6. Two equal uniform heavy bars AB, AC, each of weight W, and a light bar BC, are smoothly jointed together and stand in a vertical plane on two supports at B, C, which are on the same level. A weight W' is carried at A. Find the reactions at the joints, and shew tension of BC is

that the

7. Investigate the position of the centre of mass of a uniform triangular lamina.

Two triangles ABC, AB'C' have a common vertex A. Shew that the distance between their c.m.'s is

√ BB'2+BC12 + CB'2 + CC2 — BC2 — B'C'. 8. Investigate the ratio of applied force to resistance in the case of a lever (a) without friction, (v) with a rough pin-fulcrum.

9. Prove Archimedes' Theorem for the buoyancy due to a heavy liquid.

The base of a right circular cone of weight W, height h, and semi-vertical angle a, is in close contact with the horizontal bottom of a tank of water of depth k>h, so that water does not penetrate between them. Find the force required to lift the cone vertically off the bottom.

10. Investigate a formula for the total pressure on a plane area immersed in heavy liquid.

In the vertical side of a tank containing water is a square valve of side a hinged along its horizontal top edge which is at a depth h below the

surface of the water. Find the force to be applied at a point of the lower horizontal edge to open the valve inwards.

11. A thin spherical vessel of radius a, having a small opening at its lowest point, is forced down into water until the air occupies only half the vessel. Find the depth of the vessel and the force required to hold it at that depth.

MIXED MATHEMATICS.-PART III.

The Board of Examiners.

1. Find an expression for the pitch of the wrench which is the resultant of a system of forces such as (X, Y, Z) at (x, y, z).

A rigid body acted on by such a system of forces has one point O(x, y, z) fixed, and touches the rough fixed plane la+my+ nz pat (5, n, 5). Write down the analytical conditions of equilibrium and determine the least possible coefficient of friction.

2. Give formulæ for the determination of the centres of mass of a homogeneous solid of revolution and of a uniform thin shell forming a surface of revolution.

Find the centre of mass of a hemisphere whose density varies inversely as the square of the distance from the centre of its base.

3. Investigate a formula for the tension of a heavy chain about to slip on a rough vertical curve.

A uniform heavy endless chain of length 21 is slung on a horizontal beam whose normal section is a square of side a < 1/2. Find equations to determine the depth of the lowest point of the

chain.

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

A chord of a circle which passes through a fixed point, rotates with uniform angular velocity. Find the acceleration of either end of the chord in any position.

5. Investigate the equations connecting the true, mean, and excentric anomalies in an elliptic orbit around a focus.

Find the orbit under a central force varying inversely as the cube of the distance from the

centre.

6. Investigate a series for the period of a simple pendulum performing small oscillations of amplitude 2a.

7. Find the equations of motion of a particle which is gaining or losing mass in a continuous manner.

Find the motion of a particle projected in a given direction on a horizontal plane and throwing off mass at a constant rate with constant absolute velocity in another fixed direction.

8. Find equations for the magnitudes and directions of the principal axes of inertia at a point, in terms of the six constants of inertia with respect to a set of rectangular axes at the point.

Shew from the definition of the moment of inertia about a line, that the least semi-axis of a momental ellipsoid must be greater than

ab√a2 + b2,

where a, b are the other semi-axes.

9. Investigate, ab initio, by means of D'Alembert's principle, the equation of motion of a compound pendulum and the forces of constraint on its

axis.

Find the time of a small oscillation of a solid homogeneous right elliptic cylinder of axes 2a, 2b, and length 21 about the major axis of one of the ends.

10. A four-wheeled carriage moves down a plane of inclination a under its own weight. The wheels are two pairs of uniform circular discs of radii r, R and masses m, M, and the body is of mass M'. Find the acceleration of the carriage, neglecting frictional resistance and assuming that the wheels do not slip.

PHYSICAL GEOLOGY AND MINERALOGY.

Professor Sir Frederick Mc Coy.

1. Give the chief theoretical recognised views relating to the formation of the principal groups of Igneous Rocks, with the probable reasons for their relative ages and order of formation as affected by their mineral constitution; with the objections to each.

2. What are the chief indications that would enable you to infer the former existence of glaciers in localities where they do not now occur?

3. What are the Chemical and Physical_actions governing the original separation of the Elementary Bodies in the gaseous first state of the Earth, and their subsequent combinations to form (a) the liquid, and (b) the solid conditions of the globe?

4. Describe as fully as you can the methods of forming geological sections from geological maps, with all the precautions which should be taken to avoid errors, and the modes of correcting the amount of "dips" on sections at other compass bearings than those at right angles to the “strike.',

5. Define and explain the following technical terms, viz. False-bedding, Thinning-out, Outcrop, Dip, Strike, Conformable, Planes of Deposition, Anticlinal and Synclinal Axes, Faults, Cleavage. Give the signs used to indicate the 4th, 8th, and 9th on geological maps.

6. Define a "Mineral Species," and state the recognised causes producing apparent exceptions to the definition in certain cases.

7. Define all the systems of crystallization in which minerals occur by their optic and geometric characters.

8. Explain the diverse notations of the faces of the primary forms of all the systems of crystals according to the methods of Weiss, Naumann, and Miller respectively.