On Maxwell's Investigation of Boltzmann's Theorem. 357 



limited within an enclosure. The fundamental assumption is 

 that, apart from exceptional cases, the particle, starting from 

 a given point, will sooner or later traverse that point in every 

 direction ; and the conclusion founded upon this assumption 

 is that in the long run all directions through the point are 

 equally favoured. I do not see that there is here anything to 

 be specially surprised at. If the premises are admitted, the 

 conclusion seems natural enough. 



In another part of his investigation Maxwell puts forward 

 under the same reserves the more general hypothesis that not 

 merely does the system pass through a given configuration 

 with every possible system of velocities consistent with the 

 energy condition, but also through every configuration which 

 can be reached without violation of the same condition. In 

 the billiard -table example this means that every part of the 

 table is reached sooner or later, and, as we have seen, every 

 part that is reached is traversed as much in one direction as 

 in another. In this case, where there is no potential energy, 

 we may indeed go further. Maxwell's equation (41) shows 

 that any part of the table is occupied in the long run as much 

 as any other ; so that all points, as well as all directions, are 

 equally probable. 



To my mind the difficulty of Maxwell's investigation lies 

 more in the premises than in the deductions*. It is easy to 

 propose particular cases for which the hypothesis is manifestly 

 untrue. For example, if the table be circular, a particle pro- 

 jected otherwise than along a diameter will leave a central 

 circular area uninvaded, and in the outer zone will not pass 

 through a given point in every direction, even when the pro- 

 jection is such that the path is not re-entrant. The question 

 is how far the considerations advanced by Maxwell justify us 

 in putting aside these cases as too exceptional to interfere 

 with the general "proposition, which, at any rate in its appli- 

 cation to physics, is essentially one of probability. 



Having found Maxwell's demonstration of the fundamental 

 theorem 



dq{ . . . dgj dp{ . . . dpj = dq ± ... dq n dp 1 ... dp n 



difficult to folio w, I have sought to simplify it by an arrange- 

 ment such that the initial and final times t' and t may be 



* The particular case for which Burnside obtained a result inconsistent 

 with Maxwell's conclusions is emphasized by Mr. Bryan. But Mr. Bur- 

 bury is of opinion that the discordant result depends upon an error of 

 calculation, and that when this is set right the discrepancy disappears 

 (Proc. Roy. Soc, November 19, 1891, p. 196). 



