12 M. R. Clausius on the Moving Force of Heat, 



This equation may be regarded as the analytical expression 

 of the above maxim applicable to the case of permanent gases. 

 It shows that Q cannot be a function of v and t as long as the 

 two lattei* are independent of each other. For otherwise, ac- 

 cording to the known princij)le of the differential calculus, that 

 when a function of two variables is differentiated according to 

 both, the order in which this takes place is matter of indiffer- 

 ence, the right side of the equation must be equal 0. 



The equation can be brought under the form of a complete 

 differential, thus : 



dQ=dV-{-A.U''-^dv, .... (Lla.) 



where U denotes an arbitrary function of v and t. This differ- 

 ential equation is of course unintegrable until we find a second 

 condition between the variables, by means of which t may be 

 expressed as a function of v. This is due, however, to the last 

 member alone, and this it is which corresponds to the exterior 

 work effected by the alteration ; for the differential of this work 

 is pdvj which, when j!> is eliminated by means of (I.), becomes 



V 



It follows, therefore, in the first place, from (II«.), that the 

 entire quantity of heat, Q, absorbed by the gas during a change 

 of volume and temperature may be decomposed into two portions. 

 One of these, U, which comprises the sensible heat and the heat 

 necessary for interior work, if such be present, fulfils the usual 

 assumption, it is a function of v and /, and is therefore determined 

 by the state of the gas at the beginning and at the end of the 

 alteration ; while the other portion, which comprises the heat 

 expended on exteiior work, depends, not only upon the state of 

 the gas at these two limits, but also upon the manner in which 

 the alterations have been effected throughout. It is shown above 

 that the same conclusion flows directly from the maxim itself. 



Before attempting to make this equation suited to the deduc- 

 tion of further inferences, we will develope the analytical expres- 

 sion of the maxim applicable to vapours at their maximum density. 



In this case we are not at liberty to assume the correctness of 

 the law of M. and G., and must therefore confine ourselves to the 

 maxim alone. To obtain an equation from this, we will again 

 pursue the com'se indicated by Camot, and reduced to a diagram 

 by Clapeyron." Let a vessel impervious to heat be partially filled 

 with water, leaving a space above for steam of the maximum 

 density corresponding to the temperature t. Let the volume of 

 both together be represented in the annexed figure by the 



