The derivation of the 4-nodal Cubic Surface and the Quartic Surface, of which it is a particular case, are well known: certain new results of interest from the point of view of symmetry, and extension to n-fold space, are provided by the symbolic algebra. In particular a simple proof is given, in § 2, of the theorem that a symmetry exists between the four vertices of the tetrahedron and the fifth point whose locus is the cubic surface, and this property can be extended to the case of n + 1 points in n-fold space with one additional point.