Examination for Honours. MATHEMATICS AND NATURAL PHILOSOPHY. TUESDAY, July 15.-Morning, 10 to 1. ARITHMETIC, ALGEBRA AND GEOMETRY. : 1. The three conterminous edges of a rectangular parallelopiped are 36, 75, 80 inches respectively find the side of a cube which shall be of the same capacity. How many gallons would either vessel hold, the gallon containing 277 274 cubic inches? 2. A person transfers £5000 stock from the 3 per cents at 98 to the 3 per cents at 90: how much of the latter stock will he hold, and what will be the difference in his income? 3. What are recurring decimals? Find the vulgar fractions that will produce the recurring decimals ⚫08123123 and 000900090009. 4. When are algebraical quantities called impossible? Which of the following results are possible and which impossible? 5. Prove the rule for finding the greatest common measure of two numbers. Why is this term inappropriate as applied to two algebraical expressions? 6. Find the number of permutations that may be formed N out of (n) things, among which there are (p) of one class, (q) of another, (r) of another. Ex. The letters in the word initiation. 7. Show how to find all the positive and integral values of x and y that satisfy the equation 9. The angle in a semicircle is a right angle, the angle in a segment greater than a semicircle is less than a right angle, and the angle in a segment less than a semicircle is greater than a right angle. 10. Inscribe in a given circle an equilateral and equiangular hexagon. 11. Define the sector of a circle. Show that in equal circles the sectors and the circumferences on which they stand have the same ratio to each other. TUESDAY, July 15.-Afternoon, 3 to 6. PLANE AND SPHERICAL TRIGONOMETRY. Examiner, Mr. JERRARD. 1. Distinguish between goniometrical and geometrical angles; and show that sin 0 = sin {+2nπ+0}, sin-0-sin 0, 2. Express the sine of half an angle in terms of the cosine of the angle itself. Find the cosine of 18°, and thence deduce the sine and cosine of 9° to four places of decimals. 3. Show how the n roots of the equation xn-1=0 may be represented trigonometrically. Take as an example n=7. 4. In a plane triangle, given a, b and A; to find B, C and c. 5. Show that in any plane triangle where R is the radius of the inscribed circle, and 6. State Napier's rules of the Circular parts. Apply and verify them when two sides are given. 7. Prove that in the ellipse the sum of the squares of any pair of conjugate diameters is constant. 8. Investigate the equation to the hyperbola and thence deduce the equation to its asymptotes. 9. Show that two tangents to a parabola from the same point of the directrix are at right angles to each other. 10. If through any point within or without a curve of the second order two straight lines be drawn parallel to given lines to meet the curve, the rectangles of the segments will be to one another in an invariable ratio. WEDNESDAY, July 16.-Morning, 10 to 1. STATICS AND DYNAMICS. Examiner, Rev. Prof. HEAViside. 1. If three forces which keep a point in equilibrium be in the direction of three lines forming a triangle, they are proportional to those lines. Will the same proportion be true, when any number of forces keeping a point in equilibrium are parallel to lines forming a polygon? 2. What is the moment of a force'? If any number of forces keep any lever in equilibrium round a fulcrum, state and prove the condition of equilibrium. Will the conditions of equilibrium be changed if the fulcrum be not fixed? 3. Find the ratio of P to W when there is equilibrium on that system of pullies, where each pulley hangs by a separate string, the strings being all parallel. Take into account the weights of the pullies which are all equal. 4. Show how to find the centre of gravity of any number of heavy points in one plane. Ex. Weights of 1, 3, 7, 5 placed in the angles of a square. 5. A roof consisting of beams forming an isosceles triangle with its base horizontal supports a given weight at its vertex: find the tension of the tie-beam. 6. What is a uniform force? Show how we determine gravity at the earth's surface to be a uniform force. Explain the relation between the measure of the force and the space fallen through in vacuo in one second. 7. A body is projected perpendicularly upwards with a given velocity: find the height it will ascend in t". 8. The curve described by a projectile in vacuo is a parabola prove this. A body is projected at an angle of 45° with a velocity of 1600 feet per second: find its place at the end of 5". 9. A weight P draws Q over a fixed pulley: show that P-Q the accelerating force is P+Q.9. 10. When are bodies said to be elastic, and what is the measure of their elasticity? A ball of glass of inch diameter moving with a velocity of 10 feet in a second, overtakes a ball of glass of one inch diameter moving with a velocity of 6 feet in a second: find the velocity of each ball after direct impact, taking the measure of elasticity for glass = 15 16' WEDNESDAY, July 16.-Afternoon, 3 to 6. HYDROSTATICS, &c., OPTICS. Examiner, Mr. JERRARD. 1. Show that the pressure of a fluid on any surface is the weight of a column of the fluid whose base is equal to the area of the surface pressed, and whose height is equal to the depth of the centre of gravity of the surface below the surface of the fluid. 2. Describe the action of the hydrostatic bellows; and investigate the relation between the pressure and the weight raised. 3. Define the term Metacentre: and determine when the equilibrium of a floating body is stable, unstable, or indifferent. Show that the positions of stable and of unstable equilibrium recur alternately. 4. The elastic force of air at a given temperature varies |