VI. General Functional and Differential Equations.

2174. A surface is generated by a straight line which always intersects the two fixed straight lines

prove that the equation of the surface generated is of the form

2175. The general functional equation of surfaces generated by a straight line which intersects the axis of z and the circle z = 0, x + y2 = a*,

2176. The general functional equation of surfaces generated by a straight line which always intersects the axis of z is

2177. The differential equation of a family of surfaces, such that the perpendicular from the origin on the normal always lies in the plane of xy, is

z (p2 + q3) + px + qy = 0.

2178. The differential equation of a family of surfaces, generated by a straight line which is always parallel to the plane of xy and whose intercept between the planes of yz, zx is always equal to a, is

(px + qy)" (p2 + q3) = a2p3q3.

2179. The general differential equation of surfaces, generated by a straight line, (1) always parallel to the plane lx + my + nz = 0, (2) always intersecting the straight line

=

ī m

n

is

(1) (m + ng)2 r − 2 (m + ng) (l + np) 8 + (l + np)3 t = 0,

(2) (ly-mx) (q'r - 2pqs+p't) + 2 (ly - mx) (nx - lz) (qr —ps)

+ 2 (ly − mx) (ny — mz) (q8 − pt)

+ (nx-lz)3r + 2 (nx − l:) (ny – m:) 8 + (ny — m2)' t = 0.

VII. Envelopes.

2180. The envelope of the plane lx+my+nz = a; l, m, n being parameters connected by the equations

is the cylinder

1 + m + n = 0, l2 + m3 + n2 = 1,

(y − 2)3 + (≈ − x)2 + (x − y)2 = 3a2.

2181. Find the envelope of the planes

y

2

cos (0 − p) + // cos (0 −4) + ≈ sin (0 + 4) = sin (0 – $),

b

У

с

cos (0 - $) + (cos + cos $) + (sin 0 + sin 4) = 1;

b

both when 0, & are parameters, and when ✪ only is a parameter.

[The envelope of (1) when both 0, & are parameters is the hyperboloid

and when only is a parameter, the plane (1) always passes through a fixed generator of this hyperboloid; the envelope of (2) when 0, & are parameters is the ellipsoid

when alone is a parameter the envelope is a cone whose vertex is the point (a, b cos p, c sin p).]

2183. The envelope of all paraboloids to which a given tetrahedron is self-conjugate is the planes each of which bisects three edges of the tetrahedron.

[More generally, if a conicoid be drawn touching a given plane and such that a given tetrahedron is self-conjugate to it, there will be seven other fixed planes which it always touches, the equations of the eight planes referred to the given tetrahedron being

±px±qy±rz + w = 0.]

2184. A prolate ellipsoid of revolution can be described having two opposite umbilics of a given ellipsoid as foci and touching the given ellipsoid along a plane curve: and this will be the envelope of one system of spheres, each of which has a circular section of the ellipsoid for a great circle.

2185. Spheres are described on a series of parallel chords of a given ellipsoid as diameter: prove that they will have double contact with another ellipsoid, and that the focal ellipse of this envelope will be the diametral section of the given ellipsoid which is conjugate to the chords. Also, if a, b, c be the axes of the given ellipsoid, and a, ß, y of the envelope,

y being that axis which is perpendicular to the focal ellipse.

2186. A series of parallel plane sections of a given ellipsoid being taken, on each as a principal section is described another ellipsoid of given form; the envelope is an ellipsoid touching the given one along a central section at any point of which the tangent plane is perpendicular to the planes of the parallel sections.

2187. The envelope of a sphere, intersecting a given conicoid in two planes and passing through the centre, is a quartic which touches the given conicoid along a sphero-conic.

VIII. Curvature.

2188. From any point of a curve equal small lengths 8 are measured in the same sense along the curve, and along the circle of absolute curvature at the point, respectively: prove that the distance between the ends of these lengths is ultimately

p, σ being the radii of curvature and torsion respectively at the point.

2189. Find the radius of absolute curvature and of torsion at any

point of the curves

(1) x=a (3t-t3),

(2) x= 2at3 (1+t),

y=3ať, z= a (3t + t3);

y=at3 (t + 2), z=at3 (t2 + 2t + 2).

2190. The radius of absolute curvature (p) at any point of a rhumb line is a cos÷√1-sin3 0 cos a, where is the latitude, and a the angle at which the line crosses the meridians; and the radius of torsion is

2191. Two surfaces have complete contact of the nth order at a point: prove that there are n + 1 directions of normal section for which the curves of section have contact of the n + 1th order; and hence prove that two conicoids which have double contact with each other intersect in plane curves.

2192. Prove that it is in general possible to determine a paraboloid, whose principal sections are equal parabolas, and which has a complete contact of the second order with a given surface at a given point.

2193. A paraboloid can in general be drawn having a complete contact of the second order with a given surface at a given point, and such that all normal sections through the point have four-point

contact.

2194. A skew surface is capable of generation in two ways by the motion of a straight line, and at any point of it the absolute magnitudes of the principal radii of curvature are a, b: prove that the angle between the generators which intersect in the point is cos

+ b

at which the indicatrix is a rectangular hyperbola lie on the cones

(1) x1 (y + z) + y1 (≈ + x) + z1 (x + y) = 0,

respectively; and in (3) these points lie on the circle

x + y + z =α, x2 + y2 + z2 = a2.

2196. A surface is generated by a straight line moving so as always to intersect the two straight lines

and A, μ are the distances of the points where the generator meets these straight lines from the points where the axis of x meets them; prove that the principal radii of curvature at any point on the first straight line are given by the equation

a2p2 sin' a -2ap sin a

dx αμ

(λ - μ cos a) √a2 + μ2 sin2 a

2

= (1/4)" (a2 + μ2 sin2 a)".

2197. A surface is generated by the motion of a variable circle, which always intersects the axis of x, and is parallel to the plane of yz. At a point on the axis of x, r is the radius of the circle, and ✪ the angle which the diameter through the point makes with the axis of ≈ prove that the principal radii of curvature at this point are given by the equation

2198. A surface is generated by a straight line which always intersects a given circle and the normal to the plane of the circle drawn through its centre; is the angle which the generator makes with this normal, and the angle which the projection of the generator on the plane of the circle makes with a fixed radius: prove that the principal radii of curvature at the point where the generator meets the normal are

and that at the point where it meets the circle, the principal radii are given by the equation

p2

+ ap cos 0 = a2.

2199. A surface is generated by a straight line, which is always parallel to the plane of xy, and touches the cylinder x+y=a: prove that, if p be a principal radius of curvature at the point whose co-ordinates are (a cos 0 + r sin 0, a sin ✪ - - r cos 0, 2)

2200.

A straight line moves so as always to intersect the circle x2 + y2 = a3, z=0, and be parallel to the plane of zx; prove that the measure of specific curvature at the point (a cos p, a sin p, 0) is

O being the angle which the generator through the point makes with the axis of z.

2201. A circle of constant radius a moves so as to intersect the axis of x, its plane being parallel to the plane of yz: prove that, at the point

(x, a sina sin -0, a cos + a cos p − 0),

--

the measure of specific curvature of the surface generated is

2202.

In a right conoid whose axis is the axis of z, prove that the radius of curvature of any normal section at a point (r cos 0, r sin 0, ≈) is

for the principal radii of curvature at the point.