{ hi ) 



which F^ is cut by the conies containing the four nodes, whilst the 

 lines connecting the pairs envelop a conic and at the same time bear 

 the groups of a fundamental / ' ^). 



If the involutorj scroll of F^ is a quadratic cone then every two 

 pairs of F^ lie in a plane through the vertex, which is at the same 

 time a point of q\ This special q^ is evidently the section of a cubic 

 surface and a quadratic cone, having a right line in common '). 



4. We shall now consider a (.)" of genus two. The involutory 

 scroll of F^ is now of order three (0')- Let q be the double line, e 

 the single director of <P'\ As q' lies on 0"^ and a plane thi-ough q 

 contains but one right line of $' which line bears a pair of F\ 

 we find that q has four points in common with q\ so it is a 

 quadrisecant. So the fundamental involution is described by the pencil 

 of planes having the quadrisecant as axis. From this is at the same 

 time evident that 9" cannot have a second quadrisecant. 



Each plane through e bears two pairs of F'\ so is a chord of (>% 

 and the pairs of F'' are connected in pairs to form the groups of a 

 particular I\ 



The planes connecting e with the two torsal right lines of l'^ 

 are evidently double tangential planes of (>". On e therefore rest 

 besides the tangents in the 6 coincidences of F^ still the 4 tangenls 

 situated in those double tangential planes and the tangents to be counted 

 double in the points of support of the chord g. The developable 5?///(j[c^ 

 of tancjents of t/ is therefore of order '14. This is evident also from 

 the fact that the quadrisecant besides b}" the tangents in its points 

 of support is intersected only by the six tangents of the coincidences. 



By central projection out of a point of e we find a special c^ with 

 eight nodes of which the pairs of F^ lie two by two on rays through 

 a node which is at the same time the point of intersection of two 

 nodal tangents. 



5. The scroll ^ of the bisecants resting on a trisecant t is, like q^ 

 of genus tioo. For, if the points B^, B,, B^ of q' lie with t in one 

 plane then we can make each point Bk to correspond to the chord 



ij A number of properties of c" are to be found in ray paper : "Ueber Curven 

 fünfter Ordnung mit vier Doppelpunklen" (Sitz. Ber. Akad. Wien, 1895, GiV, 4-6— 

 59). The curves p° and c° are treated by H. E. Timf.rding "Ueber eine Raumcurve 

 fünfter Ordnung" (Journal f. d. r. u. a. Math., 1901, GXXIII, 284-311). 



~) The central projection of this ^^ has been treated in my paper quoted before 

 page 03. it is generated by stating a projective correspondence between llie rays 

 of a pencil and the pairs of an involution, formed of the conies of a pencil. 



