172. If lines be drawn from the centre of an ellipsoid (whose semiaxes are a, b, c) parallel to the generating lines of an enveloping cone, the conical surface S so formed will intersect the ellipsoid in two planes parallel to the plane of contact. The locus of the vertex of the enveloping cone which causes one of the planes to coincide with the plane of contact is 173. Of all cones which envelope an ellipsoid, have their bases in the tangent plane at a given point P, and are of the same altitude, that is the least which has its vertex in the diameter through P; and of all which have their vertices in this diameter, that is the least whose axis is twice that diameter. 174. If a globe be placed upon a table the breadth of the elliptic shadow cast by a candle (considered as a luminous point) will be independent of the position of the globe. 175. In the preceding example, if an ellipsoid having its least axis vertical be substituted for the globe, determine the condition of the shadow of the globe being circular. It may be shewn that the locus of the luminous point must be an hyperbola, and that the radius of the circular shadow is independent of the mean axis of the ellipsoid. 176. Of a series of cones enveloping an ellipsoid, the vertices lie on a concentric ellipsoid, similar to the given one and similarly situated. Prove that any two cones of the series intersect one another in two planes. 177. Prove that if, in the preceding example, the vertices are supposed to lie also on a third ellipsoid concentric with the other two and similarly situated, and whose axes are respectively as the squares of theirs, these two planes are at right angles to one another. 178. A cylinder is circumscribed about an ellipsoid, and at the extremities of the diameter parallel to the generating lines of the cylinder tangent planes are drawn: shew that the volume of all cylinders so shut in is constant. 179. The locus of the vertex of a cone which envelopes a given ellipsoid (4) is a straight line passing through the centre of (4); an ellipsoid similar to A and similarly placed has the vertex of the cone for centre and cuts the cone in a curve (B). If the major axis of this ellipsoid vary as the distance of its centre from A's, prove that the locus of B is an elliptic cylinder. 180. Suppose a cylinder to envelope an ellipsoid, and suppose a tangent plane to be drawn to the ellipsoid at one extremity of the diameter which is parallel to the axis of the cylinder. Let a line be drawn from the centre of the ellipsoid to meet the ellipsoid, the above tangent plane, and the enveloping cylinder; and suppose r, s, t to denote the respective distances of the points of intersection from the centre of the ellipsoid. Shew that 1 1 181. Suppose a cone to envelope an ellipsoid; let R' be the distance of the vertex from the centre of the ellipsoid, 2R the length of the diameter of the ellipsoid which is in the direction of the line joining the centre of the ellipsoid with the vertex of the cone; and suppose a tangent plane to be drawn to the ellipsoid at one extremity of this diameter. Let a line be drawn from the centre of the ellipsoid to meet the ellipsoid, the above tangent plane, and the enveloping cone; and suppose r, s, t to denote the respective distances of the points of intersection from the centre of the ellipsoid. Shew that 182. Let (x, y, z) = 0 be the equation to a surface of the second order; if tangent lines be drawn to it from the point (a, B, y), shew that the equation to the plane which contains all the points of contact is 183. Let (x, y, z) = 0 be the equation to a surface of the second order; then the equation to the enveloping cone which has its vertex at the point (a, B, y) is 184. Let (x, y, z) = 0 be the equation to a surface of the second order; then the equation to the enveloping cylinder which has its generating lines parallel to the line 7=2 = 2 is y m n {p d'u 2u12 dx2 + m2 d'u dau dz2 dxdy +2mn +m7 +n Ex. Determine the enveloping cylinder of the surface y2, z2 + +=1, which has its generating lines parallel to the 185. Shew that the surface (ax+by+ cz− 1)2 + 2a'yz +2b' zx + 2c'xy = 0 touches the co-ordinate axes; and find the equation to the cone which has its vertex at the origin and passes through the curve of section of the surface by the plane through the points of contact. 186. Lines are drawn through the origin perpendicular to the tangent planes to the cone ax2 + by2 + cz2 + 2a'yz + 26'zx + 2c'xy = 0; shew that they will generate the cone which has for its equa tion + 2 (b'c' — aa') yz + 2 (c'a' — bb') zx +2 (a’'b' — cc') xy = 0. Shew also that if lines be drawn through the origin perpendicular to the tangent planes to the second cone they will generate the first cone. III. Surfaces in general. 187. If a sphere pass through the origin of co-ordinates and its centre is on the surface defined by the equation {3 (x2 + y2 + z2) — b2}2 = (x + y + z)2 (x2 + y2+z2), the sum of the spherical surfaces cut off by the co-ordinate planes is constant and = 2πb". 188. A straight line drawn from the centre of an ellipsoid meets the ellipsoid in P and the sphere on the diameter 2a in Q; shew that the tangent planes at P and Q contain a constant angle a if the co-ordinates of P satisfy the equation α 189. If a straight line be drawn from the centre of an hyperboloid whose equation is = x2 y2 22 = 1 to meet the surface in P, P', and a point Q be taken in CP produced such that CP-1 (QP+QP'), where 7 is a constant, shew that the locus of Q is defined by the equation 190. The shadow of a given ellipsoid thrown by a luminous point on the plane which passes through two of the principal axes has its centre on the curve in which the same plane intersects the ellipsoid; shew that the equation to the focus of the luminous point is 191. If light fall from a luminous point whose co-ordinates are a, B, y, on a surface whose equation is xyz = m3, the boundary of light and shade lies on an hyperboloid of one or two sheets according as the product of a, B, y, is negative or positive. |