E O. Koenig 283 



whence 



/=J^(1) A=-J^('!K) (5-0 (5-2) 



ixi{di) J F, \ddjv ^i(9i) J i\ \(:dji' 



(6) 



in which Q now denotes the heat absorbed in the reversible expansion of any 

 uniform substance at the temperature 6i from P^ and V^ to P^ and V^. Eq. 

 (5.1) (5.2) (6) are the most general thermodynamic formulas deducible from 

 Carnot's theorem without the first law. We may in particular take 6i to be 

 the temperature measiaed on the perfect gas scale; denoting this temperature 

 by t and the corresponding Carnot's function by ix{t) we obtain of course 



/=^(^) , h=-^i'A (7.0(7-) 



m(/) \6'/A- ix{t) \dt)p w M/ ; 



with corresponding formulas for (7. Eq. (7.1) (7.2) were first obtained by 

 Clapeyron (1834). If on the other hand we introduce the customary absolute 

 temperature defined, as explained above, by 



M{T)dT = -dT^ ^j.i{d,)ddi 

 we obtain 



/ = i(^), k=-U'Z) (8.0(8..) 



with corresponding formulas for Q. 



Application of these results in the form of ecp (7.1) (7.2) to a perfect gas 

 defined by 



PF = nRt 



where n is the number of moles of gas in the sample and R is the gas constant, 

 yields 



, nR , nR 



/ = , h = — 



^ nR , Vo nR , P, , . 



Q== -rrlog — = — -log— (9) 



/x(/) Vi tx{t) Po 



Eq. (9) expresses Carnot's forgotten theorems 1 and 3 in mathematical form. 

 The fact that jx(t)^i/t (and therefore the further fact that i = T) cannot be 

 deduced from Carnot's theorem alone but follows only from comparison of 

 eq. (g) with the equation 



Q = nRt log — 



