70 KANSAS UNIVERSITY SCIENCE BULLETIN. 



point a there correspond nz(n — 1) points A, hence to B there 

 correspond mn(m— 1) (n — 1) points y4. It follows that A and 5 

 are in a correspondence [45 m 8 n , mn(m—l)(n — l)], and since if 

 .1 is arbitrary on Rm, one of the points B coincides with it, the num- 

 ber of coincidences is [45m 8n-\-mn(m — 1) (n — 1)]. It is seen at 

 once that when .1 coincides with #, A a is a common bisecant, and 

 further that if A' is the other end of the latter, then two points 

 B coincide with A, and two of them with A', so that to a common 

 bisecant there corresponds four coincidences in the above corres- 

 pondence, and therefore the number of common bisecants is 



. . . mn (to — 1) (n — 1) 



0m On + 



4 



In particular suppose m = n=3, then we know that 8 m =8 n = 1, 

 and therefore 



mn (ct-1) (n-1 ) 



Oiii On f- - — 1U. 



4 



In other words fa#o twisted cubics have ten common bisecants. 



3. Let now Ri and R' 2 be two twisted cubics having a common 

 point 0. They still have ten common bisecants. For the only thing 

 that could change this circumstance, would be multiple coincidences 

 in 0, and that this is not the case can be shown very simply by con- 

 sidering two twisted cubics having five common points, and ten 

 common bisecants, namely the lines joining these points two by two. 

 Now the quadric cones projecting the two curves from have four 

 generators in common which are common bisecants, and therefore: 

 When two twisted cubics go through the same point, they have six common 

 bisecants not going through this point. ( 3 ) We will denote these lines 



by Oi, o 6 . Let now P be an arbitrary plane through 0, OD 



any line in P, Si, and S 2 the two quadrics defined respectively by 

 R\, and OD, R' 2 and OD. Their intersection is composed of OD 

 and a twisted cubic r 3 . When OD describes the plane P, the two 

 quadrics describe two projective pencils of quadrics, and therefore 

 their intersection has for its locus a quartic surface. But P is a part 

 of the latter, hence the locus of r 3 is a cubic surface S 3 , which will 



contain R 2 , R\ and the lines a x a t] as having four points 



in common with each of them. This surface S 3 goes through a 

 fixed curve of order 3+3+6 = 12, when P varies, and therefore is 

 absolutely independent of the plane P. Hence two twisted cubics 

 that meet in a point define one and only one cubic surface. 



3. R. Sturm. Die Lehre von den Geometrischeu Verwandschaften, Vol. II, p. 18S. 



