250 Prof. Clausius on the Application of the Mechanical 



was an external pressure equally intense at all points of the sur- 

 face, and directed everywhere at right angles to the same ; and 

 further, that this pressure always changed so slowly, and con- 

 sequently at each moment differed so little from the opposite 

 expansive force of the body, that in calculation the two might be 

 considered equal. Let then p be the pressure, v the volume, 

 and T the absolute temperature of the body. We introduce the 

 last instead of /, the temperature counted from the freezing- 

 point, because thereby the formulae assume a simpler form. The 

 equations already established in this case are, 



dT\dv) dvUTj-^'dT • • • ^^^^^ 



These equations shall next be applied to the still more special 

 case of vapours at their maximum density. 



11. Let M be the mass of the matter whose vapour is to be 

 considered, and which is placed in a perfectly closed expansible 

 vessel. Let the part m be in a vaporous, and the rest, M— m, in 

 a liquid state. This mixed mass shall be the changing body to 

 which the foregoing equations are to be referred. 



The condition of the mass, as far as the same here enters into 

 consideration, is perfectly determined as soon as its temperature 

 T and its volume v, i. e. the volume of the vessel, are given. 

 For, according to hypothesis, the vapour is always in contact 

 with the liquid, and therefore remains at its maximum density ; 

 so that its condition, as well as that of the liquid, depends only 

 upon the temperature T. It only remains to be seen, therefore, 

 whether the magnitude of each of the parts in different conditions 

 is perfectly determined, from the condition that both parts 

 together exactly fill the space enclosed by the vessel. Let s 

 represent the volume of the unit of weight of vapour at its maxi- 

 mum density where the temperature is T, and a that of the unit 

 of weight of liquid, then 



v=wi.*4-(M— m)(r 



The magnitude s never occurs hereafter except in the combina- 

 tion s—a, so that we will introduce another letter for this differ- 

 ence, and make 



u=s-—a; (5) 



^n consequence of which the foregoing equation becomes 



01 liii* i;=7ww-f-M<7, (6) 



