330 Models of the Aether. 



Taking the ^-average of this, we observe that the first term of 

 the first member disappears, since A . v is zero, and the first 

 term of the second member disappears, since A . 3 (u'v]fix is 

 zero. Denoting by %R Z the average value of u z , v z , or w 1 , so 

 that R may be called the average velocity of the turbulent 

 motion, the equation becomes 



Let p be written (jp' + TO), where y denotes the value which p 

 would have if / were zero. The equations of motion immediately 

 give 



and on subtracting the forms which this equation takes in the 

 two cases, we have 



which, when the turbulent motion is fine-grained, so that 

 f(y t t) is sensibly constant over ranges within which u' t v, w 

 pass through all their values, may be written 



Moreover, we have 



. , , tyu'v) d(uv) 

 



for positive and negative values of u, v, w are equally probable ; 

 and therefore the value of the second member of this equation 

 is doubled by adding to itself what it becomes when for u', v, w 

 we substitute - u', -v, -w, which (as may be seen by inspection 

 of the above equation in V 2 ^>) does not change the value of p'. 



