( :^^- ) 



Tu lliese belong tiro bisecants of ,>' and tico of ,:J''; the remaining 

 seven aie singnlar bisecants of the second species. 



So the siiujular biaecants of I form two congruences (2.6) and 

 one congruence (7,3). 



5. The section of TI' witli a phine through P is a curve witii a 

 triple point, hence of class fonrteen sending eight tangents throngh 

 P. The tangents of the q'^ form therefore a complex of order eight. 



The points of contact of the tangents drawn out of P lie evi- 

 dently on a twisted curve of order nine. 



This can be confirmed as follows. The points of contact of the 

 tangents out of F to the surfaces Q- lie on a cubic surface, the "-pohir- 

 sur/ace" of F with respect to the pencil {Q-). A second cubic sur- 

 face contaijis the points of contact of the tangents out of 7^ to the 

 surfaces of [Q")' . Each point of intersection of the two polar 

 surfaces determines a q\ of which the tangent passes through F; so 

 the points of contact of the tangents drawn out of F lie on a (j\ 



6. The quadrics {Q^) and {Q-)' are arranged in a correspondence (2,2) 

 when two surfaces intersecting each other on the line / are made 

 to correspond. This causes the points of a line ni to be arranged in 

 a correspondence (4,4) ; so m contains eight points each bearing two 

 surfaces intersecting each other on /. From this it is clear that the 

 curves o'' intersecting / form a surface A\ 



On a line intersecting ,i' the (4,4) is replaced by a (2,4) ; we con- 

 clude from this that ,J^ and ,i'^ are nodal curves of .P. 

 The c)\ too, having / as bisecant is a nodal curve of A^. 



7. A plane / through / cuts A^ still according to a curve ?.' 

 passing through the points of intersection of the nodal curve q\ 

 having / as bisecant, with this line. In each of the remaining five 

 points of intersection of / with ?.' the plane ?. is touched by a q\ 

 The locus of the points of intersection of a given plane with curves 

 of r is theiefore a curve of order five, ?.\ 



Evidently /.^ is the curve of coincidences of the quadruple involution 

 determined by F and it passes through the eight points, in which 

 p^ and ,.^'^ are intersected by X ^). 



This involution containing fifteen quadruples in which three points 

 coincide ^), an arbitrary plane is osculated by fifteen curves of r. 



Furtheron four quadruples consist each of two coincidences-); 

 so each plane is bitangential plane for four curves of F. 



1) See loc. page 82. 2) ggg Jqc. page 83. 



18* 



