the perpendiculars on two opposite sides is equal to the rectangle under the other two perpendiculars.

14. Show how to draw a right line passing through the point of intersection of two circles, so that the portion of the line intercepted by the circles be a maximum.

15. The middle points of the diagonals of a complete quadrilateral lie in a right line.

16. The distances of any two points from the centre of a circle have the same ratio as the distances of each, from the polar of the other, with respect to the circle.

17. If A, B, C be three points on a line, and AP, BQ, CR the three lines connecting them with any point P, prove that

BC. AP2+ CA. BP2 + AB. CP2

is constant for all finite points; regard being had to the signs of the three segments involved.

18. Describe a circle bisecting the circumferences of three given circles.

C.

MR. WILLIAMSON.

1. The sides of a triangle are 25, 39, 56; calculate the lengths of the radii of its inscribed and circumscribed circles.

+

y

x2+yz―xy-xz y2+xz- -xy-zy z2 + xy-xz - yz

4. If £300 be laid out at simple interest for a certain number of years, it will amount to £360. If the same sum be allowed to remain two years longer, and at a rate of interest 1 per cent. higher, it will amount to £405. Required the rate of interest and the number of years in the former case.

5. Solve the simultaneous equations,

x2+ y2 = 2482, + √/y = 10.

7. Find by the method of indeterminate coefficients the sum of n terms of the series

1.32 +2.42+3 · 52 + 4.62 + &c.

8. If r1 r2 r3 be the radii of three circles which touch, prove that the area of the triangle formed by the lines joining their centres is

9.

= √ r1 r2 r3 (r1 + r2 + r3)·

A steamer sails, in 8 hours, 40 miles against the current of a river, and 48 miles with it; and on another occasion she sails, in 13 hours, 56 miles against the current, and 96 with it. Calculate the velocity of the stream, and the rate of the steamer in still water.

10. A number consists of two digits, the difference of whose squares is 40, and if it be multiplied by the number consisting of the same digits taken in reversed order, the product will be 2701. Find the number.

MR. BURNSIDE.

11. Solve the simultaneous equations,

x (y + z) = a,

y (z + x) = b,

z (x + y) = c.

12. Prove that (Byd + yda + daß + aßy)2 – aßyd (a + B + y + d)2

=(By ad) (ya – ẞd) (aß — yd).

-

13. The arithmetical mean of two numbers exceeds the geometrical mean by 13, and the geometrical mean exceeds the narmonical mean by 12; determine the numbers.

14. Given the area m2 of a right-angled triangle whose sides are in continued proportion; determine the sides.

15. Given the lengths of the three lines drawn from the vertices of a triangle bisecting the sides; determine the sides.

D.

MR. WILLIAMSON.

1. Given of a triangle its circumscribed circle in position, the centre of its inscribed circle, and the length of one of its sides; construct the triangle.

2. Inscribe a square in a given quadrilateral; and find when the problem is indeterminate.

3. Find the locus of a point, the sum of the squares of whose distances from the sides of a regular polygon is given.

4. Given the four sides of a quadrilateral, and its area; construct it. Find also when its area is a maximum.

5. If d1, d2, d, da represent the distances of the centre of the circumscribed circle of a triangle, from the centres of its inscribed and exscribed circles, prove that

d12 + d22 + d22 + d42 = 12 R2,

where R is the radius of the circumscribed circle.

6. The sides of a triangle are in arithmetical progression; prove that ac = = 6 Rr,

where a and c are the greatest and least sides, and R, r the radii of the circumscribed and inscribed circles.

DR. TRAILL.

7. If on the bisector of the vertical angle of a triangle a point be taken, such that the angles made with the base by lines joining this point to the extremities of the base shall have a maximum difference, prove that their sum is in such case equa to half the vertical angle.

8. Four right lines drawn at random form four triangles; prove that the circumscribing circles of these triangles intersect in a point, and that their centres lie on another circle passing through the same point.

9. If A, B, C be the centres of three coaxal circles, whose radii are ri, r2, rs, and the distances between their centres BC= 1; AC=X2; AB = 3; and if the tangents to these circles from any point be ti, ta, ts, prove the two following criteria of coaxality:

10. Prove that the circle which passes through the centres of the inscribed, and of two of the escribed circles of a triangle, is equal to the circle which passes through the centres of the three escribed circles, and that each of them is equal to four times the circumscribing circle of the triangle.

11. Construct a quadrilateral of given species, whose sides shall pass through four given points.

12. The rectangles under the distances of any pair of inverse points with regard to a circle, from any other pair on the same diameter, are as their distances from the centre of the circle.

MR. BURNSIDE.

