MORENO. — ON RULED LOCI IN W-FOLD SPACE. 153 



lent to r independent equations, the order of the system is — - (I + m — 



r ! 



r + 1) (Z + m — r) . . . . (Z -f m — 2 r -f- 2). 



Whether n is odd or even we have finally a curve 2i of multiplicity 

 n — 1, whose equations are found by expressing the conditions that 

 the equations of the generating (n — 2) -flat have n — 1 roots in the 

 parameter in common. We derive a restricted system of the same 

 form as (II) in which however there are I + m — 2 (n — 2) rows and 



I + m — (n — 2) columns. The order of this system is — 



V ; * (n - 1)1 



(Z+ m — n + 2) (Z + m — n + 1) . . . . (I + m — 2 n + 4). This curve 

 has M-tuple points on it whose equations are fouud by expressing the con- 

 ditions that the equations of the generating (n — 2)-flat have n roots in 

 common. We again have a restricted system of the same form as (II), 

 in which, however, there are I -\- m — 2 (« — 1) rows and I -f- m — n + 1 



columns. The order of this system is — - (I -\- m — n -\- 1) (I + m — n) 



. . . . (I + m . 2 n + 2), which is the number of points in question. For 



n = 3 these formulae for the order agree with those given in Salmon.* 



A very special case is where the parameter enters only linearly in one 



of the equations of the generating (n — 2)-flat. Let the equations of the 



flat be 



A = a t + b = 0, 



B = a' t m + V r- 1 + . . . . = 0, 



where we make the same suppositions regarding a, b, a', b', . . . , as 

 before. The S a _ t in this case is a ruled spread with m sheets through 

 the (n — 2)-flat, whose equations are 



a = 0, b = ; 



it has no other multiple locus on it at all. Consecutive generating -F„_ 2 ' s 

 of the system intersect in the flat, whose equations are, 



9 B 

 a = Q,b = 0,B= 0, V- = 0. 



at 



All the F^s of the system lie in the same (« — 2)-flat ; they generate a 

 developable (n — 3)-spread «S'„_ 3 in this flat. S' n ^> is the section by this 

 flat of the developable (n — l)-spread enveloped by the (n — l)-fl;it B. 

 Consecutive generating F^'a of S n ^ intersect in generating -F„_ 4 's of 



* Salmon, Geometry of Three Dimensions, p. 428. 



