codrdinates of the center, and let a be the radius. Then the equation may be 
derived from the fundamental equations 
tan m’ = cos & tan m, tan £’ = cos m tan &, 
tan 7’ = cos ~ tan 7, tan &’ = cos 7 tan &, 
and the polar equation 
cos a@ = sin 7m’ sin 7’ + cos ni’ Cos 7’ cos (E — £1), 
by the elimination of &’, 7’ and ’, 7’. 
The resulting equation is 
(tan — tan £)? + (tan 7— tan m)?+ (tan € tan m —tan & tan 2 
= tan’ a (1 + tan — tan & + tan 7 tan 7)”. 
When & = 7; = 0, this equation reduces to that given in (A) above. 
ms, 5 
4. The Spheric Hyperbola. This spherical curve may be defined as the 
locus of a point which moves so that the difference of its distances from two fixed 
points is constant, p — p’ = 2a. 
Using the notation of Fig. 4, but with p — p’ = 2 a, this definition leads 
to the equation : 
tan? & tan? 7 
= ae 
tan? @ tan? B 
which is the spheric hyperbola. The locus does not intersect the OY-axis; 
the conjugate spheric hyperbola may be defined by 
