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ellipse

It is the envelope of circles described on the central radii of an ellipse as diameters.

5240regular ellipse about 22 m.

4632regular ellipse about 22 m.

4632But 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.

4532The largest is an ellipse of about 60 by 66 ft., but most of the sesi have a diameter of 20-25 ft.

2828The largest is an ellipse of about 60 by 66 ft., but most of the sesi have a diameter of 20-25 ft.

2829Hence the path is approximately, an ellipse, and the period is 2sr ~/ (l/g).

2020The 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.

1811The star thus appears to describe a small ellipse in the sky, and the nearer the star, the larger will this ellipse appear.

1819The 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.

1711of the fluid, equal to the weight vertically upward through the movement of a weight P through a distance c will cause the ship to heel through an angle 0 about an axis FF' through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes a 2 -hV/A, b 2 - hV/A, (I) h denoting the vertical height BG between C.G.

159of the fluid, equal to the weight vertically upward through the movement of a weight P through a distance c will cause the ship to heel through an angle 0 about an axis FF' through F, which is conjugate to the direction of the movement of P with respect to an ellipse, not the momental ellipse of the water-line area A, but a confocal to it, of squared semi-axes a 2 -hV/A, b 2 - hV/A, (I) h denoting the vertical height BG between C.G.

15935 The Ellipse and the Ellipsoid.

1414If 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.

1417This 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.

1312In the case of an ellipse described about the centre as pole we have ~=aI+b2_r2; (12)

1316which may be called the momental ellipse at 0.

1215msh2(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.

1115It 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.

911of 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.

914Representing by P this position, it follows that the area of that portion of the ellipse contained between the radii vectores FB and FP will bear the same ratio to the whole area of the ellipse that t does to T, the time of revolution.

89This merely shows that a particular ellipse may be described under the law of the direct distance provided the circumstances of projection be suitably adjusted.

811An 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.

812Now 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.

812The greatest displacement of the star from its mean position (the semi-axis major of the ellipse) is called its parallax.

711The 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.

610The 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.

612We put e for the eccentricity of the ellipse, represented P, by the ratio M CF: CA.

59Thus the core of a circle or an ellipse is a concentric circle or ellipse of one quarter the size.

59the 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.

59We put e for the eccentricity of the ellipse, represented P, by the ratio M CF: CA.

59within 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.

510If 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.

48Proposition 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.

48Thus 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.

49It possesses thi property that the radius of gyration about any diameter is half thi distance between the two tangents which are parallel to that diameter, In the case of a uniform triangular plate it may be shown that thi momental ellipse at G is concentric, similar and similarly situatec to the ellipse which touches the sides of the triangle at their middle points.

415If 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).

37Again, 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.

37The pole 0 of the hodograph is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola.

37In 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.

38Over any ellipse n, moving with components U and V of velocity, =i+Uy-Vx=[msh(n-a) cos (3+Ucshn] sin k -[msh(n-a) sin (3+Vcchn] cos h; (7) so that ' =o, if U c sh n cos R, V = c ch n sin a, (8) m sh(n - a) m sh(n - a).

38having 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.

38If 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.

38The 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.

311EQUATION OF THE CENTRE, in astronomy, the angular distance, measured around the centre of motion, by which a planet moving in an ellipse deviates from the mean position which it would occupy if it moved uniformly.

322Sca, 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.

26- For elementary mensuration the ellipse is to be regarded as obtained by projection of the circle, and the ellipsoid by projection of the sphere.

26Hence 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.

26It is clearly the form of the fundamental property (expressed in the terminology of the "application of areas") which led him to call the curves for the first time by the names parabola, ellipse, hyperbola.

26An 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.

27(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).

27He 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.

27In 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.

27(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).

27He 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.

27Similarly, 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).

28They are probably dispersed pretty evenly along a very extended ellipse agreeing closely in its elements with comet 1862: III.

15The 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.

15Elliptic 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.

16From the properties of the ellipse, A is the pericentre or nearest point of the orbit to the centre of attraction and B the apocentre or most distant point.

16Next 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.

16In 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.

17While resembling the parabola in extending to infinity, the curve has closest affinities to the ellipse.

00The passage of the maximum turgidity round the stem may vary in rapidity in different places, causing the circle to be replaced by an ellipse.

00(9) Turning the axes to make them coincide with the principal axes of the area A, thus making f f xydA = o, xh = - a 2 cos a, y h = - b 2 sin a, (io) where ffx2dA=Aa2, ffy 2 dA= Ab 2, (II) a and b denoting the semi-axes of the momental ellipse of the area.

00Ellipse >>

00The ratio of the axes of the ellipse is sec a, the longer axis being in the plane of 0.

