24G THE MECHANICS OF THE EARTH'S ATMOSPHERE. 



temperature we have the length P W>P N' ; that is to say, the poten- 

 tial temperature T", as attained by adiabatic change after passing into 

 the condensation stage and after precipitation of some water, is higher 

 than the potential temperature T of the initial condition and of all the 

 conditions previously passed through in the dry stage 0. 

 Analytically this may be proved in the following manner : 

 For the transition from a to b the following equation obtains 



If this equation remains in force after crossiug over the curve of sat- 

 uration, then we obtain for the pressure proper to the volume v e a value 

 Py<Pc where p c is the pressure that in the condensation stage actually 

 corresponds to the volume v c . 

 But 



p.v e * =jW =C" 

 and since 



P y »c K =iW = C" 



and since also 

 therefore 



Pr < Pc 



G'<G". 



But from this it further follows that v">v' and T">T' where v' and v" 

 are the volumes corresponding to the normal pressure p on the adia- 

 batics ab and cd ; hence, 



p v' K = C 

 and 



p v"*=zC"\ 

 beside which the following equation holds good: 



v':v"=T':T". 



Thus we attain to the theorem 



In adiabatic changes of condition in moist air the potential tempera- 

 ture remains unchanged so long as the dry stage continues, but it rises with 

 the occurrence of condensation and so much the more in proportion as more 

 water is discharged. 



Since in the free atmosphere, in general, evaporation does not occur 

 and since also the carrying aloug of all the water that is formed, at 

 least in the case of heavy condensation, must be considered as an ex- 

 ceptional case only, therefore, this theorem can also be brought into 

 the following form : 



Adiabatic changes of condition in the free atmosphere, assuming that 

 there is no evaporation, either leave the potential temperature unchanged 

 or elevate it. 



