140 Mr. Ivory on a new Method of deducing 
(t'p cos. X sin. (c — w) -j- (Tp' C'^s. x' sin. (^:'~ 
rr (r-\-r) 
2 
• ;/ ) — (t"p° cos. X° 
. R° sin. (6^°— n). 
Conceive a great circle to be drawn on the surface of the 
sphere through the geocentric places of the comet at the two 
extreme observations; and let n, in the foregoing equations, 
denote the longitude of one of the intersections of that great 
circle with the ecliptic: also let h and h’ denote the arcs of the 
great circle intercepted between the same intersection, and the 
places of the comet abovementioned ; then these arcs will be 
the hypothenuses of two right-angled triangles of which the 
sides, perpendicular to the ecliptic, are the arcs x and x' \ and 
the other sides, in the ecliptic, are the arcs [c^n) and i^c' —n): 
further, let i denote the inclination of the same great circle to 
the ecliptic; then v/ill i be an angle common to the two right- 
angled triangles abovementioned, opposite to the sides x and 
a'; and, by the properties demonstrated in spherical trigono- 
metry, we shall get these formulas, viz. 
cos. X cos. (r — n) = cos. h 
cos. x' cos. {d — n) = cos. Id 
cos. X sin. (r — n) = cos. i sin. h 
cos. x' sin. (r' — n) — cos. i sin. h' 
sin. X = sin. i sin. h 
sin. x' = sin. i sin. h'. 
Now let these values be substitut<='d in the two foregoing 
equations, and likewise in the last of the equations (7), and 
we shall get, 
cos. h -j- o-p' cos. h* — (t"p° cos. cos. — w) = 
fr' (t -{- t') 
• [is .R°cos. (t-“— ;;) 
2 
( 8 ) 
