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Encyclopedia  A parabola  A parabola obtained as the intersection of a cone with a plane.

In mathematics, the parabola (pronounced /pəˈræbələ/, from the Greek παραβολή) is a conic section, the intersection of a right circular conical surface and a plane to a generating straight line of that surface. Given a point (the focus) and a corresponding line (the directrix) on the plane, the locus of points in that plane that are equidistant from them is a parabola.

The parabola has many important applications, from automobile headlight reflectors to the design of ballistic missiles. They are frequently used in physics, engineering, and many other areas.

History  Parabolic compass designed by Leonardo da Vinci.  Parabolae are conic sections.

The earliest known work on conic sections was by Menaechmus in the fourth century B.C.. He discovered a way to solve the problem of doubling the cube using parabolae. (The solution, however, does not meet the requirements imposed by compass and straightedge construction.) The name "parabola" is due to Apollonius, who discovered many properties of conic sections. The focus–directrix property of the parabola and other conics is due to Pappus.

Galileo showed that the path of a projectile follows a parabola, a consequence of uniform acceleration due to gravity.

The idea of using a paraboloid in a reflecting telescope is due to James Gregory in 1663 and the first to be constructed was by Isaac Newton in 1668. The same principle is used in satellite dishes and radar receivers.

Equation in Cartesian coordinates

Let the directrix be the line x = −p and let the focus be the point (p, 0). If (xy) is a point on the parabola then, by Pappus' definition of a parabola, it is the same distance from the directrix as the focus; in other words: $x+p=\sqrt{(x-p)^2+y^2}.$

Squaring both sides and simplifying produces $y^2 = 4px\,$

as the equation of the parabola.

By translation, the general equation of a parabola with a horizontal axis is $(y-k)^{2}=4p(x-h)\,$

and interchanging the roles of x and y gives the corresponding equation of a parabola with a vertical axis as $(x-h)^{2}=4p(y-k).\,$

The last equation can be rewritten $y=ax^2+bx+c\,$

so the graph of any function which is a polynomial of degree 2 in x is a parabola with a vertical axis.

More generally, a parabola is a curve in the Cartesian plane defined by an irreducible equation of the form $A x^{2} + B xy + C y^{2} + D x + E y + F = 0 \,$

such that $B^{2} = 4 AC,\,$

where all of the coefficients are real, neither A nor B is zero and more than one solution exists, defining a pair of points (x, y) on the parabola. That the equation is irreducible means it does not factor as a product of two not necessarily distinct linear equations.It is very difficult to solve.

Other geometric definitions

A parabola may also be characterized as a conic section with an eccentricity of 1. As a consequence of this, all parabolae are similar, meaning that while they can be different sizes, they are all the same shape. A parabola can also be obtained as the limit of a sequence of ellipses where one focus is kept fixed as the other is allowed to move arbitrarily far away in one direction. In this sense, a parabola may be considered an ellipse that has one focus at infinity. The parabola is an inverse transform of a cardioid.

A parabola has a single axis of reflective symmetry, which passes through its focus and is perpendicular to its directrix. The point of intersection of this axis and the parabola is called the vertex. A parabola spun about this axis in three dimensions traces out a shape known as a paraboloid of revolution.

The parabola is found in numerous situations in the physical world (see below).

Equations

(with vertex (h, k) and distance p between vertex and focus - note that if the vertex is below the focus, or equivalently above the directrix, p is positive, otherwise p is negative; similarly with horizontal axis of symmetry p is positive if vertex is to the left of the focus, or equivalently to the right of the directrix)

Cartesian

Vertical axis of symmetry $(x - h)^2 = 4p(y - k) \,$ $y =\frac{(x-h)^2}{4p}+k\,$ $y = ax^2 + bx + c \,$ $\mbox{where }a = \frac{1}{4p}; \ \ b = \frac{-h}{2p}; \ \ c = \frac{h^2}{4p} + k; \ \$ $h = \frac{-b}{2a}; \ \ k = \frac{4ac - b^2}{4a}$. $x(t) = 2pt + h; \ \ y(t) = pt^2 + k \,$

Horizontal axis of symmetry $(y - k)^2 = 4p(x - h) \,$ $x =\frac{(y - k)^2}{4p} + h;\ \,$ $x = ay^2 + by + c \,$ $\mbox{where }a = \frac{1}{4p}; \ \ b = \frac{-k}{2p}; \ \ c = \frac{k^2}{4p} + h; \ \$ $h = \frac{4ac - b^2}{4a}; \ \ k = \frac{-b}{2a}$. $x(t) = pt^2 + h; \ \ y(t) = 2pt + k \,$

General parabola

The general form for a parabola is $(Ax+By)^2 + Cx + Dy + E = 0 \,$

This result is derived from the general conic equation given above: $Ax^2 +Bxy + Cy^2 + Dx + Ey + F = 0 \,$

and the fact that, for a parabola, $B^2=4AC \,$.

