Energy and Entropy of a Body. 9 



The foregoing equation is thus transformed into 



dT t dS_dT .^S =E (5) 



dy dx dx dy xy ' 



This is the differential equation resulting from equation (1) which 

 serves to define S. 



In order now to eliminate the magnitude S from the two equa- 

 tions (2), we will write them as follows : — 



dS_l rfU 1_ dw 

 dx~T' dx + T' dx' 



f*S_l dV I dw 

 dy~T' dy + 'V'dy' 



Of these equations, again, we will differentiate the first with 

 respect to y, and the second with respect to x, whereby we get 



i!?L I i"£_ ]_ dT dV ±(\ dw\ 

 dxdy ~~ T * dxdy T 2 ' dy' dx + dy\T ' dx)' 



dydx ~ T ' dydx T 2 ' dx' dy + dx\T ' ~dy)' 



Subtracting the second of these equations from the first, bring- 

 ing all the terms of the resulting equation in which U occurs to 

 the left-hand side, and multiplying the whole equation by T 2 , 

 we have 



dT aV__dT dV™[d/l dw\_ ±(\ dw\\ , fi * 



dy ' dx dx' dy ldy\T ' dx) dx\T ' dy)\ ' K) 



For the magnitude which here stands on the right-hand side 

 we will likewise employ a special symbol, putting 



E '- T2 [|(r£)-£(rS)]-- • • ® 



The last equation then becomes 



d_TdV__dTdV =E , ^ 



dy dx dx dy ** v ' 



This is the differential equation, resulting from equation (1), 

 which serves to define U. 



§5. 



Before pursuing further the treatment of the two differential 

 equations (5) and (8), it will be advisable to direct attention 



