HYPER'110LA. If two similar cones be placed apex to apex, and with the lines joining the apex and center of base in each, in a straight line; then if a plane which does not pass through the apex be made to cut both cones, each of the two sections will be a hyperbola, as PBN, P'AN'. It is, viewed ana lytically, the locus of the point to which the straight lines EP, FP dif-' fering by a constant quantity are drawn from two given points, E and F. These given points are called the foci, one being situated in each hyper bola. The point 0, midway between the two foci, is called the center, and the line EF the transverse axis of theltyperbola. A line through G perpendicular to the transverse axis is called the conjugate axis; and a circle described from center 13, with a radius equal to FG, will cut the conpgate axis in C and D. If G be taken for the origin of co-ordinates, and EM and E'F for the axis, the hyperbola is expressed by the equation b ' 7 — =1. (GB=a, GC=b). The hyperbola is the only conic section
which has asymptotes (q.v.); in the figure these are GT, GT'; GS, GS'. It also appears that if the axis of co-ordinates be turned at right angles to their former position, two additional curves, IICK, H'DK', will be formed, whose equation is X2 = 1.
These two are called conjugate hyberbotas, and have the same asymptotes as the original hyperbolas. These asymptotes have the following remarkable property: if (starting from G) the asymptotes be divided in continued proportion, and from the points of section lines be drawn parallel to the other asymptote, the areas contained by two adjacent parallels and the corresponding parts of the asymptote and curve are equal; also lines drawn from the center to'two adjacent piiints of section of the curve, inclose equal areas. The equation to the hyperbola when referred to the asymptotes is