566 PROF. RANKINE ON THE THERMAL ENERGY OF MOLECULAR VORTICES. 



dynamical value of the specific heat of the gas at constant volume ; and conse- 

 quently, xfj (r) = Jc hyp. log. r ; and the same is the value for any substance 

 which, at the temperature r, is capable of approaching indefinitely near to the 

 perfectly gaseous condition. There is some reason for believing that all substances 

 may have that property;* but to provide for the possibility, pointed out by 

 Clausius ("Poggendorff's Annalen," vol. xcvi. p. 73), of the existence of substances 

 which at certain temperatures are incapable of approaching indefinitely near to 

 the perfectly gaseous condition, we may make (as that author does), 



^r( T ) = Jc hyp. log. t - x(t); 



where x ( T ) * s a function of the temperature, which becomes = at all tempe- 

 ratures at which an indefinitely close approximation to the perfectly gaseous 

 state is possible; thus giving, for the complete value of the thermodynamic 

 function, 



= Jc hyp. log. t + x (j) +J -far &n + J ~fa d l/ + & c. . (19). 



That expression may be abbreviated as follows : — Let U be the potential energy 

 of the elastic stress of unity of mass of the body at constant temperature ; then 



p = Jc hyp. log. r + x (r) + -fa . . . (20); 



and the corresponding form of the general equation of thermodynamics is as 

 follows : — 



dQ= {Jc + T^(r)}dT + Td . ^ . . . (21). 



§ 12. Conclusion. — In conclusion, then, it appears that the special supposi- 

 tions as to matters of detail, introduced into the hypothesis of molecular vortices 

 in the paper of 1849-50, are not essential to the deduction from that hypothesis 

 of the principles of thermodynamics, but that such matters of detail may be left 

 open to be determined by future investigations. 



* See Phil. Mag. December 1865. 



