482 Mr, S. H. Burbury on certain 



external disturbance bappens to tbe system, the positions and 

 velocities of all the spheres at any time t are determinate 

 functions of t and the initial positions and velocities, and of 

 no other quantity whatever. The condition of independence 

 cannot therefore strictly hold for all time, though it may hold 

 approximately. For any two of the velocities, as u and u', 

 nre functions of the same set of variables. It cannot there- 

 fore be assumed at least without further investigation that 

 they are independent of one another. 



19. In order therefore to maintain the condition of inde- 

 pendence without doubt, let us assume that after the lapse of 

 time dt, and again after every subsequent interval dt, some 

 external cause, say Maxwell's demons, effects a redistribution 

 of the positions of the spheres without altering any of their 

 velocities. And this is done in such a way as always to 

 maintain the independence. 



20. Since on this assumption the independence exists at 

 every instant, the same reasoning may be applied at any sub- 

 sequent instant which we applied at the initial instant. Then 



—r- is at every instant on average zero or negative. If the 



demons, in addition to their other services, would insure that 



-^ should always have its mean value in which k=M=l, 



H would continuously diminish. In fact, -=— may, as we 



have seen, be occasionally positive. But as long as many of 

 the factors Y'f^Ff have a great value, it is very improbable 



that -j- should be positive. The diminution of H, and also 



the diminution of every F'/' — F/, will therefore go on with 

 little interruption until every F'/' — F/ is very nearly zero, 

 and H nearly minimum. But when this state is attained! 



7TT ' 



-j— is nearly as likely to be positive as to be negative. 



21. Now, the condition of independence being maintained, 

 let us consider the chance that H, having attained its 

 minimum H at time t, shall subsequently again increase and 

 say at time t + t', attain a value H 7 , considerably greater than 

 H . We may suppose the time t' to be divided into n intervals 

 each equal to dr. Let V r be the chance that in the rth 

 interval H shall receive the increment d r K. Then, by the 

 condition of independence assumed to exist, the chance that H 

 shall during the time l' increase from H to H' by the series 

 of increments ^H, . . dr&, . . . djl is the continued product 



i 



