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



131 



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

 stricted system equivaleut to n — 1 iudepeudent equations, the equation 

 of aS'j whose order is n (m — n -\- 1). 



We can find the equations of those exceptional points where n -\- I 

 consecutive i^„_i's intersect iu a point, if we eliminate the parameter from 

 the n -{- 1 equations 



a r-" + (/« — n) b"'-"-^ +.... = 



b r-" + (m — n) c"'-"-^ +.... = 



+et+f=0. 



The result is a restricted system equivalent to « 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 (« -f 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 that cause this resultant to vanish. These values of 

 t give the special points in question.* 



Any double point on *^„_j must lie on two i^„_2's. We may find the 

 equations of the total double spread on »S'„_i, by expressing the conditions 

 that the equations of an i^„_2 regarded as equations in the parameter, 

 have two roots in common. These conditions are f 



(I) 



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



a, {771 — 1) b, 



e 



b, (/7l — 1) c, 



b, 



("« — 1) e,/ 



= 0, 



* 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. 



