NEWSON: PROJECTIVE TRANSFORMATIONS. 65 
A projective transformation of the plane which leaves invariant 
four points of the plane, no three of which lie on a line, is an iden- 
tical 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, and hence every line 
through the opposite vertex is an invariant line. The resulting 
figure, which consists of all the points on a line c and all the lines 
through a point C not on the line c, is self dualistic. This is the 
invariant figure of a transformation of type IV, which is called a 
perspective transformation. The one dimensional transformations 
along all lines through C and in all pencils with vertices on c are 
of the first kind. 
A special case of the last figure is when the vertex of the inva- 
riant pencil is on the line c of invariant points. This special case 
is also obtained when we assume a third invariant point on the line 
c of the invariant: figure of type II; likewise when we assume an- 
other invariant point on the invariant line of the linear element of 
type ilI. The resulting figure is self dualistic and is the invariant 
ficure of a transformation of type V, which is called an Elation.* 
The one dimensional transformations along all invariant lines and 
in all invariant pencils are of the second kind, leaving one element 
invariant. 
This completes the list of types of projective transformations of 
the plane; for if we modify these invariant figures in all possible 
ways we can get no new self dualistic figures. 
There are five types of projective transformations tn the plane; each 
type ts characterized by one of the self dualistic invariant figures of 
Iie. I. 
Lie in Vorlesungen ueber Continuierliche Gruppen, pp. 65-6 and 
510-12, has determined all the types of projective transformations 
in the plane with the same results as given above. See also a 
paper by the writer in this Quarrerty Vol. IV, page 243-49. The 
method here employed is important because it lends itself immedi- 
ately to the determination of all types of projective transformations 
in space. 
$3. Types of Projective Transformations in Space. 
All projective transformations in space are self dualistic, for they 
transform points into points and planes into planes. Therefore the 
conditions of dualism employed in last section for determining 
*Lie: Vontinuierliche Gruppen, 5. 262. 
