Integrating the second term under the integral by parts, and omitting the superficial terms (which may be at infinity, or wherever energy enters the space under consideration), we have
.
Hence it appears that the quantity Σ, which is of the dimensions of energy, must be proportional to the energy per unit-volume of the medium—a result which shows that there is a pronounced similarity between the dynamics of a vortex-sponge and of Maxwell's elastic aether.
A definite vortex-sponge model of the aether was described by Hicks in his Presidential Address to the mathematical section of the British Association in 1895.[1] In this the small motions whose function is to confer the quasi-rigidity were not completely chaotic, but were disposed systematically. The medium was supposed to be constituted of cubical elements of fluid, each containing a rotational circulation complete in itself: in any element, the motion close to the central vertical diameter of the element is vertically upwards: the fluid which is thus carried to the upper part of the element flows outwards over the top, down the sides, and up the centre again. In each of the six adjoining elements the motion is similar to this, but in the reverse direction. The rotational motion in the elements confers on them the power of resisting distortion, so that waves may be propagated through the medium as through an elastic solid; but the rotations are without effect on irrotational motions of the fluid, provided the velocities in the irrotational motion are slow compared with the velocity of propagation of distortional vibrations.
A different model was described four years later by FitzGerald.[2] Since the distribution of velocity of a fluid in the
