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



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



r + 1) (/ + m — r) . . . . (Z + m — 2 r + 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 (w — 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 (w — 2) rows and 



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



(n - 1)1 



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

 has w-tuple points on it whose equations are found l)y expressing the con- 

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

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

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



columns. The order of this system is — - (I -{- m — n -{- 1) (I -{- m — ?^) 



. . . . (I -{■ tn . 2 n -{- 2), which is the number of points in question. P'or 

 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 oi.e 

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

 fiat be 



B=a' r + b' t'"'^ +.... = 0, 



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

 before. The S^^i in this case is a ruled spread with m sheets through 

 the (?^ — 2)-flat, whose equations are 



a = 0,b = 0; 



it has no other multiple locus on it at all. Consecutive generating F„_2S 

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



9 B 



d t 



All the F„_iS of the system lie in the same (n — 2)-flat ; they generate a 

 developable (n — 3) -spread *S''„_3 in this flat. *S''„_3 is the section by this 

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

 Consecutive generating F^?, of S,i_i intersect in generating ^„_4's of 



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



