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Volume 85, 1957-58
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– 473 –

Properties Concerning the Hessian Complex in Relation to Quadrics, Conics, and a Twisted Cubic

Abstract

This paper is concerned mainly with the interpretation and properties of the hessian complex; how it arises from a quadric defined by three generators, its relationship to a conic as a plane section of the quadric, and to a twisted cubic. Geometrical interpretations of many invariants and covariants arise naturally, some of which appear to be new.

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If the three generators are A, B, C the equation of the quadric may be written* as [x ABC x] = o or ×.Q.× = o where Q = A.B.C — C.B.A, that is, the symmetric part of A.B.C. The antisymmetric part leads naturally to the hessian complex H defined by H = Σ (A.B.C.+ C.B.A), where Σ denotes summation over cyclic permutations of A, B, C, which in turn reduces to H = (BC) A + (CA) B + (AB) C.

It is clear that all generators of the opposite system of the quadric to A, B, C belong to the complex. It is also clear that all generators of the quadric of the system A, B, C are given in terms of a parameter λ by

G = (1—λ) (BC) A + λ (CA) B + λ (λ—1) (AB) C,

and of these lines, two only belong to the hessian complex, namely those for which λ = — ω, and λ = — ω2, (where ω3 = 1) These two lines

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h1 = (BC) A + ω (CA) B + ω2 (AB) C,

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h2 = (BC) A + ω2 (CA) B + ω (AB) C,

are called the hessian pair of lines. In terms of H, h1, h2 the quadric takes a point equation of the form [x H h1 h2 x] = o and a line equation, (h1 h2) (pH)2 = (HH) (ph1) (ph2), where p is the variable line co-ordinate.

The hessian complex also gives harmonic properties: For if A1 is the polar of A with regard to the complex H, then A, A1 separate B, C harmonically, in particular A1 is given by

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A1 = — ½ (BC) A + (CA) B + (AB) C.

If B1, C1 are formed in the same way, it is of importance that the hessian complex formed from A1, B1, C1 is the original hessian complex H. The hessian line pair are the double lines of the involution defined by the pairs AA1, BB1, CC1. Alternatively the hessian complex may be defined as containing all the generators of the second system and two lines h1 h2 of the first system. The lines h1, h2, together with any pair of generators of the second system, form a skew quadrilateral whose diagonals are polar lines both with regard to the hessian complex and the quadric.

Now take a section of the quadric by an arbitrary plane π and, on the conic, denote the points where the generators A, B, C meet the plane by the same letters and similarly for the polars A1, B1, C1 and for h1, h2 Let H represent the pole of the plane π with respect to the complex H. Points on the conic will be given para-

[Footnote] * [x ABC x] ≡ xmu; Aμ ν Bνσ Cσρ xρ = (x a B) (x a′ C) − (x a′ B) (x a C) A.B.C. ≡ Aμν Bνσ Cσρ (BC) ≡ (b b′ c c′)

[Footnote] † It is realised that this complex is represented by a prime in 5 dimensional space, but it is of no special significance in the properties developed here.

– 474 –

metrically by the above λ and the parameters for the triad A, B, C will be the roots of a binary cubic. Similarly the triad A1. B1, C1 leads to the cubic covariant of this cubic, and h1, h2 correspond to the roots of the hessian quadratic. The lines AA1, BB1, CC1 meet in the point H which is also the pole with respect to the conic of the hessian line h1 h2. Since the same hessian complex arose from the lines A1, B1, C1, the hessian pair of its triad on the conic, derived from the cubic covariant, will be the same as the hessian pair derived from the triad A, B, C. The configuration in the plane is completed by the introduction of the three complexes (ABCp), (BCAp), (CABp), where* (CABp) = — (BC) (Ap) + (CA) (Bp) + (AB) (Cp). This complex contains the lines A, C and all the generators of the opposite system of the quadric. Taking the traces of these lines and complexes in the plane π, it is found that the tangents to the conic at B, C will belong to the complex (CABp) and hence these tangents will intersect in the pole O1 of the plane π with respect to this complex, and this point lies on the line AA1. Similarly for the other two complexes (ABCp), (BCAp), giving poles O2, O3; thus the lines AHO1, BHO2, CHO3, contain respectively the points A1, B1, C1, a well-known configuration for the conic. This follows immediately, since the four complexes are linearly related.

