138 PROCEEDINGS OF THE AMEEICAN ACADEMY. 



The equation of tl^e locus of all the lines that meet the spread twice at 

 (1) is A C^i = 0. 



From the analogy of three-fold space, this locus of lines is called the 

 tangent {n — l)-flat to the {n — ])-spread, at the point (1).* At each 

 point of an {n — l)-spread there is in general a unique tangent (n — I)- 

 flat. 



A j9-spread is given by the equations, 



V=0, 



w=o, 



a restricted system equivalent to n — p independent equations. In a 

 similar manner the equations of the locus of all lines that meet the 

 jt)-spread twice at any non-singular point ( 1 ) are, 



A ?7i zzr 0, 



A Fl = 0, 



Since these equations are linear we may select any n — p that are inde- 

 pendent and the rest are superfluous.! We have then a jo-flat which 

 from analogy is called the tangent p-flat to the /^-spread at the point (I). 

 At any point of a /(-spread there is in general a unique tangent p-^att 



We define a tangent r-flat at a given point of the jO-spread where 

 r < p as an r-flat that lies in the tangent p-flat at that point and con- 

 tains the point. If r > p, we define a tangent ?--flat at a given point 

 as an r-flat that contains the tangent p-i\at at that point. The locus of 

 tangent lines then to a />-spread is simply the locus of tangent jo-flats to 

 the spread. The locus of tangent planes, 3-flats, . . . , (p — l)-flats is 

 this same locus. If then there are developables that arise from a 

 jo-spread, where 1 < /) their number is not so great as n — p — 1, for 



* This proof is given in Dr. Story's Lectures on Hyperspace. 



t Some of these equations may be satisfied identically ; this will be the case 

 when (1 ) is a multiple point on any of the [n — l)-spreads, but not a multiple point 

 on the /^-spread. 



I Dr. Story, Lectures on Hyperspace. 



