476] 
ORBIT OF A PLANET FROM THREE OBSERVATIONS. 
405 
and the ray as rigidly connected with the axis, we make the ray to rotate about this 
axis, so as continually to intersect the orbit-plane. But in this last case the ray 
describes about the axis a hyperboloid of revolution, and the orbit-plane, as an axial 
plane, meets this surface in a hyperbola having the axis for its conjugate axis; which 
hyperbola is the required locus of the trivector-extremity. It is moreover easy to see 
that if the angle of position of the variable orbit-plane, or (what is the same thing) 
the angle of position of the orbit-pole in the great circle which it describes be = q 
(where q is measured from any fixed plane or point), and if the coordinates x' and y' 
be measured from S in the direction of and perpendicular to the axis of rotation, 
then thg coordinates of the point on the hyperbola are expressed in the form 
a! = a + a tan (q + /3), y' = b sec (q 4- /3), where a, a, b, /3, are constants depending on the 
position of the ray in regard to the axis of rotation : see as to this post, No. 49. 
12. Considering the orbit-pole as describing a given curve, the value for the 
several positions thereof of any parameter of the orbit may be exhibited by means 
of a “ diagram,” viz., we may take for abscissa any quantity serving to fix the position 
of the orbit-pole on the described curve, and for ordinate the value of the parameter in 
question. In the particular case where the orbit-pole describes a great circle passing 
through the axis of the stereographic projection, and which is consequently in the 
spherogram represented by a diameter of the ecliptic or bounding circle, it is natural 
to take for the abscissa the distance (from the centre) of the representation of the 
orbit-pole; the diagram will then fit on to the diameter, and for any position of the 
orbit-pole on such diameter give at once the value of the parameter to which the 
diagram relates. 
13. It is right to remark that the construction of planograms and diagrams is 
merely subsidiary to that of the spherograms; the information given by any number 
of planograms or diagrams would be all of it embodied in a spherogram for the same 
parameter. And theoretically the construction of a spherogram is a mere matter of 
geometry; for a given position of the orbit-pole we construct the trivector, thence the 
orbit, and in relation thereto any parameters which it is desired to consider; and so, 
for a sufficient number of points on the spherogram, determine the value of the ' 
parameter, or parameters; and lay down the isoparametric lines. The construction of 
the orbit from a given trivector, and in particular the selection of the orbit as one 
of the four conics given by the trivector, has not yet been explained: in connexion 
herewith we have the discontinuity of orbit which arises when the orbit-pole is upon 
a separator, and which is a leading circumstance in the theory; until it is gone into, 
there is little more to be said in the way of general explanation as to the spherogram, 
or the isoparametric lines thereof. 
14. It may however be noticed that for any parameter whatever, the points A of the 
spherogram are common points, through which pass in general the lines belonging to 
any value whatever of the parameter; the reason of course is that the orbit-plane 
then passing through the ray, and the orbit itself being indeterminate, the value of 
any parameter belonging to the orbit is also indeterminate. Moreover, for some 
parameters the curve belonging to any particular value of the parameter not only