MORENO. — ON RULED LOCI IN n-FOLD SPACE. 145 



n-3 



^ 9 A 9'^' A 



^ = 0,^ = 0,.. .^;:^, = 0. 



If we eliminate the parameter from these equations we derive a 

 restricted system equivalent to n — 2 independent equations. The locus 



is a surface /S'g ruled bj the -Fi's ; it is an ( — - — J-tuple surface on S^^.^. 



Consecutive i^i's do not in general intersect for the n -\- 1 equations 



^ = 0, ^- = 0, . . . — —^ --= 0, 

 ^^ 9x^ 



have not in general any common solutions. If we regard these m + 1 

 equations as homogeneous in the n -\- \ coordinates and form their result- 

 ant, the values of the parameter that cause it to vanish will give points 

 where consecutive lines meet. The equations of these points may be 

 formed by eliminating the parameter from the n + 1 equations, which 

 gives a restricted system equivalent to n independent equations. These 



points are double points on ^'2 and f — - — | -tuple points on S„_x. 



If n is even then — consecutive i^„_2's intersect in a point Fq, whose 



At 



equations are, 



A = 



9k 



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



5/V 



2 



^^ 9 XT 



The elimination of the parameter from these equations gives a restricted 

 svstem equivalent to n — 1 independent equations. The locus is a curve 



10 



