## Ellipse Sentence Examples

- That the explanation fails in detail is undoubted: it does not account for the ellipticity of the planets; it would place the sun, not in one focus, but in the centre of the
**ellipse**; and it would make gravity directed towards the centre only under the equator. - It is the envelope of circles described on the central radii of an
**ellipse**as diameters. - In the course of constructions for surfaces to reflect to one and the same point (1) all rays in whatever direction passing through another point, (2) a set of parallel rays, Anthemius assumes a property of an
**ellipse**not found in Apollonius (the equality of the angles subtended at a. - If the law of attraction is that of gravitation, the orbit is a conic section -
**ellipse**, parabola or hyperbola - having the centre of attraction in one of its foci; and the motion takes place in accordance with Kepler's laws (see Astronomy). - Elliptic orbits, and a parabolic orbit considered as the special case when the eccentricity of the
**ellipse**is 1, are almost the only ones the astronomer has to consider, and our attention will therefore be confined to them in the present article. - In all the cases which have yet arisen in astronomy the extraneous forces are so small compared with the gravitation of the central body that the orbit is approximately an
**ellipse**, and the preliminary computations, as well as all determinations in which a high degree of precision is not necessary, are made on the hypothesis of elliptic orbits. - We put e for the eccentricity of the
**ellipse**, represented P, by the ratio M CF: CA. - The latter completely encloses a large area of ground in a semicircle of which Besancon itself is the centre, and the whole of the newer works taken together form an irregular
**ellipse**of which the major axis, lying north-east by south-west, is formed by the Doubs. - This shows that the C.P. is the antipole of the line of intersection of its plane with the free surface with respect to the momental
**ellipse**at the C.G. - Within which the C.P. must lie when the area is immersed completely; the boundary of the core is therefore the locus of the antipodes with respect to the momental
**ellipse**of water lines which touch the boundary of the area. - Thus the core of a circle or an
**ellipse**is a concentric circle or**ellipse**of one quarter the size. - An inclining couple due to moving a weight about in a ship will heel the ship about an axis perpendicular to the plane of the couple, only when this axis is a principal axis at F of the momental
**ellipse**of the water-line area A. - Of the wedge of immersion and emersion, will be the C.P. with respect to FF' of the two parts of the water-line area, so that b 1 b 2 will be conjugate to FF' with respect to the momental
**ellipse**at F. - The varying direction of the inclining couple Pc may be realized by swinging the weight P from a crane on the ship, in a circle of radius c. But if the weight P was lowered on the ship from a crane on shore, the vessel would sink bodily a distance P/wA if P was deposited over F; but deposited anywhere else, say over Q on the water-line area, the ship would turn about a line the antipolar of Q with respect to the confocal
**ellipse**, parallel to FF', at a distance FK from F FK= (k2-hV/A)/FQ sin QFF' (2) through an angle 0 or a slope of one in m, given by P sin B= m wA FK - W'Ak 2V hV FQ sin QFF', (3) where k denotes the radius of gyration about FF' of the water-line area. - (6) Then =o over the
**ellipse**n = a, and the hyperbola t = (, so that these may be taken as fixed boundaries; and %,1. - Having a resultant in the direction PO, where P is the intersection of an
**ellipse**n with the hyperbola 13; and with this velocity the**ellipse**n can be swimming in the liquid, without distortion for an instant. - At infinity U = -me a cos (i = a m b oos (3, V= -me a sin 1 3 - C7,1 sin 0, (9) a and b denoting the semi-axes of the
**ellipse**a; so that the liquid is streaming at infinity with velocity Q = m/(a+b) in the direction of the asymptote of the hyperbola (3. - An
**ellipse**interior to n = a will move in a direction opposite to the exterior current; and when n = o, U = oo, but V = (m/c) sh a sin 13. - Similarly, the streaming velocity V reversed will give rise to a thrust 27rpmV in the direction xC. Now if the cylinder is released, and the components U and V are reversed so as to become the velocity of the cylinder with respect +m /a) 2 - U2 The components of the liquid velocity q, in the direction of the normal of the
**ellipse**n and hyperbola t, are -mJi sh(n--a)cos(r-a),mJ2 ch(n-a) sin (E-a). - (io) The velocity q is zero in a corner where the hyperbola a cuts the
**ellipse**a; and round the**ellipse**a the velocity q reaches a maximum when the tangent has turned through a right angle, and then q _ (Ch 2a-C0s 2(3). - Next consider the motion given by = m ch 2(77a)sin 2E, tii= -m sh 2(na)cos 2E; (I) in which > ' =o over the
**ellipse**a, and =1'+IR(x2+y2) =[ -m sh 2(7 7 -a)+4Rc 2 ]cos 4Rc2 ch 2n, (2) which is constant over the**ellipse**n if 4Rc 2. - Msh2(n-a); (3) so that this
**ellipse**can be rotating with this angular velocity R for an instant without distortion, the**ellipse**a being fixed. - The velocity of a liquid particle is thus (a 2 - b 2)/(a 2 +b 2) of what it would be if the liquid was frozen and rotating bodily with the
**ellipse**; and so the effective angular inertia of the liquid is (a 2 -b 2) 2 /(a 2 +b 2) 2 of the solid; and the effective radius of gyration, solid and liquid, is given by k 2 = 4 (a 2 2), and 4 (a 2 For the liquid in the interspace between a and n, m ch 2(0-a) sin 2E 4) 1 4Rc 2 sh 2n sin 2E (a2_ b2)I(a2+ b2) = I/th 2 (na)th 2n; (8) and