MORENO. — ON RULED LOCI IN W-FOLD SPACE. 125 



a restricted system equivalent to n — 1 independent equations. "We 

 have then the case of an (n — l)-flat whose equation involves n para- 

 meters connected by n — 1 independent relations ; this is equivalent to 

 the case of a single equation containing one arbitrary parameter. We 

 may, in general, consider the system of developables as given by an 

 (» — l)-flat whose equation contains a single arbitrary parameter or 

 k parameters connected by k — 1 equations.* 



3. The tangent (n — \)-flats that are common to n — 1 (n — V)-spreads 

 envelop a developable. 



The equation in homogeneous coordinates of any (n — l)-fiat may be 



written 



x = ay-\-j3z-\-.... J ! -yw. 



This equation involves n independent parameters; if we connect them 

 by any n — 1 independent equations we shall have the equation of an 

 (n — l)-flat that contains but a single independent parameter, so that 

 the 1-fold infinite system of (n — 1) -flats represented by it envelop a 

 developable. The tangent (n — l)-flat at any non-singular point of a 

 developable S,^ contains the generating F n _ 2 through that point and 

 touches the *S'„_ 1 all over this flat, t We may speak of this developable 

 S n _! as enveloped by its tangent i^-i's. If then we impose on an 

 arbitrary (n — l)-flat any conditions that give rise to n — 1 independent 

 equations between the coefficients in its equation, the (n — l)-flat will 

 envelop a developable S n _i. 



Let IT= 



be the equation of an (n — 1) -spread. The equation of the tangent 

 (n — l)-flat at any ordinary point P' is 



9U< 3U> 9U> A 



If we impose on the equation of the arbitrary (n — l)-flat the condi- 

 tions that it shall be this tangent (n — l)-flat, the coefficients in the two 

 equations must be proportional. We must have then 



9U[ 9U[ 9U< 



9 x' = 9 y' = . . . . 9 w' 



— la y 



From these equations by means of the equation 



W = Q, 



* Salmon, Geometry of Three Dimensions, p. 286. t Killing, loc. cit. 



