emch: projective groups. 9 



(^). If K and K^ have two parallel common tangents, and 



k^ — I, i. e., if the collineation is dilation and involution (Fig. 5), 



there is: 



( cx> LAAi)=— I, or 



LAi=— LA; (claai)=— I. 



Fig. 5. 



c=^ 



C s 00 



"T^^^ 



Each pair of corresponding points lies in a fixed direction and 

 is equidistant from the axis. The ranges in corresponding straight 

 lines are symmetric with the centre of symmetry in the axis. Cor- 

 responding triangles have the same area. This collineation is 

 called oblique, or orthogonal symmetry in regard to the axis, accord- 

 ing as the direction of the centre is oblique, or orthogonal to the 

 axis. 



((•). K and K^ have two parallel common tangents and k=+i. 

 In this case one of the conies of involution is revolved about the 

 axis through 180 <^, (Fig. 5). The centre is at infinity and its 

 direction is parallel to the axis. In such a collineation corres- 

 ponding figures have equal areas and it is therefore called the affinity 

 of figures of equal areas. 



