Solid Geometry, Volume 1 |
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Common terms and phrases
algebraical projection asymptotic ax² axes axis bisects by² central conicoid ch² chords circular section cone confocal conjugate diameters constant coordinate planes corresponding cosines cr² curve of intersection cyclic sections cylinder cz² direction-cosines directrix drawn ellipse ellipsoid envelope find the equation finite fixed plane fixed point focal conic four-plane four-point coordinates given plane given point given straight line Hence hyperbolic paraboloid hyperboloid infinite distance infinite number Let the equation line joining locus middle points normals obtain parabola parallel parallelepiped perpendicular plane containing plane of xy plane passing plane section position principal planes projection quadric ratio right angles satisfy the equation second degree semi-axes sheet shew similarly sphere surface tangent plane tetrahedral coordinates tetrahedron transformation values vertex
Popular passages
Page 26 - PQ, may be interpreted in a manner similar to that adopted in the case of the radius vector in polar co-ordinates in Plane Geometry.
Page 217 - 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 289 - 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 118 - An ellipsoid may be generated by the motion of a variable ellipse, which moves so that its plane is always parallel to a fixed plane, and which changes its form so that its vertices lie in two ellipses having a common axis traced on planes perpendicular to each other, and to the fixed plane.
Page 219 - 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 340 - 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 410 - 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 d£==~di?=di? ax by cz ' *" y" z
Page 110 - A conic section is by definition the locus of a point whose distance from a fixed point is in a constant ratio to its distance from a fixed straight line.
Page 375 - A point moves on an ellipsoid so that its direction of motion always passes through the perpendicular from the centre of the ellipsoid on the tangent plane at any point ; shew that the curve traced out by the point is given by the intersection of the ellipsoid with the surface xm~" yn~l zl~m = constant, I, m, n being inversely proportional to the squares of the semiaxes of the ellipsoid.
Page 418 - ... of the phenomenon which require for their explanation a more refined theory. 9. Two surfaces touch each other at the point P ; if the principal curvatures of the first surface at P be denoted by a±b, those of the second by a...