NON-EUCLIDEAN GEOMETRY 9 



Riemann, the former when the conic is real, the latter when it 

 is imaginary. There are obviously other cases for example, 

 when the conic degenerates to two distinct lines and there 

 will be corresponding systems of geometry. Most of these 

 geometries are very bizarre. In one, for example, the peri- 

 meter of a triangle is constant. The only ones which at 

 all resemble the geometry of experience are the three just 

 mentioned. 



4. We have now to obtain the expressions for the distance 

 between two points and the angle between two straight lines. 

 As the absolute in ordinary geometry is less degenerate as an 

 envelope than as a locus (the equation in line-coordinates 

 being of the second degree) it will be simpler to take first the 

 angle between two lines. 



The expression must be such as to admit of extension to 

 the case of a proper conic. Now Laguerre x has shown that 

 the angle between two straight lines can be expressed in terms 

 of a cross-ratio. Consider two lines yx tan 6, y=x tan 6', 

 passing through 0. We have also through the two (isotropic) 

 lines, y=ix, y=ix, which pass through the circular points. 

 The cross-ratio of the pencil formed by these four lines is 



, , ,. t&nOi . tan0+i 

 (uu , tow )= ... .-=- 

 tan0'- 



Hence 6' 6=$ilog(uu', too/). 



We can now extend this to the general case. Through the 

 point of intersection L of two straight lines p, q there are two 

 lines belonging to the absolute considered as an envelope, 

 viz., the two tangents from L. Call these x, y. The angle 

 (pq) is then denned to be 



klog(pq, xy) 

 where k is a constant depending upon the angular unit employed. 



1 E. Laguerre, ' Note sur la theorie des foyers,' Nouv. Ann. Math., Paris. 12 (1863). 



B 



