124 THEORY OF COLLINEATIONS. 
The axis of the elation is the line at infinity and the vertex 
is the point at infinity on the z-axis. 
Equations (34) and (35) constitute the two essentially differ- 
ent canonical forms of type V, when the vertex is respectively 
in the finite and infinite part of the plane. 
$6. Real Collineations in a Plane. 
So far in the present chapter the development of the theory 
of collineations has proceeded on the assumption that in the 
defining equations both coefficients and variables are complex 
numbers. Also in the geometric constructions the points, 
lines and conics employed were not limited to real elements. 
We shall now go on to the consideration of the special case of 
real collineations, 7. e., collineations that transform real points 
and lines into real points and lines. The defining equations 
of areal collineation are real in both variables and coeffi- 
cients; the conics used in the construction are always real 
conics. 
153. Sub-types of Type 1. If the defining equations (1) of 
a collineation are real, then the cubic equation, 
ae+ Bee+ya+o=0, (7) 
whose roots are the three «-coordinates of the vertices of the 
invariant triangle, has real coefficients. In the general case 
when the three roots are distinct, one root is always real and 
the other two may be either real or conjugate imaginary. 
We therefore distinguish two sub-types of type I; in case the 
invariant triangle is real in all of its parts the collineation is 
said to be hyperbolic; in case the invariant triangle has one 
real and two conjugate imaginary vertices, and hence one real 
and two conjugate imaginary sides, the collineation is said to 
be elliptic. 
