Problems

Moving axes as applied to the geometry of curves and surfaces.

The uniform convergence of series.

The theory of Riemann integration.

Doubly periodic functions.

Frobenius' method for the solution of differential equations. Illustrate your account by discussing fully Bessel's equation \[ x^2y''+xy'+(x^2-n^2)y=0. \]

Give a general account of the theorems connecting the Volume, Surface and Line integrals of mathematical physics, showing for example how they are applied in Electromagnetic and Hydrodynamical theory.

Write a short account of the principal energy exchanges which occur during the production of a steady current by a voltaic cell or accumulator, during the charging of the latter, and during electrolytic decomposition of (say) water. Prove the following general theorems on steady currents in linear conductors.

1. [(i)] If \(F=\Sigma C^2R\), the summation being taken over all conductors in a network, and if no accumulation of charge is going on at any point, then the currents must actually be distributed so that \(F-\Sigma EC_s\) is a minimum, \(E_s, C_s, R_s\) being the E.M.F., current and resistance of the \(s\)th conductor.
2. [(ii)] If a unit current is led into the network at A and out at B, the potential difference between C and D is the same as the potential difference between A and B when the unit current is led in at C and out at D. There is no cell producing an E.M.F. in any conductor.
3. [(iii)] If a unit E.M.F. exists in the conductor AB, the current produced in the conductor CD is equal to the current produced in the conductor AB by a unit E.M.F. in CD.

The stability of floating bodies.

The general theory of forces, couples and wrenches in three dimensions.

The vibrations of uniform strings, or plane waves of sound.