66 KANSAS UNIVERSITY QUARTERLY. 
types of projective transformations in the plane apply equally well 
to the determination of types in space. A projective transforma- 
tion in space leaves invariant certain points, lines and planes. | In 
an invariant plane the transformation is two dimensional and must 
be one of the five types enumerated above or an identical transfor- 
mation. Along an invariant line and in an invariant pencil of 
planes the transformation must be either of the first or second kind 
or identical. 
We know that the projective transformation of the most general 
kind in space leaves a tetrahedron invariant. From this starting 
point, by the aid of the principles just stated, it is easy to enumer- 
ate all the special forms of the invariant figure and hence all the 
types of projective transformations in space. We shall find in this 
way thirteen types (Fig. 2, Pl. 1), of which type I is characterized by 
the tetrahedron itself. The two dimensional transformations in the 
four invariant planes are all of the first kind leaving a triangle in- 
variant. The one dimensional transformations are all of the first 
kind leaving two elements invariant. 
If two vertices of the tetrahedron coincide, then two faces also 
coincide. The modified figure then consists of three invariant 
points, three invariant planes, three invariant lines in a plane and 
three invariant lines through a point. ‘The resulting figure is thus 
self dualistic and characterizes type II. The two dimensional 
transformation in the plane ABC is of the first kind leaving a tri- 
angle invariant; those in the planes ABI and ACI are both of the 
second kind. 
Let the vertices of the tetrahedron coincide two and two; i. e. let 
D coincide with A, and C with B. The resulting figure then con- 
sists of two points, two planes and three lines not lying in a plane. 
It is self dualistic in every respect and characterizes type III. The 
two dimensional transformations in the planes ABI and ABm are 
both of second kind. 
Next let three vertices of the tetrahedron coincide at A, then 
must three faces of the tetrahedron also coincide. The resulting 
figure characterizes type IV. It consists of two points, two planes, 
and two lines. ‘The two dimensional transformation in the plane 
ABI is of the second kind, that in the plane z is of the third kind. 
Finally let all four vertices and all four faces of the tetrahedron 
coincide. The single invariant point hes in the single invariant 
plane, and there is an invariant line in this plane through the inva- 
riant point. This is best seen if considered as a special case of 
type IV. If in type IV A be made to coincide with B, the plane 7 
