APPENDIX B. 247 



dilatation. According to the reasoning of page 69, 

 Sr will be the power developed by the fall of the 

 quantity e of heat from the degree t -f- td to the 

 degree t. If we call u the motive power developed 

 by the fall of unity of heat from the degree t to the 

 degree zero, as, according to the general principle 

 established page 68, this quantity u ought to de- 

 pend solely on i, it could be represented by the 

 function Ft, whence u = Ft. 



When t is increased it becomes t + td, u be- 

 comes u + du ; whence 



Subtracting the preceding equation, we have 

 du = F(t + df) - Ft = F'tdt. 



This is evidently the quantity of motive power 

 produced by the fall of unity of heat from the 

 temperature t + dt to the temperature t. 



If the quantity of heat instead of being a unit 

 had been e, its motive power produced would have 

 had for its value 



edu = eF'tdt ..... (4) 

 But edu is the same thing as dr\ both are the 

 power developed by the fall of the quantity e of 

 heat from the temperature t -j- dt to the tempera- 

 ture t; consequently, 



edu = dr, 