As an illustration of the application of the hessian complex, we may interpret the vanishing of the invariant,

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(BC) (AD) + (CA) (BD) + (AB) (CD)

that is, the line D belongs to the hessian complex H, and, incidentally, each of the four lines A, B, C, D belong to the hessian complex of the remaining three lines This implies that the two generators of the same system as A, B, C which meet D, say with parameters λ1, λ2, will separate harmonically, the two special lines h1, h2 whose parameters are —ω, —ω2. The two generators which meet D are given by the equation.

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(1 − λ) (BC) (AD) + λ (CA) (BD) + λ (λ − 1) (CA) (BD) = o,

or.

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λ2(CA) (BD) − λ{(BC) (AD) − (CA) (BD) + (CA) (BD)} + (BC) (AD) = 0

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or (λ − λ1) (λ − λ2) = o.

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Thus the cross-ratio {λ12; — ω, — ω2} = — 1 provided 2λ1 λ2 — λ1 — λ2 + 2 = o, that is, provided (AB) (CD) + (BC) (AD) + (CA) (BD) = o.

Further, the vanishing of the ring product (ABCD) implies that these two generators, harmonically separate the lines A, C. Again these two generators coincide if

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√(BC) (AD) + √(CA) (BD) + √(AB) (CD) = o,

which is the condition that D is a tangent or a generator of the quadric. Further properties have been derived by similar processes. We consider next the relationship between the twisted cubic through three points a, b, c whose tangents at these points are the lines A, B, C, the quadric which has the lines A, B, C as generators, and the corresponding hessian complex. An explicit equation for the quadric is obtained in terms of the plane [abc] and the three fundamental quadrics containing the twisted cubic. Interesting configurations and an interpretation of several invariants arise from a consideration of the two quadrics derived from the tangents at two triads of points on the twisted cubic, in particular, the cases when the two triads are apolar and when the triads form covariant sets. A particularly simple invariant arises when the two quadrics have a common generator, and some properties of the residual cubic curve are derived.

These properties lead to interpretations of all the fundamental invariants and covariants of the binary cubics defining the two triads; several of these are investiated.

[Footnote] *(CABp) is a ring product given by Cλμ Aμν Bνρ Pρλ

– 475 –

Any point k, of the twisted cubic, with parameter t is given by k = a + bt + ct2 + dt3 where the points a, d correspond to values o, ∞ of the parameter, the osculating plane at is [k k̇ k̈] where [k k k] = [a b c] − 3 [a d k] + [b c d] t3, (the upper dots denoting differentiation with respect to the parameter t); thus the osculating planes at a and d are [a b c] and [b c d] respectively, and clearly the point of intersection of the osculating planes at a, d, k is the point b + ct and this point also lies in the plane [a d k]. This point will be shown to be the pole (or trace) in the plane [a d k] of the hessian complex formed from the tangents to the cubic at a, d, k and also of the nul complex to which all the tangents to the cubic belong.

The tangent to the twisted cubic at a point k may be written as [k k̇] where [k k] = [a b] + 2t [a c] + 3t2 [a d] + t2 [b c] + 2t3 [b d] + t4 [c d], and hence all such tangents belong to a nul complex [b c] — 3 [a d]. Further, the tangents at a, d, k are [a b], [c d], [k k̇] respectively, and the hessian complex formed from these three lines is.

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H = (a b c d) { [a b] + [k k̇] + [c d] t4} (i)

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and its pole in the plane [a d k], namely [H a d k] reduces to b + t c. Hence the hessian pole H is the intersection of the three osculating planes at a, d, k and lies in the plane [a d k]. Also each osculating plane is the polar of a, d, k with respect to the nul complex and hence their point H of intersection is the pole of the nul complex with respect to the plane [a d k].

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More generally, the hessian complex H of the tangents at the points p, q, r of the cubic, with parameters λ, μ, ν is

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H = (μ − ν)4 [p p] + (ν − λ)4 [q q] + (λ − μ)4 [r r] (ii)

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and this contains the tangent at the point s with parameter σ provided Σ (λ — μ)4 (ν — σ)4 = o, and in virtue of the identity Σ (λ − μ) (ν − σ) = o this implies that if A, B, C, D are the line co-ordinates of the four tangents at the points λ, μ, ν, σ, then the invariant.

