Two small heavy rings connected by a light elastic string can slide without friction one on each of two fixed straight wires \(OA, OB\), which lie in a vertical plane through \(O\), the highest point, and are both inclined to the vertical at \(45^\circ\). Prove that there is only one configuration of equilibrium, and that if the weights of the rings are \(\frac{1}{2}\) and \(\frac{2}{3}\) of the modulus of elasticity of the string, the length of the string is twice its natural length. Investigate the stability of this configuration.
Prove the identity \begin{align*} &\cos 2(\beta+\gamma-\alpha-\delta)\sin(\beta-\gamma)\sin(\alpha-\delta) \\ &+ \cos 2(\gamma+\alpha-\beta-\delta)\sin(\gamma-\alpha)\sin(\beta-\delta) \\ &+ \cos 2(\alpha+\beta-\gamma-\delta)\sin(\alpha-\beta)\sin(\gamma-\delta) \\ &= -8\sin(\beta-\gamma)\sin(\gamma-\alpha)\sin(\alpha-\beta)\sin(\alpha-\delta)\sin(\beta-\delta)\sin(\gamma-\delta). \end{align*}
Prove that if \(\tan \alpha, \tan \beta, \tan \gamma\) are in arithmetic progression, then so are \(\cot (\alpha - \beta)\), \(\tan \beta\), \(\cot (\gamma - \beta)\).
Shew that the reciprocal of a conic, with respect to a focus \(S\), is a circle, and that, if the conic is a parabola, the circle passes through \(S\). A variable parabola touches a fixed conic and has its focus at one of the foci of the conic. Prove that its directrix touches a fixed circle.
A particle of mass \(m\) is suspended from a fixed point by a light string which is blown from the vertical by a steady horizontal wind of uniform velocity \(V\). Assuming that the force exerted by the wind on each element of length \(\delta s\) of the string is normal to the element and of magnitude \(\kappa \delta s \, v^2\), \(v\) being the component of \(V\) normal to the element, shew that the string hangs in a catenary. If the wind pressure on the particle is negligible, prove that the depth of the particle below the point of suspension is \[ c \log\left[\frac{l+\sqrt{l^2+c^2}}{c}\right], \] where \(c=mg/KV^2\), and \(l\) is the length of the string.
Prove that the radius of curvature \(\rho\) of a curve \(f(x,y)=0\) is given by the formula \[ \frac{1}{\rho} = (f_{xx}f_y^2 - 2f_{xy}f_xf_y + f_{yy}f_x^2)/(f_x^2+f_y^2)^{3/2}, \] where suffixes denote partial differentiation. The equation \(f(x,y,a)=0\) represents a family of curves. Prove that, if \(\rho, \rho'\) denote the radii of curvature of a particular curve and of the envelope of the family at the point where they touch, then \[ f_{aa}\left(\frac{1}{\rho} - \frac{1}{\rho'}\right) + (f_y f_{ax} - f_x f_{ya})(f_x^2+f_y^2)^{-3/2} = 0. \]
Prove that the line joining the circumcentre and the orthocentre of the triangle \(ABC\) makes with \(BC\) the angle \[ \tan^{-1}\left(\frac{\tan B \tan C - 3}{\tan B - \tan C}\right). \]
Find the harmonic conjugate of the line \(y=mx\) with respect to the pair of lines \[ ax^2 + 2hxy + by^2 = 0. \] A variable line through the fixed point \((\alpha, \beta)\) cuts the lines \[ ax^2 + 2hxy + by^2 = 0 \] in \(P\) and \(Q\). Prove that the locus of the mid-point of \(PQ\) is the conic \[ (ax+hy)(x-\alpha) + (hx+by)(y-\beta) = 0. \]
A uniform beam of length \(2l\) rests symmetrically on two supports which are a distance \(2a\) apart in a horizontal line; prove that the beam is least liable to break if \(a=l(2-\sqrt{2})\), it being assumed that the beam is liable to break if a definite bending moment is exceeded at any point.
Prove that, if \(u_n = \int_0^\pi \frac{dx}{(a+b\cos x+c\sin x)^n}\), then for integral values of \(n\) \[ (n-1)(a^2-b^2-c^2)u_n - (2n-3)au_{n-1} + (n-2)u_{n-2} + c\left\{\frac{1}{(a-b)^{n-1}} + \frac{1}{(a+b)^{n-1}}\right\} = 0. \] Obtain the values of \(u_1\) and \(u_2\) when \(a^2>b^2+c^2\), \(a\) being supposed positive.