MIXED MATHEMATICS.-PART II.

The Board of Examiners.

1. Two particles are projected at the same instant from the same level and in the same vertical plane with velocities v, v and elevations a, a', the points of projection being at distance d. The resistance of the air being neglected, find the relations between these quantities that the particles may collide, and where the collision takes place.

2. Investigate the time of a small oscillation of a simple pendulum.

3. A particle of mass m hangs at one end of a light elastic string suspended from a fixed point, the string being double its unstretched length 7. The string being now stretched to a length 47, the particle is let go. Find the time of a complete oscillation, assuming that when the length of the string is not greater than l it exerts no force on the particle.

4. Find an expression for the velocity in a planetary orbit, in terms of the distance from the centre of force, the major axis, and the strength of the

centre.

5. AB, BC, CD are three equal smoothly jointed at B and C. are joined by light inelastic

heavy stiff rods, A, C and B, D strings of length

√3 times that of a rod, and the frame rests in a

vertical plane with A, D on a smooth horizontal floor. Find the reactions at the joints and the tension of the strings.

6. Find necessary and sufficient conditions for the equilibrium of a given system of forces on a rigid body,

7. Summarize the chief results (i) in the theory of couples, (ii) in the reduction of a system of forces to simple equivalents.

8. A sphere of radius r whose centre of mass is at a distance a from its centre, rests on two rough planes, each making an angle 45° with the horizontal. Shew that when slipping is about to take place the line a makes an angle with the vertical, where √2a sin 0 = r sin 2λ and is the angle of friction.

MIXED MATHEMATICS.-PART III.

The Board of Examiners.

1. A particle moves in a plane under a central force varying as the distance from the centre, and a force in a fixed direction, varying as a simple harmonic function of the time. Find the complete integrals of the motion.

2. In a planetary orbit find expressions for the radius vector and true anomaly in terms of the time correct to the second power of the eccentricity.

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3. Investigate completely the motion of a particle under gravity sliding on the smooth curve y = c sin2 s/a in a vertical plane, y being the vertical ordinate and s the arc, and the velocity zero when y = c.

4. Investigate the moment of inertia of the tetra-
hedron ABCD, the planes through A being at

right angles, about a line through A and the
centre of volume.

5. Find the time of a small oscillation under gravity of a thin rod of length placed inside a fixed

smooth vertical hoop of radius r.

6. Two discs, with given centres of gravity, masses, and moments of inertia, are moving in a plane in a given manner. Suddenly they become rigidly attached to each other. Find the general formulæ for the motion immediately afterwards.

7. Find the centre of mass of one of the two similar portions of a solid sphere cut off by two planes through the centre.

8. Investigate the tension of a heavy string about to slip on a rough vertical curve.

9. A smooth uniform elliptical disc slides in a vertical plane between two rods, each inclined at 45° to the horizon in that plane.

Shew that there are four positions of equilibrium, and determine their stability.

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PHYSICAL GEOLOGY AND MINERALOGY.

The Board of Examiners.

1. How are igneous and aqueous rocks generally distinguished? Discuss carefully the more important exceptions to the general rules.

2. In ordinary stratified rocks, how are the planes of deposition determined in the more difficult cases? How are the strata differentiated, and what differences may circumstances produce in the period of time represented by each?

3. What is "false bedding," and its significance?

4. Explain Babbage's theory of the rising of isothermal planes in the crust of the earth.

5. Describe the usual characteristics of metamorphism in rocks.

6. Enumerate as many as you can of the more common rock-forming minerals.

7. How are the different systems of crystallisation defined and recognised?

8. What are pseudomorphs? How do they originate, and how are they recognised?

9. What are the chemical and physical characters of the more common kinds of felspars found as constituents of igneous rocks?

10. How are the different oxides of silicon characterised? How do the different kinds of quartz originate; and how are the "acidic" and "basic" igneous rocks characterised and supposed to arise?

STRATIGRAPHICAL GEOLOGY AND

PALEONTOLOGY.

The Board of Examiners.

1. Illustrate by sketches the mode of formation of geological sections from geological maps, mentioning the methods of using the instruments employed.

2. What is understood by the term "Geological Formations"? What are the chief differences between the views of the older and of the more modern geologists as to the grounds of accepting certain groups of strata as Formations"?

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3. What Oolitic genera of Coal Plants are present in, and which are absent from, carboniferous coalLearing rocks?

4. Give the generic characters of the more common Polyzoa of the Coralline Crag, and state the range in time of each.

5. Write down all the subdivisions usually adopted by geologists of the Tertiary and of the Lower Palæozoic Formations in the chronological order of their successive depositions.