476] 
ORBIT OF A PLANET FROM THREE OBSERVATIONS. 
453 
so that the orbits in the planogram are the whole series of conics having a given 
focus, S, and passing through two fixed points, 2, 3, having the common abscissa 
x = — 1, and at equal distances 
2 
V3 
(= 1-15470) on opposite sides of the axis. 
The axis 
of x is obviously the common transverse axis for all the orbits; that is, the equation of 
the orbit will be of the form r = Ax + B; and writing x = — 1, we have V|= — A + B, 
viz. the equation is r — Vf = A (« +1); the value of A will be determined if we 
assume for the point 1 a determinate position on the line x = 1, say its ordinate is = y x ; 
for then if r. 2 — Vl + y x - we have r x — V| = 2A, and the equation is r — Vf = \ (r x — V|)(«+ 1). 
In particular if 2/i = 0, we have r x = l, and the equation of the orbit is r—= J (1—V|) («+1): 
this is the orbit, eccentricity (V| — 1), = -264, belonging to the point A as a point 
in planogram No. 1: for the value of y, being in that planogram originally assumed 
= 0, is of course = 0 when the orbit-pole comes to be the point A. 
96. We may conversely take the equation of the orbit, or say the value of 
A(=±e) in the equation r — V| = A (« +1), to be given; and then writing x=x t = l, 
we have 
for 
i\ = + 2A, that is yd — (V| + 2A) 2 — 1 ; 
r x = 1 or y x = 0, A = |(1 — Vp = — -264, 
and as r x increases to r x = , or y x increases to + , A diminishes from — '264 to 0; viz., 
- 2 g 
for r x = , or y 1 = + , the orbit is a circle; as r x increases from Vor y r from + , 
A increases from 0 positively; for r 
, = V| + 2, =3 527, or ÿ. = ±V ' 1 ^ ± l 
V21 
, = ± 2-896, 
A becomes =1; that is, the orbit is a parabola; and for larger positive values of r 1} 
or positive or negative values of y lt the orbit is a hyperbola (concave) ; and ultimately 
for r a = oo or y x = ± oo, the orbit is the right line « + 1=0. Thus A extends from 
— ‘264 to 0, and thence from 0 positively to + co. 
97. In further illustration, suppose that the orbit-pole, instead of being at A, is 
a point in the immediate neighbourhood of A, say that the rectangular spherical 
coordinates, measured from A in the direction of the meridian and perpendicular 
thereto, are £ and y ; the colatitude and longitude of the orbit-pole being thus 
c = 60° + f, and 6 = 270° + -= y ; 
V3 
we have then, f, y being indefinitely small, 
«,/3,7 = "I > -¿¿V, 
/3', y' = y, - 2 + "Y £, 
o, 
ß", 1 = V 
V3 
2 
i 
1 v 
2 9 '