SENIOR FRESHMEN. Mathematics. A. DR. STUBBS. 1. If n and n' be the lengths of two intersecting normals of an ellipse, p and p' the corresponding central perpendiculars on tangents, b' the semidiameter parallel to the chord joining the two points on the curve; prove that np + n'p′ = 2b2; and hence find the expression for the radius of the osculating circle. 2. Prove, completely, that any tangential equation of the second order in λ μv represents a conic whose trilinear equation is found from the tangential by exactly the same process by which the tangential is found from the trilinear. 3. Given in position two conjugate diameters of an ellipse, and the sum of their squares; find its envelope. 4. Expand, by M'Laurin's theorem, &* sin nx in a series of ascending powers of x. 6. Find the sum of the fourth powers of the roots of the equation x5 - 3x3 - 5x + 1 = 0. MR. TOWNSEND. dy 7. Given y = tan-1 (x + √1 − x2) + tan-1 (x−√1 - x2), find dx 8. If u=yz, where y and z are both functions of x, prove the general formula for dru 9. Investigate, for a spherical triangle, the formula for the radius of the circumscribing circle in terms of the three sides. 10. If a triangle be self-reciprocal with respect to a conic, show that an infinite number of triangles may be inscribed to either, and circumscribed to the other. II. Find the envelope of a variable line, the distances of which from two fixed points shall have―(a) a constant rectangle, (b) a constant sum of squares. 12. If a variable line touch in every position a fixed circle, find geometrically the locus of its pole with respect to another fixed circle. MR. LESLIE. 13. Prove that if a'ß'y' denote the co-ordinates of the centre of the conic S = 0, and hence find the locus of the centre of the conic touching the four lines la mẞ±ny. 14. Find the co-ordinates of the pole of λa + uß+vy with respect to Iẞy + mya + naß; and hence determine the locus of the pole of a right line with respect to a conic passing through four points. 15. Find the condition that the pairs of lines aa2+2haß + bẞ2 a'a2+2h'aß +b'ß2 18. If the angles of a spherical triangle be together equal to four right angles, prove that cos 2a + cos 26+ cos 2c = 1. B. DR. STUBBS. 1. Find the equation of the conic touching the lines a, ß, y at their middle points. 2. Conic sections are circumscribed to a triangle, the equations of the sides of which are a=0, B=0, y = o, and the angles A, B, and C, in such a manner that the three normals drawn to the curve at the angles meet in a point; show that the locus of the point is expressed by the equation ẞ (a2 – y2) (cos A cos C− cos B) + a (y2 – ẞ2) (cos B cos C'— cos A) +y (Ba-a2) (cos A cos B - cos C) = 0 3 Prove the formula for the following symmetric function of the roots of an equation: sin (s + σ) sin (s' + o') sin(s" +σ") where σ, o', σ" are the segments of three arcs between the angles and any point inside the triangle, and s, s', s" their productions to meet the opposite sides. 6. Expand sin (m sin -1x) by Maclaurin's Theorem. 7. Given the equations of two conics referred to their common self-reciprocal triangle, find that of the polar reciprocal of either with respect to the other. 8. Given, of a conic, a self-reciprocal triangle, and a point and line pole and polar to each other; determine geometrically its intersections with the line, and its tangents through the point. 9. MR. LESLIE. Find the condition that (la) + (mẞ) + (ny) should represent a parabola, and determine the co-ordinates of its focus and the equation of its directrix. 