and 



492 BEPOUT— 1887. 



Hence we find (with suffixes &c. dropped), 



ay ay in^ + ?i'' + q^ 



.zav.o(#, + f,)A-'.o=J255222^f3^1^^:. . (47). 

 \dx^ dz- J m^ + n^i-q^ 



Now, in virtue of the average uniformity of the constituent terms implied 

 in isotropy and homogeneousness (§§ 7, 8, 9), the second member of (46) 



is equal to — 122225S ^, and therefore (§ 9) equal to — ^R" ; and 



b 



similarly we see that the second member of (47) is equal to +-y-R-. Hence, 

 finally, by (44), 



Qo=-W^^ .... (48); 

 and (36) for ^^0, with ^R- for V^ on account of isotropy, becomes 



{I— ("■)}„.=-»«' {^*},. • ■ <«)• 



The deviation from isotropy, which this equation shows, is very small, 

 because of the smallness of dfjdy ; and (27) does not need isotropy, but 

 holds in virtue of (30). Hence (49) is not confined to the initial values 

 (values for i=0) of the two members, because we neglect an infinitesimal 

 deviation from |R- in the first factor of the second member, considering 

 the smallness of the second factor. Hence, for all values of t, unless so 

 far as the 'random ' character referred to at the end of §13 may be lost 

 by a reai'rangement of vortices vitiating (27), 



-^xzav(u.) = -fR^iqg^ . . . (50). 



21. Eliminating the first member from this equation, by (34), we find 



Thus we have the very remarkable result that laminar disturbance is 

 propagated according to the well-known mode of waves of distortion in 

 a homogeneous elastic solid ; and that the velocity of propagation is 



— -^R, or about "47 of the average velocity'' of the turbulent motion of the 

 o 



fluid. This might seem to go far towards giving probability to the vortex 



theory of the luminiferous aether, were it not for the doubtful proviso at 



the end of § 20. 



22. If the undisturbed condition of the medium be a stable symme- 

 trical distribution of vortex-rings the suggested vitiation by ' rearrange- 

 ment ' cannot occur. For example, let it be such as is represented in. 



' Here and henceforth an averaging through y-spaces so small as to cover no 

 sensible difEerences of f(jy,t), but infinitely large in proportion to ■nr\ is im- 

 plied. 



