LEFSCHETZ: CUBIC SURFACES AND NODES. 71 



4. We will now proceed to show rapidly how the fundamental 

 properties of the surface can be deduced from the preceding gen- 

 eration. If i^k then o ( and (ik have no point in common, since they 

 are bisecants of a twisted cubic. No quadric can contain four of 

 the lines a, for all its generators of the other system would then be 

 on S 3, which would then decompose. Let 2 be the quadric defined 

 by a i, a o, a 3 . A plane through a i, is tangent to S 3 in the two points 

 where a 3 is met by its residual conic, and to 2 in one point. The 

 points of contact of such a plane with the two surfaces are therefore 

 in a (1, 2) correspondence, and its 3 coincidences show us, that 

 the residual intersection of 2 and S 3 is formed of three straight lines. 

 On the other hand there are two generators of 2 that meet a 4, and 

 similarly for a 5 and a 6 , while such generators are common to S 3 and 

 2, as having four points on S. The only way to satisfy these two 

 conditions, and those obtained by a cyclic substitution on the as, 



is to have six lines b u b 6 , such that each meets all the lines 



a, of different index. We have thus obtained the double six of 

 Schlaefli.( 4 ) The plane aibk meets S 3 in a third line at, while it meets 

 cik in a point that must be on dk> since it is neither on a% nor bk. 



We have thus 15 new lines dk (i = l, 6, k — 1, 6), since dk 



coincides with cm, and therefore the 27 lines of the surface. Since 

 ai meets bk it does not meet dk, which therefore cuts the plane 

 di bi in a point necessarily on cu. 



The cubics r 3 do not cut ai, for the latter meets 2, in two points 

 on /?!, and therefore nowhere else. But bi cuts 2 1 in two points 

 which are not on R\, or else, being a bisecant of R\ it could not 

 meet cik, and therefore these points are on r 3 . Finally the plane 

 (dibkCik) cuts Ri in three points on S 3 , and since one is on bk, and 

 only two on ai, the third must be on cik, which therefore meets i?i, 

 and as well R 2 . Then jR'i cuts 2 2 in six points one of which is 0, 

 therefore it meets r 3 in five points. The curves r 3 form a linear 

 series on S 3 , which is 00 2 like OD. Through any point of the surface 

 there go °° 1 such curves, and since r z is residual of Rj on 8 3 their 

 relation is entirely reciprocal, and shows the existence of two sets 

 of 00 - twisted cubics on S 3 corresponding to the double six (ai)(6fc). 

 The system (R\, Rl) depends upon 23 arbitraries, since each one 

 of these curves is determined by six points, and that is variable 

 on one of them. On the other hand is arbitrary on S 3, and through 

 it there go °° J curves R 3 , hence S s depends upon 23 — 2 — 2.1 = 19 

 arbitraries. ( 5 ) If the absolute invariants are taken as parameters, 



4. Quarterly Journal. Vol. II, p. 115. 



5. R. Sturm. Die Lehre von den Geom. Verwandschaften. p. 193. 