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{ (AB) (CD) }¼ + { (BC) (AD) }¼ + { (CA (BD) }¾

vanishes. However, as will be shown later, by fixing λ, μ, ν in (ii), this quartic in σ degenerates and gives the two tangents to the twisted cubic at the two hessian points of the cubic whose roots are λ, μ, ν.

Now consider the quadric Q whose generators are the tangents [p p], [q q], [r r] at the points p, q, r of the twisted cubic. A generator G of this quadric, with the parameter l is given by

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G = l (μ-ν)4 [pp] + l (l − 1) (ν-λ)4 [qq] + (l-1) (λ-μ)1 [rr](iii)

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and since [p p. p. q.r] = − (λ − μ)2 (λ − ν)2 (μ − ν) p with similar expressions for [q q . p q r], [r r . p q r]; the hessian pole H of the plane [p q r] is H = Σ (μ − ν)4 [pp . pqr] ≡ (μ − ν)3 p + (ν − λ)3 q + (λ − μ)3 r and G will meet the plane [p q r] in the point g where

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g = l (μ − ν)3 p. + l (l − 1) (ν − λ)3 q + (l − 1) (λ − μ)3 r ≡ × p + y q + z r

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where x-1 (μ − ν)3 + y-1 (ν − λ)3 + z-1 (λ − μ)3 = o(iv)

This gives the equation of the conic in which the quadric Q meets the plane [p q r] referred to the points p, q, r as base points.

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Further, we may obtain an explicit expression for the quadric whose generators are the tangents to the twisted cubic at the points p, q, r in terms of the plane ax3 ≡ a3 (x p) (x q) (x r) = o which meets the cubic in the points with parameters p, q, r. We have, in terms of the base points a, b, c, d,

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p = a + pb + p2 c + p3 d

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x = x0 a + x1 b + x2 c + x3 d ≡ a + xb + x2 c + x3 d

– 476 –

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then (p p. q x) = − (p-q)2 (p-x)2 (q-x),

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(p p. q̇ x) = 2 (p-q) (p-x)2 (q-x) − (p-q)2 (p-x)2

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Now Q = [x ṗ. p . q q . r ṙ x] = (x p. ṗ q̇) (q r ṙ x) − (x p. ṗ q̇) (q̇ r ṙ x), whence substituting in the values of the bracket factors and after some reduction, the quadric Q takes the simple symbolic form

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Q = 4 ax3 bx3 − 9 (x y)2 (a b)2 ax by (v) in which a, b and x, y are equivalent symbols respectively. The second term involves only the coefficients in the hessian quadratic ht2 = (a b)2 a̖ b̖ Written in full Q = (ax3)2 − 9 (h0 Q0 + h1 Q1 + h2 Q2) where a̖3 = a3t + 3 a2 z + 3 a1 y + a0 x. h0 = a0 a2 − a12, 2 h1 = a0 a3 − a1 a2, h2 = a1 a1a3 − a22, Q0 = × z − y2, Q1 = × t − y z, Q2 = y t − z2, and it is noticed that Q0 = 0, Q1 = 0, Q2 = 0 are three independent quadrics each containing the twisted cubic.

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Specialising these results, consider the points a, k, d on the twisted cubic, with parameters o, t, ∞ and the binary cubic τ (τ-t) = 0 (vi) with these parameters as roots. The hessian pair of the binary cubic is given by the roots − ωt, − ω2t and will give the hessian points H1, H2 on the twisted cubic. The join of the hessian pair H1 H2 will have the equation

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[H1 H2] ≡ [a b] + [a c] t + [b c] t2 + [b d] t3 + [c d] t4 (vii)

The generators h1, h2 of the quadric (v) formed from the tangents to the cubic at the points a, k, d and belonging to this system of generators and to the hessian complex, are found by assigning the values —ω, —ω2 to the parameter l in the equation (iii), namely

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G = l (c d k k) [a b] + l (l-1) (a b c d) [k k] + (1-l) (a b k k) [c d] ≡ l [a b] + l (l-1) [k k] + (1-l) [c d] t4

which meets the plane [a k d] in the point

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g = [a k d . G] ≡ l a + l-1) (-k) + (1-l) (t3 d);

hence substituting −ω, −ω2 for l it is found that the line joining the points in which the hessian pair h1, h2 meets the conic of intersection of the quadric with the plane [a k d] is.

