the density of the surrounding fluid varies as the nth power of the depth : prove that the weight of the solid is to the weight of the displaced fluid as n − 1 : 3 (2n+1), whereas if the solid can rest in this position the ratio must be n-1: n + 1. Also prove that the generating curve of the solid will be

where a is the radius of the base and h the height.

[If n =

2 the solid is a paraboloid, if n = 3, an ellipsoid.]

2757. A hollow cylinder with vertical axis contains a quantity of fluid whose density varies as the depth and into it is allowed to sink slowly, with vertex downwards, a solid cone the radius of whose base is equal to the radius of the cylinder; the cone rests when just immersed : prove that the density of the cone is equal to the initial density of the Huid at a depth equal to one-twelfth of the length of the axis of the cone. If the cone be allowed to sink freely into the fluid, starting with its vertex at the surface and just sinking until totally immersed, the density of the cone will be to the density of the fluid at the vertex of the cone in its lowest position as 1 : 30.

2758. A tube of fine bore whose plane is vertical contains a quantity of fluid which occupies a given length of the tube; a given heavy particle just fitting the tube is let fall through a given vertical height find the impulsive pressure at any point of the fluid; and prove that the whole kinetic energy after the impact bears to the kinetic energy dissipated the ratio of the mass of the particle to the mass of the fluid.

[If m, m' be the masses of the particle and fluid, V the velocity of the particle just before impact, the impulsive pressure at a point whose distance along the arc from the free end is s will be

the whole length of arc occupied by the fluid.]

2759.

mm' V s

m+m' i '

where is

A flexible inextensible envelope when filled with fluid has the form of a paraboloid whose axis is vertical and vertex downwards and whose altitude is five-eighths of the latus rectum: prove that the tension of the envelope along the meridian will be greatest at points where the tangent makes an angle with the vertical.

π

4

[In general, if 4a be the latus rectum and h the altitude, the tension per unit of length at a point where the tangent makes an angle ✪ with σa (2h + a a the vertical will be

of the fluid.]

2

sin 0 sin3 e),

where σ is the specific gravity

2760. Fluid without weight is contained in a thin flexible envelope in the form of a surface of revolution and the tensions of the envelope at any point along and perpendicular to the meridian are equal: prove that the surface is a sphere.

2761. A quantity of homogeneous fluid is contained between two parallel planes and is in equilibrium in the form of a cylinder of radius 6 under a pressure ; that portion of the fluid which lies within a distance a of the axis being suddenly annihilated, prove that the initial pressure, at a point whose distance from the axis is r, is

2762. A thin hollow cylinder of length h, closed at one end and fitted with an air-tight piston, is placed mouth downwards in fluid; the weight of the piston is equal to that of the cylinder, the height of a cylinder of equal weight and radius formed of the fluid is a, the height of fluid which measures the atmospheric pressure is c, and the air enclosed in the cylinder would just fill it at atmospheric density: prove that, for small vertical oscillations, the distances of the piston and of the top of the cylinder from their respective positions of equilibrium are of the form A sin (λt + a) + B sin (ut + B), λ, μ being the positive roots of the equation

2763. A filament of liquid PQR is in motion in a fixed tube of small uniform bore which lies in a vertical plane with its concavity always upwards; on the horizontal ordinates to P, Q, R at any instant are taken points p, q, r, whose distances from the vertical axis of abscissæ are equal to the arcs measured to P, Q, R from a fixed point of the tube: prove that the fluid pressure at Q is always proportional to the area of the triangle pqr.

2764. A centre of force, attracting inversely as the square of the distance, is at the centre of a spherical cavity within an infinite mass of liquid, the pressure in which at an infinite distance is w, and is such that the work done by this pressure on a unit of area through a unit of length is one half the work done by the attractive force on a particle whose mass is that of a unit of volume of the liquid as it moves from infinity to the initial boundary of the cavity: prove that the time

of filling up the cavity will be #α √(2-()); where a is the initial

radius of the cavity and p the density of the fluid.

GEOMETRICAL OPTICS.

2765. Three plane mirrors are placed so that their intersections are parallel to each other and the section made by a plane perpendicular to their intersections is an acute-angled triangle; a ray proceeding from a certain point of this plane after one reflexion at each mirror proceeds on its original course: prove that the point must lie on the perimeter of a certain triangle.

2766. In the last question a ray starting from any point after one reflexion at each mirror proceeds in a direction parallel to its original direction: prove that after another reflexion at each mirror it will proceed on its original path, and that the whole length of its path between the first and third reflexions at any mirror is constant and equal to twice the perimeter of the triangle formed by joining the feet of the perpendiculars.

2767. A ray of light whose direction touches a conicoid is reflected at any confocal conicoid: prove that the reflected ray also touches the first conicoid.

