516 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



Under this latter condition we have 



«*, {t( a A + u a JL 



dt J /1 \ dt dx 



J 



Jt du dv , dw \ ,, 

 p __ + —+_) dk 

 \ dx dy dz I 



+ w^+ . . . )dk 



dt 



/dt 



= ~ \ P 



J 



dy 



u t+v-^ + w-t) dk. 



dx dy dz 



The first and second of these equations state that — dAdt is the 

 sum of the works of expansion done by all the elementary masses 

 within the volume k in a unit of time ; the third equation states that 

 this is also the work done in the same unit of time by the pressural 

 forces. In general these two quantities of work differ for each 

 individual elementary mass; it was therefore important to demon- 

 strate that starting with the work of expansion we arrive finally 

 at a correct expression for the potential energy of the distribution 

 of pressure. 



We can also deduce the value of A by another route, i. e., by 

 computing the work done by the pressural forces during the pas- 

 sage from the initial to the final stage. We thus arrive at the 

 expression 



A- (jkl'"j,A" J r 



J J/(„ Jp p 



whose identity with equation (I) can easily be proved: both of these, 

 by partial integration under the assumption of the constancy of 

 the mass of gas within the volume k, lead to the following: 



A = f fidk P dp - f dk (p - p ) 



Now the simple assumptions that have been made the founda- 

 tion of the preceding computation of A do not obtain in the atmos- 

 phere. If we assume the atmosphere to pass adiabatically from 

 any initial condition in which we happen to find it over into a con- 

 dition of equilibrium, then this will not be possible unless some 

 masses of air pass from some one horizontal layer over into another 

 layer. If an interchange of heat take place, then, except in the 



