5. Prove Jacobi's theorem that an ellipsoid with three unequal axes is a possible form of equilibrium of a rotating gravitating mass of liquid. Investigate whether the theorem can be extended to a case where there is a solid ellipsoidal core of greater density than the surrounding liquid. 6. Contrast the aims and the utilities of the static and the kinetic theories of the tides. Investigate the modification that would be produced by the Earth's rotation in the tidal phenomena in a circular basin of uniform depth with its centre at the North Pole. SECOND PAPER. PROF. BERGIN; PROF. CONWAY; MR. LARMOR; 1. Develop an explanation of the general features of the system of waves that are produced by a ship in rapid motion through still water. Contrast with the wave-system produced by a bullet travelling through air. 2. Investigate the motion of a small sphere through viscous fluid, such as a globule of water falling under gravity through air, determining, in the latter case, the terminal velocity, and enumerating the conditions which must be satisfied that the solution may be applicable in practice. It is observed that a small indiarubber balloon in air much more readily acquires a motion of spin than a motion of translation: compare quantitatively the efficiencies of the various causes that may contribute to this result. 3. Give an account of what is known as to the possibilities of stability in linked vortex-rings. 4. Discuss the problem whether it is possible for a hollow vortex column of given cross-section and given vorticity to exist in a stationary position in an infinite mass of frictionless liquid bounded by two rigid walls intersecting at right angles. Consider specially the case when the cross-section is small. 5. Give an account of the method of solution of problems in the discontinuous two-dimensional flow of liquids; work out the rate of discharge in the case of a liquid flowing through a slit in a plane wall. Discuss the stability of a jet of liquid projected into another liquid. 6. A liquid film connects two equal circles having a common axis: find the form of the film (1) when the circular ends are closed, (2) when they are open; and find conditions for stability in the two cases. ELECTRICITY AND MAGNETISM. PROF. BERGIN; PROF. CONWAY; MR. LARMOR; 1. Give an account of Schwarz's transformation as applied to the solution of two-dimensional problems in electrostatics. If denotes the stream function, and the potential function, examine the physical meanings of the relations 2. The charge on a conducting sphere is held displaced as it would be in a uniform electric field. The charge being suddenly released, obtain expressions for the external disturbance transmitted into the surrounding region, and show that it almost ceases after a few oscillations. 3. Show that the problem of finding the steady distribution of electricity on a system of conductors moving with high uniform velocity may be reduced to the general problem of electrostatics, when the conductors are elongated in a certain ratio. Examine to what approximation the effects of this elongation could be compensated by assuming that the internal forces of the material system are due to electric forces between the electrons which constitute the atoms. 4. Assuming that the Hall effect can be represented by an electromotive force which is perpendicular to the total current and to the magnetic induction, and is proportional in magnitude to the product of these quantities into the sine of the angle between them, show that plane electromagnetic waves falling upon a plane face of a magnet give rise to two circularly polarized waves in the magnetic substance moving with different velocities. Discuss the value of the above hypothesis. 5. In the electromagnetic theory of light, show that the disturbance at any point may be considered to be due to simple vibrators spread over any closed surface which contains the sources of the radiation. Determine the character of the disturbance produced in the ether by the motion of a point-charge of electricity; prove that, at any instant, it is losing energy by radiation at the time-rate (ef)/V, where e is its electromagnetic value, fis its acceleration, and is the velocity of radiation. 6. Discuss the phenomena of residual discharge in a condenser, on the theory that the dielectric is of heterogeneous character and possesses slight conductance. SECOND PAPER. PROF. BERGIN; PROF. CONWAY; MR. LARMOR; 1. Investigate the distribution of a given charge of electricity on a nearly spherical conductor of given form when no external electric force acts on it; show that its electric capacity depends only on its volume. 2. Express the magnetic potential at a point, due to a circular electric current, in a series of spherical harmonics. Obtain an approximate expression for the coefficient of mutual induction of two rings of radii a and a c, on the same axis with their planes a distance b apart (i) when b and c are small compared with a, (ii) when b is great compared with a. Describe a method of experimental con firmation of the result. 3. A solid ellipsoid of magnetizable material is introduced into a uniform field of magnetic force, the lines of the undisturbed field being parallel to the principal section (a, b) of the ellipsoid, and making an angle with the axis a. Find the couple acting on the ellipsoid, showing that it tends to turn the longer axis towards the direction of the force, whether the material is diamagnetic or paramagnetic. If the ellipsoid be pivoted on the axis c, show that, in the absence of hysteresis, it will oscillate as a quadrantal pendulum, and find an expression for the period. 4. Discuss the evidence in favour of taking the electrokinetic energy in any region to be proportional to the volumeintegral of the square of the magnetic force. If the vectors in an electromagnetic field are periodic functions of the time, show that the difference of the mean values of the magnetic and electric energies in any region is equal to the mean value of the flux, across its boundary, of the vector whose components are (Gy- HB), (Ha - Fy), (FB - Ga), where (aßy) is the the magnetic force, and (FGH) the vector-potential. Deduce under what circumstances these mean values are equal, and give a general explanation. 5. Discuss the conditions on which success in the problem of telephoning through submarine cables must depend. In the transmission of signals along a cable whose selfinduction may be neglected, show (i) that the speed of propagation of a harmonic disturbance is proportional to the square root of the frequency, (ii) that the time necessary to establish a given potential at a given point along the line is jointly proportional to the capacity per unit length, the resistance per unit length, and the square of the distance of the point. 6. Investigate the laws of the scattering of electromagnetic waves by small particles, applying the results to the problem of the blue colour of sky light, and attending to the polarisation effects observed. (a) Huius ego alienus consiliis consul usus sum: tu, sororis filius, ecquid ad eum umquam de re publica rettulisti? At ad quos refert? Di inmortales! ad eos scilicet, quorum nobis etiam dies natales audiendi sunt. Hodie non descendit Antonius. Cur? Dat natalicia in hortis. Cui ? Neminem nominabo. Putate tum Phormioni alicui, tum Gnathoni, tum Ballioni. (b) Proficiscitur in Hispaniam Caesar, paucis tibi ad soluendum propter inopiam tuam prorogatis diebus. Ne tum quidem sequeris. Tam bonus gladiator rudem tam cito accepisti? Hunc igitur quisquam, qui in suis partibus, id est in suis fortunis, tam timidus fuerit, pertimescat ? (c) Tua illa pulchra laudatio, tua miseratio, tua cohortatio. Tu, tu, inquam, illas faces incendisti, et eas, quibus semiustulatus ille est, et eas, quibus incensa L. Bellieni domus deflagrauit. Tu illos impetus perditorum hominum et ex maxima parte seruorum, quos nos ui manuque reppulimus, in nostras domos inmisisti. (d) Qui chirographa Caesaris defendisset lucri sui causa, is leges Caesaris, easque praeclaras, ut rem publicam concutere posset, euertit. |