538 Lord Rayleigh on Convection Currents in 



supposition d 2 wjdz 2 = 0, which (with w = 0) is satisfied by 

 taking as before w proportional to sin sz with *=?7r/f. This 

 is equivalent to the annulment of lateral forces at the wall 

 For (Lamb's 'Hydrodynamics,' §§ 323, 326) these forces are 

 expressed in general by 



___dw du d w dv _ rN 



while here w = at the boundaries requires also dw/dx = Q, 

 dw\dy = 0. Hence, at the boundaries, d 2 u/dxdz, d 2 vjdydz 

 vanish, and therefore by (5), d 2 w/dz 2 . 

 Equation (15) remains unaltered : — 



/3u; + {tt + /<Z 2 + m 2 + s 2 )}0 = O, . . . (15) 

 and (16) becomes 



{n + v{l 2 + m* + s 2 )}(l 2 + m 2 +s 2 )w-ry(l 2 + m 2 )0 = O. (36) 



Writing as before <r = Z 2 -f-m 2 -f-s 2 , we get the equation in n 



(n + /^(n + vo> + /3 7 (Z 2 + m 2 ) = 0, . . (37) 



which takes the place of (IT). 



If ry = (no expansion with heat) the equations degrade, 

 and we have two simple alternatives. In the first n-j-K<r = Q 

 with w = 0, signifying conduction of heat with no motion. 

 In the second n + va = 0, when the relation between w and 6 

 becomes 



j3w + or(u:-v)0 = O (38) 



In both cases, since n is real and negative, the disturbance is 

 stable. 



If we neglect /c in (37), the equation takes the same 

 form (20) as that already considered when v = 0. Hence 

 the results expressed in (22), (23), (24), (25), [26\ (27) 

 are applicable with simple substitution of v for k. 



In the general equation (37) if ft be positive, as 7 is 

 supposed always to be, the values of n may be real or 

 complex. If real they are both negative, and if complex 

 the real part is negative. In either case the disturbance 

 dies down. As was to be expected, when the temperature 

 is higher above, the equilibrium is stable. 



In the contrary case when /3 is negative (—/3 r ) the roots 

 of the quadratic are always real, and one at least is negative. 

 There is a positive root only when 



(3'y(l 2 +m 2 )>Ki>c z (39) 



