LEFSCHETZ: CUBIC SURFACES AND NODES. 73 



and we obtain a binode B 3 . This will happen when for a certain 

 position, 2 i and 2 2 are tangent in D. If 3 is their common generator 

 through 0, cutting their common tangent plane T J)T 2 in Q, then 

 Z)Q is on both quadrics and therefore on S 3. Furthermore on either 

 2 1 or 2 2 , DQ is of the other system than OQ, hence it is a bisecant of 

 Ri, and of R 2 as well. Conversely if DQ common bisecant of R\ 

 and i?|, is in the plane T \DT 2 , and if Q is any point on DQ, then 

 DQ is on both quadrics and therefore on *S 3. Furthemore on either 

 2 1 or 2 2, DQ is of the other system than OQ, hence it is a bisecant of 

 R\, and of R' 2 as well. Conversely if DQ common bisecant of R% 

 and R 2 , is in the plane T \DT ' 2 , and if Q is any point on DQ, then 

 there exists a surface 2 1 going through OQ, and it will contain DQ, 

 since it meets it in three points. The same will be true of 2 2 , and 

 therefore these two surfaces will be both tangent in D to the plane 

 T \DT 2 which will correspond to itself in the projectivity of 

 the tangent planes. Hence: the necessary and sufficient condition for 

 D to be a biplanar node of S 3 , is that the plane of the tangents of R\ 

 and R 2 in D contain one of their common bisecant s through D. 



Therefore if we propose to construct a cubic surface having a 

 biplanar node we can give ourselves three coplanar and collinear 

 lines DT 1, DQ, DT 2 , two points M 1 and M 2 on DQ, and an arbitrary 

 point 0. R\ will then be subjected to the condition of passing 

 through 0, D, M 1, and be tangent to DT 1, and R 2 to that of passing 

 through 0, D, M 2 and be tangent to DT 2 . This offers no difficulty 

 whatsoever. In particular if M 1 is made to coincide with M 2 then 

 we have a surface (B 3 + C 2 ) that is having a conic node and a binode. 

 By making both Rf and R 2 pass through a fourth point quite 

 arbitrary, the construction is still possible and we have the surface 

 (.B3+2C2) having two conic nodes and a binode. 



Consider now the points 0, D 1, D 2 . Through D 1 draw two lines 

 D X T' \, and D \T' \ such that their plane contain DiD 2 , then 

 through D 2 two arbitrary lines D 2 T /', D2T2" and in their plane a 

 line D 2 Q on which we will take two arbitrary points M 1 and M 2 . 

 We know how to construct a twisted cubic through 0, D 1, D 2 , M \ 

 and tangent to D \T \", and D 2 T x " — it is only a special case of the 

 construction of a twisted cubic through 6 points — let it be jRi. 

 Similarly we can construct R 2 through OD \D 2 M 2 , and tangent 

 to D 2 T 2 and D 2 T 2 ". The two curves define a surface (22? 3 ) that is 

 with two binodes. If now M 1 and M 2 coincide, which does not 

 affect the construction of the surface, we have the case (%B 3 + C 2 ), 

 or a surface with two binodes and one node. 



