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, b, 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 -con jugate with regard to the absolute. 

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



X# 2 -f- M 2 + z 2 + z(px+ qy] =0 



where X, p, 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 

 {ji are both positive or both negative. 

 The discriminant in this case is 



If X, {x 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 



7? o /y A 

 "^ i T-\ ~_^ i*_J it i 4/ -I- i v \2 . *y2 



/x I w^^ rr & I (^ W. I ti\^ ci~ ' } ~~~~ ~A ^ * 



2X 2(x 4X{ji 



Hence when X and \L 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 X Y. Parabolic geometry, with repre- 

 sentation by straight lines. 



(3&) A pair of imaginary lines Z X , Z Y. Parabolic geometry, 

 with representation by conies passing through Z. 



