of a Mass of Matter. 93 



remains applicable, if a series of successive alterations of condi- 

 tion be considered instead of a single one. 



§ 6. The differential equation (4), whence equation (II.) is 

 derived, is connected with a differential equation which results 

 from the already known principles of the mechanical theory of 

 heat, and which transforms itself directly into (4) for the par- 

 ticular case in which the body under consideration is a perfect 

 gas. 



We will suppose that there is given any body of variable 

 volume, acted upon, as by an external force, by the pressure 

 exerted on the surface. Let the volume which it assumes under 

 this pressure, p, at the temperature T (reckoned from the abso- 

 lute zero) be v, and let it be supposed that the condition of the 

 body is fully determined by the magnitudes T and v. If we now 



denote by -=— dv the quantity of heat which the body must 



take up in order to expand to the extent of dv, without alteration 

 of temperature (for the sake of conformity with the mode in 

 which the signs are used in the other equations occurriDg in this 

 section, the positive sense of the quantity of heat is here taken 

 differently from what it is in equation (4), in which heat given 

 up by the body, and not heat communicated to it, is reckoned 

 positive), the following well-known equation, from the mecha- 

 nical theory of heat, will hold good : — 



dv dT 



Let us now suppose that the temperature of the body is changed 

 by dT, and its volume by dv, and let us call the quantity of heat 

 which it then takes up dQ ; we may then write 



dQ,= ^dT+^dv. 



di dv 



For the magnitude here denoted by -j^, which represents the 

 specific heat with constant volume, we can put the letter c, and 

 for -=- the expression already given. Then we have 



dQ=cdT + AT ^dv (6) 



The only external force, which the body has to overcome on 



expanding, being p, the work which it performs in so doing is 



dj) 

 pdv, and the magnitude -~ dv indicates the increase of this work 



with the temperature. 



