42 CONCRETE REPRESENTATIONS OF 



The inverse of a conic is in general a curve of the fourth 

 degree, but if the conic passes through X, Y the inverse is 

 also a conic passing through X, Y. In fact, taking OXY as 

 the triangle of reference and representing the fixed conic by 

 the equation xy=z 2 , the equation of any conic passing through 

 X, Yis 



cz 2 +fyz+ gzx+ hxy=Q, 

 and this is transformed into 



hz 2 +fyz+gzx+cxy=Q. 

 Also any conic whose equation is of the form 



z 2 +fyz+ gzx+ xy=0 



is transformed into itself. One of these is the absolute. 

 Let its equation be 



z 2 + ayz-i- bzx+ xy=0. 



(The coefficients of yz and zx cannot be zero since the fixed 

 conic does not in general touch the absolute.) 



The point Z is the pole of XY, i.e. z=0, with respect to 

 the absolute, hence its coordinates are (a, 6, 1). 



The absolute and a conic of the system have a pair of 

 common chords, one of which is z=0. To find the other we 

 have to make the equation 



X (z 2 + ayz+ bzx+ xy)+cz z +fyz+ gzx+ hxy=0 

 break up into z=0 and another line. Hence >.= h, and the 

 equation of the other chord is 



(g-bh)x+ (f-dh)y+ (c-h)=0. 



But this chord is the polar of Z with respect to the conic. 

 The equation of the polar of Z is 



(g-bh)x+ (f-ah}y+ (2c-ga-fb)=0. 

 Hence ga+fb=h+c, 



which is the condition which must be satisfied by the co- 

 efficients in order that the conic 



cz 2 +fyz-\- gzx+ hxy = 



may be a conic of the system. Since the relation is symmetrical 

 in c and h the inverse is also a conic of the system. 



Hence by quadric inversion with regard to any conic of 



