ON AN ENSEMBLE OF SYSTEMS. 155 



which differs by a small quantity of the second order from 

 the minimum value which characterizes the state of statistical 

 equilibrium. And the diminution in the average index result- 

 ing in the course of time from the very small change in the 

 external coordinates, cannot exceed this small quantity of 

 the second order. 



Hence also, if the change in the external coordinates of an 

 ensemble initially in statistical equilibrium consists in suc- 

 cessive very small changes separated by very long intervals of 

 time in which the disturbance of statistical equilibrium be- 

 comes sensibly effaced, the final diminution in the average 

 index of probability will in general be negligible, although the 

 total change in the external coordinates is large. The result 

 will be the same if the change in the external coordinates 

 takes place continuously but sufficiently slowly. 



Even in cases in which there is no tendency toward the 

 restoration of statistical equilibrium in the lapse of time, a varia- 

 tion of external coordinates which would cause, if it took 

 place in a short time, a great disturbance of a previous state 

 of equilibrium, may, if sufficiently distributed in time, produce 

 no sensible disturbance of the statistical equilibrium. 



Thus, in the case of three degrees of freedom, let the systems 

 be heavy points suspended by elastic massless cords, and let the 

 ensemble be distributed in phase with a density proportioned 

 to some function of the energy, and therefore in statistical equi- 

 librium. For a change in the external coordinates, we may 

 take a horizontal motion of the point of suspension. If this 

 is moved a given distance, the resulting disturbance of the 

 statistical equilibrium may evidently be diminished indefi- 

 nitely by diminishing the velocity of the point of suspension. 

 This will be true if the law of elasticity of the string is such 

 that the period of vibration is independent of the energy, in 

 which case there is no tendency in the course of time toward 

 a state of statistical equilibrium, as well as in the more general 

 case, in which there is a tendency toward statistical equilibrium. 



That something of this kind will be true in general, the 

 following considerations will tend to show. 



