ON AN ENSEMBLE OF SYSTEMS. 157 



of D - 1J\p ,) the positive values of D - U\ p caused by the 

 second change will be in part superposed on negative values 

 due to the first change, and vice versa. 



The disturbance of statistical equilibrium, therefore, pro- 

 duced by a given change in the values of the external co- 

 ordinates may be very much diminished by dividing the 

 change into two parts separated by a sufficient interval of 

 tune, and a sufficient interval of time for this purpose is one 

 in which the phases of the individual systems are entirely 

 unlike the first, so that any individual system is differently 

 affected by the change, although the whole ensemble is af- 

 fected in nearly the same way. Since there is no limit to the 

 diminution of the disturbance of equilibrium by division of 

 the change in the external coordinates, we may suppose as 

 a general rule that by diminishing the velocity of the changes 

 in the external coordinates, a given change may be made to 

 produce a very small disturbance of statistical equilibrium. 



If we write r[ for the value of the average index of probability 

 before the variation of the external coordinates, and iff' for the 

 value after this variation, we shall have in any case 



as the simple result of the variation of the external coordi- 

 nates. This may be compared with the thermodynamic the- 

 orem that the entropy of a body cannot be diminished by 

 mechanical (as distinguished from thermal) action.* 



If we have (approximate) statistical equilibrium between 

 the times if and if' (corresponding to rf and ??"), we shall have 

 approximately 



which may be compared with the thermodynamic theorem that 

 the entropy of a body is not (sensibly) affected by mechanical 

 action, during which the body is at each instant (sensibly) in 

 a state of thermodynamic equilibrium. 



Approximate statistical equilibrium may usually be attained 



* The correspondences to which the reader's attention is called are between 

 t\ and entropy, and between and temperature. 



