THERMODYNAMIC ANALOGIES. 167 



having explained the phenomena of nature with respect to 

 these laws. For, as compared with the case of nature, the 

 systems which we have considered are of an ideal simplicity. 

 Although our only assumption is that we are considering 

 conservative systems of a finite number of degrees of freedom, 

 it would seem that this is assuming far too much, so far as the 

 bodies of nature are concerned. The phenomena of radiant 

 heat, which certainly should not be neglected in any complete 

 system of thermodynamics, and the electrical phenomena 

 associated with the combination of atoms, seem to show that 

 the hypothesis of systems of a finite number of degrees of 

 freedom is inadequate for the explanation of the properties of 

 bodies. 



Nor do the results of such assumptions in every detail 

 appear to agree with experience. We should expect, for 

 example, that a diatomic gas, so far as it could be treated 

 independently of the phenomena of radiation, or of any sort of 

 electrical manifestations, would have six degrees of freedom 

 for each molecule. But the behavior of such a gas seems to 

 indicate not more than five. 



But although these difficulties, long recognized by physi- 

 cists,* seem to prevent, in the present state of science, any 

 satisfactory explanation of the phenomena of thermodynamics 

 as presented to us in nature, the ideal case of systems of a 

 finite number of degrees of freedom remains as a subject 

 which is certainly not devoid of a theoretical interest, and 

 which may serve to point the way to the solution of the far 

 more difficult problems presented to us by nature. And if 

 the study of the statistical properties of such systems gives 

 us an exact expression of laws which in the limiting case take 

 the form of the received laws of thermodynamics, its interest 

 is so much the greater. 



Now we have defined what we have called the modulus (O) 

 of an ensemble of systems canonically distributed in phase, 

 and wha't we have called the index of probability (77) of any 

 phase in such an ensemble. It has been shown that between 



* See Boltzmann, Sitzb. der Wiener Akad., Bd. LXIIL, S. 418, (1871). 



