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hyperbola

But Landen's capital discovery is that of the theorem known by his name (obtained in its complete form in the memoir of 1775, and reproduced in the first volume of the Mathematical Memoirs) for the expression of the arc of an hyperbola in terms of two elliptic arcs.

31The hyperbola which has for its transverse and conjugate axes the transverse and conjugate axes of another hyperbola is said to be the conjugate hyperbola.

32The same name is also given to the first positive pedal of any central conic. When the conic is a rectangular hyperbola, the curve is the lemniscate of Bernoulli previously described.

32it appears that the orbit is an effipse, parabola or hyperbola according as v2 is less than, equal to, or greater than 2/sir.

21Referred to the asymptotes as axes the general equation becomes xy 2 obviously the axes are oblique in the general hyperbola and rectangular in the rectangular hyperbola.

22that it is the first positive pedal of a rectangular hyperbola with regard to the centre.

10A solution by means of the parabola and hyperbola was given by Dionysodorus of Amisus (c. 1st century B.c), and a similar problem - to construct a segment equal in volume to a given segment, and in surface to another segment - was solved by the Arabian mathematician and astronomer, Al Kuhi.

10But if the pressure-curve is a straight line F'CP sloping upwards, cutting AM behind A in F', the energy-curve will be a parabola curving upwards, and the velocity-curve a hyperbola with center at F'.

10A relation which is of historical interest connects the logarithmic function with the quadrature of the hyperbola, for, by considering the equation of the hyperbola in the form xy=const., it is evident that the area included between the arc of a hyperbola, its nearest asymptote, and two ordinates drawn parallel to the other asymptote from points on the first asymptote distant a and b from their point of intersection, is proportional to log bla.

10When the conjugate axis of the hyperbola increases without limit, the loops of the nodoid are crowded on one another, and each becomes more nearly a ring of circular section, without, however, ever reaching this form.

10If the asymptotes be perpendicular, or, in other words, the principal axes be equal, the curve is called the rectangular hyperbola.

00A second took place when Vieta pointed to Apollonius's problem of taction as not yet being mastered, and Adriaan van Roomen gave a solution by the hyperbola.

00These problems were also attacked by the Arabian mathematicians; Tobit ben Korra (836-901) is credited with a solution, while Abul Gud solved it by means of a parabola and an equilateral hyperbola.

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.

00This problem, which is sometimes known as the Apollonian Problem, was proposed by Vieta in the 16th century to Adrianus Romanus, who gave a solution by means of a hyperbola.

00In one solution of the former problem is the first recorded use of the property of a conic (a hyperbola) with reference to the focus and directrix.

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.

00Thus the boundary of the geometric shadow is a portion of a circle on the roof, but a portion of an hyperbola on the vertical wall.

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.

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

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

00The resultant of the internal pressure and the surface-tension is equivalent to a pressure along the axis equal to that due to a pressure p acting on a circle whose diameter is the conjugate axis of the hyperbola.

00If the second medium be more highly refractive than the first, the secondary caustic is a hyperbola having the same focus and centre as before, and the caustic is the evolute of this curve.

00At such time I found the method of Infinite Series; and in summer 1665, being forced from Cambridge by the plague, I computed the area of the Hyperbola at Boothby, in Lincolnshire, to two and fifty figures by the same method."

00The theorem of the m intersections has been stated in regard to an arbitrary line; in fact, for particular lines the resultant equation may be or appear to be of an order less than m; for instance, taking m= 2, if the hyperbola xy - 1= o be cut by the line y=0, the resultant equation in x is Ox- 1 = o, and there is apparently only the intersection (x 110, y =0); but the theorem is, in fact, true for every line whatever: a curve of the order in meets every line whatever in precisely m points.

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 epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively.

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.

00The genera may be arranged as follows: 1,2,3,4 redundant hyperbolas 5,6 defective hyperbolas 7,8 parabolic hyperbolas 9 hyperbolisms of hyperbola To „ II „ „ parabola 12 trident curve 13 divergent parabolas 14 cubic parabola; and thus arranged they correspond to the different relations of the line infinity to the curve.

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.

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.

00A body moving in a parabola or hyperbola would recede indefinitely from its centre of motion and never return to it.

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.

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.

00In modern notation, if we denote the ordinate by y, the distance of the foot of the ordinate from the vertex (the abscissa) by x, and the latus rectum by p, these relations may be expressed as 31 2 for 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.

00rectangular hyperbola A demand curve that displays unitary elasticity along its length.

00These two lines may be pictured in the in solido definition as the section of a cone by a plane through its vertex and parallel to the plane generating the hyperbola.

00Analytically the hyperbola is given by ax2+2hxy+by2+2gx+ 2fy+c=o wherein ab>h 2 .

00In the rectangular hyperbola a =b; hence its equation is x 2 - y 2 = O.

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).

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

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.

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).

00When the conic is a hyperbola the meridian line is in the form of a looped curve (fig.

00If we assume that the bolograph of solar energy is simply a graph of amorphous radiation from an ideal radiator, so that the con- Temperature stants cl, c 2, of Planck's formula determined terrestrially apply to it, the hyperbola of maximum intensity is XO = 2, 921 X 10 7; and as the sun's maximum intensity occurs for about X =4900, we find the absolute temperature to be 5960Ã‚° abs.

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."

00HYPERBOLA, a conic section, consisting of two open branches, each extending to infinity.

01The geometry of the rectangular hyperbola is simplified by the fact that its principal axes are equal.

01Denoting them by x, y, so that AB is axis of y and a perpendicular through A the axis of x, and rationalizing (26), we have 2 ax 2 - V 2 Xy 2 - V 2 aAy = o, which represents a hyperbola with vertices at 0 and A.

01He then supposed this cylindrical column of water to be divided into two parts, - the first, which he called the " cataract," being an hyperboloid generated by the revolution of an hyperbola of the fifth degree around the axis of the cylinder which should pass through the orifice, and the second the remainder of the water in the cylindrical vessel.

01having 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.

01Similarly, 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).

01(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).

01"In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr Wallis, and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola (for instance, in this series of curves whose common base 0 or axis is x, and the ordinates respectively (I -xx)l, (i (I &c), I perceived that if the areas of the alternate curves, which are x, x 3x 3, x &c., could be interpolated, we should obtain the areas of the intermediate ones, the first of which (I -xx) 1 is the area of the circle.

01And hence I found the required area of the circular segment 2 x3 A x5 il-A to be x - 5 - 7, &c. And in the same manner might be 3 produced the interpolated areas of other curves; as also the area of the hyperbola and the other alternates in this series (1 - (i+xx) 1, (1 --xx) I, &c....

01Sca, 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.

01When the conjugate axis of the hyperbola is made smaller and smaller, the nodoid approximates more and more to the series of spheres touching each other along the axis.

01If we assume that the bolograph of solar energy is simply a graph of amorphous radiation from an ideal radiator, so that the con- Temperature stants cl, c 2, of Planck's formula determined terrestrially apply to it, the hyperbola of maximum intensity is XO = 2, 921 X 10 7; and as the sun's maximum intensity occurs for about X =4900, we find the absolute temperature to be 5960° abs.

01One 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.

01

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