Energy and Entropy of a Body. 7 



generally by x and y. We can afterwards put in the place of 

 these undefined variables definite magnitudes, such as tempera- 

 ture, volume, pressure, or any others appropriate to the particular 

 investigation in view. 



If the condition of the body is denned by the two variables x 

 and y, all magnitudes which are defined by the actually existing 

 condition of the body, independently of the way in which the 

 body came into this condition, must be capable of being expressed 

 by functions of these variables, in which the variables themselves 

 may be regarded as independent of each other. Accordingly 

 the magnitudes S and U must likewise be regarded as functions 

 of the independent variables x and y. 



This, however, does not hold good respecting the magnitude w, 

 since, even if it is agreed that only reversible processes are to occur, 

 the external work (Werk), which is performed during the passage 

 of the body from a given initial to a given final condition, depends 

 not only on what the initial and final conditions are, but also 

 upon the succession of intermediate conditions through which it 

 passes, or upon the way in which the change takes place. The 



differential coefficients — - and -y- are indeed definite functions 

 ax ay 



of x and y ; but if the former is differentiated with respect to y, 

 and the latter with respect to x, the resulting differential coeffi- 

 cients of the second order, -j-(-j— ) and -7-1 -f- )> are not equal 



to each other, as must be the case if, x and y being variables in- 

 dependent of each other, the magnitude w could be expressed as 

 a function of them. 



If now in equation (I) we put 



dS= - 1 -dx+ -j-dy. 

 dx dy * 



dV = -7— dx H — — dy. 

 dx dy * 



, dw 7 dw , 

 it is transformed into 



T — d T — d —(— ~\j /^U dw \j 

 dx dy \dx dx / \dy dy J 



Since this equation must be true for any values of the differen- 

 tials dx and dy, and therefore for that case, amongst others, in 

 which one or other of the differentials is made equal to nought, 



