University of Saint Andrews five hundredth anniversary. Memorial volume of scientific papers contributed by members of the university

Abstract

CONCRETE REPRESENTATIONS OF which antipodal points are distinct. 1 In the Cayley-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. ' Next to determine the distance function ; let P, Q become P', Q' (Fig. 4). The orthogonal circle PQXY becomes a straight line P'Q'X'Y', and OPP', OQQ', etc., ar collinear since angles at are unaltered.