J; 



Quantities occur ring in the Mechanical Theory of Heat. 265 

 or, otherwise written, 



In the first form, integrated over an entire cyclical process, 

 it is _— 



Jf =0 (II. B) 



3. 



If we now compare these dynamic theorems with the second 

 proposition of the mechanical theory of heat, which is referred 

 to reversible cyclical processes, we find 



S=°> < IL > 



where a + t expresses the temperature reckoned from absolute 

 zero of gases ; while of clQ it is asserted that it denotes the 

 differential of the heat communicated to the body. Our dyna- 

 mic equation (II. b) indeed agrees with the thermodynamic 

 equation (II.), but only on the hypothesis that the absolute 

 temperature is proportional to the mean value of the vis viva 

 of the body, referred to the duration of an oscillation, and that 

 the thermodynamic clQ is proportional to the dynamic clQ. 

 We will first examine the latter relation, designating, for the 

 sake of distinction, the thermodynamic dQ by d<$l, so that 



d(^ — adQ. 



And since both d(£i (heat) and dQ denote energy, the propor- 

 tionality-factor a is a pure number, the quantity of which 

 depends only on the units chosen. Selecting the unit of heat 

 equal to the unit of energy, we get a = 1 and 



or 



4tda 



dt. 



From this it follows that what is called the heat-differential, 

 d<&, is not the differential of the energy, but denotes only the 

 mean value of such differentials during a complete oscillation. 

 One can now easily explain why the first principal equation of 

 the theory of heat cannot be directly integrated. It is 

 because difii is not properly a differential, but only an in- 

 finitesimal average quantity, and because the sum of the 

 average quantities 



