Shew that the differential equation for the form. of a string in two dimensions is

5. Investigate the small oscillations of a simple pendulum, assuming that the air resistance varies as the velocity.

6. Investigate the motion of a heavy particle on a surface of revolution with vertical axis. Examine the case of steady motion, including its stability.

7. Prove the theorem

I(l, m, n) Al2 + Bm2 + Cn2

- 2Dmn 2 Enl-2 Flm

for the moment of inertia about the line (1, m,
through the origin.

n)

If in Question 2 the axes Or, Oy are the principal axes of the section z = 0 of the helicoidal solid, shew that the moment of inertia of the solid about the axis of x is

} Mh2 + §(A + B)ph + P(A — B) sin 2h/p

where M is the mass, p the density, and A, B are the moments of inertia of the section z = 0 about Ox, Oy.

8. Give the theory of the ballistic pendulum.

9. A uniform heavy beam is supported horizontally at its two ends. One of the supports is suddenly removed. Shew that the pressure on the other is instantaneously halved.

DEDUCTIVE LOGIC.
Professor Laurie.

TO BE USED AS PASS PAPER, AND ALSO AS HONOUR
PAPER No. 1.

1. Examine the statements (a) that names of attributes may in some circumstances be used as concrete; (b) that if adjectives are classified as names at all, they are concrete and general.

2. Explain the nature of the opposition between each pair of the following propositions:-None but the patient are contented; Among those who

F

are contented are some who are patient; It is untrue that all who are contented are patient.

3. Should categorical propositions be regarded as logically implying the existence of their subjects? Discuss this question with special reference to propositions in A and I.

4. Explain the distinction drawn by Keynes between hypothetical and conditional propositions. If this distinction be accepted, what are the possible moods of hypothetical syllogisms?

5. Give the Canons or fundamental principles of the Syllogism as stated by Jevons, and show their connection with the primary logical laws.

6. Construct a syllogism in Bokardo, and reduce it, both directly and indirectly, to the first figure. May this syllogism be reduced to any figure other than the first?

7. What is the general character of semi-logical fallacies? What fallacies may result from a neglect of the distinction between the collective and the distributive use of names?

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

(a) This war is necessary for the defence of the empire, for it is just; and every war is just which is necessary for the defence of the empire. (b) All slaves are wronged, for they are coerced; but those who agree to work for low wages are not wronged, since they are not slaves and are not coerced.

(c) Large armies are guarantees of peace, for if armies were greatly reduced it would be easier for ambitious governments to bring about hostilities. (d) "The effect of labour-saving implements will be to increase the production of wealth. Now, for the production of wealth, two things are required-labour and land. Therefore, the effect of labour-saving implements will be to extend the demand for land."

9. B is always accompanied by AC or ac.

Wherever

there is ac, there is D. b and d are co-extensive. What can be inferred (a) about BC; and (b) about aB. Work out this question by Jevons's Method of Indirect Inference.

INDUCTIVE LOGIC.

Professor Laurie.

TO BE USED AS PASS PAPER, AND ALSO AS HONOUR
PAPER No. 1.

1. Point out the connection between Mill's enumeration of nameable things and his subsequent analysis of the import of propositions.

2. Is the syllogistic process a true process of inference? Consider critically Mill's answer to this question.

3. Is the Law of Causation properly described as "the familiar truth that invariability of succession is found by observation to obtain between

every fact in nature and some other fact which has preceded it"?

4. In what sense may a plurality of causes be admitted? And to what extent does such an

admission interfere with the Methods of Induction as stated by Mill?

5. Examine Mill's statement of the Method of Residues (a) as a canon of proof, (b) as an instrument of discovery.

6. Show the value of empirical laws in the verification of inductive inferences or hypotheses.

7. What do you understand by the explanation of laws of nature? And what are the limits of such explanation? Give examples.

8. Under what circumstances is it allowable to extend derivative laws to adjacent cases?

MENTAL PHILOSOPHY.

Professor Laurie.

To be used as Pass Paper and Honour Paper No. 1 for Second Year Students, and as Pass Paper No. 1 for Third Year Students.

1. Show the connection between Descartes' theory of the union of mind and body, and the subsequent speculations of Malebranche.