ABCD is a quadrilateral figure. Forces act along BA, BC, DA, and DC proportional to them, show that their resultant is a single force represented by four times the straight line which joins the middle point of BD to the middle point of AC. 2. Show how to find the resultant of any number of forces acting on a rigid body. A cube has forces proportional to 1, 2, 3, 4 acting along the edges of one face taken in order. Forces proportional to 4, 1, 2, 3 act along the corresponding edges of the opposite face in the opposite direction. Find the resultant. 3. Show how to find the C. of G. of a solid of revolution, and find it in the case of a hemisphere. 4. State the laws of friction. Show how to calculate the total friction in the case of a rope stretched round a rough cylinder. 5. Explain the principle of the arch, and show how you would calculate the curve required for a given arrangement of the load upon it. 6. What are the laws of motion? Give an experimental illustration of each. Investigate formulæ for the motion of a particle on an inclined plane under the action of gravity. 7. Calculate the motion of a body projected obliquely and acted on by gravity. A building 20 feet high, 20 feet wide, and 30 feet long is surmounted by a gable roof rising 20 feet higher. A smooth stone is projected horizontally with a velocity of 2 feet per second just along one side of the ridge from one end of it. Find where it will strike the ground. 8. Calculate the motion of a particle acted on by a central force varying as the distance, A weight hangs from a peg by an elastic string which it stretches to double its unstretched length. If the weight be slightly displaced, find the time of a small vertical oscillation. 9. A block of wood thrown on ice with a velocity of 10 feet per second is brought to rest after passing over 30 yards. A bullet of equal weight with the block is then shot into it with a velocity of 100 feet per second. Determine the subsequent motion. 10. A ball is dropped from a height of 10 feet on a plane inclined at an angle of 30°; the coefficient of elasticity is ; find the points where the ball will again strike the plane. 11. Two perfectly elastic particles are revolving in the same direction and in the same plane round a center of force varying inversely as the square of the distance. One is moving in a circular orbit, the other in a parabola whose latus rectum equals the diameter of the circle. They collide as the second particle is approaching the center of force. Determine the subsequent motions. APPLIED MATHEMATICS: SECOND PAPER. Examiner.-Prof. C. NIVEN. Students in naval architecture and marine engineering of second and and third years 1. Determine the general equations of equilibrium of a fluid; and show that, when the external forces are such as arise from a potential, the surfaces of equal potential, of equal density, and of equal pressure coincide. A heavy liquid is contained in a vessel and is also under the action of two centers of force which are in the same vertical line, and which exert equal forces at equal distances, but one of which is repulsive and the other attractive. The law of force being directly as the distance, prove that the free surface is a horizontal plane, and find the pressure at any point. 2. Define the whole pressure and resultant pressure on a surface immersed in a fluid; and show how to calculate them. Prove that the total normal pressure on a spherical surface immersed to any depth in water is the same as that on the circumscribed cylinder immersed to the same depth. 3. Find the center of pressure of a circle immersed in water to any depth. 4. Find the form of the free surface of a fluid which rotates uniformly, in relative equilibrium, round a vertical axis. 1 777 A cylindrical jar whose weight is th of the weight of water which it would contain, is filled (1—1)th full and is then placed, mouth downwards, on a horizontal table which is made to rotate uniformiy round a vertical axis coinciding with the axis of the jar. Prove that the angular velocity necessary to cause the fluid to escape is the same as if the jar weighed th of the water it would hold and were (1—1)th full; and find this angular velocity. 5. Investigate the conditions of stability, for small displacements, of a body floating in water. A pyramid on a square base, whose other faces are equilateral triangles, floats in water with its vertex immersed and base horizontal, find the condition of stability. How will the stability be affected by tilting it round different axes? 6. Investigate the law of density of a vertical column of still air of uniform temperature. Find the law of density on the hypothesis that the temperature diminishes in harmonical progression as the height increases in arithmetical progression, the variation of gravity in ascending being disregarded. 