# Quartic Sentence Examples

- The name lemniscate is sometimes given to any crunodal
**quartic**curve having only one real finite branch which is symmetric about the axis. - 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 Binary
**Quartic**.-The fundamental system consists of five forms ax=f; (f,f')2=(ab) 2axbx=Ax; (f,f')4=(ab) 4= 2; (f, 0)1= (ao) azsi = (ab) 2 (cb) a:b x c5 =1; (f 4) 4 = (as) 4 = (6) 2 00 2 (ca) 2 = j, viz. - The
**quartic**has four equal roots, that is to say, is a perfect fourth power, when the Hessian vanishes identically; and conversely. - 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. - 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. - For it is easy to establish] the formula (yx) 2 0 4 = 2f.4-2(f y 1) 2 connecting the Hessian with the
**quartic**and its first and second polars; now a, a root of f, is also a root of Ox, and con se uentl the first polar 1 of of q y p f? - 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. - We have A +k 1 f =0 2, O+k 2 f = x2, O+k3f =4) 2, and Cayley shows that a root of the
**quartic**can be xpressed in the determinant form 1, k, 0.1y the remaining roots being obtained by varying 1, k, x the signs which occur in the radicals 2 u The transformation to the normal form reduces 1, k 3, ? - The
**quartic**to a quadratic. The new variables y1= 0 are the linear factors of 0. - The transformation to the normal form, by the solution of a cubic and a quadratic, therefore, supplies a solution of the
**quartic**. If (Xï¿½) is the modulus of the transformation by which a2 is reduced to 3 the normal form, i becomes (X /2) 4 i, and j, (Ap) 3 j; hence? - There are four invariants (i, i')2; (13, H)6; (f2, 151c.; (f t, 17)14 four linear forms (f, i 2) 4; (f, i 3) 5; (i 4, T) 8; (2 5, T)9 three quadratic forms i; (H, i 2)4; (H, 23)5 three cubic forms (f, i)2; (f, i 2) 3; (13, T)6 two
**quartic**forms (H, i) 2; (H, 12)3. - 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. - F= ai; the Hessian H = (ab) 2 azbx; the
**quartic**i= (ab) 4 axb 2 x; the covariants 1= (ai) 4 ay; T = (ab)2(cb)aybyci; and the invariants A = (ab) 6; B = (ii') 4 . - 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. - 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. - From the invariant a2 -2a 1 a 3 -2aoa4 of the
**quartic**the diminishing process yields ai-2a 0 a 21 the leading coefficient of the Hessian of the cubic, and the increasing process leads to a3 -2a 2 a 4 +2a i a 5 which only requires the additional term-2aoa 6 to become a seminvariant of the sextic. A more important advantage, springing from the new form of S2, arises from the fact that if x"-aix n- +a2x n-2. - Again, for the cubic, we can find A3(z) - -a6z6 1 -az 3.1 -a 2 z 2.1 -a 3 z 3.1 -a4 where the ground forms are indicated by the denominator factors, viz.: these are the cubic itself of degree order I, 3; the Hessian of degree order 2, 2; the cubi-covariant G of degree order 3, 3, and the
**quartic**invariant of degree order 4, o. - Similarly for the
**quartic**A 4 /z) - -a s z 1 -az4.1 -a2.1-a2z4.1-a3 .1 establishing the 5 ground forms and the syzygy which connects them. - These latter forms are enumer ated by I - z 24 I -z 4; hence the generator of
**quartic**perpetuants must be z4 z4 z7 1-z 2.1 -z 3.1z 4 1-z 2.1-s 4 1-22.1-z3.1-z4' and the general form of perpetuants is (4 K+ 1 3A+1 2ï¿½). - 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. - Clebsch to take the form T= 2p(x12 +x22)+2p'x32 + q (xiyi +x2y2) +q'x3y3 +2r(y12+y22)+2r'y32 so that a fourth integral is given by dy 3 /dt = o, y = constant; dx3 (4 y) (q + y) _ (y y) dt - xl 'x2 xl Y Y x l 2 - 1, y2 () = (x12 +x22) (y12 + y22) = (X 1 2 + X 2) +y22)-(FG-x3y3)2 = (x 1 y32-G2)-(Gx3-Fy3) 2, in which 2 = F 2 -x3 2, x l y l +x2y2 = FG-x3y3, Y(y1 2 +y2 2) = T -p(x12 +x22) -p'x32 -2q(xiyi 'x2y2)- 2 q ' x = (p -p') x 2 + 2 (- q ') x 3 y 3+ m 1, (6) m1 = T 2 i y 3 2 (7) so that dt3) 2 =X3, (8) where X3 is a
**quartic**function of x3, and thus t is given by an elliptic (8) (6) (I) integral of the first kind; and by inversion x 3 is in elliptic function of the time t. - If all the roots of the
**quartic**in FIG. - In the third-order complex the centre locus becomes a finite closed
**quartic**surface, with three (one always real) intersecting nodal axes, every plane section of which is a trinodal**quartic**. The chief defect of the geometrical properties of these bi-quaternions is that the ordinary algebraic scalar finds no place among them, and in consequence Q:1 is meaningless. - 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. - A
**quartic**curve has 24 inflections; it was conjectured by George Salmon, and has been verified by H. - 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. - A very remarkable theorem is established as to the double tangents of such a
**quartic**: distinguishing as a double tangent of the first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double tangents of the first kind of a non-singular**quartic**is =4; it follows that the**quartic**has at most 8 real inflections. - Similarly a cubic through the two circular points is termed a circular cubic; a
**quartic**through the two points is termed a circular**quartic**, and if it passes twice through each of them, that is, has each of them for a node, it is termed a bicircular**quartic**. Such a**quartic**is of course binodal (m = 4, 6= 2, K = o); it has not in general, but it may have, a third node or a cusp. Or again, we may have a**quartic**curve having a cusp at each of the circular points: such a curve is a " Cartesian," it being a complete definition of the Cartesian to say that it is a bicuspidal**quartic**curve (m= 4, 6 = o, K= 2), having a cusp at each of the circular points. - 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. - There will be from each circular point X tangents (X, a number depending on the class of the curve and its relation to the line infinity and the circular points, 2 for the general conic, 1 for the parabola, 2 for a circular cubic, or bicircular
**quartic**, &c.); the X tangents from the one circular point and those from the other circular point intersect in X real foci (viz. - If D =t, then the transformed curve is a cubic; it can be shown that in a cubic, the axes of co-ordinates being properly chosen, 4) can be expressed as the square root of a
**quartic**function of 0; and the theorem is that the co-ordinates x, y, z of a point of the bicursal curve can be expressed as proportional to rational and integral functions of 0, and of the square root of a**quartic**function of 0. - 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. - 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.