346 Sir W. Thomson on the Propagation of Laminar 



formations which the distribution of turbulent motion will 

 experience in an infinite liquid left to itself with any distribu- 

 tion given to it initially. If the initial distribution be homo- 

 geneous through all large volumes of space, except a certain 

 large finite space, $, through which there is initially either 

 no motion, or turbulent motion homogeneous or not, but not 

 homogeneous with the motion through the surrounding space, 

 will the fluid which at any time is within S acquire more and 

 more nearly as time advances the same homogeneous distri- 

 bution of motion as that of the surrounding space, till ulti- 

 mately the motion is homogeneous throughout? 



11. If the answer were yes, could it be that this equaliza- 

 tion would come to pass through smaller and smaller spaces 

 as time advances ? In other words, would any given distri- 

 bution, homogeneous on a large enough scale, become more 

 and more fine-grained as time advances ? Probably yes for 

 some initial distributions ; probably no for others. Probably 

 yes for vortex motion given continuously through all of one 

 large portion of the fluid, while all the rest is irrotational. 



12. Probably no for the initial motion given in the shape 

 of equal and similar Helmholtz rings, of proportions suitable 

 for individual stability, and each of overall diameter consider- 

 ably smaller than the average distance from nearest neighbour. 

 Prodably also no, though the rings be of very different volumes 

 and vorticities. But probably yes if the diameters of the rings, 

 or of many of them, be not small in comparison with dis- 

 tances from neighbours, or if the individual rings, each an 

 endless slender filament, be entangled or nearly entangled 

 among one another. 



13. Again a question : If the initial distribution be homo- 

 geneous and a>olotropic, will it become more and more isotropic 

 as time advances, and ultimately quite isotropic? Probably 

 yes, for any random initial distribution, whether of continuous 

 rotationally-moving fluid or of separate finite vortex rings. 

 Possibly no for some symmetrical initial distribution of vortex 

 rings, conceivably stable. 



14 If the initial distribution be homogeneous and iso- 

 tropic (and therefore utterly random in respect to direction), 

 will it remain so ? Certainly yes. I proceed to investigate 

 a mathematical formula, deducible from the answer, which 

 will be of use to us later (§ 18). By (22) and (24) we have 



xzav uv = 0, for all values of t . . . (25). 



But by (2) and (3) we find 



d , \ J d(uv) d(uv) d(uv) dp , dp~) 



^(xzav uv) = -xza { „__ + »_ + „_ +v£ + u±^ (26). 



