II.

Translate the following passages into English Prose: 1. From ὦ θεώμενοι, κατερώ πρὸς ὑμᾶς....

EURIP., Bacch., 610-636.

2. From .. ὁ δ ̓, ὡς ἐσεῖδε, δώματ' αἴθεσθαι δοκών..... ....ἀσκεῖν σώφρον εὐοργησίαν.

to

THEOCR., Id., XI., 45-56.

Translate into Greek Tragic Iambics :
He comes, and with a port so proud,
As if he had subdu'd the spacious world:
And all Sinope's streets are fill'd with such
A glut of people, you would think some God

Had conquer'd in their cause, and them thus rank'd,
That he might make his entrance on their heads!
While from the scaffolds, windows, tops of houses,
Are cast such gaudy show'rs of garlands down
That ev'n the crowd appear like conquerors,
And the whole city seems like one vast meadow,
Set all with flow'rs as a clear heav'n with stars.

III.

LEE.

Translate into English Prose:

1. From Videte, judices, quantæ res his.... ...denique ei nuntiabatur ?

to

Cic. pro Mil. c. 18.

2. From Inclinatis semel in concordiam.... to ....vis ingens æris alieni sit.

LIV. VII. 21.

3. From Sed ubi diem ex die prolatabant... to ....an bellum cunctatione tractaret.

TAC. Ann. VI. 42-44.

Translate into Latin Prose:

Nor let any one suppose that this conflict of evidence renders the attainment of certainty impossible. Doubtless there are many points both in sacred and in common history, both in civil and ecclesiastical records, where we must be content to remain in suspense. History will have left half its work undone, if it does not teach us humility and caution. But essential truth can almost always be found: truth of all kinds can with due research be usually found: she lies, no doubt, in a well; but we may be sure that she is there if we dig deep enough. In this labour teachers and students must all work together. What one cannot discover, many at work on the same point can often prove beyond doubt. Like Napoleon and his comrades, when lost in the quicksands of the Red Sea, let each ride out a different way, and the first that comes to firm ground bid the others halt and follow him.

Translate:

IV.

From A. Adeon' rem redisse, ut qui mihi.... ....animus qui modeste istæc ferat.

to

TER. PHORM. I. 3.

From Desine, Paulle, meum lacrimis. .. ....In mea sortita vindicet ossa pila.

to

From Ardet, inexcita, Ausonia,........ to ....fidoque accingitur ense. Translate into Latin Elegiacs :

PROPER, V. 11.

VIRG. En. VII. 623.

In dark Thermopyla they lie ;
Oh death of glory there to die!
Their tomb an altar is, their name
A mighty heritage of fame :

Their dirge is triumph-cankering rust,
And time that turneth all to dust,
That tomb shall never waste nor hide,
The tomb of warriors true and tried,
The full-voiced praise of Greece around
Lies buried in that sacred mound,
Where Sparta's king Leonidas,

In death eternal glory has.

From the Greek Anthology.

V.

1. The opposite sides and angles of parallelograms are equal to one another, and the diameter bisects them.

If the diagonals of a quadrilateral figure bisect each other, it must be a parallelogram.

2. The opposite angles of a quadrilateral figure inscribed in a circle are together equal to two right angles.

The opposite sides of a quadrilateral inscribed in a circle are produced to meet in P and Q, and about the triangles so formed without the quadrilateral circles are described meeting again in R; shew that P, R, Q are in one straight line.

3. If the angle of a triangle be divided into two equal angles by a straight line which also cuts the base, the segments of the base shall have the same ratio which the other sides of the triangle have to one another; and conversely.

Given the base of a triangle and the point where the line bisecting the exterior vertical angle cuts the base produced, find the locus of the vertex of the triangle.

4. Find the locus of a point without a square such that straight lines being drawn from it to the angular points of the square, the angle contained by the two extreme lines is divided into three equal parts by the other two.

5. Shew how to reduce any recurring decimal to its equivalent vulgar fraction. Divide 1.13 by .000132.

6. What is the half-yearly interest obtained if £385. 7s. 31d. be invested in the purchase of 3 per cent. stock at 94 ?

7. The carpeting of a room twice as long as it is broad at 5 shillings per square yard, cost £6. 2s. 6d.; and the painting of the walls at 9 pence per square yard, cost £1. 6s. 3d. What is the height of the room?

