4. State the changes introduced into the character of the imperium under Sulla, Marius, Pompeius, and Cæsar, respectively. How was the use of the term imperator restricted by the best instructed Romans?

5. What error in the calendar was rectified by Augustus, and how had it arisen?

6. Describe the political characteristics of the several families of nations known to the Romans.

7. In what respects was slavery an element of strength in the empire ? 8. To what circumstances does Mr. Merivale trace the success of the Romans in assimilating the barbarian races? Why were they more successful in assimilating the Westerns than the Easterns? 9. Explain the allusions in the following:

Qui dissimulat metum

Marsæ cohortis Dacus.

Præsens divus habebitur
Augustus, adjectis Britannis
Imperio gravibusque Persis.

MR. GRAY.

Translate the following passage into Greek Iambic Verse:—

Come, do not hunt,

And labour so about for circumstance,

To make him guilty whom you have foredoom'd:
Take shorter ways, I'll meet your purposes.
The words were mine, and more I now will say;
Since I have done thee that great service, Cæsar,
Thou still hast fear'd me; and, in place of grace,
Return'd me hatred; so soon all best turns,
With doubtful princes, turn deep injuries
In estimation, when they greater rise
Than can be answer'd. Benefits, with you,

Are of no longer pleasure, than you can

With ease restore them; that transcended once,

Your studies are not how to thank, but kill.

It is your nature, to have all men slaves

To you, but you acknowledging to none.

The means that make your greatness, must not come
In mention of it; if it do, it takes
So much away, you think.

BEN JONSON.

Translate the following passage into Latin Hexameters :

She said and vanished; they, with rushing sound,
Rose, and before them drove the hurrying clouds :
Soon o'er the sea they swept; the stirring breeze
Ruffled the waves; the fertile shores of Troy

They reach'd, and falling on the funeral pyre,
Loud roared the crackling flames; they all night long
With current brisk together fanned the fire.

All night Achilles with a double cup
Drew from a golden bowl the ruddy wine,
Wherewith, outpoured, he moistened all the earth,
Still calling on his lost Patroclus' shade.
As mourns a father o'er a youthful son,

Whose early death hath wrung his parents' hearts:
So mourned Achilles o'er his friend's remains,
Prostrate beside the pyre, and groaned aloud.
But when the star of Lucifer appeared,
The harbinger of light, whom following close
Spreads o'er the sea the saffron-robèd morn,

Then paled the smouldering fire, and sank the flame;
And o'er the Thracian sea, that groaned and heaved
Beneath their passage, home the winds returned;
And weary, from the pyre a space withdrawn,
Achilles lay, o'ercome by gentle sleep.

LORD DERBY's Homer.

Translate the following passage into Latin Prose :

The moment has at length arrived, men of Rome, later indeed than became the dignity of the Roman people, but yet so opportune that it cannot be put off for a single hour. Hitherto a kind of fatality has pursued us, and we have borne it as best we could. Henceforth, if we suffer, it will be our own fault. It is not right for the Roman people to be slaves, whom the immortal gods destined to command all nations. Matters have now come to the last extremity. The struggle is for freedom. You must either be victorious-as surely you will be with so much piety and concord--or suffer anything rather than be slaves. Other nations may endure slavery; but freedom is the attribute of the Roman people.

Translate the following passage into Greek Prose:-—

I adjure you then, by the gods of our fathers, and as you value my own happiness, that you continue to love and cherish one another. For let it be far from you to imagine, that when I have passed the term of this human existence, I shall cease to live. Even in this life my soul has never been visible to you, and your knowledge of its existence is derived from its acts alone. But you cannot have failed to observe the terrors with which the spirits of injured men inspire guilty consciences, or the avenging dæmons which they send to torment the impious. Nor can you surely believe, that the custom of paying honour to men after their death would have become so inveterate, if their spirits had no perception of those honours. For myself, I never could be persuaded that the soul lives only so long as it dwells in a perishable body, but dies in the moment of its emancipation from that body. When I see that even mortal bodies, while the soul remains within them, are preserved alive, how can I believe that the soul itself, when separated from a lifeless body, becomes lifeless?

n

FELLOWSHIP EXAMINATION.

