11. When three circles are coaxal, the several pairs of tangents to the same two of them from all points of the third have the same ratio; prove this generally.

12. Every two circles, their circle of similitude, and their two circles of antisimilitude, are coaxal; prove this generally.

MR. LESLIE.

13. Upon the sides of a triangle ABC equilateral triangles are constructed, whose vertices are A', B', C'; prove that AA', BB', CC' are equal, and meet in a point.

14. Divide an arc of a circle into two parts such that the sum or difference of their chords may be equal to a given line.

15. Prove that the point of intersection of the lines which join the middle points of the opposite sides of a quadrilateral is the middle point of the line which joins the points of bisection of the diagonals; and construct a quadrilateral, being given its sides, and the line which joins the middle points of a pair of opposite sides.

16. Determine the maximum triangle of given species whose sides pass through three fixed points; and construct a quadrilateral of given species whose sides pass through given points.

17. Construct a triangle, being given the three perpendiculars; and express the area of the triangle in terms of these perpendiculars.

18. Prove that the ratio of the distances of any point on a circle from a given pair of inverse points is constant; and find the locus of a point such that the angles subtended at it by two segments of the same line may be equal.

B.

DR. STUBBS.

1. A quadrilateral field has its four sides 36, 28, 40, and 38 perches respectively, and the diagonal which forms the common base to the first two and the last two sides 45 perches; calculate its contents in acres, roods, and perches.

2. What should be the price of English standard silver, 37-40ths fine, in order that the par of exchange between England and France should be 25 francs 22 cents.; 200 francs being coined from one kilogramme of silver 9-10ths fine? [1 kilogramme = 15,434 grains.]

MR. TOWNSEND.

5. Determine, in finite terms, the square root of the binomial 2 +√3. 6. Calculate, to five decimal places, the two values of x from the equation

7. State and prove the formula for the conversion of a mixed circulating decimal into its equivalent ordinary fraction, in any scale of notation 8. For any three pairs of corresponding numbers x and x', y and y', z and z', verify by actual multiplication the identity

(x2 + x ́2) (y2 + y22) (z2 + z′2) =

(xyz + yz'x' + zx'y' — xyz)2 + (x'yz + y'zx + z'xy — x'y'z′)2.

MR. LESLIE.

9. A person buys 100 shares in a company for £3500; after receiving four half-yearly dividends of 15s. 4d., 20s. 10d., 30s. 4d., and 388. 9d. per share, he sells at a profit of 43 per cent.; reckoning the simple interest of money at 4 per cent., how much above that interest has he gained?

10. A person bought a number of £20 railway shares, when they were at a certain rate per cent. discount for £1500; and afterwards, when they were at the same rate per cent. premium, sold them all but 60 for £1000; how many did he buy, and what did he give for each of them?

II. Prove that

a3 (b2 - c2)+b3 (c2 − a2) + c3 (a2 − b2) = (a − b ) (b − c) (a−c) (ab+be+ ac), and that

• (a + b) ab + (b + c) bc + (a + c) ac > 6abc.

1. The perpendicular from the vertex on the base of an equilateral triangle ABC is produced to a point below the base, equally distant from the base with the vertex; through the point so found a line MN is drawn cutting the sides produced in M and N; the lines MC and NA are drawn to the opposite extremities of the base; show that they intersect on the circumscribing circle of the given triangle.

2. Four common tangents are drawn to two circles, and from the points of intersection of each pair perpendiculars P and P' are let fall on one of the other pair; if R and R' be the radii of the circles, prove that

3. If from 0, the centre of the inscribed circle of a triangle ABC, perpendiculars be let fall on the lines joining the middle points of sides, and produced to A'B'C' until the rectangle under each perpendicular and the whole perpendicular produced equals the square of the radius of the inscribed circle; the area of the triangle A'OB' will be to that of

ABC (sc): s

where s is half the sum of the sides of the given triangle, and similarly for A'OC' and B'OC'; and hence the area of A'B'C' equals that of ABC. 4. Draw a chord of a circle parallel to a given straight line, which shall be cut in a given ratio by a given diameter.

MR. TOWNSEND.

5. A circle intersecting another passes through its centre; the chord of contact with the former of the real pair of common tangents to both touches the latter.

6. The two centres of perspective of any two parallel chords of a circle are inverse points with respect both to the circle itself, and also to the circle touching the two chords at their middle points.

7. When two circles intersect at right angles, the pole of either centre of similitude with respect to either circle is the pole of the other centre of similitude with respect to the other circle.

8. The three circles, coaxal each with two of three arbitrary circles, and orthogonal each with the third, are coaxal.

MR. LESLIE.

9. Prove that the centres of the circles circumscribing the triangles formed by four right lines lie upon a circle.

10. Prove that the intersections of the perpendiculars of the four triangles which can be formed by connecting four points upon a circle lie upon a circle.

