Find the coordinates of the centre of a conic whose equation in trilinear coordinates is \(l\beta\gamma + m\gamma\alpha + n\alpha\beta = 0\). If this conic passes through the centre of the circle inscribed in the triangle of reference, prove that its centre lies on the conic \[ a\alpha^2+b\beta^2+c\gamma^2 - (b+c)\beta\gamma - (c+a)\gamma\alpha - (a+b)\alpha\beta = 0. \]
Find the first and second differential coefficients of \(e^{ax}\cos bx\), and deduce that the \(n\)th differential coefficient is \[ (a^2+b^2)^{n/2}e^{ax}\cos\left(bx+n\tan^{-1}\frac{b}{a}\right). \]
Explain the meanings of the partial differential coefficients \(\frac{\partial r}{\partial x}\) and \(\frac{\partial x}{\partial r}\), where \(x,y\) are the rectangular coordinates of a point and \(r, \theta\) are its polar coordinates. Prove that \(\frac{\partial^2 r}{\partial x^2} + \frac{\partial^2 r}{\partial y^2} = \frac{1}{r}\left[\left(\frac{\partial r}{\partial x}\right)^2 + \left(\frac{\partial r}{\partial y}\right)^2\right]\).
If \(p\) and \(q\) are the lengths of the perpendiculars from the origin on the tangent and normal to a curve, prove that the radius of curvature is \(p+\frac{d^2p}{d\psi^2}\), where \(\psi\) is the inclination of the tangent to the axis. Shew that in the curve \begin{align*} x &= 3a\cos t + a\cos 3t, \\ y &= 3a\sin t + a\sin 3t, \end{align*} \(p=4a\cos\left(\frac{\psi}{2}-\frac{\pi}{4}\right)\) and that the radius of curvature is \(\frac{2}{3}p\).
Homographic correspondence in Plane Geometry, with applications.
Ruled surfaces, both developable and otherwise.
Determinants.
The employment of the Calculus of Residues
Infinite integrals.
Two linear segments \(AB, CD\) in a plane being given, prove that there is one, and only one, direction such that the projections of \(AB\) and \(CD\) on this direction are equal in magnitude and sign. Prove that, if four points are given in a plane, it is possible to construct two squares the sides of which (produced if necessary) pass through the four points in a definite order.