The invariants of a system of two conics.
Envelopes of plane curves.
Curvilinear coordinates.
Differentials.
Series of complex constants.
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.