The cones whose vertices are on the given line all pass through eight fixed points and envelop a surface of the fourth degree; the conics whose planes contain the given line all lie on a surface of the fourth class and touch eight fixed planes. It is easy to see by elementary geometry that these two surfaces are identical. Further, the given line contains four singular points A1, A2, A3, A4, and the planes into which their cones degenerate are the eight common tangent planes mentioned above; similarly, there are four singular planes, a1, a2, a3, a4, through the line, and the eight points into which their conics degenerate are the eight common points above. The locus of the pole of the line with respect to all the conics in planes through it is a straight line called the polar line of the given one; and through this line passes the polar plane of the given line with respect to each of the cones. The name polar is applied in the ordinary analytical sense; any line has an infinite number of polar complexes with respect to the given complex, for the equation of the latter can be written in an infinite number of ways; one of these polars is a straight line, and is the polar line already introduced. The surface on which lie all the conics through a line l is called the Plücker surface of that line: from the known properties of (2, 2) correspondences it can be shown that the Plücker surface of l cuts l1 in a range of the same cross ratio as that of the range in which the Plücker surface of l1 cuts l. Applying this to the case in which l1 is the polar of l, we find that the cross ratios of (A1, A2, A3, A4) and (a1, a2, a3, a4) are equal. The identity of the locus of the A′s with the envelope of the a′s follows at once; moreover, a line meets the singular surface in four points having the same cross ratio as that of the four tangent planes drawn through the line to touch the surface. The Plücker surface has eight nodes, eight singular tangent planes, and is a double line. The relation between a line and its polar line is not a reciprocal one with respect to the complex; but W. Stahl has pointed out that the relation is reciprocal as far as the singular surface is concerned.

To facilitate the discussion of the general quadratic complex we Quadratic complexes. introduce Klein’s canonical form. We have, in fact, to deal with two quadratic equations in six variables; and by suitable linear transformations these can be reduced to the form

a1x12+ a2x22+ a3x32+ a4x42+ a5x52+ a6x62= 0
x12+ x22+ x32+ x42+ x52+ x62= 0

subject to certain exceptions, which will be mentioned later.

Taking the first equation to be that of the complex, we remark that both equations are unaltered by changing the sign of any coordinate; the geometrical meaning of this is, that the quadratic complex is its own reciprocal with respect to each of the six fundamental complexes, for changing the sign of a coordinate is equivalent to taking the polar of a line with respect to the corresponding fundamental complex. It is easy to establish the existence of six systems of bitangent linear complexes, for the complex l1x1 + l2x2 + l3x3 + l4x4 + l5x5 + l6x6 = 0 is a bitangent when

l1 = 0, and l2²+ l3²+ l4²+ l5²+ l6²= 0,
a2 − a1 a3 − a1a4 − a1 a5 − a1a6 − a1

and its lines of contact are conjugate lines with respect to the first fundamental complex. We therefore infer the existence of six systems of bitangent lines of the complex, of which the first is given by

x1 = 0, x2²+ x3²+ x4²+ x5²+ x6²= 0,
a2 − a1 a3 − a1a4 − a1 a5 − a1a6 − a1

Each of these lines is a bitangent of the singular surface, which is therefore completely determined as being the focal surface of the (2, 2) congruence above. It is thence easy to verify that the two complexes Σax2 = 0 and Σbx2 = 0 are cosingular if br = arλ + μ/arν + ρ.

The singular surface of the general quadratic complex is the famous quartic, with sixteen nodes and sixteen singular tangent planes, first discovered by E.E. Kümmer.