00When both are near the lowest point the horizontal projection of the path is approximately an ellipse, as shown in 13; a closet investigation shows that the ellipse is to be regarded as revolving about its centre with the angular velocity ~ab~~/l2, where a, b are the semi-axes.

00To illustrate this subject, it may be mentioned that an ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis of the ellipses, and the velocity ratio varying from to I ~eccentricitY an hyperbola rotating about its further focus rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hypereccentricity + I

00Its earliest form is a rough ellipse transfixed by an upright line, cp. In various Semitic alphabets this has been altered out of recognition, apparently from the writing of the symbol in cursive handwriting without lifting the pen.

00When the conic is an ellipse the meridian line is in the form of a series of waves, and the film itself has a series of alternate swellings and contractions as represented in figs.

00When the ellipse becomes a circle, the meridian line becomes a straight line parallel to the axis, and the film passes into the form of a cylinder of revolution.

00In all these cases the internal pressure exceeds the external by 2T/a where a is the semi-transverse axis of the conic. The resultant of the internal pressure and the surface-tension is equivalent to a tension along the axis, and the numerical value of this tension is equal to the force due to the action of this pressure on a circle whose diameter is equal to the conjugate axis of the ellipse.

00- The cylinder is the limiting form of the unduloid when the rolling ellipse becomes a circle.

00When the ellipse differs infinitely little from a circle, the equation of the meridian line becomes approximately y = a+c sin (x/a) where c is small.

00tive than the first, is an ellipse having the luminous point for a focus, and its centre at the foot of the perpendicular from the luminous point to the refracting line.

00The evolute of this ellipse is the caustic required.

00Arrow heads at the ends of an axis of an ellipse indicate tension as distinct from compression, and the semi-axes in magnitude and direction represent the principal stresses.

00Above this entrance it widens into an ellipse a mile long, half a mile broad and 15 ft.

00Newton replied promptly, "an ellipse," and on being questioned by Halley as to the reason for his answer he replied, " Why, I have calculated it."

00Similarly any other property might be used as a definition; an ellipse is the locus of a point such that the sum of its distances from two fixed points (the foci) is constant, &c., &c.

00The Greek geometers were perfectly familiar with the property of an ellipse which in the Cartesian notation is x 2 /a 2 +y 2 /b 2 =1, the equation of the curve; but it was as one of a number of properties, and in no wise selected out of the others for the characteristic property of the curve.

00Thus 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.

00The 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.

00Secondly, if two of the intersections coincide, say if the line infinity meets the curve in a onefold point and a twofold point, both of them real, then there is always one asymptote: the line infinity may at the twofold point touch the curve, and we have the parabolic hyperbolas; or the twofold point may be a singular point, - viz., a crunode giving the hyperbolisms of the hyperbola; an acnode, giving the hyperbolisms of the ellipse; or a cusp, giving the hyperbolisms of the parabola.

00As regards the so-called hyperbolisms, observe that (besides the single asymptote) we have in the case of those of the hyperbola two parallel asymptotes; in the case of those of the ellipse the two parallel asymptotes become imaginary, that is, they disappear; and in the case of those of the parabola they become coincident, that is, there is here an ordinary asymptote, and a special asymptote answering to a cusp at infinity.

00It is to be remarked that the classification mixes together non-singular and singular curves, in fact, the five kinds presently referred to: thus the hyperbolas and the divergent parabolas include curves of every kind, the separation being made in the species; the hyperbolisms of the hyperbola and ellipse, and the trident curve, are nodal; the hyperbolisms of the parabola, and the cubical parabola, are cuspidal.

00Using a powerful and elaborate analysis, Adams ascertained that this cluster of meteors, which belongs to the solar system, traverses an elongated ellipse in 334 years, and is subject to definite perturbations from the larger planets, Jupiter, Saturn and Uranus.

00This circle, projected in Q perspective as an ellipse, is shown in X the figure.

00the axes and eccentricity of the ellipse, and the position of the plane in which it lies.

00==Problem of Three Bodies== As soon as the general law of gravitation was fully apprehended, it became evident that, owing to the attraction of each planet upon all the others, the actual motion of the planets must deviate from their motion in an ellipse according to Kepler's laws.

00(I) The motion of such a planet may take place not only in an ellipse but in any curve of the second order; an ellipse, hyperbola, or parabola, the latter being the bounding curve between the other two.

00The ellipse is therefore the only closed orbit.

00If projected with this velocity in any direction the point of projection will be at the end of the minor axis of the orbit, because this is the only point of an ellipse of which the distance from the focus is equal to the semi-major axis of the curve, and therefore the only point at which the distance of the body from the sun is equal to its mean distance.

