5. Find, in magnitude and direction, the shortest distance between two straight lines whose equations are given.

Determine the shortest distance between two non-intersecting diagonals of two adjacent faces of a rectangular parallelopiped.

6. Investigate formula for the transformation of rectangular coordinates with a common origin, and prove the principal relations connecting the direction cosines.

7. Find the circular sections of a hyperboloid of one sheet, and shew that the surface has no real umbilics.

8. Find a formula for the radius of curvature at any point of a curve in space, and apply it to the

curve

x = 4a cos3 0, y = 4a sin3 0, z = 3c cos 20.

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

A surface is formed by the revolution of a catenary about its directrix. Shew that the principal curvatures at any point are equal and opposite.

10. Shew how to find the orthogonal trajectory of

the family of curves

f(x, y, c) = 0.

Find the orthogonal trajectory of the family x2 + y2 + c2 = 1 + 2cxy.

11. Shew how to find the complementary functions and the particular integrals of two simultaneous linear differential equations with

coefficients.

dx

dt

dy

It

Integrate completely

constant

+(x cos a + y sin a) cos a = cos ẞ + et cos a,

+(x cos a + y sin a) sin a = sinß + et sin a.

12. Show how to integrate the homogeneous equation f(D, D')z (ax + by)

=

MIXED MATHEMATICS.-PART I.

The Board of Examiners.

PASS AND FIRST HONOUR PAPER.

1. A particle moves in a straight line with uniform acceleration. Prove the fundamental formulæ connecting the time, the velocity, and the distance traversed.

A mass of 1 kgm. moves in a straight line, and is observed to traverse 3 metres in one second, and 12 metres in the next two seconds. Find (a) its acceleration (supposed uniform), (b) its velocity at the beginning of the first second, (c) the force on it, (i) in dynes, (ii) in kgms. wt.

2. Explain what is meant by the Principle of the Independence of Forces.

Find the acceleration of a heavy particle of mass m moving up a line of greatest slope on a plane of inclination a, (a) when the plane is smooth, (b) when the plane is rough, and a force Falong the plane upwards acts in addition to the weight. Find the pressure on the plane in each case.

3. Prove the Equation of Work for a single particle.

Two balls, each of mass 1 lb., are placed on a smooth horizontal plane, and connected by a light elastic string of natural length 6 ft., and modulus 12 lbs. wt. The balls are drawn apart to a distance of 8 ft., and then released. Find

(a) the velocity of the balls when the string becomes slack [by the equation of work], (b) their velocity after impact (coefficient e), (c) the greatest subsequent elongation of the string.

4. Investigate the acceleration of a particle moving with uniform speed v in a circle of radius r.

It is required to keep a heavy particle moving with given uniform velocity v in a horizontal circle of given radius by means of a string attached to the particle and to a fixed point vertically above the centre of the circle. Shew that this fixed point must be a distance gr2/v2 above the circle.

5. Shew that the small oscillations of a simple pendulum are very nearly a Simple Harmonic Motion, and hence prove the formula 27 lg for the period.

Shew that a pendulum beating seconds where g=980 gains 44 secs. a day, nearly, where g= 981.

6. Shew how to express algebraically the conditions that a given system of forces on a particle in one plane may be in equilibrium.

A mass of 100 kgms. is supported on a rough plane of inclination 30° by means of a rope at an inclination of 45° to the horizontal, the coefficient of friction being 1/6. Find the least and greatest tensions of the rope consistent with equilibrium. Find also the work done per metre in pulling the body up the plane by means of the rope, kept at the same inclination.

7. Prove the formula z Em Emz for the distance of the c.m. of a system of particles from a given plane.

A uniform lamina in the shape of a rightangled triangle having one side half the hypothenuse, is suspended by the right angle. Shew that the additional weight to be placed at the larger base angle to keep the hypothenuse horizontal is two-thirds the weight of the lamina.

8. Shew that it is necessary and sufficient for the equilibrium of a rigid body acted on by forces in one plane, that the sums of the moments of the forces about three non-collinear points should separately vanish.

A uniform square board of side a and weight W rests with one edge on a rough horizontal floor, and with its plane at an inclination a to the horizontal; being supported in this position by a light rod, of which one end is hinged to the board at a distance b along the middle line of slope from its lowest edge, and the other rests on the ground. If ẞ is the inclination of the stick to the horizontal, find (a) the thrust of the stick, (b) the pressure of the board on the ground, (c) the least permissible coefficient of friction with the ground.

9. Investigate the ratio of applied force to resistance in the case of a square-threaded screw of pitch p, radius a, and arni b, when the screw is (a) smooth, (b) rough.

Find the greatest pitch that the screw can have without its running back when the applied force ceases to act.