MORENO. — ON RULED LOCI IN W-FOLD SPACE. 157 



x = °, 



• t • • 



a restricted system equivalent to n — 1 independent equations. If we 

 eliminate the coordinates between the equations of the flat and curve, we 

 derive a restricted system equivalent to k — 1 independent equations in 

 the parameters alone. These are the conditions that must be satisfied 

 for the (n — &)-flat to meet the curve. In a similar way we may derive 

 a restricted system equivalent to k — p independent equations in the 

 parameters alone which are the necessary and sufficient conditions for 

 the (n — &)-flat to meet a certain ^-spread where 1 < p < k — 1. We 

 may have then curves, surfaces, . . . , or ^-spreads where 1 < jt> < & — 1 

 for the director loci of a ruled (« — k + l)-spread. The numbers of loci 

 of each kind that must be taken are A, p., ... v, p, namely, non-negative 

 integers chosen to satisfy the equation 



A (k — 1) + ix (k — 2) + . . . . + v . 2 + p ■ 1 = k (n — k + 1) — 1. 



If we consider a group of one or more points as a director locus of the 

 spread, we have to select integers to satisfy 



k . k + A (k — 1) + . . . . + p ■ 1 = k (n — k + 1) — 1. 



We may apply this to special cases. The director loci of a ruled surface 

 in three-fold space are three curves. We may take one curve and a 

 group of k points, in which case the ruled surface is reducible and has for 

 its components k cones whose vertices are the k points and whose 

 common base is the curve in question. In four-fold space the director 

 loci of a ruled surface may be five surfaces, three surfaces and one curve, 

 or one surface and two curves. The ruled surface in each case consist- 

 ing of all the lines that can be drawn to meet all the director loci. In 

 the same space the director loci of a three-spread ruled by planes may 

 be taken to be five curves. 



If the director loci be all taken on any S n _ k+1 , then the locus of all the 

 (n — £)-flats that can be drawn to meet these director loci will include 

 as one of its components the S n _ k+ i in question ; it may or may not 

 have other components. 



There are several special cases illustrative of these methods that can 

 be worked out in still greater detail. Some of these I hope to make the 

 subject of another paper. 



