80. If S be a portion of a curve surface referred to three prove that if a right cone be cut by a cylinder having a base of given area with its axis parallel to that of the cone, the intercepted portion of the conical surface is constant. 81. Shew that ax2 + by2 + cz2 + 2ayz + 26'xz + 2c'xy = d (where d denotes a positive quantity) is the equation to an ellipsoid, hyperboloid of one, or hyperboloid of two sheets, according as the cubic (s - a) (s — b) (s — c) — d'2 (s — a) — b'2 (s — b) — c'2 (s — c) = 2a'b'e, gives for s three positive values, two or one. When the surface is an ellipsoid, shew also that its volume 4rd 3abcaa2 - bb'2 - cc'2 + 2a'b'c' 82. Having given that (y2 — Cx2) {ß (a − y) + Ca (ß − y)} + Caß (a — ẞ) = 0 is the equation to the projection of the lines of curvature of the ellipsoid whose equation is ẞyx2 + ayy2 + aßz2 αβγ; prove that according as the plane of xy does or does not contain the mean axis, the arbitrary constant C admits, for each point of the ellipsoid, two values of contrary signs, or two negative values; so as to give an ellipse and hyperbola in the former, and two ellipses in the latter case. 83. Find the equation to a plane considered as generated by a straight line which moves in a direction parallel to itself along a straight line given in position. Express the result in terms of the perpendicular from the origin, and the angles which it makes with the axes of the coordinates. 84. Three chords of an ellipsoid are drawn through a given point, each at right angles to the plane containing the other two; prove that if a rectangle be formed by the segments into which each chord is divided at the given point, the sum of the reciprocals of these rectangles is constant. 85. Determine the line of intersection of the tangent planes drawn at two consecutive points of the surface of an ellipsoid. 1834 1835 If a series of such points be taken in one plane, prove that the corresponding lines of intersection will all pass through one point; and determine the coordinates of that point when the equation to the plane is z = Ax + By + C. 86. Through every point of the surface whose equation is n222 n2x2= y2 - 2ax, two straight lines can be drawn coinciding with it in all their points. 87. Determine the surface, every point of which is the intersection of three normals to an oblate spheroid. 88. Shew how to find the equation to the section of a surface made by a plane perpendicular to one of the coordinate planes, and deduce the equation to a plane which, passing through the mean axis of an ellipsoid, cuts it in a circle. 89. Define a diametral surface relative to a given surface, and shew that for a surface of the second order it is a plane. If a surface be defined by an equation of the nth order, of what order is the equation to its diametral surface? 90. Find the position of a plane on which the sum of the projections of any number of plane areas is a maximum. 91. Mention the different ways in which developable surfaces may be supposed to be generated, and find their dif ferential equation. Shew that surfaces generated by the motion of a straight line may be distinguished into two classes, in one of which the tangent planes at points in the same generating line are all coincident, and in the other all distinct. 92. Find the equation to the normal plane at any point of a curve of double curvature; and shew how the equation to the surface formed by the continual intersection of such planes may be determined. 93. The area of a section of an ellipsoid, made by a plane passing through the centre and inclined at angles a, ß, y to the principal axes, 94. Find the equation to the surface, in which lie all the evolutes to the curve formed by the intersection of the surfaces, y2 = 4a (x + z), z2 = 4a (x + y); and determine the equations to that evolute which cuts the axis of x at a distance 7a from the origin. 95. Having given the equation to a plane, determine the constants in terms of the distances from the origin of the intersections of the plane with the coordinate axes, rectangular or oblique. In the former case, what are the equations to a line perpendicular to the plane, and passing through the origin? 96. Let any two chords AA', BB' in a surface of the second order be drawn through a fixed point; the locus of the points of intersection of AB, A'B' is a plane. 97. Let SY be drawn from the origin S perpendicular to the tangent plane at any point P of a surface; then will SY2 SP be the perpendicular on the tangent plane at the corresponding point of the surface which is the locus of Y. 98. Find the equation to a plane which passes through a given straight line, and through the shortest distance between the line and the axis of x. 99. Investigate formulæ for the transformation of coordinates from one system of three rectangular axes (x, y, z) to another (x, y, z), the position of the latter being determined by the angle which the axis of ' makes with the plane xy, and that which its projection on the plane xy makes with the axis of x, the axis of y' being in the plane xy. 100. Investigate the form of the surface whose equation is x2 y2 22 + a2 1, and find the equation to the asymptotic surface. 101. Investigate the differential expression for the area of a curve surface referred to three rectangular axes; and apply it to find that portion of the surface of a cone which is included between two planes perpendicular to the axis, and at a given distance from each other. 1836 134 ANALYTICAL GEOMETRY OF THREE DIMENSIONS. 102, Shew that the equation of the surface swept out by a line cutting the directrix of a parabola at right angles, and touching the paraboloid generated by revolution of the parabola about its axis, is the axis of revolution, directrix, and a perpendicular to them being taken for the coordinate axes; and being the latus rectum. 103. The orthogonal projection of a straight line, whose length is L, upon any straight line in space L. cosine of their mutual inclination. Prove this, and apply it in finding the equation of a sphere referred to oblique coordinates. 104. Determine the equation to the surface traced out by the circumference of a circle, whose centre is a fixed point, and which always passes through each of two given straight lines at right angles to each other, and not in the same plane. 105. A quadrilateral figure is inscribed in a small circle of a sphere, determine the position of an eye on its surface when the stereographic projection of the quadrilateral is a rectangle. 106. Find the angle between a straight line and a plane whose equations are given, and thence the conditions that they may be at right angles to each other. 107. Find the general equation to conical surfaces. Apply it to prove that the stereographic projection of any plane section of a paraboloid of revolution upon a plane perpendicular to its axis is a circle, the eye being placed in the vertex. 108. Determine the radii of curvature at any point of a surface in terms of the coordinates of that point; and shew how to find those points of a surface at which the radii of curvature are equal and have the same sign. 109. Determine the volume of the solid, the equation of whose surface is 1, contained between the coordi x2 y2 22 nate planes and that whose equation is X y + 1. |