00These conditions are: - (I) The smallness of the masses of the planets in comparison with that of the sun, in consequence of which the orbit of each planet deviates but slightly from an ellipse during any one revolution; (2) the fact that the orbits of the planets are nearly circular, and the planes of their orbits but slightly inclined to each other.

00This is called the osculating orbit: The essential principle of Lagrange's elegant method consists in determining the variations of this osculating ellipse, the co-ordinates and velocities of the planet being ignored in the determination.

00He long adhered to the traditional belief that all celestial revolutions must be performed equably in circles; but a laborious computation of seven recorded oppositions of Mars at last persuaded him that the planet travelled in an ellipse, one focus of which was occupied by the sun.

00Bessel announced, in December 1838, the perspective yearly shifting of 61 Cygni in an ellipse with a mean radius of about one-third of a second.

00The orbit of the moon around the earth, though not a fixed curve of any class, is elliptical in form, and may be represented by an ellipse which is constantly changing its form and position, and has the earth in one of its foci.

00ELLIPSE (adapted from Gr.

00Definitions in piano are generally more useful; of these the most important are: (I) the ellipse is the conic section which has its.

00eccentricity less than unity: this involves the notion of one directrix and one focus; (2) the ellipse is the locus of a point the sum of whose distances from two fixed points is constant: this involves the notion of two foci.

00The relation of the ellipse to the other conic sections is treated in the articles Conic Section and Geometry; in this article a summary of the properties of the curve will be given.

00To investigate the form of the curve use may be made of the definition: the ellipse is the locus of a point which moves so that the ratio of its distance from a fixed point (the focus) to its distance from a straight line (the directrix) is constant and is less than unity.

00The most important relation between the co-ordinates of a point on an ellipse is: if N be the foot of the perpendicular from a point P, then the square on PN bears a constant ratio to the product of the segments AN, NA' of the major axis, this ratio being the square of the ratio of the minor to the major axis; symbolically PN2= AN.NA'(CB/CA) 2.

00From this or otherwise it is readily deduced that the ordinates of an ellipse and of the circle described on the major axis are in the ratio of the minor to the major axis.

00If the tangents be at right angles, then the locus of the point is a circle having the same centre as the ellipse; this is named the director circle.

00In analytical geometry, r the equation axe+2hxy+bye+2gx+2fy+ c = o represents an ellipse when ab > h 2; if the centre of the curve be the origin, the equation is a 1 x 2 +2h 1 xy+b i y 2 =C 1, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax e +By 2 = C. The simplest form is x 2 /a 2 +y 2 /b 2 = 1, in which the centre is the origin and the major and minor axes the axes of co-ordinates.

00It is obvious that the co-ordinates of any point on an ellipse may be expressed in terms of a single parameter, the abscissa being a cos q4, and the ordinate b sin 43, since on eliminating 4 between x = a cos and y = b sin 4) we obtain the equation to the ellipse.

00The angle cp is termed the eccentric angle, and is geometrically represented as the angle between the axis of x (the major axis of the ellipse) and the radius of a point on the auxiliary circle which has the same abscissa as the point on the ellipse.

00The area of the ellipse is 7rab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the calculus.

00An ellipse can generally be described to satisfy any five conditions.

00Of practical importance are the following constructions: - (I) Given the axes; (2) given the major axis and the foci; (3) given the focus, eccentricity and directrix; (4) to construct an ellipse (approximately) by means of circular arcs.

00If now the strip be moved so that the point a is always on the minor axis, and the point b on the major axis, the point P describes the ellipse.

00The intersections of the lines drawn from corresponding points are points on the ellipse.

00The thread is now stretched taut by a pencil, and the pencil moved; the curve traced out is the desired ellipse.

00Let P be the intersection of the line SL with the line RAM, then it can be readily shown that P is a point on the ellipse.

00Then with centre 0 1 and radius OJ, =OA 1, describe an arc. By reflecting the two arcs thus described over the centre the ellipse is approximately described.

00The path of the extremity of the vector is then in general an ellipse, traversed in a right-handed direction to an observer receiving the light when a - (3 is between o and 7r, or between - 7r and - air, and in a left-handed direction if this angle be between 7r and 27, or between o and - 7r.

00In ancient geometry the name was restricted to the three particular forms now designated the ellipse, parabola and hyperbola, and this sense is still retained in general works.

00One definition, which is of especial value in the geometrical treatment of the conic sections (ellipse, parabola and hyperbola) in piano, is that a conic is the locus of a point whose distances from a fixed point (termed the focus) and a fixed line (the directrix) are in constant ratio.

00This ratio, known as the eccentricity, determines the nature of the curve; if it be greater than unity, the conic is a hyperbola; if equal to unity, a parabola; and if less than unity, an ellipse.

