JUNIOR FRESHMEN. Mathematics. A. DR. STUBBS. 1. In the equation xn+pan-1 + +qxn2 + rx2 + sx+t=o, find the value of a2ß + a2y + ẞ2a + ẞ2y + &c., where a, B, and y, &c., are the roots of the equation. 5. Find the locus of the foot of a perpendicular from x'y' on a chord of a circle which subtends a right angle at that point. 6. Given the base and the difference of the sides of a triangle, find the equation of the locus of the intersection of the internal bisector of the vertical angle, and a perpendicular raised at one extremity of the base to the conterminous side. MR. TOWNSEND. 7. Investigate, by any method, the condition that the equation ax3 + bx2+ cx + d = o should have a pair of equal roots. 8. Investigate, by any method, the condition that the two equations a1x2 + b1 x + c1 = 0, and a2x2 + b2 x + C2 = 0 should have a pair of common roots. 9. Investigate, by any method, the condition that the three right lines x cos a1 + y sin a1 = p1, x cos a2+ y sin a2 = p2, x cos a3 + y sin a3 = P3, should pass through a common point. 10. A variable chord of the fixed circle x2 + y2r2 passes through the fixed point x'y'; required the equation of the circle locus of its middle point. 11. Given that ea±b-1 = xy-1, determine, as finite functions, the values of x and y in terms of a and b. 12. For a plane triangle express the values of the functions a cos A+ b cos B+c cos C and a sin A+ b sin B + c sin C in terms of the sides. MR. LESLIE. 13. Find the equation which will represent the lines bisecting the angles between the lines represented by the equation Ax2 + Bxy + Cy2 = 0. 14. Reduce the equation Ax + By + C = 0, referred to oblique coordinates, to the form x cos a + y cos ẞ=p. 15. Deduce analytically the harmonic properties of a complete quadri lateral. 16. Find the equation of the line joining the intersections of the perpendiculars and the bisectors of the sides of a triangle. 17. Find the roots of x3 - 3ax2 + 3 (a2 — bc) x − (a3 + b3 + c3 − zabe) = 0. 18. Find the roots of x4 - 2 (a2 + b2 + c2) x2 + 8abcx + a + b2 + c1 − 2b2 c2 — 2a2 c2 — 2a2 b2 = 0. B. DR. STUBBS. 1. Apply Sturm's theorem to the investigation of the numbers of the real and impossible roots of the equation 2x4 — 13x2+10x — 19 = 0. 2. If a, ß, y, d be the roots of the equation find the value of (a − ẞ)2 ( y − d)2 + ( a − y)2 (B − d)2 + ( a − d)2 (B − y2) in terms of the coefficients. 3. Find the square root of A+√, and apply it to the case of 3+2 2√2. 4. If the cubic equation x3- mx2 + nx-r=o have two equal roots, one of them is gr mn ; prove this, and apply the same method to the 6n 2m2 biquadra ic x + ̧qx2 — rx + 8 = 0. MR. TOWNSEND. 5. Assuming that the three roots a, ß, y of the cubic x3+px + q = 0 may be exhibited, when all real, in the form a = =2p cos &w, ẞ= 2p cos (w +π), y = 2p cos} (w + 2π), required the values of p and w in terms of p and q. 6. Prove that the function log (1 − 2x cos w + x2)-, if expanded in a series of ascending powers of x, becomes to infinity. x cos + x2 cos 2w + 3x3 cos zw + &c., 7. If, for a plane triangle, r and k be the radii of the circumscribing and polar circles respectively, prove the relation k2=4r2.cos A. cos B. cos C. 8. If, for three circles having a common radical axis, A, B, C be the three centres, a, b, c the three radii, and a, ß, y the three angles of intersection with any arbitrary circle, prove the relation BC. a cos a + CA. b cos ß + AB. ccos y = o. MR. LESLIE. 9. Deduce the expression for the area of the triangle formed by three lines whose equations are given; and apply it to find the area of the triangle formed by the lines 10. Find the condition that the lines la + mß + ny = 0 and l'a+m'ß + n'y = 0 may be perpendicular; and prove that the line joining the intersections of perpendiculars and bisectors of sides is perpendicular to the line joining the points where the sides are intersected by lines joining the feet of the perpendiculars. 11. Form the equations of the polar circle, and the nine-points circle of a triangle; and prove that these and the circumscribing circle are coaxal. 