546 



SMITHSONIAN MISCELLANEOUS COLLECTIONS 



VOL. 51 



where n is the direction of the normal directed inward to the surface 

 of the space k. 



In a similar way we obtain 



CldF dF\ CFd^i C 



J \Tt-Tt) dk = J lTt dk ~ J Fc ^(n,c)dO 



The surface integrals disappear in our case when the mass of air 

 is bounded by fixed walls; and so also when we extend the space 

 integral or the mass integral over the whole atmosphere, or over a 

 mass of air that is bounded by the ground and fixed vertical walls 

 but is open above; assuming that both the pressure and also the 

 product of pressure by vertical velocity diminish to zero as the 

 altitude increases. 



The equation for the energy of the whole mass of a closed system 

 as obtained from equation (1) is as follows: 



a 

 dt 



) (/o + MW)dk + I -£ dk - ) X.ccos(R,c)fidk=*0. (4) 



We would remark that here in closed systems the work done by the 

 pressural forces is equal to the sum of the work done by the expan- 

 sions of all the elementary masses or 



-Sl(t~ d m)« dk = S { 'i{l) f ' dk 



In a similar way there follows from equation (2) 



J 



dQ d 



~dt^ dk = C vJ t 



i T ^-S p ^ dk -S 



d 



/j. dt 



dk 



pt dt 



dk 



(5) 



From equations (4) and (5) there results the following equation 

 (6) which we call the equation of energy for a mass of air in a closed 

 system 



CdQ a f 



a 



d~t 



J( 



— I Re cos (R c) /idk 



{i- + fiW jdk - 



} ■ (6> 



