52 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



tude may exist locally under extraordinary conditions, as above a hot chimney or 

 above a volcano in action. They may perhaps exist also for a short while over a 

 heated area before the formation of a tornado. But the corresponding state of 

 equilibrium can not endure. For it is seen from the second equation (b) that as 

 long as Ry is comprised between co and i there will be increase of density upward. 

 The state of equilibrium is therefore completely unstable. 



The limiting case 

 (a) i?7 = i 



corresponding to a fall of temperature of 3.48 C. for every 100 meters, is interest- 

 ing from a mathematical point of view. In this case the equations (a) and (b) re- 

 duce to the simple forms 



(') * = *.(' "W o *) p = p > >-a(i-jb[*) 



These are all linear, those for the specific volume, a a , and for the density, p =p , 

 showing that specific volume and density are constant. As the pressure and the 

 temperature thus both decrease with the height, they compensate each other in 

 their influence upon the density of the air, the result being a perfectly homoge- 

 neous atmosphere. 



Also, in the case of the homogeneous atmosphere the equilibrium is unstable. 

 For if a mass of air be moved upwards, the adiabatic cooling will not suffice to 

 bring it down to the temperature of the higher strata, to which it has been moved. 

 Therefore, if once given the slightest displacement upwards, it will continue mov- 

 ing upwards, remaining always lighter than the adjacent air. 



The height <f> L ' of this homogeneous atmosphere has, according to (a) and sec- 

 tion 43 (tf), the value 



(') 4>,' = M 



It merits attention that we may introduce the two limiting heights <f> L and <j> L ' as fun- 

 damental parameters in our formulae. To do this we have the expressions 



The ratios on the right side being independent of the units of gravity potential, we 

 may also write 



measuring the height H and the limiting heights H L and H in dynamic meters. 

 Equations (5) or (b') give thus a perspicuous sense to expressions appearing in the 

 equations (a) and (b). 



