a first Approximation to the Orbit of a Comet, 
(t'R cos. e + o-R' cos. e' — o-"R° cos. = 
T- ~ • { — •"■'‘Rcos.e — T®R'cos.e'+ (T+T')®R°cos.e°} 
or, which is the same thing, 
r'R COS. e + < tR' cos. e'— <r"R'’ cos. • (p • R°cos.e* 
— T • — R^) • ^ 
— T • (;s — i^) ■ (^' “®- ^°) ■• 
but (R cos. e — R° cos. e°) and (R' cos. e' — R° cos. e°) are 
quantities of the first order ; consequently the two terms that 
contain them, will be of the fourth order, and may therefore 
be rejected : thus we get 
<r'R cos. e 4- o-R' cos. e* — o-"R° cos. == - IL. — . 
1 z \ ro* Ro* / 
R° COS. e°. 
And, in the very same manner it may be shewn that 
o-'R sin. e <rR' sin. e' — o-"R° sin. e° =a . 
R° sin. e°, ^ 
Let the values just obtained be substituted in the two first 
of the equations (7), and they will become 
€t'p cos. X cos. r + a-p' cos. x' cos. c* — <r"p® cos. x® cos. r* = 
o-'p COS. X sin. c + o-p' cos. x' sin. r' — a-”p° cos. x° sin. c° = 
- - 573) • 
Further let n denote an angle, to be afterwards determined; 
then, by combining the two last equations we shall readily 
obtain, 
<r'p cos. X cos. [c — 7l) o-p' cos. x' cos. (c' — ;z) — (t"p° cos. X® 
X cos. (6°- n)=- IlM . (± _ . RO cos. 
T 2 
\ 
r 
i 
