HYPERBOLA, a conic section, consisting of two open branches, each extending to infinity.
The hyperbola which has for its transverse and conjugate axes the transverse and conjugate axes of another hyperbola is said to be the conjugate hyperbola.
The geometry of the rectangular hyperbola is simplified by the fact that its principal axes are equal.
Analytically the hyperbola is given by ax2+2hxy+by2+2gx+ 2fy+c=o wherein ab>h 2 .
In the rectangular hyperbola a =b; hence its equation is x 2 - y 2 = O.
Referred 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.
That it is the first positive pedal of a rectangular hyperbola with regard to the centre.
The 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.
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).
Denoting 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.
He 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.
(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.
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).
"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.
And 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....
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.
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.
A 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.
But 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'.
A 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.
In 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.
Contains 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).
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.
Thus 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.
It appears that the orbit is an effipse, parabola or hyperbola according as v2 is less than, equal to, or greater than 2/sir.
To 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
When the conic is a hyperbola the meridian line is in the form of a looped curve (fig.
When 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.
When 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.
If 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.