ALGEBRAIC FORMS - APOLARITY For geometry, co-ordinates or variables all of one kind are not sufficient. The theory of duality demands, in a plane, point co-ordinates and line co-ordinates; and, in three-space, points and planes are dual, while lines, dual to lines, require a third kind of co-ordinates. A polarity of two forms is then an important relation. Ternary forms in contragredient sets of variables (x) and (u) are defined as apolar by the aid of an operator (for all x and u).
Important applications of this notion to binary forms are connected with canonical or normal forms of quantics of odd order, and with a special kind of quantic of even order. A binary quantic of even order is equivalent to a sum of n perfect 2nth powers of linear forms if its catalecticant vanishes. A (2n+ i)-ic is reducible to the sum of n-}- i perfect (2n+ i)th powers of linear forms, the linear factors of the canonizant. As to the latter, if the given form is and (33x) are the selected covariants. For a binary form of even order three quadric covariants are needed, and these can be found if the order is foregoing four; call them and the form Then the expansion of a four-rowed determinant gives an identity in the symbol and an nth power yields the desired typical representation:— The linear covariants constitute a most interesting meeting point of geometry and algebra. (F. Morley in Math. Ann. vol. 49.) The foregoing example of an even form and quadric covariants is also an example of geometric picturing of a binary form and its comitants upon a rational algebraic curve in two or more di mensions,—in this case on a curve of the second order in a plane. For, if we set = elimination of gives the equation of a conic in 63, while is trans formed into a ternary n-ic, and this, equated to zero, cuts out, on that conic, a set of 2n points, the picture of the 2n zero points of in the binary domain. Further, a unique curve of order n, can be determined, cutting out those 2n points on the conic, and itself apolar to the conic. Some use has been made of this style of picturing both on the plane conic and on the gauche curve of order three in three-space. (F. Meyer, Apolaritat and rationale Curven, Tubingen, 1883.) A sample of the relations, more easily discoverable in such geometric pictures than in algebraic symbols, is this theorem on three binary quadrics. Any three linearly independent quadrics being given (no one of which is the Jacobian of the two others), every fourth quadric is linearly related to them, with constant co efficients. But there is a unique fourth quadric, determined by the requirement that its square shall be a linear combination of the squares of the first three, with constant coefficients. Then any three of these four quadrics, taken in pairs, determine three Jacobians or functional determinants, covariant quadrics, which are related to the fourth of the original set in the same way as are the first three. (Meyer, Apolaritdt and rationale Curven, p. 244.) That is, if we use f /2, /3, /4 to denote the four quadrics, and write J12, for example, for the Jacobian of f 1 and /2,— and 2i— is identically = o on account of n+ r zero factors in which i= k. From the n+ z implied equations of condition, and from o, elimination of the n+2 coefficients of yields the canonizant in the form of a determinant. This must be resolved into its linear factors, which must all be distinct, and then the n-}- i coefficients can be determined from linear equations.
For a binary form of even order, similar reasoning shows that it is reducible to the sum of n+ z perfect 2nth powers of linear forms, one of them an arbitrary linear form; or to the sum o? n such powers if a determinant of n rows, the catalecticant, is equal to zero.
The so-called typical representation of binary forms expresses forms of odd order in terms of any two linear covariants, with coefficients automatically invariant. The symbolic identity [(a13) (ax) ]n= [(a(3) (ax) — (aa) (!3x) In exhibits this, when (ax) the in's denoting numbers, then there will be also an identical relation This is more interesting when it is noticed that the f's are neces sarily proportional to the Jacobians of the J's: J(J12, J23) : J(J23, J31) etc., for then it exhibits connections in a finite closed system.