50 GEOMETRIC THEORY. 
This gives a constant relation between P and P,, a pair of 
corresponding points. This constant is of course the recipro- 
cal of the segment AQ,, where Q, is the point into which the 
point of infinity is transformed. 
61. Number of Transformations. Every conic touching 
the lines / and l’ determines a projective transformation. It 
is therefore possible to construct as many different transfor- 
mations of the points on the line / as there are conics touch- 
ing ? and l’. We know that o* conics can be drawn 
touching any two lines; hence we infer that there are ©’ pro- 
jective transformations of the points on a line. Among the 
co* conics touching / and l’ are ~* hyperbolas having one 
asymptote perpendicular to the line OX. Hence we infer 
that there are ~* parabolic transformations each of which 
leaves only one point invariant. 
62. Continuous System of Transformations. Our next ob- 
ject is to subdivide and to classify these ~’ transformations 
of the points on the line 1. We consider first the quadrilat- 
eral ABB’A’ (Fig. 3). A range of 7 conics may be de- 
scribed touching the sides of this quadrilateral. Call this 
range R. Each of these conics determines a hyperbolic 
transformation which has A and B for its invariant points. 
Each conic of the range FR touches the line / at a different 
point ; and every point of the line / is the point of contact of 
some conic of the range Rk. If C be the point of contact of 
conic K, the characteristic cross-ratio of the transforma- 
tion produced by K is given by the cross-ratio of the four 
points A,B,C,O. The points A,B,O are fixed, while the 
point C varies for different conics of the range R. From the 
continuity of the point system on the line /, we infer the con- 
tinuity of the system of «/ transformations which leave A 
and 6 invariant. 
63. One-Parameter Groups. The range of conics inscribed 
ellipse cuts it in two conjugate imaginary points, and a parabola cuts it in two coin- 
cident points. The unique appropriateness of the names is shown by the results 
of art. 72. 
