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asymptotic axes axis bisect central conicoid centre chord circle circular sections coefficients coincident confocal conies consecutive constant corresponding cubic surface curve of intersection cyclic sections cylinder direction-cosines directrix drawn ellipse ellipsoid elliptic coordinates equal find the equation finite fixed plane fixed point focal conic focal lines geodesic given plane given point hence hyperbola hyperbolic paraboloid hyperboloid infinite distance infinite number Let the equation line joining line of curvature line of intersection locus number of points obtain osculating plane parabola perpendicular plane at infinity plane containing plane curve plane of xy plane passing plane section point of contact points of intersection polar plane pole position principal section prove radii of curvature radius reciprocal respect revolution right angles second degree semi-axes sheet shew shewn singular point sphere straight line surface of revolution tangent plane tetrahedron torse touch umbilic values vanish vertex
Page 146 - Conic, is the locus of a point which moves so that its distance from a fixed point is in a constant ratio to its distance from a fixed straight line.
Page 198 - An annular surface is generated by the revolution of a circle about an axis in its own plane; prove that one of the principal radii of curvature, at any point of the surface, varies as the ratio of the distance of this point from the axis to its distance from the cylindrical surface described about the axis and passing through the centre of the circle.
Page 148 - Find the equation of the locus of a point the square of whose distance from a given line is proportional to its distance from a given plane.
Page 120 - The locus of the middle points of a system of parallel chords in a parabola is called a diameter.
Page 33 - To determine the conditions which must subsist in order that a straight line may be parallel to a plane. Let the equation of the plane be Ax + By + Cz + D = 0, and the equations of the straight line...
Page 296 - At any point of a geodesic on a central conicoid, the rectangle contained by the diameter parallel to the tangent at that point and the perpendicular from the centre on the tangent plane at the point is constant. The differential equations of a geodesic on the conicoid aa? + by* + cz3 = 1 are (Px d*y tfz dŁ==~di?=di? ax by cz ' *" y" z
Page 249 - ... sides. The treatise on Spirals contains demonstrations of the principal properties of the curve, now known as the Spiral of Archimedes, which is generated by the uniform motion of a point along a straight line revolving uniformly in one plane about one of its extremities. It appears from the introductory epistle to Dositheus that Archimedes had not been able to put these theorems in a satisfactory form without long-continued and repeated trials; and that Conon, to whom he had sent them as problems...
Page 374 - Along the normal at a point P of an ellipsoid is measured PQ of a length inversely proportional to the perpendicular from the centre on the tangent plane at P; prove that the locus of Q is another ellipsoid, and that the envelope of all such ellipsoids is the "surface of centres," that is the locus of the centres of principal curvature.