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



Let 



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

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

 are Xi, yi^ . . . w^, x^, y^, . . . lOo, and \ Xi + fi x^, A3/1 + ^^25 • • •> 

 Xwi + juWgi respectively, so that A (1) + /n (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 Uhj fJi, and Uo 

 respectively. We use the symbols 



/ 9 9 9 \ 



\ d Xi dx^ dtoi) 



(9 9 9 \ 



\ a X " d y 9w) 



/ 9 9 9 \ 



\ 9 X.2 9 y-i 9w^j 



(9 9 9 x" 



A*2 Wi^ix^^ h ^2 -^ h • . . + ^2 ::: Ui. 



\ 9 Xi, oyi 9iOil 



In the last case the operator is to be applied Tc times to Ux. Now 

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

 {ii — l)-spread, it must satisfy the equation of the spread. Substitute 

 the cooi'dinates of A (1) + /x (2) in £7 and we have 



A- U, + A-i i, A2 L\ + ^^' ^, l\ + .. . 

 , . . -\- —, ^."' Ui = 0. 



m ! 



The m values of A : ju that satisfy this eq^iation determine the m points 

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

 the spread then 



Ui = 0. 



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



Z7i = 0, A, Ui = 0. 



