-9777r 7 49'4 -+- 9 1638 -+- 9. 1 I 26 -t- 92653 »-2553 [;/;/. 4] :^ 0.1829 ["'■'''] = 0.1829 New unknowns defined by ■^' = [8. 5200], V .)'' == [7.76oo]j' were introduced to secure homogeneity and the resulting series was solved by least squares. The normal equations for .v' and 1' were (numerical coefficients): (C) -+- 2.9052 -+- 2.8490 2-6490. 2.8292 -+■ 0.2745 + 0.2601 Here again the similarity in coefficients denoted un- certainty in the solution, but as r' appeared to be the more uncertain of the two, .v' was expressed in terms of r' giving (logarithmic coefficients) : ■v' = 9. 99i52„.i'' -h 8.97544,, (D) This value for .v' substituted into the equations of condition for ,\ ' and 1' gave the following series for the determination of r' (numerical coefficients): l) — 0.08s i.r — 0.0988 2) -h 0-1193 -H 0.0659 3) — 0.0001 — 0.0034 4) — 0.0482 — 0.0172 5) — 0.0647 -+- 0 0970 6) -+- 0.0095 -1- 0.0736 7) H- 0.0104 — 0.1261 8) — 0.0732 -+- o-'595 9) -t- 0.0217 — 0.0004 0) -t- 0.0293 — 0.1832 i) — 0.0031 — 0.12 10 2) — 0.0230 -1- 0-1433 3) + 0.0056 -1- 0.1328 [" '•5] = 0.157 0 [n"^," A new unknow 0.1569 was introduced such that 10 I'" = r ' and the series solved for r" giving whence log.." logj' 0-40572 9-40572 The residuals for the normal places were found by substituting j' into the above equations of condition. When squared and added ['■?•) = o-'547 while from the elimination, as a check, [;/;/■ 6] = 0.1546 Then by successive substitution of 1' into (D), of .v and )■' into (C), and finally of .v and r into (B) the most probable values of the unknowns were found to be log .V = 1.0167,1 log 1' = 1.6457 log: = 1.4584 log// log /c' 9-3244 Restoring the original unknowns by (A) and reintro ducing the second of arc as the unit of measurement the following corrections to the elements chosen for the cal culation were obtained. log d(p = 1 .9009,1 log dX = 9.6306 logdj' = 0.1744 nd U were derived fronc log dx = 1.7967,, logdyl/o = 1.7557 logd^M == 9.9284 The corrections to < (ix, dk and dr by d/ = cos 0) dv -+- sin oj dX sin / dQ = sin co dv — cos co dX d {i,l -^ co) == dx -+- tg V2 /■ sin / dQ d (,Q, — w) = — dx -H ctg '/2 / sin / dil As thus determined the final corrections to Chandler's elements are dJ/o = -t-5 7:'o dfi = -f-o'.'8479 d(p = — 7 9'.'6 d/ = -t-o'.'3 ohl = —29.5 djr = — 6 2'.'7 dm = —33-2 dZ = -5'.'7 ifhence the definitive osculating elements: Epoch 1894 Dt yl/n = a = »" 22 5».'2 345 23 III 48 48 23.4 2 57 558 34 51 37-3 605-9999 ± Osculation 1894 Dec. 10. o. "■■ ± 4-2 ± 4-4 I ±27.7 ± >-4 + 7-' '.'0665 190C The a[ipended quantities are the mean errors an( are based upon the standard value for a single observatioi of unit weight computed from the residuals of the equations of condition To test the accuracy of the least square solution thi definitive corrections were substituted into the origina equations of condition ; the resulting residuals were squared multiplied by the proper weight, and added with the resol [''7'] = 54 -o The value of [////• 6] from the least square solutioi was 0.1546. Exjjressed in seconds of arc [// )i ■ 6] 53-6 3607 The agreement is satisfactory in view of the fact that jnly four places of decimals have been used in the solution. The reduction in the sum of the squares of the weighted ■esiduals is from "349''3 to 54''o Finally the definitive elements were used to compute the undisturbed positions of the comet for the dates of the normal places. To these the perturbations were applied and the results compared with the observed positions. The out- standing differences, in the sense obs. — comput., resulting from a six place calculation are tabulated below, together with the residuals obtained by direct substitution of the definitive corrections to the elements into the equations of condition. Aa cos 8 A8 Def. Elem. Difif. Form. Def. Elem Diff. Form. 1894 Nov. 23.5 -+-o?09 -l-o!o7 -o'.'9 — 1^1 30-5 — 0.04 — 0.03 0.0 0.0 Dec. 16.5 -t-o.oi 0.00 ■+■ 1.2 i -t-i.o 20.5 0.00 -t-0.01 -4-0.8 -(-0.8 27-5 — 0.05 — 0.04 — 1.2 — I.I 1895 Jan. 19.5 — 0.03 — 0.08 — — 27s -+-0.07 -f-0.07 -«3 — I.I In order to determine the effect of small variations in djU upon the sum of the squares of the weighted residuals the values of the increments to the other elements were substituted into the weighted observation equations and the numerical terms were summed. The resulting equations of condition for dfi were found to be : i) 2.7522„d^< -+- 1.4085 = o 2) 2.7929,, H- 1-4527 3) 2.440511 -^- 1.0990 4) 2-3979.1 -t- 1.0549 5) 2.1099,, + 0.7741 6) 1.4056 -+- 0.0341, 7) 1.9086 -t- 0.581 ii 8) 2.3516,, -+- .0159 9) 2.4048,, -+- 1.0634 0) 2.1304,, -1- 0.7761 •) iQ944n -h 0.6405 2) 1.6494,, + 0.3368 3) ,.7506 + 0.3860 which the coefficients are logarithmic, and the logarithm of the unit of measurement is 1.2700. The definitive value for log dfi was found by the least square solution to be 9.9284. The variations 1899 August. were successively applied to this logarithm and the resulting values were substituted in the above series of equations. The residuals found by each substitution were scjuared and added. The following table exhibits the relation between the sums and the variations assumed for d/t . \ogd(i 99384 9-9304 9 9284 9.9264 g.9184 Adfi + ef.'oi99 +o'.oo4i aoooo — q.0037 — oi.0192 455" 7' 54 68 427 My acknowledgements f<.r information and assistance are due to many — to Professors de Ball, Bruns, Kortazzi, Pickering and Searle for observations from the unpublished zones of the Astronomische Gesellschaft Catalogue ; to Pro- fessors Barnard, Davis, Holdcn, Howe and Javelle for in- formation concerning observations and star catalogue dan, I to Prof. Harkness for special | meridian circle observations of comparison stars; and especially to Prof Leuschner for much assistance and many valuable suggestions concerning j the arrangement and prosecution of the work. Frederick H. Scares. r-'v YD ina^ H^