P, D, the ends of two conjugate semi-diameters, prove that, b being the reciprocal of the semi-axis minor, 2. If forces P, Q, R, acting at the centre O of a circular lamina along the radii OA, OB, OC, be equivalent to forces P', Q', R', acting along the sides BC, CA, AB, of the inscribed triangle, prove that 3. A fine thread just encloses, without tension, the circumference of an ellipse supposing a centre of force, attracting inversely as the square of the distance, to be placed at one of the foci, prove that the sum of the tensions of the thread at the ends of any focal chord is invariable, and that the normal pressure on the ellipse at any point varies inversely as the cube of the conjugate diameter. 4. Prove that the eccentricity of a section of an ellipsoid, made by a plane through its least axis, varies inversely as the distance, from this axis, of the point in which it cuts a centric circular section. 5. OA', OB', are two quadrants on the surface of a sphere, at right angles to each other: a great circle cuts them in A, B, respectively: from A', B', through any point P of the great circle, are drawn arcs B'PM, A'PN, cutting OA', OB', in M, N, respectively; if PN=4, PM=¥, LOAB=X, LOBA=μ, prove that sin2. cos2 - 2 cosλ cos μ sin o sin + sin2 μ cos2 = 1. 6. If a polygon of a given number of sides be inscribed in the orbit of a planet, such that all its sides subtend equal angles at the Sun, prove that the sum of the angular velocities of the planet about the Sun, at the angular points of the polygon, is independent of the position of the polygon. 7. A uniform homogeneous wire PAP', of which A is the middle point, is bent into the form of an arc of a loop of the lemniscate of which A becomes the vertex: prove that the resultant attraction on the wire, arising from a centre of force at the node O, attracting according to the law of the inverse square, varies as 8. A small light is placed at the focus of a perfect reflector in the form of a paraboloid of revolution: prove that the brightness, due to reflection, at any point within the volume of the paraboloid, varies inversely as the square of the focal distance of the end of the diameter through the point. 9. A hollow homogeneous cylinder, of given material, which is perfectly brittle and incompressible, is partially inserted into a fixed horizontal tube just wide enough to admit it: prove that the greatest length which the free portion of the cylinder can have, without snapping off, varies as the square root of the radius of its external surface. 10. A centre of force, repelling inversely as the square of the distance, lies below the surface of a homogeneous inelastic fluid, which is also acted on by gravity and is at rest: the intensity of the force, at a point in the surface of the fluid vertically above its centre, is equal to that of gravity : prove that the external surface of the fluid has a horizontal asymptotic plane, and that the centre of force is environed by an internal cavity, the summit of which is at the external surface of the fluid. Find the volume of the cavity in terms of its length. 11. A carriage is travelling along any level road: prove that the sum of the squares of the shadows cast on the ground by any two spokes of a wheel, which are at right angles to each other, varies during the journey as the square of the secant of the Sun's zenith distance. Prove also that, if the road run due east and west, tan 20 a being the azimuth and ≈ the zenith distance of the Sun, and the corresponding inclination of a spoke to the horizon when its shadow is greatest or least. 12. OA, OB, OC, are meridians on a surface of revolution, passing through three points A, B, C, which are connected together by the shortest arcs BC, CA, AB: BC cuts OB, OC, at angles λ1, λ2; CA cuts OC, OA, at angles λ3, 4; and AB cuts OA, OB, at angles λ5, λ: prove that sin λ. sin λ. sin λ5= sin λ 2. sin λ4. sin λ ̧. 13. A little animal, the mass of which is m, is resting on the middle point of a thin uniform quiescent bar, the mass of which is m' and the length 2a, the ends of the bar being attached by small rings to two smooth fixed rods at right angles to each other in a horizontal plane: supposing the animal to start off along the bar with a velocity V, relatively to the bar, prove that, being the inclination of the bar to either rod, the angular velocity initially impressed upon the bar will be equal to 14. A narrow tube, in the form of a common helix, is wound round an upright cylinder, initially at rest, which is pierced by two smooth fixed rods, parallel to each other and horizontal: supposing a molecule to be placed within the tube, at a point of which the distance from the axis of the cylinder is parallel to the rods, find the velocity of the cylinder when the molecule arrives at any proposed point of the tube. Prove that, m, m', being the masses of the molecule and cylinder, the velocities which the cylinder has acquired, at the successive arrivals of the molecule at points most distant from the plane in which the axis of the cylinder moves, will have their greatest values when, a being the inclination of the helix to the horizon, THURSDAY, January 22, 1857. 1...4. 1. ONE plane curve rolls on another, the planes of the two curves coinciding, and their convexities being opposed to each other: if r, r', are the radii of curvature of the fixed and rolling curves respectively, at their point of contact, p the distance of any point P in the moving plane from the point of contact, a the angle between p and the common normal to the two curves, prove that the corresponding radius of curvature of P's path is equal to 1 1 Shew also that the directions of motion of all the points in the moving plane, fixed relatively to the rolling curve, which at any instant are going through points of inflection in their respective paths, pass through a single point. 2. Prove the following relation between the sides and angles of a spherical triangle, 5. Determine the change in the position of the axis and in the eccentricity of the Moon's orbit indicated by the terms in the expression for the Moon's parallax. 6. A free rigid body, the mass of which is m, is at rest: its moments of inertia about the principal axes through its centre of gravity are A, B, C: supposing the body to be struck by an impulsive force R through its centre of gravity, and by an impulsive couple G, prove that it will revolve for an instant about an axis, the velocity of which is in the direction of its length and is equal to X, Y, Z, being the components of R, and L, M, N, of G, along the principal axes. If be the inclination of R's direction to the spontaneous axis, prove that L.X M. Y N.Z 7. Two luminous points which emit light of the same colour and of equal intensity, are placed very near to each other before a plane screen and at exactly equal distances from it: investigate the appearance on the screen. 8. Prove the following equation for the determination of the major axis of the orbit of a disturbed planet, FRIDAY, January 23, 1857. 9 to 12. 1. IF a=0, ẞ=0, y=0, be the equations of the sides of a triangle, shew that the equation of a conic touching the sides of the triangle is be the equations of the sides of a hexagon which circumscribes a conic, shew that α1 (b2c3−c2b3)+α2 (b3c1 −c3b1)+a3(b1c2 − c1b2)=0. 2. Transform the triple integral Sff (a, ß, y) dadẞdy into one in which x, y, z, are the independent variables, having given If shew that a = F1(x, y, z), ß=F2(x, y, z),_y=F3(x, y, z). ax=yz, ẞy = zx, yz=xy, Y≈ SSSƒ(a, ß, y) dadßdy =4 SSƒƒ (,,) dzdy dz. 3. Shew how to integrate the equation of differences, Ux+n+P1Ux+n−1 +... +Pnux=ƒ (x), where P1, P2,...P, are independent of x. Shew that a solution of the equation +U2+1+Us), where a is one of the imaginary (n+1)th roots of unity, the n + 1 constants being subject to an equation of condition. 4. Shew how to find the differential equation of a class of surfaces, which cuts at right angles all the surfaces represented by the equation f(x, y, z, a) = 0, where a is an arbitrary parameter. If the class of surfaces have an envelope, shew how we may find it without solving the differential equation. |