220-223] General Dynamical Theory 187 



below a finite range of small values of F, becomes approximately independent 

 of the upper limit of integration, so that the stream lines on which <E> is 

 a maximum become identical for all such upper limits. Thus so soon as 

 a value of the energy is reached so small that it is possible for F to remain 

 very small over a large range of values of E, we see that the positions of the 

 clusters tend to become independent of their starting points. In other 

 words, all clusters tend to combine into a single cluster ; or again, all stream 

 lines tend to combine into a single stream line. 



The values of the energy at which the condition just stated is satisfied, 

 are the only values which are of much importance in nature. They corre- 

 spond to temperatures at which a substance can remain in appreciably the 

 same state for a considerable interval of time ; and in considering this ques- 

 tion it must be borne in mind that the unit of time appropriate for 

 comparison is the interval between successive collisions of the same molecules. 

 We have already seen ( 8, p. 8) that under normal conditions this interval is 

 comparable with 10~ 10 seconds. And for obvious reasons the state of a 

 system which changes appreciably in 10~ 10 seconds is not of much interest 

 to us. 



When the condition in question is satisfied, the system may be said to be 

 in an " approximately steady state." It is then infinitely probable that it will 

 be represented by a point on the single cluster (or family of clusters) just 

 discussed. In this case it may be said to be in the "normal state," this 

 state being analogous to the normal state previously found for a conservative 

 gas. 



223. It has already been assumed that the system possesses an infinite 

 number of degrees of freedom. The application of the results must therefore 

 be to " Statistical Mechanics," rather than to dynamical problems of the 

 ordinary type. Leaving out of mention the infinitesimal probability of error, 

 it being understood, for reasons explained in 104, that this is inherent to 

 any problem of statistical mechanics, it may be said that we have found that 

 of our non-conservative systems, started from unknown configurations, all 

 will, by the time a sufficiently low value of the energy has been reached, be 

 represented in our generalised space by points on a single cluster or family of 

 clusters. 



In the case of a gas it is obvious that we must take the second alter- 

 native, and say " family of clusters," rather than " single cluster." For, as 

 has already been pointed out in 218, by interchanging the roles of any two 

 molecules we get two distinct clusters which are identical in so far as 

 the elements which are of importance in the dynamical problem are con- 

 cerned. In other ways, too, it is clear that we shall get further distinct 

 clusters which are identical as regards the important elements of the 

 problem. Let M be the number of clusters in any such family and S the 