Latus rectum, semi-latus rectum, and polar coordinates

In polar coordinates, a parabola with the focus at the origin and the directrix parallel to the y-axis, is given by the equation $r (1 + \cos \theta) = l \,$

where l is the semilatus rectum: the distance from the focus to the parabola itself, measured along a line perpendicular to the axis. Note that this is twice the distance from the focus to the vertex of the parabola or the perpendicular distance from the focus to the latus rectum.

The latus rectum is the chord that passes through the focus and is perpendicular to the axis. It has a length of 2l.

Gauss-mapped form

A Gauss-mapped form: (tan2φ,2tanφ) has normal (cosφ,sinφ).

Derivation of the focus  Parabolic curve showing directrix (L) and focus (F). The distance from a given point Pn to the focus is always the same as the distance from Pn to a point Qn directly below, on the directrix.  Parabolic curve showing arbitrary line (L), focus (F), and vertex (V). L is an arbitrary line perpendicular to the axis of symmetry and opposite the focus of the parabola from the vertex (i.e. farther from V than from F.) The length of any line F - Pn - Qn is the same. This is similar to saying that a parabola is an ellipse, but with one focal point at infinity.

To derive the focus of a simple parabola, where the axis of symmetry is parallel to the y-axis with the vertex at (0,0), such as $y = a x^2\,\!$

then there is a point (0,f)—the focus, F—such that any point P on the parabola will be equidistant from both the focus and the linea directrix, L. The linea directrix is a line perpendicular to the axis of symmetry of the parabola (in this case parallel to the x axis) and passes through the point (0,-f). So any point P=(x,y) on the parabola will be equidistant both to (0,f) and (x,-f).

FP, a line from the focus to a point on the parabola, has the same length as QP, a line drawn from that point on the parabola perpendicular to the linea directrix, intersecting at point Q.

Imagine a right triangle with two legs, x and f-y (the vertical distance between F and P). The length of the hypotenuse, FP, is given by $\| FP \| = \sqrt{ x^2 + (f - y)^2 }\,\!$

(Note that (f-y) and (y-f) produce the same result because it is squared.)

The line QP is given by adding y (the vertical distance between the point P and the x-axis) and f (the vertical distance between the x-axis and the linea directrix). $\| QP \| = f + y\,\!$

These two line segments are equal, and, as indicated above, y=ax², thus $\| FP \| = \| QP \| \,\!$ $\sqrt{x^2 + (f - a x^2 )^2 } = f + a x^2\,\!$

Square both sides, $x^2 + (f^2 - 2 a x^2 f + a^2 x^4) = (f^2 + 2 a x^2 f + a^2 x^4)\,\!$

Cancel out terms from both sides, $x^2 - 2 a x^2 f = 2 a x^2 f\,\!$ $x^2 = 4 a x^2 f\,\!$

Divide out the from both sides (we assume that x is not zero), $1 = 4 a f\,\!$ $f = {1 \over 4 a }\,\!$

So, for a parabola such as f(x)=x², the a coefficient is 1, so the focus F is (0,¼)

As stated above, this is the derivation of the focus for a simple parabola, one centered at the origin and with symmetry around the y-axis. For any generalized parabola, with its equation given in the standard form $y=ax^2+bx+c\,\!$,

the focus is located at the point $\left (\frac{-b}{2a},\frac{-b^2}{4a}+c+\frac{1}{4a} \right)\,\!$

which may also be written as $\left (\frac{-b}{2a},c-\frac{b^2-1}{4a} \right)\,\!$

and the directrix is designated by the equation $y=\frac{-b^2}{4a}+c-\frac{1}{4a}\,\!$

which may also be written as $y=c-\frac{b^2+1}{4a}\,\!$

Reflective property of the tangent  A diagram showing the reflective property, the directrix (solid green), and the lines connecting the focus and directrix to the parabola (blue)

The tangent of the parabola described by equation y=ax2 has slope ${dy \over dx} = 2 a x = {2 y \over x}$

