466 X. STATICS OF DEFORMABLE BODIES. 



air to change its place convective equilibrium is established. The 

 principles of thermodynamics give us the relation for adiabatic com- 

 pression 



17) p = !)(>*, 



where % is the ratio of the specific heat at constant pressure to that 

 at constant volume, whose numerical value is about 1.4. We then have 



18) 





Since x > 1, p diminishes as 8 increases and is equal to zero when 

 gz = c f so that on this hypothesis the atmosphere has an upper limit, 

 which may be calculated when the value of Q for a single value of 8 

 is known. 



It is obviously improper to consider the equilibrium of the 

 atmosphere to an infinite distance without taking account of the 

 variation of gravity as the distance above the surface of the earth 

 increases. Considering the earth to be a sphere with a density that 

 is a function only of the distance from the center, we have^ with 7 

 positive 1 ), as in 123, 149, instead of equation 8), 



Xdx+ 

 so that on the hypothesis of equal temperature 



19) - - = const. a log p, 



20) 9 = 9 ,e^. 



On this hypothesis the density decreases as we leave the earth, but 

 not so fast on account of the diminution of gravity, so that at infinity 

 the density is not zero but equal to the constant 



In this example we have neglected the attraction on the gas of 

 those layers lying below. From equation 20) the barometric formu 

 is obtained. 



Proceeding in the same manner for convective equilibrium, we have 



21) - 



- 



Here again Q decreases as r increases, giving an upper limit to the 

 atmosphere for Q = for a finite value of r. 



1) It is to be observed that in equation 8) the forces are taken as the 

 negative derivatives of the potential, but as in the following examples involving 

 the earth's attraction they are obtained by multiplying ilae positive derivative byy, 

 we must in the integral equation change the sign of V and multiply by y. 



