476] 
ORBIT OF A PLANET FROM THREE OBSERVATIONS. 
425 
58. I return to the equation of the hyperbola written in the form 
(x cos b + y sin 6) 2 — (x sin b — y cos 6 — If tan 2 8 = m 2 : 
being (as was shown) a hyperbola passing through the point ^ where its plane 
is met by the ray, and touching at this point the projection 
-A 
y 
B 
If in the equation we consider 6 as variable, we have a series of hyperbolas, viz., 
these are the intersections of the plane of xy with the hyperboloids of revolution 
obtained by making the ray rotate successively round the several lines ¿»cos 6 +y sin6 = 0 
through the focus S. 
And, as just seen, these hyperbolas all of them touch at 
the projection of the ray. 
59. The hyperbola to any particular angle 6 is the hyperbola belonging to the 
ray, in the planogram for an orbit-plane rotating about the axis x cos 6 +y sin 6 = 0; 
so that the system of hyperbolas would be useful for the construction of any such 
planogram. And there is another series of curves which, if they could be constructed 
with moderate facility, would be very useful for the same purpose; viz., reverting to 
the equations 
x : y' : 1 = (a cos 6 + b sin 6) cos c — c sin c 
: — a sin 6 + b cos 6 
(f cos 6 + g sin 6) sin c + h cos c, 
which determine in the orbit-plane the coordinates x, y' of the intersection thereof with 
the ray: imagine as before that the point is marked on the orbit-plane, and let it by a 
rotation of the orbit-plane be brought into the plane of xy; so that x', y', will be 
the coordinates in the direction of and perpendicular to the line of nodes of a point on 
the hyperbola y /2 — (x' — Z) 2 tan 2 8 = m 2 , or (x cos 6 + y sin 6) 2 — (x sin 6 — y cos 6 — Z) 2 tan 2 8 = m-. 
viz., of the point corresponding to an orbit-pole, colatitude c. Suppose that x, y, are 
the coordinates of this same point referred to the fixed axes, we have 
x = x sin 6 + y cos 6, 
x = — x cos b + y' sin 6, 
and thence 
x : y : 1 = (a cos 6 + b sin 6) sin 6 cos c — c sin 6 sin c + (— a sin 6 + b cos 6) cos 6 
: — (a cos 6 + b sin 6) cos 6 cos c + c cos 6 sin c + (— a sin 6 + b cos 6) sin 6 
: (f cos 6 + g sin 6) sin c + h cos c, 
the coordinates of the point just referred to. Now, if from these equations we could 
eliminate 6, we should have a series of curves containing the variable parameter c, 
intersecting the series of hyperbolas; and thus marking out on each of these hyperbolas 
the points which belong to the successive values of the parameter c; we should thus 
have in the plane of xy the point corresponding to an orbit-pole longitude 6 and 
c. vn. 54