( 260 ) 



).P^tinJ^yR = {} (18) 



contains two quadruples of points of contact of four times touching 

 conies ^2- 



The corresponcing parameter-values follow out of: 



V hi k.2 = X anrl r {k\ -\- /-o) =^ /' • 



4. To determine the pairs of line belonging to the system we 

 substitute y.:X for k into the condition that the discriminant of (13) 

 disappears. After reducing we find 



(«23 A — ;^)2 A3 (fl„2 «33 A3 — ;f2) ~ .... (19) 



Evidently for A=:0 we have the right lines D^D^ to be counted 

 twice; for y. = l[^a2^^a^^ we have the four times touching conic 



(«23 — l/«23 «33) •'•] + «13 '^2 + «13 -''3 = =t (-'C t^ «11 «33 + *.. ^'^«H «22) • (2^) 



breaking up into two double tangents. 



Likewise y. = — A 1/ «22 «33 gives two double tangents. 

 Finally y = a^^X produces the conic 



(«11 «23— «12") -'S^ + - («11 «23— «12 «13) ^2 -'s + («11 «33— «13*^) ■''2^ = 0, (21) 



composed of the two tangents drawn through D^ to the curve F^. 



On any right line I the conies ^2 determine a (2,2) - correspon- 

 dence ; the point of section of I with D^ D^ belonging to the coin- 

 cidences of Uiis system, there are three curves -C>2 touching a given 

 right line. These agree with the three manners in which the points 

 common to I and F4 can be divided into two pairs. Each of these 

 pairs determines an involution, i. e., a curve Sp„. This contradicts 

 in appearance only the fact that an 73 jg determined by 2 pairs, 

 for the two points lying in />j form a pair common to all 7^3. 



Evidently a system of four times touching conies is conjugated to 

 each pair of nodes of F^. 



5. If the conies f/>2 and W^ alluded to in § 2 are identical, it 

 follows out of the equation 



</>23 - r, = .ra rg .Qj (22) 



that the conic .Qo drawn through Dc, and D-. touches the curve Fa 



