MORENO. — ON RULED LOCI IN W-FOLD SPACE. 127 



on each, tangent to cr (n — l)-spreads at points of cr (n — k -\- l)-spreacls 

 that lie one on each, tangent to t (n — l)-spreads at points of t (n — 2)- 

 spreads that lie one on each, and finally tangent to v other (n — 1)- 

 spreads, where p, cr, . , . t, v, are non-negative integers connected by 

 the relation 



w— l=;^.p + (^— 1)o-+.... + 2t + v, 



envelop a developable »S'„_i. 



Similar cases occur in three-fold space where we have the tangent 

 planes that are common to two surfaces enveloping a developable surface 

 as do the tangent planes to a surface at the points of a curve' on that 

 surface.* 



4. Some additional properties of developables ; sections. 



Other properties of an »S'„_i may be deduced by regarding it as the 

 envelope of an (n — l)-flat whose equation involves a single parameter.! 

 Through any point in space can be drawn a definite number of tangent 

 i^„_i's to the Sn^y For substitute the coordinates of the point in the 

 equation of the variable {n — l)-flat and there is a certain finite number 

 of values of the parameter that satisfy the equation. 



Any i^„_i of the system meets its consecutive i^„_i in a definite i^„_2, a 

 generator of S^-^ whose equations are, 



Any three consecutive -F„_i's meet in a definite -P„_3, a generator of *S';^2> 



whose equations are, 



, ^ 9A ^ 9^A ^ 



^ = ''9X = ''9^^ = '' 



Any n — 1 consecutive jP„_i's meet in a definite line Fi, a generator of 

 S2, whose equations are, 



^ = 0,^ = 0,. ..^^, = 0. 



Finally, any n consecutive i^„_/s meet in a definite point of the curve of 

 regression of »S'3. The equations of the Fq are, 



* Salmon, Geometry of Three Dimensions, p. 547. 



t Salmon, Geometry of Three Dimensions, p. 289 et seq. 



