192. Tangent planes to an ellipsoid are drawn at a given distance from the centre; find the projections on the principal planes of the curve which is the locus of the points of contact. In what case will one of these projections consist of two straight lines?

193. Shew that the surfaces whose equations are

(x2 + y2) (a2 — z2) — c2y2 = 0,

and (x −c)2 + y2 + z2 — a2 = 0,

touch one another; and that the projection of the curve of contact on one of the co-ordinate planes is a circle, and on another a parabola.

194. Two circles have a common diameter AB and their planes are inclined to each other at a given angle; on PP' any chord of one of them parallel to AB is described a circle with its plane parallel to that of the other circle; shew that the surface generated by these circles is an ellipsoid the squares of whose axes are in arithmetical progression.

195. Find the equation to the surface generated by straight lines drawn through the origin parallel to normals x2 y2 z2

to the surface

by the surface

3+2+1 at points where it is intersected

a2

than c and less than a.

196. An ellipse of given excentricity moving with its plane parallel to the plane of (y, z) and touching at the extremities of its axes the planes of (x, y) and (x, z) always passes through the curve whose equations are y = x, cz = x2; find the equation to the surface generated, and determine the volume bounded by the surface and two given positions of the generating ellipse.

197. Shew that the surface determined by

is cut by planes parallel to the plane of (x, y) in straight lines.

198. A tangent line to the surface of the ellipsoid

passes through the axis of z, and a given curve in the plane of (x, y); shew how to find the equation to the surface genex2, rated by it. Ex. The curve being +32=m2, shew that

a2 + b2

the surface consists of two cones of the second order.

199. Find the differential equation to the surface generated by a straight line which passes through two given curves and remains parallel to the plane of (x, y). Shew that the equation x = -6) tan represents such a surface.

200. Determine the surface generated by a straight line which moves parallel to the plane of (x, y), and passes through the axis of z and through the curve given by

201. Determine the surface generated by a straight line which moves parallel to the plane of (x, y) and passes through the following curves (1) and (2):

202. A surface is generated by a straight line which always passes through the two fixed straight lines

y=mx, z=c; and y=-mx, z=

prove that the equation to the surface generated is of the

203. If a surface be such that at any point of it a straight line can be drawn lying wholly on the surface and intersecting the axis of z, then at every point of the surface

204. The general equation to surfaces generated by a straight line which is always parallel to the plane

is

lx+my+nz= 0,

dz

dz d2z

(m + n de) d = − 2 (m+nde) (1+ n da) d'a

dy

dx dy

205. Find the equation to the surface generated by a straight line which always passes through each of two given straight lines in space, and also through the circumference of a circle whose plane is parallel to them both, and whose centre bisects the shortest distance between them.

206. A surface is generated by a straight line which X α y-b always intersects the line

m

n

and is parallel to the plane x + μy+vz = 0; find the functional equation X to the surface; also find the differential equation.

207. Find the general functional equation to surfaces generated by the motion of a straight line which always intersects and is perpendicular to a given straight line.

If a surface whose equation referred to rectangular axes is ax2+by2+cz2+2a'yz+2b'zx+2c'xy +2a′′x+2b′′y+2c"z+1=0 be capable of generation in this manner, shew that

a+b+c=0, aa"+bb'2 + cc'2 = 2a'b'c' + abc.

208. Shew that xyz = c(x2- y2) represents a conoidal surface.

T. A. G.

3

209. Shew that the condition in order that

ax2 + by2 + cz2 + 2a"x+2b"′′y + 2c"z =ƒ

may represent a conical surface is

210. Shew that in order that the equation.

ax2+by2+cz2+2a'yz +2b'zx+2c'xy +2a′′x + 2b′′y+2c"z+f=0 may represent a cylindrical surface, it is necessary that

Is this condition sufficient as well as necessary?

211. Explain the two methods of generating a developable surface; find the differential equation to developable surfaces from each mode of generation.

212. Are the following surfaces developable?

(1) xyz= a3;

(2) z-c=√(xy).

213. Find the differential equation to a surface whose tangent plane at any point includes with the three co-ordinate planes a pyramid of constant volume; shew that the surface is generally developable, the only exception being the surface determined by xyz:

= a constant.

214. Find the equation to the surface in which the tangent plane at (x, y, z) meets the axis of z at a distance from the origin equal to that of (x, y, z) from the origin.

215. Find the equation to the surface in which the tangent plane at (x, y, z) meets the axis of z at a distance

n

from the origin. If n = 1, give that form to the arbitrary function which will produce the equation to an ellipsoid. 216. A plane passes through (0, 0, c) and touches the circle

x2 + y2= a2, z=0;

determine the locus of the ultimate intersections of the plane,

217. Three points move with given uniform velocities along three rectangular axes from given positions; shew how to find the surface to which the plane passing through their contemporaneous positions is always a tangent.

218. Spheres of constant radius r are described passing through the origin; find the envelope of the planes of contact of tangent cones having a fixed vertex at the point (a, b, c).

219. Find the locus of the ultimate intersections of a series of planes touching two parabolas which lie in planes perpendicular to each other and have a common vertex and axis.

220. The sphere x2+y+2=2ax+2by+2cz is cut by another sphere which passes through the origin and has its centre on the surface

shew that the equation to the envelope of the planes of intersection is

221. From each point of the exterior of two concentric ellipsoids, whose axes are in the same directions, tangent planes are drawn to the surface of the interior ellipsoid; shew that all the planes of contact corresponding to the several points of the exterior surface touch another concentric ellipsoid.

222. From any point P in the surface of an ellipsoid straight lines are drawn so as each to pass through one of three conjugate diameters, and be parallel to the plane containing the other two; these straight lines meet the surface again at P1, P, P.; find the equation to the plane which passes through these points, and the locus of the ultimate intersections of all such planes, the diameters remaining fixed while P moves; and shew that its volume = ellipsoid.

3

35

of that of the