CONTINUOUS GROUPS OF PROJECTIVE TRANSFORMATIONS. 



245 



of the plane tt. Such a projective transformation is determined 

 and completely constructed by means of two conies K and K' in the 

 pland TV and touching the line 1. 



TlicorcDi I. A projective traiisfoniiatioii of the points of a p/aiie is 

 completely tleteniii/ied l>y means of two conies K and K^ touching a fixed 

 line 1 of the plane. 



It is clear from this method of constructing a projective transfor- 

 mation that any .point P outside the conic K; i. e. a point from 

 which two real tangents can be drawn to K, is transformed into a 

 point Pi outside of K^ Similarly a point within K is transformed 

 into a point within K*. For the tangents from P to K in this case 

 are a pair of conjugate imaginary lines, and their corresponding 

 lines are conjugate imaginar}? tangents to K^, and hence intersect 

 in a real point inside of K'. And finall}^ it ma}^ be seen that any 

 point P on K has its corresponding point P^ on K^. 



In order to construct P^ on K' corresponding to any point P on 

 K, draw the tangent to K at P; from the point where this tangent 

 to K cuts 1 draw the tangent to K'; its point of contact v/ith Ri 

 will be the point Pi desired. It follows from this last construction 

 that the polar of any point P with respect to K corresponds to the 

 polar of P^ with respect to K'. This fact suggests a method for 

 constructing a point P^ within K^ which corresponds to a point P 

 within K. 'Construct p the polar of P with r<^spect to K; then con-, 

 struct pi the line corresponding to p; and finally construct the pole 

 of pi with respect to K^; this will be the point Pi desired. 



The transformation determined by K and K^ transforms the line 

 1 into some line l^. Since 1 is a tangent to K, 1' must be a tangent 

 to Ri. The line P is easily seen to be the tangent to Ri from the 

 point of contact of R and 1. The tangent to R from the point of 

 contact of R^ and 1 is transformed into 1. 



L By C/ L R 



The chief results may be concisely stated as follows: 



