4 KANSAS UNIVERSITY QUARTERLY. 



B^, the tangents in A and B to the conic K intersect in a point D 

 of the polar p and those in A^ and B^ to the conic K' in a point 

 D' of the polar p^. As is well known from synthetic geometry 

 the points D and D^ lie on a ray through C. Moreover, the tan- 

 gents in A and A^ and in B and B^ meet in the line 1, hence, the 

 points D and D^ are obtained by our construction of the collinea- 

 tion. To every point corresponds one and only one point and both 

 lie in a ray passing through the centre C. Two corresponding 

 straight lines always meet in a point of the line 1. These two 

 conditions, however, constitute perspective collineation and hold 

 for any point, or line of the plane and their corresponding ele- 

 ments. 



In the next chapter we shall make those constructions which will 

 be necessary in the study of group-properties of perspective 

 collineations. 



§2. Classification of Perspective Collineations. 



The two conies, K and K^, determining the perspective colline- 

 ation, being given we can ask for the line q^ which corresponds to 

 the infinitely distant straigiit line q. Drawing the two parallel 

 tangents to the conic K from each point at infinity, and from their 

 intersection-points with the axis 1 of perspective collineation tan- 

 gents to the conic K^, the points of q^ are obtained by the intersec- 

 tion of each such pair of tangents to the conic K^. Conversely, 

 there exists a straight line r whose corresponding line r^, or what 

 is the same, q^, is at infinity. The lines qi and r may be called 

 counter-axes (German " Gegenaxen ") of perspective collineation, 

 and are parallel to the axis of collineation. 



In central projection and perspective the constructions are usu- 

 ally made by aid of the centre and axis and the counter-axes of 

 perspective collineation. A perspective collineation is determined 

 by centre and axis and any one of the counter-axes, and, as imme- 

 diately follows by construction, also by centre and axis and two 

 corresponding points of the collineation. 



It is now of great importance to state the connection between 

 these two determinations. 



Each perspective collineation transforms a ray through the centre 

 into itself and also each point of the axis into itself. In other 

 words, it leaves the points of the axis and the rays through the 

 centre invariant. Each ray through the centre represents two 

 coincident projective point- ranges, and each pencil of rays through 

 a point of the axis two coincident projective pencils of rays. 



