NON-EUCLIDEAN GEOMETRY 37 



REPRESENTATION BY A NET OF CONICS 



30. We have next to consider a generalisation of the 

 representation by circles, in which the circles are replaced by 

 conies. The conies must form a linear system depending 

 upon two parameters, i.e. a net. Further, to make the 

 system correspond as closely as possible to the system of 

 circles, which are conies passing through the two circular 

 points, we shall suppose the net to be a special net passing 

 through two fixed points, X, Y. The general equation of a 

 system of conies passing through two fixed points may be 

 written 



S+ (px+ qy+ rz)a=0 



where S is an expression of the second degree, a of the first 

 degree, and p, q, r are parameters. The parameters must 

 be connected by a linear homogeneous relation, hence the 

 variable line px+qy+rz=Q must pass through a fixed point Z. 

 Taking X YZ as the triangle of reference, the equation reduces 

 to the form 



axy+ bz 2 + z(px+ qy)=Q 

 where p, q are now the two parameters of the net. 



The conic degenerates to two lines, one through X, the 

 other through Y, if pq=ab. It degenerates to the line z=0 

 and a line y=mx 9 passing through Z, for infinite values of the 

 parameters. 



31. Consider a line ymx through Z. This cuts a conic 

 of the system where 



amx 2 + bz 2 -{- zx(p+ qm)=0. 



By choosing p and q suitably it may be made to touch the 

 conic. The condition for this is 



(p+ qm) 2 =4:abm. 

 Eliminating p+qm we obtain 



(amx 2 + bz 2 ) 2 =4abmz 2 x 2 , 

 or (axy-bz 2 ) 2 =0. 



The locus of points of contact of tangents from Z to the 



