64. KANSAS UNIVERSITY QUARTERLY. 
A projective transformatfon which leaves invariant three elements 
of a one dimensional space is an identical transformation and leaves 
every element of the space invariant. (Lie: Continuierliche Grup- 
pen, S. 117.) Inthe case of a transformation of the second type 
leaving two coincident elements invariant one other invariant ele 
ment suffices to render the transformation identical. 
$2. Types of Projective Transformations in the Plane. 
A projective transformation of the plane is a self dualistic trans- 
formation 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 
projective transformation must be a self dualistic figure. This nec- 
essary condition will often enable us to determine whether any 
given plane figure can be the invariant figure of a projective trans- 
formation. 
‘very type of transformation in the plane is characterized by its 
invariant figure. We assume the well known theorem that the 
most general form of projective transformation in the plane leaves 
a triangle invariant. Proceeding from this fact it is easy to enu 
merate all the special forms of the invariant figure, and thus all the 
types of projective transformations inthe plane. We shall in this way 
determine five types (Fig. 1, Pl. I,) of transformations in the plane, 
of which type I is the kind of transformation whose invariant figure 
is a triangle. In this case the one dimensional transformations 
along the invariant sides and through the invariant vertices are all 
of the first kind leaving two elements invariant. 
If two vertices of the invariant triangle of type I coincide, then 
two sides must also coincide. The change isa self dualistic change 
and the modified figure is also a self dualistic figure. This is the 
invariant figure of type II. The one dimensional transformations 
of the range along AB and of the pencil through A are both of the 
first kind; those along b and through B are both of the second 
kind. 
If the two points A and B of the invariant figure of type II coin 
cide while the lines b and c do not coincide, the resulting figure is 
not self duatistic; the same is true if the two lines c and b coincide, 
but not the points A and B. Neither of the resulting figures is 
self dualistic, and hence there are no types of transformations in 
the plane characterized by these figures. But if A and B coincide 
and at the same time b and c, the change is self dualistic and also 
the modified figure. This gives us type III. The one dimensional 
transformation along the invariant line is of the second kind; so 
also is that of the pencil through the invariant point. 
