Quartic Sentence Examples
The quartic has four equal roots, that is to say, is a perfect fourth power, when the Hessian vanishes identically; and conversely.
The expression (ab) 4 properly appertains to a quartic; for a quadratic it may also be written (ab) 2 (cd) 2, and would denote the square of the discriminant to a factor pres.
For the quartic (ab) 4 = (aib2-a2b,) alb2 -4a7a2blb2+64a2 bib2 - 4a 1 a 2 b 7 b 2 + a a b i = a,a 4 - 4ca,a 3 +6a2 - 4a3a3+ aoa4 = 2(a 0 a 4 - 4a1a3 +e3a2), one of the well-known invariants of the quartic.
The vanishing of the invariants i and j is the necessary and sufficient condition to ensure the quartic having three equal roots.
On the one hand, assuming the quartic to have the form 4xix 2, we find i=j=o, and on the other hand, assuming i=j=o, we find that the quartic must have the form a o xi+4a 1 xix 2 which proves the proposition.Advertisement
The quartic will have two pairs of equal roots, that is, will be a perfect square, if it and its Hessian merely differ by a numerical factor.
The simplest form to which the quartic is in general reducible is +6mxix2+x2, involving one parameter m; then Ox = 2m (xi +x2) +2 (1-3m2) x2 ix2; i = 2 (t +3m2); j= '6m (1 - m) 2; t= (1 - 9m 2) (xi - x2) (x21 + x2) x i x 2.
It is on a consideration of these factors of t that Cayley bases his solution of the quartic equation.
Certain convariants of the quintic involve the same determinant factors as appeared in the system of the quartic; these are f, H, i, T and j, and are of special importance.
Thus the ternary quartic is not, in general, expressible as a sum of five 4th powers as the counting of constants might have led one to expect, a theorem due to Sylvester.Advertisement
This is of degree 8 in the coefficients, and degree 6 in the variables, and, for the canonical form, has the expression -9m 6 (x 3 +y 3 +z 3) 2 - (2m +5m 4 +20m 7) (x3 +y3+z3)xyz - (15m 2 +78m 5 -12m 8) Passing on to the ternary quartic we find that the number of ground forms is apparently very great.
A relative stream line, along which 1/,' = Uc, is the quartic curve y-c=?![2a(r-x)], x = 4a2y2-(y g)4, r- 4a2y2 +(y c) 4, 7) 4 a (y-c) 4a(y and in the absolute space curve given by 1', dy= (y- c)2, x= 2ac_ 2a log (y -c) (8) 2ay y - c 34.
A curve of the third order is called a cubic; one of the fourth order a quartic; and so on.
By means of Pliicker's equations we may form a table - The table is arranged according to the value of in; and we have m=o, n= r, the point; m =1, n =o, the line; m=2, n=2, the conic; of m = 3, the cubic, there are three cases, the class being 6, 4 or 3, according as the curve is without singularities, or as it has 1 node or r cusp; and so of m =4, the quartic, there are ten cases, where observe that in two of them the class is = 6, - the reduction of class arising from two cusps or else from three nodes.
We may further consider the inflections and double tangents, as well in general as in regard to cubic and quartic curves.Advertisement
A quartic curve has 28 double tangents, their points of contact determined as the intersections of the curve by a curve II = o of the order 14, the equation of which in a very elegant form was first obtained by Hesse (1849).
The theory of the invariants and covariants of a ternary cubic function u has been studied in detail, and brought into connexion with the cubic curve u = o; but the theory of the invariants and covariants for the next succeeding case, the ternary quartic function, is still very incomplete.
A non-singular quartic has only even circuits; it has at most four circuits external to each other, or two circuits one internal to the other, and in this last case the internal circuit has no double tangents or inflections.
The circular cubic and the bicircular quartic, together with the Cartesian (being in one point of view a particular case thereof), are interesting curves which have been much studied, generally, and in reference to their focal properties.
And so if D =2, then the transformed curve is a nodal quartic; 4 can be expressed as the square root of a sextic function of 0 and the theorem is, that the co-ordinates x, y, z of a point of the tricursal curve can be expressed as proportional to rational and integral functions of 0, and of the square root of a sextic function of 0.Advertisement
Observe that the radical, square root of a quartic function, is connected with the theory of elliptic functions, and the radical, square root of a sextic function, with that of the first kind of Abelian functions, but that the next kind of Abelian functions does not depend on the radical, square root of an octic function.
Zeuthen in the case of curves of any given order establishes between the characteristics pc, v, and 18 other quantities, in all 20 quantities, a set of 24 equations (equivalent to 2 3 independent equations), involving(besides the 20 quantities) other quantities relating to the various forms of the degenerate curves, which supplementary terms he determines, partially for curves of any order, but completely only for quartic curves.
Quartic gage boson couplings are implemented according to the Standard Model only (no anomalous couplings ).
Ferrari managed to solve the quartic with perhaps the most elegant of all the methods that were found to solve this type of problem.
The final book presents the solution of cubic and quartic equations.Advertisement
They were unanimous in their distaste for the Quartic steering wheel.
This is the case for all objects that involve the solution of a cubic or quartic polynomial.
Zevenbergen and Thorne (1987) use a partial quartic expression to model a 3 by 3 local neighborhood.
The name lemniscate is sometimes given to any crunodal quartic curve having only one real finite branch which is symmetric about the axis.