PARABOLA, one of the conic sections, is produced by a plane not passing through the vertex, which cuts the cone in a direction parallel to that of a plane touching the convex surface of the cone. A little consideration will show that a section so produced cannot be a closed curve, but its two branches, though continually widening out from each other, do not divel•ge so rapidly as in the' yperbola (q.v.). The nearer the cutting plane is to that which touches the cone, the less do the two branches diverge; and when the two planes coincide, the branches also coincide, forming a straight line, which is therefore the limit of the parabola. It may otherwise be considered as a curve, every point of which is equally distant from a fixed straight line and a given point; the fixed straight line is called the directrix, and the given point the focus. Thus (see fig.) PAP' is a parabola, any point P in which is equally distant from the focus S and the directrix CB, or PS = PD.
If, from S, a perpendicular, SE, be drawn to the direc trix, and produced backward, this line, AO, is the axisor principal diameter of the parabola, and the curve is symmetrical on both sides of it. As A is a point iu the parabola, AS = AE, or the vertex of a parabolas bisects the perpendicular from the focus to the directrix. All lines in a parabola which are parallel to the axis cut the curve in only. one.point, and are called diameters. All lines, such as PP', which cut the curve in two points, ; are ordinates, and the diameter to which they are ordi nates, is that one which bisects them; the portion of this diameter which is intercepted between the ordinate and the curve, is the corresponding abscissa. From the property of the parabola that PS = PD, the equation to the curve may be at once deduced; for PS = PD = EN, therefore PS' (which = NS') = EN'; hence P