EXERCISES. 129 
co? different pairs of conics K and K’. Onesuch pair of con- 
ics is associated with every line / in the plane. 
8. When the line J is the line at infinity in the plane, the 
two conics K and K’ are parabolas; determine their axes. 
9. Show that there are ~’ collineations of type I leaving 
the same triangle invariant. 
10. Show that there are ~” collineations of type II having 
the same fundamental invariant figure. 
11. Show by this geometric method that there are ~* col- 
lineations of the plane. 
12. What conditions must the conics satisfy in order that 
the transformation be a pseudo-transformation. 
13. Discuss the cases when K and K’ are one or both de- 
generate conics. 
Using the second method for constructing a collineation by 
means of two conics K and K, intersecting in S,: 
14. Show that there are ~* collineations in the plane. 
15. Show that every collineation leaving the triangle A BC 
invariant can be constructed by using the same point S, as a 
vertex. 
16. Discuss the method of construction when one or both 
of the conics K and K, are degenerate conics. 
17. When K and K, are similar and similarly placed conics, 
the line at infinity is invariant and parallel lines are trans- 
formed into parallel lines. 
18. When K and K, are both circles, show that angles are 
transformed into equal angles and all figures into similar 
figures, the ratio of areas being that of K to K,. 
19. When K and K, are circles of equal radii, show that the 
resulting collineation is a rotation of the plane about the one 
real invariant point through an angle equal to the angle be- 
tween the radii. 
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