If the asymptotes be perpendicular, or, in other words, the principal axes be equal, the curve is called the rectangular hyperbola.
We may observe that the asymptotes intersect this circle in the same points as the directrices.
If the tangent at P meets the asymptotes in R, R', then CR.CR' = CS 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.
The isothermals are approximately equilateral hyperbolas (pv= constant), with the axes of p and v for asymptotes, for a gas or unsaturated vapour, but coincide with the isopiestics for a saturated vapour in presence of its liquid.
58) has two vertical asymptotes x = ~ 3/4irx; this shows that however the thickness of a cable be adjusted there is a limit irA to the horizontal span, where A depends on the tensile strength of the material.
The orbit has therefore two asymptotes, inclined at an angle lr/m.
His discussion of curves of the third order turned mainly on the nature of their asymptotes, and depended on the fact that the equation to every such curve can be put into the form pqr-hus = o.
The work falls into two parts, which treat of the asymptotes and singularities of algebraical curves respectively; and extensive use is made of the method of counting constants which plays so large a part in modern geometrical researches.
The two legs of a hyperbolic branch may belong to different asymptotes, and in this case we have the forms which Newton calls inscribed, circumscribed, ambigene, &c.; or they may belong to the same asymptote, and in this case we have the serpentine form, where the branch cuts the asymptote, so as to touch it at its two extremities on opposite sides, or the conchoidal form, where it touches the asymptote on the same side.
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.
There are in some cases points termed centres, or singular or multiple foci (the nomenclature is unsettled), which are the intersections of improper tangents from the two circular points respectively; thus, in the circular cubic, the tangents to the curve at the two circular points respectively (or two imaginary asymptotes of the curve) meet in a centre.
The first book deals with the generation of the three conics; the second with the asymptotes, axes and diameters; the third with various metrical relations between transversals, chords, tangents, asymptotes, &c.; the fourth with the theory of the pole and polar, including the harmonic division of a straight line, and with systems of two conics, which he shows to intersect in not more than four points; he also investigates conics having single and double contact.