* “Sulle superficie dell' n ^mo ordine immerse negli spazi di n dimensioni,” Rendiconti Circolo Matematico di Palermo, Tomo I (1887).Google Scholar

† “Sulle intersezioni delle varietà algebriche…,” Memorie Torino (2), 52 (1903) p. 102.Google Scholar

‡ “Surfaces on which lie Five Systems of Conics,” Proceedings London Mathematical Society, Vol. 24 (12 1924), p. v.Google Scholar

§ In this paper, unless otherwise stated, ‘quadric’ will mean ‘quadric fourfold’: the symbol Q_n will be used to denote a quadric n-fold.Google Scholar

* Cf. Enriques-Chisini, , Teoria Geometrica delle Equazioni…, Vol. 3, p. 106;Google Scholar also Enriques, , “Sulle curve canoniche di genere p dello spazio a p − 1 dimeneioni,” Rendiconti Bologna, 23 (1919), p. 80 (not “Atti Bologna” as given in Vol. 3 above quoted).Google Scholar

† l _ik = l ^ki.Google Scholar

* The configuration has other interesting properties: for example, there are 145 triads of the fifteen points, each triad containing no pair of points lying on one of the ten lines. These triads define planes which pass by fives through certain points.Google Scholar

* Writing , the quartics are

all of which pass through the point (1, 1, − 1 ), have double points at (1, 0, 0) and (0, 1, 0) and y = 0, x = 0 as tangents at (1, 0, 0), (0, 1, 0) respectively. Taking (1, 0, 0), (0, 1, 0) and (1, 1, − 1) as the triangle of reference and applying the transformation x′:y′:z′ = YZ : ZX : XY, the system is transformed into a system of cubics through four fixed points.

* Otherwise, the line l ₁₂ is the locus of points (4.1) for which λ₁λ₂ = 1, or from the plane representation as shown later.Google Scholar

† These are the only points in common since the seven quadric twofolds are linearly independent.Google Scholar