9. Shew that the distance r from the focus at which the rate of increase of the speed in a planetary orbit is a maximum is found from the equation

2p3- Sar2+(8a2 + 362)r-5ab20.

a, b being the semi-axes.

10. A particle under gravity is projected from a given point in a given direction with a given velocity, and moves in a uniform medium whose resistance varies as the nth power of the velocity. Determine the motion.

11. A particle is describing an elliptical orbit about a gravitational centre of force. If the strength of the central force be suddenly slightly altered in the ratio 1 + k 1, shew that the decrements of the semi-axes a, b of the orbit are

k a2v2/μ, kb(av2 + μ)/2μ,

where v is the velocity at the instant, and μ is the force per unit mass at unit distance.

MIXED MATHEMATICS.-PART II.

SECOND PAPER.

The Board of Examiners.

1. Find the maximum product of inertia at a point (i) for a disc; (ii) for a general distribution of mass, when the principal moments of inertia at the point are known.

2. Obtain the three equations which give the motion of the centre of mass and the rotation in the plane motion of a rigid body.

3. The door of a railway carriage stands open at right angles when the train begins to move with uniform acceleration f. Shew that, neglecting resistances, it will slam to with angular velocity

afk where a is the breadth of the door, and k its radius of gyration about the hinge.

4. Find an expression for the kinetic energy of a body in plane motion in terms of the velocity of the centre of mass and the angular velocity.

5. A sphere of radius a, with its centre of mass at a distance b from its centre of figure, rolls in a straight line on a rough horizontal plane. Shew that it will jump, if its angular velocity when the centre of mass is lowest is greater than

√g{(a+b)2 + 4b2 + K2}

√b{(a — b)2 + K2}

where K is the radius of gyration round the centre of mass.

6. A cube on a rough horizontal plane receives a horizontal impulse in a central plane parallel to two of the vertical faces, and at a height h > a above the base. If the cube begins to turn over without sliding, shew that the coefficient of (impulsive) friction is not less than 8a/3h-1 where 2a is an edge of the cube.

7. A string of length l is tied at one end to a stick of length 2a, and at the other to a fixed point, around which the string and stick rotate uniformly with angular velocity w, gravity being neglected. The system being slightly disturbed, shew that the time of a small oscillation is

8. Obtain general equations for the pressure in a fluid at rest, and find the condition to be satisfied by the forces.

A sphere of weight W and radius a floats in air at uniform temperature. Shew that the density of the air at the level of the centre of the sphere is

where Ag/k, and k is the ratio of the pressure of the air to its density.

9. Investigate general formulæ for the centre of pressure of a plane area in heavy liquid.

Two equal thin hemispherical shells are hinged at one point, and suspended by it. Shew that the sphere can be filled with water if the weight of the shell is three times the weight of water.

10. Investigate a formula for the pressure of the air at any point, taking the variation of gravity into account, and supposing the relation between pressure and density to be p = kpY.

11. An ellipsoid of semiaxes a, b, c is free to turn about one end of the greatest axis 2a which is fixed at a depth a below the surface of heavy liquid. Shew that the upright position is stable if the ratio of the densities of liquid and ellipsoid is greater than 16a2/(5a2 + 3c2).

PHYSICAL GEOLOGY AND MINERALOGY.

SECOND HONOUR PAPER.

The Board of Examiners.

1. How are igneous rocks generally classified?

2. Taking quartz and the felspars as the commonest constituents of igneous rocks, give the chief distinctive characters of quartz. Supposing silicon to be quadrivalent, what would be the constitution of its normal or orthic acid, its meta-acid, and its condensed acid, and how are the many of each of these related to and derived from the others? And what felspars might co-exist with free quartz in a rock, and which of them could not.

3. Explain clearly the difference between the glassy, the cryptocrystalline, the colloid, and the crystalloid conditions of rock materials.

4. Give the fullest explanation you can of all the physical and chemical characters used for the definition of mineral species.

R

5. Write down the notation, according to the methods (a) of Miller, (b) of Naumann, and (c) of Weiss, for as many of the faces as you can of the Holohedral developments of fundamental forms of crystals in the Rhombohedral or Hexagonal system of crystallization.

DEDUCTIVE LOGIC.

PAPER No. 2.

The Board of Examiners.

1. What is the problem of Intellectual Philosophy? Show the connection (a) of Formal Logic, and (b) of Inductive Logic, with this problem.

2. How does the distinction between abstract and concrete names arise? Consider the statement that "an abstract idea is not necessarily a general idea, or an idea regarded as applicable to more than one object."

3. Distinguish fully between intuitive and symbolical thought, and show any danger to be guarded against in the latter.

4. Is it possible to prove the primary Logical Laws? Give reasons which have been alleged for and against regarding them as ultimate.