504 SMITHSONIAN MISCELLANEOUS COLLECTIONS VOL. 51 



Then for an isothermal change we have 



e = a (a) 



For an adiabatic change we have 



r (r - i) 

 e = y° + — ^j — °* + (&) 



And always 



J* adk = 



Hence we obtain for isothermal change 



a = p.jdk ( r ^ - ri + §^4 " ) = * J* (2 ■ ■ •) ^ 



and for adiabatic change 



(2- r ) q , (2- r )(3- r ) t 



= rPojdk (^ 



+ 



2 2.3 2.3.4 



-7 J 



r2 



J rf *U' 



These forms, like the fundamental equation (I), show that the 

 contribution of each elementary volume whose density deviates 

 from the average value, is positive. The contribution of a volume 

 in the low-pressure region is indeed somewhat larger than that of 

 the same volume in a high-pressure region having the same absolute 

 value of a. 



For very small changes of pressure the first term of the develop- 

 ment is sufficient. This solution was first given by Lord Rayleigh 

 (see Vol. II, page 2 2, of his Theory of Sound, German edition (Bruns- 

 wick, 1880). For equal values of a the potential energy of any dis- 

 tribution of pressure is y times greater under adiabatic conditions 

 than it is under isothermal; but with equal values of eit is only 1/7- 

 times as large. 



The work stored in a very large volume of gas, when only a 

 small portion of it is disturbed 



Let k indicate the volume that suffers a disturbance of its equi- 

 librium; k' ', the remaining far greater volume whose density is not 

 appreciably changed by transfer of any mass to or from k; a and 



