NON-EUCLIDEAN GEOMETRY 45 



with the fixed conic, so that the representation is by the 

 Cayley-Klein projective metric. 



The close connection between the representation by dia- 

 metral sections of a quadric surface and that by diametral 

 sections of a sphere is now apparent. 



There is an apparent gain in the generality of the repre- 

 sentation if the centre of projection be chosen arbitrarily. 

 The tangent planes through to the asymptotic cone project 

 into two straight lines cutting in Z, the projection of the 

 centre. These lines are tangents to the conic which corre- 

 sponds to the points at infinity, and the points of contact are 

 X, Y. A plane section projects into a conic passing through 

 X, Y, and its asymptotes project into the tangents at the 

 points of intersection with the fixed conic. For a diametral 

 section these tangents pass through Z. Thus we obtain once 

 more the same representation by a net of conies through two 

 fixed points, and there is no gain in generality by this general 

 projection. 



The extension of this representation to non-Euclidean 

 geometry of three dimensions requires Euclidean space of 

 four dimensions. The representation is by diametral sections 

 of a fixed quadratic variety, which must not be ruled, i.e. a 

 tangent 3-flat must cut the variety in an imaginary cone. 

 The geometry is Hyperbolic or Elliptic according as the variety 

 cuts the 3-flat at infinity in a real or an imaginary quadric. 

 DUNCAN M'LAREN YOUNG SOMMERVILLE 



