Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/352

 for we may neglect an infinitesimal deviation from in the first factor of the second member, in consideration of the smallness of the second factor. Hence for all values of t we have the equation

which, in combination with (1), yields the result

the form of this equation shows that laminar disturbances are propagated through the vortex-sponge in the same manner as waves of distortion in a homogeneous elastic solid.

The question of the stability of the turbulent motion remained undecided; and at the time Thomson seems to have thought it likely that the motion would suffer diffusion. But two years later he showed that stability was ensured at any rate when space is filled with a set of approximately straight hollow vortex filaments. FitzGerald subsequently determined the energy per unit-volume in a turbulent liquid which is transmitting laminar waves. Writing for brevity

the equations are

If the quantity

is integrated throughout space, and the variations of the integral with respect to time are determined, it is found that