Beginning, Eodem anno ab Etruscis adversus indutias... Live x. 10, 11. Tacitus, Agricola, 39, 40. 1. SOLVE the equations: (1) 4x? +6x + V2.x2 + 3x +11 = 33. 6x2 – 17xy +12y2 = 0. 2. Five hundred persons voted on a measure, which was rejected by a certain majority; but being brought forward on a future occasion it was carried by a majority double that by which it was formerly thrown out, and the number who voted in favour of the measure the second time was to the number who voted for it the first time as 9 to 8; required the number of voters who changed their mind. 3. If a, b...l be the prime factors of an integer N, and N=a0bB..., find the number of divisors of N. Example. Find the number of divisors of 504504. 4. Express log.(n+1) – log.n, and loge(n+1) – log.n- {logen-log. (n-1)} by series which converge rapidly when n is large. Calculate log. 10 to 4 places of decimals. 5. Prove the formulæ, А = $V1+ sin A + V1 - sin A, 2 cos A F and determine the signs of the radicals when A=200°. 6. From Demoivre's theorem deduce the formulæ, 03 05 + 02 04 1.2 1.2.3.4 + 7. Required to determine the distance between two rocks in the sea by observations made at two points on the shore, the distance between which is known. 8. In the ellipse, if PI be the chord of curvature passing through the centre, CP: CD :: 2CD : PI. 9. Prove by Analytical Geometry, or otherwise, that the perpendiculars let fall from the angles of a triangle on the opposite sides meet in a point. 10. Find the locus of the centre an ellipse inscribed in a triangle, and having a pair of conjugate diameters parallel to two of the sides. 11. Given the base of a triangle, and the difference of the angles at the base, prove that the locus of the vertex is a hyperbola, and find its asymptotes. 1. Any two sides of a triangle are together greater than the third side. If a polygon with only salient angles be situated inside another polygon, the perimeter of the former will be less than the perimeter of the latter. 2. Given two finite straight lines in a plane, required the locus of a point which is the common vertex of two equal triangles having the given lines for their bases. 3. The angle at the centre of a circle is double the angle at the circumference upon the same base. If a circle touch internally another of twice the diameter, and from the centre of the latter two radii be drawn cutting the former, the arcs of the inner and outer circle comprised between the radii will be equal in length. 4. Required to describe a circle through three given points. If three circles touch each other internally, a circle described through the points of contact will be inscribed in the triangle formed by joining the centres of the three circles. 5. If two triangles be equal, their bases will be inversely as their altitudes. Prove this, as well as the portion of a proposition of Euclid on which it depends. 6. If two straight lines be parallel, and one of them be perpendicular to a plane, the other also shall be perpendicular to the same plane. 7. If 10 men reap 4 acres of corn in 4 days, working 10 hours a day, how many men would be required to reap 21 acres in 12 days, working 12 hours a day, supposing that a reaper who works 10 hours a day does 3 th part more work in an hour than one who works 12 hours a day? 8. Prove the rule for finding the greatest common measure of two integers, and shew that it is a multiple of all other common measures. Find the greatest common measure of 24990, 30030, and 36225. 9. Shew generally how to reduce a circulating decimal to a vulgar fraction; and find the value of £0.714285. 10. Prove that every decimal which neither terminates nor circulates represents an incommensurable quantity. 11. If the first term of an arithmetical series be 5, the common difference 2, and the sum of the series 21, find the number of terms, and interpret the second solution of the problem. TRANSLATE into Latin PROSE : THERE is no sight in nature more elevating than the dawn even to us, whom philosophy has taught that 'nil admirari' is the highest wisdom. Yet in ancient times the power of admiring was the greatest blessing bestowed on mankind; and when could man have admired more intensely, when could his heart have been more gladdened and overpowered with joy than at the approach of the Lord of light, Of life, of love, and gladness! The darkness of night fills the human heart with despondency and awe, and a feeling of fear and anguish sets every nerve trembling. There is man like a forlorn child fixing his eye with breathless anxiety upon the East, the womb of day, where the light of the world has flamed up so many times before. As the father waits the birth of his child, so the poet watches the dark heaving night who is to bring forth her bright son, the sun of the day. The doors of heaven seem slowly to open, and what are called the bright flocks of the Dawn step out of the dark stable, returning to their wonted pastures. Who has not seen the gradual advance of this radiant procession—the heaven like a distant sea tossing its golden waves -when the first rays shoot forth like brilliant horses racing round the whole course of the horizon—when the clouds begin to colour up, each shedding her own radiance over her more distant sisters ! Not only the East, but the West, and the South, and the North, the whole temple of heaven is illuminated, and the pious worshipper lights in response his own small light on the altar of the hearth, and stammers words which express but faintly the joy that is in nature and in the human heart * Rise ! our life, our spirit is come back! the darkness is gone, the light approaches!' Max MULLER. (Oxf. Ess. 1856. p. 58.) TRANSLATE into English Prose, adding brief notes where requisite : Beginning, Προστάξας δε ταύτα, είπε, καλέσας ές όψιν, κ.τ.λ. HEROD. v. 106. Plato, Repub. p. 611. C-p. 612. A. What are the principal significations of the Greek middle voice, and what are the forms of the principal tenses adopted to express it ? Beginning, Καίτοι, ώ άνδρες δικασταί, επί των δημοσίων, κ.τ.λ. HYPERID. pro Euxenip. coll. 20—22. What are the grammatical conditions of the right use of apiv? TRANSLATE into ENGLISH Prose, adding brief notes where requisite: Beginning, Sag. Hoc age. opusnest hac tibi empta ?... Liquidumst auspicium: tace. PLAUT. Pers. 584-607. LUCRET. III. 632_661. a. abstraxe. Illustrate by examples of different words the tendency of Latin pronunciation to fuse syllables together. b. State precisely the cases in which 'cum' takes the indicative and subjunctive moods. Translate into ENGLISH VERSE or PROSE: Beginning, Hujus ut adspicerent opus admirabile, sæpe... Ovid. Metam. VI. 14–25; 53–69. Natural Sciences Tripos. March, 1857. PROF. BOND, M.D. Corpus Christi College. PHYSIOLOGY. 1. In the respiration of fishes, some physiologists have supposed that the oxygen is obtained by decomposition of the water in which the fishes live. Give observations which shew that this supposition is erroneous, and which establish the real nature of the process. How does the swallowing of atmospheric air by fishes assist the respiratory process ? 2. Explain the observation of Dr Edwards, that cold blooded animals can live for a much longer period in a limited quantity of water at a low temperature than at a higher temperature. 3. What is the effect of diminished temperature upon the amount of carbonic acid generated in a given time by warm-blooded animals? 4. Describe the way in which inspiration is effected in the frog that has no ribs, and again in the tortoise that has no moveable ribs. 5. In the perenni-branchiate reptiles how is respiration effected ? 6. Shew that as to physiological characters there is no correspondence between the swimming-bladder of fishes and the lungs of air-breathing vertebrates. Does this, however, prove that they are not homologous organs ? 7. In what order of insects are phosphorescent individuals found ? What is the usual situation of the luminous points? What is supposed to be the cause or explanation of the phenomenon ? 8. Frogs cannot swallow liquids : how then do they obtain the quantity of Muid necessary for the maintenance of their life? To whose experiments do you refer? |