348 SirW. Thomson on the Propagation of Laminar 



no other condition respecting (u, v, w) than (30) ; no iso- 

 tropy, no homogeneousness in respect to y ; and only homo- 

 geneousness of regime with respect to y and z, with no mean 

 translational motion. 



The ^-translational mean component of the motion is wholly 

 represented by 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 

 Prof. James Thomson's theory of the "Flow of Water in 

 Uniform Regime in Eivers and other Open Channels''''*. In 

 endeavouring to advance a step towards the law of distri- 

 bution of the laminar motion at different depths, I was 

 surprised to discover the seeming possibility of a law of pro- 

 pagation as of distortional waves in an elastic solid, which con- 

 stitutes the conclusion of my present communication, on the 

 supposition of §15 that the distribution U , v , w is isotropic, 

 and that df(y, t)/dy, divided by the greatest value off(y, t), is 

 infinitely small in comparison with the smallest values of m, n, q, 

 in the Fourier-formulas (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 f{y). To find whether 

 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 V , by (2) and (3), as follows: — 



OdJJ 



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

 find 



_ UiM + v<m +w i^ +v ± +u f). (35 ). 

 C ax ay dz ax 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 find, with V 2 , as in §§ 8, 9, to 

 denote the average y-component- velocity of the turbulent 

 motion, 



^ S zav(u^ = -V*^|i)-Q . . (36), 



* Proceedings of the Royal Society, Aug. 15, 1878. 



