NON-EUCLIDEAN GEOMETRY 43 



the system the absolute is transformed into itself, and any 

 conic of the system is transformed into a conic of the system. 

 A single quadric inversion is thus analogous to a reflexion, 

 while the general motion is produced by a pair of quadric 

 inversions. 



These results could also be obtained by projection, for 

 quadric inversion, in the case where the points X, Y are 

 imaginary, can be compounded of ordinary inversion in a 

 circle and a collineation. 



By a quadric inversion the pencil of lines passing through 

 Z, which, together with the line XY, form a pencil of conies 

 of the system, is transformed into a pencil of conies passing 

 through 0. Hence we may extend the result of 31 and say 

 that every conic of the system is cut in involution by any 

 pencil of conies of the system, the double points being the 

 points of intersection with the absolute. 



Like the representation by circles, this representation 

 admits of immediate extension to three dimensions. Planes 

 are represented by quadric surfaces passing through a fixed 

 conic, C. Two such quadrics intersect again in another conic. 

 The linear metric is referred to an absolute quadric also passing 

 through C, such that, if Z is the pole of the plane of C with 

 respect to the absolute, any quadric which represents a plane 

 cuts the absolute in a plane section, which is the polar of C 

 with respect to the quadric. 



REPRESENTATION BY DIAMETRAL SECTIONS OF A 

 QUADRIC SURFACE 



36. We shall briefly describe one other representation, 

 due to Poincare. 1 In this representation straight lines are 

 represented by diametral sections of a quadric surface. 



1 H. Poincare, ' Sur les hypotheses fondamcntales de la geometric,' Paris, Butt. Soc. 

 math., 15 (1887), 203-216. Cf. also H. Jansen, ' Abbildung der hypcrbolischen Geo- 

 metrie auf ein zweischaliges Hyperboloid,' Hamburg, Mitt. math. Ges,, 4 (1909), 409-440. 