This line intersects the y-axis at the point (0,-y) = (0, - a x²), and the x-axis at the point (x/2,0). Let this point be called G. Point G is also the midpoint of points F and Q: $F = (0,f), \quad$ $Q = (x,-f), \quad$ ${F + Q \over 2} = {(0,f) + (x,-f) \over 2} = {(x,0) \over 2} = \left({x \over 2}, 0\right).$

Since G is the midpoint of line FQ, this means that $\| FG \| \cong \| GQ \|,$

and it is already known that P is equidistant from both F and Q: $\| PF \| \cong \| PQ \|,$

and, thirdly, line GP is equal to itself, therefore: $\Delta FGP \cong \Delta QGP$

It follows that $\angle FPG \cong \angle GPQ$.

Line QP can be extended beyond P to some point T, and line GP can be extended beyond P to some point R. Then $\angle RPT$ and $\angle GPQ$ are vertical, so they are equal (congruent). But $\angle GPQ$ is equal to $\angle FPG$. Therefore $\angle RPT$ is equal to $\angle FPG$.

The line RG is tangent to the parabola at P, so any light beam bouncing off point P will behave as if line RG were a mirror and it were bouncing off that mirror.

Let a light beam travel down the vertical line TP and bounce off from P. The beam's angle of inclination from the mirror is $\angle RPT$, so when it bounces off, its angle of inclination must be equal to $\angle RPT$. But $\angle FPG$ has been shown to be equal to $\angle RPT$. Therefore the beam bounces off along the line FP: directly towards the focus.

Conclusion: Any light beam moving vertically downwards in the concavity of the parabola (parallel to the axis of symmetry) will bounce off the parabola moving directly towards the focus. (See parabolic reflector.)

The same reasoning can be applied to a parabola whose axis is vertical, so that it can be specified by the equation

y = ax2 + bx + c.

The tangent has then a generic slope of

mtan = 2ax + b.

Reflection derivation, together with trigonometric angle addition rules, leads to the result that the reflected ray has a slope of $m_{ref} = {m_{tan}^2 - 1 \over 2m_{tan}}$.

When b varies

The x-coordinate at the vertex is $x=-\frac{b}{2a}$, so substitute it into the equation y = ax2 + bx + c $y=a\left (-\frac{b}{2a}\right )^2 + b \left ( -\frac{b}{2a} \right ) + c$

Simplifying: $=\frac{ab^2}{4a^2} -\frac{b^2}{2a} + c$ $=\frac{b^2}{4a} -\frac{2\cdot b^2}{2\cdot 2a} + c\cdot\frac{4a}{4a}$ $=\frac{-b^2+4ac}{4a}$ $=-\frac{b^2-4ac}{4a}=-\frac{D}{4a}$

Thus, the vertex is at point $\left (-\frac{b}{2a},-\frac{D}{4a}\right )$

Parabolae in the physical world  A bouncing ball captured with a stroboscopic flash at 25 images per second. Note that the ball becomes significantly non-spherical after each bounce, especially after the first. That, along with spin and air resistance, causes the curve swept out to deviate slightly from the expected perfect parabola  Parabolic trajectories of water in a fountain

In nature, approximations of parabolae and paraboloids are found in many diverse situations. The best-known instance of the parabola in the history of physics is the trajectory of a particle or body in motion under the influence of a uniform gravitational field without air resistance (for instance, a baseball flying through the air, neglecting air friction).

The parabolic trajectory of projectiles was discovered experimentally by Galileo in the early 17th century, who performed experiments with balls rolling on inclined planes. He also later proved this mathematically in his book Dialogue Concerning Two New Sciences. For objects extended in space, such as a diver jumping from a diving board, the object itself follows a complex motion as it rotates, but the center of mass of the object nevertheless forms a parabola. As in all cases in the physical world, the trajectory is always an approximation of a parabola. The presence of air resistance, for example, always distorts the shape, although at low speeds, the shape is a good approximation of a parabola. At higher speeds, such as in ballistics, the shape is highly distorted and does not resemble a parabola.

Another situation in which parabolae may arise in nature is in two-body orbits, for example, of a small planetoid or other object under the influence of the gravitation of the sun. Such parabolic orbits are a special case that are rarely found in nature. Orbits that form a hyperbola or an ellipse are much more common. In fact, the parabolic orbit is the borderline case between those two types of orbit. An object following a parabolic orbit moves at the exact escape velocity of the object it is orbiting, while elliptical orbits are slower and hyperbolic orbits are faster.  Hercilio Luz Bridge, Florianópolis, Brazil. Suspension bridge cables follow a parabolic, not catenary, curve.  Parabolic bridge in Newcastle u.T.

