38 CONCRETE REPRESENTATIONS OF 



system is therefore a double conic, which touches ZX and ZY 

 at X and Y. 



Every line through Z is therefore cut in involution by the 

 system of conies, and the double points of the involutions lie 

 on the conic axy=bz 2 . Further, on each conic of the system 

 there is an involution formed by the pencil with vertex Z, 

 and the double points of these involutions are the points of 

 intersection of the conies with the conic axy=bz 2 . We have, 

 then, what we require, two absolute points on every conic 

 which represents a straight line, and these absolute points lie 

 on a fixed conic. We may therefore call the conic axy=bz 2 the 

 Absolute. 



32. The conic 



axy+ bz z + z(px+ qy)=Q 

 cuts the absolute where 



(2axy) 2 b=axy(px+ qy) z , 

 which gives x=0 or t/=0 or 



4tabxy= (px-\- qy) 2 - 



According as the points of intersection are real, coincident, or 

 imaginary, the conic represents a line with hyperbolic, para- 

 bolic, or elliptic metric. The condition that the points of 

 intersection be coincident is 



&=ab(ab+pq)=0. 



If a or b vanishes all lines are parabolic. 



When a=0 the absolute becomes a double line z 2 =0, and 

 every conic of the system breaks up into this line and a 

 variable line px+qy+bz=Q. The representation is then by 

 straight lines, and if X, Y are an imaginary point-pair the 

 geometry is Parabolic. If X, Y are the circular points the 

 geometry is Euclidean, and the representation is identical. 



When 6=0 the absolute breaks up into two lines #=0, y=0 9 

 and every conic of the system passes through the three points 

 X, Y, Z. If X, Y are an imaginary point-pair the geometry 

 is again Parabolic, and if X, Y are the two circular points the 

 representation is by circles passing through a fixed point. 



