TYPES OF COLLINEATIONS. 87 
$3. Types of Plane Collineations. 
106. We have thus far dealt only with the most general 
form of plane collineations and have avoided the considera- 
tion of special cases. These special cases must now be con- 
sidered, and they lead us to the fundamental conception of 
types of collineations. We shall show that there are five dis- 
tinct types of plane collineations each of which is characterized 
by its invariant figure. A plane collineation is a self-dualis- 
tic transformation in the sense that it is both a point-to-point 
and a line-to-line transformation, and hence every plane figure 
invariant under a collineation must be a self-dualistic figure. 
This necessary condition will often enable us to determine 
whether any given figure can be the invariant figure of a col- 
lineation. 
107. Type I. It has been shown in articles 98 and 103 that 
the most general form of a plane collineation leaves invariant 
atriangle. This figure, consisting of three points and three 
lines, is a self-dualistic figure. If a collineation leaves inva- 
riant more or less than three points and three lines forming a 
non-degenerate triangle, it ceases to be a collineation of 
type I. 
A collineation T of type I leaves invariant a triangle ABC. 
All points on the side ¢ of the invariant triangle undergo a 
one-dimensional projective transformation whose two inva- 
riant points are A and B. The same is true of the points on 
the other two sides a and b. Likewise the pencils of lines 
through A, B, and C, respectively, undergo one-dimensional 
projective transformations, and in each pencil there are two 
invariant lines. Hence it is evident that the properties of a 
plane collineation of this type depend in an intimate manner 
on the properties of a one-dimensional transformation of the 
first type. 
108. Type II. If two vertices of the invariant triangle of 
type I coincide, then two sides must also coincide; for the 
