534 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



— (R) is the work done by the frictional forces, or + (R) is 

 the loss of kinetic energy by the action of friction and 

 other resistances. 



Of these quantities the first two depend only on the initial stage 

 and final stage, the other two depend on the nature of the motion. 



In a closed system the work of the pressural forces [or — dA] is 

 equal to the whole work done by the expansion of the air. Let 

 (Q) be the increase of heat, and dl the increase of internal energy 

 then 



(0) = dl - dA 



For motions that occur without any general increase of heat (but 

 in which internal exchanges of heat or even external additions and 

 withdrawals that balance each other are allowable) , the value of dA 

 has also this same property, since dl depends only on the final stage. 

 The general equation of energy for a closed system as deduced 

 from the preceding considerations, 



(0) = d(K + P+T) + (R) 



tells us that that part of the added heat that does not serve to 

 increase the internal energy, represents the increase in kinetic 

 energy and in the potential energy of position and in the consump- 

 tion of energy in overcoming friction. If there be no increase of 

 heat then the increase of mechanical energy takes place at the 

 expense of the internal energy already present. 



§ (2) By the help of this last equation we will first seek for a closed 

 dry air system that without any increase of heat can develop such 

 great kinetic energy as we observe in storms. 



Let the air be initially at rest but not in equilibrium. It starts 

 in motion and tends to attain a condition of stable equilibrium. 

 In general we know the characteristics of this final condition : every 

 horizontal layer is a surface of equal pressure and equal tempera- 

 ture, the entropy (or the potential temperature) increases with the 

 altitude. In order to completely determine this final stage we 

 will assume that every part of the mass behaves adiabatically, or 

 isentropically, during the motion. We now construct the final 

 stage by the following process: we seek first the masses having the 

 least entropy at the initial stage; these will form the lowest stratum, 

 the other masses will arrange themselves proceeding upward, i 



