186 CO M E T p =0-7269562, obtained from the errors of the second and third trial in the same way that the third value of p was inferred, we may substitute seven-figure logarithms and work more closely; it will thus be found that the error in interval corresponding to the fourth assumption for p is reduced to + d 00167, or less than 2| minutes, and if we are only seeking an approximate knowledge of the orbit, the direct calculation of the elements might proceed with this fourth value of p. However, to make the computation in this example a little more complete, we work out two further hypotheses, and finally adopt for the correct value of p ... 0-7268994, with which the calcu lation is as follows : -0-2296858 -fO 4944306 R 2 .... 0-969880 Log. p ... 9-8614743 Log. p 2 .. 97229486 No.(2.).. +1 649602 No. (6.)... 4- 0-1 83555 R " 2 966903 Constant... + 0-109114 No. (4.)... 1 576893 + 0-292669 2-543796 No. (5.) ...- 0-132660 No. (3.) ... -0-312487 P.... + 0-160009 r " 2 ... 2-231309 Log. 7<; 2 .... 9-2041444 Log. r " 2 ... 348559S Log. & 9-6020722 Log./ 1742799 Tc 0-400011 / ... 1177251 Ik . . 0-200006 r - l ^ 6l >< iff fff (r +?)... 1 335604 k ... 0-200006 B = i(r + /")+i/<: 1-535510 D = 4(r / + r" )-4/fc 1-135498 Constant 1 4378117 Constant 1-4378117 J 52-14235 (1.) -0-0911601 (2.) +0-2173792 2-619482 - No (1.)..- 1-233560 Log. B 0-1862526 Log. D 0551 863 z" 33 15822 iLog. B 0-0931263 4Log. D 0-0275932 , ,,~ 9-6333570 + 4748536 - ^ 1 " Log, p... 9-8614743 Log. p 2 .. 97229486 r 2 1-385922
- * _*" 1S-PS.11 3
Log. iS 17171906 Log. ?! 1-5205912 t " - 1 18 98412 which may be considered a perfect agreement. We have thus for the direct calculation of the orbit (3.) -9-4948313 (4.) + 1978022 Log. r 2 . 1417388 -9-2612653 + 9 54081S6 Log. /.. 0-0708694 Log. p ... 9-8614743 Log. p 2 .. 97229486 Log. p ...9 8614< 43, and, since p Mp, we find Log. p "... 9-5314543. Log. r ...0 0708694, and Log. /"... (5.1 -9-1227396 (6.) + 9 2637672 With the aid of the formula (VI.) the heliocentric longitudes and latitudes of the comet at the first and third observations are found as follows : 9-5314543 Log. p 9-8614743 Log. sin. (a - A ) -97055368 (1.) -9-5670111 Log. p 9-8614743 Log. cos. (a -A ) + 9-9352968 (2.) + 97967711 No. to (2.) + 0-6262837 Subtract R ... , 9848248 Log. p ".... Log. sin. (a" -A ") - 9 9464841 Log. p " 9-5314543 Log. cos. (a "- A ") + 9-6696554 (5.) + 9-2011097 No. to (5.) + 0-1588948 Subtract R " 9833124 (3.) No -0-3585411 (6.) No . -0-8244176 Log. (3.) -9-5545389 Log. (6.) ..-9 9161473 . L O(T fin (0 A ) I 0147 00 ^ -^ -.fnrr tin In " " + 9-5617011 Log. (6.) fi "- V" e -A .... 225 49 21" 4 200 l 50"-8 27452 15" Add A 255 32 49" -0 Add A " " Log. sin. (0 " - A ") 121 29 10" 4 114 54 5" 8 Low. sin. (0 -A ) -9 8556316 . - 9 53469 9 1 (1.) iQ.QJQO/fiQ Log. sin. (fl -A ) Ll0gl < C0&- X) J lii "" ) > Log. sin. (0" -A ") ^ fe v/ wa " n . Log. (p . tan. # ) Log. (/ sin. A ) +0 0248453 Lof. to ", tnn. R" Locr. (/". sin. " + 0823706 . . Lo"-. tan. A " . . Lo<* tan. A +0 3134658 .. +0 1391243 A " A + 64 5 7" 9 . + 54T27" Lof. sin. A " Log. sin. A + 9 9539758 . 9-9080907 Log. (/.sin. A ) 0-0708695 . 1749 - Log. sin. A Log>) The equations gave Log. / 070869 4 . g :. s !" .. A . Log./" Log... -9-8243819 Log. sin. (0 + ")... +9-0517209 .. 0-1742799 BO that our first verification is complete. The comet s heliocentric longitude at the third observa tion (0 "} being less than that at the first observation ($ ), the motion in the orbit is retrograde or contrary to the order of the signs, and we therefore proceed to determine the longitude of the ascending node ( ft ) and the in clination (*)by the second set of equations in (VII.); thus ...121 22 10"-4 0" ...114 54 5" 8 ft... 282 12 48"-l 6 "... 114 54 5 8 -0-7726610 Log. tan. A ... +0-3134658 ft-0 "... 16718 42"-3 Log. tan. (ft -0 )... -9-5408048 ft -0 ...16050 37"7 Add ...121 22 10" 4 ft. ..282 12 48" l Then for the arguments of latitude at first and third observations (u , u "] Log. tan. (ft -0 )...- 9-5408048 Log. tan. (ft - ")... - 9-3524609 Log. cos. i... 9-1971480 Log. cos. i... 9-1971480 -0 "... 6 28 4" 6 Log. cos. (0 - ")... +9-9972269 Log. tan. A ... +0-3134658 +0 3134658 (1.)... + n.aino97 sm.(ft-0)... +9 51606ol No. to(l.)... +9-0440079 tan.*.... 0/9/400, Log. tan. u ... -0 3436568 Log. tan u " 1553129 Nat. tan. A "... +1-3776038 i. ... 80 56 27" 5 u ..11 4 22 57" 6 u " - it. 10 Sfi 9"-ft mill Ifti. " " ... 124 57 59" 6 ,, K e i7 5?i"-n _
Nat. tan. "- No. (I.)... -0 6673934