a first Approximation tojhe Orbit of a Comet, 353 
the terms of the final equation, which are of the fourth order, 
being always inconsiderable ; they may be omitted at first, 
and then only taken into the account when a near value of p 
has already been obtained. 
It has already been shown that ^ ^ whence ^ 
= — • cos. x°; therefore, since cos. is in every case 
positive, the quantity will have the same sign as f : 
and thus, from what is proved in No. 5, we infer that the 
comet's distance from the sun will be less or greater than the 
earth's distance, according as the sign of is positive 
or negative; and the two distances from the sun will be equal 
when — r-x — = 0. These observations will often enable us 
oos » 
to assign such first values of p as will lead to a solution with- 
out many trials. 
The preceding method will always give one solution. To 
prove this : let B denote the chord of the earth's orbit drawn 
between the places of the planet at the two extreme observa- 
tions ; and further, let 
R+R'+B 
cos. u = 1 i ^ 
■ cos. = 1 
R+R^^B ^ 
2 
then if we apply to the earth's orbit the formula for finding 
the time of describing an arc of an ellipse by means of the 
chord of that arc, and the two radii vectores drawn to its ex- 
tremities we shall get, 
$ =: (^u — sin. u) — (w' — sin. «'). 
Let cos. m = 1 — • then cos. u = cos. w — j B ; and 
* Mec. Celeste, Prem. Part. Liv. II. § 27 formulc a. 
MDCCCXIV. X 
