( 320 ) 



"By takiug on the normal curve Co,, the n points ij , //o,...L„ 

 "arbitrarily we determine on it an involution J^ of the dimension 

 "« — 1 and the order 2», of which each of those n points taken 2n 

 "times forms a group. The osculating" spaces belonging to the 2n 

 "points of any group of that involution intersect in a point A of 



"the linear space S^ , containing the n given points; if this group 

 "describes the involution I^^^ the point A generates in S ^ a linear 

 "system in projective correspondence with 1^^^ ." 



As far as I am aware of up to now a polydimeusional interpre- 

 tation suiting all values of n is known of three general invariants, 

 namely of the discriminant 2?, of the invariant {ab)^", and of the 

 invariant of Sylvester dealt with here. If a" ^ is again the equa- 

 tion of the osculating space of the normal curve a- =r A'', {k — 0,\,...n) 

 in the space with n dimensions, corresponding to the parametervalue 

 A, then i> = represents as is known the curved space \( _,.^ 

 with )i — 1 dimensions of the order 2 {n — 1) which is enveloped by the 

 osculating space if A varies; leaving alone the supposition n = 2, which 

 has no sense, we get that n^3 gives in the ordinary space the develo- 

 pable surface having the cubic normal curve of that space as cuspidal 



line. According to Clifford (1. c.) (ab)-" = is in the space S'" 

 the quadratic curved space -S'^ with 2» — 1 dimensions repre- 

 senting the locus of the point lying in a space S"" with the points 

 of contact of the 2n osculating spaces of the normal curve of that 

 space 'S' " passing through this point; whilst the corresponding inva- 

 riant (a?>)2"-' of the normal curve of the space S'"~ vanishes iden- 

 tically and the indicated particularity presents itself there, compare 

 the case of the skew cubic in our space, for any point. 



For the case « = 4 the invariant (a&)''' = is identical with « = 

 (Clebsch-Lindemann I.e.) and at the same time the condition that 

 the four points of contact of the osculatingspaces through any point 

 of the locus form an equianharmonic quadruple. Moreover D is a 

 linear combination of i-^ and ƒ, from which finally ensues that any 

 plane cuts the space 7? = according to a curve of the sixth order, 

 having the six points of intersection with the surface of intersection 



