126 PROCEEDINGS OF THE AMERICAN ACADEMY. 



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

 a, (5, .... y. For an (n — l)-flat to be tangent to an (?i — l)-spread, 

 one relation between the coefficients that enter into their equations must 

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

 (n — l)-spreads envelop an S n _ 1 . 



Let us consider only those tangent (n — l)-flats to an (n — 1)- 

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



Let £7=0 



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



U=0, V=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 



9U' 9 IP 9U' 



9x' =. 9 y' = . . . . 9 to' 



— la y 



and IP = 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, (3, ... y. For an (n — 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„-v 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 cr other (n — l)-flats, where 

 n — 1 = 2 p -\- cr, 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 imposes three con- 

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

 To be tangent to the (n — l)-flat at a point of an (n — 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 {n — 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 — &) -spreads that lie one 



