418 PROCEEDINGS OF THE AMERICAN ACADEMY 



from one state to the other, it is probably best to introduce surface 

 energy into the equations of the external work. 



Let the state of any body be represented as a function of the two 

 independent variables x and T, — where T is the absolute temper- 

 ature. If the body is an homogeneous solid, its mechanical state is 

 represented as a function of six independent variables. The equa- 

 tions, however, deduced by considering only the variations of x and T 

 can be extended to the case of a solid by adding five other equations 

 of the same form. 



When the body undergoes any small transformation, the quantity of 

 heat absorbed — expressed in mechanical equivalents — is: 



dQ = adx-^hd T. (A.) 



The coefficient "a" depends upon the external and the internal works. 

 It is the mechanical equivalent of the latent heat relative to the vari- 

 able X. represents the specific heat of the body for a constant me- 

 chanical state. 



The external work is done against superficial tension and external 

 pressure : hence " a " depends upon these two quantities. 



The value of the external work in the above transformation is : 



dW = pdx-\-Sdx. (B.) 



/S' is a function of the superficial tension and of the shape of the body. 

 The work done against superficial tension is equal to the coefficient of 

 capillarity — or the coefficient of superficial energy — into the incre- 

 ment of area. Let k ■=:. the coefficient of capillarity, and let 



dA 



where A equals the area of the surface of the body. Then for S may 



be substituted 



S=kf{x). 



Let F represent the total energy of the body : 



dF=adx^bdT-{-pdx-{-Sdx. TIL 



For any closed cycle the total variation of F is zero. The value 

 of an increment of 7^ depends merely upon the initial and final states 

 of the body, d F is therefore an exact differential. 



d(a-\-p-{.S) _db , . 



dT ~ dx ^ '^ 



