EMCH: PROJECTIVE GROUPS. II 



Systems related in such a manner are said to be in central symme- 

 try- 



The value k=-f-i together with an infinitely distant axis gives no 

 coUineation in the proper sense of the word. In this case the two 

 conies K and K^ coincide and determine what is called an identical 

 collinear transformation. We shall meet this conception in one yet 

 of the following chapters. 



Assuming the conies K and K^ as coaxial parabolas, and 

 tangent at their vertices, two collineations arise according as 

 the line at infinity or the finite common tangent is taken as the 

 axis of collineation. In the first case the finite point of tangency 

 of the parabolas is the centre of collineation. As is easily seen the 

 relation of corresponding points becomes that of similar systems 

 in similar position. In the second case the centre of collineation 

 is the infinitely distant point of the common axis of the parabolas, 



C= CO 



and its direction is orthogonal to the axis of collineation (the com- 

 mon finite tangent of the parabola). This, however, is orthogonal 

 affinity. Adding the condition k;= — i the two cases represent 

 central and orthogonal symmetry. 



