( 265 ) 



also B„_, D^ 1). In particular everv cliord B-^ B^ determines one pair 

 of the Ic^ and reversely. 



3. In connection with (8) the curve of involution jpo is fully 

 determined by the value of (^j 'o + Z»3 cj) • Substituting 2 k for this 

 parameter we can represent S^-i by the equation 



«11 («33 ■'■2" + 2 't .('o .rg + 022 .rjï) = (^ «23 »'l — ^ ^\f ' • (13) 



which is q'ladratic w. r. to the parameter k\ therefore the conies -p.^ 

 form a system witli index two. 



Generalizing' for shortness the equation (13) to 



Pk"- -\-2 Qk-\-R = (i (14) 



where P, Q, R denote known quadiatic forms, the envelope of tlie 

 system is given by 



PR— Qi — (15) 



By mere reckoning this equation proves to be identical with (1); 

 so the envelope of the system (14) is the given trinodal F^. 

 Out of the identity 



{Pk''' -Y2Qk-\- R)R — k"~{PR— q~) = {Qk + Rr . (16) 



follows, that each conic .^2 touches the curve F^ in the points 

 common to S^o and Q^-f 72=0, or, what comes to the same, in 

 the points common to the conies Q ^- 4- 7? = and Pk -{- Q =^ Q. So 

 the four points of contact of 7"^, with the conic .(^2 indicated by 

 ki are also situated on 



^•2(^^-1 + Q) + (QX-i + 7?) = (17) 



The equation (17) being symmetrical in the parameters ki and 

 k.2 the conic represented by it will also contain the points of con- 

 tact of the conic .'qz indicated by k^ . 



Hence we raav conclude that each conic of the net 



1) In n paper published in Vol. XIV of the //Nieuw Archief voor Wiskunde" 

 (pages 193 — 2U0) I have pointed out that suchlike comjilementary systems of pairs of 

 points present themselves on the binodal qnartics. Likewise the special involutions 

 of ^ 5 have th?ir analoara on these curves. 