12. Tangents to the circumscribing circle at the vertices of a triangle form another triangle; prove that its circumscribing circle, the ninepoints circle, and the circumscribing circle are coaxal. Classics. HERODOTUS. MR. PALMER. Translate the following passages : I. Beginning, Ὁ μὲν δή οἱ ἔλεγε τά περ ἐπώπεε· ὁ δὲ, κ. τ. λ. น I. 61. 2. Beginning, Ατὰρ πρὸς εὐτελείην τῶν σιτίων τάδε σφι, κ. τ. λ. Ending, ἐν κλιβάνῳ διαφανεϊ πνίξαντες, οὕτω τρώγουσι. II. 92. 3. Beginning, ἐπειρήθη δὲ καὶ τοὺς ἐν τῇ λίμνῃ κατοικημένους, κ. τ. λ. Ending, ἀνασπᾷ πλήρεα ἰχθύων. 4. Beginning, πειθομένοισι δὲ ταῦτα τοῖσι Αθηναίοισι, κ. τ. λ. Ending, καὶ αἵδε συνεστήκυῖαι μάλιστα. v. 16. VII. 141. 1. What is the significance attached by Herodotus to the word ioropin? 2. What preceding writer has been supposed to have been most made use of by Herodotus? 3. In what is the Unity of Herodotus' history generally held to consist? Rawlinson takes a different view? 4. With what event does the history really begin according to Mure ? 5. Rawlinson and Mure differ as to the latest event alluded to by Herodotus ? 6. What is the main defect of Herodotus as a historian? 7. As a writer he is distinguished for pathos; what instances occur to you? 8. How did the constitution of Clisthenes differ from that of the fullgrown Athenian democracy? 9. What is known of Miltiades before the battle of Marathon ? 10. What importance is attributed by Herodotus to the war between Ægina and Athens ? 11. Sketch the chronological succession of the early Eastern Empires. 12. Give some account of the historian Ctesias. LIVY. DR. DICKSON. Translate the following passages into English : 1. Beginning, Hæc ubi Asello nunciata sunt in castra, . Ending, stupentibus miraculo hostibus, intactum evasisse. Lib. xxiii. cap. 47. 2. Beginning, Index unum ex conjuratis Theodotum, . Ending, tantumque ipsi Theodoto virium ad arcana occultanda. Lib. xxiv. cap. 5. 3. Beginning, Ab hac adhortatione dimissi corpora curant... Ending, alii ad exitus theatri conferti obsistunt. ...... Lib. xxiv. cap. 39 4. Beginning, Scipioni ut nunciatum est,. Lib. xxix. cap. 7. 1. Write an essay on the following sentence which occurs in Mommsen's history:"Though anger and envy and meanness have written Hannibal's history, they have not been able to mar the noble image which it presents." 2. Describe Hannibal's operations from his descent into Italy until the battle of Cannæ. 3. Draw an outline map of Italy, and mark upon it the following roads and localities:-Via Appia, Flaminia, Aurelia, Emilia, Latina; Mutina, Placentia, Thrasymenus, Roma, Cannæ, ▲riminum, Beneventum, Tarentum, Brundisium. 4. Give some account of the ancient authors to whom Livy alludes in the 21st book of his history. 5. Write a sketch of the literary history of Rome during the Punic Wars. 6. Compare the constitutions of Carthage and Rome. MR. MAHAFFY. Translate the following passage into Latin Prose: Such was the story told by the Argeians themselves, but seemingly not credited either by any other Greeks, or by Herodotus himself. The prevalent opinion was, that the Argeians had a secret understanding with Xerxes, and some even affirmed that they had been the parties who invited him into Greece, as a means both of protection and of vengeance to themselves against Sparta after their defeat by Kleomenês. And Herodotus himself evidently believed that they medised, though he is half afraid to say so, and disguises his opinion in a cloud of words which betray the angry polemics going on about the matter, even fifty years afterwards. It is certain that in act the Argeians were neutral, and one of their reasons for neutrality was, that they did not choose to join any Pan-hellenic levy expect in the capacity of chiefs; but probably the more powerful reason was, that they shared the impression then so widely diffused throughout Greece as to the irresistible force of the approaching host, and chose to hold themselves prepared for the event.-GROTE. Translate the following passage into Latin Verse: Soon as the flocks shook off the nightly dews, The dawn now blushing on the mountain side, Daphnis. Hear how the birds, on every bloomy spray, |