490 KEPOET— 1887. 



Take now xzav of both members. The second term of the first 

 member and the second term of the second member disappear, each in 

 virtue of (30). The first and last terms of the second member disappear, 

 each in virtue of (18) alone, and also each in virtue of (30). There 

 remains 



• '•' ' = — xzav ' " -L"' -^ "" 



/ f7u , (Iv. , chi \ .oc\ 



dt \ dx dij 



To simplify, add to the second member [by (1)] 



r, ( dw dv , d/r\ .„.-,. 



= — xzav u — + II +U-— . . . (33); 

 \ dx dij dz J V / ' 



and, the first and third pair of terms of the thus-modified second member 



vanishing by (18), find 



df(ii,t) d(uv) ,oi\ 



_\.ii^.— —xzav- , ^- . . . . (34), 

 dt dy 



It is to be remarked that this result involves, besides (1), no other 

 condition respecting (u, v, w) than (30) ; no isotropy, no homogeneous- 

 ness in respect to ?/ ; and only homogeneousness of recjime "with respect to 

 y and z, with no mean translational motion. 



The a^-translational mean component of the motion is 'wholly repre- 

 sented ^J f(y,t), and, so far as our establishment of (34) is concerned, 

 may be of any magnitude, great or small relatively to velocity-components 

 of the turbulent motion. It is a fundamental formula in the theory of 

 the turbulent motion of water between two planes ; and I had found 

 it in endeavouring to treat mathematically my brother Professor James 

 Thomson's theory of the ' Flow of Water in Uniform Regime in Rivers 

 and other Open Channels.' ' In endeavouring to advance a step towards 

 the law of distribution of the laminar motion at different depths, I was 

 surprised to discover the seeming possibility of a law of propagation as of 

 distortional waves in an elastic solid, which constitutes the conclusion of 

 my present communication, on the supposition of § 15 that the distribu- 

 tion ito, t'o, Wn is isotropic, and that df(ij,t) Idy, divided by the greatest 

 value oif(y,t), is infinitely small in comparison with the smallest values 

 of ??i, n, q, in the Fourier-formula? (6), (7), (8) for the turbulent motion. 



17. By (34) we see that, if the turbulent motion remained, through 

 time, isotropic as at the beginning, f(y,t) would remain constantly at its 

 initial value /"(y). To find whetlier the turbulent motion does remain 

 isotropic, and, if it does not, to find what we want to know of its deviation 



from isotropy, let us find xzav-—-'^, by (2) and (3), as follows : — First^ 



by multiplying (31) by v, and (3) by u, and adding, we find 



/f(y.t),d(nv __ r ., ;^^t^) , ^2 'Ml!:*) \ 

 ' dt dt V^-" ^ dx dy J 



r (](uv^ , d(nv) , d(iiv) , dp , dp'i .r,K\ 



l dx ay dz dx dy J 



Taking xzav of this, and remarking that the first term of the first member 

 disappears by (30), and the first term of the second member by (18), we 



' Proceedings of the Eoijal So&iefy, Aug. ]5, 1878. 



