NON-EUCLIDEAN GEOMETRY 39 



If the absolute is not degenerate we may get lines of all 

 three forms. If X, Y are real the absolute is real. We may 

 suppose a, 6, which are real, to have the same sign, then the 

 conic represents an elliptic or a hyperbolic line according as 



pq^-ab. 



If X, Y are imaginary the triangle of reference has two 

 imaginary vertices, but we may take as real triangle of 

 reference a triangle self -conjugate with regard to the absolute. 

 The equation of a conic of the system may then be written 



X# 2 + p.y 2 + z a + z(px+ qy) = 



where X, /A have the same sign, and the equation of the absolute, 

 found by the same method as before, is 



The absolute is therefore real or imaginary according as X and 

 (A are both positive or both negative. 

 The discriminant in this case is 



If X, [i are both negative this is negative, and all lines are 

 therefore elliptic when the absolute is imaginary. 



The equation of a conic of the system may be written 



Hence when X and ji are both positive the conic is real only 

 when A >0, so that, when the absolute is real and X, Y are 

 an imaginary pair, all real conies represent hyperbolic lines. 



The following is a summary of the results : 



X, Y are imaginary, and the absolute is 



(1) A real proper conic, with the point Z in its interior. 



Hyperbolic geometry. 



(2) An imaginary conic. Elliptic geometry. 



(3a) A double line XY. Parabolic geometry, with repre- 



sentation by straight lines. 

 (36) A pair of imaginary lines ZX, Z Y. Parabolic geometry, 



with representation by conies passing through Z. 



