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



131 



The elimination of the parameter from these equations gives a re- 

 stricted system equivalent to n — 1 independent equations, the equation 

 of S x whose order is n (m — n -f 1). 



We can find the equations of those exceptional points where n -f- 1 

 consecutive F n _ x s intersect in a point, if we eliminate the parameter from 

 the n + 1 equations 



a t m ~ n + (m — n) b m ~ n - 1 +.... = 



b t" 1 -" + O — n) c'"-"- 1 + .... = 



+ et + f=0. 



The result is a restricted system equivalent to n independent equa- 

 tions; it is of order (n + 1) (m — n), which is the number of such 

 points, cusps on Si. We may verify this result by forming the resultant 

 of these (« + 1) equations. If we eliminate the variables from these 

 equations we have a determinant of order n + 1. If we expand this 

 result t enters to the degree (n + 1) (m — n) so that there are (n + 1) 

 (m — n) values of t tnat cause this resultant to vanish. These values of 

 t give the special points in question.* 



Any double point on S n _ x must lie on two i^_ 2 's. We may find the 

 equations of the total double spread on £„_!, by expressing the conditions 

 that the equations of an F n _ 2 regarded as equations in the parameter, 

 have two roots in common. These conditions are t 



a, (m - 1) b, i ^j '- c, 



(I) 



a, 



('» - 1) 



6, 



b, (m — 1) 



h, 



(m — \)e,f 



* For n — 3, these results agree with those of Salmon, Geometry of Three 

 Dimensions, p. 296. Neither the results there nor these hold when the system has 

 stationary (n — l)-flats. 



t Salmon, Higher Algebra, art. 275. 



