14. Find, for the biquadratic ao x + 4α1x3 +6α2x2 + 4α3x + a1 = 0, . the value of the symmetric function Σ (a – B)2 (y – 8)2. 15. Prove that tangents at the extremities of a focal chord meet the latus rectum at points equidistant from the focus. 16. Given two chords of a parabola, find the direction of its axis. 17. Given base, and sum of cosines of sides of a spherical triangle, find the locus of its vertex. prove that the equation in y derived by means of the homographic transformation 2. Apply Sturm's method to determine the number of real and of imaginary roots of the equation 4. Prove that the three bisectors of the area of a spherical triangle drawn through the vertices pass through the same point. 5. If normals be drawn at the extremities of a focal chord of an ellipse, prove that the line through their intersection parallel to the major axis bisects the chord. 7. Prove that the rectangles under the segments of parallel chords of a conic through fixed points bear to one another a constant ratio. 8. Investigate the number of real and imaginary roots of the quintic ax5+5ex+ƒ= 0, and find the condition that two roots should be equal. 9. Show that if ABC be a spherical triangle, O the middle point of the arc of the circumscribing small circle opposite to A, the G. c. arc 40 will divide the base BC into segments whose sines are proportional to the sines of half the sides. 10. A system of conic sections have a given focus and directrix, show that the locus of the extremity of the axis minor is a parabola, and determine its focus and directrix. MR. W. R. ROBERTS. II. Show that the circle circumscribing a triangle self-conjugate with regard to an ellipse cuts the director circle orthogonally. 12. Prove, by reciprocation, that confocal conics cut orthogonally. 13. If H and J are both positive, prove that all the roots of the biquadratic are imaginary, where H = ao a2-a12, J = αoα2α4 + 2A1 A2 A3 -- α12 α4 - αo aз2 — a23. 14. Find the conditions that the biquadratic αox+4α1x3 +6а2x2 + 4а3x + α = 0 may have two distinct pairs of equal roots. 15. Given two sides of a spherical triangle, find when the area is a maximum. Classics. MR. GRAY. Translate: I. Beginning, Μετὰ δὲ ταῦτα θυσίῃσι μεγάλῃσι τὸν ἐν, κ.τ.λ. HERODOTUS, i. 50. 2. Beginning, Ὁ δὲ 'Αράξης λέγεται καὶ μέζων καὶ, κ τ. λ. Ending, τούτων μὲν αὕτη λέγεται δίαιτα εἶναι. Ibid., i. 202. 3. Beginning, Μαθόντες δὲ Λακεδαιμόνιοι Κλεομένεα ταῦτα, κ. τ. λ. Ending, τὰ περὶ Δημάρητον γενόμενα λέγειν. Ibid., vi. 75. Ibid., vii. 118-119. 4. Beginning, οἱ δὲ ὐποδεκόμενοι Ἑλλήνων τὴν στρατιὴν, κ. τ. λ. Ending, ὅσα ἐπὶ τράπεζαν τίθεται, πάντα. 1. Write a note on the following forms:—προηδέατο, ἐννώσας, ἐνένωτο, κεχωρίδαται, ἀναμάξας. 2. Translate επιψηφίζειν ἀπόλι ἀνδρί. Give the meaning of καταβαίνω as used by Herodotus in a metaphorical way. 3. State what you know about the Alcmæonidæ. 4. Give some account of the invasion of Scythia by Darius. 5. What rivers did Xerxes cross in his march from Sardis to Thermopylæ ? 6. What evidence have we to fix the date of the common use of writing in Greece? 7. State what you know of the life and literary position of Hecatæus. 8. Point out the bearing of the subjugation of Ionia on Greek prose literature. 9. What were the literary and other sources of information accessible to Herodotus ? 10. Who were the principal hostile critics of Herodotus? Give such particulars as you can. MR. L. C. PURSER. Translate the following passages into English: 1. Beginning, Literis recitatis, nemo omnium erat, quin et, Ending, privataque pecunia respublica administrata est. .... LIVY, xxiii. 48, 49. 2. Beginning, Quum censores ob inopiam aerarii se iam, Ending, ac viduis foret, a quaestore perscribebatur. .... Ibid., xxiv. 18. .... 3. Beginning, Graccho, priusquam ex Lucanis moveret, Ibid., xxv. 16. 4. Beginning, Marcellus, consilio advocato, et adhibitis etiam, Ending, quam capere Syracusas potuisse.' Ibid., xxv. 31. 5. Beginning, Inenarrabilis labor descendentibus cum ruina, Ending, parva manu deleri omnem exercitum potuisse. Ibid., xliv. 5. I. Translate and explain : (a) Purgato inde capite, ut mos est iis, calvam auro caelaverunt. (b) Minime sis cantherium in fossa. (c) Sitella lata est ut sortirentur tribus ac dein suffragium ferrent. 2. What is the meaning of-actuaria navis, concubia nox, hereditatem cernere, pileatus, platea, remulcus, rubus, seria, stips, stramentum, volones, vomica ? 3. Who were the Liby-Phoenicians? 4. Briefly institute a comparison between Rome and Carthage in their (1) economy; (2) constitution; (3) treatment of subjects; (4) finance; (5) military system. 5. Give some account of the organization of the first Roman provinces. 6. Summarize the results of the Second Punic War. 7. What was the original character of the Roman Senate? Show how was converted into a governing corporation. 8. What restrictions were there on the power of the Censors ? 9. Give an account of the remodelling of the Comitia Centuriata. 10. What was the policy of the demagogues? How far did they succeed? II. A modern author has described Livy as 'inhistoria orator'. Explain and criticize this remark. MR. MAHAFFY. Translate into Latin Prose : Nearly a century before this event, in obedience to the injunctions of the Delphian oracle, the Spartans had brought back from Tegea to Sparta the bones of the "Laconian Orestês," as Pindar denominates him: the recovery of these bones was announced to them as the means of reversing a course of ill-fortune, and of procuring victory in their war against Tegea. The value which they set upon this acquisition, and the decisive results ascribed to it, exhibit a precise analogy with the recovery of the bones of Thêseus from Skyros by the Athenian Kimôn shortly after the Persian invasion. The remains sought were those of a hero properly belonging to their own soil, but who had died in a foreign land, and of whose protection and assistance they were for that reason deprived. Translate into Graek Verse: Kleisthenês, despot of Sikyôn, wishing to banish the hero Adrastus and abolish the religious solemnities celebrated in honour of the latter by the Sikyonians, first applied to the Delphian oracle for permission to carry this banishment into effect directly and forcibly. That permission being refused, he next sent to Thêbes an intimation that he was anxious to introduce their hero Melanippus into Sikyôn. The Thêbans willingly consented, and he assigned to the new hero a consecrated spot in the strongest aud most commanding portion of the Sikyonian prytaneium. He did this (says the historian) "knowing that Adrastus would forthwith go away of his own accord; since Melanippus was of all persons the most odious to him, as having slain both his son-in-law and his brother." Kleisthenês moreover diverted the festivals and sacrifices which had been offered to Adrastus to the newly-established hero Melanippus; and the lyric tragedies from the worship of Adrastus to that of Dionysus. Translate into Latin Verse : Stranger, thou hast heard my story, In the old monks' Latin book. It may help thee in thy dying, |