MORENO. — ON RULED LOCI IN W-FOLD SPACE. 151 



The equations then are of S,^_2r+i foi'ni a restricted system equivalent 

 to 2 ?• — 1 iudepeudent equations whose order is r {I -\- m — 2 7-4- 2). 

 As we have seeu, there are two cases according as n is odd or even. 



If 71 is odd we come down finally to an ( — - — |-tuple surface S„. 



The equations of S^ are found by eliminating the parameters from the 

 equations 



n-3 H-3 n— 3 



n-3 — ^'' »-5 — ^) • • • J n~S — ^> 



n-3 n-3 n-3 



9^ B_ 9^'B 9^ B 



n-3 — ^> n-5 — ^5 • • • 5 n-:i — ^• 



9X ' 9 ^ ' 9 fj. 9 jj. ~ 



The equations of S2 form a restricted system equivalent to « — 2 inde- 



71 — 1 



pendent equations, whose order is — - — (I -\- 771 — 7i +3). 



There are also f — - — j-tuple points FqS> on ;S'„_i, though in general 



n + 1 



— - — consecutive i^„_2's do not intersect. If we form the resultant of 



the n -\- 1 equations 



n— 1 n— 1 n— 1 



n-l — "> ,j-3 — ^) • • • J n-1 — ^> 



5a. ^ 9 X.' 9 fJI. 9 fX ^ 



n—1 «— 1 n— 1 



9^'B _ 9"'B 9'^- B _^ 



n-l — ^i n-3 — ^> • • • J n-l — ^> 



9X^' 9x:^'9fJi 9fji''' 



we have a determinant of the (n + l)-st order, in which the parame- 



n -\- 1 

 ters enter to the degree — - — (I -{- tti — ti -\- 1). There are then 



71 -{■ I 



— - — (Z 4- OT — « -f 1) values of the parameters that cause this 



determinant to vanish, and so this is the number of points F^. We 

 can find the equations of these points by eliminating the parame- 

 ters from these n -\- 1 equations. The result is a restricted system 

 equivalent to 71 independent equations. The order of the system is 



n + I 



- — - — (I + m — n + 1). This is another proof of the number of 



points Fq on S„_i. 



