Six variables \(x, y, z, u, v, w\) are connected by three relations, and (e.g.) \[ x_{w}^{u,v} \] denotes the partial differential coefficient of \(x\), expressed as a function of \(u, v\), and \(w\), with respect to \(w\). Show that \[ x_{w}^{u,v} = -x_{u}^{v,w} u_{w}^{x,z} - x_{v}^{w,u} v_{w}^{x,z}. \]
Prove by contour integration or otherwise that \[ \int_0^\infty \frac{\sin x}{x} dx = \frac{\pi}{2}, \quad \int_0^\infty \frac{e^{-x}\sin x}{x} dx = \frac{\pi}{4}. \]
Show that if two points are conjugate with respect to the three of the confocals \[ \frac{x^2}{a^2+\theta} + \frac{y^2}{b^2+\theta} + \frac{z^2}{c^2+\theta} = 1, \] whose parameters are \(\lambda, \mu, \nu\), then they will also be conjugate with respect to the coaxal quadric the squares on whose semiaxes are \[ a^2 + \frac{\lambda\mu\nu}{b^2c^2}, \quad b^2 + \frac{\lambda\mu\nu}{c^2a^2}, \quad c^2 + \frac{\lambda\mu\nu}{a^2b^2}. \] \item[A.] The axes of the molecular magnets in a mass of iron are originally disposed equally in all directions. Assuming that each magnet is acted upon by a local magnetic force \(D\) along its own particular direction, and that an external magnetic force affects the internal forces only by way of vector addition of its own amount, show that the intensity of magnetization produced by a uniform magnetizing force \(X\) will be \(\frac{1}{2}mn\frac{X}{D}\) if \(X \le D\), or \(mn(1-\frac{1}{3}\frac{D^2}{X^2})\) if \(X>D\). Here \(m\) and \(n\) denote respectively the magnetic moment of a molecule and the number of molecules per unit volume. \item[B.] Rays from a point source of light \(P\) pass through a plate of thickness \(t\) and refractive index \(\mu\), and enter the eye at \(E\). The plate is normal to \(PE\). The radius of the pupil is \(r\), and \(PE=c\). If \(Q\) is the geometrical focus of the rays after passing through the plate, show that the extreme rays entering the eye meet the axis in a point \(Q'\) such that \[ QQ' = \frac{1}{2}\frac{(\mu^2-1)r^2 t}{\mu^2 c^2 - (\mu-1)t^2}. \] Taking \(Q\) as the origin of \(x\), show that the rays passing through the eye envelope the surface of revolution generated by the curve \[ 8\mu^2 x^3 = 27(1-\mu^2)ty^2. \] \item[C.] Liquid of density \(\rho\) is contained between two confocal elliptical cylinders, the inner one of which is solid and of density \(\rho-\sigma\). The outer shell is moved with velocity \(u\) impulsively from rest; show that the initial relative velocity of the inner cylinder is \[ \frac{\sigma u}{\rho\frac{b_2(a_1^2-b_1^2)}{a_1(a_2 b_2 - a_1 b_1)} - \sigma}. \] The equation of the cylinders are \(\frac{x^2}{a_i^2}+\frac{y^2}{b_i^2}=1\). \item[D.] A light uniform rod \(AOB\) can turn freely about the point \(O\). At \(B\) is attached a massless extensible string \(BCD\), the end \(D\) of which is fixed. At \(C\) a particle of mass \(m\) is attached, and the string is subjected to a tension \(T\). In the equilibrium position, when \(AOBCD\) is a straight line, \(AO=CD=a, OB=BC=2a\). The mass of the rod is \(3m\). At time \(t=0\), the portion \(BC\) of the string is parallel to \(OD\), the distance between the two lines being \(\eta\). The particle at \(C\) is at rest, but the rod is turning in the plane \(BOD\) with an angular velocity \(k(T/ma^2)^{1/2}\), the angle \(BOD\) increasing. Find the distance of \(B\) from the line \(OD\) at any subsequent time \(t\), to the first order in \(k\) and \(\eta\). \item[E.] A straight massless rod of length \(2r\) can freely turn horizontally about its centre, which is fixed. At one end it carries a particle of mass \(m\), while the other end is tangential to a circular wire of mass \(\mu m\) and radius \(\lambda r\), which is suspended with freedom to swing about this tangent. The circle is drawn on one side so that its plane makes an angle \(\alpha\) with the vertical, and is then released, the system being at rest. Prove that the rod will oscillate through an angle \[ 4\sqrt{\frac{\mu}{3(2+\lambda^2\mu+2\mu)}} \tan^{-1}\left\{\frac{3\lambda^2\mu}{2+\lambda^2\mu+2\mu}\sin\alpha\right\}. \] \item[F.] Show from the following data that the eclipse of the moon on 1898 July 3rd was only partial, and calculate its magnitude. \begin{tabular}{l c l l} Moon's latitude at opposition in longitude & \dots & \(-30'~40''\) \\ Moon's hourly motion in latitude & \dots & \(+~3~~40\) \\ Moon's hourly motion in longitude & \dots & \(~~38~~2\) \\ Sun's hourly motion in longitude & \dots & \(~~~2~~22\) \\ Moon's equatorial horizontal parallax & \dots & \(~~61~~21.4\) \\ Sun's equatorial horizontal parallax & \dots & \(~~~~~~8.7\) \\ Moon's true semidiameter & \dots & \(~~16~~43\) \\ Sun's true semidiameter & \dots & \(~~15~~44\) \\ \end{tabular}