PARA'BOLA. The probable origin of this name, as applied to one of the conic sections, may be soeu in Mara:sem As in the case of the other Corm SECTIONS [ELLIPSE and II YPEHBOLA), we shall here give a small collection of the most remarkable properties of this curve.
I. Let a point P move in such a way that its distance a P from a fixed point a is always the same as its perpendicular distance I'M from a given line at L. Thie point r describes what is called a parabola.
2. The line L s, perpendicular to Is at, produced, is the axis, or principal diameter; and any lino P9 parallel to it is called a dianteter.
The point s is the focus, and the line L at the directrix. The ordinate S K drawn through the focus is called the semi-latus-rectum, and double of it is the fates rectum. The same phrase is used for the ordinate drawn through the focus of an ellipse or hyperbola.
ing to the balloon. It is in shape like an umbrella, and its construction may be understood by supposing the umbrella to be large and strong, to be provided with ropes or stays fastened to the extremities of the whalebones, and brought down to the handle, where they must be fixed, so as to prevent the umbrella from turning inside outwards. Instead of the stick, suppose a metal tube to be fixed in the centre, with a rope passing through it, attached by its upper extremity to the balloon, and by its lower end to a tub or car. This machine is a para chute : while ascending, it will be like a closed umbrella, but it may at any moment be detached from the balloon by cutting the end of the rope which is tied to the car ; the resistance of the air will tben cause it to expand, and will at the same time retard the velocity of descent.
The idea of using such a machine to break the fall from a high place is not new : it has been frequently experimented with ; has been found utterly useless for any desirable purpose ; and has caused the death of many who have attempted to descend by its means.
Three formulas have been given for calculating the velocity of descent of a parachute. They are— 3. S A is equal to A L, and 8 K to twice A S.
4. li an ellipse be described with the vertex A and the focus s, the farther the centre is from the point s, the more nearly will the part of the ellipse which falls within any given ordinate P N coincide with the corresponding part of the parabola : and the same of an hyperbola drawn with the vertex A and focus S. And a parabola may be con sidered as an ellipse or hyperbola with a given focal distance A. S, and a centre at an infinite distance.
5. The tangent r r bisects the angle x e s, and a is is equal to s r, and A N to A T. The line a v drawn perpendicular to the tangent from the focus always meets PT in a point of the tangent drawn through A.
6. If the normal en be drawn perpendicular to er, then s r is equal to s o, and N 0 is always equal to twice a s.
7. The [square on P N is equal to four times the rectangle under A N and A S.
8. The area AN P is two-thirds of N Z, the rectangle under as and N P.
9. If p p' be drawn parallel to er through any point v of the diameter r v, Q p is bisected in v, and the square on p v is equal to four times the rectangle under sP and Pr.
10. The square on s T is equal to the rectangle under AS and as'.
In applied mathematics the parabola was formerly of great im portance, both as being the curve in which a comet was supposed to move, and as that in which a cannon-ball or other projectile would move were it not for the resistance of the air. It is still sometimes used as the approximation to the elongated ellipse in which a comet moves. For the more accurate investigations which deduce the real path of a projectile, see GUNNERY.