Motion through a turbulently moving Inviscicl Liquid, 345 



6. Bemark ; as a general property of this kind of averaging, 



*-§=° (18) ' 



if Q be any quantity which is finite for infinitely great values 

 of a. 



7. Suppose now the motion to be homogeneously distri- 

 buted through all space. This implies that the centres of 

 inertia of all great volumes of the fluid have equal parallel 

 motions, if any motions at all. Conveniently, therefore, we 

 take our reference lines OX, OY, OZ, as fixed relatively to 

 the centres of inertia of three (and therefore of all) centres of 

 inertia of large volumes ; in other words, we assume no trans- 

 latory motion of the fluid as a whole. This makes zero of 

 every large average of u and of v and of w ; and, in passing, 

 we may remark, with reference to our notation of § 3, that it 

 makes, as we see by (10), (11), (12), 



= «(0, n, q) = «(m, 0, q) = «(m, n, 0) = &0, n, 2 ) = &C. &C. = <Y( m> n , 0) (19) 



Without for the present, however, encumbering ourselves 

 with the Fourier-expression and notation of § 3, we may 

 write, as the general expression for nullity of translational 

 movement in large volumes, 



= ave w = ave v = aveiv .... (20); 



where ave denotes the average through any great length of 

 straight or curved line, or area of plane or curved surface, or 

 through any great volume of space. 



8. In terms of this generalized notation of averages, homo- 

 geneousness implies 



avew 2 = U 2 , avev 2 =Y 2 , avew 2 = W 2 . (21), 



ave vw = A 2 , ave ivu = B 2 , ave uv= C 2 . (22) ; 



where U, Y, W, A, B, C are six velocities independent of 

 the positions of the spaces in which the averages are taken. 

 These equations are, however, infinitely short of implying, 

 though implied by, homogeneousness. 



9. Suppose now the distribution of motion to be isotropic. 

 This implies, but is infinitely more than is implied by, the 

 following equations in terms of the notation of § 8, with 

 further notation, R, to denote what we shall call the average 

 velocity of the turbulent motion : — 



U 2 =Y 2 =W 2 = JR 2 (23), 



0=A = B = C ....... (24). 



10. Large questions now present themselves as to trans- 

 Phil. Mag. S. 5. Yol. 24. No. 149. Oct. 1887. 2 A 



