DAM

By a uming first one of the given points F, and then the other f, as that to which the moving point is nearest, the difference of the lines I) F and D / in both cases being the same, two hyperbolas DAM and D' A' M' will be described opposite to each other; so that the curve consists of two branches. The points F and/ are the foci of the hyperbola; and C, which bisects the distance between the foci, is its centre. The line A A' is the major or transverse axis: and a straiglit line B B', pa ing through the centre perpendicular to A A', and of such a length that the square of its half C B or C B' is equal to the difference between the squares of C F and C A, is the minor or conjugate axis. The curve may be described mechanically as follows: — Let one end of a string be fastened to F, and the other to K, the extremity of a ruler /D K; and let the difference between the length of the ruler and of the string be equal to A a. Let the other end of the ruler be iixed to the point/, and let the ruler be made to revolve about /as a centre in the plane in which the axes are situated, while the string is stretched by means of a pin D, so that the part of it between K and D is applied close to the edge of the ruler: the point of the pin will by its motion trace a curve line DAM upon the plane, which is one of the hyperbolas required: and If the ruler be made to revolve about the other focus F while the end of the string is fastened to/, the opposite hyperbola will be described by the pin D'. 2. The hyperbola may also be defined as follows: — Let F be a given point, and P Q a straight line given in position; If another point D move in the same plane, so tliat Its distance D F from F shall have always to the perpendicular D E, or its distance from the given line P Q, the constant ratio of two given lines X and Y of which X is greater than Y, the locus of the point D will be a hyperbola. The line P Q is called the directrix and Its distance C G from the centre C is such that C G IS a third proportional to C F and C A. It is obvious that FA is to A G in the given ratio of X to Y. 3. Another distinguishing property of the hyperbola is that the rectangle under A H and H A' is to the square ot the ordinate H D in the ratio of the square of C A to the square of C B. Let C A = a, C B = 6, C H = iand H D = y; then A II • H A': H D2:: a^: 62 that IS, (x — a){x + a): y'^:: a'^: b^, or x"^ — a^: y^:: a^ b^; whenc ey^ ^2 (•''^ — «^). an equation which may be put under this form, 1. «2 62 4. Like the ellipse, the hyperbola may also be defined by a polar equation. Let F D = >•, G A = a, C F the eccentricity = s, and the angle A F D = a; then e'i — rt2 r = ■ —. a + e cos. « The hyperbola has two infinite branches, and it has 579 HYPERCRITICISM. also two asymptotes. Through A, one of the vertices of;; the transverse axis, let a straight line U A h he drawn equal and parallel to B b, the conjugate axis, and bisected at A; the straight lines OH, C h, drawn through the centre, and the extremities of that parallel, are asymptotes, and if produced indefinitely do not meet the curve, though their distance from it becomes le than any as^ ^ ^ signable line. The asymptotes of two opposite hyperbolas are common to both: and they are likewise the asymptotes of two other hyperbolas, L B E', eb e', whose transverse axis B 6 is the conjugate axis of D A D', and whose conjugate axis is the transverse axis of D A D'. These two hyperbolas, L B L' and ebe', are called conjugate to the former pair: and, in general, whatever property belongs to the opposite hyperbolas D A D', d a d', the same belongs also to the conjugate hyperbolas EBE' and e b e'. One of the most remarkable properties of the asymptotes IS the following: — If one of the asymptotes C M (3.) be divided in continued proportion in the points D, E, G, ., and straight lines be drawn from the points of section parallel to the other asymptote, meeting the curve in the points P, Q, R, .; the spaces DPQ E, E Q R G, ., without the curve, are equal. The hyperbolic — =rjp sections C P Q, C Q R, ., ** -"^ are also equal to each other, and to the spaces D P E Q, E Q R G, . Hence, from the nature of logarithms, the sectors C P Q, C P R, ., or the equal spaces D P Q E, D P R G, ., represent the logarithms of the ratios of C D to C E, to C G, .; and if C D represent the unit of the arithmetical scale, the sector C P Q, or the space D P Q E, will expre the logarithm of C E on any logarithmic system depending on the angle of the asymptotes. From the points P, Q, R, . let F d, Q e, II g, be drawn parallel to C M; the ordinates C d, Ce, Cg, .are also in geometrical progre ion decreasing; and wherever the points D, E, G, . are situated, all the parallelograms C dV D,^ e QE, C ^ R G, . are equal. Hence if C M be taken as the line of the absci a, and the ordinates be taken parallel to C N, and if we make at the same time C E = a, C e = b, the nature of the curve will be expre ed by this equation, xy = ab. See Conic Sections. [s. 592]