Models of the Aether. 329 



there results 



W dp 



- w - . 



dz dx 



Take now the #2-averages of both members. The quantities 

 du'/dt, du'/dx, v, dp/dx have zero averages; so the equation 

 takes the form 



df(y*t) ( ,W M 



- = - A . [u -- + v + w 

 dt \ dx dy 



if the symbol A is used to indicate that the xz- average is to be 

 taken of the quantity following. Moreover, the incompressi- 

 bility of the fluid is expressed by the equation 



whence 



du' dv dw 



+ ~ + = ' 



f\ A I / *** / t/* 1 ' ^ \JWJ 



1 aaT" 1 * ^ + l 9z 



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 dQ/dx vanishes if Q is finite for 

 infinitely great values of x ; and the equation thus becomes 



a) 



From this it is seen that if the turbulent motion were to remain 

 continually isotropic as at the beginning,/ (T/, t) would constantly 

 retain its critical value /(y). In order to examine the deviation 

 from isotropy, we shall determine Ad (u'v)/dt, which may be 

 done in the following way : Multiplying the u- and ^-equations 

 of motion by v, u' respectively, and adding, we have 



-. 



' fa ty dx 



d (u'v) d (u'v) dp , dp 



-V- - ~ w ~V -v^--u f ^- 



dy dz dx ty 