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a − k + ω2 t3 d . ω2 a-k + ω t3 d] ≡ [k + t3 d -a + t3 d] thus [h1 h2]conte = t [a b] + t2 [a c] + 3 t3 [a d] + t4 [b d] + t5 [c d] (viii) and by comparison with [H1 H2]cubic it follows that,

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[H1 H2]cuble − [h1 h2] conte = t2 { [b c] − 3 [a d] } (ix)

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Hence these two hessian lines in space are pole and polar with respect to the nul complex of the tangents; in the same manner it may be shown that [H1 H2] + [h1 h2] = H and hence these two lines are also conjugate with respect to the hessian complex. Further, these two complexes are apolar and the two lines are therefore the directrices of the congruence common to the two complexes.

It is well known that, in general, four tangents of a twisted cubic belong to a general linear complex, but in the case of the hessian complex formed from the tangents at a, k d only two tangents of the cubic belong to this complex, and these are the tangents at the two hessian points H1, H2 on the cubic. Since these two tangents belong both to the nul complex and the hessian complex, they will intersect the directrices of their linear congruence, and consequently will also intersect the line joining h1 h2 on the conic section of the quadric by the plane [a k d]. (It is shown later that these two tangents pass through the points h1 h2.)

Consider now the cubic covariant of the cubic (vi) where for convenience, the parameter of k is taken as unity. This cubic covariant will be

2 τ3 − 3 τ2 − 3 τ + 2 = 0 (x)

It is well known that the three osculating planes at the points on the cubic curve whose parameters satisfy (x) will intersect in a point K such that the cross-ratio {H1 H2, H K} = -1. Denote by Q1, Q2 the quadrics whose generators

– 477 –

are the tangents to the twisted cubic at the three points (i) a, k, d whose parameters satisfy (vi) and (ii) l, m, n whose parameters satisfy (x). It is found that the hessian complex corresponding to the tangents at a k d is the same as the hessian complex corresponding to the tangents at l m n, and that the planes [a k d], [l m n] meet in the line h1 h2 and further that K is the pole of the hessian complex in the plane [l m n].

Now denote the conics of intersection of the quadrics Q1 Q2 with the plane [a k d] by C, C′ respectively; the hessian pole h of this plane will be the same for both conics. If the generators at l, m, n to Q2 meet the conic C′ in l′, m′, n′ then the symmetry between the cubic and the cubic covariant persists, namely, that the points given on the first conic C by a, k, d and their cubic covariant points a′ k′ d′ are such that the lines [a a′], [k k′],[d d″] are concurrent in h, but also l′ a′ h a l” are collinear, where l″ lies on the conic C′; similarly for m′, m″, n′, n” and l′ m′ n′ would correspond parametrically to a binary cubic on the conic C′ whilst l″ m” n″ would represent its cubic covariant.

Exactly the same property holds if the quadrics Q1, Q2 are intersected by the plane [l m n], a very interesting reciprocity between the binary cubic on the twisted cubic and the two pairs of conics; incidentally this leads to a configuration similar to that in the above section.

In particular from (v) the quadrics Q1, Q2 reduce to the simple forms.

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Q1 = Qx + 2 P12, Q2 = 27/4 Q̖ + ½ P22,

where P1 P2 are the planes [a k d], [l m n] respectively, so that.

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P1 = y-z, P2 = 2 × − 3 y − 3 z + 2 t

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and Q̖ = 2 (t y − z2) − (t x-y z) + 2 (xz − y2)

Thus the quadrics Q1, Q2 have double contact with the quadric Qx along the two conics C1, C2 in which Qx is cut by the planes P1, P2.

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By considering the characteristic equations the following properties arise, (a) Cones of the pencil Q1 + λ Q2 have vertices (i) on the line [h, h2], (ii) at the points H, K. For the pencil 8 Q2 + 2 λ Q1, the characteristic roots are (1) λ = − - 27 (twice) giving the line h1 h2, (ii) λ = − 3 giving H, and (iii) λ = − 234 giving K.

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(b) The pencil 2 Qx + λ P1 P2 contains the conics C1, C2 and specializes, for λ = ± √3 into two cones. Let O1, O2 be the vertices of these two cones, then by suitable weighting,

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O1 + O2 = K where K = 2a + b − c − 2d,

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O1 − O2 = H. H.= √3 (b + c)

and the hessian pair H1 H2 on the cubic are given by.

