In 1871 Cremona gave a method of determining all the birational transformations in which a given surface becomes a plane, when one representation of it on a plane is known; he added a great many examples of cubic transformations, but did not aim at completeness. For the non-singular cubic surface there exist seven transformations, which have been given by Loria and Sturm. This paper deals with singular cubic surfaces.
The method is briefly as follows. It is required to discover all the homaloidal families of cubic surface (Φ1) containing a given member Φ, and having certain singularities in common with it. Suppose Φ represented on a plane α by an auxiliary transformation T; all the cubic families here concerned have at least one common node, and T is taken to be projection from the first of these. Then to the family of curves of intersection of Φ with (Φ1) there corresponds a homaloidal family of proper or degenerate curves in the plane α, satisfying certain conditions; and conversely, to every such plane family there corresponds a homaloidal family (φ1), each member of which contains a certain fixed fundamental system H consisting of curves, simple points, points of contact of various orders and multiple points. The different plane homaloidal families of the earlier degrees are known, and by examining each in turn we are led back to all the existing families (Φ1) of the required kind.
The following investigation treats of cubic surfaces having 1, 2, 3, 4, nodes or a nodal line; there are further subdivisions according as the tangent cone at any node has a fixed part of order 0, 1 or 2, presenting 4, 7, or 9 conditions to the surface.