NON-EUCLIDEAN GEOMETRY 31 



25. Finally, this representation may be transformed 

 projectively (distances and angles being unaltered as they are 

 functions of cross-ratios), and we get the usual generalised 

 representation in which the fixed circle or absolute becomes 

 any conic ; straight lines are represented by straight lines, 

 and distances, and angles in circular measure, are expressed 

 by the formulae 



XY) 



where X, Y are the points in which the straight line PQ cuts 

 the conic, and x, y are the tangents from the point of inter- 

 section of the lines p, q to the conic. 



GEODESIC REPRESENTATION ON SURFACES OF 

 CONSTANT CURVATURE 



26. It has been seen that both the Cayley-Klein represen- 

 tation and the conf ormal representation by circles are derivable 

 by projection from a sphere, real or imaginary, on which the 

 non-Euclidean straight lines are represented by great circles. 

 By Gauss' Theorem the sphere may be transformed, or limited 

 portions of the surface may be deformed, into a surface of 

 constant measure of curvature, in such a way that geodesies 

 remain geodesies and are unaltered in length. The effect 

 is that of bending without stretching ; the geometry therefore 

 remains the same. To Beltrami 1 is due this representation 

 of non-Euclidean geometry upon a surface of constant cur- 

 vature, and it is the only representation in which distances 

 and angles are represented unchanged. 



27. While this representation is of the first importance in 

 non-Euclidean geometry, it has to be distinctly understood 



1 Loc. tit., p. 5, foot-note 2. 