10. The distances from the origin of two pairs of points on the axis of z being given by the equations prove that their anharmonic ratio will be given if (ab' + a'b — 2hh)2 be in a given ratio to (ab — h2) (a'b' — h'3). Classics. HOMER. MR. PALMER. Translate the following passages :— 1. Beginning, Σήμα δέ τοι ἐρέω μάλ' ἀριφραδές, οὐδέ σε λήσει, κ. τ. λ. Ending, "Ολβιοι ἔσσονται· τὰ δέ τοι νημερτέα ἔιρω. Od., xi. 126-137. 2. Beginning, Οἱ μὲν ἔπειτα δόμονδε θοῶς κίον, αὐτὰρ Οδυσσεύς, κ.τ.λ. Ending, Στὰς ἄρ ̓ ὑπὸ βλωθρὴν ὄγχνην κατὰ δάκρυον εἶβεν. Ibid., xxiv. 220-234. 3. Beginning, Ως φάτ' ̓Αθηναίη· τῷ δὲ φρένας ἄφρονι πεῖθεν, κ. τ. λ. Ending, αἶψα δ' ἐπὶ νευρῇ κατεκόσμει πικρὸν ὀϊστόν. Il., iv. 104-118. 4. Beginning, Εν δ' ἐτίθει νειὸν μαλακήν, πίειραν ἄρουραν, κ. τ. λ. Ending, δεῖπνον ἐρίθοισιν, λεύκ ̓ ἄλφιτα πολλὰ πάλυνον. Ibid., xviii. 541-560. 1. Write notes on the following words:—ἀδινός, ἄωτος, δαίφρων, ἀμολγῷ, τηλύγετος, διάκτορος, θοός, οὖλος, νήδυμος. 2. Mr. Grote disagrees with the main position of Wolf that the Iliad and Odyssey were originally distinct ballads first put together by Pinistratus; state his chief arguments. 3. What parts of the Iliad does Mr. Grote consider to be subsequent additions made to an original Achilleis? 4. What is Mr. Grote's conclusion as to the unity of authorship of the Iliad and Odyssey? 5. Mure, defending the unity of the Iliad, remarks on the consistency shown throughout in the delineation of the chief characters; especially in the case of Agamemnon, Achilles, and Dimede ? 6. There is internal evidence, according to Müller, that the Odyssey was written after the Iliad? What parts of the Odyssey does he mention as being probably interpolations? 7. What events happened in Grecian history in the years 464, 447, 445, 421, 418, 412, 406, B. C. ? 8. Write a short account of the life of Alcibiades. DR. DICKSON. Translate the following passages into English:1. Beginning, Interea videt Æneas in valle reducta... Ending, Suscipit Anchises, atque ordine singula pandit. VIRGIL, En., vi. 702. 2. Beginning, Talibus Alecto dictis exarsit in iras.... Ibid., vii. 445. 3. Beginning, Interea Æneas socios de puppibus altis..... Ending, Impediunt, retrahitque pedem simul unda relabens. 4. Beginning, Illa nocte mihi Troja victoria parta est: Ending, Pectora sunt potiora manu: vigor omnis in illis. Ibid., x. 287. OVID, Met., xiii. 348. 1. Draw a map of Latium, and mark upon it the situations of the towns of the old confederacy. 2. When were the coast towns first included within its limits? 3. When, and by what law, was the Roman Franchise extended to all the Latin towns? 4. Write a short account of the origin and nature of Clientela? 5. At what dates did Tiberius, Caius, Claudius, and Nero, assume the imperial purple? 6. What were the limits of the Roman Empire at the death of Augustus ? 7. Write notes upon such Archaisms as you may have observed in the Eneid. 8. Explain the structure and laws of the metre of the Æneid. 9. Give your observations upon the appropriate use of Quivis, Quilibet, Quisquam, and ullus. MR. MAHAFFY. Translate the following passage into Latin Prose : It seems strange that you should not have heard the great news sooner. D. Ahenobarbus has been taken prisoner at Corfinium, But that is not all. The conqueror has let him free. Instead of murders and confiscations we have clemency and liberality. But I doubt if he will convert Domitius by any course of policy. His heart is as firm as his ancestor's beard is supposed to have been. In these exciting times I have no time for literature. Still you might let your copyists write out Theophrastus' treatise for me. I once borrowed it, and had it with me for a |