282 Thermodynamic Theorems o£ Carnot 



form of F is arbitrary because the form of its derivative, M, is arbitrary. Hence 

 the vahie of T' — T'' is an invariant of all scales i. The functions T=<ifi($i) 

 thus have the properties required by an absolute temperature. Hence eq. (3) 

 may be taken as a general definition of an infinity of absolute temperature 

 scales depending on the choice of M. The scale universally used is defined by 

 the choice M = i/T. 



Thomson's original and unsatisfactory definition of absolute temperature 

 (1848) was based upon Carnot's theorem and the caloric theory, while his later 

 definition (1854), which continues to enjoy general use, follows only from the 

 combination of Carnot's theorem with the first law. In the present paper the 

 contribution of Carnot's theorem alo?ie to the definition of absolute tempera- 

 ture is explicitly set forth for the first time to our knowledge. 



It is very instructive to note— we must omit proof for lack of space— that 

 Carnot's theorem alone yields no information as to the form of the function 

 ^i{9i', 6i') in eq. (1), and that it is only with the help of the first law that we 

 can prove this function to be of the form 



^iie/, en = 





2. Formulas for Heat of Isothermal Change of State. Consider an infinitesi- 

 mal Carnot cycle; let dOt denote the temperature interval and dV and dP the 

 changes of volume and pressure respectively along the higher isothermal. Let 

 the working substance be a single phase. The area of the corresponding in- 

 finitesimal parallelogram in the P— V plane is seen on inspection to be given by 



8JV = ( — ) ddidF 

 \ddjv 



By an obvious transformation we may replace F by P as an independent varia-. 

 ble and obtain the alternative expression 



6W^ - f — ^ ddidP 

 \ddi/p 



Corresponding to these two expressions for 8FF we may write the heat hQ_ 

 absorbed along the higher isothermal in the alternative forms 



^Q = ldV=hdP (4.1) (4.2) 



where / and h are both definite functions of the state of the working substance. 

 Carnot's theorem then gives 



io" Idv MP = '"'''^'''' 



