Shew how to make use of this to determine the reactions at the two points of support of a plane vertical framework at one of which the direction of the reaction is known.

9. Find the centre of mass of a uniform circular sector.

MIXED MATHEMATICS.-PART III.

The Board of Examiners.

1. A smooth rigid body acted on by a given system of forces defined by means of rectangular coordinates is kept in equilibrium by contact at six points (1, 1, 1), (X2, Y2, Z2),.... with fixed bodies, the normals at the points of contact being (l, m, n1), (l2, M2, N2), .....

Find equations to determine the reactions at the six points.

2. Investigate a general formula to obtain the c.m. of a homogeneous solid of revolution.

Shew that the c.m. of the homogeneous solid of revolution obtained by revolving the cardioid ra (1 + cos 0) round the axis = 0 is at a distance a from the origin.

3. Investigate general equations of equilibrium of an inextensible string under given forces in one plane.

Shew that if gravity is the only force on a string of varying density, the mass between the points at which the tangents to the string have inclinations, to the horizontal is proportional to tan 2 tan 1.

4. An endless heavy string consists of two parts of lengths 1, l' and densities m, m' <m respectively. The string is slung over a small rough peg with uppermost. Shew that if the string is about to slip the difference of the lengths of l on the two sides of the peg is

5. Investigate expressions for the components of acceleration of a particle in a plane in polar coordinates.

6. A particle is attached to an inextensible string which wraps on a curve. Shew that the accelerations of the particle along and perpendicular to the string are

- w2r

dr

dt

r + 2w + K

where r is the length of the straight part of the string, w its angular velocity, and the curvature of the curve at contact.

A particle is attached to a string which wraps on a circle. There being no external force, shew that the decrease of the square of the distance of the particle from the centre of the circle is proportional to the time elapsed.

7. Investigate an expression for the central force under which a particle can describe freely a given plane curve.

8. The point of attachment of a simple pendulum of length performs small simple harmonic oscillations of period 2/p and amplitude a in a horizontal straight line. Find the equation of motion of the pendulum, and shew that, properly started, it can execute a vibration about the vertical of period 2/p and of angular amplitude

ap2
9 - lp3

9. Prove that there are three rectangular axes at every point of a mass-system, with respect to which the products of inertia vanish.

10. Prove ab initio the equations for the motion of the c.m. and for the rotation of a rigid body in two dimensions.

11. Two equal uniform rods AB, AC are smoothly jointed together at one end A of each. Cis smoothly jointed at a fixed point and B is pushed in a smooth guide with uniform velocity V towards C.

Shew that the reaction at the joint A is

where M is the mass of each rod, a its length, and the angle ABC. Gravity is neglected.

12. A rigid body is turning about a horizontal axis. When the c.m. is vertically under the axis, the angular velocity being w, the axis is suddenly seized and carried forward with a velocity V, so that the rotation is permanently stopped. Shew that Vol where is the length of the equivalent simple pendulum.

PHYSICAL GEOLOGY AND MINERALOGY.

The Board of Examiners.

PASS AND FIRST HONOUR PAPER.

1. Mark down and explain all the symbols in use for denoting physical characters of rocks on geological maps. What portions of geological formations are represented on geological maps?

2. Illustrate the modes of forming geological sections from geological maps.

3. What explanations have been given of the jointing, cleavage, and foliation of rock masses? Describe the physical characteristics of each.

4. Explain the shape of the flexions of stratified rocks, and the relations of their parts to the force producing them.

5. Explain the mode of origin, movements, and geological effects of glaciers.

6. How has the ascertained?

mean density of the earth been How does this bear on the controversies as to the constitution of the interior of the earth?

7. Describe the characters of the different sorts of concretionary action found in aqueous and igneous rocks.

8. Explain the nature of the action of the chief cause of changes of position of great rock masses.

9. Enumerate the chief groups of minerals arising in classification from giving the main value to the electro-negative constituents.

10. Explain clearly the nature of crystallographic axes and of optic axes, and the mode of using each for the definition of systems of crystals.

STRATIGRAPHICAL GEOLOGY AND

PALEONTOLOGY.

The Board of Examiners.

1. How may Aqueous rocks be distinguished from Igneous rocks generally? Explain the chief exceptions to the general rules relating to this

matter.

2. In what respect does Dinotherium differ from the more recent Elephants, Mastodons, and Tapirs respectively? Give the generic characters and geological range of each.