536 Lord Rayleigh on Convection Currents in 



no conductivity falls within the scope of a former investi- 

 gation where the fluid was supposed from the beginning to 

 be incompressible but of variable density *. 



Returning to the consideration of a finite conductivity, w r e 

 have again to distinguish the cases where /3 is positive and 

 negative. When 3 is negative (higher temperature below) 

 both values of n in (17) are real and one is positive. The 

 equilibrium is unstable for all values of P + m 2 and of s. 

 If /3 be positive, n may be real or complex. In either case 

 the real part of n is negative, so that the equilibrium is 

 stable whatever l 2 + m 2 and s may be. 



When /3 is negative ( — /3'), it is important to inquire for 

 what values of l 2 + m 2 the instability is greatest, for these 

 are the modes which more and more assert themselves as 

 time elapses, even though initially they may be quite 

 subordinate. That the positive value of n must have a 

 maximum appears when we observe it tends to vanish both 

 when i 2 -\-m 2 is small and also when l 2 + m 2 is large. Setting 

 for shortness I 2 + m 2 + s 2 = <x, we may write (17) 



?i 2 o- + n K o 2 -l3'ry{(T-s 2 )=~-0, . . . (20) 



and the question is to find the A r alue of cr for which n is 

 greatest, s being supposed given. Making dnjda = 0, we 

 get on differentiation 



n 2 + 2nfca-/3'ry = 0; ..... (21) 

 and on elimination of n 2 between (20), (21) 



-3? <-'--> 



Using this value of n in (21), we find as the equation for v 



2s 2 _/3'7S 4 /OQX 



o- «V K ~° } 



When k is relatively great, a = 2s 2 , or 



l* + m 2 = s 2 (24) 



A second approximation gives 



S'y 



The corresponding value of n is 



- &{*-&} ™ 



* Proc. Lond. Math. Soc. vol. xiv. p. 170 (1883) ; Scientific Papers, 

 vol. ii. p. 200, 



P + m W+££ (25) 



