Fundamental Theorem in the Mechanical Theory of Heat. 95 



other temperatures may be attributed to the reservoirs of heat 



without producing thereby any change in the expression j -~ 



which shall be prejudicial to the validity of the foregoing equation. 

 As with this signification of t the several reservoirs of heat need 

 no longer enter into consideration, it is customary to refer the 

 quantities of heat, not to them, but to the changing body itself, 

 by stating what quantities of heat this body successively receives 

 or imparts during its modifications. If hereby a quantity of 

 heat received be again counted as positive, and a quantity im- 

 parted as negative, all quantities of heat will of course be affected 

 with a sign opposite to that which was given to them with 

 reference to the reservoirs of heat, for every quantity of heat 

 received by the changing body is imparted to it by some reser- 

 voir of heat ; nevertheless, this circumstance can have no influ- 

 ence upon the equation which expresses that the value of 

 the whole integral is zero. From what has just been said, it 

 follows, therefore, that when for every quantity of heat dQ, 

 which the body receives or, if negative, imparts during its modi- 

 fications the temperature of the body at the moment be taken 

 into calculation, the equation (II) may be applied without 

 further considering whence the heat comes or whither it goes, 

 provided always that the process is reversible. 



To the equation (II) thus interpreted we can now give a 

 more special form, as was formerly done to equation (I), in 

 which form it shaU express a particular property of the body. 

 . We shall thus obtain an equation essentially the same as the 

 well-known one deduced by Clapeyron from the theorem of 

 Carnot*. With respect to the nature of the modifications, we 

 shall assume the same conditions as before led to the deduction 

 of the equations (2) and (3) from (I), and which also suffice for 

 the fulfilment of equation (II). Hence, the condition of the 

 body being defined by its temperature t and volume r, we have 



.i ,>9ii -^mv dQ, = -TT dt -\- -r- dv,as&'fi yhod '^m^imdrj o/^ 

 >. .M «B 8- ^ ^ lijoifijjiijxfloo iw hmo^lq jt 



Inasmuch as by (II) I -y^ must always equal zero, whenever 



t and V assume their initial values, the expression under the 

 integral sign, which by the foregoing equation becomes 



\ dQ, ,, \ d(i^ ^^ i^^5« y^^' 



must be a complete differential, if t and v are independent vari- 

 ables ; and the two terms of the expression must consequently 



/iiii JiuiJf ^' * Journ. de VEcole Polytechnique, tome xix. - Jvi wvOiJiia,. 



