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3. State and explain the allegory of the cave, and apply it to exhibit the progressive discipline which Plato had in view.
4. Explain the relation of the ideas to the human reason.
5. Show how Plato's ethics are connected with his conception of the soul.
1. Give some account of the principles on which Hobbes was opposed by Cumberland, Cudworth, and Clarke, respectively.
2. What philosophers before Butler approached the truth on the subject of Disinterested Affection, and in what respects ?
3. Butler, in several passages of his Sermons, countenances the opinion that the character of God is one of simple goodness. In connexion with what arguments is this opinion introduced ?
4. State the theory of Butler's method of ethical inquiry, and the principles of its practical application.
5. "Must we, forgetting our own interest, go out of ourselves, and love God for His own sake?” Give Butler's reply to this question.
6. Discuss briefly the question concerning a primitive revelation.
7. Describe accurately the train of argument by which Butler shows the credibility that this life is a state of discipline for a future.
UNDERGRADUATE HONOR EXAMINATION PAPERS.
1. How is an observatory clock set to the true sidereal time? If there were reason to believe that the right ascension of no star had yet been determined with sufficient accuracy, how should we proceed ?
2. How is the time of the Moon's rising on a given day found ? How does the problem differ from that of finding the time of rising of a star?
3. The deviation of a ray of light passing through a thin lens is constant?
4. Find when the deviation of a ray of light is a maximum or minimum, which enters a sphere, and emerges after n internal reflexions.
5. Find the centre of gravity of a shell whose bounding surfaces are non-concentric spheres whose radii are a, b, the distance of the centres being D.
6. Find principal parameter of the parabola described by a body projected along a smooth plane of given inclination, with a given velocity, and at a given elevation.
7. The Sun was observed to pass the meridian at 11h 59m 18.78 by chronometer, the equation of time being + 13m 22.5$. Required the error of the chronometer both for apparent and for mean time.
8. What are the principal quantities given in the Nautical Almanack?
9. A body falls from a given point towards a centre of force, the attraction at any distance r being To find the whole time of descent.
10. The centrifugal force of the Moon in her orbit is 0.00889 feet per second, while the centrifugal force of bodies at the Earth’s équator is 0.11126 feet per second. Hence, and from the periodic time of the Moon, viz., 27 days, 7 hours, 45 minutes, calculate her mean distance from the Earth, in terms of the Earth's mean radius.
11. Compare the whole pressures on the curved surface and plane base of a solid hemisphere, radius r, immersed in water with its base horizontal, and at a depth r, the hemisphere being placed successively with its base down and up.
12. Find the magnifying power of a refracting astronomical telescope whose object-glass is of 12 feet focal length, and eye-glass of į inch, used by a person whose least distance of distinct vision is 4 inches.
13. Find how much the time of a star's rising is altered by refraction. 14. Find when the equation of time caused by the obliquity of the ecliptic is a maximum; and also its maximum value arising from the unequal motion in the ecliptic.
15. Determine the positions of equilibrium of an isosceles triangular prism floating with its vertex downwards and its edges horizontal; and investigate whether they are stable or unstable.
16. Determine the dispersion of a ray of light produced by transmission through two prisms placed in any manner towards each other.
17. A train moving at the rate of 20 miles an hour comes to the foot of an incline of 1 in 250; find how far it will ascend, steam being shut off, and resistance 8 lb. per ton.
18. Find the angle at which a body must be projected from a point in an inclined plane, in order to impinge upon the plane at right angles; the plane of projection being perpendicular to the inclined plane.
1. A cistern of height b is filled with water, and a hole is made in the side at a height z; find the point at which the issuing fluid strikes the floor on which the cistern stands. 2. Prove that when a prism is in the position of minimum deviation,
sin } (0+)
also, if o', 0" be the minimum deviations of the extreme red and violet rays, explain by what measurements ', " are found.
3. Given the lengths of a degree of the meridian commencing latitude L and commencing latitude L'; find the equatoreal and polar diameters of the Earth.
4. Deduce formulæ for the effects of aberration on the right ascension and on the cleclination of a star.
5. Write down the differential equations of the motion of a projectile, the resistance of the air being supposed proportional to the square of the velocity, and integrate them once.
6. Prove the following formulæ for the latitude (~) and longitude (1) of a star S:
sin (0 - w) sind
cos (0-w) tan a
w=obliquity of ecliptic. 7. Determine the coefficient of friction between a rough body and an inclined plane, in terms of the angle of elevation, from observing that the least force that will suffice to move the body up the inclined plane is n times the least force that will sustain it on the plane.
8. Show that the condition that two prisms, each of which is placed in the position of least deviation, should form an achromatic combination, is
u'au tan $' + pau' tan 0" = 0. 9. Show that the longitudinal aberration of parallel rays, in a spherical refracting medium is
2 (m – 1)2 fi x being the semiaperture, f the focal length, and m the ratio of sini: sin r.
10. A rectangle has one side in the surface of a fluid ; show that the depths of the horizontal lines which divide it into n parts, on which the pressures are equal, are
11. If a plane area be immersed in a vertical position in a fluid, prove that the co-ordinates of its centre of pressure, referred to the principal axes of its area at centre of gravity, are
h where k and k' are its principal radii of gyration, h depth of centre of gravity, and 0 angle made by one of the principal axes with the vertical line.
12. A body is describing a conic round a centre of force at the focus ; if, when it arrives at the extremity of its axis minor, the central force be suddenly doubled in intensity, show how to construct the subsequent path described.
13. Prove that the aberration of an inferior planet is a maximum when in conjunction; and give a geometrical construction for the case in which it vanishes.
14. A bent lever, whose sides are inclined at a given angle, makes oscillations in its own plane about the angular point; find the length of the isochronous simple pendulum.
15. Find the caustic of a parabola when the rays are incident in a direction perpendicular to its axis.
Experimental and Natural Science,
ELECTRICITY AND MAGNETISM.
1. Let ten jars of equal dimensions be charged by cascade, and then placed in battery arrangement; compare the power of the battery with what it would have if charged directly.
2. Explain the theory of the condenser, and show how its condensing power may be found numerically by means of a proof-plane and a torsionbalance.
3. In comparing the accumulation of electricity at different points of an insulated conductor by means of the proof-plane and torsion-balance, how is the loss of charge eliminated from the result by means of alternate contacts ?
4. What is the distribution of free electricity at the surface of elliptical conductors ? Derive this from the ordinary hypothesis of a compound fluid.
5. State Faraday's view of electro-chemical decomposition, and his objections to Grotthus's theory founded on the specific difference of electrical character of chemical atoms.
6. State and explain Faraday's proposed nomenclature for expressing the parts of a battery, and the results of electro-chemical decomposition.
7. Give a diagram explanatory of Grove's gas battery.
8. Describe and explain the curvilinear arrangement which iron filings assume under the joint action of the poles of a magnet.
9. Deduce the geometrical property of these magnetic curves, and state the method of constructing them by points.
10. Pouillet gives the following experiment of Coulomb’s:-A small proof-needle oscillates under the action of the earth 15 times in a minute ; when brought under the influence of a magnetic pole at a distance of 4in. it makes 41 oscillations in a minute; at a distance of 8 inches, under the influence of the same pole, it makes 24 oscillations; from this derive the law of magnetic force.