B.

MR. CATHCART.

1. If points D, E, F be taken in the sides of a triangle ABC, such that BD = aBC, CE = BCA, AF = 7AB, show that the ratio of the areas DEF: ABC

2. If the sides of a quadrilateral be given in length, the difference of squares of the lines joining the middle points of opposite pairs is con

stant.

3. The line joining intersection of tangents to a circle to the opposite extremity of the diameter through one point of contact bisects the perpendicular let fall on that diameter from the other point of contact.

4. Determine the triangle of given species whose sides pass through three fixed points, and whose area is a maximum.

5. Prove that the triangle formed by joining the centres of equilateral triangles described on the sides of any triangle is equilateral.

MR. BURNSIDE.

6. Through a given point to draw a right line so that the intercepts made by two given circles shall be equal.

7. Determine the radius of the circumscribed circle of the triangle formed by joining the feet of the perpendiculars of a given triangle.

8. If, ', ' be the lengths of the perpendiculars drawn from the centre of the circumscribing circle to the sides, prove the relation

« + r' + x' = R + r.

9. Through a given point to draw a right line so that the area of the triangle formed by it and two given lines may be given.

10. Given the base of a triangle, the difference of the sides, and the locus of the vertex a right line; construct the triangle.

MR. PANTON.

11. The base AB of a right-angled triangle ABC is bisected in 0; prove that the circles circumscribing the triangles AOC and BOC cut orthogonally.

12. Given base and ratio of sides of a triangle, the locus of the vertex is a circle whose centre divides the base in the duplicate ratio of the constant ratio of sides.

13. Prove the following properties of a self-conjugate triangle with respect to a circle:

(a). Any side is equal to twice the tangent from its middle point to the circle.

(3). The square of any side is equal to the sum of the squares of the tangents from its extremities to the circle.

14. A square is described on the hypotheneuse of a right-angled triangle, and lines are drawn connecting the remote corners of this square with the vertex: prove that these lines divide the base into three segments which are in continued proportion.

15. The sides of a triangle are 20, 15, and 11; calculate the radius of the inscribed circle, and the distances of its centre from the three angular points.

C.

MR. CATHCART.

1. Given the hypotenuse of a right-angled triangle, if equilateral triangles be described on the sides, required the locus of the middle point of the line joining their vertices.

2. In every triangle, the polars of the middle points of the sides with respect to the inscribed circle form a triangle whose area equals that of the original triangle?

3. Through any point outside a circle, draw a chord such that the intercept on the diameter through the point between perpendiculars let fall on it from the extremities of the chord shall be a given quantity.

4. Triangles are inscribed in a given triangle having their vertices at one given point, and their bases passing through another given point in one of the sides; determine that one of them whose area is a maximum. 5. Construct a triangle, being given vertical angle, altitude, and

(a). Sum of sides;

(8). Difference of sides.

MR. BURNSIDE.

6. From the extremities of a diameter of a given circle perpendiculars are drawn to the sides of an inscribed triangle: prove that the two right lines passing through the feet of the perpendiculars intersect at right angles on the nine-point circle of the triangle.

7. If a triangle inscribed in a given triangle is copolar thereto, the triangle formed by joining the middle points of the sides of the inscribed triangle is also copolar to the given triangle ?

8. Find the locus of the intersection of the perpendiculars of a triangle of given species whose sides pass through three fixed points.

9. Prove that the centre of the circle passing through the centres of the exscribed circles of a triangle lies on the right line joining the centres of the inscribed and circumscribed circles.

10. If ABC be any triangle, and A'B'C' its polar with regard to a circle whose centre is at O, prove the following formula:

(BOC) (COA) (A0B) (B'OC') (C'OA') (A'OB')

where (PQR) signifies the area of the triangle whose vertices are P, Q, R.

MR. PANTON.

11. If A, B, C be the centres of three coaxal circles, and t1, t2, tз the three tangents to them from any point, prove the relation

BC.t2+ CA. t22 + AB. t32 = 0.

12. A right line is drawn across two given circles cutting the first in A, B, and the second in A', B', so that the ratio of the chord AB to A'B' is constant; prove that the locus of intersection of tangents at A and B' is a circle coaxal with the given circles.

13. Given of a triangle the base and ratio of the area to the sum of squares of sides, find the locus of the vertex.

14. A quadrilateral ABCD is inscribed in a circle; through the middle points of a pair of opposite sides AB and CD, and through O, the intersection of diagonals, a circle is described; prove that this circle is touched by the line joining O to the intersection of AD and BC.

15. Construct a triangle, given base, vertical angle, and ratio of sum of sides to perpendicular.

4. The radius of a circle inscribed in a triangle is 12 feet, and the distance from its centre to that of the circumscribed circle 5 feet; what is the radius of the latter? Prove the formula employed.

