15. Find the envelope of a system of lines the sum of the square roots of whose intercepts on the axes is constant.

16. A cube is placed on a rough inclined plane, two of the opposite edges of the face in contact with the plane being horizontal; find the conditions necessary for its equilibrium.

17. If a body describe an orbit under the action of a central force F, where r and are the polar co-ordinates

show that Fr

de \2 d2v

dt

of a point on the orbit.

dt2'

18. Show that a projectile would describe a parabola if there was no air, and find the equation of the curve in terms of the initial velocity and angle of projection.

B.

MR. W. ROBERTS.

1. Given the length of a man's foot, and the distance between his heels; find the position of his feet when he stands the firmest.

2. Denoting by a force directed to a fixed centre, by u the reciprocal of the radius vector, by w the polar angle, and by h the area described in the unit of time; prove that the differential equation of the orbit described will be

3. Write, in its algebraic form, the integral of the equation

5. A material point is projected with a given velocity v from a point in a plane which is inclined at an angle a to the horizon; find the angle of projection which will give the maximum range on the inclined plane, and find the range.

6. Show that in the orbit

a (1 − e2)

7=

I + e cos (w - a)

described by a body round a centre of force, the constants a and e are determined by the equations

where m is the force at the unit of distance, p the initial distance, & the initial angle of direction, and the initial velocity.

7. A flexible string AOB fixed at two points A and B is acted on by gravity; the unit of mass at any point P varies inversely as the square root of the length OP measured from the lowest point 0. Show (a) that the equation to the catenary is

2®y = } (x + 2ß) (x2 + 4ßx)} − 2,3o log {*

́x + 2ß + (x2 + 4ßx)* ́

23

and that the tension at any point of the curve is given by the equation

ẞ being a constant depending on T the tension at the lowest point.

8. Prove dx. tan-1x = x tan-1x — log (1 + x2)1.

MR. TARLETON.

9. Show how to find f cax sin "x dx in general, when n is a positive integer, and find thus fear sin 3x dx.

10. A cylinder is placed on a rough inclined plane, with its axis horizontal; a string is coiled round the middle of the cylinder, and passed over a pulley in the same vertical plane with the centre of gravity of the cylinder; a weight is attached to the string; find the conditions of equilibrium.

11. If a particle move under the action of any number of forces, and hdt be the area described round a fixed point in an infinitely short time dt, show that at any time is equal to the moment of the forces

dh dt

round an axis through the fixed point, and at right angles to the plane through the fixed point, and the direction of the particle's motion at the time.

12. Show that the integral by means of which the time of vibration in a simple pendulum is expressed, can be reduced to the form

αφ
1- c2 sin 20

Experimental Physics.

DR. HAUGHTON.

1. If 100 lbs. of water be cooled down to 22°, and then stirred, a portion of it will be immediately converted into ice; determine the quantity of ice formed, without assuming either that all the ice is formed at 32o,

or at 22°.

2. The specific heat of calespar is known to be o. 20, and that of rock crystal o. 19; and the specific heat of soils not containing much humus may be assumed as 0.20; while that of soils containing humus rises to o. 4 and 0.5, and may be assumed fully equal to 0.40.

State clearly what effect will be produced by this difference of specific heats under the following circumstances:-Two acres of freshly ploughed and sown land are composed, the one of soil free from humus, and the other of soil largely mixed with turf mould; their temperature is 40°, and they are wetted by rain at 50° to the depth of a quarter of an inch.

3. A man, travelling at the rate of 4.353 miles per hour, inclines his body forward through an inclination of 7° 40′; calculate, from this observation, how many miles he must advance in a horizontal direction before he has done as much work as would be involved in raising his body through one mile of vertical height.

4. Calculate the centrifugal force at the equator of Mercury, his radius being 1570 miles, and his time of rotation 24h.5m.

5. A heavy ball, suspended by a fine wire, vibrates in a small arc, and 48 vibrations are counted in 3 minutes; find the length of the wire.

Classics.

EURIPIDES.

MR. FERRAR.

Translate the following passages into English Prose :—

1. Beginning, εὐθὺς δὲ πώλοις δεινὸς ἐμπίπτει φόβος κ. τ. λ. Ending, τροχῶν ἐπήδων ἀξόνων τ ̓ ἐνήλατα.

2. Beginning, ΤΡ. ὦ παῖ, τί θροεῖς; κ. τ. λ. Ending, ἐν χειρὶ βέλος.

3. Beginning, ἄγετέ με τὰν Ιλίου, κ. τ. λ. Ending, Μυκηναῖαί τ' ἐμαὶ θεράπναι. d

Hippolytus, 1218-1235.

