REAL COLLINEATIONS. 125 
154. The Hyperbolic Case. The invariant triangle of a 
hyperbolic collineation is real in all its parts; hence the one- 
dimensional transformations along the invariant sides and 
through the invariant points are real and hyperbolic. The 
cross-ratios k and k’ along the lines AB and AC respectively 
Bee : 
are real numbers. The cross-ratio =~ through A is also real. 
155. The Elliptic Case. An elliptic collineation leaves 
invariant a triangle with one real vertex, A, and two conju- 
gate imaginary vertices, B and C, Fig. 23. The one-dimen- 
Fic. 23. 
sional transformation along BC is elliptic. So, also, is that 
through A. The character of the one-dimensional transfor- 
mations along the imaginary lines AB and AC cannot be in- 
ferred from anything developed in Chapter I. & and k’ 
cannot be two independent complex numbers, for = must be 
a complex number of the form e’*. From equations (10) it 
follows that if (A’, B’) and (A”, B”’) are conjugate imaginary 
points, then & and k’ differ only in the sign of 7 and hence 
must be conjugate imaginary numbers. 
THEOREM 32. Real collineations of type lina plane are either 
hyperbolic, with three real invariant points, or elliptic with one real 
and two conjugate imaginary invariant points. 
