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



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

 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 



n — 1 = k. p + (k — 1) o- + •••• + 2 t + v, 



envelop a developable S n _ v 



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 devehpables ; sections. 



Other properties of an S n _ x 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^n-j's to the S n _ v 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 F n _ x of the system meets its consecutive F„-\ in a definite F n _ 2 , a 

 generator of S n _ t whose equations are, 



„ = o,fi = <, 



Any three consecutive i^-^s meet in a definite F n _ z , a generator of *^„_ 2 , 



whose equations are, 



. 9 A PA . 



A==0 >9X=°>W =0 - 



Any n — 1 consecutive 2 ?T B _ 1 's meet in a definite line F x , a generator of 

 $2, whose equations are, 



. n 9A 9->A 



^ = 0,_=:0,. ..^=0- 



Finally, any n consecutive i^_ 1 's meet in a definite point of the curve of 

 regression of S 2 . The equations of the F are, 



. . 9A 9^A 



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



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



