398 Intelligent* and Miscellaneous Articles. 



I purpose to demonstrate that this relation is far from being pos- 

 sibly arbitrary- — that the pressure supported by any body ivhatever, so 

 long as the body does not change its state, can only be a linear func- 

 tion of its temperature. In other terms, under a physical form, If 

 any body ivhatever be heated under a constant volume, the pressure 

 which it exerts upon the immovable sides of the enclosure which con- 

 tains it cannot but increase in rigorously exact proportion to its tem- 

 perature. 



I say that this proposition is an absolutely rigorous corollary of 

 the two fundamental propositions of the mechanical theory of heat, 

 and of the hypothesis that the reciprocal actions of the atoms of 

 bodies are directed along the lines which join their points of appli- 

 cation, and depend only on the distances of these points from one 

 another. 



To demonstrate the law above enunciated, let dQ be the quantity 

 of heat necessary in order to modify infinitesimally the volume v, 

 the pressure p, and the temperature T of a body without its 

 changing its state. The first principle of the mechanical theory of 

 heat gives the classical equation 



dQ=dU+Apdv, (1) 



A= — being the thermal equivalent of the work, and XJ the func- 



E 

 tion which is often called the internal heat. Let us take v and T 

 for the independent variables, so that 



dr dv 



The signification of each of the two terms of the second member is 

 obvious : the first represents the quantity of heat necessary to pro- 

 duce an increment dT of the temperature without change of volume; 

 consequently, and since there is no change of state, the second ne- 

 cessarily represents the quantity of heat equivalent to the work of 

 the molecular actions during the increase of volume dv. Now, if 

 we represent by mm'f(r) Lhe amount of the mutual action of two 

 molecules the masses of which are m and m', placed at the distance 

 r the one from the other, this work is represented by an expression 

 of the form %mm'f(r)dr, so that we have identically 



2mm'f(r)dr = E — — - dv. 



The first member not containing the letter T, it is the same with the 



second. Therefore — depends only on the variable v alone ; and 



consequently tJ is of the form E(T)+/(V). Hence this first con- 

 sequence : — The internal heat of a body cannot be any function of the 

 specific volume and the temperature of that body ; it can only be the 

 sum of two functions, — the one, of the volume alone, the other, of the 

 temperature alone. 



