a first Approximation to the Orbit of a Comet. 
cr'p cos.i sm.h-\- o-p' cos. z sm.h' — o-"p° cos. sin. = 
o-'p sin. i sin. h + (rp' sin. i sin. /z' — c-"p° sin. x° = o: 
and finally if we subtract the two last of these equations after 
having divided them by cos. i and sin. f, we shall get 
//of cos.x°siti.(c° — 7l) 
(T P ) : • 
‘ L cos. I 
sin.?.*7 (t-I-t)tt' j 1 
sin. i ^ 2 
;o3/ COS. I 
5. In the last equation, all the terms being of the third 
order, we may consider cr" as equal to (r + t') • therefore, by 
division, we shall get 
P ^ sin. (c*-«) • “'0 “ tan. z \ — 2 * \r«3 1 » 
or, by introducing a new letter, we shall have these two for- 
mulas, equivalent to the last one, viz. 
^ cos. A* V • / O N tan. ?P I 
r =z — : . j sin. (6°-- n) — r \ 
^ sin. (<r° — «) t '■ ^ tan. z 3 . . 
If we suppose all the three geocentric places of the comet 
to be situated in the same great circle of the heavens ; then 
sin. (6°— • n) . tan. i = tan. x°: in this case therefore ^ = o, 
which requires that ~ ^ Thus we 
learn that a comet and the earth are equally distant from the 
sun, when the comet's apparent motion continues for a short 
time to be performed in one great circle of the heavens. Again 
if ^ be positive, then J vvili be positive too, and will 
be less than R°: but if T be negative, tlien (T — ^vill like- 
wise be negative, and will be greater than R°. But there are 
two cases when f vn ill be indeterminate, and the preceding rules 
