Examples of Analytical Geometry of Three DimensionsIsaac Todhunter Macmillan and Company, 1878 - 88 Seiten |
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a² b2 absolute curvature ax² b2 c² bisect by² centre circle co-ordinate planes concentric ellipsoid conical surface cos² cosines Crown 8vo curve of intersection cylinder cz² d²u developable surface distance edges ellipse ellipsoid equal Find the equation find the locus fixed point fixed straight lines given ellipsoid given plane given point given straight line hyperbola hyperbolic hyperboloid line of intersection lines of curvature lx+my+nz normal plane normal section origin osculating plane parabola paraboloid plane of contact plane parallel plane passing plane perpendicular plane which passes planes are drawn points of contact principal radii principal sections radii of curvature represent right angles right cone second order sheet shew sphere surface of revolution surface whose equation surface x² tangent plane tetrahedron touch the surface u₁ vertex y²+z²
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Seite 41 - ... the radius of curvature of the section of the surface made by the osculating plane of the curve at that point.
Seite 9 - I.) the equation of a curve in space, the equation of the cone which has its vertex at the origin, and passes through this curve, is of the form, I.
Seite 48 - 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...
Seite 11 - Show also that if the friction exceed a certain quantity the disc will not come to rest at all. 11. A sphere touches each of two right lines which are inclined to each other at a right angle, but do not intersect; prove that the locus of its centre is a hyperbolic paraboloid. 12. A slender rod suspended horizontally by two equal parallel strings attached to two points equidistant from its ends oscillates round a vertical line; find the time of a small oscillation. If in the position of equilibrium...
Seite 59 - ... the square of the distance from the centre, find the absolute force when the space originally occupied by the fluid is left a vacuum. 14. A circle always touches the axis of z at the origin, and passes through a fixed straight line in the plane of xy ; find the equation to the surface generated. Shew that the origin is a singular point, and that in its immediate neighbourhood the surface may be conceived to be generated by a circle having its plane parallel to that of xy, and its radius proportional...
Seite 26 - ... shadow. 42. A person standing beside a river near a bridge observes that the inverted image of the concavity of the arch receives his shadow exactly as a real inverted arch would do if it were in the place where the image appears to be. Explain this. 43. If a globe be placed upon a table, show that the breadth of the elliptic shadow cast by a candle (considered as a luminous point) will be independent of the position of the globe. 44. What is the length of the cone of the umbra thrown by the...
Seite 60 - C prove that a sphere can be described through the curve of contact provided P lie in a certain right line passing through the origin. 18. The equations to a system of right lines in space contain two arbitrary parameters; prove that when the roots of a certain quadratic are real and unequal, there are two planes passing through a given line of the system which contain consecutive lines. 19.
Seite 60 - The equations to a system of lines in space, straight or curved, contain two arbitrary parameters; show how to find whether the lines can be cut at right angles by a system of surfaces, and when they can, show how to find the equation to that system. Examples. (1) Let the lines be a system of right lines each of which intersects two given right lines which are perpendicular to each other but do not intersect. (2) Let the equations to the system of lines...