128 PROCEEDINGS OP THE AMERICAN ACADEMY. 



In general n + 1 consecutive F n _^s do not have any common inter- 

 section, for the n + 1 equations, 



have no common solutions. If we regard these equations as homo- 

 geneous in the n -f 1 coordinates we may form their resultant, and the 

 values of the parameter that cause this determinant to vanish, give 

 special points where n + 1 consecutive F^s intersect. These points 

 are cusps on the curve Si. 



Reciprocally there will, in general, be a finite number of F n _^a that go 

 through n + 1 consecutive points of S^ 



Veronese has shown that a curve in n-fold space has 3 n singularities 

 which are connected by 3 (n — 1) relations, an extension of the Pluecker- 

 Cayleyan characteristics of a twisted curve in three-fold space.* 



In this we have assumed that the variables that enter into the equation 

 of the enveloping (n — l)-flat cannot be expressed in terms of fewer 

 than n + 1 independent linear functions of the variables alone. If they 

 could be expressed in terms of v such linear functions, where v < n, the 

 developable »S n _ 1 is a conoid with an (n — v)-way head, a case to be con- 

 sidered later. 



The developable S k oi the series is ruled by (k — l)-flats, F k _ r 'a. The 

 S u , where 2 < k ^ n — 1 can be given by means of its enveloping F k 

 whose equations involve a single parameter. The n — k equations of the 

 F k must however be of the form 



, n 9A n d n ~ k ~ x A n 



as we have previously seen. Even the S x may be represented in this 

 manner. 



Any (n — l)-flat B = 



cuts the £„_, in a developable (n — 2)-spread, for it cuts the system of 

 F n _i& in a system of (n — 2)-flats that intersect consecutively in (n — 3)- 

 flats. We may see this in another way. By means of this new equa- 

 tion we can eliminate one variable from the equation of the enveloping 

 (n — l)-flat. The resulting equation in n variables may evidently be 

 considered as the envelope of an (?i — 2)-spread in a new (n — l)-fold 

 space. The (n — l)-flat cuts any S k of the system in a (i — l)-way 



* Veronese, Ioc. cit. ; Killing, loc. cit. p. 197 et seq. 



