Motion through a turbulently moving Inviscid Liquid. 343 



du __ I 



u ^ +v— +iv— + C ^\ (2) 



dx dy dz dx) ' 



dv ( dv dn , dv , dp\ , ox 



<% = _( u ^ +v p +w <%L + ±) . . . (4). 



dt \ dx dy dz dz J 



From (2), (3), (4) we find, taking (1) into account, 



2 _du 2 dv 2 dio 2 /dv dw div du du dv\ r 

 ~ V P= d? + df + ~d? +l \dz dy + dx dz + dy dz) ' [D) ' 



3. The velocity-components u, v, iv may have any values 

 whatever through all space, subject only to (1). Hence, on 

 Fourier's principles, we have, as a perfectly comprehensive 

 expression for the motion at any instant, 



7^^222S«^f;^ ) sin(7^ + 6)cos(7^+/)cos(^ + ^) . (6), 

 v ^tSttXP^^ cos (mx+e) sin (ny+f) cos (qz + g) . (7), 

 w = 222222^ ^ : cos (mx + e) cos (ny+f) sin (qz + g) . (8); 



where a^^, @( m , n , q )> 7 (fB ,», 2 ) ai 'e any three velocities satis- 

 fying the equation 



Q = ™V,^) + ? ^(™,^) + 2V,n, 2) ... (9); 



and 222222 summation (or integration) for different values 

 of ?72, n, q, e,f, g. The summations for e,f, g may, without 

 loss of generality, be each confined to two values : e = 0, and 

 e=^7r ; /=0, and /= ^ir ; g = 0, and g = \ir. We shall admit 

 large values, and infinite values of m~ l , ?i~ 1 , q~ 1 , under certain 

 conditions [§ 4 (10), (11), (12), and § 15 below], but other- 

 wise we shall suppose the greatest value of each of them to be 

 of some moderate, or exceedingly small, linear magnitude. This 

 is an essential of the averagings to which we now proceed. 



4. Let xav, xzav, xyzav denote space-averages, linear, 

 surface, and solid, through infinitely great spaces, defined and 

 illustrated by examples, each worked out from (6), (7), (8), 

 as follows, L denoting an infinitely great length, or a very 

 great multiple of whichever of m" 1 , n~ l , q~ l may be con- 

 cerned : — 



