256 THEORY OF COLLINEATIONS. 
two real collineations is also real and hence all real collinea- 
tions in a plane form a real group RG,. This real group is a 
subgroup of G,, the general projective group of the plane. 
A list of the subgroups of RG, will be found to be nearly 
identical with the list of subgroups of G, given in § 6. 
THEOREM 60. All real collineations ina plane form a group 
ENG 
310. Hyperbolic and Elliptic Collineations of Type I. A 
collineation of type I leaves a triangle invariant, as was 
shown in chapter II, Arts. 98, 103. The coordinates of the 
vertices of the invariant triangle were found by solving the 
cubic equation 
Ge enh yt ok Op 
In the case of a real collineation the coefficients of this 
equation are all real, since they are rational functions of the 
real coefficients of the collineation. The roots of this equa- 
tion, when unequal, may be all real or one real and two con- 
jugate imaginary; the same is also true of the equation 
giving the y-coordinates of the invariant points. Hence there 
are two varieties of real collineations of type 1; one whose 
invariant triangle is real in all of its parts, and the other 
whose invariant triangle has one real and two conjugate im- 
aginary vertices, one real and two conjugate imaginary sides. 
We shall call these Hyperbolic and Elliptic collineations re- 
spectively. 
THEUREM 6. A real collineation of type Lis either hyperbolic 
or elliptic. 
311. The Hyperbolic Group hG,(AA'A”). All collinea- 
tions leaving a real triangle invariant form a group 
hG,(AA’A”). The one-dimensional transformations along 
the three sides are all hyperbolic, each with two real inva- 
riant points and a real cross-ratio k (Chap. I, Art. 39). 
Hence the hyperbolic collineation h T has a real invariant tri- 
angle and real cross-ratio parameters k and k’. 
