3. Surfaces of the Second Order, or Quad ric Surfaces.— The earliest investigations were connected with the surfaces of the second order, namely, those defined by the general equauon of the second degree: Asa+ By' + Cz1-1-2Fyz-1-2Gzx +2Hxy +2My-E2Nz+ P="0. This equation contains 10 coefficients which enter homogeneously. However, only the nine ratios of the coefficients are essential, as the equation may be divided through by any coefficient that is not zero. From this fact comes an important theorem. The substitu tion of the co-ordinates of a given point in the general equation imposes one equation of condition upon the coefficients; nine such equations determine the ratios of the coeffi cients, and herewith the equation, and with it the surface. The theorem follows: A sur face of second order is in general determined by nine points through which it is to pass.
4. Classification of Quadric Surfaces.— There are in all 16 surfaces of the second order, when the purely imaginary and degen erate cases are included in the numeration. The grouping of the individual surfaces varies with the principle employed. The principle of division may be based on analytical criteria or on geometrical characteristics. Four differ
ent varieties of geometrical classification are lcnown. In one the surfaces arc divided into (a) the surfaces with centre or central sur faces. (6) the non-central surfaces. A second classification gives, (a) ruled surfaces with real generating lines (see 16), (b) non-ruled sur faces (analytically these latter surfaces are ruled surfaces with imaginary generating lines). A third classification rests upon the presence or absence of vertices on the surface. For example, a cone has a vertex and two inter secting planes are a degenerate form of a sur face of second order with the line of intersec tion as a line of vertices. An ellipsoid is with out a vertex. The fourth classification is based upon the nature of the conic that is cut from the surface by the plane at infinity.
We now present a classification based upon analytical criteria. This is effected by means of the values of two polynomials .1 and D, functions of the coefficients, and of the roots k =X, v, of a cubic equation in k called the discriminating cubic. The polynomials may be conveniently put in the determinant form, as also the cubic equation: