HYPERBOLA, a geometrical curve; as first conceived, probably by Menaechmus (c. 35o B.c.), a section of an obtuse angled circular cone made by a plane orthogonal to a cone-ele ment. About 2 20 B.C. Apollonius generalized the notion into that of a section of any circular cone, made by a plane cutting one element and diverging from the opposite at an angle greater than the vertical, so as to cut also the other nappe of the cone, the double section extending indefi nitely both ways without meeting the opposite cone-edge. This is of course indicated in the central rectangular equation of the hyperbola I, in which, for x (or y) real and infinite (oo ), y (or x) is also real and oo ; i.e., the curve has two real points at 00 . This equation of the hyperbola is seen to differ from that of the ellipse only in the sign of and accordingly the hyperbola is the counter part of the ellipse, reflecting its properties, mutatis mutandis, although we must suppose the branches continuous through 00 to detect any likeness in form. Thus (see fig.) the ratio of the distances of any point of the hyperbola from either focus (F. F') and the corresponding directrix (DR), the polar of the pole F, is a constant, the eccentricity e= but the changed sign of makes e> I. Similarly, the difference (not the sum, as in the ellipse) of the focal distances FP, F'P of any point P (x y) of the hyperbola is a constant, the transverse axis is aa; Either of these properties may be taken as a definition of the hyperbola. Again, it is the changed sign of that marks the two points at 00 (imaginary in the ellipse) as real in the hyperbola. The tangent and normal still bisect the angles between the focal radii, r, r', to any point of the hyperbola, but they exchange positions, the tangent lying within, between F and F', the normal without. Thus, an ellipse and a hyperbola that are confocal intersect orthogonally, a property that extends to confocal coni coids, making possible important orthogonal co-ordinate systems of such surfaces. The ends (±a, o) of the transverse axis, AA', are real, but these (o, -!- ib) of its conjugate, BB', are imaginary in the hyperbola—a relation exactly reversed in the conjugate hyperbola — by' a2 i In the ellipse, the asymptotes, or tangents at 00 , are imaginary, = x/a = ±i y/b but in the hyperbola both are real, — = o, x/a = ±y/b, plainly common to the hyperbola and its conjugate and forming the most striking feature of the curve. The analogue to the cir cle in relation to the ellipse, I, is the rectangular (equilateral, equiaxial) hyperbola in relation to i (both obtained by putting b=a). Every other hyperbola is obtained from this, as every ellipse from the circle, by vertical compression (or expansion) in the ratio b/a. On taking the asymptotes, ay±bx=o, as coordinate axes, the equation of the hyperbola becomes depiction of various physical phenomena in which two tudes vary inversely, their product being a constant, as in xy = By the general law, the equation of the tangent at any point x'' y) is xy'-kx'y= hence, for y=o, and for x=o, i.e., the tan gent-intercept between the asymptotes is bisected at the tangent point. Hence the triangle of asymptotes and tangents is constant in area. This tangent-intercept equals the parallel diameter con jugate to the diameter through the tangent-point (P). Obviously the two intercepts (CA, C'A') of any chord (being parallel to a tangent), between the hyperbola and its asymptotes, are equal (see fig.) .