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



The equations then are of S n _o r+1 form a restricted system equivalent 

 to 2 r — 1 independent equations whose order is r (/ -f- m — 2r + 2). 

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



If n is odd we come down finally to an f — - — J-tuple surface S. 



The equations of S 2 are found by eliminating the parameters from the 

 equations 



n-3 n-3 n-3 



9~*'A _ 9'*' A 9~*'A 



n = 3 — ^' n-5 — O, . . . , n _ 3 — 0, 



9X' 2 ' QK'V'Qfi d/i r 



n-3 n-3 n-3 



n-3 — ^, n-S — "»•••} n-3 — 0. 



2 A" 2 "" , 9\'* dp 9fi Y 



The equations of $ 2 form a restricted system equivalent to n — 2 inde- 



n — 1 



pendent equations, whose order is — - — (I + m — n 4-3). 



Li 



There are also f — — J-tuple points jP 's on S n _ u though in general 



n 4- 1 



— - — consecutive F n ^ 2 's do not intersect. If we form the resultant of 



the n -f- 1 equations 



n-l n— 1 



5 2 ~J rt 9^'A 



—^i = 0, — ^ = 0, . 



3 A 2 9 k 2 9 ft 



n—l n—1 



5A. 2 3 A. 2 5 //. 



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

 ters e 

 n + 1 



n 4- 1 

 ters enter to the degree — — — ( l 4- m — n -\- 1). There are then 



(I + m — n -\- 1) valujs of the parameters that cause this 



Ld 



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 «4 1 equations. The result is a restricted system 

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



ii 4- 1 

 — - — (I 4- m — ii 4- 1). This is another proof of the number of 



points F on S n _i. 



