Give the theory of the reduction of a three dimensional system of forces, and the various conditions for the equilibrium of such a system. Prove that a line distribution of couple of amount \(H\) per unit length of a plane closed curve \(s\), the axis of the couple at any point being normal to, and in the plane of the curve, is statically equivalent to a line distribution of force of amount \(-\dfrac{\partial H}{\partial s}\), the direction of the force at any point being at right angles to the plane of the curve.
Discuss the theory of the small oscillations of a dynamical system which is slightly disturbed from a position of stable equilibrium.
The stability of floating bodies.
Define the coefficients of potential, capacity and induction of a system of conductors, and give an account of their properties. Find the electrical energy of such a system, and prove that it is diminished by the introduction of a new conductor.
Prove that \[ \iint_S (lu+mv+nw)d\sigma = \iiint_T \left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z}\right) dx dy dz, \] where \(l,m,n\) are the direction cosines of the outward drawn normal to the boundary \(S\) of \(T\), and give some of the applications of this result either in electrostatics or in the theory of the irrotational motion of a liquid.
Give the theory of two dimensional surface waves on a liquid under no force but gravity, considering in particular waves at the common surface of two liquids.
Two pairs of points \(A, B\) and \(A', B'\) lie on an axis \(Ox\), and their abscissae are given by the equations \(ax^2+2bx+c=0\) and \(a'x^2+2b'x+c'=0\) respectively. Find an equation with rational coefficients which has \(AA' \cdot BB'\) for one of its roots. Give the geometrical interpretation of the relations obtained by equating the various coefficients in the equation to zero.
Three spherical balls, two of which have a radius of 1 inch and the third a radius of 2 inches, rest on a table, the points of contact being corners of an equilateral triangle of side 6 inches. A fourth ball rests on the table and touches each of the other three; prove that its radius is slightly greater than 2.4 inches.
In a quadrilateral \(ABCD\) the sides are \(AB=a, BC=b, CD=c, DA=d\); and the angle \(DAB=\theta, ABC=\phi\). Prove that \[ 2bd \cos(\theta+\phi) - 2ad \cos\theta - 2ab \cos\phi + a^2+b^2+d^2-c^2 = 0. \] Shew also that, if the quadrilateral is slightly deformed so that its sides remain of constant length, \[ \frac{\delta A}{\Delta BCD} = -\frac{\delta B}{\Delta CDA} = \frac{\delta C}{\Delta ABD} = -\frac{\delta D}{\Delta ABC}. \]
Discuss the general form of the curve \(y=x-a \log(x/b)\), where \(a\) and \(b\) are positive, and give a rough sketch of the curve. Find the asymptote. Prove that at any point \(P\) the chord of curvature, parallel to the asymptote, is proportional to the square of the length \(PT\) of the tangent intercepted between \(P\) and the asymptote. Give geometrical constructions for the point \(T\) and for the centre of curvature at \(P\).