574 



SMITHSONIAN MISCELLANEOUS COLLECTIONS 



VOL. 51 



increases with the altitude and it can happen that the entropy at 

 the altitude c x in the cooler mass 1 is as large as that at the base 



of the mass 2, i. e., at the alti- 

 tude (h — c 2 ) ; the upper layers 

 of 1 may one after the other ser- 

 ially have respectively the same 

 entropy as the layers in 2 up to 

 the altitude (h — c 2 ). In the 

 final stage the lower part of 1 

 will become spread out at the 

 base; over it will lie strata con- 

 sisting of 1 and 2 mixed; above this will rest that portion of 2 

 that initially lay between (h — c 2 ) and h. At the boundaries be- 

 tween these three layers the temperature changes are continuous. 

 If the temperatures diminish linearly as in the equations 



fig. 4. 



T, - T hl + 



n x C. 



(*-«)■ T 2 = T h2 +- g c {h-z) 



then we have to seek the altitude at which the entropies are equal 

 or S 1 =S 2 for the same value of p h in the equations 



S, = K + C p K log T hx - K - 1) log rj 

 5 2 = K + C p [n 2 log T h2 - (« 2 - 1) log T 2 ] 



Let us assume n x = n 2 = n; let 8 t be the temperature of the mass 1 

 at the altitude c x and 2 the temperature of the mass 2 at the alti 

 tude h — c, then we have 



log-. = log— = 



C p 



0x) 



* , T h2 



^1 log tZ 



c, 



C 2 " J' (^ - T h2 ) 



Hence for n = 2 and a vertical temperature gradient of about 

 0.5 per 100 meters and for h = 3000 meters T hl = 263 , T h2 = 273 , 

 we find c x = 2154 meters and c 2 = 2090 meters. 



In this case the greater part of the masses 1 and 2 remain unmixed, 

 the available kinetic energy will not be much smaller than if in the 

 final stage the whole of mass 1 lies below and the whole of 2 above. 



Again, for h = 6000 meters T hl = 248 , T h2 = 258 , we find c x = 

 2390 meters and c 2 = 2096 meters. 



