( 492 ) 



bearing the last three points will in general not pass throngli 

 P^ = P^. So then there are no lines a^ and a^ to be drawn through 

 the point of intersection P^^ of the planes a^ and c..^ cutting s\^ and 

 5,' in cori-esponding triplets. 



6. We shall now consider the more general case of two projectivelj 

 related curves ó\^ and .s-^'' Ijing in such a way in a^ and «^ that 

 through the point of intersection P^^ no triplets of corresponding 

 points bearing lines a^ and a^ are to be draw^n. The argument leading 

 to the order of the scroll, which is the locus of the line connecting 

 the corresponding points of those curves, retains here its force. So 

 we have but to determine the number of nodes. Of course 0^ and 

 0^ are nodes. If farthermore A^A^ and B^B^ are two generatrices 

 cutting each other outside a^ and «^, then A^B^^ and A^B^ pass 

 through Pi2> ^s they must cut each other. So we consider the central 

 triple involution {A^ B^ C\) marked by the pencil of rays with P^^ 

 as vertex in s^^ and the non-central triple involution {A^ B^ C^) of 

 the corresponding triplets of ,s\^ ; then the latter furnishes as envelope 

 of the sides of the triangles A^ B^ C\ a definite curve of involution 

 which makes us acquainted by the number of its tangents through 

 Pi2 with the number of nodes not lying in «^ and «, of the new 

 surface 0\ Now the class of the indicated curve of involution is 

 four ; for evidently four tangents pass through the node 0^ of ó\/. 

 If to the two points of s^^ coinciding in O^ the points M^, N^ on 

 s^^ correspond, and if P^^ M^ and P^^ N^ cut the curve s^^ still in 

 the point il//, J//' and N^, N^', then the lines connecting Pj, with 

 the corresponding points M^, J//, N,', N'^' are the only tangents of 

 the curve of involution passing through P^^. So 0^ has here also 

 six nodes. 



7. It is now easy to see that the first surface 6*^ of the three 

 projective pencils of rays is found back, if the correspondence of 

 the curves .s\^ and ,s^^ is given in such a wa}^ that through the point 

 of intersection P^^ of «^ and «., lines n^ and a^ pass bearing two 

 triplets of corresponding points. The plane a/i^ is then again a plane 

 ft through three generatrices of 0^ and the line a^ represents three 

 of the four tangents to be draw^n through P^„ to the above found 

 curve of involution, whilst the fourth tangent causes us to iind a 

 node not lying in «j, «^ oi' "• ^f we now cut the surface by the 

 space determined by « and this node, the section will consist of the 

 three generatrices in a and a curve of order three with a node, 

 i. e. a rational plane cubic curve. The plane of that curve is then 



