126 KANSAS UNIVERSITY QUARTERLY. 
ponding rays (planes). The second type leaves invariant a single 
ray (plane) m’. x and x, represent a pair of corresponding rays 
(planes), such that cot(x,,m’)—cot(x,m’)=«. If x’ is the ray 
(plane) which is transformed into the perpendicular to m’, then 
| a==—cot(x’,m’). 
An analogous theory may be developed for the projective trans- 
formations of the planeand of space. The equations of the projec- 
tive transformation in the plane are given in cartesian coordinates 
by 
es ax-+by-+c snare _a4X+biy rey 
a,x+b,y+c, 1 a,x+b,y+c, (2) 
There are five types of such transformations, each type being 
characterized by its invariant figure. Corresponding to each type 
is a normal form in which the constants are the essential parame- 
ters of the transformation. 
The equations of the projective transformations in space are 
given in cartesian coordinates by 
ry Hi ax+by-+cz+d __a,x+b,y+c,z+d, 
1 a,x+b,y+c,z4+d, (ame ae hare e 3) 
ia, 8- bey e524 Gs 
a 
1 a,x+b,y+c,z+d, 
There are thirteen types of projective transformations in space; 
each is characterized by its invariant figure.* (See Fig. 1.) 
The object of this paper is to give the normal forms of the five 
types in the plane and of the thirteen types inspace. These forms 
have been arrived at from geometrical considerations rather than by 
purely algebraic processes. These forms are given for the most 
part without proof. The detailed proof is given for the first and 
second types in the plane; the other types are derived by analog- 
Part 1—The Plane. 
ALN GEN se 
ous processes. 
The most general form of projective transformation in the plane 
leaves invariant a triangle, the cartesian coordinates of whose ver- 
tices may be represented by A(a,b) B(a,,b,) and C(a,,b,). In a 
previous papert it has been shown that a projective transformation 
leaving a triangle invariant is characterized by the position of the 
triangle and three anharmonic ratios taken along the invariant 
*See Kan. Univ. Quar., Vol. VI, p. 63. 
+See Kan. Univ. Quar., Vol. V, p. 8. 
