An Elementary Treatise on Cubic and Quartic Curves |
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Common terms and phrases
angle asymptote axis bicircular quartic biflecnode cardioid Cassinian centre of inversion circle of inversion circular cubic circular points cissoid coincident points conjugate point cos² crunode cusp cuspidal cubic cuspidal tangent double point double tangent ellipse fixed circle flecnodes focal conic follows harmonic polar Hence Hessian hyperbola hypocycloid imaginary points lemniscate limaçon line at infinity nine-point circle nodal cubic nodal tangents node obtain origin orthoptic orthoptic locus oval pair parabola passes pedal perpendicular plane point of intersection points at infinity points of contact points of inflexion points of undulation polar conic projection quartic curves r₁ radius real points reciprocal polar respect shown sin² single foci singularities stationary tangents straight line tacnode tangential equation theorem triangle of reference trilinear coordinates trinodal quartic triple focus triple point u₁ u₂ vertex whence whilst
Popular passages
Page 162 - Hence, the hyperbola is often defined as the locus of a point which moves so that the product of its distances from two fixed lines is constant.
Page 205 - Cycloid. The cycloid is a curve generated by a point on the circumference of a circle which rolls on a straight line tangent to the circle.
Page 156 - O'; shew that either O or O' is the orthocentre of the triangle ABC. Distinguish between the two cases. 9. Three equal circles pass through the same point A, and their other points of intersection are B, C, D : shew that of the four points A, B, C, D, each is the orthocentre of the triangle formed by joining the other three. 10. From a given point without a circle draw a straight line to the concave circumference so as to be bisected by the convex circumference. When is this problem impossible ?...
Page 24 - J\ but the theorem of Newton's, which has hitherto guided us, is perhaps insufficient immediately to furnish a second relation. Such a relation, however, may be obtained by the following considerations. It is well known that the polar conic of a point of inflexion breaks up into two lines : one of these is the tangent at the point of inflexion, the other will be found to be the locus of the harmonic centres of the n - 1 points in which the curve is cut by a transversal through the point. Exactly...
Page 202 - Hence the polar may be defined as the locus of the points of intersection of tangents at the extremities of chords through a fixed point.
Page 172 - conic' is the locus of a point P which moves so that its distance from a fixed point S, called the 'focus,' is in a constant ratio to its distance from a fixed line called the 'directrix.
Page 231 - Such a curve, in which the tangent at any point makes a constant angle with the radius drawn to that point from a fixed point, is called an equiangular spiral. As the dog at B' is moving at right angles to A'B', the distance A'B
Page 145 - Hence the radical axis of two circles is perpendicular to the line joining their centres. This is geometrically obvious when the circles cut in real points. 84. The three radical axes of three circles taken in pairs meet in a point. If 8= 0, 8' = 0, 8 " = 0 be the equations of three circles (in each of which the coefficient of a?
Page 197 - We have, incidentally, a proof that the pedal of a circle with respect to a point on its circumference...
Page 254 - This is a conic passing through the origin, that is, through the intersection of one of the three pairs of straight lines which can be drawn through the four points.