48 GEOMETRIC THEORY. 
ratio (A,B,PP’)=(ABPP,). But since the first cross-ratio 
is constant for all tangents to K it follows that the second is 
constant for all pairs of corresponding points; hence for every 
projective transformation which has two real invariant points 
we have the theorem that any pair of corresponding points 
and the two invariant points have a constant cross-ratio. A 
transformation with two real invariant points is called a 
hyperbolic transformation. In this case the constant cross- 
ratio k is a real number. 
Fig. 4. 