7. State the hypotheses upon which the equation of fluid motion = C + 9 z −' is founded; and prove the equation. R 8. Define the component velocities at any point of a fluid in motion; and, in the case of motion in one plane, find an expression for the quantity of fluid which flows, in given time, in through the boundary of a circle of radius a whose center is at the origin. 9. Given a plane figure of any form; find the line round which it has the least moment of inertia. The diagonals of a square plate being drawn, the two opposite triangles are cut out; find the principal axes and moments of inertia of the remaining figure, and the moments of inertia about each of the edges of the figure, 10. State D'Alembert's principle, and investigate any conclusions which can be drawn from it for the motion of a rigid body under no forces. 11. State and prove the principle of the convertibility of the centers of suspension and oscillation of a pendulum. A pendulum is formed of two uniform rods of equal lengths, but of different materials and thicknesses, connected at one end so as to be in the same straight line. Their masses are m, m', and the axis of suspension passes through the middle point of m; find the time of oscillation of the pendulum. 12. State and prove the equation of Vis Viva. A rod AB is capable of turning round A in a vertical plane, the other end being attached to an elastic string BC which is fastened to a fixed peg vertically above 4, and such that AC=AB. The elasticity of the string is such that a weight equal to that of the rod would stretch it to three times its natural length AB. If the rod be started from its position of stable equilibrium with an angular velocity find the subsequent motion until the string becomes slack. NOTE H. ADMIRALTY CIRCULAR IN REGARD TO PRIVATE STUDENTS IN NAVAL ARCHITECTURE AND MARINE ENGINEERING. A limited number of students unconnected with the naval service will be permitted to receive instruction at the Royal Naval College, in the course laid down for acting second-class engineers and dockyard apprentices. The full course will be for three sessions, of nine months each. The fee (payable in advance before entry) is £30 for each session, or £75 for the full course. Students who have already paid one fee of £30 will be allowed to compound for the next two sessions by a payment of £50 at the commencement of the second session. Proportionate fees will be paid by students attending special classes only. Students not connected with the naval service will reside outside the precincts of the college. Facilities for visiting the royal dockyards will be offered to all private students, being British subjects. Applications for admission should be addressed to the secretary of the Admiralty, Whitehall. My lords reserve entire discretion in the selection of the candidates to be admitted. ENTRANCE EXAMINATIONS. Private students will be examined before entrance, in accordance with the programme laid down in the general regulations established for the admission of students to the Royal Naval College, as follows, viz: 1. The ordinary rules of arithmetic. 2. Algebra up to quadratic equations, the three progressions, the binomial theorem, and the theory of logarithms. 3. The subjects of the first four books of Enclid's Elements; proportion and similar figures, or the definitions of the fifth book and the proportious of the sixth book of Enclid's Elements. 4. The definitions and fundamental formulæ of plane trigonometry, including the solution of plane triangles. De Moivre's formula and its principal applications. 5. Elements of statics, dynamics, and hydrostatics. 6. Co-ordinate geometry, up to the equations of the conic sections. 7. Geometrical drawing. ANNUAL EXAMINATIONS. All private students will be examined at the end of each session. Certificates of proficiency in the various subjects they may have studied will then be awarded. NOTE I. EXAMINATION PAPERS, GUNNERY SHIP EXCELLENT. SUB-LIEUTENANTS. (July, 1876.) 1.-Explain fully how to divide an arc into degrees. What is the angle between the axis of the gun and the keel line, when the pointer on the slide coincides with the zero mark? S. Ex. 51-20 2.-Show how to calculate the correction which has to be applied to the bearing at any particular gun in order that its fire may be directed on the same point as the center gun. Give the form in which these corrections are tabulated. Is this method absolutely correct for all bearings? If not correct, express the error in terms of (d) the distance from, and (4) the angle of training of, the center gun. 3.