714 BOLTMAJO*'* THEOREM ON THE AVERAGE DISTRIBUTION 



olid*. for in these bodies the particles are never free from the action of 

 neighbouring particles. It is true that in following the steps of the investi- 

 gttrn. M giTen either by Boltzmann or by Watson, it is difficult, if not 

 impossible, to see where the stipulation about the shortness and the isolation 

 of the encounters is made use of. We may almost say that it is introduced 

 rather f-r the sake of enabling the reader to form a more definite mental 

 image of the material system than as a condition of the demonstration. Be 

 tfrfo M it may, the presence of such a stipulation in the enunciation of the 

 problem cannot fail to leave in the mind of the reader the impression of a 

 oomeponding limitation in the generality of the solution. 



In the theorem of Boltzmann which we have now to consider there is no 

 such limitation. The material points may act on each other at all distances, 

 and according to any law which is consistent with the conservation of energy, 

 and they may also be acted on by any forces external to the system provided 

 also are consistent with that law. 



The only assumption which is necessary for the direct proof is that the 

 system, if left to itself in its actual state of motion, will, sooner or later, pass 

 through every phase which is consistent with the equation of energy. 



v it is manifest that there are cases in which this does not take place. 

 The motion of a system not acted on by external forces satisfies six equations 

 besides the equation of energy, so that the system cannot pass through those 

 phanra, which, though they satisfy the equation of energy, do not also satisfy 

 these su equations. 



Again, there may be particular laws of force, as for instance that according 

 to which the stress between two particles is proportional to the distance between 

 them, for which the whole motion repeats itself after a finite time. In such 

 caws a jKirtictilar value of one variable corresponds to a particular value of 

 each of the other variables, so that phases formed by sets of values of the 

 variables which do not correspond cannot occur, though they may satisfy the 

 general equations. 



But if we suppose that the material particles, or some of them, occasionally 

 encounter a fixed obstacle such as the sides of a vessel containing the particles, 

 tlion, except for special forms of the surface of this obstacle, each encounter 

 will introduce a disturbance into the motion of the system, so that it will 



