54:0 Lord Rayleigh on Convection Currents in 



Also g — 980, and thus 



P2—P1 '033 



pi r 



(47) 



For example, i£ f=l cm., instability requires that the 

 density at the top exceed that at the bottom by one-thirtieth 

 part, corresponding to about 9° 0. of temperature. We 

 should not forget that our method postulates a small value 

 of (p2— P\)\p\- Thus if kv be given, the application of (46) 

 may cease to be legitimate unless f be large enough. 



It may be remarked that the influence of viscosity would 

 be increased were we to suppose the horizontal velocities 

 (instead of the horizontal forces) to be annulled at the 

 boundaries. 



The problem of determining for what value of Z 2 + w 2 , 

 or cr, the instability, when finite, is a maximum is more 

 complicated. The differentiation of (37) with respect to cr 

 gives 



n 2 + 2ncr(Ar + j/)+3/cv(7 2 -/3'7 = 0, . . . (48) 

 whence 



expressing n in terms of cr. To find <r we have to eliminate 

 n between (44) and (45). The result is 



o *kv{k -v) 2 + a 4 ^y( K + v ) 2 - <r' . 2/3 V (k 2 + v 2 ) - /3 /2 7 V = 0, 



. . . (50) 



from which, in particular cases, a could be found by 

 numerical computation. From (50) we fall back on (23) 

 by supposing v = 0, and again on a similar equation if we 

 suppose k = 0. 



But the case of a nearly evanescent n is probably the 

 more practical. In an experiment the temperature gradient 

 could not be established all at once and we may suppose the 

 progress to be very slow. In the earlier stages the equi- 

 librium would be stable, so that no disturbance of importance 

 would occur until n passed through zero to the positive side, 

 corresponding to (44) or (46). The breakdown thus occurs 

 for s = 7r/£ and by (42) / 2 + m 2 = 7r' 2 /2^. And since the 

 evanescence of n is equivalent to the omission of djdt in the 

 original equations, the motion thus determined has the 

 character of a steady motion. The constant multiplier is, 

 however, arbitrary ; and there is nothing to determine it so 

 long as the squares of u, v, to, are neglected. 



