a first Appro xiniatio 7 i to the Orbit of a Comet, 143 
will correspond two arcs, differing fro.n one another by 180®, 
which will determine both the points of intersection of the 
great circle and the ecliptic ; but it will be sufficient to take 
that one which immediately precedes the comet in longitude. 
The arc n being thus found, we have these formulas for 
finding f, viz. 
tan. i == 
tan, A 
sin (c — n) 
tan. x" 
sii).(c — «) ’ 
and the double value will serve to prove the accuracy of the 
calculation. 
But we may determine whether r° is greater or less than 
R°, without any calculation. Let 1 f denote the arc of a great 
circle drawn from the intersection whose longitude is n, to the 
geocentric place of the comet at the middle observation: then 
/i° will be the hypothenuse of a right-angled triangle, having 
the arc (c° — n) of the ecliptic for one side, and x°, perpendi- 
cular to the ecliptic, for the other side : and if we put i° to 
denote the angle of the triangle opposite to x°, we shall have 
cos. x° sin. (6° — n) — cos. sin. If 
sin. x° = sin. sin. If. 
Substitute these values in the equation (9), and we shall get, 
by division. 
Q sin. (i— -i®) . sin tt / i i \ -po 
^ * sin. /■ sin. (c®— m) 2 * 
Conceive two arcs to be drawn perpendicular to the great 
circle which passes through the geocentric places of the comet 
at the two extreme ob.servations ; one, denoted' by tt, drawn 
from the extremity of If, that is, from the geocentric place of 
the comet at the middle observation ; and the other denoted 
by n, drawn from the place in the ecliptic occupied by the 
earth at the same time: then, observing that tt will be one 
side of a right-angled spherical triangle, of which If is the 
