MORENO. — ON RULED LOCI IN W-FOLD SPACE. 147 



Any F n _o contains two consecutive F„_ 4 's, three consecutive i^„_ 6 ' s > o con- 



secutive FqS. Any F„__ 2r contains two consecutive -^ fT n _ 2 (r+i)'s and any 

 two consecutive I , n _ 2r , s determine one F H _ 2{r _i ) except in the case that 



r = -. We cannot then start with a curve and retrace our steps ; two 



consecutive points of the curve Si do not determine uniquely a plane of 

 the system. The i'Vs of the system in general intersect consecutively 

 in the points of S v Starting with such a system of planes we may 

 retrace our steps. Through any two consecutive planes of the S a we 

 may pass a four-flat. These four-flats are generators of S 5 . Through 

 any two consecutive B^s we may pass six-flats ; they are the generators 

 of S 7 . Finally through any two consecutive i^_ 4 's pass (n — 2)-flats ; 

 they are generators of S n _ t . We may retrace our steps only in case we 

 do not begin with S v 



9. Director curves of the ruled (n — \)-spread. 



Let the equation of such a ruled (n — l)-spread S n _ x be 



= 0. 



"We shall show that the equations of the generating flats of the spread 

 may be represented by linear equations involving a single parameter. 

 The equation in homogeneous coordinate of an arbitrary (?i — 2)-rlat in 

 n-fold space may be written 



x = a x z + fix «+.... + 71 w 



y — a 2 z + /? 2 s + . . . . + 72 w - 



In this form the equations of the (n — 2)-flat, which we may call the 

 (n — 2)-flat AB, involve 2 {n — 1) independent arbitrary parameters. 

 These parameters must be connected by 2 (n — 1) — 1 equation to make 

 A B a generator of such an (n — 1) -spread. We wish to connect these 

 parameters in such a way that A B will be a generator of the S n _i in 

 question. The equations of a curve on <£ are 



<£ = 0, Ui = 0, u 2 = o,... u n _ 2 =o. 



If we eliminate the coordinates between these equations and the equa- 

 tions of A B we derive a single equation in the 2 (n — 1) parameters. 

 This resulting equation is the necessary and sufficient condition for A B 

 to meet the curve. In a similar way we may derive 2 (« — 1) — 1 

 such conditions and make -A B meet 2 (n — 1) — 1 curves on </>. If 

 from these 2 (n — 1) — 1 equations and the equations of A B we elimi- 