00is projected depends upon the relation of the "vanishing line" to the circle; if it intersects it in real points, then the projection is a hyperbola, if in imaginary points an ellipse, and if it touches the circle, the projection is a parabola.

00the line at infinity intersects the hyperbola in real points, the ellipse in imaginary points, and the parabola in coincident real points.

00A conic may also be regarded as the polar reciprocal of a circle for a point; if the point be without the circle the conic is an ellipse, if on the circle a parabola, and if within the circle a hyperbola.

00An important property of confocal systems is that only two confocals can be drawn through a specified point, one being an ellipse, the other a hyperbola, and they intersect orthogonally.

00The definitions given above reflect the intimate association of these curves, but it frequently happens that a particular conic is defined by some special property (as the ellipse, which is the locus of a point such that the sum of its distances from two fixed points is constant); such definitions and other special properties are treated in the articles Ellipse, Hyperbola and Parabola.

00When the cutting plane is inclined to the base of the cone at an angle less than that made by the sides of the cone, the latus rectum is greater than the intercept on the ordinate, and we obtain the ellipse; if the plane is inclined at an equal angle as the side, the latus rectum equals the intercept, and we obtain the parabola; if the inclination of the plane be greater than that of the side, we obtain the hyperbola.

00Pappus in his commentary on Apollonius states that these names were given in virtue of the above relations; but according to Eutocius the curves were named the parabola, ellipse or hyperbola, according as the angle of the cone was equal to, less than, or greater than a right angle.

00His proofs are generally long and clumsy; this is accounted for in some measure by the absence of symbols and technical terms. Apollonius was ignorant of the directrix of a conic, and although he incidentally discovered the focus of an ellipse and hyperbola, he does not mention the focus of a parabola.

00He also considered the two branches of a hyperbola, calling the second branch the "opposite" hyperbola, and shows the relation which existed between many metrical properties of the ellipse and hyperbola.

00We may also notice Kepler's approximate value for the circumference of an ellipse (if the semi-axes be a and b, the approximate circumference is ir(a+b)).

00The major axis of the ellipse will be along the East-West line and the minor axis of the ellipse will be along the East-West line and the minor axis will be North-South.

00The mean is trimmed since some incorrect ellipse fits produce extremely deviant parameter estimates that have a large influence on the mean.

00draughtiddle entrance can emit a strong draft from its sizeable ellipse.

00earth's orbit around the sun is elliptical with the sun at one focus of the ellipse, not the center.

00eccentricity of an ellipse is a measure of how fat (or thin) it is.

00Your ruler should look like this: Figure 9: A ruler for drawing the ellipse.

00Using the ellipse Tool, create a circular ellipse roughly 20 pixels in diameter on the left edge of the stage.

00The tricky part is making an ellipse at a point different from the origin.

00The shadow on the Earth's surface was a very elongated ellipse in these places.

00ellipse at a point different from the origin.

00The tendrils are moderately thin and long; one made a narrow ellipse in 5 hrs.

00Parameters: w - The overall width of the full ellipse of which this arc is a partial section.

00The smaller ellipse include the position of thin disk stars, the larger one that of thick disk stars.

00Figure 6 - Start with a vertical ellipse Choose Arrange, Convert to Curves (CTRL + Q ).

00ellipse layer should be still switched off) to select it.

00ellipse parameter data is followed by the associated Fourier descriptor values for each profile.

00ellipse tools and tools for drawing joined and radial lines.

00ellipse lifeline to cover 30 frames in the Timeline.

00This quantity is equal to the width of the error ellipse orthogonal to the visibility vector.

00Therefore, all that is needed to produce a sensitive refractive index sensor is a sensitive measure of the rotation of the polarization ellipse.

00BBC BASIC for Windows can draw only axis-aligned ellipses; to draw an angled ellipse use the ELLIPSE library routines.

00Principal strains, strain ellipsoid and strain ellipse; special types of strain ellipsoid and Flinn plots.

00The shadow on the Earth's surface was a very elongated ellipse in these places.

00Stars outside the thick disk ellipse belong to the galactic halo.

00major axis of the ellipse will be along the East-West line and the minor axis will be North-South.

00That 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.

00While resembling the parabola in extending to infinity, the curve has closest affinities to the ellipse.

00Thus it has a real centre, two foci, two directrices and two vertices; the transverse axis, joining the vertices, corresponds to the major axis of the ellipse, and the line through the centre and perpendicular to this axis is called the conjugate axis, and corresponds to the minor axis of the ellipse; about these axes the curve is symmetrical.

00The passage of the maximum turgidity round the stem may vary in rapidity in different places, causing the circle to be replaced by an ellipse.

