LEFSCHETZ: CUBIC SURFACES AND NODES. 75 



(1, 1) correspondence, and the two coincidences of the latter show 

 that there are two lines meeting R'l, R 2 , DT and u fi . Of these 

 one goes through 0, and the other a transversal t meets the surface 

 S3 in four points and therefore belongs to it. The plane (DE, a 4 ) 

 meets R\ and R'-> in the two points not on a 2 and defining a line 

 e on S3. Similarly the plane (Dh , « 2 ) defines a line / on S3. 

 both these lines meeting t in the points where it cuts the planes 

 (DE, a 4 ) (Db , a 2) respectively. Finally the plane (a ■>, a 4 ) meets 

 a B and t in two points defining a third line g. This system of lines 

 is identical with that found by Schlaefli as we should expect. 



8. Consider now the three points 0, D 1, D 2, and three coplanar 

 lines D 2 T 1, D 2 T 2 and D 2 M 1 M 2 . We could define an S 3 having a 

 C 2 in D 1 and a 7i 3 in D 2 by considering the twisted cubics 7?i and 

 R 2 , the first passing through OD 1 D 2 M x and tangent to D 2 T 1, 

 and the second 0, 7J 2 , 7J, 2 -17 2 and tangent to D 2 T 2 . We wish to 

 find what will happen when D 2 is made to approach D 1 indefinitely. 

 If we consider the two conies f 1 and £ 2 going through D 1 and 7) 2 

 and tangent to R\ and 7? 2 respectively, in D 2 , there are °° x quadrics 

 going through them, all tangent to the plane T 1 D 2 7% in D 2 . At 

 the limit Pi and T 2 will osculate 7? x and 7?^, and the plane tan- 

 gent to the quadrics containing both of them will contain a common 

 bisecant of R\ and 7?| through D. We can also say that the tan- 

 gent plane in D 1 to any quadric osculating both R\ and 7?| at 

 that point contains a common bisecant. Hence the following con- 

 struction. Let DT be a tangent to a given quadric 2, through DT 

 consider two planes cutting 2 in the conies £ 1 and £ 2 , and in the 

 plane P tangent to 2 in D take two points M 1, M 2 collinear with D. 

 If is an arbitrary point in space, 7?'i will go through 0, D, M 1, and 

 osculate £ 1 in D, while R' 2 will go through 0, D, M 2 and osculate 

 r 2 in D. The construction presents no difficulty — R\ for example 

 is determined by its two projecting cones from D and 0, the first 

 being tangent to the plane of Ti along DT and going through M 1 

 and 0, while the second osculates £ 1 in D and goes also through M \. 

 We thus hare the surface with a binode R 5 equivalent to (C 2 + 73 3). 



In particular suppose that the planes of £ 1 and £ 2 coincide, 

 or that 7?i and R'l have the same osculating plane, in D. Then 

 M 1 and M 2 will be the points infinitely near D on the respective 

 cubics, and can still be considered as being collinear with D. It 

 is clear that the preceding case is obtained as the limit of the one in 

 which D 1 is a binode R 3 as well as D 2 , for then D 1 D 2 is in the 

 plane of the tangents to the curves in D 1, and the latter becomes 



2— Univ. Sci. Bull., Vol. IX, No. 6. 



