1)70 



ciu'xe foi'iii tlio only in\ oliitioii i>l' pairs that can e\js( on a ciirNC 

 of gemis liuo ; the straiglit lines y> enveloj) a conic ^). 



If r contains a singulai- point A, ()■' degenerates into the line a 

 and a o' ; the hitter will Cnrlhei- degenerate as it must possess four 

 nodes, consequently is composed of two conies. 



On a shujulai- line a lie two coijicidences of the involution 

 P ^lEi {P' , P") ; they are at (he same time coincidences of tlie (i-"). 

 The curve of coincidences y is of the third order, so a must contain 

 another coincidence. Let it be Q' ^ Q ; Q' forms a triangle of 

 involution A with A and a {)oint Q' oï </. hut moreover a A with 

 Q and a point Q* lying outside (f. Consequently Q' is a singular 

 point viz. a point B, for the pairs A,Q'' ^^'kI Q,Q^ tl() "Ot lie on 

 07ie line. 



The cur\'e (/' belonging to o consists lirst of a itself and a conic 

 {By ; the completing curve must also have arisen from singular 

 points. No second point B lies on a, for this line would then contain 

 four points of the curve of coincidences. Hence two more singular 

 points of the first order lie on a, A*, and ^1**. Each singidar line a 

 contains therefore tivo jioints A and one point B. If a* cuts the 

 line a in S, then A* and >S form a pair of the involution lying in 

 a ; so that AA^^S is a triangle of involution. Hence A is the point 

 of intersection of the singular lines «*, a**. 



7. The connector of two singular points At and Ai is not always 

 a singular line a. Let A[ lie on *■//,., Ai- then forms with Ai and an- 

 other point Q of eik «^ triangle A, so that A^-Q is the line ai. If 

 Ai lies on a.k, <ii passes consequently through Ak. 



Let us now consider the line that connects the centres of invo- 

 lution belonging to C'l and C'j. It contains a pair of points forming 

 a triplet with B^, and a pair that is com))leted into a triplet by 5, . 

 Hence it is a sineiular line h ; we call it h^. The axis of involution 

 Cj belonging to it, is apparently the line />, />'.^ ; the three lines c 

 form the triangle BJi,^B^. 



In the transforuiafion {1\B') c^ corresponds with the figure com- 

 posed of h^ and the conies (Z>),', {Bj\ With y, it has in common 

 the coincidences lying in />*, and B.,, its third point of intersection 

 with y^ lies apparently in h^c^ . The singular line b^ is transformed 

 by {P,P') into a ligure of the fifth order ; to this belongs /;, itself 

 and the line c^ twice. As no point Ji lies on A, it must connect 



1) The quadrinodal curves -, ''• I have treated in " Ueber Curven fünfter Ord- 

 nunrj mit vier Doppelpunktevi" (Sitz. ber. der Akad. d. Wiss. in Wien, vol. CIV, 

 p. 46-59). 



