On the Distribution of Surfaces of the Third Order into Species, in reference to the absence or presence of Singular Points, and the reality of their Lines

The theory of the twenty-seven lines on a surface of the third order is due to Mr. Cayley and Dr. Salmon; and the effect, as regards the twenty-seven lines, of a singular point or points on the surface was first considered by Dr. Salmon in the paper On the triple tangent planes to a surface of the third order 2). The theory as regards the reality or non-reality of the lines on a general surface of the third order, is discussed in Dr. Schläfli’s paper An attempt to determine the twenty-seven lines upon a surface of the third order, and to divide such surfaces into species in reference to the reality of the lines upon the surface 3). This theory is reproduced and developed in the present memoir under the heading, I. General cubic surface of the third order and twelfth class; but the greater part of the memoir relates to the singular forms which are here first completely enumerated, and are considered under the headings II., III., etc. to XXII., viz. II. Cubic surface with a proper node, and therefore of the tenth class, etc., down to XXII. Ruled surface of the third order. Each of these families is discussed generally (that is, without regard to reality or non-reality), by means of a properly selected canonical form of equation; and for the most part, or in many instances, the reciprocal equation (or equation of the surface in plane-coordinates) is given, as also the equation of the Hessian surface and those of the Spinode curve; and it is further discussed and divided into species according to the reality or non-reality of its lines and planes.