72 KANSAS UNIVERSITY SCIENCE BULLETIN. 



and if we consider only classes with respect to the general projective 

 group, we can say that there are only °° 19 ~ 15 = m 4 classes of S 3. 



5. ( ')Suppose now that R* and Ri have, besides 0, another 

 common point D. Let DT 1 and DT 2 be their tangents in D. Then 

 2 1 and 2 % pass both through D, which will be a fixed point of the 

 curve r 3 generating the surface. The tangent planes to the two 

 quadrics in D\ pass respectively through DT 1 and DT 2 , and are 

 protectively related. Therefore their intersection, which is tangent 

 to r 3 in D, describes a quadricone. Hence D is a conic node of the 

 surface. Two twisted cubics cannot have more than five points in 

 common, since only one goes through six points. Hence a cubic sur- 

 face has at most four nodes. The construction of surfaces with 1 , 2. 

 3, 4 nodes follows at once. 



If the surface has only one node say D, three of the lines a go 

 through D, and are obtained by projecting R\ and R% from D, and 

 taking the three common generatrices of the projecting cones other 

 than OD. Let a u a 2 , a s be these lines, then a 4 , a 5 , a 6 are the three 

 remaining common bisecants or : Two twisted cubics having two 

 points in common have three common bisecants not going through either 

 common point. It follows readily that three of the lines b pass through 

 D, and also that all these lines through D are binary. 



Let now the surface have two nodes, D 1 and D 2 . Then it is seen at 

 once that of the lines a, for example a 1 coincides with D 1 D 2 , a 2 and 

 a 3 go through D u a 4 and a 5 through D 2 , hence only a 6 is left as 

 common bisecant of i?i and R2, going through one of their common 



points. But 6 e must meet a,\ a 5, hence it coincides with 01, 



and b x meets a 2 a 6 , hence it is the intersection of the planes 



(a 2 , a 3) and (o 4 , a 5 ) and meets a 6 . Hence this proposition : When 

 tiro twisted cubics have three common points they have one and only 

 one common bisecant going through none of them. Their G common 

 bisecants going through only one of the common points form three planes 

 that meet on the preceding bisecant. D x J) 2 which is called the axis 

 is easily seen to be a quaternary line, the 4 other lines through either 

 nodes are binary, so that there are 27 — 4 — 4.2.2 = 7 unary or simple 

 lines. 



Nothing particular can be said about surfaces with three or four 

 nodes, except that their properties are readily deduced from our 

 construction, but it is not worth while entering into any details. 



0. Suppose now that the plane T \DT 2 corresponds to itself, in the 

 projectivity between the tangent planes of 2 1 and 2 2 in D. Then 

 the tangent cone in D decomposes into the plane T \DT 1 and another, 



6. For the surfaces with nodes, we will follow the usual relation. See Salmon, 

 Geometry of Three Dimensions, 4th ed., p. 489. 



