5. Forces defined in Cartesian co-ordinates act at fixed points, and in fixed directions, on a body at rest. Shew that the time of a small oscillation around an axis through the origin whose direction cosines are l, m, n is

2π √ MK2 / √l2Σ (yY+ zZ)

Σmn (yZ+zY).

6. Prove Poinsot's theorem that a body under no forces with one point fixed moves so that an ellipsoid in the body rolls on a plane in space.

7. A rigid body has one point of an axis of symmetry fixed; the intersection of this axis with a sphere of unit radius describes a curve whose arc is s, and curvature in the tangent plane к. Shew that the couple on the body along the tangent to the curve is Cws + Aks2 where w is the angular velocity round the axis, and A, C are the principal moments of inertia. Deduce the condition

of steady motion of a top.

8. A plane sheet can undergo simple expansion or contraction at each point, but no shear. Obtain the general solution for a small deformation, and shew that if the boundary of a circular disc of radius a of the material receives a radial displacement a am+1 cos me, the radial and transversal displacements throughout the disc are

a pm +1 cos mo, a pm+1 sin mo respectively.

9. Investigate St. Venant's solution of the torsion of a cylinder.

10. A straight rod of length l is clamped at one end

and a string is attached to the other, and to a point in the line of the rod a distance a beyond

D

the clamped end. Shew that the greatest tension T of the string compatible with stability of the rod in the straight form is the least root of

11. A spherical shell of finite thickness and radii a, b, is submitted to pressures p1 p1 inside and outside. Find the stress at each point.

MATHEMATICS.-PAPER V.

The Board of Examiners.

1. The position of a system of rigid bodies is defined by n coordinates 0, and the work of the external forces due to a small displacement is EP80. There are smooth contacts, the breaking of each being determined by a condition Lade positive. Assuming the principle of virtual work, obtain all the conditions of equilibrium.

2. Each of the joints of a plane quadrilateral frame, ABCD requires a couple L to move it. Show that the pull along the diagonal AC required to deform the frame is

3. Investigate the condition that a system of forces should reduce to a single force, and determine its magnitude and position.

4. A heavy body of any form rests on two rough vertical curves in parallel planes touching them at the points A, B. The line AB is horizontal and perpendicular to the planes of the curves which have horizontal tangents at A and B. The vertical GM through the centre of gravity of the body meets AB in M. Show that the condition of stability is

GM <

AM 'R'

+

AB p' + R' AB P + R

where p, p' are the radii of curvature of the curves at A, B, and R, R' those of the sections of the body by the planes of the curves.

5. Investigate the Cartesian equation of the catenary of a heavy chain of variable section which is at the point of breaking at every point.

6. Two fine smooth extensible strings, one resting on the other, are attached to the same two points. Show that the curve assumed under gravity is given by

T

m'

2

= mg (1+ 2) (1+) sec2

λ

where Ρ is the radius of curvature,

the angle

the tangent makes with the horizontal, and T, T

the tensions of the strings.

7. Find the potential of a solid homogeneous ellipsoid as a single integral, using that of a homœoidal shell if necessary.

8. A homogeneous gravitating solid is bounded by a single closed surface, which is an equipotential, and all the internal equipotentials are similar, and similarly situated concentric surfaces. Shew that the solid is a sphere.

9. Prove Thomson's theorem that in a perfect fluid

dp

1

B

B

[ -f d2 - √ + } q2 ] = [ " (ude+vdy+wdz),

ρ

2

δι A

and deduce the ordinary expressions for the pressure in the cases of irrotational and steady

motion.

10. A long canal of rectangular section leaks slightly from a small crack extending across its bottom. If p, are the velocity and stream functions, prove that to a first approximation + ¿ ¥ = —

π

Q

log

cosh (x + iy) and that the form of the sur2h

Q2 face is given by y = tanh2 where x is 8gh2

2h

measured along the canal, h is its depth and Q is the leakage per unit length per unit time.

11. Obtain an expression for the velocity potential due to a system of vortex filaments in an infinite liquid.

MATHEMATICS.-PAPER VI.

The Board of Examiners.

1. Shew that if one central orbit is the orthogonal projection of another, and their centres correspond, the positions of the particles also correspond.

A particle describes a circle of radius, a under a centre of force (taken as origin), at a distance b from the centre along the axis of x.

Shew that the force

∞ (x2 + y2)} / {a2x2 + (a2 — b2)y2}},

2. Obtain the general equations of motion of a particle referred to axes moving with the earth, and demonstrate Foucault's property of the pendulum.

3. A heavy rotating disc of radius a is placed gently in a vertical plane on two rough pegs at the same level, and at distance 2c apart. If λ is the angle of friction, shew that the disc will remain on the pegs if sin λ <ca, and that it comes to rest in time √a2. c2 w/g sin 21, where w is the initial angular velocity.

4. Investigate the general equations of motion of a dynamical system in the Hamiltonian form.

5. A body whose surface is F(x, y, z) = o is slightly deformed, so that the point P (x, y, z) comes to