LEFSCHETZ: CUBIC SURFACES AND NODES. 77 



obtained for U e directly the well known system of lines existing on 

 the S 3 having such a point. But it will be well to verify our con- 

 clusions by analysis, for we have shown here that U 6 on a cubic- 

 surface is not equivalent to 3C 2, but is a special singularity. 



If 



W(X + Y + Z) 2 + XYZ = (1) 



is the equation of an S 3 

 having an Z7 6 in (0, 0, 0, 1), then it is seen at once that: 



X[h(X +Y + Z) - kY] - (X + Y + Z) 2 = (2) 



W[k(X + Y + Z) - k>Y] + YZ = (3) 



represent °° 2 twisted cubics R 3 on S3. The cone projecting them 

 from the £7-node are prepresented by (2) and they are tangent to 

 X = 0, where it is met by X+Y+Z = 0. These R 3 meet X+F 

 +Z = and Z= kW the first being a point of contact as can easily 

 be seen. The line X = W = is effectively on the surface, and it is 

 the intersection of the fixed tangent plane X = to the cones (2), 

 with the plane W = which is the plane containing W = Y = 0, and 

 PT = Z = 0, the only other two lines of the surface, not going through 

 the node. Our conclusions obtained by direct considerations are 

 thus verified. 



10. If in the preceding construction for U 6 , R\ and i?l have 

 the same osculating plane in D, and still the same tangent D Tat 

 that point, then a 2 will coincide with DT and we icill have obtained a 

 surface with a node U -, such that the ?7-plane contains only two 

 lines, while the only other line on the surface will be a 6 , which meets 

 the common osculating plane in a point E, situated on DE, the second 

 line of the [/-plane. We will not carry out the analytical verification, 

 which is as easy as before. 



The surface with a node Us is the only one which cannot be ob- 

 tained by the method used so far. The reason of it, is that this surface, 

 as we shall prove now, contains no twisted cubics. For as is well 

 known, there is only one straight line D on such a surface, and it 

 is obtained by the coincidence of the 27 lines of the ordinary one. 

 Hence a twisted cubic R 3 on our surface must meet D in two points. 

 Now there are °o 1 quadrics going through R 3 and D, and they must 

 meet S 3 in <x> 1 conies. Anyone of the latter has only D for residual 

 intersection, and is therefore necessarily in a plane going through D, 

 the latter meeting the corresponding quadric in a line of the third 

 order, is part of it. Hence the co l quadrics considered are necessarily 

 decomposed into two planes, which shows that ii 3 ona cubic surface 



