The proof of Cauchy's theorem.
Graphical methods in statics and their geometrical applications.
Theorems on the changes in motion of a system of bodies produced by impulses.
The "instantaneous ellipse" for a particle moving under a gravitational force to a fixed centre and a perturbing force.
The motion of a top.
The "uniqueness theorems" of electrostatics and their applications in the methods of images and of inversion.
The properties of the velocity-potential \(\phi\) and the stream-function \(\psi\) in the hydrodynamics of an incompressible fluid in two and three dimensions.
The porous-plug experiment and the determination of absolute temperature.
Solve the equations: \begin{align*} x+y+z+w &= 1, \\ ax+by+cz+dw &= \lambda, \\ a^2x+b^2y+c^2z+d^2w &= \lambda^2, \\ a^3x+b^3y+c^3z+d^3w &= \lambda^3. \end{align*} Prove that, if \(a, b, c, d, \lambda\) be all real and unequal, at least two and not more than three of \(x, y, z, w\) are positive.
If \(\phi(x)\) be a function such that \(\phi(x)+\phi(y) = \phi\left(\frac{x+y}{1-xy}\right)\) for all values of \(x, y\) such that \(xy \neq 1\), shew that \[ (1+x^2)\phi'(x) = \phi'(0). \] If \(\phi'(0)=1\) and \(x>0\), shew that