( '267 ) 



in both points common to £2^ and the conic <i>n passing through the 

 three nodes. 



It is therefore rational to put 



b.2= ± \/'^ en 63 = l/ör~ (23) 



Of the two systems pointed out hereby we shall consider only the 

 system determined by the upper sign. All conies (p^ belonging to 

 this are defined by 



03 = J-i ,r2 l/(^+ .'■! .'3 \/a^+ A 3-2 j-g l/a^= . . . (24) 



So we get 



S2o = (1— A-) an .1-0, .ro + 2 («12—/ l/au «22) ■''i ''3 + 



+ 2 («13- A 1/ «n 033) .ri .r2 + 2 («„3 - 1/ «22 «33) •''i'^ = , (25) 



which proves that these conies drawn through P^ and D.^, touching 

 two times elsewhere, form a system with index two. 



For the connecting line of the points of contact B^, B^ we find 

 with the aid of (G) 



(1 + A^) (./-J l/aii «33 + .^3 l/a„ 022) =: 2 A (^ ai2 .r^-x^ [/ a.^ «33) • (26) 



So these chords of contact form a pencil whose vertex is the 

 point common to the lines 



:r.y i/o^ (7o3 + ^'3 l/aii a22 = , 2 a^^ -'"a = -^1 ^ "i-z «33 • (27) 



Each ray bears two pairs of the involution formed by the pairs 

 i?i , B.2. The values of A corresponding to the pairs borne by the 

 ray 



,« (•'•2 l^«ii 033 + •'■3 y^'n «22) + (-^ «J3 -^3 — '«1 1^022 033) = • (23) 

 are found out of 



A2 + 2 /< A + 1 = (29) 



For ^< = ± 1 we get two rays (/g and d-^, on which the two pairs 

 coincide J so these are double tangents. 



Likewise is proved that the negative sign in (23) produces a 



