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 (f). It includes possibly other flats besides the generators of <^, but in 

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

 component is cJ3. This is the case in three-fold space. 



The spreads Ui, f^, . . . £4-2 i^^J ^^ each case be taken to be flats ; 

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

 of (j); 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 n-ioh\ 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 surfiice. 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 intei'sect 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 — \)-spread. 



Any generator of the (n — l)-spread is an (n — 2)-flat i^„_2 ; it is met 

 by any other generating F„_2 in an {n — 4)-flat. If then 4 ^ w 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^_2 contains a single infinity of (n — 4)-flats where it 

 is met by the other i^„Vs. These are evidently double flats on »S'„_i. On 

 *S„_i there are in general a 2-fold infinite system of such (n — 4)-flats 

 constituting a double (n — 2)-spread, 2„_2 on S„_i. In general, then, any 

 (n — l)-spread *S'„_i ruled by {n — 2)-flats i^,Vs 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^_i& that arise from the intersec- 

 tion of consecutive F^^^^s. These F„_iS generate *S'„_3, which therefore 

 lies on S„_2 and forms but an infinitesimal part of it. 



Any three F^^^^ intersect in an (« — 6) -flat; there are in general 

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



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



