which some of the forces maintaining equilibrium are suddenly removed.

A heavy uniform circular disc is suspended in a vertical plane by two equal light strings fixed at its lowest point, and passing round the disc to the same point vertically above the disc's centre. If one string is cut, shew that the tension of the other is suddenly decreased in the ratio 2 cos2a: 3, where 2a is the angle the disc subtends at the point of suspension.

7. Investigate Lagrange's Method for the small oscillations of a system about a stable position of equilibrium.

An endless string is placed around a disc of any shape and over a small smooth peg, the whole being on a smooth horizontal plane. The disc rotates steadily around the peg with angular velocity w. Shew that the time of a small oscillation about steady motion is 2π/p where

and k is the radius of gyration about the c.m., a its distance from the peg, 20 the angle between the lengths r, r' of straight string.

8. Investigate the general equations of equilibrium of a fluid and the conditions that the force must satisfy.

Shew that the rate at which the cross-section A of a thin tube of force increases in going along it is given by the equation

where R is the resultant of the component forces (X, Y, Z).

9. Investigate the connexions between the surfaces of equal pressure and density and the lines of force in a fluid in equilibrium.

Shew that the angle between the direction of the maximum rate of change do/dn of the density p and the line of force is given by

where R is the resultant force and w the curl of the forces.

10. Investigate the small oscillations of a body of any form floating in liquid, neglecting the motion of the liquid.

A ship swells out suddenly and symmetrically at the water line so that the corresponding breadths immediately below and above the water are 2y, 2y' respectively. Shew that, with the usual notation, the condition of stability is

}{S (y' + y)dx}{S (y3 + y3)dx } -+{(y2-y3)dx}'>V· HG f(y+y)dr. {S v

PHYSICAL GEOLOGY AND MINERALOGY.

SECOND PAper.

Professor Sir Frederick Mc Coy.

1. Write out the chemical, physical, and geometric or crystallographic characters of Quartz, including

the characteristic angle of the normal of 1,0,0, on 1,1,1.

2. Explain clearly why free quartz is not to be expected in igneous rocks with Orthosilicates or Metasilicates, but abounds in those containing the Anhydro-silicates. Write out full formula to illustrate your statements.

3. Explain and illustrate Tschermak's theory of the formation of several species of Felspar, as accepted by many mineralogists, from a few of those species.

Write a brief essay on the various geological characteristics of Ice-action.

5. Write a brief treatise on fissures in the earth, giving the technical names and brief descriptions of each kind with their relations; and the characters of the fillings, and of the movements of the sides of those exhibiting such characteristics.

DEDUCTIVE LOGIC.

SECOND PAPER.

Professor Laurie.

1. What is the relation of Formal Logic (a) to Metaphysics, (b) to Material or Inductive Logic.

2. Explain the importance attached to Formal Logic by the schoolmen. Sketch briefly the history of the science since their time.

3. Trace the origin of concepts in the human mind. Is it strictly correct to speak of the act of judgment as the joining or disjoining of concepts?

4. Discuss the following statement:-"Thought is the representation, through imagination, of a whole class of individual objects-actual or possible. This is the proper doctrine of Nominalism, at once true and self-evident."

5. What rules may be laid down as to the introduction of terms into a complex proposition? Justify these rules.

6. (a) If A is BD or BCE or BCF, what can you infer about the terms Ace, cf, BCD?

(b) Give the full contrapositive of the proposition, Whatever is ACd or acd is AB or bc.

7. What information concerning aD, b, b Cd, can you gather from the following propositions:-All Ab is b Cd or c; all bd is A or bC or abc; whatever is a or B is c or D; whatever is Ab or bc is bD or cD or e; everything is A or ab or Bc or Cd?

8. Test the equivalence between the two following sets of propositions:-(1) All Ab is cd; all aB is Ce; all D is E; (2) Whatever is A or e is B or d; all a is bE or bd or BCe; all bC is a; all D is E.

INDUCTIVE LOGIC.

SECOND PAPER.

Professor Laurie.

1. On what grounds has it been maintained that all deductive sciences are inductive? Test this question by reference to the science of geometry.

2. May the Law of Causation be correctly 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"? Discuss this question.

3. Is it possible to state, in a satisfactory manner, any universal axiom of co-existence? Compare the views of Mill and Venn on this question.