( 270 ) 



of tbc conic {0^ D.^ I\ /P^ Ru) is fouml (accoiding to § 2) the 

 equation 



A .<-3 «3 + fl23 -^1 ■''3 + «33 XyX^ — O . . . . (40) 



for the pencil of conies complementary to ^g ^'i'^'^ respect to D.,, D^. 

 Consequently to this pencil belongs 



C3 = «13 ■'■2 ''^ + "33 -''l ^3 + "33 •''1 '''2 = . . . (41) 



The points D^, Do, E^, Bu, B^, Bm can be Joined by a conic. 

 By applying (10) we find that this conic is indicated by 



(^23 ^ ^ ]] •'^1^ — ^^ 12 ■^l •'"2 — ^13 •'■! ■^S + ^23 ■'2 ■''3 = • (^2) 



So it belongs to the pencil represented by (32). This was easy 

 to forest e, the pairs of tangents through .Pji -^^3 furnishing one of 

 the inscribed quadrangles alluded to in § 6. 



In a similar manner by considering the pencil complementary 

 to (>i we deduce that D.2, D^, A'j and Ei lie in a conic with the 

 pair of points C", C" determined by 



"11 "23 •■''2 •'■3 + "22 «31 •■'■3 *'l + "33 "12 ■«I •'•2 = . . . (43) 



Hence out of the symmetiy of this equation follows that C', C' 

 are also joined by conic -i with the quadruples D-^, P^, /j's, h'm 

 and y)i, Z>3, R.2, Rn. 



For tiie conic {Do, D^ R^ Ri C' C") the equation 



7l = («12 ^ll2 + «13 ^^ 13) ■''l' + ("13 -•'23 — «12 ^123) •'■j •'^2 + 



+ ("12 -^-3 — «13 ^33) -''l -'s + «11 ^^23 ■''2 ^3 = • • (44) 



is furnished by (10). 



According to a wellknown property the six antitangential points 

 R lie in a conic. This is confirmed by the identity 



^23^ A — (J-Zi 7l ^ -^1^ (^123 1 (45) 



where /^jja represents a quadratic form. 



But moreover (45) proves that this conic has still the points C' 

 and G" in common with r^. 



