616 Mr. J. H. Jeans on the 



try to calculate the average value of H throughout the motion 

 o£ the gas. Since nothing is known ahout the initial con- 

 figuration, nothing is known as to which trajectory the 

 representative point is describing. Thus, relatively to the 

 knowledge which we possess, and to our arbitrary basis of 

 probability, the representative point is at any instant equally 

 likely to be at any point of our generalized space for which 

 B has the assigned value. Hence the expectation of the 

 average value of H throughout the motion (lasting through 

 any interval we please) is exactly the same as the average 

 value of H throughout the whole of the region for which E 

 has the assigned value, and therefore is equal to the minimum 

 value of H throughout this region. It follows that all the 

 physical properties of the gas (i.e., properties which depend, 

 firstly, only on the statistical law of distribution, and not on 

 the individual molecules, and, secondly, only on the values 

 integrated through an interval of time, and not on values at 

 any single instant, e. g., the pressure of the gas) may be 

 calculated on the supposition that the gas is in the normal 

 state throughout. 



This must be the interpretation of the second theorem of 

 § 24. There is a theoretical possibility of failure, for it is 

 possible (although infinitely improbable) that the value of H 

 may differ from its minimum value by a finite amount through 

 the whole of a stream-line. This question will be continued 

 in a later section (§ 38). 



§ 32. We now examine the third proposition of § 24. If 

 we consider all the points on the various stream-lines for a 

 given value of e, which have a value of H, say H7, different 

 from the minimum, we see (from the proposition analogous 

 to (iii.) of § 24) that only for an infinitesimal fraction of 

 these can H increase in either direction to a value of H which 

 is greater by a finite amount than J3/. Thus of the points 

 for which H = H', this value of H is a maximum on the par- 

 ticular stream-line to which it belongs, for all except an 

 infmitesimally small proportion of these stream-lines. Hence 

 if we start a gas in any configuration for which H is greater 

 than its minimum value, it is infinitely probable that in the 

 initial motion dH/dt will be zero or negative. 



Here we have the solution of the apparent paradox men- 

 tioned in § 17. There is no real irreversibility in the motion 

 of the gas on the whole, but there is an apparent irreversi- 

 bility if we start from a point at which H is different from 

 its minimum value. 