-What adjustments are necessary when placing the director in position? Explain fully how to test their accuracy. 4.—What are the essential points of the Fraser system of gun construction? Why is steel preferable to wrought iron for the inner barrel? 5.-Draw a diagram of the 12" 35-ton M. L. R. gun, distinguishing between the coiled, forged, and steel portions. State generally how a double coil is formed. 6. The deflection scale is marked to 30', calculate the maximum speed for which it can be used with the 7" M. L. R. gun when firing at an object distant 1,000 yards; the time of flight being 2.2 seconds. 7.-Define the following terms, and give diagram explaining: Line of fire. Line of sight. What forces act upon a projectile during flight? 8. What are the advantages of elongated over spherical projectiles? 9.-Why is it necessary to give rotation to elongated projectiles? Upon what does the velocity of rotation required by a projectile depend ? 10.-Describe the manner in which a charge of powder is consumed in the bore of a gun. How does the size of the grain affect the action? 11.-How is the perforating power of a projectile measured? 12. Find the thickness of armor plate which can be penetrated by a projectile whose weight is 250 lbs. and diameter 8.92 inches, when moving with a velocity of 1,200 feet per second. Given R3.138 T 3. Where Renergy in foot tons per inch of shot's circumference. 13.-Explain, with diagram, the general distribution of the armor in the Hercales and Shannon. SUB-LIEUTENANTS. (January, 1877.) 1.-In what plane should racers be laid? Give full reasons for your answer. Explain how to calculate the correction which has to be applied to the bearing at each gun, except the center one, when concentrating a broadside. Give the form of the table. 2.—The director is exactly over the center gun and the broadside is converged on the beam for a distance of 400 yards, but when the guns are fired the object is distant 200 yards. Will the shot from the center gun pass above or below the point aimed at, and at what approximate vertical distance? Given axis of telescope above axes of guns = 10 feet. Elevation for 200 yards = 13'. 3. With reference to what plane is a gun given elevation when laid by scale, as in broadside firing-gun directing? Explain, with a figure, why the heel scale is required in addition to the elevating scale when laying by director. How would you test the accuracy of the elevating scales? 4.-Explain the meaning of the term "steel." In what does it essentially differ from wrought iron? What are its advantages and disadvantages as a gun material? 5.-Explain, with a diagram, the principal parts of a 9" M. L. R. gun with two double coils. Describe the vent bush with which the Woolwich guns are vented. 6-A gun is laid, with the sight close down, for an object distant 120 feet. How far below the point aimed at will the shot strike? Initial velocity 1,200 feet per second. Tip of center fire-sight to axis of gun, 25 inches. 7.-Does the axis of a rifled elongated projectile remain parallel to itself during flight? If it does not, give some idea of the motion, together with its cause. -Compare the lengths of the dangerous spaces, with battering and full charges, when firing a 9-inch M. L. R. gun at an object of which the height is 20 feet and the distance 1,500 yards. Given Explain, with a figure, the meaning of the term "dangerous space.” 9-When a charge of powder is exploded, are the products of combustion liquid, solid, or gaseous? And in what proportion by weight? By which products is the shot propelled from a gun? 10.-How are combustion and ignition affected by the size of the grain and by the density and hardness of the powder? 11.-Compare the perforating powers of the following guns at 1,000 yards: Gun. 9-in. M. L. R .... 8-in. M. L. R Total energy at 1,000 yards. 2,648 feet tons. 12-Compare the resisting power of a 6-inch plate, when fired at direct, with that of a 9-inch plate inclined at an angle of 60° with the line of fire. 13-Describe generally the armored side of the Warrior. SUB-LIEUTENANTS. (April, 1878.) -Explain fully how to divide any arc into degrees. No. 1 of the 1st gun (5th gun being the center gun) in firing an electric broadside applies his correction the wrong way; where will the shot from his gun fall?* 2-Explain how to test the accuracy of the racers and the director without using a spirit-level. -What are the principal points of the English gun manufacture? Give a diagram of the 9-inch R. 4-The deflection scale is marked to 30'; for what speed can it be used, distance of object 1,000 yards, time of flight 3"? Data for first question: distance of object 300 yards on the beam; distance between each pivot 20 feet. |