124 PROCEEDINGS OF THE AMERICAN ACADEMY. 



rive i'Vi's intersect in an F n _ s , three in an F n _ 3 , r in an F n _ r , n — 2 

 in an F. 2 or plane, n — 1 in an F 1 or line, n in an F or point. There 

 is a 1-fold infinite system of these i^,_ 2 's which are generators of £„_!, 

 a 1-fold infinite system of F n _ 3 s, generators of S n _ 2 , a 1-fold infinite 

 system of lines generators of S 2 , the developable surface. 



Through any F n _ 2 there pass two consecutive F^s, through any F„_ 3 

 there pass three consecutive -F„_i's, through any F , n consecutive F^s. 

 Tlirough any F n _ z there pass two consecutive F„_ 2 's, through any F„_ 4 

 there pass two consecutive -F n _ 3 's and three consecutive F„_2S,&nd so on. 



"We may then reverse this process and start with the curve of the 

 system. Through any two consecutive points of the curve there passes 

 a line, an F u through any three consecutive points an osculating plane, 

 an F 2 , through any four consecutive points an osculating 3-flat, an F 3 , 

 through any n — consecutive points an osculating (n — l)-flat, an F n _ v * 



That these operations may give unique results this curve must lie in 

 the n-fold space and in no flat space of a less number of ways. If the 

 curve lie in a £-flat, where k < n — 1, all the £-flats through h + 1 con- 

 secutive points coincide and definite (k -f l)-flats are not determined at 

 all. By a theorem of Clifford, such a curve must be of an order as 

 great as ra.f 



This theorem has been generalized by Veronese.^ 



Let us consider any curve in n-foh\ space whose equations are, 



= 0, x = 0, . . . . if, = 3 



a restricted system equivalent to n — 1 independent equations. The 

 equations of the tangent at any point P' of this curve are linear equa- 

 tions whose coefficients are functions of the n non-homogeneous co- 

 ordinates, x', y', . . . . v'. The same thing is true of the equations of 

 any of the osculating flats at the point P. The osculating (n — l)-flat 

 is given by a single equation, the coefficients of which are functions of 

 these n quantities x', y', . . . v'. If we regard these as n parameters 

 they are connected by the equations, 



^ = 0, x ' = 0, . . . . ip' = o,§ 



* We shall say a /t-flat osculates a curve if it contains k + 1 consecutive 

 points of it. Killing, loc. cit. 



t Clifford, Classification of Loci; Mathematical Papers, pp. 305-331. 



i Veronese, Behandlung der projectivischen Verhaltnisse der Baume von ver- 

 schiedenen Dimensionen durch das Princip des Projicirens und Schneidens, 

 Mathematische Annalen XIX. 



§ <p' = <p (x>, y', . . . v>), etc. 



