a Horizontal Layer of Fluid. 535 



so that (8), (5), (9) become 



il'uy irrfur 1 d'ur A N 



nu=- , nv= , nw= tyu, . (11) 



p p p dz 



ilu-\-imv + dw/dz = 0, . . . . (12) 



n0 + /3w = fc(d 2 ldz 2 -l 2 -m 2 )0, . . . (13) 



from which by elimination of u, v, w, we derive 



n d 2 w 



l 2 + m 2 dz 2 



= nw — y0 (14) 



Having regard to the boundary conditions to be satisfied 

 by w and 6, we now assume that these quantities are pro- 

 portional to sin sz, where s = q7r/£, and q is an integer. 

 Hence w 



/3w+\n + K (l 2 + m 2 + s 2 )}0 = O, . . . (15) 



n(l 2 + m 2 + s 2 )w-ryQ 2 + m 2 )0 = O, . . (16) 



and the equation determining n is the quadratic !,, 



7i 2 (l 2 + m 2 + s 2 )+?7 K (l 2 + m 2 + i > 2 ) 2 + l3v{l 2 +m 2 ) = 0. (17) 



When ac = 0, there is no conduction, so that each element of 

 the fluid retains its temperature and density. If ft be 

 positive, the equilibrium is stable, and 



±iy/{fiy(P + m*)} 

 n ~ y/\P + m* + *} ' ■ • ■ ■ (18) 



indicating vibrations about the condition of equilibrium. 

 If, on the other hand, /3 be negative, s;iy — /3', 



When n has the positive value, the corresponding disturbance 

 increases exponentially with the time. 



For a given value of l 2 + m 2 , the numerical values of n 

 diminish without limit as s increases — that is, the more sub- 

 divisions there are along z. The greatest value corresponds 

 ith q = l or s = 7rj£. On the other hand, if s be given, 



w 



|w| increases from zero as 1 2 + jh 2 increases from zero (great 

 wave-lengths along x and y) up to a finite limit when P-\-m 2 

 is large (small wave-lengths along x and y). This case of 



