[327 
327] ON THE STEREOGRAPHIC PROJECTION OF THE SPHERICAL CONE. 
107 
SPHERICAL 
and thence putting 2 = 0, x, y will be the coordinates of a point of the projection, 
and we have 
x y 1 
rv = W ; 
or, what is the same thing, 
£=<b(1-£), V = y0-~0; 
the equations of the spherical conic may be written 
W 2 = p + 9?2j 
£2 = c 2 (I 2 - rf cot 2 /3); 
and by eliminating £, y, £ from the four equations, we obtain the equation of the conic. 
Substituting for £ and y their values, we find 
-353.] 
i + ?=(v+</ s )(i-?), 
'C = o'(x- - f cot*/3) (1 — £■)-: 
herical geometry, 
lal conic. To fix 
per, and through 
f vertical, in the 
of the plane of 
t its intersection 
X, Y, and Z are 
rection, meet the 
nd Z'. The eye 
ne of the paper, 
principal axes of 
f, tj, £ as the 
or, observing that the first equation gives 
x 2 + y 2 — 1 
x 2 + y 2 + 1 ’ 
and that thence 
w-* + ;. +1 . +*■-!), 
the equation is 
(x 2 + y 2 — l) 2 = 4c 2 (x 2 — y 2 cot 2 /3). 
It is now very easy to trace the curve. We see first that the curve is symmetrical 
with respect to the axes, and that it meets the axis of y in four imaginary points, 
but the axis of x in four real points, the coordinates whereof are 
x = + (V1 + c 2 ± c), 
Lane conic which 
3 centre of the 
so that the two points on the same side of the centre are the images one of the 
other in regard to the circle radius unity. Moreover the curve touches the lines 
y = ± x tan /3 
le projecting line 
irical conic), the 
at their intersections with the circle. By developing in regard to y, the equation becomes 
y i + 2 (x 2 — 1 + 2c 2 cot 2 ¡3) y 2 + (x 2 — l) 2 — 4c 2 «' 2 = 0 ; 
and putting 
x = + (Vl + c 2 + c),