Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/348

 mitting waves through a medium consisting of an incompressible fluid in which small vortex-rings are closely packed together. The wave-length of the disturbance was supposed large in comparison with the dimensions and mutual distances of the rings; and the translatory motion of the latter was supposed to be 30 slow that very many waves can pass over any one before it has much changed its position. Such a medium would probably act as a fluid for larger motions. The vibration in the wavefront might be either swinging oscillations of a ring about a diameter, or transverse vibrations of the ring, or apertural vibrations; vibrations normal to the plane of the ring appear to be impossible. Hicks determined in each case the velocity of translation, in terms of the radius of the rings, the distance of their planes, and their cyclic constant.

The greatest advance in the vortex-sponge theory of the aether was made in 1887, when W. Thomson showed that the equation of propagation of laminar disturbances in a vortex-sponge is the same as the equation of propagation of luminous vibrations in the aether. The demonstration, which in the circumstances can scarcely be expected to be either very simple or very rigorous, is as follows:—

Let (u, v, w) denote the components of velocity, and p the pressure, at the point (x, y, z) in an incompressible fluid. Let the initial motion be supposed to consist of a laminar motion {f(y), 0, 0}, superposed on a homogeneous, isotropic, and finegrained distribution (u&prime;0, v0, w0): so that at the origin of time the velocity is {f(y) + u&prime;0, v0, w0}: it is desired to find a function f(y, t) such that at any time t the velocity shall be {f(y, t) + u&prime;, v, w}, where u&prime;, v, w, are quantities of which every average taken over a sufficiently large space is zero.

Substituting these values of the components of velocity in the equation of motion