88 THEORY OF COLLINEATIONS. 
change is a self-dualistic change and the modified figure must 
be a self-dualistic figure. This modified figure consists of 
two invariant points, A and B, and two invariant lines, / and U’. 
Two of the invariant points are on one of the invariant lines ; 
and two of the invariant lines pass through one of the inva- 
riant points (Fig. 14, II). This is the invariant figure of 
type I]. The one-dimensional transformations along the line 
AB and through the point A are of the first type; those 
along the line / and through the point B are parabolic. 
109. Type III. If the two points A and B of the invariant 
figure of type II coincide while the lines / and l’ do not coin- 
cide, the resulting figure is not self-dualistic; the same is 
true if the two lines / and l’ coincide but not the points A and 
B. Neither of the resulting figures is self-dualistic, and 
hence there are no types of collineations in the plane charac- 
terized by these figures. But if A and B coincide and at the 
same time / and l’, the change is self-dualistic, and also the 
modified figure. The invariant figure (14, III) consists of a 
single invariant line and a single invariant point on the inva- 
riant line. Such a figure is called a lineal element. This 
gives us type III. The one-dimensional transformation 
along the invariant line is parabolic; so also is that of the 
pencil through the invariant point. 
110. Type IV. A collineation of the plane which leaves 
invariant four points of the plane, no three of which lie on a 
line, is an identical transformation, and leaves every point of 
the plane invariant. It may happen, however, that a third 
invariant point is situated on one of the sides of the invariant 
triangle of type I. In that case every point on this side is an 
invariant point, art. 8, and hence every line through the 
opposite vertex is an invariant line. The resulting figure 
(14, IV), which consists of all the points on a line / and all 
the lines through a point A not on the line J, is self-dualistic. 
This is the invariant figure of a collineation of type IV, which 
is called a perspective collineation. The one-dimensional 
transformations along all lines through A and in all pencils 
