212 



KANSAS UNIVERS^^^• (HIARTERLV. 



To the plane z=o corresponds the plane at infinity; thus every 

 pencil of rays having its apex in the XY-plane is transformed into 

 a pencil of parallel rays, and a pencil of planes with its carrier in 

 the XY-plane is transformed into a pencil of planes. The invari- 

 ant elements of the involution are 



y ^ ._ , , ^ 



rl a* 



1 a--r be 

 c 



be 



^=— 1 a^ 



be. 



X a— I a2-fbc 

 Both lines intersect the z-axis and are parallel to the xy-plane. 

 Designating b}' A and B the angles which these rays make 

 with the positive part of the xz-plane there is 



c 



tgA= 



tg:B=- 



1 a2-|-bc 

 c 



a— i/as-^bc 



therefore the trigonometric tangent of the angle under which the 

 lines cross each other 



tg (A-B) =^; 



be 



The condition for a right angle is b=c. 



rit 3. 



Designating in Fig. 3 the lines of invariant points by g and 1, it 

 is clear that a ray connecting any point of g with any point of 1 is 



