EXERCISES. NDT 
Exercises in Chapter 2. 
A. ANALYTIC THEORY. 
1. Obtain equations (3) by solving equations (1) for # and y. 
2. Show that the determinant of (3) is the square of the 
determinant of (1). 
3. Discuss the collineation (1) when A= 0. 
4, Show that the collineation (1) transforms a conic into a 
conic and a curve of the nth degree into a curve of the nth 
degree. 
5. Find the coordinates of the point which is transformed 
into the origin by (1); also the coordinates of the point into 
which the origin is transformed. 
6. Find the equation of the line into which the line at in- 
finity is transformed by (1). 
7. What values do the coefficients have in (1) when it rep- 
resents an identical transformation? 
8. Give a direct analytic proof that the cross-ratio of four 
collinear points or four concurrent lines is unaltered by the 
collineation (1). 
9. Prove the theorem of Menelaus quoted in Art. 127. 
10. In the normal forms of types I, II and III, show that T 
and its inverse 7 differ only in this, that k and k’ are 
changed into 1/k and 1/k’, k and t into 1/k and —t, and t 
into —t, respectively. 
11. Show that the determinants of the normal forms of 
7\ Jay il ARE aL 
types II, IV, III and V are respectively k |4’ B 1], k|A4’ B 1), 
Gt QO dd 7 
ih, ehaxel he 
12. If both terms of the equations (11) of the normal form 
of T are multiplied (or divided) by any factor M0 or ~, 
the determinant of T will be multiplied (or divided) by M°. 
