# Serie VII, Nr 24a

## Surface of order three. Hessian surface for models VII, 2 and VII, 5.

Mathematical description
Let S be a surface in the projective space given by a homogeneous polynomial equation f(x1, x2, x3, x4) = 0. The Hessian surface is given by:
 H = | δ2f | δxiδxj
Let d be the degree of f, then the Hessian surface will have degree 4(d-2). In this case, the surface is the Hessian surface H, where f is a homogeneous polynomial of degree 3 describing the cubic surface with 4 A1 double points. The surface H=0 has degree 4. The Hessian surface of a smooth cubic surface has 10 double points. For every double point in the cubic surface, the Hessian surface acquires an additional double point, so H=0 has 14 double points and they are all real.

Title model and translation
Fläche dritter Ordnung. Hesse'sche Fläche zu modelle VII, 2 und VII, 5 Fläche dritter Ordnung. Cayley'sche Regelfläche mit (unendlich fernen im Endlichen gelegenen) Cuspidalpunkte.
SSurface of order three. Hessian surface for models VII, 2 and VII, 5.

Text on Sticker and translation
unclear sticker.

Designer
Dr. Carl Rodenberg
Darmstadt in 1881

Company
Martin Schilling in Halle a.S.

Original price
Mark 21-25

Dimension
13-15 cm

Material
Plaster

Literature
Schilling, M., Catalog mathematischer Modelle für den höheren mathematischen Unterricht, Leipzig: Verlag von Martin Schilling, 1911