MORENO. — ON RULED LOCI IN W-FOLD SPACE. 133 



The order of the curve 2 is, 



— - (2 m — n) (2m — n — 1) (2 m-2n + 2) — n (m — n+ 1).* 



(n — 1)1 



The equations of all the w-tuple points on S n _ x are given by means of 

 a rectangular system similar to (I), in which, however, there are only 

 2 (jn — n) rows and 2 m — n — 1 columns. They form a restricted 

 system equivalent to n independent equations, whose order is 



—. (2 m — n — 1) (2 m — n — 2) . . . . (2 m — 2 m) ; 

 n ! 



this is the number of w-tuple points. The number of the rc-tuple points 

 other than the cusps on S x , are 



— (2 m — n — 1) (2 m — n — 2) . . . . (2 m — 2 n) — (n + 1) (w — n). 



These points necessarily lie on Si ; they are either n-tuple points on 2i, 

 or else they are n-tuple points on the combined curves Si and 2i. In 

 three-fold space the double curve on the developable may have tripl 

 points on it ; it can have no double points off of the cuspidal curve. 



If m = n, then the order of S x is n, and there are no cuspidal points 

 on the curve ; this is the rational normal curve of Veronese. f The 

 order of S n _ x in this case is 2 (n — 1) ; no developable S n _ x can be of 

 lower order unless it is a cone or conoid, for no curve of lower order 

 than n can lie in the n-fold space without at the same time lying in a 

 space of fewer than n ways. 



Let us consider the case where m = p < n, where p is an integer. 

 Any p -\- 1 consecutive F n _i& intersect in an F n _p_i whose equations are 



. A 9 A n 9 p A . 



If we use two homogeneous parameters X and /x instead of the single 

 parameter t, these equations may be written 



* For n = 3, this result agrees with that in Salmon, Geometry of Three Dimen- 

 sions, p. 296. 



t Veronese, loc. cit. 



