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



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 a restricted system equivalent to re — 1 independent equations. We 

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

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

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

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

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

 /' parameters connected by X; — 1 equations.* 



3. The tangent (n — \)-Jlats that are common to w — \ (n — l)-spreads 

 envelop a developable. 



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

 written 



x = ay + fiz-{-....-{-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 (k — 1) -flats represented by it envelop a 

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

 developable S„_i contains the generating F'„_2 through that point and 

 touches the S„_i all over this flat, t We may speak of this developable 

 Sn_i as enveloped by its tangent i^,_i's. If then we impose on an 

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

 equations between the coefficients in its equation, the (?2 — l)-flat will 

 envelop a developable 'S'„_i. 



Let U= 



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

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



9U' 9U' 9U' ^ 



dx' dy' dw' 



If we impose on the equation of the arbitrary (yi — 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>_ 91P 



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



— la y 



From these equations by means of the equation 



U = Q, 



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



