( 541 ) 



of the three quadrangles h or in one of the four triangles e, what 

 was assumed follows immediately. 



To the case treated here of c 12 broken up into four lines to be 

 counted three times corresponds the parabolic curve of the surface 

 S' with four nodes. 



5. In the third place we consider still the special case of six 

 basepoints lying on a conic, in which the linear system of cubics 

 contains a net of curves degenerating into a conic and a right line; 

 in this net of degenerated curves the conic is ever and again the 

 conic c 2 through the six basepoints and the right line is an arbitrary 

 right line of the plane. 



This case can in a simple way be connected with a surface 

 *S' with a node 0. If we project this surface out of this node 

 O on a plane a not passing through this point, then the plane 

 sections of the surface project as cubics passing through the six 

 points of intersection of « with the lines of the surface passing- 

 through 0; because these six lines lie on a quadratic cone, the six 

 points of intersection with a lie on a conic. Besides, the sections 

 with planes through O project as right lines; therefore the completing 

 conic c 2 must evidently be regarded as the image of the node 0. 

 Of course we must here again think that c 2 corresponds point for 

 point to the points of 0* lying at infinite short distance from 0'; 

 for c 2 is the section of « with the cone of the tangents to S 9 in 0. 



As c 2 with one of its tangents represents a curve of the linear 

 system, this conic belongs at least twice to the locus of the cusps. 

 Here too this locus of cusps improper with continuing branches must 

 be accounted for three times, so that the locus proper is a curve 

 c' of order six, touching c a in the six basepoints. 



Let us suppose that c 2 is a circle and that the six basepoints on 

 that circle (fig. 4) form the vertices of a regular hexagon, then the 

 curve c" has the shape of a rosette with six leaves having the centre 

 0' of the circle and the points at infinite distance of the diameters 

 A X A K , A^At, A t A t as isolated points. Of the ten ovals there are four 

 contracted to points, whilst the six remaining ones have joined into 

 the^circle of the basepoints and the curve c\ 



If we take point 0' as origin and the line O'^^as t *>axis of a 

 rectangular system of coordinates, then if 0'A 1 is unity of length 

 we find for the equation of c' 



4r/ 2 (r/ 2 - 3a; 2 ) 2 + 9 (.r 2 + </ 2 ) 2 - 9 (a; 2 + y 2 ) = 0. 



It is evident from this equation that the curve c 8 can really stand 



