1. If the four sides of a quadrilateral be bisected, the lines joining the points of bisection form a parallelogram whose area is half that of the quadrilateral?

2. Given of a triangle the base, one base angle, and the sum of the sides: construct it.

3. Bisect a triangle by a line drawn from a given point in the base. 4. Calculate the value of

(a + b + c)3 + (b + c − a)3 + (c + a − b)3 + (a + b − c)3.

5. If a men or b women can do a piece of work in c days, how long will it take a men and B women?

6. Find the least common multiple of

x2 − a2, x2 + a2, (x − a)2, (x + a)2, x3 − a3, and x3 + a3.

MR. BURNSIDE.

7. Find a point in an indefinite right line such that the lines joining it to two given points on the same side of the right line shall be equally inclined to it.

8. Inscribe a square in a triangle.

9. Prove that the perpendiculars drawn from the vertices of a triangle to the opposite sides meet in a point.

13. The sides of a triangle are 3.3, 4.4, 5.5; find the length of the perpendicular from the intersection of the first two on the last.

14. Determine the amount by which the square of the side subtending the obtuse angle of a triangle exceeds the sum of the squares of the sides containing it.

15. Prove that the lines drawn from the angles of a triangle to bisect the opposite sides meet in a point.

1. Beginning, Quocunque adspicio, nihil est nisi pontus et aether; .... Ending, nescit agi ventis: nescit adesse necem.

OVID, Trist., lib. i. el. 2.

2. Beginning, Post mortem autem Crassi eo mihi etiam...... Ending, etiam per alios eum videbam id consequi posse.

CICERO, Ep. ad Diversos, lib. xiii. ep. 16.

MR. PALMER.

Translate the following passage into Greek Prose :

But they revolted with indignation against the idea of complying with laws by which they were to be stripped at once of all they had earned so hardly during many years of service and suffering. As the account of the new laws spread successively through the different settlements, the inhabitants ran together, the women in tears, and the men exclaiming against the injustice and ingratitude of their sovereign in depriving them, unheard and unconvicted, of their possessions.

:

Translate the following passages into English :

I. Beginning, ΧΟ. καὶ μὴν ὅδ ̓ ἀλλόχρως τις ἔκδημος ξένος, κ. τ. λ.
Ending, δόμων ἄνασσαν τήνδε Μενέλεω κόρην ;

EURIPIDES, Androm., 878-895.

2. Beginning, Μαθὼν δέ μιν ̓Αρτάβανος ὁ πάτρως, κ. τ. λ. Ending, φθονερὸς ἐν αὐτῷ εὑρίσκεται ἐών.

HERODOTUS, lib. vii. c. 47.

GREEK AND LATIN GRAMMAR AND HISTORY.

PROFESSOR BRADY.

1. Decline (in singular only) гpaûs and Aeλuкús.

2. Give the 1st person future of—Πλέω, Βοάω, Ἐλαύνω, Πυνθάνομαι, Αφίημι.

3. Distinguish in meaning-Φοβέω, Φοβοῦμαι; ̓Αποδοῦναι, ̓Αποδόσθαι; Παῦσαι, Παύσασθαι; Αμύνω, ̓Αμύνεσθαι.

4. Explain the difference between Ou and M.

5. Give some account of Peisistratus or Epaminondas.

6. Mention the principal enactments of Draco or Cleisthenes.

7. What was the duration and what were the causes of the overthrow of the Spartan and Theban Hegemonies, respectively?

8. Decline in combination (in singular only) Alia species.

9. Give the 1st person pluperfect active of-Tondeo, Luceo, Pango, Reperio, Uro.

10. Mention the different meanings and constructions of Quin.

II. Write a short account of the Carthaginian Constitution.

12. Give the laws proposed by Caius Gracchus.

13. Mention the dates and results of the following battles:-Mycale, Mantinea, Granicus, Lake Regillus, Sentinum, Actium.

14. Name the most famous Roman Epic, Elegiac, and Dramatic Poets.

man.

LATIN COMPOSITION.

MR. TYRRELL.

