on the determination of the Orbits of Comets. 73 



which, probably, have not occurred to the distinguished mathema- 

 ticians, Mho have laboured on the theoretical difficulties of the 

 problem. 



" Every method," the author remarks, " which I have yet seen 

 requires that the observed geocentric places of the comet be reduced 

 to longitude and latitude. The places mnst, however, in the first 

 instance, be observed in right ascension and declination. Now, 

 the conversion of right ascension and declination into longitude and 

 latitude is one of the most troublesome operations that commonly 

 occurs. It requires the use of 7-figure logarithms, and is liable to 

 errors. An alteration in one original JR, or declination, requires a 

 complete repetition of the calculations ; and when all is done, the 

 elements of the comet's orbit are obtained as referred to the ecliptic ; 

 and, for convenience of calculating predicted places, it is generally 

 necessary to refer them back to the equator. For these reasons, 

 it has long since appeared to me desirable that the orbits should 

 be deduced at once from the right ascensions and declinations. 

 Since 1 have become familiar with the instruments used for obser- 

 ving comets, an additional reason has suggested itself. It is known 

 that on the assumption of a parabolic orbit, the equation given by 

 three complete observations, or by observations which furnish the 

 JR and declination at a certain time, and their first and second 

 differential coefficients, are one more than are necessary ; and, 

 therefore, it rests with the computer to use his discretion in reject- 

 ing one of the observations. Now, it often happens that the instru- 

 mental or observing errors in right ascension are of an order quite 

 different from those in declination ; and, if the method of computa- 

 tion proceeds at once from right ascensions and declinations, the as- 

 tronomer can at once determine which of the observations ought to 

 be rejected in the calculation, on the score of possible inaccuracy in 

 the observation." 



The principal objection which has been made to Laplace's 

 method is the trouble of investigating the differential coefficients of 

 the spherical co-ordinates. It must be avowed, that the process 

 pointed out by Laplace is very laborious ; but it may also be as- 

 serted, that the principal part of the labour is introduced without 

 any necessity. Three observations, made at proper intervals, are 

 sufficient to give the motion of the comet in either direction, and its 

 two diiferential coefficients, with an amount of labour that is quite 

 insignificant ; or a great number may be introduced by a simple 

 jDrocess well known to every computer, and involving very little 

 trouble. In the present paper it is shown, that by adopting for 

 epoch the middle time between the first and second observations, 

 the great mass of the calculations of every kind may be made imme- 

 diately after the second observation ; and the operation, therefore, 

 completed in a very short time after the third. 



The author divides his paper into three sections. In the first he 

 gives the " theory," or analytical solution of the problem. On sub- 

 stituting, in the general equations of motion, the right ascension 

 and declination of the comet at the epoch with their first and second 



