138 PROCEEDINGS OF THE AMERICAN ACADEMY. 



The equation of the locus of all the Hues that meet the spread twice at 

 (1) is A U x = 0. 



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

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

 point of an (n — l)-spread there is in general a unique tangent (n — 1)- 

 flat. 



A ^-spread is given by the equations, 



V=0, 



W=Q, 



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

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

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



A U x = 0, 



AV 1 =0 ) 



A W l = 0, 



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

 pendent and the rest are superfluous. t We have then a ^?-flat which 

 from analogy is called the tangent p-flat to the p-spread at the point (1). 

 At any point of a ^-spread there is in general a unique tangent p-fl&t.t 



We define a tangent r-flat at a given poiut of the jo-spread where 

 r < p as an r-flat that Jies in the tangent />-flat at that point and con- 

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

 as an r-flat that contains the tangent ;>flat at that point. The locus of 

 tangent lines then to a ^-spread is simply the locus of tangent p-flats to 

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

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

 jo-spread, where 1 < p 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 p-spread. 



t Dr. Story, Lectures on Hyperspace. 



