TYPES OF COLLINEATIONS. 89 
with vertices on I are of the first type with two invariant 
elements. 
111. Type V. A special case of the last figure is also ob- 
tained when we assume a third invariant point on the line 
AB of the invariant figure of type 11; likewise when we as- 
sume another invariant point on the invariant line of the lineal 
element of type III. The resulting figure (14, V) is self- 
dualistic and is the invariant figure of a collineation of type 
V, which is called an Elation. The one-dimensional trans- 
formations along all the invariant lines and in all the invari- 
ant pencils are parabolic, having one element invariant. 
This completes the list of types of collineations of the 
plane ; for if we modify these invariant figures in all possible 
ways we get no new self-dualistic figures. 
THEOREM 18. There are five types of collineations in the plane; 
each type is characterized by one of the self-dualistic invariant fig- 
ures of Fig. 14. 
C | 
A | BA Te e 
Fic. 14. 
112. Analytic Determination of the Five Types. Let the 
collineation be given analytically in the homogeneous form : 
Thus 
Ot, — 0,0 Dy 1 ¢,2), 
py, — a,c + by + C22, (2) 
i, = ORY SU Gye 
Putting x,=2, y,=y, and z,=z we have three linear equa- 
tions from which to determine the coordinates of the inva- 
riant points. These equations are 
(a,—p)«+by+ez2=0, 
av +(b,—p)yt+e2=0, (8) 
a,¢+b.y+(c,—e)z=0. 