Approximations of parabolae are also found in the shape of the main cables on a typical suspension bridge. Freely hanging cables as seen on a simple suspension bridge do not describe parabolic curves, but rather hyperbolic catenary curves. Under the influence of a uniform load (such as a horizontal suspended deck), the otherwise hyperbolic cable is deformed toward a parabola. Unlike an inelastic chain, a freely-hanging spring of zero rest length takes the shape of a parabola.

Paraboloids arise in several physical situations as well. The best-known instance is the parabolic reflector, which is a mirror or similar reflective device that concentrates light or other forms of electromagnetic radiation to a common focal point. The principle of the parabolic reflector may have been discovered in the 3rd century BC by the geometer Archimedes, who, according to a legend of debatable veracity, constructed parabolic mirrors to defend Syracuse against the Roman fleet, by concentrating the sun's rays to set fire to the decks of the Roman ships. The principle was applied to telescopes in the 17th century. Today, paraboloid reflectors can be commonly observed throughout much of the world in microwave and satellite dish antennas.  Parabolic shape formed by a liquid surface under rotation. Two liquids of different densities are contained in a narrow rectangular tank

Paraboloids are also observed in the surface of a liquid confined to a container and rotated around the central axis. In this case, the centrifugal force causes the liquid to climb the walls of the container, forming a parabolic surface. This is the principle behind the liquid mirror telescope.

Aircraft used to create a weightless state for purposes of experimentation, such as NASA's “Vomit Comet,” follow a vertically parabolic trajectory for brief periods in order to trace the course of an object in free fall, which produces the same effect as zero gravity for most purposes.

Generalizations

In algebraic geometry, the parabola is generalized by the rational normal curves, which have coordinates $(x,x^2,x^3,\dots,x^n);$ the standard parabola is the case n = 2, and the case n = 3 is known as the twisted cubic. A further generalization is given by the Veronese variety, when there are more than one input variable.

In the theory of quadratic forms, the parabola is the graph of the quadratic form x2 (or other scalings), while the elliptic paraboloid is the graph of the positive-definite quadratic form x2 + y2 (or scalings) and the hyperbolic paraboloid is the graph of the indefinite quadratic form x2y2. Generalizations to more variables yield further such objects.

Notes

1. ^ Dialogue Concerning Two New Sciences (1638) (The Motion of Projectiles: Theorem 1); see 
2. ^ However, this parabolic shape, as Newton recognized, is only an approximation of the actual elliptical shape of the trajectory, and is obtained by assuming that the gravitational force is constant (not pointing toward the center of the earth) in the area of interest. Often, this difference is negligible, and leads to a simpler formula for tracking motion.
3. ^ Middleton, W. E. Knowles (December 1961). "Archimedes, Kircher, Buffon, and the Burning-Mirrors" (GIF). Isis 52 (4): 533–543. doi:10.1086/349498. Retrieved 2006-08-08.

References

• Lockwood, E. H. (1961): A Book of Curves, Cambridge University Press

1911 encyclopedia

Up to date as of January 14, 2010

From LoveToKnow 1911

PARABOLA, a plane curve of the second degree. It may be defined as a section of a right circular cone by a plane parallel to a tangent plane to the cone, or as the locus of a point which moves .so that its distances from a fixed point and a fixed line are equal. It is therefore a conic section having its eccentricity equal to unity. The parabola is the curve described by a projectile which moves in a non-resisting medium under the influence of gravity (see Mechanics). The general relations between the parabola, ellipse and hyperbola are treated in the articles Geometry, Analytical, and Conic Sections; and various projective properties are demonstrated in the article Geometry, Projective. Here only the specific properties of the parabola will be given.

