8. Find the centre of pressure in heavy liquid (1) Of a triangle with its base in the surface. (2) Of an isosceles triangle with its plane vertical, vertex downwards, and base horizontal, at a depth h below the surface. 9. Define carefully the term metacentre, and show the connexion between the position of the metacentre and the stability of a floating body. Investigate the expression for the height of the metacentre above the centre of gravity of the liquid displaced when the section of the body at the water-line is rectangular, 2a being the breadth of the rectangle in the plane of displacement, its length, and V the volume of liquid displaced. 10. Find the form of the free surface, and the pressure at each point, of a heavy liquid revolving as a rigid body about a vertical axis. MIXED MATHEMATICS.-PART II. SECOND PAPER. The Board of Examiners. 1. A triangle, ABC, is formed by three rods smoothly jointed at their ends. Couples L, M, N are applied to the sides in the plane of the triangle. Find the condition of equilibrium and show that the reaction at the joint A is cosec A J M2 N2 2MN cos A bc a, b, c being the lengths of the sides. 2. A rough hemisphere rests with its base on a The rough lower end smooth vertical tube Show that the hemi rough horizontal plane. W' tan (a x) = (W + W') tan X where is the angle of friction and W, W are the weights of the hemisphere and stick respectively. 3. Prove the principle of virtual work for a system of coplanar forces acting on a rigid body. 4. Find the attraction of a thin spherical shell at a point outside it. 5. If u, v are the components of velocity of a particle in the direction of two rectangular axes which revolve with angular velocity w, show that the component accelerations in the same directions wv, v + wu. are u A rod of length 7 moves so that its ends are always on the revolving axes. Find the acceleration of a point dividing the rod in a given ratio. 6. Investigate the polar equation of a central orbit d2u de2 + u = P h2 u2 Show that the attracted particle is approaching the centre most rapidly when P = h2 u3, and find where this is in an elliptic orbit with the centre of force at the focus. 7. A string is attached to a point of a circular disc and carries a particle on its other end. Initially the whole is at rest, and the string is a prolongation of a radius of the disc, and is equal to it in length. If now the disc begins to revolve uniformly about its centre, show that the string will begin to wrap on the disc when the latter has turned through an angle 8. Investigate the general formulæ for the centre of pressure of a plane area immersed in a heavy liquid using rectangular co-ordinates. A sphere of radius a is totally immersed in a liquid with its centre at a depth 6 below the surface. Find the whole, vertical, and horizontal pressures on an eighth part of the sphere below the centre bounded by the horizontal and two vertical great circles. 9. Prove that the question of the equilibrium and stability of a floating body is the same as for a body bounded by the surface of buoyancy resting on a horizontal plane, the centre of gravity being the same in the two cases. 10. Two equal uniform thin rods are joined at one end of each and float partially immersed and juncture uppermost in liquid, the rods being equally inclined to the vertical. Show that the equilibrium is unstable for displacement perpendicular to the plane of the rods, and also in that plane if the ratio of the part not immersed to the rest is less than cot2 A where 24 is the angle between the rods. PURE MATHEMATICS.-PART I. FIRST PAPER. The Board of Examiners. 1. If f (a), f(b) have opposite signs, one root at least of the equation ƒ (x) = Ō lies between a and b. Shew that the roots of each of the equations (x − b) (x − c) — ƒ2 = 0 (x (x-a) (x - b) — h2 0 separate the roots of the equation (x − a) (x — b) (x — c) — ƒ2 (x — a) · g2 (x — b) — h2 (x — c) — 2fgh = 0. 2. State and prove Newton's theorem concerning the sums of the powers of the roots of an equation. x8 + y8 + 28 = } (x2 + y2 + 22)4 + ‡ (x2 + y2 + z2) (x3 + y3 + z3)2. 3. Prove the binomial theorem for any exponent. Shew that the remainder after the first r terms of the expansion of (1 x), where m and x are positive and m x < 1, is greater or less than the (r+1)th term divided by 1x according as m> or < 1. 4. Find the sum of n terms of the series whose nth term is the reciprocal of (a + nb) (a + n + 1b)........ (a + n + m − 1 b). Sum to n terms and to infinity the series 5. State and prove the rule for forming convergents to a continued fraction of the form If Pn/qn be the nth convergent to the continued fraction q2n = P2n+1, bq2n+1 = аpån + abp2n+1• |