518 F. II. Bigelow — Thermodynamics of the 



The reduction of the adiabatic system to the nonadiabatic 

 system through the ratio of the temperatures is, 



i — JL J^ 1 fc-i _fc-i a i \ 



n k nk k ' n ) 



\"T7/ adiab - = \~PT/ adiabatic. = \~T7/ 



nonadiab- 



We may now differentiate equation (6) for the nonadiabatic 

 formula, with P, T, n variables, and P , T , k constants, using 

 the type for the unit mass, 



clu dx/ 



u = y x , and = log y . dx -f- x — — ' 



the result being, as given in (35), (36), M. W. E., March, 1906, 



dP r 



(17) gdz = — ' 1- Cp T log Tdn (for nonadiabatic system). 



r 



dP 



Since gdz = — in the adiabatic system, it follows that 



the term + Cp T log Tdn transforms the adiabatic into the 

 nonadiabatic system. If a similar set of combinations, as for 

 (1) (2) (3) to produce (4), be made with (IT) (2) (3), we find, 



dT h - 1 dV Cp 



(18) —^- = — — — a log Tdn, 



which leads again to the difference between the two systems, 

 as given in (16). 



Circulation and radiation in the nonadiabatic system. 



In the adiabatic system the fundamental relation is contained 

 in the formula for no circulation, and no radiation of heat in 

 the atmosphere, 



(19) gdz = - Cp. dT = - .^l- 



Pa 



If there is circulation we have by the differential general 

 equation of motion, 



dP 



(20) gdz = — qdq. 



P 

 If in addition there is radiation of heat, it is 



dP 



(21) gdz = ¥h - d Q- Integrating, 



(22) g {z - z ) = - ^—^ - \ {q\ - q\) - (Q, - Q n ). 