11. Lines connecting a point on one of two circles with their centres of similitude intersect the other in a, a′ and b, b′; prove that ab and ab’ intersect the line joining their centres in fixed points.

12. Prove that the sum of the perpendiculars dropped from the centre of the circumscribing circle of a triangle upon the sides is equal to the sum of the radii of the inscribed and circumscribed circles.

Classics.

DEMOSTHENES.

MP. MAHAFFY.

Translate the following passages :

1. a. Beginning, Δεῖ δὲ μηδένα ὑμῶν, ὦ ἄνδρες δικασταί, κ. τ. λ. Ending, οὐχ ὡς ὅδε Φωκέας ἀπώλεσε καθ' ἑαυτόν, πόθεν ;

b. Beginning, Δεινὸν οὖν ψεύσασθαι καὶ προέσθαι, κ. τ. λ. Ending, ἀποδημίαν πάντα τἀναντία ἔπραττον.

c. Beginning, Οὐκ ἴσασιν οὗτοι τὸ μὲν ἐξ ἀρχῆς τὰς βίβλους, κ. τ. λ. Ending, παρ' ὑμῖν ἐπὶ πορνείᾳ.

d. Beginning, Τοῦτο δὲ τὸ δρᾶμα οὐδὲ πώποτε οὔτε, κ. τ. λ. Ending, δικαστὰς εἶπεν. λέγε.

2. Beginning, Οἶμαι τοίνυν καὶ τοῦτον τὸν λόγον, κ. τ. τ.λ. Ending, ποιοῦσα χάριν τισὶν οὐκ ἀποδώσει.

3. Beginning, Οὐ τοίνυν μόνον τὰ δικαστήρια ἄκυρα, κ.τ.λ. Ending, τῷ καταστάσει τῶν ἐγγυητῶν τὸν δεσμὸν ἀφαιρῶν.

1. Explain the causes of the rise of mercenary forces in Greece, and state the most celebrated instances of mercenary armies in the 3rd century B. C.

2. The general condition of Greece was peculiarly favourable to the development of Philip's power?

3. What is the dispute about the order of the Olynthiac Orations? 4. Discuss the character and policy of Phokion.

5. What were the date, scope, and result of Demosthenes' Oration περὶ παραπρεσβείας

6. Explain the expressions, ἀνάκρισις, εἰσαγγελία, ἔρημον καταδιαιτᾶν τινος, προβολαί, δέχεσθαι τὰ πρυτανεῖα, διαιτητης, ἀτίμητος ἀγών. 7. Describe the route taken by Alexander in his invasion of Asia. Its peculiarities are easily explained on strategical grounds?

8. Give some account of the orators contemporary with Demosthenes (omitting Æschines).

CICERO.

MR. PALMER.

Translate the following passages :—

1. Beginning, Itaque, excussis tuis vocibus,...

Ending, in suis fortunis, tam timidus fuerit, pertimescat?

Phillippica, II. cap. xxix.

2. Beginning, Quoties ego hunc Archiam vidi,. Ending, permulti alii præterea pugnant inter se atque contendunt. Oratio pro Archia Poeta, cap. viii.

3. Beginning, Reliqua pars accusationis duplex fuit: una,. Ending, vel quia non nosset, vel si nosset, contemneret.

Pro rege d. ad C. Cæsarum, cap, viii.

4. Beginning, C. Malleolo, quæstore Cn. Dolabellæ, occiso, Ending, pupillo Malleolo retulit.

In Verrem, act II. lib. i. cap. 36.

1. What were the chief measures of C. Gracchus ?

2. Give a short sketch of Sulla's constitution.

3. Give the date of the following battles, with the names of the contending parties-Aqua Sextiæ, Žela, Munda, Thapsus, Tigranocerta, Colline Gate, Charonea.

4. What is Mommsen's conception of the character of Pompeius?

5. Mommsen animadverts on the government of the Sullan restoration, speaking of the wars that took place during that period?

6. Mommsen remarks that the Gabinio-Manilian laws terminated one political struggle, and began another?

7. What, according to Mommsen, was the real aim of the Catilinarian conspiracy?

8. What is the true historical significance of the conquests of Cæsar in Gaul?

9. On what occasion did Cicero defend, or purpose defending Catiline, and on what charge?

10. Contrast Mr. Forsyth's estimate of Cicero's character with that entertained by Mommsen.

MR. FERRAR,

Translate the following passage into Greek Prose :

:

In this situation, man has called in the friendly assistance of philosophy, and Heaven, seeing the incapacity of that to console him, has given him the aid of religion. The consolations of philosophy are very amusing, but often fallacious. It tells us that life is filled with comforts, if we will but enjoy them; and, on the other hand, that though we un