Smithsonian miscellaneous collections . ntrinsic equation of a planecurve is one giving the radius of curvature, p, as a function of the arc, s, If r is the angle between the a:-axis and the positive tangent ():(Is d-T m pds ?•*£ X = Xo + y cos T-dssin T-ds. The general equation of the second degree: anX^ + 2avixy + a^^.f + + 2any + 033 = O an ayi flis (h\ 0-22 ^23 Q31 O32 OssAhk = Minor of Ohk-Criterion giving the nature of the curve: dhk = dkh ^33 + 0 ^33 = 0 ^+0 ^330 Parabola Hyperbola QnA0 Ellipse ImaginaryCurve A =0 .4330 An0 An = A22= 0 Pair of RealStraight Lines Inters
Smithsonian miscellaneous collections . ntrinsic equation of a planecurve is one giving the radius of curvature, p, as a function of the arc, s, If r is the angle between the a:-axis and the positive tangent ():(Is d-T m pds ?•*£ X = Xo + y cos T-dssin T-ds. The general equation of the second degree: anX^ + 2avixy + a^^.f + + 2any + 033 = O an ayi flis (h\ 0-22 ^23 Q31 O32 OssAhk = Minor of Ohk-Criterion giving the nature of the curve: dhk = dkh ^33 + 0 ^33 = 0 ^+0 ^330 Parabola Hyperbola QnA0 Ellipse ImaginaryCurve A =0 .4330 An0 An = A22= 0 Pair of RealStraight Lines Intersecti( Pair of ImaginaryLines )n Finite RealPair of ImaginaryParallel Lines DoubleLine (Pascal: Repertorium der hoheren Mathematik, II, i, p. 228) GEOMETRY 45 Parabola (Fig. 3). 0, Vertex; F, Focus;ordinate through D, Direc-trix. Equation of parabola,origin at O, f= 4<7X X = OM, y = MP,OF = 0D = aFL = 2a = semi latus = DP. FP = FT = MD= X + Fig. 3 NP = 2Va(a + x), TM = 2X, MN = 2a, ON = x + = \l- (x + 2a), OQ = xJ-^, OS = (x + 2a)J- a -\- X FB perpendicular to tangent TP. FB = Va{a + x), TP = 2TB = 2Vx(a + x).Tb = FT X FO = FP X tangents TP and UP at the extremities of a focal chord PFP meeton the directrix at U at right angles. T = angle XTP. tan ^ = N/f The tangent at P bisects the angles FPD and FUD. Radius of curvature: 2(x + a) _ I WF P = Coordinates of center of curvature: ^ = 3a: + 2a, rj = - 2x\/^ Equation of Evolute: 270/ = 4{x - 2a)^ 46 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS Length of arc of parabola measured from vertex, 5 = Vx{x + a) +a log (y I +~ + y - j • Area OPMO = - Polar equation of parabola: r = FP, e = angle XFP, 20 I — cos d Equation of Parabola in terms of p, the perpendicular from F upon thetangent, and r, the radius vector FP: p rI = semi latus rectum. Ellipse (Fig. 4). ?^ { P f /n P V^\ 1 F 0\^^^ N ^ A F Ia T Fig. 4 O, Centre; F,
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