476] 
ORBIT OF A PLANET FROM THREE OBSERVATIONS. 
467 
given by 0 = +1 and 6 = — 1 respectively; the dotted curve on the base C'B (= C'B') 
is merely the upper curve on the base C'B' transferred to the base C'B; and the 
curve composed of the lower curve on the base AC' and of the dotted curve gives 
by its ordinates the value of the eccentricity as the orbit-pole moves along AB'B 
within the triangle B'BBthe upper curve on the base AB' gives by its ordinates 
the value of the eccentricity as the orbit-pole moves along AB' on the other side 
thereof, that is, within the convex region. 
The base of the diagram is graduated not for the value of H, but for that of 
the angular distance (or distance in longitude) of the orbit-pole from the point A 
(or A'); viz. this is the angle opposite II in a right-angled spherical triangle, the sides 
and hypothenuse of which are 60°, H, c; writing (3 for the angle in question we have 
cos c = 
and any position of the orbit-pole on the separator may be conveniently laid down by 
means of this angle ¡3. The values of /3 corresponding to the before-mentioned values 
A = -9592 and A = 8'073 are ¡3 = 47° 54' and /3 = 83° 53' respectively. 
Article Nos. 114 and 115. The Spherogram and Isoparametric Lines—General 
Considerations. 
114. We first construct a blank spherogram, as already explained (and see also 
Plates IV. and V.), viz., we draw on the stereographic projection a hemisphere—say 
the northern hemisphere : the meridians being radii and the parallels of colatitude 
circles with the pole as centre; the parallel of 60° is the regulator circle, and the 
separators are great circles touching this at the points A, A, A, in longitudes 30°, 150°, 
270° respectively ; the separators intersect in the points B, B, B, in the northern hemi 
sphere, and they are produced to meet again in the points B, B, B, of the southern 
hemisphere ; but instead of taking the whole northern hemisphere, we omit portions 
thereof, and take in the opposite portions of the southern hemisphere ; the spherogram 
being thus bounded by portions of the separator circles, and consisting of the inner 
spherical triangle B, B, B, and three surrounding triangles B, B, B. The inner triangle 
contains the regulator-circle, touching its sides at the points A, A, A respectively, and 
dividing it into an inner circular region and three surrounding regions A, B, A ; these 
last are the loci in quitus of the orbit-poles which correspond to convex orbits; and 
to mark them off from the other regions, it is proper to shade them in the sphero 
gram. Excluding them from consideration, we have the inner circular region and the 
outer triangular regions separated off from each other by the shaded regions, except 
at the points A, where these are thinned away to nothing. The points A are positions 
of the orbit-pole for which the orbit is indeterminate ; and consequently any parameter 
belonging to the orbit is also indeterminate. Hence the isoparametric line for any 
given value of the parameter will always pass through the points A ; that is, all the 
isoparametric lines will pass through these points, which are thus points of connexion 
59—2