Then, C is the centre, CS or CH the eccentricity, Pp is a diameter, P and 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 diameter drawn parallel to the tangent at the vertex of any other diameter, is called the conjugate diameter to the latter, and the two diameters are called a pair of conjugate diameters. 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 is divided by one of its own ordinates are called the abscissa 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, cutting Pp in v. Through S draw Rr parallel to Tt. Qg is the ordinate to the diameter Pp, corresponding to the point Q. Pv, vp are the abscissæ of the diameter Pp, corresponding to the point Q. Rr is the parameter of the diameter Pp. R Q T Н ย a 14. Any straight line drawn at right angles to the major axis, and terminated both ways by the curve, is called an ordinate to the axis. 15. The segments into which the major axis is divided by one of its own ordinates are called the abscissæ of the axis. 16. The ordinate to the axis which passes through either focus is called the 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 tun gent, and the straight line the directrix. Thus, from P any point in the curve, draw PMP perpendicular to Aa, cutting BP Then, Pp is the ordinate to the axis, corresponding to the point P LT is the focal tangent. Nn is the directrix. 18. A straight line drawn at right angles to a tangent from the point of contact, and terminated by the major axis, is called a normal. The part of the major axis intercepted between the intersections of the nor nal and the ordinate, is called the subnormal. Let Tt be a tangent at any point P. From P draw PG perpendicular to Tt meeting Aa in G. From P draw PM perpendicular to The sum 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, sected in C, S H .. Pp is a straight line, and a diameter, and is bisected in C. And in like manner, it may be proved that every other diameter is bisected in C. PROP. III. The distance of either focus from the extremity of the axis minor is equal to the semi-axis major. Cor. 2. The square of the eccentricity is equal to the difference of the squares of the semi-axes; PROP. IV. To draw a tangent to the ellipse at any point. Let P be the given point. Join S,P; H,P; produce SP. Bisect the exterior angle HPK by the straight line Tt. Tt is a tangent to the curve at P. For, if Tt be not a tangent, let Tt cut the curve in some other point p. Join S, p; H,p; make PK≈ PH; join p,K; H,K cutting Tt in Z. Since HP PK, PZ common to the triangles HPZ, KPZ, and the angle HPZ = angle KPZ by construction, .: HZ = KZ, and the angle HZP = angle KZP. K Again, since HZ = KZ, Zp common to the triangles HZp, KZp, and angle HZp angle KZp, .. pK = pH. But, since any two sides of a triangle are greater than the third side, But we have just proved that pK = pH, which is absurd, .. p is not a point in the curve, and in the same manner it may be proved that no point in the straight line Tt can be in the curve except P. .. Tt is a tangent to the curve at P. Cor. 1. Hence, tangents at A and a, are perpendicular to the major axis, and tangents at B and b are perpendicular to the minor axis. Cor. 2. SP and HP make equal angles with every tangent. Cor. 3. Since HPK, the exterior angle of the triangle SPH, is bisected by the straight line Tt, cutting the base SH produced in T .. ST: HT:: SP: HP. K PROP. V. Tangents drawn at the vertices of any diameter are parallel. Let Tt, Ww, be tangents at P, p, the vertices of the diameter PCp. Join S, P; P, H; S, p; p, H; Then, by Prop. 2, SH is a parallelogram, and since the opposite angles of parallelograms are equal, :: ≤ SPH = angle SpH supplement of SPH = supplement of ← SpH or, < SPT +< HPt = < SpW+ < Hpw But < SPT = < HPt } W And <SpW = <Hpw by Prop. 4. Cor. 2. Hence, these four angles are all equal, .:: < SPT = <Hpw. And since SP is parallel to Hp, SPpPpH, ... whole TPp = whole wpP, and they are alternate angles, .. Tt is parallel to Ww. Cor. Hence, if tangents be drawn at the vertices of any two diameters, they will form a parallelogram circumscribing the ellipse. PROP. VI. If straight lines be drawn from the foci to a vertex of any diameter, the distance from the vertex to the intersection of the conjugate diameter, with either focal distance, is equal to the semi-axis, major. That is, if Dd be a diameter conjugate to Pp, cutting SP in E, and HP in e, PE or Pe Draw PF perpendicular to Dd, and HI parallel to Dd or Tt, cutting PF in O, Then, since the angles at O are right angles, the IPO = ≤ HPO, and PO common to the two triangles HPO, IPO, .. IP HP. Also, since SC HC, and CE is parallel HI, the base of A SHI, ... SE EI. AC. Hence, 2 PE 2 EI + 2 IP = SE+ EI + IP + HP = SP + HP = 2 AC .. PE = AC. Also, PEe = < PeE. Pe AC. .. PE Pe, and |