24 PROCEEDINGS OF THE AMERICAN ACADEMY. 



where f -^ 1 and f -7-^ j represent the change of internal energy 



and volume respectively, with a change of temperature at constant pres- 

 sure, P. 



\dTJ^-\dTj, • \dv)XdT); 



from (37), 



^''~ \dT Jp 



( 



dm\ f dv 



dv J x\d T I p 



^^\ -c -p(^ 



dv 

 ~dTjp 



(dm\ _p(d_v\ 

 \dv)^ \dT); 



dv\ r„ , (d'^\ 

 a T 



(38) 



C„ — Cp 



We have found for a gas or a liquid, -y = — . Substituting in (38), 



Since ( y— , j is the measure of thermal expansion, equation (39) 



contains only quantities capable of experimental determination, and we 

 are able to calculate r,, from the experimentally found value of Cp. The 

 calculation is further simplified by the fact that in the case of most va- 

 pors — is negligible compared with P, and in liquids P is negligible 



compared with — . For close approximations, therefore, we may write 



for liquids, 



a ( dv\ ,, s 



and for vapors, 



c^ = cp-p(^) =Cp-R, (406) 



dT,p 



that is, the equation for the specific heats of perfect gases applies to 



