476] 
ORBIT OP A PLANET FROM THREE OBSERVATIONS. 
455 
Article Nos. 99 to 103. Planogram No. 4, the Orbit-pole in the Ecliptic. 
99. When the orbit-pole describes the circle of the ecliptic, the orbit-plane passes 
through the axis of z, or polar axis. We have c = 90°, and consequently 
a , ß , 7 = sin b, 
— cos b, 
0, 
a '> ß' > y = 0 , 
0 , 
- 1, 
a", ß\ 7" = cos b, 
sin b, 
0. 
Reverting for a moment to the general case where the six coordinates of the ray are 
(a, b, c, f, g, h), the formulae for the intersection by the orbit-plane are 
x' : y : 1 = (a, b, c$V, ß', y) = — c 
: — (a, b, c§a , ß , 7) : — a sin 5 + b cos b 
: ( f > g> h$a", ß", y") ■ f cos 6 + g sin b, 
that is 
and thence 
consequently 
If Sf 
— + - cos b + - sin b = 0, 
X c c 
+ - cos b — - sin b = 0 ; 
x c c 
1 : cos b : sin b = 
-af-bg _ gy' -f a 
c 2 * ex' 
b 
ex' 
ha/ : gy' + a : -iy' +b; 
h#' 2 = (gy’ + a) 2 + (%' - b) 2 , 
or, what is the same thing, 
ha/ 2 = (f 2 + g 2 ) y' 2 + 2 (ag — bf) y + a 2 + b 2 , 
or, in particular, if (as in the special symmetrical case) ag — bf = 0, then 
ha/ 2 = (f 2 + g 2 ) y' 2 + a 2 + b 2 . 
100. For the symmetrical system of rays we have as before 
a l> 
bi, Ci, fi, gi, h 2 = 0, V.3, 
-1, 
0, 1, 
Vs, 
a 2 , 
bo, C 2 , fo, g 2 , h 2 = 3, V3, 
2, 
Vs, 1, 
- 2 V3, 
a», 
bs> C 3 , f 3 , g 3 , h 3 = 3, V3, 
2, 
V3, 1, 
- 2V3, 
x-l 
• Vi ' 1= 1: 
n/ 3 cos & 
: sin b 
J 
x% 
: y 2 ' : 1 = — 2 : — 3 sin b + V3 cos b 
: sin 6 + V3 cos b, 
a? 3 
: y 3 ' : 1 = — 2 : 3 sin b + V3 cos b 
: sin b - 
V 3 cos 6, 
and thence