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curve lies on ^\ so it is the partial section of tV* with the involu- 

 tory scroll 4»'. As q is nodal line of <P'^ and single i-ight line of ^\ 

 the two surfaces have another line r in common. This r cannot 

 coincide with the single directrix e, for then each right line of <P^ 

 would have four points in common with ip', viz: its points of inter- 

 section with q\ q and e; the surface ip^ would then however coincide 

 with 01 



Inversely we can regard ^' as section of a cubic scroll 0^ with 

 nodal line q and a cubic surface \f having with ^'^ the right line q 

 in common and a right line r resting on the former one. A plane :t 

 through q cuts ?' in a right line, i('^ in a conic, so it contains besides 

 q two points of the curve of intersection, from which is evident 

 that q is a quadrisecant ; its points of support are coincidences of 

 the (l,4)-correspondences between the points of contact of rr with the two 

 surfaces; one of the five coincidences is the point of intersection of 

 q and r. That the single directrix of $'^ is a chord of 9*, is evident 

 f om the fact that it cuts ip^ on r, thus two times on o«. 



8. If 0' is replaced by a scroll of Cayley so that q is single 

 directrix and at the same time generating line, then the conic 

 of ij'% b'i"o "^ ^^1^ torsal tangential plane of 0' determines on q two 

 points each of which replaces in each plane rr through q two points 

 of intersectio]! with rj" ; so they are nodes of q\ On this special 

 curve the groups of F^ are not arranged in pairs; for e coincides 

 with q. 



We obtain an other special q" by taking instead of 4>' a cone 

 with nodal edge q. The conies of tp' situated in the planes touching 

 4>' along the nodal edge cut q in the points of support of the 

 quadrisecant. Each edge of 4>' bears a pair of F\ so that a plane 

 through the vertex T contains three pairs. 



The tangential cone out of T to tp' has q and r as nodal edges; 

 the six single edges which it has in common with $"' are evidently 

 tangents of 9' and contain the coincidences of F\ 



Through an arbitrary point pass four tangential planes to 0^; 

 the central projection of 9' furnishes a plane curve c" with four 

 nodal tangents meeting in a single point C The six single tangents 

 out of C contain the coincidences of the fundamental involution, 

 each ray of which through C bears three pairs. These are separated 

 if we describe on i^' a pencil of cubic curves having the eight 

 nodes of c" as base-points. 



