106 
[327 
327. 
ON THE STEREOGRAPHIC PROJECTION OF THE SPHERICAL 
CONIC. 
[From the Philosophical Magazine, vol. xxv. (1863), pp. 350—353.] 
In order to the tolerable delineation of some figures relating to spherical geometry, 
I had occasion to consider the stereographic projection of the spherical conic. To fix 
the ideas, imagine a sphere having its centre in the plane of the paper, and through 
the centre three rectangular axes, that of x horizontal and that of y vertical, in the 
plane of the paper, and the axis of z perpendicular to and in front of the plane of 
the paper. The radius of the sphere is taken equal to unity (so that its intersection 
by the plane of the paper is the circle radius unity), and the points X, Y, and Z are 
taken to denote the points where the axes, drawn in the positive direction, meet the 
surface of the sphere; and the opposite points are called X', Y', and Z'. The eye 
is supposed to be at Z, and the projection to be made on the plane of the paper. 
This being so, and supposing that the axes of coordinates are the principal axes of 
the spherical conic, the axis of x being the interior axis, and taking £, y, £ as the 
coordinates of a point on the spherical conic, its equations are 
I 2 + f = L 
-f 2 + 77 2 cot 2 /3 + 4 2 = 0; 
c 
where it may be remarked that tan /3, c are the semiaxes of the plane conic which 
is the gnomonic projection (i.e. the projection by lines through the centre of the 
sphere) of the spherical conic on the tangent plane at X or X'. 
Taking, for a moment, x, y, z as the coordinates of a point on the projecting line 
(that is, the line through the eye to a point (£, y, £) on the spherical conic), the 
equation of this line is 
x y z — 1