502 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 5 1 



PART I. THE WORK EQUIVALENT TO A GIVEN DISTRIBUTION 



OF PRESSURE 



(1.) IN GENERAL FOR ANY GAS 



Given air in a definite volume k on which no outer forces are 

 acting. 



The initial condition is uniform constant density ft and uniform 

 constant pressure p . The final condition is ft and p. During the 

 transition we have ft t and p r 



For a small change of condition the elementary mass dm performs 

 a work of expansion expressed by 



dmp t dl - ) = - dm P ( d/n ( 



and from beginning to end the total work is 



— , d !h 



i 



Mo lh 



The air which when brought to the density ft is contained in the 

 elementary volume dk, has the mass ftdk, therefore the work of 

 expansion done by the whole mass is 



-S" dk L 



'Mo Mf 



If the relation between pressure and density is independent of 

 the path followed by the particle of air, if for instance it is arranged 

 that the transition or change of position of the particle of air shall 

 take place under constant temperature (isothermal), or that it shall 

 take place without increase of heat and without exchange of heat 

 (adiabatically), then the value of a will be determined by the initial 

 and final conditions. 



For the final distribution of pressure p the gas has a store of 

 energy A that is equal and opposite to a. It is demonstrable by 

 means of the aerodynamic equations that this represents the poten- 

 tial energy of the pressural forces for the given distribution of 

 mass, ft, or 



A = { fidk P* Pt dfi t (I) 



■ J Jflo ft t 2 



