Prove that Laplace's Equation, transformed to a, 8, y as coordinates becomes

3.

Prove that the superficial area of an ellipsoid is represented by.

where μ and v are elliptic coordinates, represent two systems of circles having double contact with the conic μ-a.

7. Prove that if a surface has on it a system of geodesics lying in parallel planes, it is either a cylinder, or a ruled surface generated by lines parallel to the planes of the geodesics.

and establish the relation between them.

The Jacobian of the two integrals vanishes conditionally at the points where (i) touches (ii). Interpret geometrically.

10. Find the equation of the curves orthogonal to the surfaces

Sinh æ. Sinh y. Sinh z = constant.

where c is an arbitrary constant, and 0 is a function of x and y only, prove that

MIXED MATHEMATICS.

Examiner-C. LITTLE, ESQ., M.A.

FIRST PAPER.

Not more than SEVEN questions may be attempted, and the selected questions must be fairly distributed.

1. Describe and illustrate the graphic method of considering the equilibrium of a system of forces in a plane. Show how, in the event of the forces not being in equilibrium, the magnitude of the resultant may be found. Particular care must be taken that accurate methods of estimating the magnitude and direction of forces are introduced.

From a point O perpendiculars Oa, OB, Oy, are let fall on the three sides of a triangle ABC. At the points a, B, y forces P, Q, R are applied. each force being perpendicular and proportional to the face at which it acts. The forces are then all turned round their points of application in the same sense so as to make equal angles with the perpendiculars Oa, OB, Oy, the lines of action of the forces in this last position intersecting in the points A, B, C. Find by any method the resultant of the three forces, showing that it is a couple proportional to the square root of the area of the triangle A'B'C'.

2. Find the centre of gravity of a system of matter irregularly distributed.

Explain how in any particular case you would choose the origin and axes of coordinates so as to simplify the analysis.

Find the centre of gravity of the matter contained between a spherical surface of radius a, and a plane at a distance d from the centre of the sphere, the distribution being uniform.

3. Find the general equations of a flexible string in equilibrium under the action of a given system of forces. Find the constraining forces which will have to be applied at the extremities.

An elastic string, in its original state, is placed symmetrically on a right circular cylinder having its axis horizontal. Find the extension of the string; and its maximum length so that there may be no part of the string not in contact with the cylinder.

4.

What is meant by (1) a screw, (2) a wrench. Find the resultant wrench to which two wrenches in intersecting rectangular screws are equivalent. Having given data sufficient to determine a cylindroid, and wrenches in two of the screws of the cylindroid, find the resultant wrench and the screw in which it acts.

5. Prove that a system of forces acting on a rigid body may be replaced by two equal forces whose lines of action are perpendicular to each other, and each inclined at an angle of 45° to Poinsot's axis. Also prove that the forces act at the ends of a line bisected by the axis; the length of the line 2 K R being and the magnitude of the force being

R

6. Assuming that the general displacement of a rigid body may be produced by a motion of translation which is the same for all its points, and a motion of rotation round an axis through an angle which is the same for all its points, find mathematical expressions for the displacement of any point of the body.

From these expressions deduce the six equations of equilibrium.

By the method of Lagrange find the conditions of equilibrium of a system

of three particles forming a rigid triangle, each particle being acted on by given forces.

7. State the general property of Stable Configurations.

Find a mathematical expression for the increase of the Potential Work of the system as a result of the introduction of a single hampering condition when the newly applied external forces are about to act, explaining carefully the meaning of the symbols and functions employed.

8. Find the attraction of a spherical shell of uniform density on an external particle, the law of force being the inverse square of the distance.

Show that the law of the inverse square is the only law of attraction for which a spherical shell of uniform thickness and density will produce no resultant attraction on any internal particle.

Explain what is meant by the constant of gravitation employed in the above investigation, and find what fraction it is of a dyne.

9. Prove that NdS=0 or-4уM, explaining the meaning of the symbols.

Deduce from the above equations that 2V = 0 or-4πуp, the coordinates employed being cylindrical.

Find the value of the potential due to an infinite circular cylindrical shell of uniform matter at a point (1) inside the shell, (2) within the matter of the shell, (3) outside the shell.

10. Show how to determine a homogeneous function of x, y, z of the most general form which satisfies the equation v2V = 0.

Find such a function of the 4th degree.

11. Show that every strain can be resolved into a pure strain and a rotation. Find the elements of the pure strain, also the resultant rotation and the axis about which it takes place.

A strain is given in the form of a shear of two known rectangular lines. Find the elements of the strain with reference to three given rectangnlar

axes.

12. Deduce from first principles the equation of the stress ellipsoid in its simplest form.

If the stress on any plane is wholly a shearing stress, prove that its line of action is the line of contact of the plane with the cone of shearing stress, and find its magnitude.

13. A uniform slighty elastic beam rests, in non-limiting equilibrium, with one end on the ground and the other against a vertical wall, the vertical plane through the beam being at right angles to the wall; find the form of the mean fibre of the beam,

14. A weight W is supported on a rough inclined plane of inclination a by a force P, whose line of action makes an angle i with the plane, and whose component in the plane makes an angle B with the line of greatest inclination in the plane : prove that equilibrium will be impossible if

μ2(1+cos 2a cos 2i - sin 2a sin 2i cos B)

7 2 sin 2a sin 28 cos 21.

MIXED MATHEMATICS.

SECOND PAPER.

Not more than SEVEN questions may be attempted, and the selected questions must be fairly distributed.

1. Explain how a case of impact is dealt with in discussing the motion of a system of several unconnected bodies acted on by given forces.

Two smooth elastic spheres, moving in the same vertical plane, under given conditions, impinge; investigate fully the subsequent motion.

2. Find the equations of motion of a particle moving under any conditions and show how they may be solved, giving special attention to the case where the process may be simplified owing to the quantities, representing the motion, being small.

A particle is at rest on a given curve under gravity. If slightly disturbed from the position of eqilibrium, investigate the subsequent motion, (1) if the curve be smooth, (2) if the curve be rough

3.

Show how to find the motion of a heavy particle under gravity, when the resistance varies as the nth power of the velocity.

Discuss briefly the result of gunnery experiments as regards the resisting influence of air on projectiles.

Find the intrinsic equation of the path of a projectile when the resistance varies as the square of the velocity.

4. Find the equation of the motion of a parcticle constrained to slide on a curve moving in its own plane.

A bead is at rest on an equiangular spiral of angle a at a distance a from the pole. The spiral begins to turn round its pole with an angular velocity w. Prove that the bead comes to a position of relative rest when ra cos that the pressure is then ma2a sin 2a.

and

Prove also when the bead is again at its original distance from the pole, the pressure is mw2a sin a (3+ sin2a).

5. Discuss the method of treating problems on initial motions and initial stresses.

A circular wire of mass M is held at rest in a vertical plane, on a smooth horizontal table, a smooth ring of mass m being supported on it by a string which passes round the wire to its highest point, and from there horizontally to a fixed point to which it is attached. If the wire be set free, show that the pressure of the ring on it is immediately diminished by

where a is the angular distance of the ring from the highest point of the wire.

6. Discuss fully the motion of a body under the influence of a central force varying directly as the distance

A portion of an epicycloid is described under a force tending to the centre of the fixed circle: prove that, if a straight line be drawn from any fixed point always parallel and proportional to the radius of curvature in the epicycloid, the extremity of this line will describe a central orbit.

7. Define an apse, and from the definition deduce the more important theorems regarding apses, apsidal distances and apsidal angles, when the law of force is a one-valued function of the distance. Careful attention should be given to the case of an orbit nearly circular.