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5. Find in terms of the major axes and focal distance the velocity at any point of a conic section described about a centre of force at the focus.

Find when possible the point in an elliptical orbit at which, if the centre of force were transferred to the empty focus, the orbit would be a parabola. Prove that such a point cannot exist unless the ®ccentricity of the elliptical orbit is greater than 5 — 2.

6. Find the law of force tending to the pole under the action of which a body may describe an equiangular spiral.

7. State the reason why the face of the moon presented to the earth is always approximately the same. Explain the full meaning of the word “approximately.”

8. Explain the method of determining the latitude and the local time by observations on the sun's altitude.

9. Investigate expressions for the aberration of a star in right ascension and declination.

At half yearly intervals the relative right ascensions of three stars will not be altered by aberration provided their right ascensions and declinations satisfy the equation

cos o sin (a' - a") + cos & sin (a" - a)

+ cos g" sin (a - al 10. Define a true solar and a mean solar day. Trace the causes to which the “equation of time” is due. Prove that it vanishes four times a year:

Assuming that if the orbit of the earth were a circle with the sun in its centre, the Sun's R A (a) and its longitude (1) would be connected by the equation tan a = tan l cos w, (w being the obliquity of the ecliptic) prove that the equation of time will have its greatest value when tan a = cos w and that if e be then the difference between the Sun's longitude and right ascension

0

sin

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tan o

COS

PROBLEMS. Examiner-MR. JOHN ELIOT, M. A. 1. A railway train of given mass is travelling dne south at a uni. form rate along a line which runs due north and south. Prove that (supposing the earth perfectly spherical) the train will exert a strain on the metal rails towards the east the magnitude of which varies as the product of the velocity of the train and the sine of the latitude of the position, and a strain towards the south the magnitude of which varies as the sine of twice the latitude. 2. Two points describe the curve r = 0 sec 3

in the same

3 direction and under the action of a force from the origin. Prove that when their directions of motion are parallel, the sum of their velocities is a constant quantity.

sin a

3. If a particle moves on a rough inclined plane prove that

ppt cos 30 = r. where p, pl are the radii of curvature of the path at the two points where the tangents are inelined at an angle o to the horizon, and a is the radius of curvature at the highest point.

4. A particle is projected from a platform with velocity V, and elevation B. On the platform is a telescope fixed at elevation e. The platform moves horizontally in the plane of the particle's motion, so as to keep the particle always in the field of view of the telescope. Shew that the original velocity of the platform must be

sin (a B)

V and its acceleration g cot a.

5. A small pencil of parallel rays is incident on a reflecting paraboloid in a direction perpendicular to the axis. If P be the point of incidence of the axis of the pend and S the focus of the paraboloid, shew that the primary focus Q will be found by drawing from S a perpendicular" to sP meeting the normal at P in Ř and from K à perpendicular KQ on the direction of the ray reflected from P.

6. A particle mass m moves in a smooth circular tube (radius a) under the action of a force (u m m distance) to a point inside the circle at a distance c from its centre. If the particle be placed very nearly at its greatest distance from the centre of force, prove that it will move over the quadrant ending at the least distance in the time -log2 + 1

uc 7. In a common pump V is the volume of the cylinder in which the piston works, V that of the water pipe above the surface of the water; prove that the water will just rise to the valve at the first stroke if the height of the water-pipe be to the height of the water barometer as V-V!: V.

8. The surface of a sphere is divided into triangles and lunes by the planes of the faces of a regular tetrahedron. Prove that the area of a triangle is three times that of a lune.

9. Two prisms whose refracting angles are right angles, and refractive indices M, u', are placed so that one face of each is in contact. Their edges are parallel and their refracting angles opposed. Prove that the minimum deviation of the compound prism is

1

sin (в° — ?). 10. A solid of revolution floats in different liquids. If the time of oscillation (t) in any liquid and the density (o) are found to be

1

connected by the equation p=shos denoting a given function,

show that the equation to the meridian section of the solid is

f (gy 2g dy

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POLITICAL ECONOMY. Examiner-Rev. DR. A. F. RUDOLF HOERNLE. 1. Explain the principles and conditions of Free Trade. Under what circumstances is a protectionist policy defensible ?

2. What are the proper limits of Government intervention in works that may be performed by private agency ? Examine the case of Railways and Telegraphs.

3. Compare the state of England and India with reference to the conditions under which a representative form of government is applicable.

4. Explain the term conscription? What are the advantages and disadvantages of a voluntary and compulsory military system respectively ?

5. Which are the systems of land tenure, most prevalent in India ? Describe their characteristics and their effects on the people.

6, What is political economy ? State, examine and criticise the various definitions that have been given of it.

7. What are usury laws ? What are their economical effects ? What are the proper remedies for the evils, against which they were enacted ?

8. What system of taxation is the most advantageous to a country ? Examine the case of the license and income-taxes. State Adam Smith's and Mill's views on the subject,

HISTORY OF GREECE AND ROME, Examiner-Rev. DR. A. F. RUDOLF HOERNLE. 1. Give some account of the political institutions of Greece during the heroic age, and of the migrations of the Greek tribes after the Trojan war.

