OX ALGfiBKAIC HYPEKCONICAL CONNEXES IN SPACE, ETC. 487 



maij he expressed linemiy in terms of these parameters, since the 

 equations of condition on the coefficients of the general poly- 

 nomial are linear. 



In particular, if m = n , the above expression reduces to 



J^il^lL^^i+'lzdli , In tliis case B\xìi)=^{) determines one and 



only one line complex in S,. and conversely. Hence: The equation 



of a line complex m b^ mvoLves -, r— v, , — --. tt— homoqeneovs 



essential parameters. The equation is linear in these parameters (*). 



4. The hypercone F(yQ ... y,) = determined by Jetting 

 {xQ...Xr), in the equation of the connex, be the coordinates of 

 a fixed point [x] will be called the connex hypercone of (x). 

 Similarly, if m >> n, the in — n points on any line {p) whose 

 connex hypercones contain (p) will be called the connex points 

 of {p). By direct extension of the reasoning of Masoni (sections 5 

 and 6 of bis previously cited article) we obtain the following 

 two theorems. 



Let A, be an invariant of degree s in the coefficients of 

 the binary form which determines the m — n connex points of 

 a line in Sr . The locus of the lines in Sr whose connex points 

 are so situateci that this invariant is zero is a line complex of 



degree -^ (m -)- n). 



Let As be an invariant, of degree s in the coefficients, of 

 an hypersurface of order n in S,._i . The locus of a point ivhose 

 connex hypercoìie is the projection from (x) of an hypersurface in 

 an S,._x for which A^ = is an hypersurface in S,. of order 



— (mr — n). 



r ^ 



As a particular case of the second theorem, let ni = n and 

 let As ;= be the condition that the hypersurface in S,._i bave 

 a node. Then s ^^ r {n — l)*""'. Hence, the sinyular hypersurface 

 of a line complex of degree n in S,. is of order (r — ^l)n(n — 1)''~\ 



(*) See. for r = 4, W. H. Young , " Atti R. Accad. delle Scienze di 

 Torino ,, voi. XXXIV (1899). 



L' Accademico Segretario 

 Corrado Segre. 



