12 CONCRETE REPRESENTATIONS OF 



which antipodal points are distinct. 1 In the Cay ley-Klein 

 representation spherical geometry is conveniently excluded, 

 since two lines only intersect once. 



7. Consider next the case where the absolute is a real 

 proper conic. This divides the plane into two distinct regions 

 which we may call the interior and the exterior, and it is of 

 no moment whether the conic be an ellipse, a parabola, or a 

 hyperbola. It is convenient to picture it as an ellipse. If 

 the points P, Q are in different regions, then (PQ, XY) is 

 negative and log (PQ, X Y) is a complex number of the form 

 a+(2n+l)iir, or simply a+iir, to take its principal value. 

 a is zero only when (PQ, X Y) = - 1. K log (PQ, X Y) also will 

 in general be complex whatever be the value of K. Of course 

 it is possible to choose K=aitr, which would make the 

 distance real, but for points in the vicinity of Q the distance 

 (PQ) would still be complex. On the other hand, if P, Q 

 are in the same region, (PQ, X Y) is either real, when X, Y are 

 real, or purely imaginary, when X, Y are conjugate imaginary 

 points. Then by taking K either real or a pure imaginary 

 we can make the distance between two points in the same 

 region real when measured along a certain class of lines, 

 purely imaginary when measured along another class : these 

 are the lines which do or do not cut the absolute. Hence we 

 are led to consider certain points and lines as ideal, 



Suppose we consider points within the absolute as actual 

 points. The line joining two actual points always cuts the 

 absolute, and we must take K real. Then all points outside 

 the absolute are ideal points, for the distance between an 

 exterior point and an interior point is complex (or purely 

 imaginary in the case of harmonic conjugates). If Q lies on 

 the absolute, while P does not, (PQ, XY) is either zero or 

 infinite and log (PQ, X Y) is infinite. Hence the absolute is 

 the assemblage of points at infinity. Two lines cutting in an 



1 Some writers have distinguished these two geometries as single or polar elliptic 

 and double or antipodal elliptic. 