00It is the envelope of circles described on the central radii of an ellipse as diameters.

00In 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.

00If 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).

00But unless the orbit is an ellipse the body will never complete a revolution, but will recede indefinitely from the centre of motion.

00Elliptic 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.

00In 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.

00From the properties of the ellipse, A is the pericentre or nearest point of the orbit to the centre of attraction and B the apocentre or most distant point.

00Representing by P this position, it follows that the area of that portion of the ellipse contained between the radii vectores FB and FP will bear the same ratio to the whole area of the ellipse that t does to T, the time of revolution.

00Nevertheless, it should be observed that our globes take no account of the oblateness of our sphere; but as the difference in length between the circumference of the equator and the perimeter of a meridian ellipse only amounts to o 16%, it could be shown only on a globe of unusual size.

00The 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.

00(9) Turning the axes to make them coincide with the principal axes of the area A, thus making f f xydA = o, xh = - a 2 cos a, y h = - b 2 sin a, (io) where ffx2dA=Aa2, ffy 2 dA= Ab 2, (II) a and b denoting the semi-axes of the momental ellipse of the area.

00This 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.

00within 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.

00Thus the core of a circle or an ellipse is a concentric circle or ellipse of one quarter the size.

00An 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.

00of 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.

00(6) Then =o over the ellipse n = a, and the hyperbola t = (, so that these may be taken as fixed boundaries; and %,1.

00Over any ellipse n, moving with components U and V of velocity, =i+Uy-Vx=[msh(n-a) cos (3+Ucshn] sin k -[msh(n-a) sin (3+Vcchn] cos h; (7) so that ' =o, if U c sh n cos R, V = c ch n sin a, (8) m sh(n - a) m sh(n - a).

00having 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.

00At 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.

00An 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.

00Similarly, 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).

00Next 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.

00msh2(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.

00The 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.

00Inside a cylinder ?1 = - I iR (' x + yi) 2a2 /(a2 +b 2 '), (Io) 92+1 i 2 i = I ZR (x + yi) 2b2 I (a2 +b 2), (21) and for the interspace, the ellipse being fixed, and a l revolving with angular velocity R (1 +11/11= - giRc 2 sh 2 (n-a+ i)(ch 2a+I)/sh 2 (a i -a), (12) 42+1,'21 = *iRc 2 sh 2(na+Ei)(ch 2a-1)/sh 2(a i - a), (13) satisfying the condition that 4/ 1 and ' // 2 are zero over n = a, and over n =a 1 constant values.

00They are probably dispersed pretty evenly along a very extended ellipse agreeing closely in its elements with comet 1862: III.

00Sca, 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.

0035 The Ellipse and the Ellipsoid.

00- For elementary mensuration the ellipse is to be regarded as obtained by projection of the circle, and the ellipsoid by projection of the sphere.

00Hence 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.

00If 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).

00If 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.

00If 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.

00It 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.

00If 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.

00It is clearly the form of the fundamental property (expressed in the terminology of the "application of areas") which led him to call the curves for the first time by the names parabola, ellipse, hyperbola.

00As the star decreases in latitude, this circle will be viewed more and more obliquely, becoming a flatter and flatter ellipse until, with A is zero latitude, it degenerates into a straight line (fig.

00The major axis of any such aberrational ellipse is always parallel to AC, i.e.

00contains also (I), under the head of the de determinate sectione of Apollonius, lemmas which, closely examined, are seen to be cases of the involution of six points; (2) important lemmas on the Porisms of Euclid (see PoRIsM); (3) a lemma upon the Surface Loci of Euclid which states that the locus of a point such that its distance from a given point bears a constant ratio to its distance from a given straight line is a conic, and is followed by proofs that the conic is a parabola, ellipse, or hyperbola according as the constant ratio is equal to, less than or greater than i (the first recorded proofs of the properties, which do not appear in Apollonius).

00Proposition 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.

00EQUATION OF THE CENTRE, in astronomy, the angular distance, measured around the centre of motion, by which a planet moving in an ellipse deviates from the mean position which it would occupy if it moved uniformly.

00The 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.

00The star thus appears to describe a small ellipse in the sky, and the nearer the star, the larger will this ellipse appear.

00The greatest displacement of the star from its mean position (the semi-axis major of the ellipse) is called its parallax.

00Thus 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.

00the 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.

00Again, 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.

00which may be called the momental ellipse at 0.

00It possesses thi property that the radius of gyration about any diameter is half thi distance between the two tangents which are parallel to that diameter, In the case of a uniform triangular plate it may be shown that thi momental ellipse at G is concentric, similar and similarly situatec to the ellipse which touches the sides of the triangle at their middle points.

00The 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.

