vi. Investigate the equations of equilibrium of a flexible string acted upon by any tangential and normal forces.

An uniform steel wire in the form of a circular ring is made to revolve in its own plane about its centre of figure. Show that the greatest possible linear velocity is independent both of the section of the wire and of the radius of the ring, and find roughly this velocity, the breaking strength of the wire being taken as 90,000 lbs. per square inch, and the weight of a cubic foot as 490 lbs.

vii. Calculate from the principle of energy the rate at which water will be discharged from a vessel in whose bottom there is a small hole, explaining clearly why the area of the vena contracta, and not that of the hole, is to be used.

A cistern discharges water into the atmosphere through a vertical pipe of uniform section. Show that air would be sucked in through a small hole in the upper part of the pipe, and explain how this result is consistent with an atmospheric pressure in the cistern.

viii. Investigate the disturbance in an unlimited atmosphere due to a source of sound which is concentrated at a single point, and whose effect is to produce an alternate production and destruction of air, given in amount and periodic according to the harmonic law.

Show that, if a given source of sound as defined above be situate at the vertex of an infinite conical tube, the energy emitted in a given time is inversely as the solid angle of the cone.