488 REPOET— 1887. 



7. Suppose now the motion to be homogeneously distributed through 

 all space. This implies that tlie centimes 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, OT, 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 translatoiy motion of the 

 fluid as a whole. This makes zero of every large average of it, and of v 

 and of ?y ; and, in passing, we may remark, with reference to our notation 

 of § 3, that it makes, as we see by (10), (11), (12), 



= "co,»,7)=acm,o,?)=a(m,«,o)=/J(o,n,j) = &C., &C. = y(;,„,„,o) • (19)- 



Without for the present, however, encumbering ourselves with the 

 Fourier-expression and notation of § 3, we may write, as the general ex- 

 pression for nullity of translational movement in large volumes, 



= are u = ave v = ave lo . . . (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 generalised notation of averages, homogeneousness 

 implies 



ave v^ =U^, ave v^ = V^ ave w^ ="W2 , . (21) , 

 ave vw = A-, ave wu = B^, ave tiv =. C^ . . (22) ; 



where U, V, W, A, B, C are six velocities independent of the positions of 

 the spaces in which the averages are taken. These equations are, how- 

 ever, 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 equa- 

 tions in terms of the notation of § 8, with further notation, R, to denote 

 what we shall call the avekage velocity of the turbulent motion : — 



U2 =Y2 =W^ =1112 .... (23), 

 =A= B = C (24). 



10. Lai'ge questions now present themselves as to transformations 

 which the distribution of turbulent motion will experience in an infinite 

 liquid left to itself with any distribution given to it initially. If the 

 initial distribution be homogeneous through all large volumes of space, 

 except a certain large finite space, S, through which there is initially 

 either no motion, or turbulent motion homogeneous or not, but not homo- 

 genous with the motion through the surrounding space, will the fluid 

 which at any time is within S acquire more and more nearly as time 

 advances the same homogeneous distribution of motion as that of the 

 surrounding space, till ultimately the motion is homogeneous through- 

 out ? 



11. If the answer were yes, could it be that this equalisation would 

 come to pass through smaller and smaller spaces as time advances ? In 

 other words, would any given distribution, homogeneous on a large 

 enough scale, become more and more fine-grained as time advances? 

 Probably yes for some initial distributions ; probably 7io for others. 

 Probably yes for vortex motion given continuously through all of one 

 large portion of the fluid, while all the rest is irrotational. 



