216 Prof. W. J. M. Rankine on the Thermal Energy 

 well-known formula being 



£=fiU^ ........ (7) 



P Po T o 



in which t is the absolute temperature of melting ice, t the 



ijn on 



actual absolute temperature, and — the value of the quotient - 



at the temperature of melting ice, for the particular substance 

 in question. 



§ 9. Temperature and Specific Heat — It is shown in the paper 

 of 1849-50, that temperature, according to the hypothesis of 

 molecular vortices, is a function of the quotient found by dividing 

 the energy of the steady circulation in a unit of mass by a con- 

 stant depending on the nature of the substance ; which constant 

 may be defined as the value which the energy of steady circula- 

 tion in a unit of mass of the given substance assumes at a 

 standard temperature, such as that of melting ice. The energy 

 of the steady circulation in a unit of mass is 



w* __ 3 p 



whence it appears that the principle stated as to absolute tem- 

 perature is expressed by equation (7), already given in §8. The 

 total energy of the thermal motions in a unit of mass is expressed 

 by dividing equation (6) of § 7 by the density p ; hence that quan- 

 tity of energy (denoted for shortness by Q) is given in terms of 

 the absolute temperature by the following equation, 



n Jew 2 3k p 3k p t , e . 



A Z p <l p Q t 



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

 paper, when expressed in units of work per degree, is 



Jc= — = -^ + Sp ° T • — (9) 



dr 2p r 2p r dr' ~* 



in which c is the real specific heat in terms of the minimum spe- 

 cific heat of liquid water, and J Joule's equivalent, or the dyna- 

 mical value of the ordinary thermal unit. 



There is one part of the specific heat which is necessarily con- 

 stant 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 a unit of mass, viz. 



«f /Q\ = w 1 Sp _ Sp (m 



dr\k) 2gr 2pr 2p r ' ' V ; 