The form of the curve is shown in fig. 1, where P is a point on the curve equidistant from the fixed line AB, known as the directrix, and the fixed point F known as the focus. The line CD passing through the focus and perpendicular to the directrix is the axis or principal diameter, and meets the curve in the vertex G. The line FL perpendicular to the axis, G D and passing through the focus, is the semilatus rectum, the latus rectum being the focal chord parallel to the directrix. Any line parallel to the axis is a diameter, and the parameter of any diameter is measured by the focal chord drawn FIG.  parallel to the tangent at the vertex of the diameter and is equal P A B to four times the focal distance of the vertex. To construct the parabola when the focus and directrix are given, draw the axis CD and bisect CF at G, which gives the vertex. Any number of points on the parabola are obtained by taking any point E on the directrix, joining EG and EF and drawing FP so that the angles PFE and DFE are equal. Then EG produced meets FP in a point on the curve. By joining the points so obtained the parabola may be described. A mechanical construction, when the same conditions are given, consists in taking a rigid bar ABC bent at right angles at B (fig. 2), and fastening a string of length BC to C B Y C and F. Then if a pencil be placed along B C so as to keep the string taut, and the limb AB be slid along the directrix, the A pencil will trace out the parabola.

Properties which may be readily de FIG. 2. duced by euclidian methods from the definition include the following: the tangent at any point bisects the angle between the focal distance and the perpendicular on the directrix and is equally inclined to the focal distance and the axis; tangents at the extremities of a focal chord intersect at right angles on the directrix, and as a corollary we have that the locus of the intersection of tangents at right angles is the directrix; the circumcircle of a triangle circumscribing a parabola passes through the focus; the subtangent is equal to twice the abscissa of the point of contact; the subnormal is constant and equals the semilatus rectum; and the radius of curvature at a point P is 2 (FP) 4 /a 2 where a is the semilatus rectum and FP the focal distance of P.

A fundamental property of the curve is that the line at infinity is a tangent (see Geometry, Projective), and it follows that the centre and the second real focus and directrix are at infinity. It also follows that a line half-way between a point and its polar and parallel to the latter touches the parabola, and therefore the lines joining the middle points of the sides of a self-conjugate triangle form a circumscribing triangle, and also that the ninepoint circle of a self-conjugate triangle passes through the focus. The orthocentre of a triangle circumscribing a parabola is on the directrix; a deduction from this theorem is that the centre of the circumcircle of a self-conjugate triangle is on the directrix ("Steiner's Theorem").

In the article Geometry, Analytical, it iS Shown that the general equation of the second degree represents a parabola when the highest terms form a perfect square.

Analytic This is the analytical expression of the projective Geometry. property that the line at infinity is a tangent. The simplest equation to the parabola is that which is referred to its axis and the tangent at the vertex as the axes of co-ordinates, when it assumes the form y 2 = 4ax where as = semilatus rectum; this may be deduced directly from the definition. An equation of similar form is obtained when the axes of co-ordinates are any diameter and the tangent at the vertex. The equations to the tangent and normal at the point x' y are yy' = 2a(x+x) and aa(y - y')+y'(x - x')=o, and may be obtained by general methods (see Geometry, Analytical, and Infinitesimal Calculus). More convenient forms in terms of a single parameter are deduced by substituting x' =am t, y' = aam (for on eliminating in between these relations the equation to the parabola is obtained). The tangent then becomes my=x+amt and the normal y +aam - am 3 . The envelope of this last equation is 27ay 2 =4(x-2a) 3, which shows that the evolute of a parabola is a semi-cubical parabola (see below Higher Orders). The cartesian equation to a parabola which touches the coordinate axes is 1 / ax+'1 / by= i, and the polar equation when the focus is the pole and the axis the initial line is r cos 2 6/2 = a. The equation to a parabola in triangular co-ordinates is generally derived by expressing the condition that the line at infinity is a tangent in the equation to the general conic. For example, in trilinear co-ordinates, the equation to the general conic circumscribing the triangle of reference is 113y+mya+naf3=o; for this to be a parabola the line as + b/ + cy = o must be a tangent. Expressing this condition we obtain mb = 1/ nc = o as the relation which must hold between the co-efficients of the above equation and the sides of the triangle of reference for the equation to represent a parabola. Similarly, the conditions for the inscribed conic 1/la+1/ ms +'? Try = o to be a parabola is lbc+mca+nab = o, and the conic for which the triangle of reference is self-conjugate la 2 +143 2 +n7 2 =o is a 2 inn--+b 2 nl+c 2 lm=o. The various forms in areal co-ordinates may be derived from the above by substituting Xa for 1, µb for m and vc for n, or directly by expressing the condition for tangency of the line x+y+z = o to the conic expressed in areal coordinates. In tangential q, r) co-ordinates the inscribed and circumscribed conics take the forms Xqr+µrp+vpq=o and 1/ X p+ 1 /µ q + V y r = o; these are parabolas when X++'=° and V X = 1 / µ 1 / v= o respectively.

