GEOMETRIC THEORY. 47 
A and A’, B and B’, are therefore corresponding points, and 
the revolution about O brings A’ to A and B’to B. The 
points A and B are the wvariant or double points of the 
transformation. 
It has just been said that generally there are two tangents 
to K perpendicular to OX. This should be examined more 
closely. When the conic K is an ellipse, two real tangents to 
K can always be drawn perpendicular to OX; and hence the 
projective transformation determined by an ellipse always has 
two invariant points. When the conic K is a parabola, there 
are still two real tangents perpendicular to OX; but one of 
them is the line at infinity : hence the projective transforma- 
tion determined by a parabola always has two real invariant 
points, one of which is the point at infinity on /. 
When the conic K is a hyperbola, there are three cases to 
be considered. If the asymptotes of the hyperbola kK make 
with the line OX angles which (measured in the same direc- 
tion) are both less than, or both greater than, a right angle, 
then two real tangents to the hyperbola can be drawn perpen- 
dicular to OX, and the transformation determined by K has 
two real invariant points. If on the other hand the asymp- 
totes to K make with OX angles one less than, and the other 
greater than, a right angle, then the tangents to K perpen- 
dicular to OX are imaginary and the transformation deter- 
mined by K has its invariant points imaginary. Butif K has 
one of its asymptotes perpendicular to OX, the transformation 
determined by K has one real invariant point. Or, since the 
asymptote to a hyperbola is the limiting position of two par- 
allel tangents, we may say in the last case that the trans- 
formation determined by K has two coincident invariant 
points. 
58. Hyperbolic Transformations. The lines AA’, BB’, |, l’ 
are four fixed tangents to the conic K. Any fifth tangent, as 
PP’, cuts these four tangents in four points whose cross-ratio 
is constant. The range A,, B,, P, P’ may be projected or- 
thogonally on / by lines drawn parallel to AA’; and the cross- 
