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 P 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, and passing through the focus, is the semilatus rectum, the latus rectum being the focal chord parallel fo 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 . I .

parallel to the tangent at the vertex of the diameter and is equal B 748 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 and F . Then if a pencil be placed along /p BC so as to keep the string taut, and the limb AB be slid along the directrix, the n pencil will trace out the parabola . F Properties which may be readily de- 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 nine-point 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 y2=4ax where 2a=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 2a(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 xi = amt, y' =tam (for on eliminating m between these relations the equation to the parabola is obtained) . The tangent then becomes my=x+amt and the normal y=mx+2am—am3 . The envelope of this last equation is 27ay2=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 co-ordinate axes is V ax+1/ by= 1, and the polar equation when the focus is the pole and the axis the initial line is r cos2O/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 l0y+mya+naf=o; for this to be a parabola the line as + b/3 + cy=o must be a tangent . Expressing this condition we obtain V la* 1/ml Vnc=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 S/la+V m#+V ny = o to be a parabola is lbc+mca+nab=o, and the conic for which the triangle of reference is self-conjugate lag+ml2+ny2=c is a2mn+b2nl+c2lm=o . The various forms in areal co-ordinates may be derived from the above by substituting Xa for 1, ub 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 co-ordinates . In tangential (p, q, r) co-ordinates the inscribed and circumscribed conics take the forms Xqr+µrp+vpq=o and V Ap+ A/,uq±V ye = o; these are parabolas when X +µ+v = o and V X= V,u* V 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 p2=ar; the first positive pedal for the vertex is the cissoid (q.v.) 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'nxn= ye.+n, thus ax=y2 is the quadratic or Apollonian parabola; a2x=y3 is the cubic parabola, a3x=y4 is Higher Orders . the biquadratic parabola; semi parabolas have the general equation axn-1=yn, thus ax2=y3 is the semicubical parabola and ax3=y4 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=ax3-+-bx2+cx+d, and it consists of two legs asymptotic to the- axis of y and two parabolic legs (fig . 3) . The simplest form is axy=x3—a3, 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 inter-sections 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=ax3+bx2+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 y2=ax3+bx2+cx+d .

Newton discussed the five forms which arise from the relations of the roots of the cubic equation . When all the 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 . . If two roots are imaginary the equation is y2 _ (x2+a2) (x —b) and the curve resembles the parabolic branch, as in the preceding case . This is some-times termed the cam paniform (or bell-shaped) parabola . If the two greater roots are equal the equation is y2-- (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 y2=(x—a)3; this curve is the cuspidal or semi-cubical 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 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 semi-cubical parabola . A plane oblique to the axis and passing below the vertex gives the first variety; if it passes through the vertex, 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 y=ax++bx8+cx2+dx+e, and consists of a serpentinous and two parabolic branches (fig . 8) . If all the roots of the quartic in 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. so results . Other forms which correspond to other relations between the roots can be readily deduced from the most general form .