170 Prof. Thomson on the Dynamical Theory of Heat. 



density, and for the case of a medium consisting of parts in dif- 

 ferent states at the same temperature, as water and saturated 

 steam, or ice and water. 



47. In the first place it may be remarked, that by the defi- 

 nition of M and N in § 20, N must be what is commonly called 

 the " specific heat at constant volume " of the substance, pro- 

 vided the quantity of the medium be the standard quantity 

 adopted for specific heats, which, in all that follows, I shall take 

 as the unit of weight. Hence the fundamental equation of the 

 dynamical theory, (3) of § 20, expresses a relation between this 

 specific heat and the quantities for the particular substance de- 

 noted by M and p. If we eliminate M from this equation, by 

 means of equation (3) of § 21, derived fi'om the expression of 

 the second fundamental principle of the theory of the motive 

 power of heat, we find 



</N _ \i),dt/ I dp 



dv ~ dt Jdt • • • • i^^)' 



which expresses a relation between the variation in the specific 

 heat at constant volume, of any substance, produced by an altera- 

 tion of its volume at a constant temperature, and the variation 

 of its pressure with its temperature when the volume is constant ; 

 involving a function, jul, of the temperature, which is the same 

 for all substances. 



48. Again, let K denote the specific heat of the substance 

 under constant pressure. Then, if dv and dt be so related that 

 the pressure of the medium, when its volume and temperature 

 are v + dv and t + dt respectively, is the same as when they are 

 V and t, that is, if 



0= -fdv+ ~dt; 

 dv dt 



we have 



Kdt-Mdv^^dt. 



Hence we find 



__ '^P 



M = -±(K-N) (15), 



di 

 which merely shows the meaning in terms of the two specific 

 heats, of what 1 nave denoted by M. Using in this for M its 

 value given by (3) of § 21, we find 



fdpV 



K-N=-^ (^^)' 



