Problems

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.

Establish Lagrange's equations of motion for a general dynamical system, including the form they reduce to for impulsive actions. Give proofs of the following general theorems:

1. [(i)] The reciprocal relation \(\dot{q}_s/P_r = \dot{q}'_r/P'_s\).
2. [(ii)] Bertrand's Theorem that the energy generated by given impulses is diminished by the introduction of any constraint.
3. [(iii)] Kelvin's Theorem that the energy acquired when the system is started with prescribed velocities is increased by the introduction of any constraint.

Illustrate these theorems by discussing the motion of a waggon of mass M carrying a pendulum of mass m and length l which can swing in the direction of motion of the waggon.

Shew that a plane section of a circular cone satisfies the focus-directrix definition of a conic, and that the eccentricity is equal to \(\cos\theta \sec\alpha\), where \(\theta\) is the angle between the plane and the axis of the cone, while \(2\alpha\) is the vertical angle of the cone: shew further that the latus-rectum of the section is equal to \(2p \tan\alpha\), if \(p\) is the perpendicular from the vertex of the cone on the plane of the section. Prove that an ellipse or parabola of given size can be cut from a cone of any assigned angle; and discuss to what extent this is possible for a hyperbola.

Two planes (\(\alpha, \alpha'\)) cut at right angles in a line \(m\) and the point \(V\) is the vertex of a conical projection from \(\alpha\) to \(\alpha'\). A third plane \(\beta\) is drawn through \(V\) to cut both \(\alpha\) and \(\alpha'\) at right angles, in the lines \(l\) and \(l'\) respectively. Shew that the formulae of projection may be written in the forms \[ x' = \frac{f'x}{x-f}, \quad y'=\frac{-fy}{x-f}, \] where \(l\) is taken as axis of \(x\), \(l'\) as axis of \(x'\), and the axes of \(y, y'\) coincide along \(m\); \(f\) is the distance of \(V\) from the plane \(\alpha'\) and \(f'\) is its distance from \(\alpha\). Shew that, if \(c=f-f'\), any line through the point \(N(x=c, y=0)\) is projected into a line through \(N'(x'=-c, y'=0)\) making the same angle with the line \(l'\) as the original line does with \(l\); and that any circle with \(N\) as centre is projected into a conic with \(N'\) as focus and the line \(x'-f'=0\) as directrix.