50 The Law of Distribution [CH. in 



except for an infinitesimal probability of error, a system selected at random 

 from the generalised space will be in the state specified by equation (102). 

 It will be convenient to refer to this state as the " Normal State." And 

 when a result is certain except for an infinitesimal probability of error, it will 

 be convenient to speak of the result as "infinitely probable." 



If, therefore, a system is selected at random, it is infinitely probable that 

 it will be in the " Normal State." Suppose that a system is selected at 

 random, and then allowed to move under its natural motion for a time t, 

 what do we know now as to its probable state after a time t ? The answer is 

 provided by the theorem proved in 37. The motion of all possible systems 

 is represented in our generalised space by the motion of the supposed fluid. 

 Instead of selecting a system at random and allowing it to move for time t, 

 we may allow the whole fluid in the generalised space to move for a time t, 

 and select a system at random after the motion has proceeded for a time t. 

 The theorem of 3T proved the motion of the fluid in the generalised space 

 to be a " steady motion." Hence selecting a system after time t is the same 

 thing as selecting a system at time 0, and it is infinitely probable that the 

 system thus selected will be in a normal state. 



59. This completes our information about the motion of the gas. At 

 any instant it is infinitely probable that it is in a normal state. In the 

 course of the motion departures from the normal state will occur, but it is 

 infinitely probable that these will only occupy an infinitesimal fraction of the 

 time occupied by the whole motion. 



There is in theory a possibility of a gas continuing throughout its motion 

 in a state different from the normal. Suppose for instance that the contain- 

 ing vessel is cubical, and that the molecules are started so that all move 

 perpendicular to one edge along a system of parallel lines, no two of which 

 are at a less distance than the diameter of a molecule. Then it is obvious 

 that the molecules will not leave the lines on which they start, and will not 

 change their velocities. In this case any law of velocities /(w, 0, 0) will be 

 permanent, where u is the velocity in the direction of the parallel lines. 



The analysis of this chapter breaks down, it will be seen, because the 

 supposition made in 38 is no longer true, that there are no functions of the 

 coordinates in the generalised space except the energy which remain constant 

 through the motion. For obviously we have u^ = constant, u b 2 = constant, 

 etc., and v a = w a = 0, vj ) = Wb= 0, etc. 



Our results shew that it is infinitely probable that a system selected 

 at random will not be of this special type. The connection between the 

 trajectories of such systems and the " periodic orbits " of abstract dynamics 

 is interesting, but cannot be discussed here. We shall return to the dis- 

 cussion of cases in which there are constants other than the energy in 

 Chapter V. 



