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4. Form the equation whose roots are the products in pairs of the roots of a biquadratic equation.

5. If a, ß, y, d be the roots of the equation

(a, b, c, d, e) (x, 1)4 = 0,


express (a-B). (y-5)2 in the form A + where a is a root of the

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equation (ax)3 − I (ax) + 2J = 0, and A, B functions of the coefficients.

6. Eliminate x, y, z from the equations

x + y + z = a,

x2 + y2+ z2 = b,

x3 + y3 + z3 = c,

x1+ y1 + z1 = d.

7. Express a determinant of the fourth order in terms of its minors of the second order.

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I. Find the sum of sin3 x + sin3 2x + sin3 3x +


+ sin3 nx.

2. Given the tangent of an angle, the tangent of its nth part can be

found by an equation of the nth degree.

Show how to reduce this to a binomial equation.

3. Eliminate x and y between

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5. Prove that if x be determined by the equation

a sin x + bi cos x = c1,

it satisfies relations of the form an sin nx + bn cos nx = cn, and determine for n = 3 the values of a3 : b3: c3 in terms of a1: b1: c1.

6. Deduce the expression for the bisector of an angle of a spherical triangle

2 sin ẞ sin y cos

tan λ =

sin (8 + y)

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8. If in a spherical triangle the side AB be produced equal to itself to A1, and AC equal to itself to A2 below the side BC, and the same be done for the other vertices, show that

cos 41 42 cos 2BC= cos B1 B2 - cos 2CA =

cos C1 C2


9. In a plane triangle the base is 2063, 6, and the base angles 106° 43' 16" and 23′ 52": solve the triangle.

10. Solve the cubic x3+8x=51 by the trigonometrical tables.

II. Solve the spherical triangle whose sides are 25° 13′ 12′′, 37° 14' 9", 58° 31′ 51′′.

12. The angle between two objects a = 52° 31' 30", one of them is 40° 12' 10" above the horizon, and the other 31° 27' 20"; reduce a to the horizon.

13. Calculate the spherical excess of an equilateral triangle each side of which is 10°, and find the difference between the area of such a triangle on a sphere of 4000 miles radius and the plane equilateral having sides of equal length.

14. Calculate the modulus of the system of logarithms whose base is 12, to five places of decimals.





Translate the following passages into English prose :—

1. Beginning, Ὡς δ ̓ ὅτε τίς τε δράκοντα ἰδὼν παλίνορσος ἀπέστη κ. τ. λ. Ending, Εἶδος ἔπ' ἀλλ ̓ οὐκ ἔστι βίη φρεσὶν, οὐδέ τις ἀλκή.

HOMER, Iliad, iii. 33-45.

2. Beginning, Ζεὺς δὲ πατὴρ Αἴανθ ̓ ὑψίζυγος ἐν φόβον ὦρσεν, κ. τ. λ. Ending, "Ηϊε, πόλλ' ἀέκων· περὶ γὰρ δίε νηυσὶν Ἀχαιῶν.

Ibid., xi. 544-557.

3. Beginning, ΕΚ. οὐδεὶς ̓Αχαιῶν ἔνδον, ἀλλ' ἡμεῖς μόναι, κ. τ. λ. Ending, ἀπολέμῳ δὲ χειρὶ λείψεις βίον.

4. Beginning, μὰ τὸν μετ ̓ ἄστρων Ζῆν” ̓́Αρη τε φοίνιον, κ. τ. λ. Ending, πειρώμεναι τὸ λοιπὸν εὐτυχοῖεν ἄν.

EURIP., Hecuba, 1018-1034.

Ibid., Phœuissa, 1oo6-1018.

Ibid., Orestes, 1426-1443.

5. Beginning, Φρυγίοις ἔτυχον Φρυγίοισι νόμοις, κ. τ. λ. Ending, πετ', οὐ πρόμαντις ὧν ἔμελλεν.

6. Beginning, τί δὴ ταῦτα νῦν λέγω, καὶ καλεῖν φημὶ, κ. τ. λ. Ending, ὀργίλους καὶ τραχεῖς ὑμᾶς ἔσεσθαι νομίζω.

DEMOS., Phil., iii. p. 73.

7. Beginning, ἔπειτα σὺ τοιαῦτα ποιῶν λέγεις, κ. τ. λ. Ending, ἐνταῦθα ἔλαττον ἔχων δίκαιός ἐστιν ἀπελθεῖν.

Ibid., De Cor., p. 268.

