84 M. R. Clausius on a modified Form of the second 



In order to give special forms to equation (I), in which it 

 shall express definite properties of bodies, we must make special 

 assumptions with respect to the foreign influences to which the 

 body is exposed. For instance, we will assume that the only 

 active external force, or at least the only one requiring consi- 

 deration in the determination of work, is an external pressure 

 everywhere normal to the surface, and equally intense at every 

 point of the same, which is always the case with liquid and 

 gaseous bodies when other foreign forces are absent, and might 

 at least be the case with sohd bodies. It will be seen that under 

 this condition it is not necessary, in determining the external 

 work, to consider the variations in form experienced by the 

 body, and its expansion or contraction in different directions, 

 but only the total change in its volume. We will further assume 

 that the pressure always changes very gradually, so that at any 

 moment it shall differ so little from the opposite expansive force 

 of the body, that both may be counted as equal. Thus the 

 pressure constitutes a property of the body itself, which can be 

 determined from its other contemporaneous properties. 



In general, under the above circumstances, we may consider 

 the pressure as well as the condition of the body, so far as it is 

 essential to us, as determined so soon as its temperature t and 

 volume V are given. We shall make these two magnitudes, there- 

 fore, our independent variables, and shall consider the pressure 

 p as well as the quantity U in the equation (I) as functions of 

 these. If, now, t and v receive the increments dt and dv, the cor- 

 responding quantity of external work done can be easily ascer- 

 tained. If any increase of temperature is not accompanied by 

 a change of volume, no external work is produced ; on the other 

 hand, if, with respect to the differentials, we neglect terms higher 

 than the second in order, then the work done during an incre- 

 ment of volume dv will be pdv. Hence the work done during a 

 simultaneous increase of / and v is 



dW=pdv, 

 and when we apply this to the equation (I), we obtain 



dQ=dV-\-A.pdv (2) 



On account of the term A .pdv, this equation can only be inte- 

 grated as soon as we have a relation between i and v, by means 

 of which t and p may be expressed as functions of v alone. It 

 is this relation which, as above required, defines the manner in 

 which the changes of condition take place. 



The unknown function U may be eliminated from this equa- 

 tion. When written in the form 



^Q^/ . ^Q^ ^U - /dV , \ , 



