CHAPTER II. 
COLLINEATIONS IN THE PLANE; TYPES AND 
NORMAL FORMS. 
1. General Analytic Theory of Plane Collineations. 
2. Geometric Construction of Plane Collineations. 
3. Types of Plane Collineations. 
4. Normal Forms of Equations of the Five Types. 
5. Canonical Forms of Equations of Collineations. 
6. Real Collineations. 
Exercises. 
75. The present chapter is devoted to the theory of project- 
ive transformations or collineations ina plane. Following 
the methods of the last chapter we shall first, in § 1, define 
a collineation analytically and develop the fundamental prop- 
erties of such transformations. We shall then develop in 
§$ 2 two mutually dualistic geometric methods of constructing 
plane collineations. In § 8 we show both analytically and 
geometrically the existence of five distinct types of plane 
collineations. In § 4 we develop the normal forms of the 
defining equations of the five types, and in § 5 the canonical 
forms of these same equations. The special case of real col- 
lineations in the plane is then discussed in § 6, and the chapter 
closes with a list of exercises supplementing the theory. 
$1. General Analytic Theory of Plane 
Collineations. 
76. Analytical Definition of a Plane Collineation. Using 
rectangular or oblique Cartesian coordinates, the transforma- 
tion of the plane which is expressed by the linear fractional 
equations, having the same denominator, 
ax+by+e 
a/x+ bly +e! 
a! +b/y+e" ( 1 ) 
vy — ax + by +c! 
ang ga— 
(65) 
