10 



M. R. Clausius o?i the Moving Force of Heat, 



I 



^ 



In applying the foregoing Fig. 2. 



conaideratious analytically, we 

 will assume that the various 

 alterations which the gas has 

 undergone have been infinitely 

 small. We can then consider 

 the curves before mentioned 

 to be straight lines, as shown 

 in the accompanying figure. 

 In determining its superficial 



content, the quadrilateral abed ^ € h f 



may be regarded as a parallelogi-am, for the error in this case can 

 only amount to a differential of the third order, while the area 

 itself is a differential of the second order. The latter may there- 

 fore be expressed by the product ef.bk, where k marks the point 

 at which the ordinate 6/* cuts the lower side of the parallelogram. 

 The quantity bk is the increase of pressure due to the raising of 

 the constant volume of from t to t, that is to say, due to the 

 differential t—T=dt. This quantity can be expressed in terms 

 of V and t by means of equation (I.), as follows : 



dp=z 



Rrf/ 



If the increase of volume ef be denoted by dv, we obtain the 

 content of the quadrilateral, and with it 



TTie work produced = (1 .) 



We must now determine the quantity of heat consumed during 

 those alterations. Let the amount of heat which must be im- 

 parted to change the gas by a definite process from any given 

 state to another, in which its volume is =« and its temperature 

 = /, be called Q ; and let the changes of volume occurring in 

 the process above described, which are now to be regarded sepa- 

 rately, be denoted as follows : efhy dv, hg by dJv, eh by 8u, and 

 fg by h'v. During an expansion from the volume oe=zv to 

 of=^V'\-dvy at the constant temperature /, the gas must receive 

 the quantity of heat expressed by 



and in accordance with this, during an expansion from vhzszv-\-hv 

 to og=iV-{-hv-\-d!v at the temperature t—dt, the quantity 



