MORENO. ON RULED LOCI IN W-FOLD SPACE. 137 



Let 



£7=0 



be the equation of an (n — 1) spread of order m. We shall use the 

 points (1), (2), A (1) + fi (2) to denote the points whose coordinates 

 are x x , y x , . . . w x , x 2 , y 2 , . . . w 2 , and \ x x -{- p x 2 , \y x -\- py., , . . ., 

 Xw x -j- \iw 2 , respectively, so that A (1) + ft (2) represents a point on the 

 line (12), i. e., the line joining (1) and (2). We denote the result of 

 substituting the coordinates of the points (1) or (2) in U by U x , and U 2 

 respectively. We use the symbols 



( 9 9 9 \ T t 



A 2 U x = x, - h y, -, h • • • + »i n — U x , 



V 9 Xi 9x x die J 



( 9 9 9 \ 



A U = [x 2 — + y 2 7 r- + . . . + w 2 ^—) Uj 

 \ a x 9 y dw) 



f 9 9 9 \ rT 



A U 2 =[X yr— + ^ h • . • + «> ^ Us, 



\ 9x 2 9y 2 9w 2 J 



(9 9 9 \ k 



^ k 2 U x ~(x 2 ^ h y 2 ■= h • • • + «>2 s — tfi- 



\ 9x x J 9y x <9wv 



In the last case the operator is to be applied h times to f^. Now 

 A (1) + /i (2) is a point on the line (12), if it is also a point of the 

 (n — l)-spread, it must satisfy the equation of the spread. Substitute 

 the coordinates of A (1) + p (2) in Z7and we have 



A"' U x + A- 1 M A 2 U x + ^ T /i - A 2 2 0i + . . . 



u" 1 



. . . + —. A 2 ra U x = 0. 

 m I 



The m values of A: /u that satisfy this equation determine the m points 

 where the line (12) meets the (n — l)-spread. If the point (1) lies on 

 the spread then 



t7 1= =0. 



If the line (12) meets the spread twice at the point (1), then 



U x = 0, A 2 U x = 0. 



