12 Prof. Clausius on the Determination of the 



These assume particularly simple forms when the temperature 

 T is taken as one of the independent variables. If, for instance, 



dT dT 



we put T=y, it follows thence that -7- =1 and — =0; and 



we have also, in place of E.^ and E'^, to write E*r and E' jT . 

 Equations (5) and (8) thus become 



£-*< w 



S= E '- w 



These equations can at once be integrated with respect to <r, and 

 we so get 



S= : JH*£+-*(T). (18) 



U = JE', T rfa ; + ^(T) ) (19) 



where <£(T) and ^r(T) are two arbitrary functions of T. 



The last two equations can of course be easily changed back 

 again by putting any other variable y in place of the variable T. 

 For this purpose we only require to substitute for T the func- 

 tion of x and y which represents this magnitude. The equations 

 hence resulting are the same as those obtained when we start 

 from the more general differential equations (5) and (8), and 

 apply to them the common process of integration, keeping in 

 mind, at the same time, that according to (14) we have to put 



J,E„=E^ and |^=EV 



We have thus in what precedes arrived, by help of the par- 

 tial differential equations deduced from equation (I), at expres- 

 sions for SandU, each of which still contains an arbitrary func- 

 tion of T. If we want to determine these functions, which are 

 there left arbitrary, we must go back to equations (la) and (II«), 

 whence equation (1) was obtained by elimination of dQ. 



it 



Let us assume that the condition of the body is determined 

 by its temperature and any other variable x, we can then give to 

 the two equations (I a) and (II a) the following form : 



dT dl + dw ax -"£ rfT dl+ T dx' 



rfU.-rfU, (dQt <fcA ,„ , /</Q dw\, 

 STdT+ ^ dx ={ M - Jt )dT+[-^-~)dx. 



