350 SirW. Thomson on the Propagation of Laminar 



20. Using now the Fourier expansion (7) for v 09 we find 

 v-,-2 4£4?v? o(e,f,g) cos (wm? + <?) sin (wy +/) cos (qz + ff) 



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



,^ Z av^- ~V"%=-iS2SSSS "*f, , (46)* 

 and <fy <fy m*+n*4-$* y 



Now, in virtue of the average uniformity of the constituent 



terms implied in isotropy and homogeneousness (§§ 7, 8, 9), 



B 2 

 the second member of (46) is equal to — -|222S22 -q-> and 



therefore (§ 9) equal to — JR 2 ; and similarly we see that the 

 second member of (47) is equal to + §R 2 . Hence, finally, by 



( U ^> df(v t) 



Q =-iR 2 ^M (48); 



and (36) for £ = 0, with ^R 2 for V 2 on account of isotropy, 



*"" {$-w},'.,'--»Bl^}, - - ■ m 



The deviation from isotropy, which this equation shows, is 

 very small, because of the smallness of df/dy ; and (27) 

 does not need isotropy, but holds in virtue of (30). Hence 

 (49) is not confined to the initial values (values for £ = 0) of the 

 two members, because we neglect an infinitesimal deviation 

 from |R 2 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 rearrangement of vortices 

 vitiating (27), 



^xzav(u,)=-|R 2 fc^ ) .... (50). 



21. Eliminating the first member horn this equation, by 

 (34), we find J2f ^ { 



%-$*% <">• 



Thus we have the very remarkable result that laminar dis- 

 turbance 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 



o 



* Here and henceforth an averaging through y-spaces so small as to 

 cover no sensible differences of f(ij, t), but infinitely large in proportion to 

 n~ 1 i is implied. 



