126 PROCEEDINGS OF THE AMERICAN ACADEMY. 



we may eliminate the coordinates of P' leaving a single equation in 

 a, /3, . . . . y. For an (n — l)-flat to be tangent to an (w — l)-spread, 

 one relation between the coefficients that enter into their equations must 

 be satisfied. We conclude then that the (n — l)-fl.ats that touch n — 1 

 (« — l)-spreads envelop an *S'„_i. 



Let us consider only those tangent (?i — l)-flat8 to an (n — 1)- 

 spread that touch it at the point of an (« — 2) -spread that lies on it. 



Let U=0 



be the equation of the (n — l)-spread and let 



U= 0, F- 0, . . . , 



a restricted system equivalent to two independent equations, be the equa- 

 tions of the (n — 2)-spread on it. We derive now the equations 



— la y 



and U' = 0, V = 0, ... 



If we eliminate the parameters from these equations there remains 

 a restricted system equivalent to two independent equations in the 

 coefficients a, /?,... y. For an (?2 — l)-flat to be tangent to an 

 (n — l)-spread at a point of an (n — 2)-spread on it requires two con- 

 ditions between the coefficients in the equation of the (n — l)-flat. 

 These two conditions may be used as part of the n — 1 conditions that 

 connect the coefficients of an (n — l)-flat that envelops a developable 

 'S'„_i. We have then the theorem that the (n — l)-flats that are tangent 

 to p (n — l)-spreads at the points of p (n — 2)-spreads that lie one on 

 each (n — l)-spread, and are tangent to a other (?z — l)'flats, where 

 n — l = 2p-fcr, envelop a developable. 



In a similar manner for an (n — l)-flat to be tangent to an (n — 1)- 

 spread at a point of an (n — 3)-spread that lies on it imjjoses three con- 

 ditions on the coefficients that enter into the equation of the (ti — l)-flat. 

 To be tangent to the (n — l)-flat at a point of an (?i — 4) -flat on it 

 requires four conditions, etc. To be tangent to an (n — l)-spread at a 

 point of a curve that lies on it requires n — 1 conditions between the 

 coefficients, which is just sufficient to make the (« — l)-flat envelop a 

 developable. 



We have then the general theorem that the (n — l)-flats that are 

 tangent to p {n — l)-spreads at points of p (n — ^)-spread3 that lie one 