2768. In a hollow ellipsoidal shell small polished grooves are made coinciding with one series of circular sections and a bright point placed at one of the umbilics in which the series terminates: prove that the locus of the bright points seen by an eye in the opposite umbilic is a central section of the ellipsoid, and that the whole length of the path of any ray by which a bright point is seen is constant.

2769. A ray proceeding from a po'nt on the circumference of a circle is reflected n times at the circle: prove that its point of intersection with the consecutive ray similarly reflected is at a distance from the centre equal to √1 + 4n (n + 1) sin', where a is the radius

a

2n + 1

and the angle of incidence of the ray: also prove that the caustic surface generated by such rays is the surface of revolution generated by an epicycloid in which the fixed circle has the radius

2n + 1

and the

2770. A ray of light is reflected at two plane mirrors, its direction ore incidence being parallel to the plane bisecting the angle between mirrors and making an angle with their line of intersection: prove the deviation is 2 sin (sin 0 sin 2a), where 2a is the angle between

<-1

the mirrors. More generally, if D, be the deviation after r successive reflexions,

cos D1 = sin 0 sin (2n − 1 a − 4), sin } D1 = sin 0 sin 2na, where is the angle which a plane through the intersection of the mirrors parallel to the incident ray makes with the plane bisecting the angle between the mirrors.

2771. Two prisms of equal refracting angles are placed with one face of each in contact and their other faces parallel and a ray passes through the combination in a principal plane: prove that the deviation will be from the edge of the denser prism.

2772. The radii of the bounding surfaces of a lens are r, s, and its thickness is

1 +

(1

+) (8-r): prove that all rays incident on the lens

from a certain point will pass through without aberration but also without deviation.

2773. Prove that a concave lens can be constructed such that the path of every ray of a pencil proceeding from a certain point after refraction at the first surface shall pass through the centre of the lens; that in this case there will be no aberration at the second refraction, and that the only effect of the lens is to throw back the origin of light a distance (1)t, where t is the thickness of the lens.

2774. What will be the centre of a lens whose bounding surfaces are confocal paraboloids on a common axis? Prove that the distance

μ - 1 between the focal centres of such a lens is μ + 1 latera recta.

(a+b), 4a, 4b being the

2775. The path of a ray through a medium of variable density is an arc of a circle in the plane of xy: prove that the refractive index at any point (x, y) must be

1

x-a

x

f

function and (a, b) the centre of the circle.

where is an arbitrary

A ray of light is propagated through a medium of variable density in a plane which divides the medium symmetrically: prove that the path is such that when described by a point with velocity always proportional to μ, the index of refraction, the accelerations of the point parallel to the (rectangular) axes of x and y will be proportional to d (μ3) d (μ3) respectively.

dx

dy

2777. A ray is propagated through a medium of variable density in a plane (xy) which divides the medium symmetrically: prove that the projection of the radius of curvature at any point of the path of the ray on the normal to the surface of equal density through the point is 'dp αμ

equal to μ:

dx

2

+

2

2778. A small pencil of parallel rays of white light, after transmission in a principal plane through a prism, is received on a screen whose plane is perpendicular to the direction of the pencil: prove that the length of the spectrum will be proportional to

(-) sin i÷cos D cos (D+i - 4) cos p';

where i is the refracting angle, p, o' the angles of incidence and refraction at the first surface, and D the deviation, of the mean ray.

2779. Prove that, when a ray of white light is refracted through a prism in a principal plane so that the dispersion of two given colours is a minimum,

where is the angle of refraction at the first surface and i the refracting angle. Hence prove that minimum dispersion cannot co-exist with minimum deviation.

2780. A transparent sphere is silvered at the back: prove that the distance between the images of a speck within it formed (1) by one direct refraction, (2) by one direct reflexion and one direct refraction, is 2pac (a−c) ÷ (a + c − μc) (μc + a − 3c), where a is the radius of the sphere and c the distance of the speck from the centre towards the silvered side.

=

2781. The focal length of the object-glass of an Astronomical Telescope is 40 inches, and the focal lengths of four convex lenses forming an erecting eye-piece are respectively,,,inches, the intervals being the first and second, and the second and third being 1 inch and inch respectively: find the position of the eye-lens and the magnifying power when the instrument is in adjustment; and trace the course of a pencil from a distant object through the instrument.

[The eye-lens must be at a distance of 41.5 inches from the objectglass.]

2

2782. Two thin lenses of focal lengths f,, f, are on a common axis and separated by an interval a; the axis of an excentric pencil, before incidence, cuts the axis of the lenses at a distance d from the first lens: prove that, if F be the focal length of the equivalent single lens,

Ff

2783. The focal length F of a single lens equivalent to a system of three lenses of focal lengths f, f, f, separated by intervals a, b, for an excentrical pencil parallel to the axis, is given by the equation

2784. Prove that the magnifying power of a combination of three nses of focal lengths f, f f on a common axis at intervals a, b will independent of the position of the object, if