All nations and cities are ruled by the people, the nobility, or by one A constitution, formed by selection out of these elements, it is easy to commend but not to produce; or, if it is produced, it cannot be lasting. Formerly, when the people had power, or when the patricians were in the ascendant, the popular temper and the methods of controlling it had to be studied, and those who knew most accurately the spirit of the Senate and aristocracy had the credit of understanding the age and of being wise men. So now, after a revolution, when Rome is nothing but the realm of a single despot, there must be good in carefully noting and recording this period, for it is but few who have the foresight to distinguish right from wrong, or what is sound from what is hurtful, while most men learn wisdom from the fortunes of others. Still, though this is instructive, it gives very little pleasure. Descriptions of coun

tries, the various incidents of battles, glorious deaths of great generals, enchain and refresh a reader's mind. I have to present in succession the merciless biddings of a tyrant, incessant prosecutions, faithless friendships, the ruin of innocence, the same causes issuing in the same results, and I am everywhere confronted by a wearisome monotony in my subject-matter.

ENGLISH COMPOSITION.

MR. MAHAFFY.

Write a short Essay on one of the following subjects :—

I. A comparison of the grammars of the Latin and English languages.

2. On the difficulty of translating the idioms of any language (with examples).

3. The theatre and dramatic performances of the Greeks.

4. Ireland in 1798.

SCIENCE SIZARSHIP EXAMINATION.

MR.

GEOMETRY.

PANTON.

1. Find, in a given arc AB of a circle, a point the sum of whose distances from the extremities A and B shall be a maximum.

(a). Find a point such that the distance intercepted on the tangent at that point by the tangents at A and B shall be a minimum.

(b). Inscribe in the given segment a rectangle of maximum area.

2. If A, B, C, be the three vertices of any triangle, and O their mean centre for any three multiples a, b, c,; prove the relation

area BOC: area COA: area AOB a: b: c.

=

(a). A tangent is drawn to the circle inscribed in the triangle ABC, and perpendiculars AP, BQ, CR, are let fall on it; prove that, for all positions of the tangent, the sum of the three rectangles

BC. AP+CA . BQ + AB. CR

is constant, and equal to double the area of the triangle.

3. If two triangles are polar reciprocals with respect to a circle, prove that the three points of intersection of corresponding sides are in one right line, and the lines joining corresponding angles meet in one point. 4. A variable line is cut harmonically by two given circles; find the locus of the middle point of the chord intercepted on it by either circle.

(a). Prove that the rectangle under the perpendiculars let fall on it from the centres of the two circles is constant.

5. If the chords intercepted on a variable line by two fixed circles are in a constant ratio, prove that the four tangents at the points of section form a quadrilateral inscribable in a circle coaxal with the given circles.

6. If two circles have each contact with two others, prove that the radical axis of the former pair passes through the external centre of similitude of the latter pair when the contacts are of the same kind, and through the internal centre of similitude when the contacts are of different kinds.

(a). Hence, or otherwise, prove that the locus of the centre of a circle touching two fixed circles is a curve of such a nature that any variable point on it subtends four fixed points on it in a constant anharmonic ratio. 7. Describe a circle cutting three given arcs of three given circles harmonically.

8. Prove that the "nine-point circles" of any four self-conjugate triangles with respect to the same circle have the same radical centre, and that their six radical axes form a pencil in involution.

9. Show that the problem to inscribe in a given circle a triangle whose sides shall touch another given circle is either indeterminate or impossible.

(a). Find, when the problem is possible, the relation connecting the radii of the circles with the distance between their centres.

(b). Find, in the same case, the locus of the intersection of perpendiculars of the triangle satisfying the required conditions.

10. The inscribed and escribed circles of a triangle ABC are 0, 0′, 0′′, 0""; and the middle points of the sides are A', B', C'; prove

(a). That the six radical axes of 0, 0', 0′′, 0′′′′ intersect in pairs at A', B', C', cutting one another at right angles.

(b). That the pair intersecting at the middle point of any side intercept, on either of the remaining two sides, a distance equal to the other of those sides.

(c). That the four radical centres of 0, 0', O", O"" are the centres of the inscribed and escribed circles of the triangle A'B'C'.

11. Apply the method of inversion to the following problem :

A chord of a given circle passes through a given point, and on its segments circles are described touching the given circle; find the locus of their intersection.

12. Given a triangle, find the locus of a point such that, letting fall perpendiculars on the sides and joining their feet, the area of the triangle thus formed shall be given.