00Hence the path is approximately, an ellipse, and the period is 2sr ~/ (l/g).

00Now 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.

00In the case of an ellipse described about the centre as pole we have ~=aI+b2_r2; (12)

00This merely shows that a particular ellipse may be described under the law of the direct distance provided the circumstances of projection be suitably adjusted.

00But 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.

00In 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.

00The pole 0 of the hodograph is inside on or outside the circle, according as the orbit is an ellipse, parabola or hyperbola.

00The ratio of the axes of the ellipse is sec a, the longer axis being in the plane of 0.

00When both are near the lowest point the horizontal projection of the path is approximately an ellipse, as shown in 13; a closet investigation shows that the ellipse is to be regarded as revolving about its centre with the angular velocity ~ab~~/l2, where a, b are the semi-axes.

00To illustrate this subject, it may be mentioned that an ellipse rotating about one focus rolls completely round in outside gearing with an equal and similar ellipse also rotating about one focus, the distance between the axes of rotation being equal to the major axis of the ellipses, and the velocity ratio varying from to I ~eccentricitY an hyperbola rotating about its further focus rolls in inside gearing, through a limited arc, with an equal and similar hyperbola rotating about its nearer focus, the distance between the axes of rotation being equal to the axis of the hypereccentricity + I

00Its earliest form is a rough ellipse transfixed by an upright line, cp. In various Semitic alphabets this has been altered out of recognition, apparently from the writing of the symbol in cursive handwriting without lifting the pen.

00When the conic is an ellipse the meridian line is in the form of a series of waves, and the film itself has a series of alternate swellings and contractions as represented in figs.

00When the ellipse becomes a circle, the meridian line becomes a straight line parallel to the axis, and the film passes into the form of a cylinder of revolution.

00As the ellipse degenerates into the straight line joining its foci, the contracted parts of the unduloid become narrower, till at last the figure becomes a series of spheres in contact.

00In all these cases the internal pressure exceeds the external by 2T/a where a is the semi-transverse axis of the conic. The resultant of the internal pressure and the surface-tension is equivalent to a tension along the axis, and the numerical value of this tension is equal to the force due to the action of this pressure on a circle whose diameter is equal to the conjugate axis of the ellipse.

00- The cylinder is the limiting form of the unduloid when the rolling ellipse becomes a circle.

00When the ellipse differs infinitely little from a circle, the equation of the meridian line becomes approximately y = a+c sin (x/a) where c is small.

00About 3 metres in front of them was arranged a pair of smaller horizontal j aeroplanes, shaped like a long narrow ellipse, which formed the rudder that effected changes of elevation, the driver being able by means of a lever to incline them up or down according as he desired to ascend or descend.

00tive than the first, is an ellipse having the luminous point for a focus, and its centre at the foot of the perpendicular from the luminous point to the refracting line.

00The evolute of this ellipse is the caustic required.

00Arrow heads at the ends of an axis of an ellipse indicate tension as distinct from compression, and the semi-axes in magnitude and direction represent the principal stresses.

00Above this entrance it widens into an ellipse a mile long, half a mile broad and 15 ft.

00Newton replied promptly, "an ellipse," and on being questioned by Halley as to the reason for his answer he replied, " Why, I have calculated it."

00Similarly any other property might be used as a definition; an ellipse is the locus of a point such that the sum of its distances from two fixed points (the foci) is constant, &c., &c.

00The Greek geometers were perfectly familiar with the property of an ellipse which in the Cartesian notation is x 2 /a 2 +y 2 /b 2 =1, the equation of the curve; but it was as one of a number of properties, and in no wise selected out of the others for the characteristic property of the curve.

00Thus 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.

00The 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.

00Secondly, if two of the intersections coincide, say if the line infinity meets the curve in a onefold point and a twofold point, both of them real, then there is always one asymptote: the line infinity may at the twofold point touch the curve, and we have the parabolic hyperbolas; or the twofold point may be a singular point, - viz., a crunode giving the hyperbolisms of the hyperbola; an acnode, giving the hyperbolisms of the ellipse; or a cusp, giving the hyperbolisms of the parabola.

00As regards the so-called hyperbolisms, observe that (besides the single asymptote) we have in the case of those of the hyperbola two parallel asymptotes; in the case of those of the ellipse the two parallel asymptotes become imaginary, that is, they disappear; and in the case of those of the parabola they become coincident, that is, there is here an ordinary asymptote, and a special asymptote answering to a cusp at infinity.

00It is to be remarked that the classification mixes together non-singular and singular curves, in fact, the five kinds presently referred to: thus the hyperbolas and the divergent parabolas include curves of every kind, the separation being made in the species; the hyperbolisms of the hyperbola and ellipse, and the trident curve, are nodal; the hyperbolisms of the parabola, and the cubical parabola, are cuspidal.

