10 Prof. Clausius on the Determination of the 



for a moment to the magnitudes E^ and E'^ which therein 

 occur. 



Between these two magnitudes the following relation exists, 

 which can be easily deduced from the expressions given in (4) 

 and (7), 



, dT dw dT dw 



^*y=™*y-Ty"te + T x %' ' • " (9) 



Both the magnitudes E^ and W xy are functions of x and y. 

 If in order to define the body we select, instead of x and y, any 

 other two variables, which we may call f and 77, and form with 

 them the corresponding magnitudes E^ and E'^, namely, 



^=!(t)-|(S>- •■•;••• ^ 



these magnitudes are of course functions of f and 97, just as the 

 foregoing magnitudes are functions of x and y. But if now one 

 of the last two expressions, e. g. the one for E^, is compared 

 with the expression for the corresponding magnitude E^, we find 

 that they represent, not merely expressions for the same magni- 

 tude with reference to different variables, but actually different 

 magnitudes. For this reason I have not called E^ simply the 

 difference of work, but the difference of work in relation to xy, 

 whereby it is at once distinguished from E^ or the difference of 

 work in relation to f and 77 The same holds good of E'^ and 

 E'^, which are also to be regarded as two different mag- 

 nitudes. 



The relation existing between the magnitudes E^andE^ may 

 be found as follows. The differential coefficients occurring in 

 the expression given for E^ in (10) may be arrived at by first 

 forming the differential coefficients in relation to the variables x 

 and y, and then treating each of these two variables as a func- 

 tion of £ and 77. In this way we obtain 



dw _ dw dx dw dy 

 d% ~ dx d£ dy d% 



dw __ dw dx dw dy 

 drj dx drj dy drj 



Let the first of these two expressions be differentiated with 

 respect to 77 and the second with respect to £, and we then get, 

 by the application of the same process, 



