MR. BURNSIDE.

13. The circle described on the axis major of an hyperbola as diameter is its polar reciprocal.

14. The four perpendiculars let fall from the intersection of two tangents to an ellipse, on the focal radii vectores to the point of contact of the tangents are equal.

17. Prove the following formula in terms of the sides and angles of a plane triangle

a2 cos? A = b2 cos? B+ c2 cos2 C + 2bc cos B cos C. Cos 2A. 18. Prove the following formula for the area of a spherical triangle

1. Apply the algebraic method of investigating maxima and minima to find the max. and min. values of the fraction

4. Given five points on a conic, show how to construct a pair of its conjugate diameters, in magnitude and position.

5. Given a pair of conjugate diameters of an ellipse in magnitude and position; show how its axes major and minor can be found.

Prove that the equation of the squares of the differences of the roots of the cubic is

ay3 + 18α ̧1Hy2 + 81a02 H2y + 27 ( G2 + 4H3) = 0.

8. Prove the following properties of numbers

9. If the roots of a biquadratic when deprived of the second term be

I

I

n + I

n+ 3

2n

10. If from two points diametrically opposite on the circle circumscribing a given triangle, perpendiculars be let fall on the sides of the triangle, prove

that the two lines on which the feet of these perpendiculars lie, intersect at right angles on the nine point circle of the triangle.

11. If P and Q be a pair of inverse points with respect to a circle, on a diameter whose extremities are A and B, and if O be the middle point of PQ, and N the foot of the perpendicular from any point X on the curve on the line PAQB, prove that

12. Prove that when its

sphere, a spherical triangle

sides are small compared with the radius of the may be treated as a plane triangle having the same sides, and each of its angles less, by one-third of the spherical excess than the corresponding angle of the given triangle.

MR. BURNSIDE.

13. The locus of the centre of a rectangular hyperbola circumscribing a triangle is the nine point circle.

14. From the intersection of two tangents to an ellipse, a line is drawn perpendicular to the chord of contact, denoting by n and n' the intercepts made on it by the axes, and by the length of the perpendicular from the centre on the chord; prove that

nw= a2

n'w = b2.

15. Deuoting by P1, P2, P3, P4 the perpendicular arcs drawn from any point of a small circle on the sides of an inscribed quadrilateral; prove that

where a1, a2, aз, a are the sides of the quadrilateral in order.

ax +4bx3+6cx2 + 4dx + e = o should have a double root; and prove that the functions ae - 4bd3c2, ace + 2bcd — aď2 — eb2 — c3

vanish if this equation has a triple root.

1. Beginning, οἱ δ ̓ ἱπποβόται ἐκαλέοντο οἱ παχέες. κ. τ. λ. Ending, Τῶν ἵππους δεκάτην Παλλάδι τάσδ ̓ ἔθεσαν. HERODOTUS, lib. v. cap. 77.

2. Beginning, οἱ δὲ Πάριοι, ὅκως μέν τι δώσουσι Μιλτιάδη, κ. τ. λ. Ending, καταθρώσκοντα δὲ τὴν αἱμασιὴν τὸν μηρὸν σπασθῆναι. Ibid., lib. vi. cap. 133.

3. Beginning, ἅτε γὰρ τοῦ αἰγιαλοῦ ἐόντος οὐ μεγάλου, κ. τ. λ. Ending, αἱ δὲ ἐς Κασθαναίην ἐξεβράσσοντο.

Ibid., lib. vii. cap. 188.

4. Beginning, Απὸ δὲ τῆς Εφέσου ὁ ̓Αλκίδας ἔπλει κατὰ τάχος, κ.τ.λ. Ending, καὶ φυλακὴν σφίσι καὶ ἐφόρμησιν παρασχεῖν.

THUCYDIDES, lib. iii. cap. 33.

1. Discuss the passage in which Herodotus refers to his critics.

2. Explain καὶ παρίζειν βουλεύουσι τοῖσι γέρουσι, ἐοῦσι δυῶν δέουσι τριήκοντα· ἣν δὲ μὴ ἔλθωσι, τοὺς μάλιστάσφι τῶν γερόντων προσήκοντας ἔχειν τὰ τῶν βασιλήων γέρεα, δύο ψήφους τιθεμένους, τρίτην δὲ τὴν ἑωυτῶν.—HERODOTUS.

3. What are Thucydides' criticisms on Herodotus ?

4. In what points is Herodotus superior to Thucydides ?

5. On what unpalatable truths does Herodotus insist, as regards the conduct of Athens in the Medic wars?

6. What are his chief points of difference from Ctesias ?

7. Write a short note on Greek historiography previous to Herodotus.

LIVY.

MR. GRAY.

Translate the following passages:

I. Beginning, Et prudentiam quidem novi ductoris extemplo timuit: .... Ending, successibus crevit, sese extollebat.

Lib. xxii. cap. 12.

Ending, Septunces auri in singulos pretium convenit.

Lib. xxiii. cap. 19.

2. Beginning, Ea ægre patiens Gracchus, .

3. Beginning, Secundum eos citati nimis callidi...... Ending, a quæstore perscribebatur.

4. Beginning, Vallo et Macedones et Græci..... Ending, et alium reponere perfacile est..

Lib. xxiv. cap. 18.

Lib. xxxiii. cap. 5.

1. What settlements constituted the class of so-called Libz-Phoenicians? Describe their political position.

2. Compare the respective resources of Carthage and Rome at the commencement of the Punic Wars.

3. How does Mommsen account for the fact that the Romans managed the first Punic war so wretchedly and with such vacillation, both in a military and in a political point of view?

4. Sketch the history and projects of Hamilcar.

5. On what occasion, and with what purpose, did Hannibal march nearest to Rome ?

6. Sketch the career of Massinissa.

7. How does Mommsen describe the result of the Licinian laws? 8. Write a short acnount of Caius Flaminius.

MR. TYRRELL

GREEK PROSE.

When all things were dispatched, the emperor sent for this man, and demanded who he was, whence he came, and of what lineage?-His answer was as follows:-I am a genuine Frank, and in the number of their nobility. One thing I know, which is, that in a certain part of the country I came from, and in a place where three ways meet, there stands an ancient church, where every one who has a desire to engage in single combat, having put himself into fighting order, comes, and there implores the assistance of the Deity, and then waits in expectation of some one that will dare attack him. On this spot I myself waited a long time, expecting and seeking some one that would arrive and fight me. But the man, that would dare this, was nowhere to be found.

The emperor, having heard this strange narrative, replied pleasantlyIf at the time when you sought war, you could not find it, a season is now coming in which you will find wars enough. I therefore give you this advice; not to place yourself either in the rear of the army, or in the front, but to keep among those who support the centre; for I have long had knowledge of the Turkish method in their wars.

LATIN PROSE.

Mucian's speech in Tacitus contains many important matters in a small compass; and in a few clear and emphatical words goes through