sect the arcs EB in K and AE in M, from which points through the centre, C, draw the conjugate diameters KN, ML; join KL, MN; draw also the chords BE, BF, BH, also the diameters fn, hi, and qt, sr. The triangles ECB, KPC, portional; and since KC = KPC the triangle BRC CP: + PK* = Hence, and or, Hence = being similar, their sides are proBC, and KP = BR, the triangle the triangle ECB. - CB2+ CE KC + CL2 = CB2 + CE AB+ ED KN2 + ML' = And since the semi-ordinate KP:ƒP::EC: FC the triangles FCB, ƒPC, are similar, and ƒPC + FCB and ƒC2 FB2 = Hence, Therefore, XV. FC' + CB' 2 fC+Ct FC2 + BC2 = fn2 + qt2 = = FG' + AB' agreeably to Prop. ¦ HB2 = Also in the triangles HCB, hPC, being for the same reason as before shown, similar, hPC = HCB, and hC2 = I HC+cB', and HC + C = HC + CB' rC2 Therefore, hi+ sr2 = HI2 + AB' agreeably to the property of the ellipse. Cor. 2. If AB, instead of being the axis major, should be the minor axis of a series of ellipses, the vertices of their equal conjugate diameters would still all be found in the lines MN, KL, produced. HYPERBOLA. DEFINITIONS. 1. AN HYPERBOLA is a plane curve, such that, if from any point in the curve two straight lines be drawn to two given fixed points, the excess of the straight line drawn to one of the points above the other will always be the same. 2. The two given fixed points are called the foci. Thus, let QAq be an hyperbola, S and H the foci. Q Pi 1 39... then. = If HP, SP, and SP', HP', .... be always equal to the same constant quantity, the points P, P, P, ... and P',, P2, P3, will lie in two opposite and similar hyperbolas QAq, Q'aq', which in this case are called opposite hyperbolas. 3. If a straight line be drawn joining the foci, and bisected, the point of bisection is called the centre. 4. The distance from the centre to either focus is called the eccentricity. 5. Any straight line drawn through the centre, and terminated by two opposite hyperbolas, is called a diameter. 6. The points in which any diameter meets the hyperbolas are called the vertices of that diameter. 7. The diameter which passes through the foci is called the axis major, and the points in which it meets the curves the principal vertices. 8. If a straight line be drawn through the centre at right angles to the major axis, and with a principal vertex as centre, and radius equal to the eccentricity, a circle be described, cutting the straight line in two points, the distance between these points is called the axis minor. Thus, let Qq, Q'q' be two opposite q hyperbolas, S and H the foci, join S.H; Bisect SH in C, and let SH cut the curves in A,a. Through C draw any straight line Pp, terminated by the curves in the points P, p. Through C draw any straight line at right angles to Aa, and with cen B a C P tre A and radius = CS describe a circle cutting the straight line in the points B, b. Then C is the centre, CS or CH the eccentricity, Pp, is a diameter, P and p its vertices, Aa is the major axis, Bb is the minor axis. 9. A straight line, which meets the curve in any point, but which, being produced both ways, does not cut it, is called a tangent to the curve at that point. 10. A straight line, drawn through the centre, parallel to the tangent, at the vertex of any diameter, is called the conjugate diameter to the latter, and the two diameters are called a pair of conjugate diameters. The vertices of the conjugate diameter are its intersections with the conjugate hyperbolas. 11. Any straight line drawn parallel to the tangent at the vertex of any diameter, and terminated both ways by the curve, is called an ordinate to that diameter. 12. The segments into which any diameter produced is di vided by one of its own ordinates and its vertices, are called the abscissæ of the diameter. 13. The ordinate to any diameter, which passes through the focus, is called the parameter of that diameter. Thus, let Pp be any diameter, and Tt a tangent at P; Draw the diameter Dd parallel to Tt; Take any point Q in the curve, draw Qq parallel to Tt and cutting Pp produced in v; Through S draw Rr parallel to Tt; Then Dd is the conjugate diame ter to Pp, R Qq is the ordinate to the diameter Pp corresponding to the point Q. Pv, up, are the abscissæ of the diameter Pp corresponding to the point Q. Rr is the parameter of the diameter Pp. 14. Any straight line drawn from any point in the curve at right angles to the major axis produced, and terminated both ways by the curve, is called an ordinate to the axis. 15. The segments into which the major axis produced is divided by one of its own ordinates and its vertices, are called the abscisse of the axis. 16. The ordinate to the axis which passes through the focus, is called the principal parameter or latus rectum. (It will be proved in Prop. IV, that the tangents at the principal vertices are perpendicular to the major axis; hence definitions 14, 15, 16, are in reality included in the three which immediately precede them.) 17. If a tangent be drawn at the extremity of the latus rectum, and produced to meet the major axis; and if a straight line be drawn through the point of intersection, at right angles to the major axis; the tangent is called the focal tangent, and the straight line the directrix. Thus, form P, any point in the curve, draw P Mp perpendicular to Aa, cutting Aa in M'; Through S draw Ll perpendicular to Aa; Let LT, a tangent at L, cut Aa in T; Through T draw Nn perpendicular to Aa: Then, Pp is the ordinate to the axis corresponding to the point P. N TAS M AM, Ma, are the abscissæ of the axis corresponding to the point P, LI is the latus rectum, LT is the focal tangent, Nn is the directrix. 18. An asymptote is a diameter which approaches the curve continually as they are both produced, but which, though ever so far produced, never meets it. 19. If the asymptotes of four opposite hyperbolas cross each other at right angles, the hyperbolas are called right angled or equilaterial hyperbolas. PROPOSITION I. THEOREM. The difference of two straight lines drawn from the foci to any point in the curve, is equal to the major axis. That is, if P be any point in the curve, For. HP-SP Aa; = Cor. 1. The centre bisects the major axis; for, since PROPOSITION II. THEOREM. The centre bisects all diameters. Take any point P in the curve; Complete the parallelogram SPHp; Then, since the opposite sides of parallelograms are equal, Hp; HP Sp, SP H S |