104. Shew that the sections of a surface of the second order made by parallel planes are similar curves. Having given the area of the section of an ellipsoid by the plane lx +my+nz = 0, find the area of the section made by the plane lx+my+nz = 8. 105. If S be the area of a section of an ellipsoid made by a plane at the distance h from the centre, S' that of the parallel section through the centre, and p the perpendicular from the centre on the parallel tangent plane, shew that 106. Tangent planes are drawn to an ellipsoid from a given point; shew that an ellipsoid similar to the given ellipsoid and similarly situated can be made to pass through the given point, the points of contact, and the centre of the given ellipsoid. 107. Normals are drawn to the ellipsoid at the points where it is intersected by the plane zh. Shew that the locus of the intersection of these normals with the plane of (x, y) is the ellipse 108. All the normals to the ellipsoid in Ex. 107 meet the plane of (x, y) within an ellipse whose equation is 109. 5y-2x2- z2 + 4xy - 6yz +8xz-1= 0. 110. 2y2- 5x2+2z2+10xy +4yz + 4y +16z + 18 = 0. 111. 4y2-9x2+ 2xy +36x-8y-4z32 = 0. 112. y2 - 4xy + 4x2 − 6x + 3z = 0. 113. x2-y2+z2 - 4xy + 6xz - 2yz=f. 114. (ay-bx)+(cx- az)2 + (bz — cy) = f. (ay + Bx — cy)2 = aß (xy — z3). 118. Find the centre of the surface x2 + y2+ z2 + 4xy — 2xz — 4yz + 2x + 4y − 2z = 0. 119. Find the centre of the surface x2+2y2+3x2 + 2 (xy + yz+xz) + x + y + z = 1. 120. Shew that the equation yz + zx + xy=a2 may be reduced to 2x2 - (y2 + z2) = 2a2 by transforming the axes. 121. Shew that the equation = a2 x2 + y2 + z2 + xy + yz + zx = represents an oblate spheroid whose polar axis is to its equatoreal in the ratio of 1 to 2, and the equations to whose axis are xy=2. 122. What is represented by the following equation? x2 + y2+z2 + k (xy +yz+zx) = f. 123. If two concentric surfaces of the second order have the same foci for their principal sections they will cut one another everywhere at right angles. T. A. G. 2 124. Find the locus of the intersection of three planes at right angles to each other, each of which touches one of the following three ellipsoids, 125. Determine the position of the circular sections of an hyperboloid of two sheets, and shew that the same plane will cut the asymptotic cone in a circle. 126. If x, y, 1 ; X q, Y 2, Z q ; Xz, Y, Z, be the co-ordinates of the extremities of a set of conjugate diameters of an ellipsoid, shew that 2 2 x2+x2+x2=a2, y,+y,2+y ̧2=b2, z ̧2+z ̧2+z ̧2= c2, 2 127. If spheres be described on three semi-conjugate diameters of an ellipsoid as diameters, the locus of their intersection is the surface determined by a2x2 + b2y2 + c2z2 = 3 (x2 + y2 + z2)2. 128. A plane is drawn through the extremities of three semi-conjugate diameters of an ellipsoid; find the locus of the intersection of this plane with the perpendicular on it from the centre. 129. Tangent planes at the extremities of three conjugate diameters of an ellipsoid intersect in the ellipsoid whose equation is x2 + y2 22 + = 3. 130. A prolate spheroid is cut by any plane through one of its foci; prove that the focus is a focus of the section. 131. Shew that the locus of the diameters of the ellipsoid which are parallel to the chords bisected by tangent planes to touch the ellipsoid 132. If three straight lines at right angles to each other + + = 1, and intersect each other in the point (x', y', z') shew that 12 12 x22 (b2+c3) +y' (c2 + a3)+z"2 (a2 + b2) = b2c2 + c2a2 + a2b2. 133. Find the greatest angle between the normal at any point of an ellipsoid, and the central radius vector at that point. 134. If four similar and similarly situated surfaces of the second order intersect each other, the planes of their intersections two and two all pass through one point. 135. If three chords be drawn mutually at right angles through a fixed point within a surface of the second order 1 whose equation is u= 0, shew that Σ will be constant R. r where R and r are the two portions into which any one of the chords drawn through the fixed point is divided by that point. Prove also that the same will be true if instead of the fixed point there be substituted any point on the surface whose equation is uc. y 136. Let = 1/2 Τ m n be the equations to a straight line; find the equation to a surface every point of which is at the same distance from this line as from the point (a, B, y); and shew that the plane lx+my+nz = 8 cuts the surface in a straight line. 137. A and B are two similar and concentric ellipsoids, the homologous axes being in the same straight line; C is a third ellipsoid similar to either of the former, its centre being on the surface of B, and axes parallel to those of A or B; shew that the plane of intersection of A and C is parallel to the tangent plane to B at the centre of C. 138. If a parallelepiped be inscribed in an ellipsoid its edges will be parallel to a system of conjugate diameters. 139. The edges of a parallelepiped are 2a, 2b, 2c; shew that an ellipsoid concentric with it and whose semidiameters parallel to the edges are a 2, b2, c√2, intersects the faces in ellipses which touch each other and the edges. 140. Two similar and similarly situated ellipsoids are cut by a series of ellipsoids similar and similarly situated to the two given ones, so that the planes of intersection of any one of the series with each of the given ellipsoids make a right angle with one another. Shew that the centres of the series of ellipsoids lie on another ellipsoid. 141. If pyramids be formed between three conjugate diametral planes of an ellipsoid and a tangent plane, so that the products of the intercepted portions of the three conjugate diameters may be the least possible, the volumes of all these pyramids will be equal. 142. If from any point in an ellipsoid three straight lines are drawn mutually at right angles, prove that the plane which passes through their intersections with the surface passes through a point which is fixed so long as the original point is fixed. And shew that if the position of the original point on the surface is changed the locus of the point is an ellipsoid whose semiaxes are |