surface. Apply the equation to determine the lines of curvature of a surface of revolution.

281.

282.

Determine the lines of curvature on the surface

xy = az.

Find the radius of curvature of any normal section of a surface at a given point.

If y' - y = m (x' — x) be the projection on the plane of (x, y) of the tangent line to the curve of section, shew that the values of m corresponding to the principal sections of the

y

surface 2=ƒ(2) at the point (x, y, z) are

283. The links of a chain are circular, being of the form of the surface generated by the revolution of a circle whose radius is one inch about a line in its own plane at a distance of four inches from the centre; apply Euler's formula for the curvature of surfaces to shew that if one link be fixed, the next cannot be twisted through an angle greater than 60° without shortening the chain.

284. An annular surface is generated by the revolution of a circle about an axis in its own plane; shew that one of the principal radii of curvature at any point of the surface varies as the ratio of the distance of this point from the axis to its distance from the cylindrical surface described about the axis and passing through the centre of the circle.

285. If p, p' be the greatest and least radii of curvature of a curve surface at a given point, 4, the angles which the normal to the surface at the given point makes with the axes of x and y, shew that

286.

Define an umbilicus. In what sense do you understand that there is an infinite number of lines of curvature at an umbilicus? And from this consideration deduce the partial differential equations which exist at such points.

287. Shew that in the surface

• (x) + x (y) + ¥ (≈) = 0,

if_p" (x) = x′′ (y) = "(z) at any point, that point is an umbilicus.

289. Find the umbilici of the surface xyz = a3.

290. Shew that the radius of normal curvature of the surface xyz = a* at an umbilicus is equal to the distance of the umbilicus from the origin of co-ordinates.

291. A sphere described from the origin with radius abc will touch the surface

ac + ab + bc

292. Shew that a sphere whose centre is at the origin and whose radius r is determined by the equation

at umbilici; and the radius of normal curvature of the surface

293. Determine whether there is an umbilicus on the surface

294. If R be the radius of absolute curvature at any point of a curve defined by the intersection of two surfaces u1 = 0, u2 = 0, and r2 be the radii of curvature of the sections of u1 =0, u,= 0, made by the tangent planes to u2 =0, u1 = 0, respectively at that point, shew that R, r1, r, will be connected by the relation

2

295. Two surfaces touch each other at the point P; if the principal curvatures of the first surface at P be denoted by a+b, those of the second by a'b', and if be the angle between the principal planes to which a+b, a+b', refer, and the angle between the two branches at P of the curve of intersection of the surfaces, shew that

296. Shew that at any point of a developable surface, the curvature of any normal section varies as the square of the sine of the angle which this section makes with the generating line, and that at different points along the same generating line the principal radius of curvature varies as the distance from the point of intersection of consecutive generating lines.

297. A surface is generated by the motion of a straight line which always intersects a fixed axis. If P be any point in this axis at a distance x from the origin, the angle which the generating line through this point makes with the axis, and the angle which the plane through the axis and the generating line makes with its initial position, shew that the principal radii of curvature at Pare

298. The shortest distance between two points on a curved surface never coincides with a line of curvature unless it be a plane curve.

VI. Miscellaneous Examples.

299. Find the condition that must hold in order that the equations

ax + c'y+b'z = 0, by +a'z + c'x=0, cz+b'x+a'y = 0, may represent a straight line; and shew that in that case the straight line is determined by the following equations :

x (aa' — b'c') = y (bb′ — c'a') = z (cc' — a'b').

300. Shew that the six planes which bisect the interior angles of a tetrahedron meet in a point.

301. Shew that the three planes which bisect the exterior angles round one face of a tetrahedron and the three planes which bisect the interior angles formed by the other three faces meet in a point.

302. If a = 0, ẞ= 0, y=0, d=0 be the equations to the faces of a tetrahedron expressed in a suitable form, A, B, C, D the areas of the respective faces, shew that

Interpret

Aa+BB+ Cy+ Dd a constant.

Aa + BB + Cy = 0.

303. If a=0, 80, y = 0 be the equations to three planes which form a trihedral angle, the equation to a cone of the second degree which has its vertex at the angular point and touches two of the planes at their intersections with the third, is y2 - kaß = 0.

=

304. Give the geometrical interpretation of the equation uu' kvv', where k is a constant, and the other letters denote linear functions of x, y, z. Hence shew that there must exist surfaces of the second order which contain straight lines.

305. If u1=0, u2=0, u=0 be the equations to three planes, interpret the equation

306.

Also interpret Au ̧ ̧+ Bu ̧μ ̧+ СÙ‚μ‚=0.

307. If u, v, w are linear functions of x, y, z shew that uvw represents a conical surface; and shew that the equation to the tangent plane is Xu - 2λw + v = 0.

308. Let v=0 be the equation to a surface of the second order; u=0 and u,=0 the equations to two planes; shew that by giving a suitable value to the constant the equation v+λuu2 = 0 will represent any surface of the second order which passes through the intersections of the two planes with the given surface,

309. If t=0, u = 0, v = 0, w = 0 be the equations to four given planes, and λ, μ, be two arbitrary constants, shew that t+Au=0 represents a plane which passes through a fixed straight line, t+Aλu + μv=0 a plane which passes through a fixed point, and tw+puv0 a surface of the second order which contains four fixed straight lines.

310. If the equation to a surface of the second order be u2+2u, +1=0, where u, and u, represent the terms of the first and second order respectively, and tangent planes be drawn to the surface from any point of the plane determined by u1+1=0, the planes of contact will all pass through the origin.

311. If a, ß, y, & be the distances of any point from the faces of a tetrahedron, shew that the general equation to a surface of the second order circumscribing the tetrahedron is

Aaß + By + Cad + Dẞy + EBS + Fyd = 0.

Determine the condition necessary in order that the straight B γ

line

α

ī

=

- =

m n

may touch the surface; and hence shew that the

equation to the tangent plane at the point (a=ẞ=y=0) is

Ca + EB + Fy = 0,

312. If A, B, C, D be the angular points of a tetrahedron; a, B, y, & the distances of any point from the faces respectively