left. Thus, it appears that all lines terminated in either branch of the curve, parallel to B B', are bisected by the production of A A', and all lines terminated in the opposite branches, and parallel to A A', are bisected by BB'. (679.) The lines A A' and B B' are therefore conjugate diameters of the curve, the line A A' being called the transverse axis, and the line BB' the conjugate axis. (680.) The leading properties of an hyperbola have a close analogy to those of an ellipse. From the symmetry and equality of the four branches of the curve, contained in the four right angles formed by its axes, it follows, that all right lines, such as E E' (fig. 239.) drawn through the centre C and terminated in opposite branches of the curve, will meet these branches at corresponding points, and that the centre C will be their common point of bisection. The ordinates to the axis EM and E'M', through the extremities of the same diameter, are equal; and it is evident, that another diameter ee', through the other extremities of these ordinates to the axis, will be equal to the former, and equally inclined to the axis, so that MM' will bisect the angles ECe and E' Ce'. (681.) Lines drawn from the foci of an hyperbola to a point P in it, make equal angles with the tangent at that point, in the same manner as was shown to be the case in the ellipse; but in the case of the hyperbola, the tangent lies between these lines FP and F' P, bisecting the angle under them, whereas, in the ellipse, it lay outside them. Tangents to the hyperbola at the vertices A and A' are perpendicular to the axis, and therefore parallel to its ordinates; and, in like manner, tangents to the curve, through the extremities of any diameter, are parallel to the ordinates to that diameter. (682.) If F (fig. 240.) be the focus of an hyperbola, it appears from what has been stated, that rays of light, or heat, or any other principle which obeys the law of reflection, will, if they diverge from F, and are reflected from the curve at P after reflection, follow directions diverging from the other focus F'; and, on the other hand, if rays X P, converging towards F', be reflected by the curve, they will after reflection converge towards the other focus F. Hence, if an hyperboloid of revolution be formed by the revolution of an hyperbola on its transverse axis, and such a surface be endowed with the property of reflection, rays converging to, or diverging from, one focus, may be made to converge to, or diverge from, the other focus. (683.) If C (fig. 241.) be the centre, A the vertex, and F the focus of an hyperbola, take CD a third proportional to CF and CA, and through D draw DK perpendicular to CA; the line DK will be the directrix of the hyperbola, and is distinguished by properties analogous to the directrix of the ellipse (647.). Let P be any points on the hyperbola, from which let the lines Pm be drawn parallel to the axis, and there : fore perpendicular to the directrix, the ratio of each of the lines FP to Pm will be the same as the ratio of FA to AD, or, what is the same, that of FC to AC. Thus the line DK is distinguished by the property of having the distances of all points in the curve from it proportional to the distances of those points from the focus. (684.) From this property, a method of determining the curve, by a series of points, follows, perfectly similar to the method of determining the ellipse by points in reference to its directrix explained in (651.). It is only necessary to draw any number of lines perpendicular to the axis, and from the focus to inflect on each of them a line which shall bear to their distances respectively the same ratio as the distance of the focus from the centre bears to the semi-transverse axis. (685.) If TP (fig. 242.) be a tangent to the hyperbola at P, and PM be an ordinate to the axis at the same point, then, by a property analogous to that of the ellipse, already explained (642.), we shall have CT to CA as CA to CM; and, therefore, the rectangle under CT and CM is always equal to the square of the semi-axis; and, therefore, as CM increases CT must diminish. But as the hyperbola extends indefinitely from A in the direction A P, consisting of an infinite branch, the distance CM increases without limit, as the point of contact P recedes, and therefore the distance CT at the same time diminishes without limit. If the point of contact P be removed to an infinite distance, then CM becomes infinite, and CT vanishes. Hence it appears that the tangent to the curve always intersects the axis between C and A, but that the farther the point of contact is removed from the vertex, the nearer the tangent approaches to the centre; and that the curve has a constant tendency to coincide with a certain line passing through the centre, although it never can actually coincide with such a line, since that would involve the condition of its being at an infinite distance from the centre. (686.) It was shown among the properties of the ellipse, that the square of the ordinate PM to the axis always bears the same ratio to the rectangle under the distances between that ordinate and the extremities AA' of the axis; this ratio being that of the square of the semi-conjugate to the square of the semi-transverse axis. In like manner the square of PM (fig. 242.) bears to the rectangle under MA and MA', the same ratio, wherever the point P is taken. Let a distance CB be taken on the conjugate axis, such that the square of CB shall bear to the square of C A, the same ratio as the square of any ordinate bears to the rectangle under the corresponding segments. This distance C B is considered as the length of the semi-conjugate axis, although it does not, as in the ellipse, meet the curve at B. The properties of the hyperbola may be expressed by the same notation as was used to express the properties of the ellipse in reference to its axes. Let A = CA, B = CB, y = PM, and x СМ. By what has been stated we shall have which is therefore the equation of the hyperbola referred to its axes. Let x = A M. Therefore A + x'= x. Hence the above equation becomes A2 y2 B2 x2 = 2 A Β2 x', which is the equation when the abscissæ are measured from A. If FC = c, we shall have c2 = A2 + B2; and if A2 y2 - A2 (e2-1) x^2 = 2 A3 (e2 – 1) x', (687.) Since the distance CT (fig. 242.) diminishes without limit as the point of contact Precedes, it is evident that when the distance of P is infinite, the tangent would pass through the centre C. Its ultimate direction, or more strictly the direction which limits its position as the point of contact recedes without limit, may be determined by finding the value to which the ratio of PM to MT, or of PM to MC tends when they both become infinite. This will be readily obtained from the equation A2 y2 B2 x2 = A2 B2. Dividing all the terms by A2 x2 we obtain |