2. Give an analytical presentation of the principle of virtual work, using generalized coordinates.

Apply the principle to obtain the general equations of equilibrium of a string.

3. Investigate the general formulæ for the determination of the centre of mass of a plane area in polar coordinates.

Find the centre of mass of the area of the curve ra0 between 0 = 0 and 0 = a.

4. Find the tension at any point of a heavy string in limiting equilibrium on a rough curve.

Find the limiting positions of equilibrium of a heavy chain placed over a rough circle in a vertical plane.

5. Investigate expressions for the component and resultant accelerations of a particle in Cartesian coordinates.

Find the general solution for the rectilinear motion of a particle acted on only by a resistance varying as the velocity of the particle.

6. Investigate the differential equation in u and for a central orbit.

Find the orbit when the law of force is (a + b)u2 where a, b are constants.

7. Investigate a series for the period of a simple pendulum in terms of the amplitude.

8. Investigate the equation for the rectilinear motion of a particle gaining or losing mass.

Complete the solution when the particle is under no external forces and throws off mass at a uniform rate with a uniform relative velocity.

9. Given the moments and products of inertia at the centre of mass, find (a) the moment of inertia around the line (x — a)|l = (y — b)|m = (z −c)/n; (b) the moment of inertia with respect to the plane la+my+nz - p=0.

10. Find the time of a small oscillation under gravity of a rigid pendulum whose axis of suspension makes an angle a with the vertical.

Find the time of a small oscillation of a uniform square of side a about one side as a horizontal axis of suspension.

11. Investigate the general equations of impulsive motion of a rigid body in two dimensions.

A lamina lying on a smooth horizontal plane receives an impulse P at a given point. Find the subsequent motion.

PHYSICAL GEOLOGY AND MINERALOGY.

Professor Sir Frederick McCoy.

1. Describe the methods used by Maskelyne and Playfair to determine the mean density of the Earth for geological purposes, and the Torsion

Balance methods of Cavendish, and the Pendulum observations in mining shafts with the same object, and explain the geological sources of error in each.

2. At about what rate does Central Heat of the Earth increase with the depth? At about what heat will ordinary Volcanic Rocks melt? And at about what depth would the present so-called Central Heat melt all known Trap Rocks if the beating had the same effect as at the surface?

3. If Mount Blanc were of uniform Trap Rock, how much would it deflect a plumb-line? About what is the actual deflection? Explain the reason of the difference.

4. Explain the methods pursued in making a geological map in the field, with the precautions to be observed, and explaining all the signs made use of to indicate the positions of strata.

5. Explain fully the following geological terms, viz.: Overflow, Intrusion, Underlying as applied to Igneous Rocks; Thinning-out, Conformity, Unconformity, Dip, Strike, and Fault as applied to strata; Lamination and Cleavage as results of changes.

6. Define a "Mineral Species." Write a brief treatise on the causes producing the chief exceptions to the usually accepted definition.

7. Enumerate, and give the chief characters of, the most common minerals forming "Rocks."

8. What are the chief chemical and geometric characteristics of the Hornblendes and Augites? What are their relations and differences, and under what circumstances may the one group change into the other?

9. Describe the principles on which the chief classifications of minerals have been founded, giving as full examples as you can of the groups recognised

in each.

10. Define the fundamental forms and each of the systems of crystals, with the proper notation for the faces of each of the fundamental and a few of the secondary forms, according to the methods of Miller, Weiss, and Naumann.

DEDUCTIVE LOGIC.

Professor Laurie.

1. How would you distinguish between abstract and concrete names ? And how would you deal with the statement that " some abstract names are certainly general"?

2. (a) What rules may be laid down for deciding which term is the subject and which the predicate of a categorical proposition? (b) How would you express, in recognised propositional forms, the meaning of the propositions-Most S's are P's, Few S's are P's, A few S's are P's?

3. Give the contrapositives, the inverses, and the obverted converses, which may be drawn from each of the following propositions:-The rule is proved by the exception; no system is perfect; some plants are not beautiful. Mention in each case whether the proposition inferred is, or is not, the full equivalent of the original propo

sition.

4. Is the ordinary categorical syllogism liable to the charge of incompleteness? May a greater amount of information be drawn from the premisses by any other process?

5. What are the aims of direct and of indirect reduction respectively? Give examples of each process in the reduction of Bocardo to the first figure.

of

6. It has sometimes been argued that the cogency the disjunctive syllogism depends on the mutually exclusive character of the alternants. Discuss this question, giving examples.

7. Wherein does the fallacy of ignoratio elenchi consist? Mention different forms of this fallacy.

8. State the following in syllogistic form, pointing out fallacies, if any:

(a) Respect should be paid to every ruler who

administers justice impartially. The Sultan fails in this duty, and therefore does not merit respect.

(b) The introduction of machinery lessens the amount of labour required to turn out a given quantity of work. It is therefore to be condemned, since everything which diminishes employment is an evil.