16 CONCRETE REPRESENTATIONS OF 



11. An apparent extension of the Cay ley-Klein theory, 

 elaborated by Fontene l for space of n dimensions, deserves 

 mention. 



The absolute conic in the Cayley-Klein theory is the double 

 conic of a transformation by reciprocal polars. If we replace 

 this transformation by the general dualistic linear transforma- 

 tion there arise two distinct conies having double contact, the 

 pole conic or locus of points which lie upon their corresponding 

 lines, and the polar conic or envelope of lines which pass 

 through their corresponding points. Consider any line Z and 

 a point A upon it. To A there corresponds a line a which 

 cuts Z in a point A'. Thus a homography is established 

 between pairs of conjugate points A, A' on the line Z. The 

 double points Qj, H 2 of this homography are the points in 

 which I cuts the pole conic. The distance (PQ) between two 

 points P, Q on I can then be defined as 



The distance between two conjugate points P, P' is constant 

 for the line Z, but it varies for different lines. It may be called 

 the parameter of the line. 



By allowing K to vary the parameter could of course be 

 made the same for all lines ; but it is impossible to adjust the 

 system so that it may represent a geometry with the necessary 

 degrees of freedom. In fact, since a motion consists of a 

 collineation which leaves the absolute invariant, and since 

 the general collineation leaves just three points invariant, 

 these points must be the points of contact of the two conies 

 and the pole of their chord of contact. The general motion 

 is therefore impossible, the only possible motion being a 

 rotation about a definite point, the pole of the chord of contact. 



E. Meyer 2 has considered a further generalisation of these 

 ideas by taking two independent conies as the absolute 



1 G. Fonten6, 'L'hyperespace It (n - 1) dimensions. Pr&pri&is mttriques de la 

 corrttaivon gentrale. Paris, Gauthier-Villars, 1892. 



2 ' tiber die Kongruenzaxiome der Geometric,' Math. Ann., Leipzig, 64 (1907). 



