332 Models of the Aether. 



for we may neglect an infinitesimal deviation from (2/9) IP 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. Fitz Geraldf subsequently determined the energy per 

 unit-volume in a turbulent liquid which is transmitting laminar 

 waves. Writing for brevity 



(2/9) R* - V\ f(y, t) = P, and A (u'v) = 7, 

 the equations are 



s ?.--h and h-.y*^ 

 dt dy' ft " 8y 



If the quantity 



p-f jVP2S 



is integrated throughout space, and the variations of the 

 integral with respect to time are determined, it is found that 





JIM- 



* Proc. Roy. Irish Acad. (3) i (1889), p. 340 ; Kelvin's Math, and Phys. Papers, 

 iv, p. 202. 



t Brit. Assoc. Rep., 1899. Fitz Gerald's Scientific Writings, p. 484. 



