524 Prof. Thomson on the Dynamical Theory of Heat. 



the temperature ; are brought forward for the purpose of pointing 

 out the importance of making the mechanical energy of a fluid in 

 different states an object of research, along with the other ele- 

 ments which have hitherto been considered, and partially inves- 

 tigated in some cases. 



84. If we consider the circumstances of a stated quantity (a 

 unit of matter, a pound, for instance) of a fluid, we find that its 

 condition, whether it be wholly in the liquid state or wholly 

 gaseous, ou partly liquid and partly gaseous, is completely defined, 

 when its temperature, and the volume of the space within which 

 it is contained, are specified (§§ 20, 53, ... 56), it being under- 

 stood, of course, that the dimensions of this space are so limited 

 that no sensible differences of density in different parts of the 

 fluid are produced by gravity. We shall therefore consider the 

 temperature, and the volume of unity of mass, of a fluid, as the 

 independent variables of which its pressure, thermal capacities, 

 and mechanical energy are functions. The volume and tempe- 

 rature being denoted respectively by v and /, let e be the mecha- 

 nical energy, p the pressure, K the thermal capacity under con- 

 stant pressure, and N the thermal capacity in constant volume ; 

 and let M be such a function of these elements, that 



dp 



K = N +35 M » 



dv 



or {§§ 48, 20), such a quantity that 



Mdv + Kdt (2), 



may express the quantity of heat that must be added to the fluid 

 mass, to elevate its temperature by dt, when its volume is aug- 

 mented by dv. 



85. The mechanical value of the heat added to the fluid in 

 any operation, or the quantity of heat added, multiplied by J 

 (the mechanical equivalent of the thermal unit), must be dimi- 

 nished by the work done by the fluid in expanding against 

 resistance, to find the actual increase of mechanical energy which 

 the body acquires. Hence (de of course denoting the complete 

 increment of e, when v and t are increased by dv and dt) we have 

 de=i{Udv + ^dt)-pdv .... (3). 

 Hence, according to the usual notation for partial differential 

 coefficients, we have 



^=m-p (4), 



dv 



|=JN (5). 



dt 



