476] ORBIT OF A PLANET FROM THREE OBSERVATIONS. 443 
or developing, this is 
6 (sin 2 c — 3 cos 2 c), 
-= i 4r\ (sin 2 c — 3 cos 2 c) cos c 
\/3 1 
— r 2 (sin c + 2 cos c) (sin 2 c — 3 cos 2 c) 
+ r 3 (sin c — 2 cos c) (sin 2 c — 3 cos 2 c)} 
+ y {— 2r x sin C COS G 
+ r 2 (— sin 2 c + sin c cos c + 6 cos 2 c) 
+ r 3 ( sin 2 C 4- sin c cos c — 6 cos 2 c)} 
i — 6 cos 2 c 
\/3 1 
+ 3r 2 (sin 2 c + sin c cos c — 2 cos 2 c) 
+ 3 r 3 (sin 2 c — sin c cos c — 2 cos 2 c){ = 0; 
(observe that the orbit will be a right line if sin 2 c — 3 cos 2 c = 0, that is if c = 60°, 
which is right, since 60° is the angular radius of the regulator circle). 
89. Putting in the equation tan c = A, and therefore cos c = — - , the equation 
becomes 
We have 
bVl + X 2 
^4rj — (X + 2) r 2 + (X — 
1 
+ 
(2X + (X + 2) (X — 3) 
2 V3 (X 2 - 3) V 
1 
+ 2 (X 2 - 3) 
^— 2r : + (X — 1) (X + 2) 
- Vi + x 2 
x x = V1 + X 2 , 
X + 2 ’ 
1 . 
1 2A+3 
^+2 ’ 
Vi+x 2 
A —2 ’ 
1 2A-3 
V3 X-2 ’ 
and thence, writing for shortness 
E 1 = Vl + f X 2 , 
E, = ~ V7X 2 + 12X + 12, 
" Vs 
jR 3 =4r V7X 2 - 12X + 12, 
V3 
56—2