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

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

${\frac {\partial }{\partial t}}A.(u^{\prime }v)={\tfrac {2}{9}}R^{2}{\frac {\partial f(y,t)}{\partial y}}$,

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

${\frac {\partial ^{2}}{\partial t^{2}}}f(y,t)={\tfrac {2}{9}}R^{2}{\frac {\partial ^{2}}{\partial y^{2}}}f(y,t)$;

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

$(2/9)R^{2}=V^{2}$, $f(y,t)=P$, and $A(u^{\prime }v)=\gamma$,

the equations are

${\frac {\partial P}{\partial t}}=-{\frac {\partial \gamma }{\partial y}}$, and ${\frac {\partial \gamma }{\partial t}}=-V^{2}{\frac {\partial P}{\partial y}}$

If the quantity

$P^{2}+\gamma ^{2}/V^{2}=2\Sigma$

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

${\frac {\partial }{\partial t}}\iiint \textstyle \sum \ dx\ dy\ dz=\iiint \left(P{\frac {\partial P}{\partial t}}+{\frac {\gamma }{V^{2}}}{\frac {\partial \gamma }{\partial t}}\right)\ dx\ dy\ dz$

$\iiint \left(P{\frac {\partial P}{\partial t}}-\gamma {\frac {\partial P}{\partial y}}\right)\ dx\ dy\ dz$

1. Proc. Roy. Irish Acad. (3) i (1889), p. 340; Kelvin's Math. and Phys. Papers, iv, p. 202,
2. Brit. Assoc. Rep., 1899. FitzGerald's Scientific Writings, p. 484.