Ibid., 213-222.

Iphigenia in Aulis, 1475-1499.

4. Beginning, ἥκεις διφρεύων βαλιαῖσι πώλοις. κ. τ. λ. Ending, καπφθίμενον Θρηκὶ μόρῳ φίλτατον ἄχθος οἴσει. Rhesus, 356-379.

5. Beginning, ΙΩ. τάδ' οὐχὶ δεινά ; ῥυσιάζομαι λόγῳ. κ. τ. λ. Ending, ΚΡ. Γοργὼν μὲν ἐν μέσοισιν ἠτρίοις πέπλων.

Ion, 1405-1421.

1. Give some account of the political and philosophical opinions of Euripides, quoting passages from his dramas in support of your views. 2. What was the cause of Aristophanes' enmity to him?

3. Compare Euripides with Sophocles and Eschylus.

4. Discuss the statement of Schlegel, that "Euripides is a precursor of the New Comedy, towards which he manifestly verges."

5. To what tetralogy did the Medea belong? When was it acted? 6. What was the political object of the Heraklidæ ?

7. How are the later plays of Euripides distinguished from the earlier? 8. Arrange in chronological order his plays that relate to the Trojan

war.

9. The worship of Bacchus was twofold? What various explanations have been given of his name Dionysus ?

10. Draw a plan of the theatre at Athens.

MR. ABBOTT.

Translate the following passages into English :

(N. B.-Special credit will be given for elegance of translation.) 1. Beginning, Me. Quid tu te solus e senatu sevocas?.. Ending, Loces efferundum: nam jam, credo, mortuost.

PLAUTUS, Aulularia, act III. sc. vi. 13-32.

2. Beginning, Sicut me hodie nimis sollicitum cliens

Ending, Meque adeo, qui hodie forum unquam oculis inspexi meis! Ibid., Menæchmei, act IV. sc. ii. 21–32.

3. Beginning, Ge. Nunc auctionem facere decretumst mihi:. Ending, Hæc venivisse jam opus est quantum potest :

Ibid., Stichus, act II. sc. i. 218-232.

4. Beginning, Nisi me animus fallit, haut permultum à me aberit infor

tunium: ...

Ending, Retraham hercle opinor ad me idem illud fugitivum argentum

tamen.

TERENCE, Heautont., act III. sc. vi.

5. Beginning, Pam. Nemini plura ego acerba credo esse ex amore homini

umquam oblata,.

......

Ending, Levia sunt, quæ tu pergravia esse in animum induxti tuum. Ibid., Hecyra, act III. sc. i. 1-12.

6. Beginning, Omnes autem hos motus subsequi debet gestus, . Ending, qui in agendo contuens aliquid pronunciaret. CICERO, De Orat., lib. iii. s. 59.

1. Write a short sketch of the rise of Dramatic Composition among the Romans.

2. Give an account of the state of Literature at Rome in the sixth century A. U. C.

3. What is the origin of the Latin perfects of the 1st and 2nd conjugations?

4. What is the origin of the perfect of the 3rd conj. of the forms scripsi and feci, respectively?

5. Explain the Oscan forms fefakust (= fecerit), pruhibust (= prohibuerit); and show how they are related to the corresponding Latin forms.

Comic Metres.

6. What is Fleckeisen's rule with respect to the elision of monosyllables?

7. In what cases is hiatus justified?

8. Enumerate the combinations of consonants (besides the case of a mute and liquid) which admit of the preceding vowel being shortened.

9. How may such metrical measurements as the following be accounted for, respectively simillume, inde, amat, žile?

10. Scan the following lines, and explain any metrical peculiarities:Hæc res vitæ me soror saturant

Ne lacruma soror neu tuo id animo} Anapastics.

Poteris agere atque is dum veniat sedens ibi opperibere.

Trochaic Septenarius.

Hæc apud nos magna turba ac magna familia est
Ego ut multo melior. Cocum ego, non furem rogo

Unde exit hæc? Unde nisi domo? Domo? Me viden? Te video.
Iamb. Septenarius.

MR. GRAY.

Translate the following passage into Greek Prose :-

I take it to be the highest instance of a noble mind, to bear great qualities without discovering in a man's behaviour any consciousness that he is superior to the rest of the world. Or, to say it otherwise, it is the duty of a great person so to demean himself, as that, whatever endowments he may have, he may appear to value himself upon no qualities but such as any man may arrive at. He ought to think no man valuable but for his public spirit, justice, and integrity; and all other endowments to be esteemed only as they contribute to the exerting those virtues. Such a man, if he is wise or valiant, knows it is of no consideration to other men that he is so, but as he employs those high talents for their use and