10. Find a formula of reduction for (x2 + a2)—" dx, and determine the value of the integral when n = 5.

11. Find the whole area of the curve

b4y2x2 (a2x2).

12. Find the volume generated by the revolution about the axis of a of the curve in the preceding question.

PURE MATHEMATICS.-PART III.

The Board of Examiners.

1. State and prove the rule for finding the differential coefficient of a function of several functions of one variable, and apply it to find the second differential coefficient.

2. Shew how to eliminate the arbitrary functions 0,, from the equation

f{x, y, z, 0(a), p(a), 4(a)} = 0 where a is a given function of x, y, z.

Eliminate 0, 0, from

z = 0 (ax + by) + xp (ax + by) + x2 ↓ (ax +by). 3. Find the polar equations of the tangent and normal at any point of a plane curve.

Find the number of tangents and the number of normals which can be drawn from an arbitrary point to the curve

rP sin? += a2 cos10

where p, q are positive integers.

4. Trace the curves

(i) x + y2 = a2xy.

(ii) r5 cos 50 = a3.

5. If k be positive and less than unity, prove that

6. Investigate formula for transforming from one set of rectangular axes to another set with the same origin, and express the coefficients of the transformation in terms of three quantities.

7. Shew that there are two systems of generating lines on a hyperboloid of one sheet.

Prove that two generators do or do not intersect according as they are not or are of the same system.

8. State and prove Euler's theorem relating to the curvatures of normal sections at any point of a surface.

Find the principal radii of curvature of the surface

2zax2 + 2hxy + by2

at the origin.

9. Shew how to find the complementary function of an ordinary linear differential equation with constant coefficients.

where X, Y, Z are given functions of x, y, z.

(bz — cy) p + (cx — az) q = ay — bx.

12. Shew how to integrate the homogeneous equation ƒ (D, D') z = 4 (ax + by),

the coefficients in f being constant.

MIXED MATHEMATICS.-PART I.

The Board of Examiners.

PASS AND FIRST HONOUR PAPER.

1. Two particles P, Q move in the same straight line. Prove that the velocity of P relative to Qis the difference of the velocities of P and Q relative to any point O in the line, and that the corresponding proposition is true for accelerations.

The two particles P, Q start from the same point 0, and move with the same acceleration f The initial velocity of P is v, that of Q is equal and opposite to the velocity which P has after a time t. Shew, from first principles, that at time t the particles are at a distance apart

(2v + ft)t,

and hence shew that the distance described by either particle in the time t is

vt + ft2.

2. Find the highest point and the horizontal range of a projectile of given initial velocity and elevation, the resistance of the air being neglected.

Shew that if two projectiles have the same initial velocity and the same range but different elevations, the product of their greatest heights is one-sixteenth of the square of the range, and the sum of those heights is the height reached in vertical projection with the same velocity.

3. Prove that the increase of kinetic energy of a particle is equal to the work done on it, the force being constant or variable and in any direction.

After a body of mass m has fallen from a height of h feet, it is required to arrest its motion with a force not exceeding P lbs. weight. Shew that the force must be applied through a distance of h(P/mg - 1) feet at least.

4. State and prove the principle of the Conservation of Momentum.

Two balls of masses 1 lb. and 3 lbs. are connected by an elastic string of modulus 10 lbs. weight, and are held apart so that the length of the string is 1 foot. The balls, being set free, come into collision, the string then having its natural length of 6 inches. Applying the equations of energy and momentum, shew that the velocity of the 3-lb. ball at collision is 5g/12, and, if the coefficient of impact is 8, find the velocities of the balls immediately after collision. Assume Hooke's Law for the string, and neglect gravity.

5. Shew that two equations are necessary and sufficient to express the condition of equilibrium of a particle confined to one plane.

A mass M is kept at rest on a smooth inclined plane of elevation a by means of a string attached to the mass and to a point of the plane. Find the tension of the string and the pressure on the plane. If the pressure is taken off the plane by means of a second string attached to the mass and pulled at an angle B with the horizontal, find the tensions of the two strings.