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



The order of the curve 2 is, 



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



(n-l)\ 



The equations of all the «-tuple points on /S„_i are given by means of 

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

 2 (w — 7i) rows and 2 m — n — 1 columns. They form a restricted 

 system equivalent to n independent equations, whose order is 



1 

 n 



— (2 m — n — I) (2 m — n — 2) . . . . {2 m — 2 n) ; 



this is the number of n-tuple points. The number of the n-tuple points 

 other than the cusps on Si, are 



— (2 ?« — w — 1) (2 m — n—2) (2 m — 2 7i) — {71 + 1) (m — n). 



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

 or else they are ?i-tuple points on the combined curves *S'i 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 Si is n, and there are no cuspidal points 

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

 order of S,^_i in this case is 2 (n — 1) ; no developable S„_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 < w, where p is an integer. 

 Any J9 + 1 consecutive i^„_i's intersect in an Fn_p_i whose equations are 



' 9t ' 9tp 



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

 parameter t, these equations may be written 



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

 sions, p. 296. 



t Veronese, loc. cit. 