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H1 = a − ω b + ω2 c − d, H2 = a − ω2 b + ω c − d

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whence H1 = H2 = K and H1 − H2 = 1 H,

verifying that H1 H2 H K are collinear and form a harmonic range.

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(c) The planes P1, P2 are conjugate with respect to the quadric Qx since the coefficient of λ in |2 Q̖ − λ P1 P2 | = 0 gives the invariant relation {Qx3, P1 P2} 0 or “θ” = 0. The whole configuration gives a very compact closed system.

Various Invariants and Covariants

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If the point × (x0 x1 x2 x3) is written symbolically as (l, x, x2, x3) then by equating the coefficients of the various powers of t the identity at3 ≡ (tx)3 defines a point x ≡ (a3, − a2, a1, − a0) if at3 = 0 is the plane of the three points p, q, r of the twisted cubic, then the new point so defined, gives the point of intersection of the osculating planes at p, q, r, it is also the pole of the hessian complex formed from the three tangents to the cubic at p, q, r in the plane [p q r] and also the pole of the nul complex in this plane.

If, further, the two cubics at3, b0t are apolar so that (a b)3 = 0 or.

a3 b0 − 3 a2 b1 + 3 a1 b2 − a0 b3 = 0

which immediately shows that the hessian pole for the plane 3 = 0 lies in the

– 478 –

plane bx3 = 0 and since the relation is symmetrical, the hessian pole for the plane bx3 = 0 lies in the plane ax3 = 0. In general there is one cubic which is apolar to three given cubics, and this cubic is clearly given by the plane through the hessian poles of the three cubics.

Taking three points a, k, d on the twisted cubic to have parameters o, k, ∞ with respect to both the twisted cubic and also the conic of intersection of the quadric, with the tangents at a, k, d as generators, with the plane [a k d] there is a (l, l) correspondence between the points of the cubic (l, λ, λ2, λ3) and the points of the conic given by k (k − λ) a + λ k (-k) + λ (λ − k) (k3 d) and it follows easily that the tangent plane to the quadric at the point λ of the come passes through the point λ on the cubic; in fact the equation of the tangent plane is (x λ) ax2 aλ = 0 and this plane cuts the twisted cubic in two further points whose parameters are given by the roots of the quadratic at2 aλ = 0 which provides an interpretation of the first polar at2 aλ. It should also be noticed that the chord of the cubic through these two points meets the plane [a k d] in the point λ of the conic. It is of interest that the quadratic in t, at2 aλ becomes a perfect square provided its hessian quadratic hλ2 = (a a′) aλ a′λ vanishes. Hence it follows that the tangent plane to the quadric at one of the hessian points h1, on the conic, will pass through the hessian point H1 on the cubic, and this tangent plane will also meet the cubic again in two points which are coincident at the second hessian point H2; hence the tangents at H1, H2 to the twisted cubic will meet the plane ax3 = 0 in the hessian points h1, h2 respectively of the conic.

Again, by expansion

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(t λ) at2 aλ = (λ a3 + a2) t3 + (λ a2 + 2 a1 − λ2 a3) t2 + (a0 − 2 λ2 a2 − λ a1) t − λ2 a1 − λ a0

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whence the point of intersection of the tangent planes to the quadric at the points λ, μ, ν of the conic is given by the three column minors (a3 a2 − a1 − a0 a2 2a1 a0 0 0 a3 2 a2 a1) and these co-ordinates are in fact those derivable from the covaniant cubic: for writing (t x)3 ≡ (a b)2 (a c) bt ct2, (a, b, c being equivalent symbols) then (x0 x1 x2 x3) takes these required values. This point is also the pole of the plane ax3 = 0 with respect to the quadric whose generators are the tangents to the cubic at the points in which the cubic is met by the plane ax3 = 0; hence given a triad of points at3 = 0, the plane of the covariant triad passes through the pole of the plane ax3 = 0 with respect to the quadric, and further this pole is the pole of the covariant plane Cx3 = 0 with respect to the hessian complex, and with respect to the nul complex. If any plane dx3 = 0 is apolat to Cx3 = 0, it passes through the pole of this plane with respect to the quadric, so that the plane dx3 = 0 and the covariant plane of Ct3 = 0 are then conjugate with respect to the quadric. If two triads are apolar, the plane of one triad and the plane of the covariant triad of the second triad are conjugate with respect to the quadric whose generators are the tangents to the cubic at the points of the second triad.

