Prof. Thomson on the Dynamical Theory of Heat. 293 



denote the specific heat of unity of mass of the fluid under the 

 constant pressures at which it exists in the lower and upper hori- 

 zontal branches of the tube in the second case; n(T), 11 (T') the 

 quantities of heat evolved and absorbed respectively by the pass- 

 age of a unit mass of fluid through the two vertical branches 

 kept at the respective temperatures T, T' ; and if F denote the 

 work done by a unit mass of the fluid in passing through the 

 engine ; the fundamental equations obtained above with refer- 

 ence to the thermo-electric circumstances, may be at once written 

 down for the case of the ordinary fluid as the expression of the 

 two fundamental laws of the dynamical theory of heat, both of 

 which are applicable to this case, without any uncertainty such 

 as that shown to be conceivable as regards the application of the 

 second law to the case of a thermo-electric current. The two 

 equations thus obtained are equivalent to the two general equa- 

 tions given in §§ 20 and 21 of the first part of this series of 

 papers, as the expressions of the fundamental laws of the dyna- 

 mical theory of heat applied to the elasticity and expansive pro- 

 perties of fluids. In fact, when we suppose the ranges of both 

 temperature and pressure in the circulating fluid to be infinitely 



small, the equation F = jl ^^ dt, reduced to the notation for- 

 merly used, and modified by changing the independent variables 

 from [t,p) to [t, v), becomes 



^^^-' i dr 



which is the same as (3) of § 21 ; and a combination of this with 



-(-) = 

 dt\t / 



(Tg 0-, 



, , gives 



^_^_1^ 

 dt dv ~ J dt' 



which is identical with (2) of § 20. It appears, then, that the 

 consideration of the case of fluid motion here brought forward 

 as analogous to thermo-electric currents in non-crystalline linear 

 conductors, is sufficient for establishing the general thermo- 

 dynamical equations of fluids; and consequently the universal 

 relations among specific heats, elasticities, and thermal effects of 

 condensation or rarefaction, derived from them in Part III., are 

 all included in the investigation at present indicated. Not going 

 into the details of this investigation, because the foi-mer investi- 

 gation, which is on the whole more convenient, is fully given in 

 Parts I. and III., I shall merely point out a special application 

 of it to the case of a liquid which has a temperature of maximum 

 density, as for instance water. 



