INVOLUTIVE 1-1 QUADRIC TRANSFORMATION IN A PLANE. 259 



§ 8. In the Hirst transformation all points on the fixed conic are self-corresponding 

 points, and the three principal points are X, the given fixed point, and the points of 

 contact Y, Z of the tangents from X to the conic. To a straight line through X 

 corresponds a line through X, but to a line through Y a line through Z, and vice versa, 

 while the numerical equations for two corresponding curves are 



n' = 2n — a — (S — y ; 

 n'-a' — n — a; n—fi' = n — y] n' — y=n — j3', 

 n = In — a — /3' — y, 



so that the Beltrami transformation is the more symmetrical. 



In the Hirst transformation the points on any line through X are paired in an 

 involution w r hich is hyperbolic or elliptic according as the line cuts or does not cut the 

 fundamental conic. 



Also in order that a conic shall transform into a conic it must pass through two of 

 the points X, Y, Z. Hence, if a conic transforms into itself it must pass through the two 

 points Y and Z, for to a conic through X and Y corresponds a conic through X and Z, so 

 that, if self-corresponding, it would pass through X, Y, Z, which is impossible. 



If the self- corresponding conic through Y and Z cut the fundamental conic again 

 in P and Q, since P and Q are self-corresponding points, it follows that X P and X Q are 

 tangents to the new conic. The points on it are paired in involution, the centre of 

 involution being, of course, the point X. 



§ 9. If we take the fundamental conic to be 



x 2 — yz = 



it is easy to prove that the correspondence gives 



1 1 1 

 y £ £ n 



and the self-corresponding conies are given by 



x 2 + yz + x(By + Gz) = 0. 



There are therefore a two-fold infinity of such conies. Such a conic is over-specified, 

 for it passes through Y, Z, P, Q, and is tangent to X P and to X Q. This suggests the 

 following theorem : — 



" If two conies cut in four points Y Z P Q, and if the pole of Y Z Avith respect to one 

 conic is on the tangent at P to the second conic, it is also on the tangent at Q to the 

 second conic and is the pole of P Q with respect to the second conic." 



This proposition may be verified analytically as follows : 



Let x 2 — ayz = be the equation in trilinear co-ordinates to one of the conies, so that 

 the pole of x = with respect to it is the point X. 



Let P Q be the line 2x + By + Cz = 0. 



The equation to any conic through the four points is 



k(x* - ayz) + x( 2* + By + Gz) = 0. 



