Motion through a turbulently moving Tnviscid Liquid. 347 



Hence 



~ f d(uv) d(uv) d(uv) dp , dp \ /c .„. 



I. dx dy dz dx dy } y 



This equation in fact holds for every random case of motion 

 satisfying (30) below, because positive and negative values of 

 u, v, iv 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, w we substitute —u, — ?;, —w, which, 

 it may be remarked, and verified by looking at (5), does not 

 change the value of p. 



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

 a laminar motion [f(y), 0, 0] superimposed on a homo- 

 geneous and isotropic distribution (lt , v , w Q ) ; so that we 

 have 



when t=0, u =/ (y) + it , v = v Q , w = w . (28) ; 



and we shall endeavour to find such a function, / (y, t) , that 

 at any time t the velocity-components shall be 



f(y,t) + u,v, W (29), 



where it, v, w are quantities of each of which every large 

 enough average is zero, so that particularly, for example, 



= xzavlt=xzav v = xza.vw . . . (30). 



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



d f{y^) dii_ $ du , df(y, t) -) '/du t du , du^dp\ 



~dT + dt- V^^ +v % }-( u S +1? % + ^ + ^) (31) 



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

 first member and the second term of the second member dis- 

 appear, 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 



^/G/>0 /. dU , du , dU\ /OON 



r*-=-- ; ™T(«5+^+*s) • • (32). 



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



A / du , dv div\ /OON 



0= - xzav ("^ + % + "^)- • • • ( 33 )- 



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

 second member vanishing by (18), find 



«._„„*) ..... (M , 



It is to be remarked that this result involves, besides (1), 



2A2 



