46 GEOMETRIC THEORY. 
this geometric construction is not necessary for the establish- 
ment of any part of the theory of such transformations, yet 
the method beautifully illustrates the theory and gives ita 
concrete geometrical form. We add in this section a brief 
outline of the geometric form of the theory, and in the exer- 
cises at the end of the chapter many problems depending on 
this construction.* 
If the lines / and l’ are real and the conic K is real, then 
real points are transformed into real points and the trans- 
formation is real. The transformation determined by the 
conic K is designated by T',. 
57. Invariant Points. We observe in the first place that 
a projective transformation T,, usually leaves two points on 
the line unaltered in position, for generally two tangents 
can be drawn to the conic K perpendicular to the bisector OX, 
Fig. 3; these cut / and /’ in A and A’, B and B’ respectively. 
iGo: 
*The whole theory of the projective transformations of the points ona line may 
be developed by the above geometric construction without any resort to analytical 
formula. It was so developed by the author in a paper entitled ‘‘Continuous Groups 
of Projective Transformations Treated Synthetically,’’ published in the Kansas Uni- 
versity Quarterly, vol. IV, No. 2, October, 1895. 
