o62 PROFESSOR RANKINE ON THE THERMAL ENERGY 



expressed by equation (7), already given in § 8. The total energy of the thermal 

 motions in an unit of mass is expressed by dividing equation (6) of § 7 by the 

 density p ; hence that quantity of energy (denoted for shortness by Q) is given 

 in terms of the absolute temperature by the following equation, 



kv 2 3k p 3k , p r 



The real specific heat of a substance, as defined in the previous paper, when 

 expressed in units of work per degree, is 



JC - dr ~ 2p r + 2 Fo T dr ' " W 



in which c is the real specific heat, in terms of the minimum specific heat of liquid 

 water, and J, Joule's equivalent, or the dynamical value of the ordinary thermal 

 unit. 



There is one part of the specific heat which is necessarily constant for a given 

 substance in all conditions ; and that is the part which expresses the rate of 

 increase with the temperature, of the energy of the steady circulation alone in an 

 unit of mass, viz., 



ir\kj ~ 2gr~ 2pr-2p r " ™- 



dr\kj ' 2gr 2pr 2p r t 



The part of the specific heat which depends on periodical disturbances is 

 expressed as follows : — 



d f (fr-l)Q ) 3(fc-l) Po 3 Po t . dk 



dr{ k j - 2 Po r + 2 Po t dr ■ (iih 



It is only by experiment that it can be ascertained whether this part of the 

 specific heat is constant or variable. Experiment has proved that it is constant 

 for the perfectly gaseous state, and nearly, if not exactly constant, for other con- 

 ditions; but that its values for the same substance in the solid, liquid, and 

 gaseous conditions are often different.* 



The apparent specific heat contains other terms, depending on the expenditure 

 of energy in performing external and internal work, according to principles of 

 thermodynamics which are now well known. 



§ 10. Examples of the Proportion in which the Total Energy of the Thermal 

 Motions exceeds the Energy of the Steady Circulation. — In the perfectly gaseous 



* According to the nomenclature used by Clausius, the phrase "real specific heat" is applied 

 to that part only of the specific heat which is necessarily constant for a given substance in all 

 conditions. Hence, if that nomenclature were adapted to the hypothesis of molecular vortices, the 

 term real specific heat would be applied to the coefficient given in equation (10) only, and that given 

 in equation (11) would be considered as part of the apparent specific heat. 



