( 271 ) 



8. According to the identity 



^4 = (^33 .T.^ + 2 «23 ^2 •''S + "J2 ^Z~) -^1 + («11 ^2 -^3 + 



+ 2 «12 .ci .^s + 2 «13 ^1 3-2) .rj ,r3 (46) 

 the tano-ential points T^ and Ti of />! lie on the conic 



Ti ^ «11 .rj ƒ3 + 2 «,2 ;r; .r, + 2 «13 .ri .''j = . . . (47) 



It Is complementary to the conic (43); so there is a conic fj 

 passing ihrough D.2, D-i, Ti, Tj^ C', C" . 



It is similarly proved that A, A, T's, Tn, To, Tm lie in a 

 conic Tos- 



And now again we can form an identity 



X r* + ro3 fi = ,ri2 ri23 (48) 



out of which follows that C' and C" are also connected with the 

 six points T by a conic t^.2s • 



Thus the conic Tjog of the six tangential points and the conic 

 Ci23 '^'f ^^'^ ^*^ antitangential points intersect in two points tying 

 on r^. 



Mr. Brill has pointed out that also the six points of inflection 

 are connected by a conic « (Math. Ann. XII, 106), intersecting 

 r^ still in two points belonging to the conic T123 (Math. Ann. 

 Xlir, 182). 



Out of the preceding follows that the complementary points are 

 lying on three remarkable conies «123, r^j and co . 



9. Evidently there are four conies <^P^, coinciding with their 

 associates, each determining the points of contact of a double tangent. 

 They are represented by 



^0— -^a^sl/^ll + ^3 '«^ll/^SZ + ^'l ^2^^033 = 0, \ 



^1 ^ —^2 •'■3 V^an + -'s '^1 \^"i2 + •''i -^2 l^'^ss — 0, I 



} • (49) 



^2 = -^2 ■'•3 l/Oll — ••'■3 -^1 I' '«'22 -h •''! •'"2 V/«33 =0, I 

 (\ = :'2 .Ts j/flu + .rg Ty l/'«22 — .«1 •'-2 1/^33 = 0. / 



