NON-EUCLIDEAN GEOMETRY 41 



xy=z*. P is found from P' as the intersection of ZP' with 

 the polar of P'. The polar of P'(x'y'z') is 



xy'+x'y=2zz', 

 and the equation of ZP' is 



xy'=x'y. 



Hence r . . ~_ 1 . 1 . 1 



*V M 2 J . J * j . 



35. We may similarly establish a quadric inversion with 

 regard to any conic of the system. Let the conic cut the 



FIG. 6 



absolute in 7, J. Draw the tangents at I, J to the absolute, 

 cutting in 0. Then is to be taken as the centre of inversion. 

 The same point is obtained by drawing the tangents to the 

 conic at X, Y. (See Fig. 6, where the absolute is the ellipse 

 and for clearness X, Y are taken to be real.) Hence XY is 

 the polar of 0. The conic and the absolute with the points 

 Z and simply exchange roles, and the conic is left invariant 

 by the transformation, while the absolute is transformed into 

 itself. 



