3. If u be a solution of the differential equation 5. Integrate of S xdx xa do (a + b cos 0 + e Sin 0) a+b2 Cos @ + c1 Sin @ 6. If up, (x, y) and v = :P2 (x, y) and if we can deduce from these equations the relations x = 41 (u, v) = dv du dv dy dy dx show then that (du du-dy d) (du do Vdxdydz be transformed so as to depend on integrations with respect to new variables u, v, w, connected with x, y, z, by the relations Find the value of 2+ Cos 2 x Sin when x = π 2 8. The locus of the extremity of the polar subtangent of any plane curve is the reciprocal of the evolute of the reciprocal curve. 9. Find the curve such that the normal is equal in magnitude and opposite in direction to the radius of curvature. 10. Integrate the equation 2. Every circle which cuts two of the three diagonals of a complete quadrilateral harmonically cuts the third also harmonically. 3. If A and A', B and B', &c., denote two homographic systems of points on the same right line, D a double point, shew that AD2 — AD (AI + AI′) + AÏ·AA' = 0 I and I' being the points of either system corresponding to a point at infinity on the other. 4. The equations being given, develop in a series proceeding according to sines of multiples of x; and log A in a series proceeding according to cosines of multiples of x. 5. Find the simplest values of the expressions Sin 10° Sin 30° Sin 50° Sin 70° and ing powers of x, there will be no even powers in the expansion: also A1x + A ̧æ3 + A, x3 + &c., shew by means of the exponential values of trigometrical functions that if u = 7. Simplify the expression 8. If abcd be the sides of a quadrilateral inscribed in a circle and if D, be the diagonal joining ab and cd and D, the diagonal joining be and ad; shew that D1: D2 = ab + cd; bc + ad. (a.) Hence write down the lengths of D, and D, in terms of the m-n to infinity. 10. Express the area of a spherical triangle in terms of the sides; also if p denote the perimeter of the triangle, and E, E,, Eg, Eg, the spherical excesses of the triangle itself, and of those formed by producing its sides, two by two: then will MIXED MATHEMATICS. Examiner-MR. JOHN ELIOT, M. A. 1. Shew that any system of forces acting on a rigid body may be replaced by a force acting at a given point and a couple. 2. Obtain formulæ for the position of the centre of gravity of a solid. Find the centre of gravity of an octant of an ellipsoid. 3. A heavy uniform flexible string rests on a given curve in a vertical plane. Obtain formulæ for the determination of the tension of the string and of the pressure at any point of the curve. A heavy uniform flexible string rests on a complete cycloid, the axis of which is vertical and vertex upwards, the whole length exactly coinciding with the whole arc of the cycloid, prove that the pressure at any point of the cycloid varies inversely as the curvature. 4. Investigate fully the laws of attraction by which the attraction of a spherical shell on an external particle is the same as that of its mass condensed at its centre of gravity. 5. State and prove the principle of virtual velocities. 6. Find the law of force tending to a given point which will constrain a particle to describe a given curve about it. Find the law of force so that a particle may describe the curve r = a sin 3 0 about the pole. Find the velocity at any point. Deduce the value of the radius of curvature at the apex. 7. A particle moves under the action of gravity on a smooth surface of revolution whose axis is vertical. Determine the motion. If the surface be a paraboloid (latus rectum 4a) with its vertex downwards, and the particle be projected along the surface in the horizontal plane through the focus with a velocity 2nag, the initial radius of curvature of the path will be 8. An impulsive tension in the direction at the tangent is applied at an extremity of a uniform perfectly flexible string lying on a smooth plane. Find the impulsive tension at any point of the string, and the initial direction of motion of each particle of the string. If all the particles start with equal velocities, prove that the string must lie in the form of a catenary or a straight line. 9. A particle is projected along a smooth circle with velocity V in a medium whose resistance a Va Prove that when the direction -kp. of motion has changed through an angle & the velocity is Ve MIXED MATHEMATICS. 1. A fluid at rest is acted on by forces. Obtain the differential expression for the pressure at any point; also the conditions which must be fulfilled by the forces in order that equilibrium may be possible. 2. Obtain equations giving the resultant pressure on any surface bounding a fluid at rest under the action of any given forces. What is the condition that there should be a single resultant pressure? 3. Define the term metacentre. Obtain an expression for the position of the metacentre of a body floating at rest in a homogeneous fluid. The solid formed by a portion of cy =z (a3 — x2) cut off by a plane parallel to that of xy floats in a fluid of n times its density. Prove if it is in neutral equilibrium for small angular displace. ments in any vertical plane 4. Prove that the surface of a mass of homogeneous fluid revolv. ing about a vertical axis in relative equilibrium is a paraboloid of revolution. A circular tube of radius a, which is very large in comparison with the radius of its bore, contains liquid filling one-twelfth of its circum. ference. The tube turns about the vertical diameter with uniform angular velocity. Prove that if the highest point of the liquid is in the horizontal diameter the angular velocity is 2√ for the pressure at any point of a moving fluid, assuming that the forces are such that Xdx + Ydy + Zdz dv, and that the velocities can give rise to a velocity function (i. e. are such that vdx + vdy + vdz аф). 6. What is meant by the focal lines of small oblique pencils? Obtain expressions giving their position in the case of a small oblique pencil refracted at a spherical surface. 7. Prove that for minimum deviation in the case of a prism, the angles of incidence and emergence are equal. 8. A ray passes through a medium, the value of μ at any point of which is a function of r, the distance from a fixed point. Find the differential equation to the path of the ray. If μ varies inversely as c2+r2, where c is constant, find the equa tion. с 9. If a pencil of light passes through two prisms, the edges of which are parallel, and the axis of the pencil passing in a principal plane of each, find the condition of achromatism. If the pencil passes through each with minimum deviation, the angles of incidence in the first and emergence from the second being and ", the two rays for which the refractive indices are μ, μ + dμ and μ', 'du' will emerge parallel if 9. What is meant by an equivalent lens ? Find the focal length (F) of a lens equivalent to a combination of two lenses (focal lengths fif) on the same axis at a distance a from each other. 10. A plane luminous ellipse throws light on a small area parallel to it and situated in the line drawn through the centre of the ellipse perpendicular to it. If a, b are the semi-axes of the ellipse, and c the distance of the small area the illumination is ab √(a2 + c2) (b2 + c2) MIXED MATHEMATICS. Examiner MR. JOHN ELIOT, M. A. 1. In a spherical triangle prove that cos a = cos b cos c + sin b sin c cos A. If from a fixed point P any great circle be drawn cutting a given PA PB 2 small circle in A and B, prove that tan tan is constant. 3. State and prove Newton's fourth Lemma. The area between two focal radii of a parabola and the curve is half the area between the curve and the corresponding perpendiculars on the directrix. 4. A particle describes a circle about a centre of force at an excentric point. Find the law of force. The sum of the reciprocals of the velocities at the extremities of any diameter is independent of the position of the centre of force and varies as the periodic time. |