Problems

If \(x,y,z\) are three variables each of which may be regarded as a function of the other two, shew that (i) \(\left(\frac{\partial x}{\partial y}\right)_z = \frac{1}{(\frac{\partial y}{\partial x})_z}\), where \(\left(\frac{\partial x}{\partial y}\right)_z\) denotes the partial differential coefficient of \(x\) with respect to \(y\), \(z\) being constant. (ii) \(\left(\frac{\partial x}{\partial y}\right)_z \left(\frac{\partial y}{\partial z}\right)_x \left(\frac{\partial z}{\partial x}\right)_y + 1 = 0\). By changing the independent variables to \(\xi=x+y, \eta=x-y\) prove that the solution of \[ \frac{\partial^2 z}{\partial x^2} = \frac{\partial^2 z}{\partial y^2} \] is given by \(z=F_1(x+y)+F_2(x-y)\), where \(F_1(t), F_2(t)\) denote arbitrary functions of \(t\).

Prove that the volume enclosed by rotating a closed plane curve about a non-intersecting coplanar axis is given by the product of the area enclosed by the curve, and the length of the path traced by the centroid of the area. Shew that the volume of the surface formed by rotating the larger part of an ellipse about the latus rectum which is its boundary, is given by \[ 2\pi\frac{l^3}{(1-e^2)^3}\left\{ e\cos^{-1}(-e) + \sqrt{1-e^2}\left(\frac{2+e^2}{3}\right) \right\}, \] where \(e\) is the eccentricity and \(l\) the semi-latus rectum of the ellipse.

If \(a, b, c, d\) are four coplanar lines, prove that

1. the circumcircles of the four triangles \(bcd, cad, abd, abc\) have a common point \(O\),
2. the orthocentres of these four triangles are collinear,
3. the circumcentres of these four triangles are concyclic with \(O\).

(i) The equation of a central conic referred to rectangular axes is \[ S = ax^2+2hxy+by^2+2gx+2fy+c=0. \] Shew that, if \(y=mx\) and \(y=m'x\) are parallel to conjugate diameters, then \[ a+h(m+m') + bmm' = 0. \] Hence, or otherwise, shew that the axes of the conic are given by \[ h\xi^2+(b-a)\xi\eta - h\eta^2 = 0, \] where \(\xi = ax+hy+g\), and \(\eta = hx+by+f\), and shew that the axes bisect the angles between the asymptotes. (ii) Shew that the equation of the director circle of the conic \(S=0\) is \[ C(x^2+y^2)-2Gx-2Fy+A+B=0, \] where \(A, B, C, F, G\) are the minors of \(a, b, c, f, g\) respectively in the discriminant of the conic.

If \(\alpha, \beta, \gamma\) are the roots of \(x^3+bx+c=0\), find an expression for \[ (\alpha-\beta)^2(\beta-\gamma)^2(\gamma-\alpha)^2. \] Hence shew that the roots of \(x^3+px^2+qx+r=0\) are all real provided \[ p^2q^2+18pqr-4q^3-4p^3r-27r^2 \ge 0. \] If this condition is satisfied, shew that the necessary and sufficient condition for all the roots to be positive is that \(p\) and \(r\) should be negative and \(q\) positive.

The polynomials \(f(x)\) and \(\phi(x)\) are of degrees \(n\) and \(m\) respectively, \(n\) being greater than \(m\). Shew that, if \(f(x)\) and \(\phi(x)\) have no common factor, it is possible to find polynomials \(F(x)\) and \(\Phi(x)\) such that \[ f(x)F(x) + \phi(x)\Phi(x) = 1. \] Shew further that it is always possible to find an \(F\) whose degree is less than \(m\), and that the polynomials \(F\) and \(\Phi\) are then unique. Hence prove that, if the polynomials \(A\) and \(B\) have no common factor, the rational function \(C/AB\) can be expressed in only one way as \[ D + \frac{P}{A} + \frac{Q}{B}, \] where \(P\) and \(Q\) are polynomials whose degrees are less than those of \(A\) and \(B\) respectively. If \(A\) is \((x-a)^n\), and \((x-a)\) is not a factor of \(C\), then \(P/A\) can be expressed as \[ \frac{\lambda_1}{x-a} + \frac{\lambda_2}{(x-a)^2} + \dots + \frac{\lambda_n}{(x-a)^n}. \] Shew that \[ \lambda_{n-r} = \frac{1}{n!}\frac{d^r}{dx^r}\left[\frac{C(x)}{B(x)}\right]_{x=a}. \]

The function \(f_n(x)\) is defined to be \[ \frac{d^n}{dx^n}\{(x^2-1)^n\}. \] Shew by integration by parts, or otherwise, that \[ \int_{-1}^1 x^m f_n(x) dx = 0, \] if \(m\) is an integer less than \(n\), and is equal to \[ \frac{2^{2n+1}(n!)^3}{(2n+1)!} \] if \(m=n\). Shew that, if \(\phi(x)\) is a polynomial of degree less than \(n\), \[ \int_{-1}^1 \phi(x) f_n(x) dx = 0, \] and by using this last result, or otherwise, prove that \(f_n(x)\) has exactly \(n\) zeros in the range \(-1 \le x \le 1\).

A system of coplanar forces acts on a rigid body, and \(A, B, C, D\), are four points in the plane of the forces. If the moments of the system about the four points are denoted by \(\alpha, \beta, \gamma, \delta\) respectively, prove that \[ a\alpha - b\beta + c\gamma - d\delta = 0, \] where \(a\) denotes the area of the triangle \(BCD\), \(b\) of \(CDA\), \(c\) of \(DAB\), and \(d\) of \(ABC\). (The areas of triangles in different senses are given opposite signs.) In connexion with this theorem

1. examine the special form taken by the theorem when three of the points are collinear;
2. prove that, if the moments of the system about three non-collinear points of the plane are equal, then the system has the same moment about all points of the plane.

Three similar uniform rods \(AB, BC, CD\) are freely hinged together at \(B\) and \(C\), and \(A, D\) are attached to light rings which slide on a rough horizontal rail. If the coefficient of friction between a ring and the rail is \(\mu\), shew that the greatest inclination that \(AB\) and \(CD\) can make to the vertical in a symmetrical position of equilibrium is \(\phi\), where \(\tan\phi = \frac{3}{2}\mu\). If the joints at \(B\) and \(C\) are not smooth but can sustain a frictional couple \(\gamma\), shew that, if \(\gamma\) is not too large, the greatest inclination is increased by \(\sin^{-1}\frac{\gamma}{aW\sqrt{9\mu^2+4}}\), where \(W\) is the weight and \(2a\) the length of a rod.

Explain what is meant by the potential energy of a dynamical system on which only conservative forces act. Shew that the potential energy is stationary in a position of equilibrium, and that if the potential energy is a minimum the equilibrium is stable. A bead slides on a smooth circular hoop fixed in a vertical plane, and is attached to a particle, whose mass is equal to that of the bead, by a light inelastic string passing through a smooth ring placed very near to the highest point of the hoop. Determine the positions of equilibrium, and discuss their stability.