KANSAS UNIVERSITY QUARTERLY. 
WOt Vals APRNs 1807. NO 2s 
Types of Projective Transformations in the 
Plane and in Space. 
BY, ely Be INEWiSOING 
With Plate I. 
The object of this paper is to determine the different types of 
projective transformations in space and to classify them according 
to their invariant figures. In order to do this it will be necessary 
to summarize the results already known for one and two dimen- 
sions, and to state the principles which will be used in the develop- 
ment of the method here employed. 
$1. Types of Projective Transformations in one Dimension. 
The one dimensional transformations* which we shall have occa- 
sion to make use of in this paper are the transformations of the 
range of points ona line, of the pencil of lines through a point, and 
the pencil of planes through a line. We know that there are just 
two types of projective transformations of a one dimensional form, 
viz: the kind that leaves invariant two distinct elements, either, 
real or imaginary; and the kind that leaves invariant two coincident 
elements. By elements of a one dimensional form are understood 
the points of a range, the rays of a flat pencil and the planes of an 
axial pencil. 
There are only two types of transformations of a one dimensional 
space. Type r leaves two distinct elements invariant; type LI leaves 
two coincident elements invariant. 
Much use will be made in this paper of the following theorem. 
*Throughout this paper the word transformation must always be understood to 
mean projective transformation, 
(68) KAN. UNIV. QUAR., VOL, VI, NO. 2, APRIL, 1897, SWRINS A. 