13. Construct a triangle of given species, so that its vertices may rest on three concentric circles.

14. If a line be drawn from the vertex of a triangle ABC to the base, dividing it into two triangles ACD and BCD, denoting by rr'" the radii of the circles inscribed in these three triangles, and by p the perpendicular from the vertex on the base; prove the relation

p (r' + r'' − r) = 2r' r''.

15. A circle intersects the sides of a triangle ABC in the points A'A", B'B', C'C"; if any three of the six lines 44' AA", BB' BB", CC CC are concurrent, prove that the remaining three are concurrent.

e

16. The intersections of the perpendiculars of the four triangles formed by four lines lie in a right line.

17. The polars of the middle points of the sides of a triangle, with respect to the inscribed circle, determine a triangle equal in area to the original.

18. If A, B, C, D be any four points on a circle, and O any fifth point taken arbitrarily, prove the relation

OAa. BCD + OC2. ABD = OB2. ACD + OD2. ABC, where BCD denotes the area of the triangle BCD, &c., &c.

Classics.

DEMOSTHENES.

MR. PALMER.

Translate the following passages, adding notes:-

1. Beginning, Ὃν τοίνυν χρόνου ἦμεν ἐκεῖ καὶ καθήμεθ', κ. τ. λ. Ending, Παναθήναια φήσας ἀποπέμψειν.

De Falsa Legat.

2. Beginning, Τοιούτοις μέντοι λόγοις ὦ κακὴ κεφαλή, κ. τ. λ. Ending, ὁ τέως προσκυνῶν τὴν θόλον.

Ibid.

3. Beginning, Καὶ γὰρ νῦν, ὅτε εἰς Ταμύνας παρῆλθον οἱ ἄλλοι, κ.τ.λ. Ending, δόξης καὶ τῶν ἔργων εἶναι..

Orat.

1. Write illustrative and explanatory notes on the following:

α. διὰ ταῦτ ἐσπαθᾶτο ταῦτα καὶ διὰ ταῦτ ̓ ἐδημηγορεῖτο.

6. διῳκισμένοι κατὰ κώμας.

c. φιλοτησίας προυπινεν.

α. σὺν αἷς ἐπορεύθημεν ὁμοῦ πεντήκονθ' ὅλας.

ε. ἀδημονούσης δὲ τῆς ἀνθρώπου.

†. ὑποκορίζεσθαι.

g. ἵν' εἰδῆθ ̓ ὅτι τὸ ψυχρὸν τοῦτο ὄνομα τὸ ἄγχρι κόρου παρελήλυθ ἐκεῖνος φενακίζων ὑμᾶς.

2. Explain the following legal terms:

εισαγγελία, προβολή, ἐκμαρτυρία διαιτήτης, ἐρήμην ἀντιλαχεῖν, ἐξουλης δίκη, παρακαταβολή.

3. On what occasion was Demosthenes' first public oration delivered?

4. Give a short account of Phocion.

5. What was the classification by symmories, and when was it instituted?

6. State Mr. Grote's views respecting the theoric fund.

The orator Demades is said to have made use of a remarkable expression concerning it?

7. Give a short chronological account of the sacred war.

8. Mark on a map the positions of Olynthus, Amphipolis, Potidæa, Pydna, Pagasa, and Methone.

CICERO.

MR. MAHAFFY.

Translate the following passages :

1. Beginning, Si vero etiam vitiosi aliquid est,.

Ending, Pericles atque Alcibiades et eadem ætate Thucydides.

2. Beginning, Ita fit, ut unum genus in eis causis,
Ending, quod totum a facti controversis separatum est.

3. Beginning, Arguta etiam significatio est,
Ending, quod Asellum ignominia levarit.

4. Beginning, In quo admirari soleo non equidem istos,. Ending, lenioribus principiis natura ipsa prætexuit.

5. Beginning, In eo autem ipso dominatus est omnis oculorum : Ending, aures ad motus animorum declarandos dedit.

1. Give the substance of the criticism which immediately follows the first of the above passages.

2. Re-write the following passage in direct narration :

Dixit: "Quæ natura aut fortuna darentur hominibus, in eis rebus sc vinci posse animo æquo pati; quæ ipsi sibi homines parare possent, in eis rebus se pati non posse vinci."

3. Distinguish the various terms for wit and humour used by Cicero. 4. Sketch the life of Antonius the orator.

5. State the controversy as to the year of Cæsar's birth.

6. What should the policy of the Romans in the East have been? How far did they deviate from it?

7. Give a brief sketch of the political attitude at Rome during the absence of Pompeius in the East.

8. How does Mommsen explain the treatment of Cicero and Cato in the year 58 B. C.?