Find the velocity of long waves in a uniform channel of rectangular section containing an incompressible fluid under gravity. Assuming that the tide in a river of depth \(h\) consists of a simple harmonic wave of period \(T\) and that the above theory applies, show that a small floating body will be carried up the river by the flow of the tide a distance \[ \frac{AT}{2\pi}\sqrt{\frac{g}{h}}, \] where \(A\) is the height of the tide. Evaluate numerically for \[ h=18 \text{ feet}, \quad A=2 \text{ feet}, \quad T=12 \text{ hours}. \]
Explain briefly the method of images for the solution of problems in electrostatics. Show that the image of a point charge in a conductor consisting of the outer segments of two orthogonal spheres consists of three point charges, and that the capacity of such a conductor freely charged is \[ a+b-ab/\sqrt{a^2+b^2}, \] where \(a,b\) are the radii of the two spheres.
Show that the mutual potential energy of two small magnets of moments \(M,M'\) is \[ MM'(\cos\epsilon-3\cos\theta\cos\theta')/r^3, \] where \(r\) is the distance apart of their centres, \(\epsilon\) the angle between their axes and \(\theta, \theta'\) the angles made by their axes with the line of centres. Three small magnets of equal moment \(M\) can rotate in a plane about their centres, which are fixed at equal intervals \(a\) along a straight line. Find the periods of the normal oscillations about the position of equilibrium in which the axes all point in the same sense along the line of centres.
A Wheatstone bridge has resistances as shown and \(A,B\) are maintained at a constant potential difference \(E\). Show that when \(S\) is large compared with \(R_1\) and \(R_2\), the current through the galvanometer is approximately \[ E(R_2-R_1)/S(R_2+R_1), \] and that the potential difference between \(A\) and \(C\) is approximately \[ ER_1/(R_1+R_2). \] The wires are uniform and made of the same material of specific heat \(\sigma\), and their resistances increase with the temperature \(\theta\) according to the law \[ R_1=r_1(1+\alpha\theta), \quad R_2=r_2(1+\alpha\theta), \] where \(\alpha\) is small. Under the influence of the currents each wire is supposed to heat up uniformly from the temperature \(\theta=0\) without loss of heat. Show that, to the first order in \(\alpha\), \(R_1\) and \(R_2\) increase linearly with the time. If the bridge is initially in balance, show that at a time \(t\) after the potential difference between \(A\) and \(B\) has been applied the current through the galvanometer is approximately \[ \frac{aE^3t}{SJ\sigma(r_1+r_2)^3}\left(\frac{r_2^2}{m_2}-\frac{r_1^2}{m_1}\right), \] where \(J\) is the mechanical equivalent of heat, \(m_1\) the mass of each wire \(AC,DB\) and \(m_2\) the mass of each wire \(AD,CB\).
Two conics touch at \(A\) and intersect in \(B\) and \(C\). A line through \(A\) meets the conics in \(P\) and \(Q\). Shew that the tangents at \(P\) and \(Q\) meet on \(BC\). State the reciprocal theorem.
Shew that there are two spheres, real, coincident, or imaginary, which pass through three given points and touch a given plane, explaining how the three cases arise. Find also how many spheres pass through three given points and touch a given line.
Shew that, if \(\tan\alpha, \tan\beta, \tan\gamma\) are all different and such that \[ \tan 3\alpha = \tan 3\beta = \tan 3\gamma, \] then \[ (\tan\alpha+\tan\beta+\tan\gamma)(\cot\alpha+\cot\beta+\cot\gamma) = 9. \] If \(\tan\alpha:\tan\beta:\tan\gamma = a:b:c\), shew that \(\tan^2\alpha = \displaystyle\sqrt{-\frac{3a^2}{bc+ca+ab}}\). Generalize the first theorem to the case of \(2n+1\) angles \(\alpha, \beta, \dots, \lambda\).
Give without proof expressions for \(\sin\theta, \cos\theta\) in terms of \(t \left(=\tan\frac{\theta}{2}\right)\). If \(\theta\) is an acute angle, shew that \[ \frac{\tan\theta}{\theta} > \frac{\sin\theta}{\theta}. \] Hence, or otherwise, prove that the equation \[ \frac{1}{\sin\theta} - \frac{1}{\theta} = k \] is satisfied by one and only one acute angle \(\theta\) if \(0 < k < 1 - \frac{2}{\pi}\), and by no acute angle \(\theta\) if \(k\) lies outside these limits.
If \(ax+by+cz=1\) and \(a,b,c\) are positive, shew that the values of \(x,y,z\) for which \(\displaystyle\frac{1}{x}+\frac{1}{y}+\frac{1}{z}\) is stationary are given by \[ ax^2=by^2=cz^2. \] Shew that this is a true maximum or minimum if \(xyz > 0\).
Define the mean value of \(f(x)\) with respect to \(x\) for values of \(x\) lying in an interval \((a,b)\). A point moves along a straight line in such a way that \[ v_t = v_s+ks, \] where \(v_t, v_s\) are the mean values of the velocity with respect to the distance travelled \(s\) and the time taken \(t\) respectively, and \(k\) is a constant. Shew that \(s,t\) satisfy the equation \[ \frac{ds}{s} = \frac{dt}{t}\{1+kt \pm \sqrt{kt(2+kt)}\}. \] Interpret this solution in the case \(k=0\), and shew on general grounds that a negative value of \(k\) is inadmissible.