00Using a powerful and elaborate analysis, Adams ascertained that this cluster of meteors, which belongs to the solar system, traverses an elongated ellipse in 334 years, and is subject to definite perturbations from the larger planets, Jupiter, Saturn and Uranus.

00This circle, projected in Q perspective as an ellipse, is shown in X the figure.

00the axes and eccentricity of the ellipse, and the position of the plane in which it lies.

00==Problem of Three Bodies== As soon as the general law of gravitation was fully apprehended, it became evident that, owing to the attraction of each planet upon all the others, the actual motion of the planets must deviate from their motion in an ellipse according to Kepler's laws.

00(I) The motion of such a planet may take place not only in an ellipse but in any curve of the second order; an ellipse, hyperbola, or parabola, the latter being the bounding curve between the other two.

00The ellipse is therefore the only closed orbit.

00If projected with this velocity in any direction the point of projection will be at the end of the minor axis of the orbit, because this is the only point of an ellipse of which the distance from the focus is equal to the semi-major axis of the curve, and therefore the only point at which the distance of the body from the sun is equal to its mean distance.

00These conditions are: - (I) The smallness of the masses of the planets in comparison with that of the sun, in consequence of which the orbit of each planet deviates but slightly from an ellipse during any one revolution; (2) the fact that the orbits of the planets are nearly circular, and the planes of their orbits but slightly inclined to each other.

00This is called the osculating orbit: The essential principle of Lagrange's elegant method consists in determining the variations of this osculating ellipse, the co-ordinates and velocities of the planet being ignored in the determination.

00He long adhered to the traditional belief that all celestial revolutions must be performed equably in circles; but a laborious computation of seven recorded oppositions of Mars at last persuaded him that the planet travelled in an ellipse, one focus of which was occupied by the sun.

00Bessel announced, in December 1838, the perspective yearly shifting of 61 Cygni in an ellipse with a mean radius of about one-third of a second.

00The orbit of the moon around the earth, though not a fixed curve of any class, is elliptical in form, and may be represented by an ellipse which is constantly changing its form and position, and has the earth in one of its foci.

00The eccentricity of the ellipse is in the general average about 0.055, whence the moon is commonly more than i' further from the earth at apogee than at perigee.

00ELLIPSE (adapted from Gr.

00Definitions in piano are generally more useful; of these the most important are: (I) the ellipse is the conic section which has its.

00eccentricity less than unity: this involves the notion of one directrix and one focus; (2) the ellipse is the locus of a point the sum of whose distances from two fixed points is constant: this involves the notion of two foci.

00The relation of the ellipse to the other conic sections is treated in the articles Conic Section and Geometry; in this article a summary of the properties of the curve will be given.

00To investigate the form of the curve use may be made of the definition: the ellipse is the locus of a point which moves so that the ratio of its distance from a fixed point (the focus) to its distance from a straight line (the directrix) is constant and is less than unity.

00The most important relation between the co-ordinates of a point on an ellipse is: if N be the foot of the perpendicular from a point P, then the square on PN bears a constant ratio to the product of the segments AN, NA' of the major axis, this ratio being the square of the ratio of the minor to the major axis; symbolically PN2= AN.NA'(CB/CA) 2.

00From this or otherwise it is readily deduced that the ordinates of an ellipse and of the circle described on the major axis are in the ratio of the minor to the major axis.

00If the tangents be at right angles, then the locus of the point is a circle having the same centre as the ellipse; this is named the director circle.

00In analytical geometry, r the equation axe+2hxy+bye+2gx+2fy+ c = o represents an ellipse when ab > h 2; if the centre of the curve be the origin, the equation is a 1 x 2 +2h 1 xy+b i y 2 =C 1, and if in addition a pair of conjugate diameters are the axes, the equation is further simplified to Ax e +By 2 = C. The simplest form is x 2 /a 2 +y 2 /b 2 = 1, in which the centre is the origin and the major and minor axes the axes of co-ordinates.

00It is obvious that the co-ordinates of any point on an ellipse may be expressed in terms of a single parameter, the abscissa being a cos q4, and the ordinate b sin 43, since on eliminating 4 between x = a cos and y = b sin 4) we obtain the equation to the ellipse.

00The angle cp is termed the eccentric angle, and is geometrically represented as the angle between the axis of x (the major axis of the ellipse) and the radius of a point on the auxiliary circle which has the same abscissa as the point on the ellipse.

00The area of the ellipse is 7rab, where a, b are the semi-axes; this result may be deduced by regarding the ellipse as the orthogonal projection of a circle, or by means of the calculus.

