506 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



(2.) THE ATMOSPHERE OF THE EARTH 



If external forces act on the air then in a condition of equilibrium 

 the pressure varies from place to place. In studying perturbations 

 the potential energy of these external forces comes into consideration 

 in addition to the potential energy of the change of distribution of 

 pressure. Still in many cases the expressions above deduced can 

 easily be applied. 



We will designate as an atmosphere, any mass of air on which the 

 force of gravity is acting. For brevity we assume the acceleration 

 of gravity, g, to be constant, and the ground to be a smooth plane 

 and the initial temperature to be a function of the altitude only. 

 If this atmosphere be divided into individual layers of indefinitely 

 small thickness dz then under simple assumptions we can carry out 

 the analysis for each layer with the formulae that apply to a mass 

 of gas of constant initial density. 



If the condition of equilibrium is disturbed in only a relatively 

 small portion of the whole mass, then it will be assumed that the 

 excess or deficit of gas in each horizontal layer of this disturbed 

 cylinder comes from or has flowed into the undisturbed portion of 

 this same layer: hence the potential of the gravitational force 

 remains unchanged. The potential energy of the distribution of 

 pressure is given by equation (I) or the formulae derived therefrom. 

 The elementary volume dk is to be replaced by the product of the 

 elementary area dS and the altitude dz. 



If we assume that the equilibrium is disturbed in the horizontal 

 direction only, and that on the other hand the vertical equilibrium 

 remains unchanged and that the hypsometric formula is still appli- 

 cable, then the integration with reference to or along the vertical 

 direction is easily executed. 



(a.) Isothermal solution 



Under isothermal conditions and according to equation (la') 

 the store of work for any layer is 



dA = dz j yp log- + p - pJdS 



where the integral extends over a surface that includes all the dis- 

 turbed portion. 



If now we assume the temperature of the atmosphere to be con- 

 stant and designate by P the pressure at the base over the elementary 



