The solid generated by the rotation of a parabola about its axis; any surface of the second order whose sections by planes parallel to a given line are parabolas.
In the ellipse this ratio is less than unity, in the parabola equal to unity, and in the hyperbola greater than unity.
The ellipse and hyperbola have each two foci, and two corresponding directrixes, and the parabola has one focus and one directrix.
The same speculations take up the parabolaonce more, imagine it rolling on an indefinite straight line and ask what course does the focus of this curve follow.
It is essential that aviators be familiar with the parabola described by the shells fired by cannon of various calibers.
T0, (14) which is the equation of the parabola in question.
This property is characteristic of a parabola whose axis is vertical.
In the intermediate case of the parabola we have a = [oo] and the last term disappears.
Hence AD, BC are parallel and are bisected by the same vertical line; and a parabola with vertical axis can therefore be described through A, B, C, D.
The graph of F is a straight line; that of M is a parabola with vertical axis.
The pole O of the hodograph is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola.
It is further of interest to note that if the weights be all equal, and at equal horizontal intervals, the vertices of the funicular will lie on a parabola whose axis is vertical.
This is a parabola with vertical axis, of latus-rectum 2u0^2/g.
In the fated evolution of Italian art, describing its parabola of vital energy, Michelangelo softened, sublimed, and harmonised his predecessor's qualities.
It is neither ellipse norparabola nor section of the circle, but an inspiration of creative fancy.
But in the case of the true parabola the body can never return; to do so it would have to double the distant focus, and as that is infinitely remote, it could not be reached except in the lapse of infinite time.
The theory of gravitation, though it admits the parabola as a possible orbit for a comet, does not assert that the path must necessarily be of this type.
If the velocity of the comet is accelerated by this disturbing influence, the orbit will be changed from a parabola into another curve known as a hyperbola, and the comet will swing round the sun and pass away never to return.
It will here be only necessary to point to the connection which exists between the parabola and the ellipse.
The fact that the comet moves in an ellipse or in a parabola proves that the two bodies act and react on each other in obedience to the law of universal gravitation.
But he never mentions how very nearly Cluvier agrees with the Greek: they only differ by 1 divided by 3n^2, where n is the infinite number of parts of which a parabola is composed.
The mechanical proof recalls that of the Method in some respects, but is more elaborate in that the elements of the area of the parabola to be measured are not straight lines but narrow strips.
The title of this work seems originally to have been On the Section of a Right-angled Cone and to have been changed after the time of Apollonius, who was the first to call a parabola by that name.
The mechanical proof with the necessary preliminary propositions about the parabola (some of which are merely quoted, while two, evidently original, are proved, Props.
This is because these Comets move in orbits which are not closed, and which are known as parabolic or hyperbolic orbits respectively, because derived from those sections of a cone which are called the Parabola and the Hyperbola.
It is in it self a very barren thing to know, that in a Parabola a Subtangant is double the corresponding Abscissæ; but yet it is a Degree of Knowledge necessary to the Art of throwing Bombs, so exactly as they can do now.
The nature of the two kinds of branches is best understood by considering them as projections, in the same way as we in effect consider the hyperbola and the parabola as projections of the ellipse.
The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively.
Thus the curve of the first order or right line consists of one branch; but in curves of the second order, or conics, the ellipse and the parabola consist each of one branch, the hyperbola of two branches.
An elegant result which Galileo also deduced from this theory, is that the form of such a beam, to be equally strong in every part, should be that of a parabolical prism, the vertex of the parabola being the farthest removed from the wall.
When the conic is a parabola the meridian line is a catenary (fig.
All diameters of a parabola are parallel, the pole to the line at infinity being the point where the curve touches the line at infinity.
Hence-- The segment PM which joins the middle point M of a chord of a parabola to the pole P of the chord is bisected by the parabolaat A.
The parabola has one focus within it, and another, the "caecus focus," which may be imagined to be at infinity on the axis within or without the curve.
In case of the parabola this becomes-- The line joining any point on a parabola to the focus and the diameter through the point, are equally inclined to the tangent and normal at that point.
But a diameter cuts the parabola once at infinity.
A parabola has one axis, if we define an axis as a diameter perpendicular to the chords which it bisects.
It follows in an ellipse the ratio between the distance of a point from the focus to that from the directrix is less than unity, in the parabola it equals unity, and in the hyperbola it is greater than unity.
In a parabola the vertex lies halfway between directrix and focus.
Hence the theorem-- The tangents to a parabola cut each other proportionally.
Any two tangents to a parabola are cut by all others in projective rows; but the line at infinity being one of the tangents, the points at infinity on the rows are corresponding points, and the rows therefore similar.
Through the dark curtain of the sky in a parabola rising from the German trenches swept the brilliant sputter of red light of a German flare.
But its scream, instead of passing in a long parabola toward the German lines, goes up in the heavens toward something as large as your hand against the light blue of the summer sky--a German aeroplane.
While resembling the parabola in extending to infinity, the curve has closest affinities to the ellipse.
If the comet of 1668 had moved in a parabola instead of the ellipse supposed above, how many years would have been required to reach its present distance from the sun?
A parabola is not so simply defined, and five numbers, called the elements of its orbit, are required to fix accurately a comet's path around the sun.
Astronomers, therefore, consider meteors as well as comets to have the parabola and the elongated ellipse as their characteristic orbits.
Unless it is something like the boomerang of the Australian aborigines I do not even know what a parabola is.
He loves to tell how he went to a bull fight down there and saw the gallant mandatory stab the charging parabola to the heart with his shining bolero or whatever you call it.
The comet of 1680, Newton found to move in an orbit so nearly a parabola that the time of describing it must be reckoned in hundreds of years at the least.
The prow of the boat is connected with the apex of the parabola by a curve whose concavity is directed upward.
The surface thus formed is a parabola whose apex is in front, the maximum ordinate behind, and the concavity directed toward the bottom of the water.
Straight and strong it flew, describing a wide parabola ere it thudded into the road.
A particle moving in a parabolaor hyperbola can only make one approach to its attracting body; after such approach it continually recedes from it.
This curve is usually some form of conic section: an ellipse, parabola or hyperbola.
And the parabola is the customary path of the cyclonic storm.
The directrix of a parabola is a line perpendicular to the axis produced, and at a distance from the vertex equal to the distance of the vertex from the focus.
Thus AB is the directrix of the parabola VED, of which F is the focus.
Examination of the cases where the theory of the parabola is applicable.
The parabola may be considered as an ellipse, in which the major axis is indefinitely increased while the distance from one focus to the adjacent summit remains constant.
All the diameters of the parabola are right lines parallel to the axis, and reciprocally.
