108 
ON THE STEREOGrRAPHIC PROJECTION OF THE SPHERICAL CONE. 
[327 
the last term vanishes, and the equation gives y~ = 0, or 
= 2 (1 — x 2 — 2c 2 cot 2 /3) 
= 4 (— c 2 + c V1 + c 2 — c 2 cot 2 /3) 
= 4c (— c cosec 2 /3 + Vl + c 2 ), 
the upper sign corresponding to the exterior values 
± x = Vl + c 2 + c, 
and the lower sign to the interior values 
+ x = Vl + c 2 — c. 
In the former case the values of y are imaginary; in the latter case they are real if 
vr+ c 2 > c cosec 2 /3, 
or, what is the same thing, if 
sin 2 /3 > == ; 
Vl + c 2 
that is, if (for a given value of c) /3 is sufficiently great, but otherwise they are 
imaginary. 
If, as in the annexed figures, c = T % (and therefore / v /, l + c 2 = |f, Vl+c 2 +c = f, 
Fig. 1. Fig. 2. 
Vl + c 2 — c = f), then for the limiting value of /3 we have 
sin 2 /3 = -^- -3846, sin /3 = ‘62, or /3 = 38° nearly. 
In the first figure /3 is less, in the second figure greater than this value: the form 
for the limiting value is obvious from a comparison of the two figures. 
I take the opportunity to mention the following theorem, which is perhaps known, 
but I have not met with it anywhere; viz. any three circles, each two of which meet,