6. Prove that subject to certain conditions

f(a + h) = f (a) + hf'(a) + }h2ƒ" (a + 0h), where 0 is a positive proper fraction, and state the conditions.

Apply this formula to justify the use of the principle of proportional parts in the case of the logarithmic tangent of an angle, except when the angle is small or nearly equal to a right angle.

7. Find the equations of the tangent and normal at any point of the curve

x = p(t), y = (t).

Find in their simplest forms the equations of the tangent and normal at any point of the

8. Define the curvature of a curve, and prove the

formula

9. State and prove the rule for integration by parts.

Integrate √2+ a2, and shew how to find the value of

and shew that its value is (b) — (a) where (x) is any function which has p(x) for its differential coefficient.

11. Find a formula for the area bounded by the curve r =ƒ(0) and by two radii vectores.

Find the area of the loop of the curve

r(cos30+ sin30) 3a sin 0 cos 0.

12. Find a formula for the volume of a solid of revolution.

The curve

y1 = x3 (a — x)

revolves about the axis of x.

Find the volume

generated.

MIXED MATHEMATICS.-PART I.

The Board of Examiners.

1. Prove the polygon law for the composition of relative velocities.

A ship is steaming north at 15 knots, and the wind is blowing north-east at 20 knots; find the direction in which a flag on the ship flies.

2. State the laws of impact, and obtain formulæ for the velocities of two elastic balls after impact.

An elastic ball is let drop on to a horizontal plane. At a height h above the plane its velocity is observed to be v in falling and v' on rising. Find the coefficient of impact and the number of impacts before the ball ceases to reach the height h.

3. Investigate the position of the highest point and the horizontal range of a projectile fired with velocity v at an elevation a.

Find the distance of the particle from the point of projection at time t after projection.

4. Prove the formula 2g for the period of a simple pendulum.

Assuming that the acceleration of gravity varies inversely as the square of the distance from the earth's centre, shew that a pendulum loses about 21.5 seconds a day for each mile of

ascent.

5. Shew that if three forces on a particle are in equilibrium, each is proportional to the sine of the angle between the other two.

A weight w rests on a smooth horizontal table, and is attached by means of an elastic string of modulus X to a fixed point whose distance above the table is equal to the unstretched length l of the string. A horizontal pull P is now applied to the weight. Shew that if the weight does not leave the table and if P = 15λ the length of the string becomes 1 251, and find the pressure on the table.

6. A uniform heavy beam AC of weight Wis smoothly jointed at a fixed point A, and is maintained pointing obliquely upwards by a tie BC to a point B on the level of A and such that AB AC. A weight W is carried at C. Shew that the tension of BC is (2W' + W) cos 2C/2 sin C, and find the reaction at A.

7. Prove the formula = mx/2m for the centre of mass of a system of particles.

A uniform lamina forms a regular hexagon ABCDEF. Find the c.m. of the lamina when

the triangle ABC is removed.

8. Find the efficiency of a square-threaded rough

screw.

Verify that the work done by the power is equal to the sum of the works done against the opposing force and friction.

9. Investigate the formula for the magnitude of the resultant pressure on a plane area immersed in heavy liquid.

Shew that if a reservoir with a dam-wall of rectangular section and specific gravity 3 can be filled without upsetting the wall, the height of the wall must not be greater than three times its thickness if the strength of the joint at the base of the wall may be neglected.

10. A barometer with an imperfect vacuum reads h' instead of h at temperature 0° C., the length of the vacuum being then a. Find the error due to the enclosed air of a reading h" at tempera

ture t.

11. Find the pressure at any point of a liquid rotating steadily as if rigid about a vertical axis.

If the liquid is contained in a long vertical circular cylindrical vessel of radius r open at top, find the depth of the hollow formed when the angular velocity about the axis of the cylinder is w.

MIXED MATHEMATICS.-PART III.
The Board of Examiners.

1. Investigate the equations giving the positions of equilibrium of a particle on the smooth surface F(x, y, z) = 0 and acted on by the external force (X, Y, Z) given as a function of the position of the particle.

Find the positions of equilibrium on an ellipsoid, the particle being acted on by three forces perpendicular to three orthogonal planes through the centre of the ellipsoid and proportional to the distances from those planes.

AA