OF MOLECULAR VORTICES. 305 



tension is equivalent, during a change of the figure and dimensions of all the 

 elementary circulating streams in a given body, may therefore be expressed by 

 multiplying the absolute temperature by the change in the value of a function, 

 to be afterwards determined, of the dimensions, figure, and temperature. If to 

 that function be added a function which is the integral of the increment of the 

 energy of steady circulation divided by the absolute temperature, the sum is 

 what I have elsewhere called the thermodynamic function. Let it be denoted by 

 <f>\ and let d Q denote the quantity of energy which must be communicated to 

 the body, in order to produce the increment d . 4> in the thermodynamic function 

 at the mean absolute temperature T ; then we have 



dQ = rd.<J> (18); 



and this, when the proper value has been assigned to the thermodynamic 

 function, is the general equation of thermodynamics. The process of finding the 

 value of the thermodynamic function is well known ; but a summary of it will 

 be given here for the sake of completeness : — 



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<f>, the external work done by unity of mass of 

 the body is 



Xdx + Ydy + Zdz + &c; 



Then, by the principle of the conservation of energy, it is necessary that the 

 following expression should be a complete differential : — 



rd(f) — Xdx — &c. ; 



whence it follows, that the thermodynamic function <\> is the integral of the 

 following set of partial differential equations :* 



dtp _ dX m dp dY dp d/L 



dx ~ dr ' dy ~ dr ' dz ~ dr ' 



that is to say, the thermodynamic function has the following value : — 



<p = xj,(r) +J -^dx+f—dy + bc; 



in which all the integrals are taken at constant temperature. 



For a perfect gas at constant volume, we have c/Q = Jc dr, in which ,)c is the 



* See Philos. Mag. for December 1865. 

 VOL. XXV. PART II. 7 G 



