Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/349

 there results Take now the xz-averages of both members. The quantities have zero averages; so the equation takes the form

if the symbol is used to indicate that the xz-average is to be taken of the quantity following. Moreover, the incompressibility of the fluid is expressed by the equation

whence

When this is added to the preceding equation, the first and third pairs of terms of the second member vanish, since the -average of any derivate vanishes if  is finite for infinitely great values of ; and the equation thus becomes

From this it is seen that if the turbulent motion were to remain continually isotropic as at the beginning, would constantly retain its critical value. In order to examine the deviation from isotropy, we shall determine, which may be done in the following way:—Multiplying the - and -equations of motion by respectively, and adding, we have