This paper is a sequel to several communications which have already appeared in the Proceedings and Transactions of the Royal Society of Edinburgh. It commences with an investigation of the circumstances under which a portion of an incompressible frictionless liquid, supposed to extend through all space, or through space wholly or partially bounded by a rigid solid, can be projected so as to continue to move through the surrounding liquid without change of shape; and goes on to investigate vibrations executed by a portion of liquid so projected, and slightly disturbed from the condition that gives uniformity. The greatest and least quantities of energy which a finite liquid mass of any given initial shape and any given initial motion can possess, after any variations of its bounding surface ending in the initial shape, are next investigated; and the theory of the dissipation of energy in a finite or infinite frictionless liquid is deduced.