72 KANSAS UNIVERSITY (QUARTERLY. 



XX,' y}',' zz.' These three lines together with 1 and 1' determine a 

 conic K touching the five lines. Any other tangent to K as PP ' 

 cuts the lines 1 and 1' in corresponding points of the projection. 



Thus by means oj a conic K touching the two lines 1 and \' a projec- 

 tion of the iifie \ on V is cotnpletely determined.* 



It now r be revolved about O until it coincides with 1. an}- point 

 P' on r will be broi^ht to P,, so that OP'— OP,. The two ranges 

 of points are then considered as existing on the same line 1. This 

 operation of projecting a range of points on 1 into a new range on 

 1 ' and then by revolution about O bringing the new range back to 

 1 will be called a Projective Trafisformation. The effect of a pro- 

 jective transformation is therefore to shift the points of a line into 

 new positions, so that there shall be a projective relation, i. e. a 

 one to one correspondence, between the old and new positions ot 

 the points. A projective transformation determined by a conic K 

 will be designated by the symbol Tk- 



We observe in the first place that a projective transformation T^ 

 usually leaves two points of the line 1 unaltered in position. For 

 generally two tangents can be drawn to the conic K perpendicular 

 to the bisector OX; these cut 1 and 1' in A, A' and B,B' respect- 

 ively. A and A,' B and B' are therefore corresponding points, 

 and the revolution about O brings A' to A and B' to B. The 

 The points A and B are called the double points or the ini'ariant 

 points of the transformation. 



We said that generally there are two tangents to K perpendic- 

 ular to OX. This should be examined more closely. When the conic 

 K is an ellipse, two real tangents to K can always be drawn perpen- 

 dicular to OX; and hence the projective transformation determined 

 by an ellipse always has two real 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 

 transformation determined by a parabola always has two real 

 invariant points, one of which is the point at infinit}' on 1. When 

 the conic K is a hyperbola, there are three cases to be considered. 

 If the asymtotes to the hyperbola K make with the line OX angles 

 which (measured in the same direction) are both less than, or both 

 greater than, a right angle, then two real tangents to the hyperbola 

 can be drawn perpendicular to OX, and the transformation deter- 

 mined by K has two real invariant points. If on the other hand 



*This principle mid t lie tlieorems involved in it arc fundamental lu all our work on 

 the projective transformations of the straight line. The principle itself is ton well 

 known to need proof or explanation. See: 



Cremona's Projective Geometry, Chap. XIY. Art. 150 (English Edition). 



Rej^e's Geometrie der Lage. 8 Band. p. 60 and 6() (3d EJdition). 



Salmon's Conic Sections. Art. D:.'!) tl'. (lit h Edition). 



