Prof. Thomson on the Dynamical Theory of Heat. 527 



known for all temperatures within that range. It follows that 

 when the values of M for different states of a fluid have been 

 determined experimentally, the law of pressures for all tempera- 

 tures and volumes (with an arbitrary function of v to be deter- 

 mined by experiments on the pressure of the fluid at one par- 

 ticular temperattrre) may be deduced by means of equation (6) ; 

 or conversely, which is more likely to be the case for any parti- 

 cular fluid, if the law of pressures is completely known, M may 

 be deduced without further experimenting. Hence the second 

 member of (4) becomes completely known, the equation assuming 

 the following form, when, for M, its value according to (6) is 

 substituted: — 



de J ap .-. 



dv = ^Jt-P < 8 >' 



The integration of this equation with reference to v leads to an 

 expression for e, involving an arbitrary function of t, for the 

 determination of which more data from experiment are required. 

 It would, for instance, be sufficient for this purpose to have the 

 mechanical energy of the fluid for all temperatures when con- 

 tained in a constant volume ; or, what amounts to the same (it 

 being now supposed that J is known), to have the thermal capa- 

 city of the fluid in constant volume for a particular volume and 

 all temperatures. Hence we conclude, that when the elements 

 J and /j, belonging to the general theory of the mechanical action 

 of heat are known, the mechanical energy of a particular fluid 

 may be investigated without experiment, from determinations of 

 its pressure for all temperatures and volumes, and its thermal 

 capacity for any particular constant volume and all temperatures. 

 90. For example, let the fluid be atmospheric air, or any other 

 subject to the " gaseous " laws. Then if v be the volume of a 

 unit of weight of the fluid, and the temperature, in the stand- 

 ard state from which the mechanical energy in any other state 

 is reckoned, and if p denote the corresponding pressure, we have 



and 



P- v {*■+**)> dt - v > 



Hence if wc denote by N the value of N when v = r 0) whatever 

 be the temperature, wc have as the general expression for the 

 mechanical energy of a unit weight of a fluid subject to the 



gaseous law*, 



<=Wo{--(l+E/))logf -rjflV/ . . (9). 



