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



H 



Any i^„_2 contaius two consecutive Fj^_^Sj three consecutive F^_i% - con- 



It 



secutive F^%. Any -F^.or contains two consecutive -?^„_2(r+i)'s and any 

 two consecutive i^„_2r's determine one i^„_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 S^ do not determine uniquely a plane of 

 the system. The FJ% of the system in general intersect consecutively 

 in the points of S-^. Starting with such a system of planes we may 

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

 may pass a four-flat. These four-flats are generators of ^S's. Through 

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

 of S-j. Finally through any two consecutive i^_4's pass {n — 2)-flats ; 

 they are generators of »S'„_i. We may retrace our steps only in case we 

 do not begin with aS'i. 



9. Director curves of the ruled (n — V)-sj)read. 



Let the equation of such a ruled {n — l)-spread S^_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 (ii — 2)-flat in 

 w-fold space may be written 



X = ai z -\- ^i s -\- . . . . + yiW 



y = ao z + /32 s + ...• + 72 w. 



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

 (n — 2)-flat^-6, involve 2 (ti — 1) independent arbitrary parameters. 

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

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

 parameters in such a way that A B will be a generator of the *S'„_i in 

 question. The equations of a curve on ^ are 



<^ =z 0, C7i = 0, ZZs = 0, . . . U„_o = 0. 



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

 tions of ^ 5 we derive a single equation in the 2 (n — V) 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 (« — 1) — 1 curves on ^. If 

 from these 2 (n — 1) — 1 equations and the equations of ^ 5 we elimi- 



