606 Mr. J. H. Jeans on tli 



certain class of systems, whereas it has just been seen that 

 the result arrived at is only true upon the understanding that 

 all conceivable systems — cieordnet as well as unaeordnet — are 

 included. It would therefore appear that the effect of this 

 assumption is simply to defeat its own ends : it is brought in 

 ostensibly to make a certain calculation of probability 

 legitimate; whereas in point of fact the calculation is illegi- 

 timate (or at any rate cannot be shown to be legitimate) if 

 we adhere to the limitations of the " ungeordnet " assumption, 

 and becomes legitimate as soon as these limitations are 

 ignored. 



The calculations which have just been given must, in a 

 logically perfect kinetic theory, replace the " molekular- 

 ungeordnet " assumption as the justification for treating the 

 possibilities of two molecules having given coordinates as 

 independent events. The fact that the motion of the fluid in 

 our generalized space is steady motion must supply the 

 further justification for supposing these probabilities to 

 remain independent throughout all time. This latter result, 

 it will be noticed, rests on Liouville's theorem, and this in 

 turn rests upon the conservation of energy. There is no 

 justification for supposing probabilities to remain independent 

 when the gas is not a conservative system. It must be noticed 

 that all our results are true only with reference to our arbi- 

 trarily chosen basis of probability. 



§ 16. We have seen that the ; ' ungeordnet " assumption 

 leads to accurate results as regards frequency of collisions; 

 and hence we infer that all results which depend upon this 

 result and upon the dynamics of collisions will be accurate. 

 The results must, however, be interpreted in a special way. 

 Take, for instance, the H-theorem which deals with a special 

 law of distribution, say/. The theorem must not be taken 

 to prove that dH/dt is negative for all possible systems 

 corresponding to the given /, but that the expectation of the 

 value of dH/c/t for a system selected at random from all 

 systems having this / is negative, in other words that the value 

 of dH/dt averaged over all systems having this given / is 

 negative. There is no justification for confining the theorem 

 to " ungeordnet " systems, and none for stating the proposition 

 to be true of individual systems : it is, so to speak, only true 

 on the average. 



§ 17. We are here confronted with a paradox. For each 

 system for which dH/dt is negative, there will be a second 

 system — the image of the former (§ 7) — for which dH/dt 

 will have an equal but positive value. These two systems 

 must of course be equally included in the average, and since 



