and others. Of cosingular complexes of higher degree nothing is known.
Following J. Plücker, we give an account of the lines of a quadratic complex that meet a given line.
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 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 formQuadratic complexes.
| 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 − a1 | a4 − a1 | a5 − a1 | a6 − 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 − a1 | a4 − a1 | a5 − a1 | a6 − 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 = ar λ + 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.
We cannot give a full account of its properties here, but we deduce at once from the above that its bitangents break up into six (2, 2) congruences, and the six linear complexes containing these are mutually in involution. The nodes of the singular surface are points whose complex cones are coincident planes, and the complex conic in a singular tangent plane consists of two coincident points. This configuration of sixteen points and planes has many interesting properties; thus each plane contains six points which lie on a conic, while through each point there pass six planes which touch a quadric cone. In many respects the Kümmer quartic plays a part in three dimensions analogous to the general quartic curve in two; it further gives a natural representation of certain relations between hyperelliptic functions (cf. R. W. H. T. Hudson, Kümmer’s Quartic, 1905).
As might be expected from the magnitude of a form in six variables, the number of projectivally distinct varieties of quadratic complexes is very great; and in fact Adolf Weiler, by whom the question was first systematically studied on lines indicated by Klein, enumerated no fewer than forty-nine different Classification of quadratic complexes. types. But the principle of the classification is so important, and withal so simple, that we give a brief sketch which indicates its essential features.
We have practically to study the intersection of two quadrics F and F′ in six variables, and to classify the different cases arising we make use of the results of Karl Weierstrass on the equivalence conditions of two pairs of quadratics. As far as at present required, they are as follows: Suppose that the factorized form of the determinantal equation Disct (F + λF′) = 0 is
where the root α occurs s1 + s2 + s3 . . . times in the determinant, s2 + s3 . . . times in every first minor, s3 + . . . times in every second minor, and so on; the meaning of each exponent is then perfectly definite. Every factor of the type (λ − α)s is called an elementartheil (elementary divisor) of the determinant, and the condition of equivalence of two pairs of quadratics is simply that their determinants have the same elementary divisors. We write the pair of forms symbolically thus [(s1s2 . . .), (t1t2 . . .), . . .], letters in the inner brackets referring to the same factor. Returning now to the two quadratics representing the complex, the sum of the exponents will be six, and two complexes are put in the same class if they have the same symbolical expression; i.e. the actual values of the roots of the determinantal equation need not be the same for both, but their manner of occurrence, as far as here indicated, must be identical in the two. The enumeration of all possible cases is thus reduced to a simple question in combinatorial analysis, and the actual study of any particular case is much facilitated by a useful rule of Klein’s for writing down in a simple form two quadratics belonging to a given class—one of which, of course, represents the equation connecting line coordinates, and the other the equation of the complex. The general complex is naturally [111111]; the complex of tangents to a quadric is [(111), (111)] and that of lines meeting a conic is [(222)]. Full information will be found in Weiler’s memoir, Math. Ann. vol. vii.
The detailed study of each variety of complex opens up a vast subject; we only mention two special cases, the harmonic complex and the tetrahedral complex.
The harmonic complex, first studied by Battaglini, is generated in an infinite number of ways by the lines cutting two quadrics harmonically. Taking the most general case, and referring the quadrics to their common self-conjugate tetrahedron, we can find its equation in a simple form, and verify that this complex really depends only on seventeen constants, so that it is not the most general quadratic complex. It belongs to the general type in so far as it is discussed above, but the roots of the determinant are in involution. The singular surface is the “tetrahedroid” discussed by Cayley. As a particular case, from a metrical point of view, we have L. F. Painvin’s complex generated by the lines of intersection of perpendicular tangent planes of a quadric, the singular surface now being Fresnel’s wave surface. The tetrahedral or Reye complex is the simplest and best known of proper quadratic complexes. It is generated by the lines which cut the faces of a tetrahedron in a constant cross ratio, and therefore by those subtending the same cross ratio at the four vertices. The singular surface is made up of the faces or the vertices of the fundamental tetrahedron, and each edge of this tetrahedron is a double line of the complex. The complex was first discussed by K. T. Reye as the assemblage of lines joining corresponding points in a homographic transformation of space, and this point of view leads to many important and elegant properties. A (metrically) particular case of great interest is the complex generated by the normals to a family of confocal quadrics, and for many investigations it is convenient to deal with this complex referred to the principal axes. For example, Lie has developed the theory of curves in a Reye complex (i.e. curves whose tangents belong to the complex) as solutions of a differential equation of the form (b − c)xdydz + (c − a)ydzdx + (a − b)zdxdy = 0, and we can simplify this equation by a logarithmic transformation. Many theorems connecting complexes with differential equations have been given by Lie and his school. A line complex, in fact, corresponds to a Mongian equation having ∞3 line integrals.
As the coordinates of a line belonging to a congruence are functions of two independent parameters, the theory of congruences is analogous to that of surfaces, and we may regard it as a fundamental inquiry to find the simplest form of surface into which Congruences. a given congruence can be transformed. Most of those whose properties have been extensively discussed can be represented on a plane by a birational transformation. But in addition to the difficulties of the theory of algebraic surfaces, a subject still in its infancy, the theory of congruences has other difficulties in that a congruence is seldom completely represented, even by two equations.
A fundamental theorem is that the lines of a congruence are in general bitangents of a surface; in fact, since the condition of intersection of two consecutive straight lines is ldλ + dmdμ + dndν = 0, a line l of the congruence meets two adjacent lines, say l1 and l2. Suppose l, l1 lie in the plane pencil (A1a1) and l, l2 in the plane pencil (A2a2), then the locus of the A′s is the same as the envelope of the a′s, but a2 is the tangent plane at A1 and a1 at A2. This surface is called the focal surface of the congruence, and to it all the lines l are bitangent. The distinctive property of the points A is that two of the congruence lines through them coincide, and in like manner the planes a each contain two coincident lines. The focal surface consists of two sheets, but one or both may degenerate into curves;
