Q(x + u, y + v, z+w). Shew that if the direction cosines of the normal to the surface at P were l, m, n, those at Q are l (1 + el2 + ƒm2 + gn2

+ amn + bnl + clm) — l

du

dx

m

dv

-

dw

dx n dx

and two similar; e, f, g, a, b, c being the elements of strain.

6. If,

satisfy Laplace's equation, and are finite and continuous through the space exterior to the surfaces S, shew that

where the integrals are taken over the surfaces.

Hence shew how to find the motion of a liquid due to given motions of bodies in it when the motion due to a source at all points in the presence of the bodies is known.

con

7. A rectangular canal of depth h and length tains liquid which is under the influence of a small uniform simple harmonic force of period 2/p, and intensity X. Shew that the velocity potential of the resulting motion is oΣ X An cosh n (y + h) cos nx cos (pt + a) where A2 = 4p/n2 (gn sinh nh - pcosh nh) x being measured along the canal, and y vertically upwards.

n

8. Express xZ, in surface harmonics over the sphere ra where Zn is a solid harmonic of order n.

A nearly spherical solid r = a (1 + aZ) moves with velocity V parallel to the axis of a in infinite liquid. Shew that the velocity potential is

a3x

m 1 a2m+1 dZm

дов

+ Ꮴ 3

m (2m + 1) g2m-1

9. A rigid body in an infinite liquid with one point fixed, moves, under any forces, as if the liquid were not present, and the form and mass of the body were different.

10. A body is floating in liquid, and is turned through a small angle 0, about a line in the surface, without altering the displacement. Shew that the change of potential energy is W 02 [Ak2 | V — HG], where Ak2 is the moment of inertia of the plane of flotation about its undisplaced line; Vis the volume immersed, H its centroid, and G that of the body. Deduce the condition of stability.

SCHOOL OF NATURAL PHILOSOPHY.

NATURAL PHILOSOPHY.

GENERAL PHYSICS AND HEAT.

The Board of Examiners.

1. Describe fully and give the theory of the method of determining by torsional vibrations the modulus of rigidity of a substance.

2. Describe a method of determining the absolute density of a gas.

Obtain a formula including all necessary corrections which shall give the absolute density in terms of the results of experiment.

3. Describe Amsler's planimeter, and give the theory of it.

4. Describe how to determine the absolute pressure of saturated water vapour for temperatures ranging between - 30° Ć. and 200° C.

5. If you were asked to investigate the correctness of a law of cooling expressed, say, by

where a is a constant and 0 and r are the absolute temperatures of the hot body and the enclosure respectively, for a range between these of 200° C., how would you proceed?

6. Find the law of the permanent temperatures in a metal bar one end of which is kept at a constant temperature.

Describe Ingenhaus's experiment, and show that if you use in it two similar bars of the same dimensions but of different materials, then their conductivities are proportional to the squares of the lengths of paraffin melted.

7. Describe Jamin's experimental method for determining the ratio of the two specific heats of air, and discuss the theory of the method.

8. Write a short account of the Second Law of Thermodynamics.

9. Describe fully how, by means of a diagram, the thermal properties of a vapour in presence of its liquid can be represented.

If m be the specific heat of the substance in the liquid state and m1 its specific heat in the state of saturated vapour at the same temperature (0); prove that

where L is the latent heat of vaporisation at the same temperature.

10. If three equal masses of the same substance at temperatures 01, 02, 03, be enclosed in a vessel impervious to heat, show that the utmost amount of work that can be derived from them, assuming that the specific heat c of the substance is constant, is

m c {0, + 02 +03 − 3 √ 0, 0, 0 }

02

where m is the mass of one of the bodies.
Show also that the final common temperature

is = 0, 0, 03.

NATURAL PHILOSOPHY.

SOUND AND Light.

The Board of Examiners.

1. Explain fully the principle of sympathetic resonance, and how it has been applied to the investigation of complex musical notes.

2. Investigate the effect on the apparent pitch of a musical note when the listener, the source of sound, and the medium, are all in motion with different velocities in the line joining the source and listener.

3. Show how to calculate the positions of loops and nodes in open and closed organ pipes, and how to calculate the velocity of sound by means of them. In this method for velocity an error comes in; what is it, and how can it be eliminated?