155 
a first Approximation to the Orbit of a Comet. 
p r= c/ 3 , 6* will be infinitely great, and ~ will be infinitely little. 
Therefore there will always be one value of p, between the 
4 . 0 * 
limits 0 and cn, that will satisfy the equation 
lo. Before applying the method here proposed to examples, 
it will be necessary to investigate the formulas by which the 
elements of the orbit are to be computed. 
Let V and v' denote the true anomalies of the comet at the 
two extreme observations; or the angles which rand r' make 
with the perihelion distance of the parabolic orbit; then in the 
triangle formed by r, r', and 6, the angle opposite to b will be 
equal to the sum or difference of u and v': therefore + 
r'^ ~ 2 rr' cos. (u ± u') : but it has already been shown that b* 
~ ^2 _|_ yn — gY . therefore V = r/ cos. (u + u'). Let the 
angle contained by r and b in the same triangle, be denoted 
by (p ; and that contained by r' and b, by (p*: then it is plain that 
cos. p 
r — r' cos. (v d: j # 
= -rf ^ ; and cos. p = 
r — r cos. 
: whence 
we get the following formulas for finding the angles p and p', 
which are the angles that r and r' make with the chord 6, viz. 
r*-V 
cos. p = r- 
^ T ,h 
cos. p' = 
r'*-V 
r',h * 
observing that regard must be had to the signs of the cosines, 
as well as to their values in numbers. Again let 
— cos. 
then the angles 4^ and 4^', the cosines of which differ only in 
their signs, will be supplements of one another ; and, as it is 
X s 
