220 On the Thermal Energy of Molecular Vortices. 



Let dx, dy, dz, &c. denote changes in the dimensions of unity 

 of mass of the body, of the nature of strain, such as dilatations 

 and distortions ; and let X, Y, Z, &c. denote the forces, of the 

 nature of elastic stress, which the body exerts in the respective 

 directions of such changes, so that while the thermodynamic 

 function undergoes the change d<j> the external work done by 

 unity of mass of the body is 



Xdz+Ydy'+Zdz+&c, 



Then, by the principle of the conservation of energy, it is neces- 

 sary that the following expression should be a complete differ- 

 ential, 



rdcj) — ILdx — &c; 



whence it follows that the thermodynamic function <£ is the inte- 

 gral of the following set of partial differential equations"*, 



d<j>__dX t d<p_dY f #_i? & 

 dx ~ dr r 3 dy ~~ dr f dz Or 3 ' ' 



that is to say, the thermodynamic function has the following 



value, 



in which all the integrals are taken at constant temperature. 



For a perfect gas at constant volume we have dQ, = JcdT, in 

 which Jc is the dynamical value of the specific heat of the gas at 

 constant volume; and consequently ^(r)=Jchyp. log-r; and 

 the same is the value for any substance which, at the tempera- 

 ture r, is capable of approaching indefinitely near to the per- 

 fectly 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 (PoggendorfFs 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 ^(t) is a function of the temperature, which becomes =0 

 at all temperatures at which an indefinitely close approximation 

 to the perfectly gaseous state is possible — thus giving for the 

 complete value of the thermodynamic function, 



lh dxJt )~aW dyJrkQ ' (19) 

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



* See Phil. Mag. for December 1865. f See ibid. 



