Energy and Entropy of a Body. 1 3 



Since these equations must be true for any values of the differ- 

 entials dT and dx, each of them splits up, as has been already 

 pointed out in a similar case above, into two equations. Of the 

 four equations so arising we will here employ only those two 



which can serve for the determination of -^ and of -p=, namely 



dT~T dT' { } 



dV __ dQ, dw -__. 



dT-dT~dT' l ] 



In order to determine the two other differential coefficients 



-7— and -y-, we will apply equations (16) and (17) deduced 



above. With the aid of these expressions of the four differen- 

 tial coefficients we can form the following complete differential 

 equations of S and U : 



rfS = I.g<fT + E lT( fo, (22) 



rfU-^-Jj^dT+EW*. . . . (23) 



Since the magnitudes S and U must be capable of being re- 

 presented by functions of T and so 7 in which the two variables 

 T and x may be looked upon as independent of each other, the 

 known equation of condition of integrability must apply to both 

 the foregoing equations. For the first equation this is 



d_(\ dQ\_ dV xT 

 dx\T'dT/~ dT' 



or, differently written, 



dAdH) l dT (M > 



For the second equation the equation of condition is 



d/dQ\_d(dw\_dW xT 

 dxXdH) dx\dT/~ dT ' " • [ } 



These two equations of condition are connected with each 

 other in such a manner that from either of them the necessity 

 of the other can be immediately deduced. Between the two 

 magnitudes E, T and E f , T , which occur in them, the following 



