CHAPTER V. 



IDEAL STATES OF EQUILIBRIUM IN THE ATMOSPHERE. 



41. Analytical Integration of the Equation of Atmospheric Equilibrium. 



The hydrostatic equation 



() dip = adp 



contains three variable quantities, <p, fi, a. Two of them, p and a, are connected 

 with a third variable # by the equation of state 



(6) fa = R& 



& being the true temperature of dry or the virtual temperature of moist air. By 

 this equation we may introduce temperature i?asa variable in (a) instead of the 

 specific volume a. This will generally be convenient, and the equation of atmos- 

 pheric equilibrium then takes the form 



(c) d<p=-R9 d 



Now, supposing a relation between temperature and pressure to be known, 



equation (c) is seen to be integrable immediately. To perform the integration we 

 may choose either of two ways. We may use (d) to eliminate the pressure from 

 the second member of (c). The integration then gives a relation between gravity 

 potential and temperature 



(c) yi{<p, ;?) = o 



Eliminating afterwards the temperature between (d) and (<?), we get the relation 

 ot equilibrium connecting gravity potential and pressure 



(/) F&, p) = o 



Or we may, on the other hand, use equation (<f ) to eliminate the temperature from 

 the second member of (c). The integration then immediately leads to the equi- 

 librium relation ( f) between gravity potential and pressure. The elimination of 

 pressure between equations (/") and (d ) will then lead to the relation (e) connect- 

 ing gravity potential and temperature. 



Again, we might have written equation (c) in the form 



(*) -} = - R 



d<p v dp 



49 



