Find the form of the equation to the surface when two of the generators are taken as two of the axes of coordinates. 10. Find the condition that the equation Xd. + Ydy + Z =0 may be derivable from a single primitive, and shew how to find that primitive when the condition is satisfied. Find the solution of yz(y2 + z2)dx + zx(z2 + x2)dy + xy(x2 + y2)dz=0. 11. Give the theory of the solution of the equations MIXED MATHEMATICS.-PART I. PASS AND FIRST HONOURS PAPER. The Board of Examiners. 1. Define the total force on a particle, and express a force of 1 pound weight in dynes. Two particles of masses m, m, moving in the same straight line are connected by a straight light inelastic string. The forces on the particles are F, F, respectively in the direction of motion (exclusive of the tension of the string). Find the acceleration of the particles and the tension of the string. 2. A train is being drawn up an incline of 1 in 50 by an engine exerting a uniform pull of 100,000 poundals on the first carriage, and a dog running with the train gains on it 5 feet a second. The mass of the carriages is 60 tons, of the engine 40 tons, and friction is neglected. Find the accelerations of the train and dog and the work of the engine in one minute from starting. 3. Two spheres of equal mass having velocities v, and vv, in the same straight line and in the same direction impinge. Assuming a coefficient of impact e shew that the velocities after impact are } (1 − e) v, + 1 (1 + e) v2 } (1 + e) v, + § (1 − e) vg. E 4. A particle of mass m describes a horizontal circle on a smooth cone with vertical axis and vertex upwards, being attached to the vertex by a string of length 1. If is the velocity of the particle and a the semivertical angle of the cone shew that the pressure of the particle on the cone is 5. Prove that it is necessary and sufficient for the equilibrium of a particle acted on by a number of forces in one plane that the sums of their resolved parts in two directions at right angles should vanish. A heavy particle of mass m rests on a horizontal plane. To the particle are attached two strings leaving at equal angles a to the plane. and in the same vertical plane. These strings pass over smooth pegs and sustain two equal masses M hanging freely. (i.) Find the pressure on the plane. (i.) If the coefficient of friction between plane and particle is u shew that one of the masses M may be increased by μ 6. Shew that a system of forces in one plane, on a rigid body, can in general be reduced to a single force at an arbitrarily chosen point and a couple. A uniform heavy rod AB of weight w and length 21 rests in contact with a smooth vertical wall at A making an angle a with it in a vertical plane. The rod rests on a smooth peg at a perpendicular distance h from the wall, and a string attaches B to a point 0 of the wall vertically over A, the angle ABO being a right angle. Shew that the pressure against the wall is w cot a and against the peg wl (2 sin 2 a) h 7. From a body of mass M whose centre of mass is at G, is cut out a piece of mass m whose centre of mass is at G. Find the position of the centre of mass of the remainder. 8. Prove the formula p = gph+II for the pressure in a heavy liquid under atmospheric pressure II at depth h, and define carefully the symbols in the formula. Find in dynes the total pressure on the horizontal base of a hemispherical vessel of 1 metre radius full of water, there being a small opening at the top of the vessel, and the barometric height being 76 cm. 9. A body of volume v and sp.g. > 1 is supported' wholly immersed, in water, by being attached by a string to a body of volume v' and sp.g. '< 1 which floats partly immersed. Find the volume immersed of the second body and the tension of the string. 10. A circular cylindrical vessel of height h and radius r, closed at the upper end, contains a perfectly fitting piston of weight w and thickness k. The lower face of the piston being flush with the lower end of the cylinder, and the length h k of the cylinder containing air at atmospheric pressure II, the cylinder is put vertically into water and pressed down until the lower face of the piston is at a depth 7 below the surface of the water outside. Shew that a length II (hk) II + gpl - w/wr2 of the cylinder is now occupied by air. MIXED MATHEMATICS.-PART II. The Board of Examiners. 1. It is required to hit a point O at a height h above the ground with a particle projected with velocity v from a point on the ground at a horizontal distance c from 0. Shew that the angle of elevation a is given by the equation gc2 tan2a 2v2c tan a + 2v2h + gc2 = 0. 2. A particle of mass m has a simple rectilinear harmonic motion of period T. Shew that the force on it must be m T 2 times the distance from the mean position. A small smooth heavy ring of mass m is slung on a light elastic string of modulus X and natural length 7, the ends of which are attached at two |