126 THEORY OF COLLINEATIONS. 
156. Type I, the Parabolic Case. When a real hyper- 
bolic collineation of type I degenerates into one of type II by 
reason of two of its invariant points coinciding, the resulting 
invariant figure consists of two real points and two real lines. 
Likewise when a real elliptic collineation degenerates by rea- 
son of its two conjugate imaginary points coinciding, the 
resulting figure again consists of two real points and two real 
lines. It is clear that a real collineation of type II leaves in- 
variant a figure real in all of its parts. It is also clear that 
this real collineation of type II stands in the same relation to 
the hyperbolic and elliptic cases of type I as the real para- 
bolic transformation in one dimension stands to the hyper- 
bolic and elliptic transformations. The one-dimensional 
transformations along the invariant lines of the invariant 
figure of type II are a real parabolic and a real hyperbolic 
transformation, 7. e., t and k are always real numbers. 
157. Type III. The invariant figure of a real collineation 
of type III consists of one real point and one real line through 
it. The one-dimensional transformations along the invariant 
line and through the invariant point are both real parabolic 
transformations. The constants a, h and ¢ are all real num- 
bers. 
158. Types IV and V. A real perspective collineation of 
type IV leaves invariant a figure consisting of all points on a 
real line BC, and all lines through a real point A. Every in- 
variant line except BC has on it two real invariant points and 
hence the one-dimensional transformations along the invariant 
line through A are all real and hyperbolic. Thus the cross- 
ratio k is always a real number. 
The invariant figure of a real collineation of type V consists 
of all points on a real line / and all lines through a real point 
A onl. The one-dimensional transformations along the real 
invariant lines are all real parabolic transformations and ¢ is 
always a real number. 
