10. A heavy beam lying on a rough horizontal plane is raised to a vertical position by applying to one end the least force necessary at each stage.

Shew that the coefficient of friction is not less than 1/8.

11. Prove the principle of virtual work for a system of rigid bodies.

12. A heavy beam is placed symmetrically across a rough fixed horizontal circular cylinder. Shew that the equilibrium is stable if the thickness of the beam is less than the diameter of the cylinder.

MIXED MATHEMATICS.-PART III.

The Board of Examiners.

1. Investigate the analytical conditions of equilibrium of a system of forces on a rigid body, using a rectangular system of coordinates.

The forces on a rigid body consist of a system of wrenches on given axes.

conditions of equilibrium.

Give the analytical

2. A uniform wire forms a circular helix of radius a, making m complete turns and 1/n of a turn. Shew that its centre of mass is at a distance

an sin n

(mn + 1)

from the axis in the plane bisecting at right angles the axial length of the helix.

3. Investigate general equations of equilibrium for a flexible string under forces in one plane.

Solve completely the case where the force is normal and varies as cos2;

being the inclina

tion of the tangent to a fixed line.

4. Define the gravitational potential, and investigate its value inside and outside a solid sphere.

5. Prove that in general, in the motion of a disc in its own plane, one point is at rest at each instant.

Prove also that there is one point at each instant having no acceleration.

6. Obtain the polar differential equation of a central orbit, and apply it to find the orbit when the force varies as the distance from the centre.

7. A heavy particle is sliding down a rough curve under an air resistance varying as the square of the velocity. Find the velocity at each point.

8. Obtain an expression for the product of inertia with respect to two planes at right angles through a point in terms of the moments and products with respect to rectangular axes at the point.

9. A pendulum consists of a cylindrical bob of radius a and length k attached to a cylindrical rod of the same material of radius b and length 1, the axis of suspension being a diameter of the end;

shew that the length of the equivalent simple pendulum is

† (a1k + b1l)

+ } a2 (k3 + 3k2l + 3 kl2) + b213. — b22 + a2(l + Įk)k

10. Prove the independence of the motions of translation and rotation in plane motion of a rigid body.

11. A solid circular cylinder of mass M and radius a is driven on a rough horizontal track by a string which is wrapped on a light barrel of radius b>a on the cylinder, and, leaving the barrel at its lowest point, passes horizontally over a pulley and carries a mass m hanging freely. Shew that the acceleration of the axis of the cylinder is

mga (b − a)|{3 Ma2 + m(b − a)2}.

PHYSICAL GEOLOGY AND MINERALOGY.

The Board of Examiners.

PASS AND FIRST HONOUR PAPER.

1. What is the proper direction for a geological section? How do you correct the dips if obliged to use any other directions?

2. Explain, according to Mallet, the chief characters of earthquake shocks, the successive order of the movements, and the methods of observing and recording them.

3. How are upheavals, contortions, metamorphism, jointing, and cleavage of stratified rocks produced?

4. Explain the conditions necessary for the formation of glaciers, their chief characters, and the effects of glacial action on rocks, and other indications of their former presence in localities in which they no longer exist.

5. Give an account of the chief theories to explain the movements of glaciers.

6. Describe the ways in which stratified rocks are formed, and the evidences of the different conditions in different cases.

7. Explain all the ways in which valleys are formed.

8. Give, as fully as you can, the main important geological characters of Faults. Define "hade," "throw," "upcast," and "downcast."

How is the inclination of a Fault differently measured from the Dip of strata?

9. What is the general form of convolutions of stratified rocks, and what relations have been

indicated between these and the position of the disturbing force?

10. Mention the various modes of determining the shape of the earth, and the relations between this and the probable earliest conditions of the earth.

STRATIGRAPHICAL GEOLOGY AND

PALEONTOLOGY.

The Board of Examiners.

1. State in detail the method of procedure for construction of geological maps in the field, and of geological sections in the study.

2. Define all the technical terms used in ordinary descriptions of stratified rocks.

3. How are the Paleozoic or first formed great group of fossiliferous rocks distinguished from those of the Mesozoic and Tertiary periods?

4. Give the generic characters of as many plants as you can, enabling you to discriminate coal-bearing strata of Paleozoic from those of Mesozoic age.

5. Write down all the divisions generally adopted by geologists of the Palæozoic rocks.

6. Give some account of the controversies as to the distinctions and boundaries of the Cambrian and Silurian systems of rocks.

7. Give the general lithological and the Palæontological characters of the Old Red Sandstone and Devonian Strata in the typical local developments in (a) the border between England and Wales, (b) Devonshire, (c) Scotland, (d) the Eifel country.