12. Prove that the sum of the fractions which are intermediate in magnitude to any two numbers m and n and have 3 for a denominator is n2 m.2

cosec.

{n

TRIGONOMETRY.

Morning Paper.

=

1. Trace the changes in the sign and magnitude of the cosecant of an angle as the angle varies from 0° to 360° and shew that cosec. 0 n π +(−1)”0 } where n is any negative or positive integer. 2. Prove that cos. A + B cos. A. cos. B-sin. A. sin. B. when A+ B is > 180° < 270°

3. In determining the value of tan a, from tan 3a given, a cubic equation results, or tan a has three values: account for this.

and cos A+ cos (7 2-A) + cos (7 2° + A) : cos (36° + A) + cos (36°—A).

5. Define a logarithm and from your definition shew that loga b loga 1 assuming the expansion of a in a series ascending by

=

powers of x find the value of ε the Napierian base for logarithms, and prove that it is an incommensurable quantity.

6. Prove that the increment of tan 0 and of log tan ✪ varies as the increment of ✪, provided ✪ be not nearly equal to (1

7. Find the cosine of any angle of a triangle in terms of its sides and explain if when a, b and A are given, there is any difficulty in finding B from

the process, where the condition that a is the circular measurement enters. 9. Expand (cos. 0)" in terms of the co-sines of multiples of 0 when n is an even number. In this case is ℗ the circular or angular measurement. 10. Find an expression for 0 in a series ascending by powers of tan 0.

Afternoon Paper.

1. Find the tangent of 6°. .Given log. 78598 = 4·8954115 and the

10

modulus equal .43429448 find log. 785982

2. Find in the equation a + b. sin. 0 + ā b. cos. 0 a2 + b2

a b

and if cos. a= cos. cos. 0 = cos. B' cos. ' and sin. a

3. How may the series

tan 50 ...... be

used for calculating the value of an angle that lies between

the tangent of the angle is given.

4. The sides of a triangle are 17·09 and 93'451, and the angle opposite to the greater of them is 93°16′ find the angle opposite to the less.

Log. 1709 3.2327421
Log. sin. 86°44' 9.9992938

Log. 93451 = 4·9705840.
Log. sin. 10°31′ = 9-2614519.

5. If a circle be described in an equilateral triangle, and again at each of the three angles, a circle be described touching two sides and the original circle, and afterwards other circles touching these and the sides of the triangle and so on ad infinitum, prove that the area of all the

11

circles = π a', a being a side of the triangle. 96

6. Sum the series cos 20 cos 220+ cos 230 for 2n+1 terms.

7. Find x & y from the equations tan. 1x-tan.~1y

8. Let A B C be any triangle inscribed in a circle, and let the arcs upon which the angles A, B, and C stand be bisected in the points a, b, c respectively. Then if a, b, and c be joined, prove that angle C. angles (a + b— c), A b+c - a and B = a + c − b.

=

9. Two opposite angles of a quadrilateral figure are each of them right angles, and the sides are as 6, 6, 8, 8, find the radius of the circle described about it.

English Essay.

"What custom wills, in all things should we do't,
The dust on antique time would lie unswept,
And mountainous error be too highly heap'd

For truth to over-peer."-SHAKESPEARE.

"They that reverence too much old times are but a scorn to the new."-LORD BACON.

Latin Essay.

Interfecto Julio, utrum Romæ profuit quod unus Augustus imperio

nactus est.

Bengali Essay.

সুকৃতি না করিলে কেহ প্রকৃত রূপে সুখী হইতে পারে না ৷ No one can be truly happy without practising virtue.

JUNIOR SCHOLARSHIPS, 1850.

Grammar.

Morning Paper.

1. What kind of elementary sound enters into the formation of every syllable; and with what other kind of sound is it commonly united?

2. To what classes do the elementary sounds in the word Births severally belong?

3. What are the plurals of court-martial, mouse-trap, spoonful, fatherin-law, analysis, nebula, appendix? What nouns form their plural by a change of the vowel?

4. Give the rules for forming the comparative and superlative degrees; and compare small, large, pretty. Explain why certain adjectives cannot be compared. What is the use of comparison?

5. Explain the meaning of the term inflection; and show how the deficiencies of inflection are supplied in the English language.

6. Give the derivation of the word Syntax and show in what it differs from Etymology. Explain and illustrate the distinction between concord and government.

7. In what circumstances does the nominative follow its verb?

What mode of a verb has its nominative generally suppressed?

8. For what purpose are words placed in apposition, and in what case must such words be?

9. What is meant by a nominative absolute? Give the rule regarding it, and construct a sentence to exemplify the rule.

10. When two verbs are connected by the conjunction that in what tense must they be?

Afternoon Paper.

1. Correct the following sentence:

There are, indeed, but very few who know how to be idle and innocent, or have a relish of any pleasures that are not criminal; every diversion they take is at the expense of some one virtue or other, and their very first step out of business is into vice or folly.

2. Write a paraphrase of the following passage; and parse the words printed in Italics :

'Tis greatly wise to talk with our past hours,

And ask them what report they bore to Heaven,

And how they might have borne more welcome news.
Their answers form what men experience call ;—

If wisdom's friend, her best; if not, worst foe.

3. Distinguish between the meanings of the following words:

Peculiar-Particular.

Reason-Cause.

Contented-Satisfied.

4. Explain the meaning of the term metre; and state in what English

metre essentially consists.

5. Mark the accented syllables in the following lines:

But vainly thou warrest,

For this is alone in

Thy power to declare

That in the dim forest

Thou heard'st a low moaning,

And saw'st a bright lady surpassingly fair.

1. What were the subdivisions of the Roman people before the time of Servius Tullius? What changes were introduced by Servius?

2. Give the particulars of the first retreat of the Plebs to the sacred mountain. What induced them to return?

3. Give an account of the Decemviri and their legislation.

4. Relate briefly the events which occurred in the second Punic war. 5. What measures did Augustus take after the death of Antony in order to secure himself in the empire?

6. Mention in chronological order the names of the most eminent Romans from the destruction of the city by the Gauls to the accession of Augustus.

India.

1. Mention the leading incident in the life of Mahmood of Ghizni.

2. Give an account of the administration of Aurengzebe.

1. Mention in chronological order the wars in which the British have been engaged since their establishment in India.

Greece.

1. State what you know respecting the government and institutions of Sparta.

2. What changes were made by Solon in the government of Athens ? 3. What caused the first Persian invasion? Mention the most remarkable events that occurred in it.

4. What was the effect on Athens of the failure of the Sicilian expedition? What changes were made in the government at this time?

5. Give a brief account of the life and character of Alexander.

6. Write down in chronological order the names of the most eminent men in Greece from the time of Solon to the battle of Choroneia.