Further, the tangent planes to the quadric at the points λ, μ of the conic will meet the twisted cubic in points with parameters λ, μ and in a further four points, it follows easily that if the cubics giving the two tangent planes are apolar, then these four points form an harmonic range.

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Again, the three tangents to the twisted cubic at the points a, k, d are the generators of a quadric: consider the three generators of the quadric of the opposite system and passing through a, k, d. They are easily found to be [a, c + 2 k d], [k, a + k b], [2 a + k b, d]

respectively, and it is of interest that their hessian complex is the nul complex itself.

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Further these three new generators define an associated twisted cubic through three points a k d which has these three generators as the tangents at a, k, d and the parameterization on this cubic can be chosen so that a, k, d have parameters o, k, ∞. Hence a correspondence is set up between points T of the original cubic and points T* of the associated cubic such that T, T* correspond to the same parameter. It is found that the join of TT* passes through a fixed point σ, the pole of the plane [a k d] with respect to the quadric (and also the hessian pole of the plane of the covariant points) and that if TT* meets the plane [a k d] in u then the range {TT*, u σ} is harmonic. An involutory collineation connects T and T* and is given by (π σ) T* = T − 2 σ (π T) where π is the plane [a k d].

Consider next the possibility that any two quadrics, one (say) with the tangents at the points p, q, r of the cubic as generators, and the other with the tangents at l, m, n of the cubic as generators, should possess a common generator. Let the parameters corresponding to these points on the cubic be p, q, r; l, m, n respectively. Since the tangent at p is given by.

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[p p] = [a b] + 2 p. [a c] + p2 {3 [a d] + [b c]} + 2 p3 [b d] + p4 [e d]

and all the tangents belong to the nul complex 3 [a d] − [b c], it follows that a linear relation exists between any six tangents, which can therefore be written, [P Q L M N] R + [Q R L M N] P + [R P L M N] Q = [L M P Q R]N + [M N P Q R]L + [N L P Q R]M (xi)

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where [P Q L M N] is the determinant |1, q, l2, m3, n4|.

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If the left side of this equation is special, then so also is the right side, and either side then gives the common generator of the two quadrics formed from the lines P, Q, R and the lines L, M, N. Writing φ (q) ≡ (q-I) (q-m) (q-n) and Δ = (l-m) (m-n) (n-l) then [Q R L M N] = Δ (q-r) φ (q) φ (r) so that we have to make special the complex.

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q-r/φ (p)) + r-p/φ (q) Q + p-q/φ (r) R, (xii)

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but a complex of the form α (QR) P + β (R P) Q + γ (P Q) R where (P Q) = (p-q)4 is special provided 1/α + 1/β + 1/γ = 0 or Σ (q-r)3 φ (p) =); Writing a3 0 (p) = ap3 so that ax3 = 0 is the plane of l, m, n the condition becomes Σ (q-r)3 ap3 = 0, and reduces immediately to.

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(q-r) (r-p) (p-q) ap ar ai = 0

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Finally, writing b13 = b3 (t-p) (t-q) (t-r) so that b13 = 0 is the plane of the points p q, r then ap aq ar = a3 p. q r + a2 Σ q r + a1 Σ p. + a0 ≡ a3 b0 − 3 a2 b1 + 3 a1 b2 − a0 b3 = (a b)3

Hence the two quadrics will have a common generator provided the two triads at3 = 0, bt3 = 0 are apolar.

Further, if the two triads at3 = 0, bt3 = 0 are apolar, then any pair of members of the pencil at3 + k bt3 = 0 will be apolar and the pair of corresponding quadrics (belonging to a single-parameter quadratic family) will have a common generator, which can be shown, by a determination of this generator, to be common to the whole family.

Any generator of one system of the quadric which has the tangents [p p], [q q], [r r] as generators, can, in terms of a parameter μ be written as.

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1/(q-r) (μ-p) (q q r r) [p p] + 1/(r-p) (μ-q) (r r p. p) [q q] + 1/(p-q) (μ-r) (p p. q q) [r r]

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and by identifying this with the right side of equation (xi) and after some calculation, it is found that the parameter μ is given by the transvectant, bμ (b h)2 = 0 where ht2 is the hessian quadratic (a a′)2 at a′t formed from the cubic at3.