8. Beginning, Οἱ δὲ Καρδοῦχοι ὡς ἑώρων τοὺς ὀπισθοφύλακας, κ. τ. λ. Ending, ὡς μὴ ἐμποδίζειν ἀλλήλους.

9. Beginning, Επεὶ δέ, κραυγῆς εἰς τὴν πόλιν Ending, μάχῃ εἰς ἔλαττον συνειλοῦντο.

XEN., Anab., lib. iv. c. 3. ἀφικομένης, κ. τ. λ.

Ibid., Hellen., lib. vii. c. 2.



1. Beginning, MI. Ausculta paucis, nisi molestumst, Demea....... Ending, Non quo dissimilis res sit, sed quo is qui facit.

TERENCE, Adelphi, 806–825.

2. Beginning, Quas ob res ita censeo: . . . Cum talis vir
Ending, uti quod optimo iure publice sepulchrum datum esset.

CICERO, Orat. Philipp., XI. c. vii. ss. 15-17.

3. Beginning, Primus equi labor est, animos atque arma videre.... Ending, Aequora, vix summa vestigia ponat arena.

VIRGIL, Georg., lib. iii. 182-195.

4. Beginning, Obsequio grassare; mone, si increbuit aura,. ..... Ending, Imperiosa trahit Proserpina; vive valeque.

HORACE, Sat., II. V. 93-110.

5. (a). Beginning, Diu non perlitatum tenuerat dictatorem, Ending, aliquot equites Romani cecidere.

(b). Beginning, Huic infensus crudelitati Iunius, Ending, benigneque omnis militi concessa.


LIVY, lib. vii. c. 8.


Ibid., lib. ix. c. 31.



1. Give a list of Greek feminine substantives in the O declension.

2. Decline νεὼς and ναῦς.

3. Write out the present optative of spáw, and the indicative oúvoida. 4. Illustrate by short phrases the meanings of rapà, with genitive, dative, and accusative.

5. Translate into Greek :

(a). The Spartans are come to (va) make the Thebans cease their insolence.

(b). If he had happened to have had a mina he would have given it. (c). Although he is an old man he shall be punished.

(d). The king said he would not be like the Athenians.

6. Give (a) the gender of―ordo, Ŏs, grex, tribus, caro, cinis, tellus. (b) the pluperfect indicative (1st person) of-cogo, comperio, gigno, urgeo, pasco, verto, audeo, faveo, fundo, lino.

7. Explain with examples the meaning of the suffixes -ina, -etum, -ile, -osus.

8. Give examples of Latin inceptive, frequentative, desiderative, and diminutive verbs.

9. Notice grammatical peculiarities in

(a). Sunt quos curriculo pulverem Olympico collegisse iuvat.
(¿). Vir bonus et sapiens dignis ait esse paratus.

(c). Vix equidem credor.

(d). Pecus egit altos visere montes.

10. Translate into Latin :

(a). Xerxes ordered the bridge to be destroyed which he had built over the Hellespont.

(b). He said that we do not live to eat, but eat to live.

(c). Having left Rome late at night he feared he would not reach Aricia before morning.


In these circumstances the Spartan commander ran his vessels ashore, where, with the assistance of Pharnabazus, he endeavoured to defend them against the attacks of the Athenians. Alcibiades having landed his men, a battle ensued, in which Mindarus was slain, the Lacedæmonians and Persians routed, and the whole Peloponnesian fleet captured, with the exception of the Syracusan ships, which Hermocrates caused to be burnt. The severity of this blow was pictured in the laconic epistle in which Hippocrates, the second in command, announced it to the Ephors: "Our good luck is gone; Mindarus is slain; the men are starving; we know not what to do."



The barbarity of the conquerors stained the glory which they acquired by this complete victory. The violence of civil rage hurried on some to slaughter their countrymen with indiscriminate cruelty; the meanness of private revenge instigated others to single out individuals as the objects of their vengeance. Orgognez, and several officers of distinction, were massacred in cold blood; above a hundred and forty soldiers fell in the field; a large proportion, where the number of combatants were few, and the heat of the contest soon over. Almagro, though so feeble that he could not bear the motion of a horse, had insisted on being carried in a litter to an eminence which overlooked the field of battle. From thence, in the utmost agitation of mind, he viewed the various movements of both parties, and at last beheld the total defeat of his own troops with all the passionate indignation of a veteran leader long accustomed to victory. He endeavoured to save himself by flight, but was taken prisoner, and guarded with the strictest vigilance.


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