( -268 ) 



system oi' couics of double contact drawn through Dj, Z).,, wliose 

 chords of contact pass through the point of intersection of the 

 remaining double tangents d^ and d.y 



6. The conies through i>2, D^ touching r,^ in B^ determine an 

 ƒ'-. Of its two double points one, B^^ is joined with B^ by means 

 of a ray passing through the point {d^ d-^)^DQ^; the second Bn 

 lies with By on the ray through D^o ^ (d^ d^). Now B^ is a double 

 point of the complementary Z^; let us indicate the second double 

 point by B^ . As any pair of the first involution can be united to 

 any pair of the second by a conic containing D^, D^, there are two 

 twice touching conies fij, touching F^ in Bi , B^ and Bi , Bu. 

 Hence Bi Bo passes through X'gg and Bi Bn through Z^oj. 



So in r^ an infinite number of quadrangles may be inscribed, 

 one pair of opposite sides passing through Dqi, the other through Z>23. 



Their vertices form the groups of a biquadratic involution. 



Each couple of pairs of the involution {B^, B„) coUinear with 

 i>oj lies in a conic containing the nodes D.^, D.^. This follows readily 

 from the equation 



«11 A — (^1 ^2 1/«33 + «'1 '^S l/«22 + ^1 •''•2 ••'■:! l/«n) (■''l ■•^2 '/«33 + ■'^l «'S '/«22+ 

 + ^2 ''Y^S l/«l]) = «11 ■''2'''3 I 2 (flog — l/a23 «gg) X-y^ + 



+ [2 «13 — (^1 + ^2) l/«ll "33] •■'■1 ■''2 + 

 + [2a]2 — (Ai+A2)l/On022l-''l-'3 + (J— '^1^2K'»3l/«22«33} • • (30) 



To find the equation of the conic joining D^ and i^g with the 

 vertices of one of the above mentioned quadrangles B^ B^ Bi Bn 

 let us consider the following identity 



(,y(Ti;'-2l/n33 + '''l'''3l/«22 + ^l'''2'''3l/«]l)0''r'2l/"33 + '''l«"3l/«22 + ^2'''2'''3l/«ll) + 

 + O (J'l .'2 1/033— •«! ■'s l/«22 + /'l •''2 ■'•3 l/«ll) ('''1 •'-'2 1/033 — -^I^'S 1/ «22 + 



4-/'2-'^2''''3l/«n) = «ii(^«]i-''2"-'»'3^ + 2^ai2,r, .r2,J-32) . , . (31) 

 We easily find that this is satisfied if we put 



(y = (l/a22f'33 + «23) : 2 l/ajjOgg and <7 =r (l/ooj «gg — «23) = 2 l/agj O33. 



By substituting the value of Aj + X^ following out of (HI) into 

 the equation of the conic indicated in the part on the right side of 



