526 Prof. Thomson on the Dynamical Theory of Heat. 



regarding its pressure and thermal capacities, but not necessarily 

 comprehending the values of each of these elements for all states 

 of the fluid. The theory of the integration of functions of two 

 independent variables will, when any set of data are proposed, 

 make it manifest whether or not they are sufficient, and will 

 point out the methods, whether of summation or of analytical 

 integration, according to the forms in which the data are fur- 

 nished, to be followed for determining the value of e for every 

 value of v. Or the data may be such, that while the thermal 

 capacities would be derived from them by differentiation, values 

 of e may be obtained from them without integration. Thus, if 

 the fluid mass consist of water and vapour of water at the tem- 

 perature t, weighing in all one pound, and occupying the volume 

 «*, and if we regard the zero or "standard" state of the mass 

 as being liquid water at the temperature 0°, the mechanical 

 energy of the mass in the given state will be the mechanical 

 value of the heat required to raise the temperature of a pound of 



water from 0° to t, and to convert — -^ of it into vapour, dimi- 



nished by the work done in the expansion from the volume X to 

 the volume v ; that is, we have 



e = j(c/ + L^)-p(r,-X) • • • • (8). 



The variables c, L, and p (which depend on t alone) in this ex- 

 pression have been experimentally determined by Regnault for 

 all temperatures from 0° to 230°; and when y is also determined 

 by experiments on the density of saturated steam, the elements 

 for the determination of e in this" case will be complete. The 

 expressions investigated formerly forM and N in this case (§ 54) 

 may be readily obtained by means of (4) and (5) of § 85, by the 

 differentiation of (8). 



89. If Carnot's function has once been determined by means 

 of observations of any kind, whether on a single fluid or on dif- 

 ferent fluids, for a certain range of temperatures, then according 



dp 



to (6) of § 85, the value of «■ for any substance whatever is 



* The same notation is used here as formerly in § 54, viz. p is the pres- 

 sure of saturated vapour at the temperature t, y the volume, and L the 

 latent heat of a pound of the vapour, X the volume of a pound of liquid 

 water, and c the mean thermal capacity of a pound of water between the 

 temperatures and t. A mass weighing a pound, and occupying the 



volume v, when at the temperature t, must consist of a weight of 



y — A 



vapour, and t — - of water. 



