ON THE VORTEX TIIEOEY OF THE LUMINIFEKOUS ^ETHEll. 489 



12. Probably no for the initial motion given in tlie shape of equal and 

 similar Helmholtz rinses, of proportions suitable for individual stability, 

 and each of overall diameter considerably smaller than the average dis- 

 tance from nearest neighbours. Probably 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 homogeneous and 

 ceolotropic, will it become more and more isotropic as time advances, and 

 ultimately quite isotropic ? Probably yes, for any random initial distribu- 

 tion, vyhether 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 isotropic (and 

 therefore utterly random in respect to direction), will it remain so ? Cer- 

 tainly 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 i , . . (25). 



But by (2) and (3) we find 



d, y r d(uv) , d(uv) , rl(vv) , rip dp\ ^c)c\ 



— (xzav ifv) = — xza < u '-, ^ +t'— — - + u- +v-^+u^ > i~o). 



at I dx cly 'h dx ay J 



Hence 



a f d(^iv) , d(uv) , d(uv) , dp , dp "1 /■o'7\ 



L dx dy dz dx dy J 



This equation in fact holds for every random case of motion satisfying 

 (30) below, because positive and negative values of u, v, w are all equally 

 probable, and therefore the value of the second member of (27) is doubled 

 by adding to itself what it becomes when for u, v, lo we substitute —it, 

 —V, —w, which, it may be remarked, and verified by looking at (5), does 

 not change the value of j;. 



15. We shall now suppose the initial motion to consist of a laminar 

 motion [f{y), 0, 0] superimposed on a homogeneous and isotropic distri- 

 bution (iig, Vq, Wq) ; so that we have 



when i=0, u=/(y) + Uo, v=Vq, iv=Wo . . (28); 



and we shall endeavour to find such a function, f(y,t), that at any time t 

 the velocity-components shall be 



f(y,t) + n,v,iu , . . . (29), 



where u, v, w are quantities of each of which every large enough average 

 is zero, so that particularly, for example, 



0=xzav u=xzav ij:=xzav la . . . (30) 



16. Substituting (29) for u, v, w in (2) we find 



df{y,t) , d\x J,, ,,f/u , df(y,t)} I dw. , d\\ , dn.dp\ ,.^,, 



dt dt V^-^'^dx dy J \ dx dy dz dx ^ ^ 



