NON-EUCLIDEAN GEOMETRY 23 



18. The distance function is thus periodic with period 

 2ip.ir. If P, P' are inverse with respect to the fixed circle 



-/* log =/* log (-D- 



and 



When Q is on the fixed circle (PQ)=cc . The fundamental 

 circle is thus the assemblage of points at infinity. 



If the fundamental circle is imaginary, k is positive and p. 

 is purely imaginary and may be put =i. Then if inverse 

 points are considered distinct their distance is TT and the 

 period is 2ir, but if inverse points are identified the period 

 must be taken as TT. 



If the fundamental circle is real, k is negative and p. is real 

 and may be put =1. Then the period must be taken as iir and 

 inverse points must be identified, otherwise we should have 

 two real points with an imaginary distance. In this geometry 

 there are three sorts of point-pairs, real, coincident, and 

 imaginary, or actual, infinite, and ultra-infinite or ideal. 



19. Now if we change x, y into x', y' with the help of an 

 additional variable z' by the equations 



x y 



then x' 



so that (x, y) is the stereographic projection of the point 

 (x', y', z') on a sphere of radius R. 



Obtaining the differentials dx', dy', dz', we find 



Hence R 2 =-p. 2 . 



Hence when k is positive and /* purely imaginary and 

 =iR, the geometry is the same as that upon a sphere of radius 