Examiners.

ANDREW S. HART, LL. D.

JOHN TOLEKEN, M. D.

JOSEPH CARSON, D. D.

JOHN A. MALET, D. D.

THOMAS STACK, M. A.

JOHN H. JELLETT, B. D., Professor of Natural Philosophy.
GEORGE LONGFIELD, D. D., Regius Professor of Hebrew.

MICHAEL ROBERTS, M. A., Erasmus Smith Professor of Mathematics.
JOHN K. INGRAM, LL. D., Regius Professor of Greek.

THOMAS K. ABBOTT, M. A., Professor of Moral Philosophy.
JOHN R. LESLIE, M. A., Professor of Experimental Philosophy.

Mathematics, and Mathematical Physics.

PURE MATHEMATICS.

DR. HART.

1. Given seven points on a cubic of the fourth class, find the locus of its double point.

2. Given the intersections of a cubic with a right line, and also the points where tangents at these intersections meet the curve again; show how to find the possible positions of a double point on the cubic.

3. Find the locus of intersection of any tangent to a given cubic with the polar, with regard to its Hessian, of the point of contact.

4. Find the equation of the curve which determines the points of contact of double tangents to the curve

x3 (ay + b) = y3 (cx + d).

5. If a double point on a quartic is a point of inflexion on each branch of the curve, prove that the tangents at the other points of inflexion may be divided into pairs whose points of intersection lie on a right line.

8. Find the equation of a surface of given area bounded by two given surfaces, so that

dS ρ

may be a minimum, p being the distance from a given point.

MR. M. ROBERTS.

1. Find the locus of the centre of a sphere which is touched by the edges of a tetrahedron which is self-conjugate with respect to the surface

Ax2+By+ Cz2 − 1 = 0.

2. Find the equation of the cone whose vertex is the centre of an ellipsoid, and whose base is the section made by the polar of any point x', y', z'.

3. Adopting the notations of the "Fundamenta nova," let u be an elliptic function of the first kind to the modulus k, K the complete function, K' the complete function to the complementary modulus; find the value of

cos am (u) cos am (u + K + i K') (i = √√-I).

4. From the following series given in the " Fundamenta nova,"

deduce the development by cosines of multiples of 20 of the function

5. Adopting Jacobi's definitions of the functions, Z(u), ✪ (u),

→ (u)

=

(1 · 29 cos 20 +92) (1 − 293 cos 2σ +96) (1 − 295 cos 20 + 910

I-2 { 9 cos 20 - q1 cos 40 + qo cos 60 -
(1 − 92) (1 − 91) ( 1 − qo) . . .

deduce the value of (o) if

=

}

→ (u) = 1 − -29 cos 20 + 29 cos 40 - 299 cos 60+....

7. From the definitions

AO (u) = ( 1 − 29 cos 20 +93) (1 − 2q3 cos 20 + qo) . . . .

AH (u) = 2 /q sin σ (1 − 292 cos 20 + 94) (1 − 29a cos 2σ + q8) .. when A does not contain σ, deduce the relation

and hence prove that e1KK' → (u) has 4i K' for a period.

8. If h is the modulus derived from k in the decreasing scale for the odd number 3, show that by putting h=u, k = v*, we have

and exhibit the four values of v corresponding to a given value of u.

9. If X denote a sextic function where skew invariant vanishes, prove that the hyperelliptic integral

(a + ẞx) dx
VX

depends on elliptic functions of the first kind.

10. K, K, K" are three quantities, unequal, greater than zero, and less than unity; let

and putting k2 = 1 − k2, k′2 = 1 − k′2, k′′2 = 1 − k”?, let

cos 20d0

✓ (1 − k2 sin 20) (1 − k’2 sin 20) (1 − k”1⁄2 sin 20) ’