Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/350

 Taking the xz-average of this, we observe that the first term of the first member disappears, since is zero, and the first term of the second member disappears, since  is zero. Denoting by the average value of, so that  may be called the average velocity of the turbulent motion, the equation becomes

where

Let be written ($$r^\prime + \bar\omega$$), where  denotes the value which 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 is sensibly constant over ranges within which pass through all their values, may be written

Moreover, we have

for positive and negative values of 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 we substitute ; which (as may be seen by inspection of the above equation in ) does not change the value of.