6. Assuming the truth of De Moivre's theorem, prove that 7. Show how to describe a conic section, whose focus, directrix, and eccentricity are given. If a series of conics pass through two fixed points, P, Q, and if a directrix of each pass through a fixed point in PQ produced, prove that the corresponding foci of the conics all lie on a fixed circle. 8. If the ordinate, tangent, and normal at any point of an ellipse meet the major axis in N, T, G, respectively, prove that 9. (i.) NG: CN:: BC2: AC2. (ii.) SC is a mean proportional between CG and CT. Prove that the latus rectum of an hyperbola is a third proportional to the transverse and conjugate axes. If SY, the perpendicular from the focus upon an asymptote, be produced to meet the conjugate axis in W, prove that YW is a third proportional to the conjugate and transverse semi-axes. 10. Show how to express the position of a point on a plane by means of polar coordinates. Indicate in a figure the four points whose polar coordinates are (a,1), (za, -=), (-a, %), 2 and ( 2π - 2a, 3 respectively, and also the locus of the equation (4 – 3 sin20) = 8a cos 0. II. Find the equations of the two straight lines drawn through the point (0, a), on which the perpendiculars let fall from the point (2a, 2a) are each of length a. Show that the equation of the straight line joining the feet of these perpendiculars is y+2x = 5a. 12. Define the terms pole and polar with respect to a conic section, and find the equation of the polar of any point with respect to a parabola. If a perpendicular be let fall from the pole upon the polar, prove that the distance of the foot of this perpendicular from the focus is equal to the distance of the pole from the directrix. 13. Find the equation of the tangent at any point of the ellipse a2y2+b2x2 = a2b2. If the tangent at P have intercepts f, g upon the coordinate axes, prove that a2 b2 = 1. 14. Investigate the condition that y = mx, y = m'x may be conjugate diameters of the hyperbola b2x2 - a2y2 = a2b2. Find the equation of that diameter of the hyperbola 2x2+xy − y2 = 9a2, which is conjugate to the diameter coinciding with the axis of x, and prove that the difference of the squares of these two conjugate diameters is equal to 16a2. VI. MECHANICS. [Full marks may be obtained for about two-thirds of this paper. Great importance is attached to accuracy. N.B.-g may be taken = 32.] I. Two equal weightless rods, AB and AC, are hinged together at A at an angle of 74°, and placed in a vertical plane, with B and C on a smooth horizontal plane. If B and C be connected by a string, and a weight of 30 lbs. be suspended at D, where AD: DC = 5 : 4, find graphically the tension of the string. 2. Four equal weightless rods are hinged together to form the rhombus ABCD, and the hinges A and C are connected by a string. If the rhombus be suspended from A, and equal weights of I cwt. each be suspended from B and D, find graphically the tension of the string. 3. Four equal weightless rods are hinged together to form the rhombus ABCD, and the hinges at B and D are joined by the weightless rod BD. The angle BAD = 64°. If the rhombus be suspended from A, and a weight of I cwt. be suspended from C, find the thrust in BD. 4. Prove that the centre of gravity of a triangle coincides with that of three equal particles at the angles. The uniform plane quadrilateral ABCD right-angled at A and obtuseangled at B is bisected by the diagonal AC. If it be placed with its plane vertical and the side AB on a horizontal plane, prove that the condition of equilibrium is 2BM < 3AB, where M is the foot of the vertical from Con AB produced. 5. Explain the construction and action of the screw, and find the relation of the power to the weight when there is no friction. 6. A uniform beam, weight W, laid on a horizontal plane, can be just moved by pushing it with a horizontal force W√3. Prove that the least force which can move it is equal to and find the direction of this force. W 2 If the beam be pulled slowly by a rope attached at one end A, prove that A will not rise from the ground unless the inclination of the rope to the horizontal be 60° at least. 7. A body is projected from a given point O, with the velocity v, at the angle a to the horizon; find its distance from O after the time t. If any number of such particles be projected from ◇ with the same velocity in different directions, prove that at any subsequent instant they will all lie on the surface of a sphere. 8. Find the horizontal and vertical accelerations of a body sliding down a smooth inclined plane. If two bodies start at the same instant sliding down two lines in the same vertical plane sloping towards the same direction, at angles a and ß to the horizon, prove that each as seen from the other will always appear to be moving parallel to a line inclined to the horizon at the angle a +ß. 9. Find the velocities of two elastic spheres of given masses and elasticities after direct impact. If two perfectly elastic smooth spheres, A and B, impinge upon each other directly or obliquely, prove that B's velocity relative to A, after impact, will make the same angle with the line of centres as the relative velocity of A to B did before impact. IO. Find the acceleration of a particle describing a circle, radius a, with velocity v. A circus horse gallops round a circle of 30 feet radius, at a speed of 15 miles an hour, prove that the least value of the coefficient of friction between feet and ground, that the horse may not slip, neglecting the distance between the feet and centre of gravity, is very nearly. ton. II. A train weighs M tons, and the resistance of friction is p lbs. per If the engine can exert a pull of P lbs., and the break a resistance of R lbs., find the distances passed over in attaining a speed of v miles per hour from rest, and in slowing down from that speed to rest respectively moving on the level. MATHEMATICAL EXAMINATION PAPERS FOR ADMISSION INTO Royal Military Academy, Woolwich, JUNE, 1893. OBLIGATORY EXAMINATION. I. EUCLID (BOOKS I.-IV. AND VI.). [Ordinary abbreviations may be employed; but the method of proof must be geometrical. Proofs other than Euclid's must not violate Euclid's sequence of propositions. Great importance will be attached to accuracy.] 1. Define right angle, trapezium, quadrilateral. If two triangles have two sides of the one equal to two sides of the other, each to each, but the angle contained by the two sides of one of them greater than the angle contained by the two sides equal to them of the other, the base of that which has the greater angle shall be greater than the base of the other. ABCD, EFGH are quadrilaterals in which the sides AB, BC, EF, FG are all equal to one another; and the angles at C, D, G, H are right angles. Show that if the angle at F is greater than that at B, then is EH greater than AD. 2. What is the magnitude of the angle of a regular figure of ʼn sides? Three regular figures, of n1, n, and ng sides, have one vertex in common; and just fill the whole space at that vertex; show that I I I I + + |