Captain Hall's comet seen at Valparaiso. 
53 
Computation of the Elements. 
The method used for the first approximation was that of 
M. Laplace.* The observations chosen were those of April 
8, 11, and 14. It considerably diminishes the length of the 
calculation to use only three observations, where it can be 
done. In the present instance it is not likely, had a greater 
number of observations been used, that the superior accuracy 
derived would have compensated for the additional trouble. 
The result from the three observations, is as exact as would 
be necessary on common occasions. Here, on account of the 
small perihelion distance and small motion of the comet in 
heliocentric longitude between the first and last observations, 
the method of M. Laplace,^ “ Determination exacte des ele- 
mens de l’orbite, &c. &c.” is inapplicable; the errors of the 
motion in heliocentric longitude being greater than the motion 
itself. 
The extension however of that method, as indicated by 
M. Delambre, £ might succeed in this instance, but the for- 
midable calculations necessary are sufficient to deter one from 
the attempt. 
The method which I gave in Vol. XIII. p. 189, of the Trans- 
actions of the Royal Irish Academy, also, in this case, requires 
to be extended. It will be necessary to introduce the squares 
of the variation of the perihelion distance, and of the cor- 
rection of the time from perihelion. 
The manner in which my method is extended to this 
case will be easily understood from the following detail ; 
and also the change I have made in deriving on common 
• Mec. Celes. Tom. i. p. 221, Sec. 
t Astron. Tom. 3. p. 384. 
t P- 225. 
