124 PROCEEDINGS OP THE AMERICAN ACADEMY. 



tive F„_{s intersect ia an i^„_2, three in an i^„_3, r in an I\_r, n — 2 

 in an F2 or plane, n — I in an Fi or line, ?i in an Fq or point. There 

 is a 1-fold infinite system of these F„_2S which are generators of aS'„_j, 

 a 1-fold infinite system of i^„_3's, generators of S,^_2, a 1-fold infinite 

 system of lines generators of S2, the developable surface. 



Through any F„_2 there pass two consecutive -^„_i's, through any F„_^ 

 there pass three consecutive i^„_i's, through any Fq, n consecutive F„_^'s. 

 Through any i^„_3 there pass two consecutive i^„_2's, through any F„_i 

 there pass two consecutive i^„_3's and three consecutive i''„_2's, and 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^, through any three consecutive points an osculating plane, 

 an F2, through any four consecutive points an osculating 3-flat, an F^, 

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



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 ^ -f 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 n.f 



This theorem has been generalized by Veronese.l 



Let us consider any curve in ?i-fold space whose equations are, 



</> = 0, X = 0, . . . . V' = 0, 



a restricted system equivalent to h — 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, . . . . .^' = 0,§ 



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

 points of it. Killing, loc. cit. 



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



t Veronese, Behandlung der projectivisclien Verhaltnisse der Eaume von ver- 

 schiedenen Dimensionen durch das Prlncip des Frojicirens und Schneidens, 

 Mathematische Annalen XIX. 



^ <l>' = <t> (x', >/', . . . v'), etc. 



