476] 
ORBIT OF A PLANET FROM THREE OBSERVATIONS. 
429 
to 
the 
we obtain the six coordinates of the three rays respectively 
( a i, 
bi, Ci, fi, gl , h x ) = ( 0, 
a/3, 
-1, 
0, 
1, 
Vs), 
(a 2 , 
o' 
p 
OQ 
11 
W 
V3, 
2, 
a/3, 
1, 
- 2 VS), 
( a 3> 
K c*, f 3 , g 8 , h 3 ) = (- 3, 
V3, 
9 
- Vs, 
1, 
- 2 VS), 
whence the intersections with the orbit-plane are given by 
Xi' 
: Vi 
1 = 
X-2 
■ Va 
1 = 
as 
X-j 
: 2// 
1 = - 
nts 
where if 
(as before) 
/3' V:3 - 
— /3 V 3 + y 
ß" + y" V3, 
- 3a - ß V3 - 2y : a" V3 + /3" - 2 V3 7 ", 
3a - /3 V3 - 2 7 : - a" V3 + /3" - 2 V3 7 ", 
longitude 6 and colatitude c of the orbit-pole, we have 
a , /3 , 7 = sin b , — cos b , 0 , 
OL , ¡3', y = cos b cos c, sin b cos c, — sin c, 
a", /3", y" = cos b sin c, sin b sin c, cos c, 
and the passage from the coordinates x', y f , to x, y, is given by 
x' = x sin b — y cos b, 
y — x cos b + y sin b, 
or conversely 
x' sin b+ y' cos b, 
— x cos b + y sin b. 
60. To develope the results, I consider the orbit-pole as passing through certain 
series of positions. The locus may be a meridian circle: by reason of the symmetry 
of the system, the results are not altered by a change of 120° in the longitude of 
the meridian; so that, by considering the two meridians 0°—180° and 90°—270°, we, 
in fact, consider twelve half meridians at the intervals of 30°. An illustration is 
afforded by Plate I.; the orbit-pole describes successively the meridians 0°, 30°, 60°, 90°, 
and the line 1, by its intersection with the orbit-plane, traces out on this plane a 
series of hyperbolas shown in the figure ; the hyperbola for the meridian 90° is a 
right line, but (except for the position where the orbit-plane passes through the 
line 1) the locus is a determinate point on this line. Planogram No. 1 (Plate II.) 
refers to the meridian 90°—270°, and Planogram No. 2 (Plate III.) to the meridian 
0°—180°. Next, if the orbit-pole be at one of the points A, that is, if the orbit- 
plane pass through a ray—though the position of the orbit-pole be here determinate, 
yet as there is a series of orbits, this also will give rise to a planogram: I call it 
Planogram No. 3. The orbit-pole may pass along a separator circle (viz. the orbit- 
plane be parallel to a ray), this is Planogram No. 4. And, lastly, the orbit-pole may 
pass along the ecliptic (or the orbit-plane may pass through the axis SZ), I call this 
Planogram No. 5. But the last three planograms are not considered in the like detail 
as the first two, and I have not, in regard to them, tabulated the results, nor given 
any Plates. 
m iii 1 
10