Find the lengths of the axes of the conic \(ax^2+2hxy+by^2=1\). An ellipse of semi-axes \(u,v\) revolves round its centre in its own plane. Show that the locus of the poles of a fixed straight line whose distance from the centre is \(c\) is a circle of radius \(\frac{u^2-v^2}{2c}\).
The equations of the sides of a triangle are \(\alpha=0, \beta=0, \gamma=0\). Show that the equation of any conic which touches the three sides is \(\sqrt{l\alpha}+\sqrt{m\beta}+\sqrt{n\gamma}=0\). Show that two triangles whose sides pass respectively through \(A, B, C\) can be inscribed in the above conic and that the equations of the lines joining corresponding angular points of the triangles are \(l\alpha+m\beta-3n\gamma=0\), and two similar equations.
Complex Numbers.
The Exponential and Logarithmic Functions of a real variable.
Starting from the existence of real numbers, and Dedekind's theorem concerning sections of real numbers, state, without proof, the chain of theorems leading to the proposition that any continuous function is integrable. Establish the principal properties of integrals. Show that if \(f(x)=0\) for irrational values of \(x\), and \(f(x)=1/q\) when \(x\) is a rational \(p/q\), where \(p/q\) is in its lowest terms, then \(f(x)\) has in any finite interval a Riemann integral, whose value is zero.
Curvature.
Green's Theorem and its applications to Electrostatics.
The potentials, charges, and energy of a system of conductors.
Lines and tubes of electrostatic force, and equipotential surfaces.
The parabolic motion of a particle under gravity.