The length of a parabolic arc can be obtained by the methods of the infinitesimal calculus; the curve is directly quadrable, the area of any portion between two ordinates being two thirds of the circumscribing parallelogram. The pedal equation with the focus as origin is p 2 =ar; the first positive pedal for the vertex is the cissoid and for the focus the directrix. (See Infinitesimal Calculus.) REFERENcEs. - Geometrical constructions of the parabola are to be found in T. H. Eagles' Plane Curves (1885). See the bibliography to the articles Conic Sections; Geometry, Analytical; and Geometry, Projective.

In the geometry of plane curves, the term parabola is often used to denote the curves given by the general equation a' n x n = ym+n, thus ax= y 2 is the quadratic or Apollonian parabola; a 2 x = y 3 is the cubic parabola, a 3 x = y4 is the biquadratic parabola; semi parabolas have the general equation ax n-1 = yn, thus ax e = y 3 is the semicubical parabola and ax 3 = y 4 the semibiquadratic parabola. These curves were investigated by Rene Descartes, Sir Isaac Newton, Colin Maclaurin and others. Here we shall treat only the more important forms.

The cartesian parabola is a cubic curve which is also known as the trident of Newton on account of its three-pronged form. Its equation is xy=ax 3 -{-bx 2 +-cx-l-d, and it consists of two legs asymptotic to the axis of y and two parabolic legs (fig. 3). The simplest form is 3 - a 3 in this case the serpentine position shown in the figure degenerates into a point of inflexion. Descartes used the curve to solve sextic equations by determining its intersections with a circle; mechanical constructions were given by Descartes (Geometry, lib. 3) and Maclaurin (Organica geometrica). The cubic parabola (fig. 4) is a cubic curve having the equation y= 3 +- 2 -+cx-+d. It consists of two parabolic branches tending in opposite directions. John Wallis utilized the intersections of this curve with a right line to solve cubic equations, and Edmund Halley solved sextic equations with the aid of a circle.

Diverging parabolas are cubic curves given by the equation y 2 = 3 -f-bx 2 -cx+d. Newton discussed the five forms which arise from the relations of the roots of the cubic equation. When all the v FIG. 3.

 FIG. 4. FIG. 5.

roots are real and unequal the curve consists of a closed oval and a parabolic branch (fig. 5). As the two lesser roots are made more and more equal the oval shrinks in size and ultimately becomes a real conjugate point, and the curve, the equation of which is y2= (x - a) 2 (x - b) (in which a > b) consists of this point and a bell-like branch resembling the right-hand member of fig. 5. If two roots are imaginary the equation is y 2 =(x 2 +a 2) (x - b) and the curve resembles the parabolic branch, as in the preceding case. This is sometimes termed the campaniform (or bell-shaped) parabola. Jf the two greater roots are equal the equation is y 2 = (x - a) (x - b) 2 (in which a<b) and the curve assumes the form shown in fig. 6, and is known as the nodated parabola. Finally, if all the roots are equal, the equation becomes y 2 =(x - a) 3; this curve is the cuspidal or semicubical parabola (fig. 7). This curve, which is sometimes termed the Neilian parabola after William Neil (1637-1670), is the evolute of the ordinary parabola, and is especially interesting as being the first Higher Orders. curve to be rectified. This was accomplished in 1657 by Neil in England, and in 1659 by Heinrich van Haureat in Holland. Newton showed that all the five varieties of the diverging parabolas may be exhibited as plane sections of the solid of revolution of the semicubical parabola. A plane oblique to the axis and passing below the vertex gives the first variety; if it passes through the vertex, FIG. 6. FIG. 7.

the second form; if above the vertex and oblique or parallel to the axis, the third form; if below the vertex and touching the surface, the fourth form, and if the plane contains the axis, the fifth form results (see Curve).

The biquadratic parabola has, in its most general form, the equa 'tion' 4 -1-cx2 -1-dx -fie, and consists of a serpentinous and two parabolic branches (fig. 8). If all the roots of the quartic in FIG. 8. FIG. 9. FIG. 10.

x are equal the curve assumes the form shown in fig. 9, the axis of x being a double tangent. If the two middle roots are equal, fig. 10 results. Other forms which correspond to other relations between the roots can be readily deduced from the most general form. (See CURVE; and GEOMETRY, ANALYTICAL.)

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