THROUGH LONG PERIODS OF TIME. 149 



t". In the part of DV" occupied by systems which at the 

 time if were in DV the value of 77 will be the same as its 

 value in D V at the time t r , which we shall call 77'. In the 

 parts of DV" occupied by systems which at if were in ele- 

 ments very near to D V we may suppose the value of 77 to 

 vary little from T?'. We cannot assume this in regard to parts 

 of DV" occupied by systems which at tf were in elements 

 remote from DV. We want, therefore, some idea of the 

 nature of the extension-in-phase occupied at tf by the sys- 

 tems which at t" will occupy D V". Analytically, the prob- 

 lem is identical with finding the extension occupied at t" 

 by the systems which at t 1 occupied DV. Now the systems 

 in D V" which lie on the same path as the system first con- 

 sidered, evidently arrived at DV" at nearly the same time, 

 and must have left D V 1 at nearly the same time, and there- 

 fore at if were in or near DV. We may therefore take T/ as 

 the value for these systems. The same essentially is true of 

 systems in DV" which he on paths very close to the path 

 already considered. But with respect to paths passing through 

 D V and D V", but not so close to the first path, we cannot 

 assume that the time required to pass from DV to D V" is 

 nearly the same as for the first path. The difference of the 

 times required may be small in comparison with "-', but as 

 this interval can be as large as we choose, the difference of the 

 times required in the different paths has no limit to its pos- 

 sible value. Now if the case were one of statistical equilib- 

 rium, the value of 77 would be constant in any path, and if all 

 the paths which pass through DV 1 also pass through or near 

 D V, the value of 77 throughout D V" will vary little from 

 ?;'. But when the case is not one of statistical equilibrium, 

 we cannot draw any such conclusion. The only conclusion 

 which we can draw with respect to the phase at t 1 of the sys- 

 tems which at t" are in DV" is that they are nearly on the 

 same patji. 



Now if we should make a new estimate of indices of prob- 

 ability of phase at the time t", using for this purpose the 

 elements D V, that is, if we should divide the number of 



