( 272 ) 



By applA-ing (G) the equations of the double tangents J,j, Jj, d^^d^ 

 can be easily found. Of course they can also be deduced from (26). 

 There are two identities of the form 



A — (^0 f% = 2 .ïi .('2 lo3 . j 



1 (50) 



A — f^l ^2 =^ 2 .rj X2 è]2 ' 1 



^0 and ^3, ^1 and S^ being oomplementaiy w. r. to D^ and D^] 

 hence it follows that />, and Bo are connected with tlte points of 

 contact of d^ and <l.^ bij a conic èo?, = and irith the points of con- 

 tact of (/j and d^ by a conic §12= 0. 

 Furthermore the identical equation 



«33 J4 — lo3 I12 = ^3^ f^* (51) 



furnishes the proof of the wellknowu property that the eight points 

 of contact of the four double tangents lie in a conic 



»9 = ^ ^n «'i^ — 2 -^ ai3 «23 3'i .rj = . . . . (52) 



We easily see that the points of contact of di and d^ determine 

 with the node D^ a conic with the equation 



'?12 = '^1 (k — («13 + l/«ll "22) §12 = («12 + y'cn O22) (-'s — 



-(/lllV-2^]2-^l.^'2 + ^'l2-2^2') = . (53) 



Out uf the second form of jjjj now follows that it also contains 

 the antitangential points of D^,. 



In the same way we can show that D^ is joined by a conic to 

 its antitangential points and the points of contact of d^ and d^. 



The six points R lying in a conic (>j23, whilst Z?o3^(R] Z?ii, iïgiïi), 

 Z>(,2 ^ (^3 -^i ' -Si^^iii)3.nd2^i^(ii2 /?iii , i?3-Sii), the inscribed hexagon 

 R] Rii R2, Ri Tti}, Rni lias the double tangent tZg for its line of 

 Pascal. Similarly is proved that the remaining three double 

 tangents are lines of Pascal for three other hexagons formed out 

 of points R. 