00An ellipse can generally be described to satisfy any five conditions.

00Of practical importance are the following constructions: - (I) Given the axes; (2) given the major axis and the foci; (3) given the focus, eccentricity and directrix; (4) to construct an ellipse (approximately) by means of circular arcs.

00If now the strip be moved so that the point a is always on the minor axis, and the point b on the major axis, the point P describes the ellipse.

00The intersections of the lines drawn from corresponding points are points on the ellipse.

00The thread is now stretched taut by a pencil, and the pencil moved; the curve traced out is the desired ellipse.

00Let P be the intersection of the line SL with the line RAM, then it can be readily shown that P is a point on the ellipse.

00Then with centre 0 1 and radius OJ, =OA 1, describe an arc. By reflecting the two arcs thus described over the centre the ellipse is approximately described.

00The path of the extremity of the vector is then in general an ellipse, traversed in a right-handed direction to an observer receiving the light when a - (3 is between o and 7r, or between - 7r and - air, and in a left-handed direction if this angle be between 7r and 27, or between o and - 7r.

00In ancient geometry the name was restricted to the three particular forms now designated the ellipse, parabola and hyperbola, and this sense is still retained in general works.

00One definition, which is of especial value in the geometrical treatment of the conic sections (ellipse, parabola and hyperbola) in piano, is that a conic is the locus of a point whose distances from a fixed point (termed the focus) and a fixed line (the directrix) are in constant ratio.

00This ratio, known as the eccentricity, determines the nature of the curve; if it be greater than unity, the conic is a hyperbola; if equal to unity, a parabola; and if less than unity, an ellipse.

00is projected depends upon the relation of the "vanishing line" to the circle; if it intersects it in real points, then the projection is a hyperbola, if in imaginary points an ellipse, and if it touches the circle, the projection is a parabola.

00the line at infinity intersects the hyperbola in real points, the ellipse in imaginary points, and the parabola in coincident real points.

00A conic may also be regarded as the polar reciprocal of a circle for a point; if the point be without the circle the conic is an ellipse, if on the circle a parabola, and if within the circle a hyperbola.

00An important property of confocal systems is that only two confocals can be drawn through a specified point, one being an ellipse, the other a hyperbola, and they intersect orthogonally.

00The definitions given above reflect the intimate association of these curves, but it frequently happens that a particular conic is defined by some special property (as the ellipse, which is the locus of a point such that the sum of its distances from two fixed points is constant); such definitions and other special properties are treated in the articles Ellipse, Hyperbola and Parabola.

00Menaechmus discussed three species of cones (distinguished by the magnitude of the vertical angle as obtuse-angled, right-angled and acuteangled), and the only section he treated was that made by a plane perpendicular to a generator of the cone; according to the species of the cone, he obtained the curves now known as the hyperbola, parabola and ellipse.

00In his extant Conoids and Spheroids he defines a conoid to be the solid formed by the revolution of the parabola and hyperbola about its axis, and a spheroid to be formed similarly from the ellipse; these solids he discussed with great acumen, and effected their cubature by his famous "method of exhaustions."

00When the cutting plane is inclined to the base of the cone at an angle less than that made by the sides of the cone, the latus rectum is greater than the intercept on the ordinate, and we obtain the ellipse; if the plane is inclined at an equal angle as the side, the latus rectum equals the intercept, and we obtain the parabola; if the inclination of the plane be greater than that of the side, we obtain the hyperbola.

00Pappus in his commentary on Apollonius states that these names were given in virtue of the above relations; but according to Eutocius the curves were named the parabola, ellipse or hyperbola, according as the angle of the cone was equal to, less than, or greater than a right angle.

00His proofs are generally long and clumsy; this is accounted for in some measure by the absence of symbols and technical terms. Apollonius was ignorant of the directrix of a conic, and although he incidentally discovered the focus of an ellipse and hyperbola, he does not mention the focus of a parabola.

00He also considered the two branches of a hyperbola, calling the second branch the "opposite" hyperbola, and shows the relation which existed between many metrical properties of the ellipse and hyperbola.

00We may also notice Kepler's approximate value for the circumference of an ellipse (if the semi-axes be a and b, the approximate circumference is ir(a+b)).

00In that year, President Calvin Coolidge's wife, Grace Coolidge, allowed the D.C. public school district to put up a tree on the Ellipse, near the Washington Monument.

00While the National tree was originally a cut tree, it is now a permanent, living fixture on the Ellipse.

00Golden Ellipse: Simple, elegant, and stylishly austere, the Golden Ellipse is a flagship of the entire Patek collection.

00

The word usage examples above have been gathered from various sources to reflect current and historial usage. They do not represent the opinions of YourDictionary.com.