( 496 ) 



continue to assume two rational curves .s\^ and .s\/ corresponding 

 to eacli otlicr in this manner, then of course to every three col- 

 linear points of .s\^ correspond three likewise collineai' points of 

 ój^ and therefore we can take for the above determined pair of 

 points Qj, Q,^ any cori'esponding |)air of points of a^, a^. In this 

 special case a plane « through tin-ee generatrices will present itself 

 already for arbitrary position in Sp^ and the position, that an infinite 

 number of those planes present themselves, will l)e able to be brought 

 about in a twofold infinite inimber of different ways; in the last 

 case however tiie three generatrices lying in a plane « pass through 

 a point, as the series of points lying on the lines of intersection 

 (t.^,a^ of this plane with «,,«2 are perspectively related, so that the 

 locus of the nodes becomes a conic instead of a k\ In both cases 

 surfaces (>" are formed differing from the above also in this respect 

 that they admit not only of a single but of a twofold infinite 

 number of spaces through three generatrices. 



12. Also wheji we start from two i)rojective rational curves 

 .yj', .y/ ii^ "ot })rojectively related fields a great number of special 

 cases are left for consideration. So the point of intersection P^^ of 

 the planes c.^, a^ can lie 



a. on one of the curves s^, 



b. on botii curves .-?^ 



c. on the two curves s^ and correspond to itself, 

 (L it can be the node of one of the curves .y', 



e. it can be the node of one of the curves and lying on the other, 



/. it can be the node of one of the curves and forming on the 

 other part of the two points corresponding to this node, 



g. it can be the node of both curves, 



h. it can be the node of both curves and in such a way that one 

 pair of points coinciding in this node has a point in common with 

 the other, 



i. it can be the node of both curves and in such a way that the 

 pairs of points coinciding in this point correspond to each other. 



Of course the number is still increased if we furtlier permit 

 the pointfields «^ , rr^ to be projectively related. We do not wish to 

 investigate more closely all these special cases. Neither do we intend 

 to investigate here the scrolls presenting themselves in both cases 

 of projective or non-projective pointfields «^ , «^ '^^ ^^^^ locus of the 

 line I\P^ connecting corresponding points F, , I\ of other curves 

 of the same genus and of the same order, which are projectively 

 related. We only wish to observe that these scrolls will lie in the 



