XEWSON : I'ROJECnVK IRAXSIORM ATIONS. 



73 



the asymptotes 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 determined by 

 K has its invariant points imaginary. But if 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 parallel tangents, we may 

 say in the last case that the transformation determined by K has 

 two coincident invariant points. 



It may happen that the three lines xx.' yy.' zz ' meet in a point; 

 let Q be such a point (Fig. 2). The conic which touches these 



three lines and 

 both 1 and 1 ' de- 

 generates into the 

 segment QO of the 

 line QO. Any line 

 -iSTof the pencil, whose 

 vertex is Q, is a 

 tangent to this 

 conic and cuts 1 

 and 1 ' in corre- 

 sponding points of 

 the projection. The two projective ranges on 1 and 1 ' are now in 

 perspective position. When 1' is brought as before into coinci- 

 dence with 1, one of the invariant points is O and the other is 

 determined by the perpendicular through Q on OX. 



Thcorciii I.- — Evcrx projective traiisfonnation of the points 0/1 a line 

 leaves two of its points invariant : these tiuo points max he real ami 

 distinct, or real and coincident, or iinaginarx. 



The lines y\.A.' BB.' 1, 1' are four fixed tangents to to the conic 

 K. Any fifth tangent as PP' cuts these four fixed tangents in four 

 points whose anharmonic ratio is constant. The range A,, Bj, P, 

 P' may be projected orthogonally on 1 by lines drawn parallel to 

 AA'; and the anharmonic ratio ( A, B , PP ' )=:=( ABPP, j. But 

 since the first anharmonic 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 corre- 

 sponding points and the two invariant points have a constant 

 anharmonic ratio. 



In the case where the two tangents to K perpendicular to OX 

 are imaginary, it still holds that the anharmonic ratio of the four 



