ORBIT OF A PLANET FROM THREE OBSERVATIONS. 
445 
476] 
but this expression is too complicated to allow of an analytical discussion of the 
series of values of e (such as was given for A, = ±e, in planogram No. 1). The 
numerical calculation gives the results mentioned ante No. 87, viz., c = 0, e = 0; c = 51°, 
g = l; c = 60°, e = oo ; c = 63° 26' — e, e = 4'912; c = 63°26' + e, 1*853 ; c = 69°, e = -628 
(min.); e = 89°20', (viz. X = 86476), e=l; c — \90°, e=r018; values which are ex 
hibited in the diagram in the preceding page. 
92. It may be further remarked, in reference to the formula 
r = Ax + By + G, 
that for c = 60°, that is X = V3, we have A finite, B and G each infinite, but equal 
and of opposite signs; viz., the equation becomes r = ‘2242x ± oo (y — 1), that is y = 1, 
orbit a right line as above. 
The abrupt change at c = 63° 26', X = 2, arises from the change of sign of R 3 ; 
viz., c = 63° 26' -e, R 3 = -~=- 2-309, but c = 63° 26' + e, R 3 = 4= = 4- 2-309 ; the two 
v3 V3 
orbits are 
c = 63° 26' — e, r = *234 * +4-906 y-3-671, e = 4-912, a= -159, 
c = 63° 26' 4- e, r = -678«-1-761 y+ 3*257, e = 1-853, a = 1-338 
For c = 90° the equation is 
4 /- 
r = j=X + f y + vf 
3V3 
= '770 x + "666 y + 1"527 
and therefore e = ^ff= 1*018 as above; a = 9 V21 = 41"243. 
It is to be added that for c nearly =90°, or X very large, we have 
and thence 
R 2 — V|-X+2V^, 
R 3 = Vf X— 2 Vf-, 
A=- T - 
3 V3 
2 1 _ 
’ V21 
•770 - 
■* 30 l 
B= f - 
5 1 _ 
V7 
•666- 
1-890 \ , 
C= V| - 
1 2 _ 
V3 ^ 
1-527 - 
1-555 ^. 
A. 
It was, in fact, by means of these expressions that the value X = 86476 (c = 89°20') 
corresponding to the last-mentioned parabolic orbit was obtained.