570 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



in the initial stage the temperature of the lowest surface is a con- 

 tinuous function of the distance % from the left-hand end of the 

 trough; therefore the temperature at the point (x, z) is expressed 

 by the equation 



T = f(x)-^- 



If the / (x) increases from the left toward the right then the entropy- 

 will also do so, since the pressure p h is again assumed to be constant 

 at the level h; hence the entropy is only a function of the distance 

 x or the length of the trough and this is equally true of the pressure 

 p 0x at the base level. 



If the mass of air under the pressure p h passes over adiabatically 

 into the condition of equilibrium, then a horizontal layer is formed 

 from each vertical column of the initial condition and the masses 

 will now succeed each other from below upward in the same order 

 as they were before arranged from the left toward the right. A 

 mass originally in a vertical column at x having an initial p and 

 T has in the final stage a p' and T' such that 



P' - Ph + 7 I (Pox - Ph) dx 



I Jl-x 



= P - (yP -Ph-\- [ JPox ~ Ph) dX j 



In this last equation we consider the member in the parenthesis as 

 small relative to p, which is true when the difference between any 

 two values of the pressure within the mass is small as compared with 

 the total pressure p h . Under this assumption we introduce an 

 approximate computation analogous to that of the last section 



§ (25). 

 r-r-r-r(J)*-«[r-g-^£ > - A )*]. 



We first seek for the mass-integral of (T — T') throughout the 

 whole mass of the unit column above the point x on the axis of 

 abscissae; in accord with the previous definition we put T* for the 

 average temperature of this column, then we have 



T* Pox ~ Ph = T* f fidz - f Tftdz 



