156 PROCEEDINGS OF THE AMERICAN ACADEMY. 



n + 1 



n 



2 k + 2% 1 or £ ^ 



For values of k that satisfy this condition there is in general a continuous 

 locus of double points. If 



rc-2&+2 = 0, ov k = ^i-= 



there is in general only a finite number of double points on the locus. If 



n — 2k+2<0,ork> n ^^- 



there are in general no double points on the locus. 



If there enter into the equations of the generating (n — £)-flat p 

 parameters connected by p — 1 equations the properties of the system of 

 related loci will be similar to those of the system just described. 



Any two consecutive F n _ k 's intersect in an F n ^ 2k while through any 

 F n __< 2k pass two consecutive F„_^a. Any three consecutive F n _ k s intersect 

 in an F n ^, k while through any F„_o k pass two consecutive F n _ 2k s and 

 three consecutive F n _ k s. Any two consecutive F n _ rk s determine in 

 general one F ll _ k(r _ l) . An exception may occur if r = a the greatest 



n 

 integer in -=• • Thus, if n = ^mod k), two consecutive points of ^ do 



rC 



not determine a (k + l)-flat where 2 < k. 



If n = 1 (mod k), two consecutive lines of S- 2 do not determine a 

 (k + l)-flat, except in the case k — 2. In the last case, however, where 

 n EE k — 1 (mod k), two non-intersecting (k — l)-flats do determine a 

 (2 k — l)-flat. Only in this last case can we retrace the steps if we 

 come down to the last spread. We can always retrace the steps if we 

 do not come down to this last case. 



14. Director spreads of the ruled spread. 



The equation in homogeneous coordinates of any (n — £)-flat, 2 < k, 



may be written 



x = ai s + & t + . . . . + y x w, 



y = a 2 * + (3o t + . . . . -f y 2 w, 



z = a k S + (3 k t +.... + y k W. 



In this form the equations of the flat contain k (n — k -f 1) independent 

 parameters. These parameters must be connected hy k(n — k + 1)— 1 

 equations for this (n — £)-flat to be a generator of such a ruled 

 (n — k -f- l)-spread. Any curve is given by the equations 