8. What sum of money will amount to £425. 19s. 4 d. in 10 years at 31 per cent. simple interest; and in how many years more will it amount to £453. 11s. 7d.?

9. Reduce the following expressions to a single fraction;

a2

+

+

1

(x − a) (a - b) (a − c) 1 (x − b) (b− a) (b − c) (x − c) (c − a) (c − b) *

1 1 1
= + +
x + a + b x α

(3) x1 − x2 + y1 – y2 = 84, x2 + x2y2 + y2 = 49.

11. Explain what is meant by Mathematical Induction. Shew by it that the sum of the cubes of the first n natural numbers is equal to the square of the sum of the numbers.

12. A person has a capital of £2000, which produces him interest at 5 per cent.; if he spend every year £180, find in how many years he will be ruined; having given

log 2 = 30103, log 3 = 47712, log7 = ·84510.

13. Shew that the arithmetical mean of any number of positive quantities is greater than the geometrical mean. Shew that (a + b + c + d) (a3 + b3 + c3 + d3) > 16abcd.

VI.

1. Define the cosine of an angle ;-express the cosine in terms of each of the other Trigonometrical Ratios.

2. Find a formula which shall include all angles which have the same co-tangent-and write down the four smallest angles which satisfy the equation 3 cot2 0 - 1 = 0.

3. Prove that

i. tan(A-B)

=

tan Atan B

1+tan A tan B'

geometrically or otherwise.

ii. cos A+ cos B+ cos2 C+2 cosA cosВ cos C= 1, if A+B+C=180°.

4. Find the value of x in each of the equations:

i. tan 1(x-1) + tan ̈1(x + 1) = π, ii. tan x + tan 3x = 2 tan 2x. 5. Find the numerical value of 1/( (242447),

having given log10 24244 = 4·3846043, diff. for 1 = 179.

6. Shew that the base of the Napierian logarithms is incommensurable.

Also if sin {0+ Ø √(− 1)} = p {cos a + √(− 1) sin a} where are real quantities, then will

p, a, 0,

tan a =

еф

e

еф теф

cote and 4p2 = e2 + e2 - 2 cos 20.

7. What is meant by the parameter of any diameter of a parabola? Prove, with the usual notation, that QV2 - 4SP.PV. 8. In an ellipse prove that CP2 + CD2 = AC2 + BC2.

Also if the normal at P meet the axis-major in G and the conjugate diameter in F, then PF.PG BC.

=

9. A circle is described upon the latus rectum of a parabola as a diameter, and any chord QPN is drawn parallel to the axis of the parabola,-meeting the circle in Q, the parabola in P, and the latus rectum in N, prove that QN2 ∞ PN.

10. Find the equation to the tangent to an ellipse

Deduce the corresponding equation to the tangent to a parabola y = 4cx.

11. If y = mx, y = m'x be conjugate diameters of the curve ax2 + 2bxy + cy2 =ƒ, m and m' must be connected by the relation a + (m + m2) b + mm'c = 0.

12. What is meant by the moment of a force with respect to a point? The algebraic sum of the moments of two forces acting in one plane about any point in the plane is equal to the moment of their resultant.

13. Explain how the statical action of a hinge or pivot may be estimated.

A plane lamina which can turn about a pivot P is acted on by two equal forces (Q, Q) in opposite directions at a distance c from each other-find the least force which can be applied at a given point A so as to maintain equilibrium— and the pressure on the pivot.

14. Shew that the path of a projectile is a parabolaand that at every point of the path the velocity is that which would be acquired by a body falling to that point from the directrix.

Particles are projected from the same point and in the same vertical plane so as to describe equal parabolas-shew that the vertices of their paths lie in a parabola.

VII.

1. Find the equation to the straight line which passes through the intersection of the lines

2. Find the polar equation to a parabola when the vertex is the pole, and the axis of the parabola the initial line. Determine the shortest chord which can be drawn in a parabola so as to subtend a right angle at the vertex.

3. Find a point in an ellipse such that the normal at the point bisects the angle between the ordinate of the point and the central radius vector of the point.