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

revolution.

Determine the volume generated by the revolution about the axis of y of the curve

y = ae

PURE MATHEMATICS.—PART III.

The Board of Examiners.

1. If u be a function of x, y, z where x, y, z are functions of t, state and prove the rule for finding the differential coefficient of u with respect to t.

Find the second differential coefficient of u with respect to t.

2. Find the maximum and minimum values of a function of three variables which are connected by a given equation.

Determine the parallelopiped of maximum volume inscribed in a given ellipsoid with its edges parallel to the axes.

3. Shew how to find the asymptotes of a curve whose equation is given in the form

where u, is a homogeneous function of x, y of r dimensions.

Find the asymptotes of the curve

x2y2(ax+by+ c) + a'x + b'y + c' = 0.

4. State and prove a rule for differentiating a definite integral with respect to any quantity involved in the function under the sign of integration.

5. Define the Beta and Gamma functions, and the formula connecting them.

prove

6. Investigate formulæ for transforming from one set of rectangular axes to another set with the same origin, and prove the principal relations among the coefficients of transformation.

7. Determine in magnitude and position the semiaxes of a given central plane section of an ellipsoid.

Find the envelop of central plane sections of the same shape.

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

Find the principal radii of curvature at any point of the surface

9. Shew how to find the orthogonal trajectory of the system of curves

f(x, y, c) = 0.

Find the trajectory of the parabolic system y" = cx.

10. Find the condition that the equation

Id + Ydy + Zdz=0

may be derivable from a single primitive, and shew how to find the primitive when the condition is satisfied.

Discuss the case in which the equation is homogeneous.

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

MIXED MATHEMATICS.-PART I.

The Board of Examiners.

PASS AND FIRST HONOUR PAPER.

1. Define the acceleration at any instant of a particle moving in a straight line, and investigate an expression for the distance described in time t with uniform acceleration f, u being the initial velocity.

A mass of 20 lbs. moves with uniform velocity down a line of slope on a rough inclined plane of elevation 45°, the coefficient of friction being 1/5. The air-resistance varies as the square of the velocity, and is 1 lb. weight when the velocity is 50 feet per second. Shew that the uniform velocity of the mass is 168 feet per second nearly.

2. Define kinetic energy and potential energy. Prove the equation of energy in as general a form as

you can.

A mass of 10 lbs. is hung up by an elastic string of natural length 1 foot, and modulus. 50 lbs. weight. A mass of 2 lbs. is let fall from the point of suspension of the string and, hitting the 10-lb. mass, becomes fixed to it. (a) the initial velocity of the combined masses, (b) the subsequent maximum elongation of the string, (c) the period of the vertical oscillation

that ensues.

Find

3. Find the acceleration of a particle moving uniformly in a circle of radius r with angular velocity w.

A mass M is in relative rest on the inside of a circular cylinder of radius r rotating about its (horizontal) axis with angular velocity w. Find the force between the mass and the cylinder in any position, and shew that the coefficient of friction must not be less than g/√w1μ‚2 — g2.

4. A mass M is suspended by an inextensible string of length 7. Find the force at right angles to the string required to hold the mass in a position in which the string makes an angle ✪ with the vertical.

If the mass is moved in a vertical plane at a constant speed v by a force always at right angles to the string, find in any position (a) the magnitude of the applied force, (b) the tension of the string, (c) the rate of work.

5. 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 points, not in one straight line, should separately vanish.

A uniform beam of length and weight w rests at an inclination 0 to the horizontal with its lower end on a rough horizontal plane, being supported by a smooth horizontal rail at a height h above the plane. Shew that the smallest coefficient of friction for which equilibrium is possible is

7 sin2 0 cos 0