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axes axis bisects central conicoid centre chord circle circular sections coefficients cone confocal conical surface conies constant corresponding cos0 cosines curve of intersection cyclic planes cyclic sections cylinder developable surface directing plane direction-cosines directrix drawn ellipse ellipsoid envelope equal find the equation finite fixed plane fixed point focal conic focal lines given plane given point hence hyperbola hyperbolic paraboloid hyperboloid infinite distance infinite number Let the equation line joining line of intersection lines of curvature locus obtain osculating plane parabola perpendicular plane containing plane of xy plane passing plane section point of contact points of intersection polar position principal curvature principal planes principal section projection prove quadric radii of curvature radius ratio right angles satisfy the equation second degree semi-axes sheet shew singular point sphere tangent plane tetrahedral coordinates tetrahedron theorem transformation umbilical values vertex
Page 423 - THE FIRST THREE SECTIONS OF NEWTON'S PRINCIPIA, ' With Notes and Illustrations. Also a collection of Problems, principally intended as Examples of Newton's Methods. By PERCIVAL FROST, MA Third Edition.
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 377 - R be the radii of curvature, torsion and spherical curvature of a curve at a point whose distance measured from a fixed point along the curve is s, prove that 8. When the polar surface of a curve is developed into a plane, prove that the curve itself degenerates into a point on the plane, and if r, p be the radius vector and perpendicular on the tangent to the developed edge of regression of the polar surface drawn from this point, prove that 9. Prove that the angle between the shortest distance...
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 18 - The orthogonal projection of a line upon a plane is the length of the line multiplied by the cosine of the angle of inclination of the line to the, plane.
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 dŁ==~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 80 - To shew that the straight lines joining the middle points of opposite edges of a tetrahedron intersect and bisect each other.