A particle \(P\) moves in an ellipse under the action of a force directed to a focus \(S\). Show that the force must vary inversely as the square of the distance \(SP\) from the focus. If \(H\) denotes the other focus, show that the speed of the particle varies as \(|HP|/SP|^{\frac{1}{2}}\). If \(A\) denotes the end of the major axis nearer to \(S\), and \(\theta\) denotes the angle \(ASP\), find an expression in terms of \(\theta\) for the time taken to travel from \(A\) to \(P\) as a proportion of the period \(T\) required to describe the whole ellipse. [It may be found helpful to use the eccentric angle of \(P\) on the ellipse.]
Let $$f(x) = 1 + \frac{x}{a} + \frac{x^2}{a(a+1)} + \ldots + \frac{x^n}{a(a+1)\ldots(a+n-1)} + \ldots,$$ where \(|x| < 1\) and \(a\) is a constant satisfying \(0 < a < 1\). Show that \(f(x)\) can be expressed in a form using only the elementary functions and a finite number of operations of addition, subtraction, multiplication, division, integration and differentiation.
Let \(N_+\), \(N_-\) be the number of positive integers of the form \(3k + 1\), \(3k - 1\), respectively, with integral \(k\), which divide a given positive integer \(n\), both 1 and \(n\) being counted. Show that $$N_+ \geq N_-.$$
On a level plain are to be seen three church steeples of different heights. Three men walk on the plain so that each man always sees two of the steeples at equal angles of elevation, no two of the men looking at the same two steeples. Prove that each man walks in a circle and that the centres of the three circles lie in a straight line.
By the 'first octant' of 3-dimensional space with a given co-ordinate system we mean the set of points \((x, y, z)\) with $$x \geq 0, \quad y \geq 0, \quad z \geq 0.$$ A line \(\lambda\) passes through the origin and contains no other point of the first octant. Show that there is a plane \(\pi\) which passes through \(\lambda\) and contains no point of the first octant except the origin.
The tangents at two points \(X\), \(Z\) of a non-singular conic \(S\) meet in \(Y\), and another non-singular conic \(S'\) touches \(XZ\) at \(X\) and touches \(YZ\) at a point distinct from \(Y\). From a general point \(P\) of \(S\) tangents are drawn to \(S'\), meeting \(S\) again in \(Q\), \(R\). Prove that \(QR\) touches \(S'\).
A set of \(m + 1\) white mice is taken at random, where \(m\) and \(n\) are positive integers. Show that at least one of the following two situations must occur: either there is a set of \(n + 1\) white mice or \((1 \leq j \leq n + 1)\) such that \(w_j\) is a parent of \(w_{j+1}\) \((1 \leq j \leq n)\) or there is a set of \(m + 1\) white mice no one of which is a parent of any other.
A particle whose horizontal and upward vertical co-ordinates are \(x\) and \(y\), respectively, moves under gravity in a resisting medium in which the retardation always acts in a direction opposite to the velocity. Show that at time \(t\) $$\frac{d^2y}{dx^2} = -g \left( \frac{dx}{dt} \right)^2,$$ where \(g\) is the acceleration due to gravity. Show also that \(\psi\), the inclination to the horizontal of the tangent to the path when the particle is at a height \(y_0\), is given by $$\tan \psi = \int_{y'}^{y_0} 2g \left( \frac{dx}{dt} \right)^2 dy,$$ where \(y'\) is the maximum height attained. Deduce that the angle at which the particle strikes the ground exceeds the angle of projection whatever the form of the retardation as a function of velocity.
A particle moves in a circle of radius \(a\) about a centre of force which exerts an attraction of magnitude \(\mu r^n\), \(r\) being the distance from the centre. By considering first-order equations for the time variation of the quantity \(\epsilon\), where \(r = a(1 + \epsilon)\) and \(\epsilon\) is considered small, discuss the stability of this motion when it is disturbed
A bead moves on a rough wire which is in the shape of the cycloid whose intrinsic equation is $$s = 4a \sin \psi.$$ The wire is in a vertical plane and its cusps point upwards, \(s\) is measured from the lowest point, and \(\psi\) is the angle between tangent and horizontal. Show that if the particle is released from rest at one of the cusps it just comes to rest again at the bottom of the wire if the coefficient of friction \(\mu\) satisfies the equation $$\mu^2 e^{4\pi} = 1.$$