F. O. Koenig 281 



SUMMARY OF THE CONSEQUENCES OF CARNOT'S THEOREM ALONE 



1. The Existence of Absolute Temperature. Let the different possible em- 

 pirical temperature scales be numbered \, 2, . . . , i, . . . and let the values of a 

 given temperature on these scales be denoted by 9^, 6.,, . . . , Oi, . . . respectively. 

 Then Carnot's theorem can be expressed in the form 



^ = 'i>,(e/,en (I) 



in which Ti^ is the work of a Carnot cycle whose higher and lower tem- 

 peratures are 6i' and ^/' respectively, Q is the heat absorbed at the higher 

 temperature, and <I>i denotes a function of ^,/ and 6/' which is independent 

 of the working substance. Since W/d is independent of the selection of 

 temperature scales, it is furthermore clear that the value of $, for a given 

 pair of temperatures is the same for all scales and that therefore the form 

 <J>, must vary from scale to scale, which is the reason for attaching the subscript 

 i to the $. If we introduce the physically obvious assumption implicit in the 

 work of Carnot and his successors, that as 6/— 6/' approaches zero, TT^ becomes 

 an infinitesimal, SIF, of the same order as di'—Oi", eq. (1) yields, for a Carnot 

 cycle working over the infinitesimal temperature interval dOi which includes 

 the temperature 6i 



-^ = ^,{edde, (2) 



in which the function ixi{6i), named Carnot's function by Thomson, is 

 independent of the working substance. Furthermore, while both the form 

 of fxi and its value at a given temperature depend on the choice of the 

 scale /, the value of the infinitesimal, ixi{6i)d6i, for a given temperature and 

 a given infinitesimal temperatuie interval, is an invariant of all scales /. 

 Hence the value of 



f 



J er 



for a given temperature interval is also an invariant of all scales /. Now con- 

 sider the functions T^^i{6i) defined by the differential equations 



M{T)dT = ui,{di)Si (3) 



in which the form of M is arbitrary but the same for all scales /. Integra- 

 tion gives 



F(r) - F{T") = <p,{d/) - ^,{d/') 



whence it appears that F{T')-F{T") is invariant under ;' with respect both 

 to its value for a given temperature interval and to the form of F. But the 



