for we may neglect an infinitesimal deviation from (2/9) R2 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[1] he showed that stability was ensured at any rate when space is filled with a set of approximately straight hollow vortex filaments. FitzGerald[2] subsequently determined the energy per unit-volume in a turbulent liquid which is transmitting laminar waves. Writing for brevity
, , and ,
the equations are
, and
If the quantity
is integrated throughout space, and the variations of the integral with respect to time are determined, it is found that
↑Proc. Roy. Irish Acad. (3) i (1889), p. 340; Kelvin's Math. and Phys. Papers, iv, p. 202,
↑Brit. Assoc. Rep., 1899. FitzGerald's Scientific Writings, p. 484.