A Treatise on the Higher Plane Curves: Intended as a Sequel to A Treatise on Conic Sections |
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Common terms and phrases
a₁ acnodal angle asymptote axis b₁ bitangents Cayleyan centre circle coincide common tangents condition consecutive points considered coordinates coresidual corresponding points covariant crunodal cusp cuspidal cubic denote determine discriminant double point double tangent envelope equation expressed fixed point foci four points function given curve given line given points Hence Hessian hyperbola imaginary intersection invariant Jacobian last article line at infinity line joining locus meets the curve multiple point nodal cubic node oval pair parameter point of inflexion points at infinity points of contact polar conic polar line pole quadric quartic quartic function ratio real points reciprocal respectively right line shew stationary tangent substituting tangent theorem three points touch transformation triangle triple point unicursal values vanishes
Popular passages
Page 312 - The feet of the perpendiculars on the sides of a triangle from any point on the circumscribing circle lie in one right line.
Page 275 - This curve is generated by the motion of a point on the circumference of a circle which rolls along a right fcline.
Page 100 - ... refraction of a circle. Let a circle be described through A, the radiant point, and R, the point of incidence, to touch OR ; then the point B is given, since OA . OB = OR1.
Page 85 - ... we shall call for shortness p, is to be found from the equation of the curve. For the tangent passes through the point xy, and makes with the axis of x an angle whose tangent is p (ArtI 38).
Page 181 - ... inscribed in a circle. 514. Find the shortest distance between two circles which do not meet. 515. Two circles cut one another at a point A : it is required to draw through A a straight line so that the extreme length of it intercepted by the two circles may be equal to that of a given straight line. 516. If a polygon of an even number of sides be inscribed in a circle, the sum of the alternate angles together with two right angles is equal to as many right angles as the figure has sides. 517....
Page 108 - OB, 08 be constant.* And the proof is the same as that already given in the case of conic sections. From the polar equation of the curve, Art. 26, we see that the product of all the values of the radius vector on a line through the origin making an angle 0 with the axis of x is = A ~ P cos"0 + Q cos"-J0 sin 0 4- &c.
Page 30 - Q, that is, if a curve have its maximum number of double points, the coordinates of any point on the curve can be expressed as rational algebraic functions of a variable parameter.
Page 12 - ... from it by the theory of reciprocal polars (or that of geometrical duality), viz. we do not demonstrate the first theorem and deduce from it the other, but we do at one and the same time demonstrate...
Page 183 - The curve may also be defined as the locus of the foot of a perpendicular let fall from the vertex of a parabola upon a tangent. The problem of "duplicating the cube" is not taken up directly by Cantor.
Page 88 - At a cusp it will be found that the radius of curvature vanishes. 103. The length of any arc of the evolute is equal to the difference of the radii of curvature at its extremities. For, draw any three consecutive normals to the original curve : let G be the point of intersection of the first and second, G...