( 494 ) 



''curves and if now we place those planes in S^ in sncli a way that 

 "Q-i and Q^ coincide in the points of intersection ]\,, of the planes, 

 "the locus of the line connecting the pairs of corresponding points of 

 "the cubic curves forms a tortuous surface with the following properties : 



a. "It is projected out of P^^ by a quadratic conic space, on 

 "which two systems of planes are lying; 



b. "It is cut by each plane of one system according to a cubic 

 "curve with a node, by each plane of the other system according 

 "to three generatrices; 



c. "The cubic curves in two planes of the first system have no 

 "point in common, neither have the triplets of lines in two planes of 

 "the second systems; each cubic curve, however, is cut by each 

 "generatrix ; 



d. "The generatrices cause a mutual projective correspondence 

 "among all cubic curves and the cubic curves among all the gene- 

 ratrices." 



9. From the preceding ensues immediately that the tortuous scroll 

 with a nodal curve /l" can he i-epresented on a plane. If in a plane 

 o we assume arbitrarily two pencils of rays with different vertices 

 7\, T^, and if we allow three arbitrary rays «j, b^, Cj of the former 

 to correspond to three rational cubic curves 5?,, .sf,,, s^ , of 0% three 



1 a) (6) [Cj 



arbitrary rays a,, b^, c^ of the second to three generatrices /(«), lb), 

 \c) of 0\ then to each rational cubic curve s^ corresponds a definite 



ra}' i\ of the first pencil, to each generatrix /(^) corresponds a definite 

 ray q^ of the second ; so we can assign the point of intersection of 

 s^ and /(r^) to the point of intersection of p^ and q^. The elements 



of exception of that representation are immediately found. If to the 

 line connecting the vertices of the pencils of rays counted with the 

 first pencil the curve s^ corresponds and counted with the second pencil 

 the generatrix /, and if >S' is the point of intersection of .s-' and /, 

 then to point 1\ corresponds the line /, to point 1\ the curve .y^* and 

 reversely to point .S' the line T^ T^. To each point F of the nodal 

 curve k'' correspond two points P', P" of <J collinear to T^, because 

 in the correspondence of .s^ to /(^) the node of s^ represents two 



different points and two points of I(p) correspond to this point. As 

 T^ forms part of two representing pairs, the pairs belonging to the 

 nodes of the generatrix / belonging to T^, this point is node and 

 the curve of order four. This is also evident when we consider the rays 

 of the other ])encil. On each ray 5- lie two points of the curve forming 



