a first Approximation to the Orbit of a Comet, i8i 
that, in every case, sin. (/ — z°) is of the second order ; and 
hence cos. (2 — f°), differing from 1 only by quantities of the 
fourth order, may generally be regarded as equal to unit. The 
last equation will therefore become, 
(Tp sin. h + sin. H — sin. If — 
I 1 
. R° cos. i sin. — n ) : 
and, if this equation be combined with the first of the three 
equations set down at the beginning of this article,* so as to 
exterminate p° and p', we shall get. 
(Tp sin. [If — h) — crp'sin. [Ii—Jf) 
tt' (t-}-t') 
I 
,-o 3 
1^3) .R°x. 
I cos. (e° — n ) sin. h° — cos. / sin. («° — n) cos. h° | ; 
I cos. (e° — 71) sin. h' — cos. / sin. (e°— ?i) cos. /i'|- 
Observing then that and nearly; if we put 
-po COS. (e° — 7i) sin. cos. i sin. (e° — n) cos. b° ^ 
/ 2 sin. b — ’ 
sin. 
sin. (/y — b*) ’ 
we shall get. 
9 
p = e . p 
sin. (Ji * — Ij) r 
4 - — • 
”1 ro3 Ro3 ’ 
“ ^ ^ sin. (/y — b*) * T ' 
the quantities of the second order being omitted in the expres- 
g 
sion of p°; because p® is only to be used in valuing ~ wliich is 
already of the second order. 
If we substitute the values of the coordinates (Mo. 3), in the 
expressions = x*~ + + y" + 2:"; ?'" =. 
j.r- ^ cr'"* ; V = xx' 4 * yy' 4 " shall get 
* In this equation cos. must be written for its equal cos. A® cos. 
