Motion through a turhulently moving Inviscid Liquid. 349 

 where 



^ (. eto dz/ dz dx dy ) v 



lo. Let /oq\ 



^ = ^ + ^7 , (38), 



where p denotes what ^ would be if / were zero. We find, 



by (5), 



-W = 2 ^>J (39), 



and, by (27) and (37), 



<* = :a "{ v &- + a *; W- 



So far we have not used either the supposition of initial 

 isotropy for the turbulent motion, or of the infinitesimalness 

 ofdf/dy. We now must introduce and use both suppositions. 



19. To facilitate the integration of (39), we now use our 



supposition that -yf(y, t) t divided by the greatest value of 



f(y, t), is infinitely small in comparison with m, n, q, which, 

 as is easily proved, gives 



OT = 2 %0 1 * (41)' 



dy — V dx v ' 



by which (40) becomes 



Q-,a^)„ T (.-«|) 7 -.* . (42). 



Now, by (x, z) isotropy, we have 



2xzav(^ + u ^) V ^ 



f /d 2 d 2 \ , / d , d\d 1 „_ 2 , AO , 



=xzav {H<^ + ^) + r^ + ^te} v v °- (43) - 



Performing integrations by parts for the last two terms of 

 the second member, and using (1), we find 



/ d . d\ d „_ 2 /dlio dw \ d ,__2 



xzav ^_ + u , _j_ V , 0= _ xzav( _o + _J>)_ V v „ 



dv d __ 2 

 dy dy 

 and so we find, by (43) and (42), 



n df{y, t) / (d* d 2 \ dv Q d\ „ 2 fAA , 