the effective k 2 of the liquid is reduced to 4c 2 /th 2 (n-a)sh 2n, (9) which becomes 4c 2 /sh 2n = s (a 2 - b 2)/ab, when a =00, and the liquid surrounds the**ellipse**n to infinity. - They are probably dispersed pretty evenly along a very extended
**ellipse**agreeing closely in its elements with comet 1862: III. - The largest is an
**ellipse**of about 60 by 66 ft., but most of the sesi have a diameter of 20-25 ft. - Sca, through,, u rpov, measure), in geometry, a line passing through the centre of a circle or conic section and terminated by the curve; the "principal diameters of the
**ellipse**and hyperbola coincide with the "axes" and are at right angles; " conjugate diameters " are such that each bisects chords parallel to the other. - Regular
**ellipse**about 22 m. - - For elementary mensuration the
**ellipse**is to be regarded as obtained by projection of the circle, and the ellipsoid by projection of the sphere. - Hence the area of an
**ellipse**whose axes are 2a and 2b is Trab; and the volume of an ellipsoid whose axes are 2a, 2b and 2c is t rabc. The area of a strip of an**ellipse**between two lines parallel to an axis, or the volume of the portion (frustum) of an ellipsoid between two planes parallel to a principal section, may be found in the same way. - If the two forks have the same frequency, it is easily seen that the figure will be an
**ellipse**(including as limiting cases, depending on relative amplitude and phase, a circle and a straight line). - If the forks are not of exactly the same frequency the
**ellipse**will slowly revolve, and from its rate of revolution the ratio of the frequencies may be determined (Rayleigh, Sound, i. - If the rod is circular in section and perfectly uniform the end will describe a circle,
**ellipse**or straight line; but, as the elasticity is usually not exactly the same in all directions, the figure usually changes and revolves. - It imitates the motions made in polishing a speculum by hand by giving both a rectilinear and a lateral motion to the polisher, while the speculum revolves slowly; by shifting two eccentric pins the course of the polisher can be varied at will from a straight line to an
**ellipse**of very small eccentricity, and a true parabolic figure can thus be obtained. - If the pressure falls off uniformly, so that the pressure-curve is a straight line PDF sloping downwards and cutting AM in F, then the energy-curve will be a parabola curving downwards, and the velocity-curve can be represented by an
**ellipse**, or circle with centre F and radius FA; while the time-curve will be a sinusoid. - He first brought the revolutions of our satellite within the domain of Kepler's laws, pointing out that her apparent irregularities could be completely accounted for by supposing her to move in an
**ellipse**with a variable eccentricity and directly rotatory major axis, of which the earth occupied one focus. - Proposition 14 shows how to draw an
**ellipse**through five given points, and Prop. 15 gives a simple construction for the axes of an**ellipse**when a pair of conjugate diameters are given. - 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. - The star thus appears to describe a small
**ellipse**in the sky, and the nearer the star, the larger will this**ellipse**appear. - The greatest displacement of the star from its mean position (the semi-axis major of the
**ellipse**) is called its parallax. - Thus Whewell mistook Kepler's inference that Mars moves in an
**ellipse**for an induction, though it required the combination of Tycho's and Kepler's observations, as a minor, with the laws of conic sections discovered by the Greeks, as a major, premise. - The line parallel to q' q-- 1 which intersects the axes of Q and Q'; the plane of the member contains a fixed line; the centre is on a fixed
**ellipse**which intersects the transversal; the axis is on a fixed ruled surface to which the plane of the**ellipse**is a tangent plane, the**ellipse**being the section of the ruled surface by the plane; the ruled surface is a cylindroid deformed by a simple shear parallel to the transversal. - Again, the locus of G is an arc of an
**ellipse**whose centre is in the intersection of the planes; since this arc is convex upwards the equilibrium is unstable. - Which may be called the momental
**ellipse**at 0. - The path is therefore an
**ellipse**of which a, b are conjugate semi-diameters, and is described in the period 24 Ju; moreover, the velocity at any point P is equal to ~ OD, where OD is the semi-diameter conjugate to OP. ~,This type of motion;,s called elliptic harmonic. If the co-ordinate axes are the principal axes of the**ellipse**, the angle ft in (I o) is identical with the excentric angle. - Hence the path is approximately, an
**ellipse**, and the period is 2sr ~/ (l/g). - Now in a conic whose focus is at 0 we have where 1 is half the latus-rectum, a is half the major axis, and the upper or lower sign is to be taken according as the conic is an
**ellipse**or hyperbola. - This merely shows that a particular
**ellipse**may be described under the law of the direct distance provided the circumstances of projection be suitably adjusted. - But since an
**ellipse**can always be constructed with a given centre so as to touch a given line at a given point, and to have a given value of ab(=h/-~ u) we infer that the orbit will be elliptic whatever the initial circumstances. - In elliptic harmonic motion the velocity of P is parallel and proportional to the semi-diameter CD which is conjugate to the radius CP; the hodograph is therefore an
**ellipse**similar to the actual orbit. - The pole 0 of the hodograph is inside on or outside the circle, according as the orbit is an
**ellipse**, parabola or hyperbola.