of the Spherical Conic. 351 



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 7 . The eye is supposed to be at Z, and the projection to 

 be made on the plane of the paper. This being so, and sup- 

 posing that the axes of coordinates are the principal axes of the 

 spherical conic, the axis of x being the interior axis, and taking 

 f , 77, f as the coordinates of a point on the spherical conic, its 

 equations are 



r-tv+t 2 =h 



where it may be remarked that tan /3, c are the semiaxes of the 

 plane conic which is the gnomonic projection (i. e. the projec- 

 tion 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 (f, 77, f) on the spherical conic), the equation of this line is 



x _ y _ z— 1 



and thence putting z = 0, x, y will be the coordinates of a point 

 of the projection, and we have 



x _y _ 1 



or, what is the same thing, 



f=*(l~0, *7=y(l-0. 

 The equations of the spherical conic may be written 



f 2 =c 2 (£ 2 -77 2 Gotl- 

 and by eliminating £, 77, f from the four equations, we obtain 

 the equation of the conic. 



Substituting for f and 77 their values, we find 



l+?=(#*+y*)(l-S), 



£* =cV-y 2 cot*/3)(l-(0*i 



or, observing that the first equation gives 



3? 2 + ?/ 2 -l 

 * .*■ + ?»+ 1 



