supposed Irreversible Processes. 481 



These values of -j- relate to the initial instant, and an interval 

 dt after it. dt 



15. Now let us consider the difference between the two 

 expressions (k'/'F 1 — k/F), which is the actual, and 

 {f'F' — /F) which is the mean value. It is possible that 

 WfF'-kfF may be of the opposite sign to f'W-fF, but 

 the greater the absolute value of the difference /'F'^/F, 

 or the ratio of that difference to fF'-\-/F, the less probable it 

 is that k'/F'-k/F shall be of the opposite sign to/F'-/F. 



Boltzmanns H Theorem, 



16. Still considering the system in its initial state, let 



H =$du dv divf(\ogf- 1). 



Then 



dB. +» , df, , 



^— $dudvdwf t logf 



— 00 



= W SW lu dv dw dV dY dW W-* P /) R l0 S/F7 



— 00 



by a known process. And on average, writing K' = K = 1, 

 ^ = W ^du dv dto d\J dV dW(F'f - F/) R log ^ , 



which is necessarily negative if not zero. It will be noted 

 that the logarithmic factors are not affected by the coefficients 

 k and k'. 



17. It is evident from the foregoing that -j— may be 



positive, if, namely, many of the factors k f F f f—k¥f differ 

 in sign from their mean values F'f'—Ff. And it is further 



evident that the chance of j— being positive is greater as the 



factors F'f ' — F/ are in absolute value less. 



18. The reasoning is based on the assumption, which in the 

 initial state is true, that the chance of two spheres being in 

 the classes/ and F is independent of their relative positions. 

 It is also independent of the past history of the system, 

 because, as we have formed it, the system has no past. If the 

 results are to hold for every subsequent state of the systems 

 the condition of independence must also hold. But if no 



Phil Mag, 8. 5. Vol. 49. No. 300. May 1900. 2 L 



