8 Prof. Clausius on the Determination of the 



it splits up at once into the two following equations : 

 dS _dV dw 



dx ~~ dx dx 



</S_<HJ dw 

 dy ~~ dy dy 



(2) 



From these equations the magnitudes S and U can be eliminated 

 by a second differentiation. 



§4. 



We will first eliminate the magnitude TJ, since the equation 

 that so results is the simpler of the two. 



For this purpose we differentiate the first of the equations (2) 

 with respect to y, and the second with respect to x. In so doing we 

 will write the differential coefficients of S and Uof the second order 



quite in the usual way. The differential coefficients -r- and -=-, 



on the other hand, we will write as above, j- 1 j- ) aad j- ( -j- j, 



in order to express formally that they are not differential co- 

 efficients of the second order of a function of x and y. Lastly, 

 we have to consider that the magnitude T which occurs in these 

 equations, namely the absolute temperature of the body, is also 

 to be regarded as a function of x and y. We thus obtain 



dT d$ +T> «PS d*U . d /dw* 



d /dw\ 

 dy\dx) 



dy dx dxdy dxdy 



dx dy dydx ~ dydx dx\dy ) 



Subtracting the second of these equations from the first, and re- 

 membering that 



rf 2 S rf*S , d*U d*V 

 and 



we obtain 



dxdy dydx dxdy dydx 3 



dy dx dx dy ~ dy\dx ) dx\dy ) ' * ' 



The difference which occurs here in the right-hand member 

 plays an important part in the mechanical theory of heat. In 

 my last published paper I have called it difference of work (Werk) 

 in relation to xy, and have denoted it by E^ ; so that we may put 



d /dw\ d /dw\ 



^"SW/ Si*/' • • • • ( 4 ) 



