148 PROCEEDINGS OP THE AMERICAN ACADEMY. 



nate the parameters, we derive a single equation in the variables alone. 

 It is the locus of all the (n — 2) -flats that can be drawn to meet the 

 curves in question, and so it necessarily includes all the generating flats 

 of <jf>. It includes possibly other flats besides the generators of <j>, but in 

 this case the general locus will break up into several components, and one 

 component is <£. This is the case in three-fold space. 



The spreads U^ U 2 , . . . U n _ 2 may in each case be taken to be flats ; 

 then the director curves are plane curves. These are the director curves 

 of <£; any or all of these curves may be plane, or they may be twisted to 

 any extent permitted by the space. Any 2 n — 3 curves in w-fold space 

 may be taken as the director curves of a ruled (n — l)-spread. In three- 

 fold space any three curves plane or twisted may be taken as the director 

 curves of a ruled surface. In four-fold space, any five curves plane or 

 twisted may be taken as the director curves of a ruled three-spread. In 

 this case the generating planes intersect consecutively in the points of a 

 sixth curve; so in four-fold space any five curves completely determine a 

 sixth. In five-fold space seven curves plane or twisted may be taken as 

 the director curves of a four-spread ruled by three flats. In six-fold 

 space nine curves determine a five-spread ruled by four-flats. Consecu- 

 tive four-flats intersect in planes and these in turn intersect consecutively 

 in points. So in six-fold space nine curves determine a tenth. 



10. Multiple loci on the ruled (n — V)-spread. 



Any generator of the (ii — l)-spread is an (n — 2)-flat F n _ 2 \ it is met 

 by any other generating F n _ 2 in an (n — 4)-flat. If then 4 < n every 

 generator is met by every other generator. If n = 3, any generator is 

 met by only m — 2 other generators, m being the order of the surface.* 



For 4 < n, any F n _ 2 contains a single infinity of (n — 4)-flats where it 

 is met by the other F n _ 2 s. These are evidently double flats on &„_!• On 

 &„_! there are in general a 2-fold infinite system of such (n — 4)-flats 

 constituting a double (?i — 2)-spread, 2„_ 2 on S n _x. In general, then, any 

 (n — l)-spread S n _ x ruled by (n — 2)-flats F n _ 2 s has on it such a double 

 (n — 2)-spread 2„_ 2 ruled by the 2-fold infinite system of (n — 4)-flats. 

 2„_4 has on it all those (n — 4)-flats, F^s that arise from the intersec- 

 tion of consecutive i^,_ 2 ' s - These i^./s generate S n _s, which therefore 

 lies on 2 n _ 2 and forms but an infinitesimal part of it. 



Any three F n _ 2 s intersect in an (n — 6)-flat; there are in general 

 a 3-fold infinite system of such (n — 6) -flats constituting an (?i — 3)- 



* Salmon, Geometry of Three Dimensions, p. 427. 



