128 THEORY OF COLLINEATIONS. 
13. Show that the characteristic equations of the normal 
forms of types II and III are respectively 
p'— (2k) Det + (2k-+ 1) D'p—kD° = (9 — D}' (p— kD) =0, 
and Pa Paes lS (=i) eS. 
14. Verify the invariance of the points A, A’, A”, by sub- 
stituting their coordinates successively in the normal form of 
type I. 
15. Show directly from the normal form of type I that 
and k’ are the cross-ratios of the one-dimensional transforma- 
tions along AA’ and AA” respectively. 
16. Solve the problems analogous to 14 and 15 for types II, 
III, IV and V. 
17. Show that k and k’ in type I are the ratios of the roots 
of the characteristic equation of T. 
B. GEOMETRIC CONSTRUCTION. 
Using the first method of § 2 for constructing a collineation 
by means of two conics, K and K’, touching a line /: 
1. Show that the line / in Fig. 12 is transformed into the 
tangent to K’ from the point of contact of K and 1. 
2. What point corresponds to the point of contact of K 
and 1? 
3. Show that the tangent to K from the point of contact 
of K’ and lis transformed into 1. What point corresponds to 
the point of contact of K’ and 1? 
4. Discuss the collineation when both K and K’ are para- 
bolas. 
5. Show that if the conics K and K’ coincide, the collinea- 
tion T is the identical collineation. 
6. Show that interchanging K and K’ changes T into its 
inverse, 7. 
7. Show that a given collineation T can be constructed by 
