THROUGH LONG PERIODS OF TIME. 147 



tion of the cylindrical coordinates r, 6 and 2, the effect of the 

 same motion continued indefinitely would be an approach to 

 a condition in which the density is a function of r and z alone. 

 In this limiting condition, the average square of the density 

 would be less than in the original condition, when the density 

 was supposed to vary with 0, although after any finite time 

 the average square of the density would be the same as at 

 first. 



If we limit our attention to the motion in a single plane 

 perpendicular to the axis of the cylinder, we have something 

 which is almost identical with a diagrammatic representation 

 of the changes in distribution in phase of an ensemble of 

 systems of one degree of freedom, in which the motion is 

 periodic, the period varying with the energy, as in the case of 

 a pendulum swinging in a circular arc. If the coordinates 

 and momenta of the systems are represented by rectangu- 

 lar coordinates in the diagram, the points in the diagram 

 representing the changing phases of moving systems, will 

 move about the origin in closed curves of constant energy. 

 The motion will be such that areas bounded by points repre- 

 senting moving systems will be preserved. The only differ- 

 ence between the motion of the liquid and the motion in the 

 diagram is that in one case the paths are circular, and in the 

 other they differ more or less from that form. 



When the energy is proportional to p 2 + q 2 the curves of 

 constant energy are circles, and the period is independent of 

 the energy. There is then no tendency toward a state of sta- 

 tistical equilibrium. The diagram turns about the origin with- 

 out change of form. This corresponds to the case of liquid 

 motion, when the liquid revolves with a uniform angular 

 velocity like a rigid solid. 



The analogy between the motion of an ensemble of systems 

 in an extension-in-phase and a steady current in an incompres- 

 sible liquid, and the diagrammatic representation of the case 

 of one degree of freedom, which appeals to our geometrical in- 

 tuitions, may be sufficient to show how the conservation of 

 density in phase, which involves the conservation of the 



