(24) only if the change in T is proportional to that of P or 

 that the temperature is constant. The second condition 

 would require that X ~ 1 be identified with the isothermal 

 compressibility, a deduction which is known a posteriori 

 to be incorrect for acoustic waves in gases. * The other 

 possibility requires that 



\ 

 dP = —dp = A dT (26) 



P 



where A is an appropriate coefficient. The three parametric 

 relations implied by (26) represent the differential equations 

 for a curve in the p, P, T diagram. Clearly this curve must 

 lie on the equation of state "surface" given by (24). Except 

 for this constraint, the curve is arbitrary. 



In order to remove the ambiguity regarding the thermo- 

 dynamic path, it is evident that a further relation between 

 the thermodynamic variables is needed. Such a relation 

 could be provided a priori by stipulating that the path is 

 adiabatic. This requires a definite relation between P and 

 T which is given by Kelvin's equation 



hT 

 dT-^-dP (27) 



pO p 



where V is the specific heat at constant pressure. This 

 implies in turn that A' 1 is the coefficient in (27) and that 



b T 

 K -^~T (28) 



Pdp 



r This would lead to the Newtonian expression for the speed 

 of sound, which is known to be less than the measured 

 values. 



38 



