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 ^>-flats through two 

 consecutive (p — l)-flats all have this same locus possihly of a certain 

 multiplicity. 



b. Intersections of consecutive tangent flats. 



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

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

 have (p — 1) -flats in common. Tangent ^o-flats at consecutive points 



fi 

 of a j9-spread where 1 < p < - do intersect in points at least. Let 



ft 



v=o, 



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

 equations of the p-spread. Let 



P' = (x 1 , y', . . . ) and P" = (x r + dx', y' + dy', . . . ) 



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

 points are 



9 x' 9 y' 



9 V 9 V 

 dx dy 



and 



A U" = A U< + x 



/<? 2 U' 9 2 U' \ 



\j* dx ' + w*js d ' + ■■■■)= "• 



{9 2 V 9 2 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 ja-flats at con- 



