166 THERMODYNAMIC ANALOGIES. 



which in general is an equation of few variables, is derived 

 from the fundamental mechanical equation, which in the case 

 of the bodies of nature is one of an enormous number of 

 variables. 



We have also to enunciate in mechanical terms, and to 

 prove, what we call the tendency of heat to pass from a sys- 

 tem of higher temperature to one of lower, and to show that 

 this tendency vanishes with respect to systems of the same 

 temperature. 



At least, we have to show by a priori reasoning that for 

 such systems as the material bodies which nature presents to 

 us, these relations hold with such approximation that they 

 are sensibly true for human faculties of observation. This 

 indeed is all that is really necessary to establish the science of 

 thermodynamics on an a priori basis. Yet we will naturally 

 desire to find the exact expression of those principles of which 

 the laws of thermodynamics are the approximate expression. 

 A very little study of the statistical properties of conservative 

 systems of a finite number of degrees of freedom is sufficient 

 to make it appear, more or less distinctly, that the general 

 laws of thermodynamics are the limit toward which the exact 

 laws of such systems approximate, when their number of 

 degrees of freedom is indefinitely increased. And the problem 

 of finding the exact relations, as distinguished from the ap- 

 proximate, for systems of a great number of degrees of free- 

 dom, is practically the same as that of finding the relations 

 which hold for any number of degrees of freedom, as distin- 

 guished from those which have been established on an em- 

 pirical basis for systems of a great number of degrees of 

 freedom. 



The enunciation and proof of these exact laws, for systems 

 of any finite number of degrees of freedom, has been a princi- 

 pal object of the preceding discussion. But it should be dis- 

 tinctly stated that, if the results obtained when the numbers 

 of degrees of freedom are enormous coincide sensibly with 

 the general laws of thermodynamics, however interesting and 

 significant this coincidence may be, we are still far from 



