128 PROCEEDINGS OF THE AMERICAN ACADEMY. 



In general « + 1 consecutive F„_j's do not have any common inter- 

 section, for the w + 1 equations, 



9A ^ d^A ^ 



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

 geneous in the w + 1 coordinates we may form their resultant, and the 

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

 special points where w + 1 consecutive F^-^s, intersect. These points 

 are cusps on the curve ^i. 



Reciprocally there will, in general, be a finite number of i^„_i's that go 

 through w + 1 consecutive points of *S'i. 



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

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

 Cayleyan characteristics of a twisted curve in tliree-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 w + 1 independent linear functions of the variables alone. If they 

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

 developable *S„_i is a conoid with an {n — v)-way head, a case to be con- 

 sidered later. 



The developable S^ of the series is ruled by {k — l)-flats, F^_^^. The 

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

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

 i^4 must however be of the form 



9A S"-*-M 



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

 manner. 



Any (n - l)-flat ^ = 



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

 F„_^s, 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 {n — 2)-spread in a new (n — l)-fold 

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



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



