254. Find the angle between the osculating planes at two consecutive points of a curve.

255. The shortest distance between the tangents at two

consecutive points of a curve of double curvature is d (ds),

12p where ds is the length of the arc between the two points, do is the angle between the osculating planes at the two points, and p is the radius of absolute curvature.

256. Shew that the edge of the developable surface formed by the locus of the ultimate intersections of normal planes of a curve of double curvature is the locus of the centres of spherical curvature of the curve.

Find the locus of the centres of spherical curvature of a helix.

257. Find the equation to the surface on which are found all the evolutes of the curve x2 = az, x=y.

258. Shew that the curve represented by the equations x2 + y2 + z2 = a2,

√x + √y + √z = c;

is cut perpendicularly by each of the series of surfaces + (μ − 1) ya = μza,

μ being an arbitrary parameter.

259. A curve is traced on a surface; shew that the radius of absolute curvature at any point of this curve is the same as the radius of curvature of the section of the surface made by the osculating plane of the curve at that point.

260. A curve is traced on a sphere; shew that generally the radius of the sphere is the radius of spherical curvature of the curve; but that this does not hold if the curve be a plane curve, or plane for an indefinitely short period at any point.

261. If the normal plane of a curve constantly touches a given sphere the curve is rectifiable.

262. If (x', y', z') be the point in the locus of the centres of curvature corresponding to the point (x, y, z) on a curve, p the radius of curvature at (x, y, z), and s' the arc of the

dp

ds'

above locus, shew that is the cosine of the angle between the tangent at (x', y', z') and the direction of ρο

263. Find the equation to the surface on 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.

V. Curvature of Surfaces.

264. Determine the conditions necessary in order that the surfaces whose equations are

ax2 + by2 + cz2 + 2a'yz + 2b'zx + 2c'xy + 2z = 0,

Ax2 + By2+ Cz2+2A'yz+2B'zx+2C'xy + 2z = 0,

may have their principal radii of curvature at the origin equal; and shew that if these conditions be fulfilled any sections of the two surfaces parallel to the plane of (x, y) will

be similar.

265. Obtain the quadratic equation for determining the principal radii of curvature at any point of the surface

$ (x) +x (y) +¥ (z) = 0 ;

and find the condition that the principal curvatures may be equal and opposite.

266. The locus of the points on the hyperboloid

for which the principal curvatures are equal and opposite, has for its projection on the plane of (x, y) the ellipse

267. Shew that the principal radii of curvature are equal in magnitude and opposite in sign at every point of the surface determined by

268. The trace on the plane of (y, z) of the locus of the extremities of the principal radii of curvature of the ellipsoid whose equation is

find that in which the principal radii of curvature are equal but of opposite sign.

270. Find the surface of revolution at every point of which the radii of curvature are equal in magnitude and opposite in sign.

271. If a, b be the principal radii of curvature at any point of a surface referred to the tangent plane at that point as the plane of (x, y) and the principal planes as planes of (x, z) and (y, z), then will the locus of the circles of curvature of all normal sections of the surface at the origin be

(x2 + y2+ 2") (2*2 + 2,5"). =2z (x2 + y2).

a

272. Find the radius of curvature in any normal section of the surface

Ax2 + By2 + Cz2+2A'yz + 2B'zx + 2 C'xy + Ez = 0,

at the origin; and shew that the sum of the reciprocals of the radii of curvature in sections at right angles to each other is

constant.

273. Required the sum of the principal radii of curvature at any point of a curved surface in terms of the co-ordinates of that point.

The equation to the surface being f(x, y, z) = 0, express the result by partial differential coefficients of f (x, y, z).

274. If r=ƒ(0, 4) be the equation to a surface referred to the tangent plane at the origin as the plane of (x, y), then

the radius of curvature at the origin of a normal section inclined at any angle to the plane of (x, z) is

275. If the surface (x-a)+(y-b)-(2-c) have contact of the second order with the surface z=f(x, y), shew that the relations

must be satisfied at the point of contact.

276. At either point at which the surface

(-1)+(-1)+1

meets the axis of z, an elliptic paraboloid may be found, which has at its vertex a complete contact of the third order with the surface.

277. Find the radius of curvature of a normal section of a spheroid made by a plane inclined to the meridian at any given angle.

278. Shew that the locus of the focus of an ellipse rolling along a straight line is a curve such that if it revolves about that line, the sum of the curvatures of any two normal sections at right angles to one another will be the same for all points of the surface generated.

279. In any surface of the second order the tangents to the lines of curvature at any point are parallel to the axes of any plane section parallel to the tangent plane to the surface at that point.

280. Obtain the differential equation to the projection on one of the co-ordinate planes of the lines of curvature of a