279 



coordinates of the center, and let a. be the radius. Then the equation may be 

 derived from the fundamental equations 



tan 771' = cos £1 tan 771, tan £/ = cos 771 tan £1, 

 tan 7]' = cos £ tan 77, tan £' = cos 77 tan £, 

 and the polar equation 



cos a = sin 171' sin 77' + cos 77/ cos 77' cos (£ — £1), 

 by the elimination of &', 77/ and £', 77'. 

 The resulting equation is 



(tan £ — tan £i) 2 + (tan 77- tan 77,) 2 + (tan £ tan 771 -tan £1 tan t?) 2 

 = tan 2 a (1 + tan £ tan £1 + tan 77 tan 771) 2 . 

 When £1 = rji = o, this equation reduces to that given in (A) above. 



00.- f 



F< s s 



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 = 2 a. 



Using the notation of Fig. 4, but with p — p = 2 a, this definition leads 

 to the equation 



tan 2 £ 



tan 2 77 



tan 2 a tan 2 /3 



which is the spheric hyperbola. The locus does not intersect the O Y-axis; 

 the conjugate spheric hyperbola may be defined by 



