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 .
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.
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.
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 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.
If 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.
At 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."
The 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.
Thus the curve of the first order or right line consists of one branch; but in curves of the second order, or conics, the ellipse and the parabola consist each of one branch, the hyperbola of two branches.
The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively.
The nature of the two kinds of branches is best understood by considering them as projections, in the same way as we in effect consider the hyperbola and the parabola as projections of the ellipse.
The 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.
Secondly, 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.
As 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.
It 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.
(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.
A body moving in a parabola or hyperbola would recede indefinitely from its centre of motion and never return to it.