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This provides a geometrical determination of the parameter μ; for if u, ν are the roots of the hessian quadratic ht2 = 0 then bμ (b h)2 = bμ bu bv = 0 so that the plane of the points on the twisted cubic, is apolar to the plane bx3 = 0. Hence the plane through the hessian points H1, H2 on the cubic and apolar to bx3 = 0, that is, also passing through the hessian pole in the plane bx3 = 0, will meet the twisted cubic again in the point whose parameter is μ. The corresponding point on the conic in the plane bx3 = 0 gives a point on the common generator.

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Also, since aμ (a h)2 ≡ 0, it follows that if at3 and bt3 are apolar, at3 + k bt3 is apolar to each of at3 and bt3 and that the quadric formed from the triad at3 + k bt3 = 0 has a generator in common with the quadric formed from at3 = 0 given by bμ (b h)2 = 0 and this generator is clearly independent of the value of k, and hence all the quadrics of the family have a generator in common.

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In particular, taking the planes ax3 = 0, bx3 = 0 as the planes [a k d], [p q r] respectively, the value of μ for the common generator reduces to − b0/ (k2 b3) and, with points a, k, d as points of reference, this generator meets the plane [a k d] in the point (x′, y′, z′) where.

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(x′)-1: (y′)-1: (z′)-1 = b0: − (b0 + k3 b3): k3 b3

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and in these co-ordinates the equation of the line of intersection of the two planes becomes x/x′ + y/y′ + z/z′ = 0, a second polar of (x′, y′, z′) with respect to the cubic × y z = 0.

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As a further development, the equation of the residual cubic of intersection of the two quadrics may be deduced. Returning to the original base points a, b, c, d the quadric derived from the tangents at a, k, d has the equation 2 k × z − k × t − 3 k y z + 2 y t = 0 and any point on this quadric is given in terms of parameters λ, μ by x/2λ-k = y/λk = z/μk2 = t/μk2 (2k−λ) where λ = μ = 0 gives a, λ = μ = k gives k, λ = μ = ∞ gives d, and μ = constant corresponds to generators of the system [a b], [k k], [c d]. Inserting these values in the quadric derived from bt3 = 0, namely.

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(b0 × + 3 b1 y + 3 b2 z + b3 t)2 − 9 [h2 (y t-z2) + h1 (x t-yz) + h0 (xz-y2)] = 0

and using the apolar condition (a b)3 = 0 it is found (after some reduction) that there is a factor b0 + μ k2 b3 whose vanishing gives of course the common generator, and leaving a (2, 1) relation between λ, μ which therefore gives the residual twisted cubic, namely.

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b0 (2μ-k)2 + 3 b2 λ k2 (2k−λ) + μ k2 [b3 (2k−λ)2 − 3 b2 (2λ-k)] = 0

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By setting λ = μ the residual cubic meets the plane [a k d] in three points whose parameters on the conic are given by (c a)33 − 4 (b c)3 aλ3 = 0 where ct3 is the cubic covariant of at3 and since bt3 are apolar, this triad of points is therefore apolar to both at3 = 0, bt3 = 0 and the plane (c a)3 bx3 − 4 (b c)3 ax3 contains the line of intersection of the planes.

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It may also be noticed that if the common generator meets the residual cubic in two points given by λ = λ1, λ = λ2 the generators with parameters μ1, μ2 of the system opposite to [a b], [k k], [c d] will meet the conic in the plane [a k d] in two points which are easily shown to be collinear with the hessian pole in the plane.

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The condition that the two hessian completes H, K formed from the tangents to the twisted cubic at the points of the two triads given by at3 = 0, bt3 = 0 bt3 = 0

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may be apolar involves the invariant of the hessian quadratics ht2 = (a a′)2 at at′, kt2 = (b b′)2 bt bt′ and may be written 4 (h h′)2 (k k′)2 = 3 { (h k)2 }2, h h′, k k′ being equivalent symbols.

References

Babbage. Journal London Math. Soc. (25). 1950.

Forder. Calculus of Extension. p. 132.

Grace and Young. Algebra of Invariants, p. 196, pp. 240–2.

Meyer. Apolarital und rationale Curven, p. 463.

Turnbull. Theory of Determmants, Matrices and Invariants, chapter XII