476] 
ORBIT OF A PLANET FROM THREE OBSERVATIONS. 
427 
the sign being taken in such manner that r\ shall be positive; viz., the sign must 
be the same as that of (f x , g 1} h x ]£a", /3", 7"). And we have the like formulse for 
r. 2 and r 3 . Substituting these values, the equation of the orbit becomes 
= 0. 
r , 
x' 
, 
y' 
, 
1 
u 1} 
( a i, bj, c x $a, 
7). 
-(a 1, b x , c^a, ß, 
yX 
(4, gl , hW, ß", y") 
^¿2 5 
(a 2 , b 2 , c 2 ][ 
f) 
X 
— (a 2 , b 2 , C 2 $ „ 
X 
(4, g 2 , h 2 $ „ ) 
U 3) 
(a- 3 , b 3 , c 3 $ 
» 
X 
-(a 3 , b 3 , c 3 $ „ 
X 
(4. g3, h 3 ][ „ ) 
62. 
Considering the 
minor 
determinants formed with the terms under the 
for instance 
this is 
( a i, b 1} c^a', /3', 7)--( a 2> b 2) c 2 ][a, ß, 7) 
+ (Al b 1? c x ][a, /3,7). (a„, b 2 , ß', 7) 
= (hiC 2 — boCi) (ßy' — ß'y) + (c x a 2 — c 2 a,) (ya! — y a) + (a,b,, - a 2 bj) (aff - cc'ß) 
= a" (b x c 2 — b^) + ß" (cja 2 — Caa^ + y" (ajb, — a 2 bj), 
or, what is the same thing, 
= (biCo — b 2 Cj, c x a 2 — c 2 aj, a x b 2 — aob^a", ß", y”); 
with the like expressions for the other two minors. And we thus obtain the following 
developed form of the equation, viz. 
\x' (a x , b x , c x $a, /3, y)+y'(a 1 , b lt c&a', ß', 7)}[-w 2 (f 3 , g 3 , h 3 $a", ß", y") 
+ u z (f 2 , g 2 , ha^a", ß", 7")] 
+ {x (& 2 , b 2 , c 2 $ „ )+2/ / (a 2 , b 2 , c 2 $ „ )} [- M 3 (f 1 , g x , h x ][ „ ) 
+ u 1 (f s , g 3 , h 3 $ „ )] 
4- (a 3 , b 3 , c,$ „ ) + y (a 3 , b 3 , C 3 £ „ )} [— Ui(f 2 , g 2 , h 2 ^ „ ) 
+ M4, g x , h x ]£ „ )] 
+ (b 2 c 3 — b g c 2) c 2 a 3 — c 3 a 2 , a 2 b 3 - agb 2 ][a", ß", y") [r (f x , g x , h^a", ß", 7") - m x ] 
+ (b 3 c x b x c 3 , c 3 a x c x a 3 , a 3 b x a x b 3 ^ „ ) [i (f 2 , g 2 > h 2 ]£ ,, ) m 2 ] 
4* (b x c 2 — b 2 c x , c x a 2 c 2 a x , a x b 2 a 2 b x ^£ ,, ) [i (f 3 > g 3 > h 3 ^£ ,, ) m 3 J 0, 
being an equation of the form fir = Ax' + By' 4- C. 
63. The coefficient of r is a quadric function of (a", ß", y"), and if this vanish 
the orbit is a right line. It thus appears that the orbit will be a right line provided 
only the orbit-axis be situate in a certain quadric cone, or (what is the same thing) 
the orbit-pole be situate in a certain spherical conic: agreeing with a preceding result, 
viz. the cone is that reciprocal to the cone, vertex S, circumscribed about the hyper- 
54—2