IDEAL STATES OF EQUILIBRIUM IN THE ATMOSPHERE. 51 



w ^ -*-()* 



Adding, finally, the equation of state 



we can find the corresponding equilibrium values of the specific volume a or of its 

 reciprocal, the density p. 



The problem is thus fully solved. Summing up the results, we shall choose 

 once the pressure and once the gravity potential as independent variable. In the 

 first case we shall represent the distribution of mass by the specific volume a, in 

 the second by the density p. Denoting by # > P<*> a o> />o the values of temperature, 

 pressure, specific volume, and density at sea-level, we easily arrive at the follow- 

 ing two schemes of formulae: 



-*(*r or -tt-ttn 



( B ) &=s ^--j.^ P = Po (i-^<t>yy-' p=p(i-} a $f> 



each of which represents the full solution of the problem. 



43. Limit of the Atmosphere in Case of Constant-Temperature Gradient. 



Temperature being a linear function of the gravity potential, and decreasing 

 upwards, absolute zero will be reached at a certain finite height 



w ** = * 



Substituting this in the two last equations (b), section 43, and remembering that y 

 is positive when temperature decreases upwards, we get 



p = o p = 



Supposing, thus, the gas laws to be true even at absolute zero, we find the atmos- 

 phere to be limited by the level surface determined by (a). 



For decreasing values of the temperature gradient y the height of the atmos- 

 phere always increases and converges towards infinity when y converges towards 

 zero, i. e., in the case of the isothermic atmosphere. 



When y is negative, and thus the temperature rises with the height, 4> L also is 

 negative. The atmosphere remains unlimited upwards, while its analytical con- 

 tinuation below sea-level has the limit 4> L determined by equation (a). 



44. States of Unstable Equilibrium. In the extreme case Ry = 00, ?'. <?., in 

 the case of an infinite decrease of temperature with the height, we get <f> L = o. The 

 atmosphere is, then, condensed to an infinitely thin sheet. For values of Ry decreas- 

 ing from 00 to 1, we get values of the temperature gradient y decreasing from 00 to 

 0.00348, this last value representing a fall of temperature of 3.48 C. for every 100 

 dynamic meters of height. Extreme falls of temperature of this order of magni- 



