50 DYNAMIC METEOROLOGY AND HYDROGRAPHY. 



The equation in this torm is seen to be integrable at once if a relation between 

 temperature and gravity potential be given, i. e., a relation of the form (e). For 

 the integration we again have the choice of either of two ways. We may use 

 equation (e) to eliminate the gravity potential from the left member of (g). The 

 integration then leads to the relation (<f) between temperature and pressure. Then 

 the elimination of the temperature between (rf) and (g) leads to the equilibrium 

 relation (f) connecting gravity potential and pressure. Or we might have used (e) 

 to eliminate the temperature from equation (g). The integration would then have 

 led directly to the equilibrium relation (_/") between gravity potential and pressure, 

 while elimination between (y~) and (e) would have led to the corresponding rela- 

 tion (a?) between temperature and pressure. 



As will be inferred from the above discussion, we have to notice two cases of 

 immediate integrability, the first characterized by a relation between temperature 

 and pressure (</), the second by a relation between temperature and gravity 

 potential (e). Between these two cases of integrability there is a full correspond- 

 ence in this sense: that to a given relation between temperature and pressure (a?) 

 there will correspond a perfectly definite relation between temperature and gravity 

 potential (e), and vice versa. 



42. Atmosphere with Constant-Temperature Gradient. Let us suppose 

 temperature to be a linear function of gravity potential 



() * - #, - Tfi 



# being the temperature at sea-level and y the temperature gradient 



which is in this case constant. 



To find the relation between temperature and pressure, corresponding to the 

 relation (a) between temperature and potential, we eliminate d<f> between equations 

 (') and section 41 {g). This gives 



and hence after integration, p being the pressure at sea-level, 



H!) 



We thus arrive at this important result: 



If temperature be a linear function of gravity pote?itiaI, with the tempera- 

 ture gradient y, it will be proportional to the power Ry of pressure, R being the 

 gas constant. And vice versa: If temperature be proportional to any power 

 Ry of the pressure, it will be a linear function of gravity potential with the 

 temperature gradient y. 



Eliminating the temperature between the equations (a) and (3), we arrive at 

 the equilibrium relation between gravity potential and pressure, namely, 



