OF ENERGY IN A SYSTEM OP MATERIAL POINTS. 715 



pass from one undisturbed path into another. The two paths must both satisfy 

 the equation of energy, and they must intersect each other in the phase for 

 which the conditions of encounter with the fixed obstacle are satisfied, but they 

 are not subject to the equations of momentum. It is difficult in a case of such 

 extreme complexity to arrive at a thoroughly satisfactory conclusion, but we may 

 with considerable confidence assert that except for particular forms of the surface 

 of the fixed obstacle, the system will sooner or later, after a sufficient number 

 of encounters, pass through every phase consistent with the equation of energy. 



I shall begin with the case in which the system is supposed to be contained 

 within a fixed vessel, and shall afterwards consider the case of a free system, 

 or of a system contained in a vessel rotating uniformly about an axis which 

 itself moves uniformly in a straight line. 



I have found it convenient, instead of considering one system of material 

 particles, to consider a large number of systems similar to each other in all 

 respects except in the initial circumstances of the motion, which are supposed 

 to vary from system to system, the total energy being the same in all. In 

 the statistical investigation of the motion, we confine our attention to the number 

 of these systems which at a given time are in a phase such that the variables 

 which define it lie within given limits. 



If the number of systems which are in a given phase (defined with respect 

 to configuration and velocity) does not vary with the time, the distribution of 

 the systems is said to be steady. 



It is shewn that if the distribution is steady, a certain function of the 

 variables must be constant for all phases belonging to the same path. If the 

 path passes through all phases consistent with the equation of energy, this 

 function must be constant for all such phases. If however there are phases 

 consistent with the equation of energy, but which do not belong to the same 

 path, the value of the function may be different for such phases. 



But whether we are able or not to prove that the constancy of this 

 function is a necessary condition of a steady distribution, it is manifest that 

 if the function is initially constant for all phases consistent with the equation 

 of energy, it will remain so during the motion. This therefore is one solution, 

 if not the only solution, of the problem of a steady distribution. 



Now we know from the empirical laws of the diffusion of heat that the 

 problem of the equilibrium of temperature in an isolated material system has 



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