358 M. CLAPEYRON ON THE MOTIVE POWER OF HEAT. 



but the temperature remaining constant during the variation of the vo- 

 lume, we have 



V dp + p dv = 0, whence dp =■ — ^- dv, 



and consequently 



dQ=m-p^''^\dv. 



\d V V dp/ 



If we divide the effect produced by this value of dQ., we shall 

 have 



Rdt 



dQ dQ 



dv dp 



for the expression of the maximum effect which can be developed by 

 the passage of a quantity of heat equal to unity, from a body main- 

 tained at the temperature < to a body maintained at the temperature 

 t — dt. 



We have shown that this quantity of action developed is indepen- 

 dent of the agent which has served to transmit the heat ; it is there- 

 fore the same for all the gases, and is equally independent of the pon- 

 derable quantity of the body employed: but there is nothing that 



proves it to be independent of the temperature ; v --— — p -7— ought 



therefore to be equal to an unknown function of t, which is the same 

 for all the gases. 



Now by the equation />v = R (267 + t), t is itself the function of 

 the product /> v\ the partial differential equation is therefoi-e 

 dQ dQ ^, s 



dv ^ dp ^^ ^ 



having for its integral 



Q =f{p-v) -F('p.v)log \_{hyp)p']. 



No change is effected in the generality of this formula by substitut- 

 ing for these two arbitrary functions of the product jt)!;, the functions 

 B and C of the temperature, multiplied by the coefficieiit R ; we shall 

 thus have 



Q = R(B- Clog 77). 



That this value of Q satisfies all the conditions to which it is sub- 

 ject may be easily verified ; in fact we have 



rflT ^ U< R '°'=-^ dt r) 



dQ J, (dY, V , dQ. V ^ 1 \ 



rf^=^(l7R-^^«^^ R-^F)' 