5. If the total number of oxen in America be m, and in Europe n, the price here being double what it is there, how many must be imported to equalise the prices, assuming that the total money value on each side is unaltered? Determine also the ratio of the new price to the old.

MR. BURNSIDE.

6.. Determine values of x, y satisfying the simultaneous equations

x2 + y2 = 3xy,

x3 + y3 = 2a3.

9. Prove that every

forms 5m, 5m ± 1,

10. If 6 white and 5

VA + VB + VT = 0.

number which is a perfect square is of one of the where m is a positive integer or zero.

black balls be thrown at random into a bag, what is the chance of drawing a white ball, and then two black ones?

MR. PANTON.

11. If a, b, c are any three unequal numbers, prove that the fraction

is less than unity.

bc + ca + ab

a2 + b2 + c2

12. Prove that the term independent of x in the expansion of

15. A capitalist borrows £5000 in order to complete a sum of £19,000 he is about to lend; he gains in one year's interest £795. He borrows in another transaction, at a rate lower by one per cent. than in the previous case, a sum of £3500 to complete a loan of £14000; he gains in this case £630, lending at the same rate as before. At what rates does

he borrow and lend?

Classics.

DEMOSTHENES.

PROFESSOR BRADY.

:

Translate, adding very short notes where necessary :

1. Beginning, οὗτος τὰ μὲν ἄλλα ὡς ὑβρίζετο καὶ, κ. τ. λ. Ending, τὸν δ ̓ ἐπιτήδειον εἶναι ταῦτα παθεῖν ἔφη καὶ ἐπέχαιρεν. Phil. iii. p. 126 (ed. Reiske).

2. Beginning, εὖ γὰρ εἰδέναι χρὴ τοῦθ ̓, ὅτι οὐ καταφρονεῖ, κ. τ. λ. Ending, ἀποτετυμπανισμένους, ἐποίησεν ἂν ταὐτὸ τῷ βασιλεῖ. De Falsa Leg., pp. 382-3.

3. Beginning, εἶτα ὑπὲρ μὲν συγγενῶν καὶ ἀναγκαίων, κ. τ. λ. Ending, τούτους οὐ κρίνεις, ἀλλὰ καὶ σώζειν κελεύεις;

Ioid., pp. 434-5.

4. Beginning, Ταύτης τῆς ἀρᾶς καὶ τῶν ὅρκων καὶ τῆς μαντείας, κ. τ. λ. Ending, τούτων ἑκάστους ἀνιάτοις κακοῖς περιβάλλειν.

I. Translate and explain :

:

Esch. in Ctes., pp. 504-5.

(α). ὡς θαυμάσι ̓ ἡλίκα πεισόμενοι διὰ τούτους ἀγαθά.

(6). εἰ γὰρ αὖ ταῦτ ̓ ἐρεῖ, σκοπεῖτ ̓, ὦ ἄνδρες δικασταί, εἰ ἐφ ̓ οἷς ὁ μηδ ̓ ὁτιοῦν ἀδικῶν ἐφοβούμην ἐγὼ μὴ διὰ τούτους ἀπόλωμαι, τί τούτους προσήκει παθεῖν τοὺς αὐτοὺς ἠδικηκότας ;

(c). τοὺς καιροὺς ἐφ ̓ ὧν ἕκαστα ἐξηπάτησθε ὑπομνήσω, ἵν ̓ εἰδῆθ ̓ ὅτι τὸ ψυχρὸν τοῦτο ὄνομα, τὸ ἄχρι κόρου, παρελήλυθ ̓ ἐκεῖνος φενακίζων ὑμᾶς.

(α). καρπουμένη τὰς τῶν χρωμένων οἰκίας. What objection is there to οἰκίας ?

2. In what meanings are the following words found in the De Falsa Leg.—σπαθᾶν, διοικίζειν, κίχρημι, παρατρέφειν, χορηγός, ἀναβεβλημένος, τυρεύειν, στρατηγιᾶν, ἀνήκοος, καθυποκρίνεσθαι ?

3. What are the principal reasons in favour of arranging the Olynthiac Speeches, I. Ἐπὶ πολλῶν, κ.τ.λ. ; ΙΙ. Αντὶ πολλῶν, κ.τ.λ. ; III. Οὐχὶ ταὐτὰ παρίσταται, κ.τ.λ. ?

4. τὸ θρυλούμενόν ποτε ἀπόρρητον (Olynth. ii.) this transaction.

Give an account of

5. Μαγνησίαν κεκωλύκασι τειχίζειν (Olynth. i.) Where was this town, and how may the value set upon it by the Macedonian kings be shown?

6. τὴν μὲν τοίνυν εἰρήνην . . . εὕροντο παρ' ὑμῶν ἄνευ Φωκέων (Fals. Leg.) How was this brought about? Point out the great change in Greek political relations caused by the peace of Philocrates.