Two transformations in the complex plane are defined by \(Tz = -\frac{1}{z}\), \(Sz = z-1\). Explain the geometrical significance of these transformations. Show that \begin{align*} TST &= STS^{-1}, \\ STSTST &= TSTSTS = I, \end{align*} where \(Iz=z\) is the identical transformation and \(S^{-1}z=z+1\) is the inverse of the transformation \(S\). \(\Delta\) is the region defined by \[ -\frac{1}{2} < x < \frac{1}{2}, \quad y>0, \quad x^2+y^2>1. \] Give a diagram showing the regions into which \(\Delta\) is transformed by the transformations \(TS, ST\).
A rectangular sheet of paper \(ABCD\) is folded over so that the corner \(A\) comes to lie on the edge \(CD\). Show that the line of the crease always touches a fixed parabola. If \(CD=2BC\), find the least area of the folded part when the folded part is a triangle.
The roots of the cubic equation \(x^3-3qx-pq=0\) are \(\alpha, \beta, \gamma\). Express \(\alpha^{-3}+\beta^{-3}+\gamma^{-3}\) in terms of \(p\) and \(q\). Show that the semi-symmetric functions of the roots \[ \alpha^2\beta+\beta^2\gamma+\gamma^2\alpha \quad \text{and} \quad \alpha^2\gamma+\beta^2\alpha+\gamma^2\beta \] are the roots of the quadratic equation \[ x^2+3pqx+9p^2q^2-27q^3=0. \]
Show that the double points of the involution determined by the two pairs of points given by the equations \begin{align*} ax^2+2bxy+cy^2 &= 0, \\ a'x^2+2b'xy+c'y^2 &= 0, \end{align*} are given by \[ (ab'-a'b)x^2 - (ca'-c'a)xy + (bc'-b'c)y^2 = 0. \] Three collinear points \((x_i, y_i)\), where \(i=1, 2, 3\), are given by the equation \[ ax^3+3bx^2y+3cxy^2+dy^3=0, \quad ad-bc\ne 0; \] the harmonic conjugate of \((x_i, y_i)\) with respect to the other two points is \((x_i', y_i')\). Show that the pair of points \((x_i, y_i)\), \((x_i', y_i')\) is given by the equation \[ (ax_i+by_i)x^2+2(bx_i+cy_i)xy+(cx_i+dy_i)y^2=0. \] Deduce that the three pairs of points \((x_i, y_i)\), \((x_i', y_i')\) for \(i=1, 2, 3\) are in involution.
Prove that there are six circles of curvature to the rectangular hyperbola \[ xy=1 \] which pass through a general point \((\xi, \eta)\). Show that the centres of these six circles lie on a conic.
A uniform flexible chain of length \(2l\) and weight \(2wl\) hangs between two points \(A\) and \(B\) on the same level and at mutual distance \(2a\). Show that the parameter \(c\) of the catenary of which the chain forms part is a root of \(l=c\sinh(a/c)\). A weight \(w'=2w\alpha\), where \(\alpha\) is small compared with \(l\), is affixed to the midpoint \(P\) of the chain. Neglecting terms of the second order in \(\alpha\), show that each half of the chain now forms part of a catenary whose vertex lies at a horizontal distance \(x\) from \(P\) and which has parameter \(c(1+\beta)\), where \[ \beta = \frac{\alpha\{\cosh(a/c)-1\}}{a\cosh(a/c)-c\sinh(a/c)}. \] Show further that to the first order the increased sag (i.e. the increase in depth of \(P\) below \(AB\)) is \[ \alpha \frac{2c\{1-\cosh(a/c)\}+a\sinh(a/c)}{a\cosh(a/c)-c\sinh(a/c)}. \]
A mass \(M\) can oscillate in the line \(Ox\), the restoring force being \(Kx\) when \(M\) is at distance \(x\) from \(O\). A second mass \(m\), also on \(Ox\), is attached to \(M\) by a spring in which the restoring force is \(k\) times the extension.
Find in terms of polar coordinates \((r, \theta)\) the radial and transverse velocities and accelerations of a particle moving in a plane. A particle is attracted to the fixed point \(O\) by a force \(\mu f(u)\) per unit mass, where \(u=r^{-1}\). Prove that the path of the particle satisfies the equation \[ \frac{d^2u}{d\theta^2}+u = \frac{\mu f(u)}{h^2u^2}, \] where \(h\) is a constant. Find the condition that the particle may move in a circle \(u=c\), and by taking \(u=c+\xi\), where \(\xi\) and its derivatives are small, show that the circular motion is stable for small disturbances provided that \(cf'(c)<3f(c)\).
A perfectly rough circular disc of radius \(a\) and radius of gyration \(k\), with centre of mass at its geometrical centre, rolls down a line of greatest slope of a plane of inclination \(\alpha\). The plane of the disc remains vertical. After rolling a distance \(l\) down the plane from rest, the disc hits the corner of an inelastic step which projects normally a distance \(b\) (where \(a(1-\cos\alpha) < b < a\)) from the plane. Prove that the disc surmounts the step if \[ l > a\frac{1-\cos(\psi-\alpha)}{\sin\alpha} \left(\frac{k^2+a^2}{k^2+a^2\cos\psi}\right)^2, \] where \(\cos\psi = (a-b)/a\).