2. Write a sketch of the history of the literature of Greece up to the time of the Macedonian conquest.

3. Trace and describe the effects of the Roman conquest of Greece on the culture of the Roman people, and compare the age of Pericles with that of Augustus.

4. What is the “Retreat of the Ten Thousand”? Who is its historiographer? Narrate it ; together with the circumstances that led to it, and the events that arose out of it.

5. What are the sources of our knowledge of the history of Rome? Give some account of them. State and discuss the various views that have been held as to their trustworthiness.

6. Name the kings of Rome in chronological order, with dates ; and enumerate the additions and improvements which they severally made to the city from time to time.

7. Write a sketch of the life of Sulla, and of his wars and laws.

8. Who are the Flavian Emperors ? Describe the events which led to the accession of their house. Which are the two great events that have made their reigns memorable in history ? Give dates.

MODERN HISTORY. Examiner-Rev. Dr. A. F. RUDOLF HOERNLE. 1. Describe the effects on Modern Europe of the invention of printing and of gunpowder. By whom, when, and where were they invented ?

2. What were the causes which led to the Reformation of the sixteenth century ? Trace its course in Germany. Wherein did the German Reformation differ from the Swiss ? Mention some of the leading men on both sides.

3. Describe the constitution of the Holy Roman Empire. What changes were made in the course of time, by whom and when ?

4. Describe the condition of France under Louis XIV. Who were his principal ministers and generals ? Enumerate his wars, and sketch the history of any one of them, giving dates.

5. Enumerate the states into which Italy was divided before the consolidation of the present kingdom ; and narrate the events which brought about the latter.

6. What were the limits of the Turkish empire in Europe at the time of its greatest extent, and what are they now according to the Berlin Treaty ? Mention in chronological order the various events which led to the gradual contraction.

7. What were the causes that led to the Greek war of indepen. dence ? Write briefly the history of it Who is the present king of Greece, and to which of the reigning houses of Europe is he directly related ?

8. Explain the terms: Salic law, Pragmatic Sanction, Chambers of Reunion, Fronde, Janissaries, Chivalry. Who were the following: Tilly, Pascal, Rienzi, Scanderbeg, Ypsilanti, Saint-Simon ?

HISTORY OF ENGLAND. Examiner-Rev. DR. A. F. RUDOLF HOERNLE. 1. Mention the various nations from which the present English people is divided, with the languages they used and the dates of their arrivals in England. Where and how did the English language come into use ? What is the oldest extant specimen of it ? Mention any words in present use of Celtic origin.

2. What was the origin of the Wars of the Roses ? How were the rival claims finally adjusted ? How did these wars contribute morally and how materially to the auymented power of the Tudor sove. reigns ? Mention any illustration of such augmented power.

3. Narrate the origin of the Puritan Sect. Into what three parties was it divided in the time of Charles I, and by what parties were these severally opposed ?

4. Describe the particular circumstances which led to the union of England and Scotland, and of Great Britain and Ireland. Mention the provisions of each act of Union.

5. To what country have the Hindus been traced ? State the grounds. To what ancient writers and travellers of foreigu nations are we indebted for the most authentic information concerning the Hindus ?

6. Write a short account of Lord Cornwallis' two campaigns against Tippoo. What were the conditions of peace ?

7. Give a short account of the arrangements made by Lord Cornwallis for the permanent settlement of the public demand on account of land revenue in Bengal. What led to the adoption of these arrangements ?

8. When, under whom, and where did the Portuguese arrive in India ? Give a brief account of the Portuguese in connection with India.

LOGIC.

Examiner-Dr. P. K. ROY. 1. State and criticise the views of Logic taken by Kant, Hegel, Ueberweg, Mill, Hamilton, and Spencer.

2. State and criticise the theories as to the import of a Proposi. tion.

3. Explain fully the meaning, and give the arguments for and against (if there be any) the truth of the proposition, “ Every phenomenon or event has a cause."

4. The aim and object of scientific classification.

5. Describe and illustrate the uses of Hypotheses ; and state the conditions of a legitimate Hypothesis.

6. What is explanation ? State and illustrate its meaning, noting diversities of opinion. 7. Explain and examine the following :

The ultimate Laws of Nature cannot possibly be less numerous than the distinguishable sensations or other feelings of our nature."- Mill.

MORAL SCIENCE.

Eraminer-DR. P. K. ROY. 1. Is a science of ethics possible ? State and criticise Sidgwick's definition of it.

2. What does Sidgwick mean by a method of ethics? Give his classification of ethical systems.

3. Define a voluntary action, and distinguish it from other kinds of action. What are Reflex actions ?

4. Describe the origin, growth, and nature of the moral senti. ments. Is there a moral sense or moral reason ?

6. State the fundamental assumptions of intuitionism, and give a critical account of that form of it which appears to be the most reasonable to you.

6. Trace the development of Utilitarianism from the time of Bentham up to the present day.

7. Give an account of Spinoza's Ethics.

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