40 CONCRETE REPRESENTATIONS OF 



When X, Y are real there are conies which represent 

 hyperbolic, parabolic, and elliptic lines, and the measure of 

 angle is hyperbolic. 



When X, Y are coincident the measure of angle is parabolic. 



33. In the representation by circles the points X, Y are 

 the circular points, while Z is the centre of the fixed circle. 

 The general representation by conies in the case where X, Y 

 are imaginary is, of course, at once obtainable by projection 

 from the representation by circles. A real conic and a point 

 inside it can always be projected into a circle and its centre. 

 All that is necessary is to make the centre correspond to the 

 given point and the line infinity to the polar of this point. 

 From this we deduce at once the distance and angle functions 

 in this representation. 



The angle between two lines is \i times the logarithm of 

 the cross-ratio of the pencil formed by the tangents to the 

 two conies at their point of intersection and the lines joining 

 this point to X, Y. 



Two points P, Q determine a conic cutting the absolute 

 in U, V ; the distance (PQ) is then ^ times the logarithm 

 of the cross-ratio (PQ, UV) of the four points on this 

 conic. 



A circle is represented by any conic passing through X, Y. 



34. In the circular representation we saw that motions 

 are represented by pairs of inversions in orthogonal circles. 

 In the representation by conies there is an analogous trans- 

 formation. Any line through Z is cut in involution by the 

 system of conies, the double points being on the absolute. 

 The transformation by which any point is transformed into 

 its conjugate is a quadric inversion. 1 The conies of the 

 system are transformed into themselves by such a trans- 

 formation, while the points of the absolute are invariant. 



To find the equations of transformation, take XYZ as 

 triangle of reference, and write the equation of the absolute 



1 On quadric inversion see C. A. Scott, Modern Analytical Geometry, pp. 230-236. 



