The function \(\mu(n)\) is defined as being equal to 0 when \(n\) contains any squared factor, to 1 when \(n=1\), and to \((-1)^{\nu}\) when \(n=p_1 p_2 \dots p_{\nu}\), \(p_1, p_2, \dots, p_{\nu}\) being different primes. Prove that \[ \sum \mu(d)=0, \] the summation being extended to all divisors \(d\) of a given number \(N\).
Prove that if \(u_n(x)\) is a continuous function of \(x\) for \(a \le x \le b\), and \(\sum_{0}^{\infty} u_n(x)\) is uniformly convergent for \(a \le x \le b\), then \[ \int_a^b \left\{\sum_0^\infty u_n(x)\right\}dx = \sum_0^\infty \int_a^b u_n(x)dx. \] Prove that, if \(\alpha, \beta\), and \(\delta\) are positive, then \[ \sum_{n=0}^{\infty} \int_0^\delta (\alpha e^{-n\alpha x} - \beta e^{-n\beta x})dx = \log\frac{1-e^{-\alpha\delta}}{1-e^{-\beta\delta}} - \log\frac{\alpha}{\beta}, \] but \[ \int_0^\delta \sum_{n=0}^\infty (\alpha e^{-n\alpha x} - \beta e^{-n\beta x})dx = \log\frac{1-e^{-\alpha\delta}}{1-e^{-\beta\delta}}. \] Explain the discrepancy in the results.
Show that if \(t=u+iv = f(x+iy) = f(z)\), where \(f\) is an analytic function, and \(F\) is a real function of \(u\) and \(v\), with continuous second derivatives, then \[ \left(\frac{\partial^2 F}{\partial x^2}\right) + \left(\frac{\partial^2 F}{\partial y^2}\right) = M^2 \left\{\left(\frac{\partial^2 F}{\partial u^2}\right) + \left(\frac{\partial^2 F}{\partial v^2}\right)\right\}, \quad \frac{\partial^2 F}{\partial x \partial y} = M^2 \left(\frac{\partial^2 F}{\partial u \partial v}\right), \] where \(M = \left|\frac{dt}{dz}\right|\).
Obtain the solution of the equation \[ \frac{d^2 y}{dx^2} + \frac{1}{x}\frac{dy}{dx} + \left(1-\frac{n^2}{x^2}\right)y=0, \] where \(n\) is not an integer, in the form \[ A J_n(x) + B J_{-n}(x), \] where \[ J_n(x) = (\frac{1}{2}x)^n \sum_{v=0}^\infty \frac{(-1)^v (\frac{1}{2}x)^{2v}}{v!\Gamma(n+1+v)}. \] Show that \[ \frac{i e^{\frac{1}{2}(n+1)\pi i}}{\sin n\pi}\{e^{-n\pi i}J_n(ix) - J_{-n}(ix)\} \] is real for all real values of \(x\).
Show that a right circular cone can be drawn to touch three consecutive osculating planes of a curve in space, that its semivertical angle is \[ \arctan(\sigma/\rho) \] and that the direction cosines of its axis are \[ \frac{l'\sigma-l\rho}{\sqrt{(\rho^2+\sigma^2)}}, \frac{m'\sigma-m\rho}{\sqrt{(\rho^2+\sigma^2)}}, \frac{n'\sigma-n\rho}{\sqrt{(\rho^2+\sigma^2)}}; \] \(l,m,n\) being the direction cosines of the tangent and \(l', m', n'\) those of the binormal.
Prove that for any triangle \(ABC\), and a point \(D\), a point \(D'\) may be found such that \(DD'\) subtends at each vertex of the triangle an angle having the same bisectors as the angle of the triangle. From a quadrangle \(ABCD\) is derived a quadrangle \(A'B'C'D'\); \(D'\) being found as above, and \(A', B', C'\) similarly. Shew that \(AD\) perpendicularly bisects \(B'C'\), and similarly for the other pairs of sides of the quadrangles.
Prove that if chords \(AA', BB', CC'\) of a circle are concurrent the products \(BC' \cdot CA' \cdot AB'\) and \(CB' \cdot AC' \cdot BA'\) are equal. Points of the compass are marked round the circumference of a circle and lines are drawn from the points N., NNE., NE. to the points ESE., S. and W. respectively. Shew that they are concurrent.
Four points \(A, B, C, D\) are marked on a straight line so that \(AB=14''\), \(AC=7''\), \(AD=6''\). Shew that they may be projected into four points \(A', B', C', D'\) equally spaced, in order, on another line. Draw a figure effecting the change.
Prove that there are four plane sections of a cube which are regular hexagons. Shew that a flexible elastic ring stretched tight round a smooth cube along one of these plane sections would be in equilibrium.
Prove that any triangle inscribed in a rectangular hyperbola has the orthocentre as another point on the curve. What theorems arise from the cases of coincidence of (i) two vertices of the triangle, (ii) all three vertices?