MORENO. — ON RULED LOCI IN W-FOLD SPACE. 139 



the tangent lines, tangent planes, tangent 3-flats, . . . , tangent jo-flats all 

 have the same locus. The planes through two consecutive lines, the 

 3-flats through two consecutive planes, etc., the jo-flats through two 

 consecutive {p — 1) -flats all have this same locus possibly of a certain 

 multiplicity. 



b. Intersections of consecutive tangent Jiats. 



We shall show further that {p -f l)-flats cannot in general be passed 

 through two consecutive tangent p-flats, for such ^-flats do not in general 

 have (^ — l)-flats in common. Tangent j9-flats at consecutive points 



71 



of a jt)-spread where 1 ^ jo ^ - do intersect in points at least. Let 



F=0, 



a restricted system equivalent to n — p independent equations be the 

 equations of the jo-spread. Let 



P' = (x', y, . . . ) and P" = (x' + dx', y' + dy', . . . ) 



be consecutive points of the spread. The tangent j9-flats at these 

 points are 



d X' d y 



9V' 9V' 



and 



A£7"=Aj/' + .(^rfx' + ^rfy + ....)=0, 



/5' V 9^ V \ 



All of these equations being linear, only n — p equations in each set can 

 be independent. In general, 2 (n — p) equations for such a value of p 

 have no common intersection. In the present case the resultant of any 

 n + 1 equations of the combined systems vanishes for any consecutive 

 points P' and P" on the ^-spread, so that no more than n equations of the 

 combined systems can be independent. Hence tangent j9-flats at con- 



