368 Dr. G. J. Stoney on the 



freedom ; but it is between certain quasi-groups of these 

 degrees of freedom that the partition takes place under the 

 Boltzmann-Maxwell Theorem. These quasi-groups, like the 

 momentoids which define them, are of the same number as the 

 degrees of freedom. 



In most cases, where the only forces intervening are inter- 

 actions between parts of the system, the energy can be ex- 

 pressed in the form required by the theorem. But this is 

 not the case when certain external forces * come into opera- 

 tion, as, for example, when the aether acts on molecules as 

 well as the molecules or parts of a molecule on one another. 



Nevertheless, the theorem is of value in the interpretation 

 of nature, because in many cases the aether intervenes some- 

 what as perturbating forces do in the case of the planets, 

 modifying but not annulling the dynamical condition which 

 would prevail if the sun's attraction alone exercised dominion 

 over them. 



On the other hand, it must be remembered that such a 

 vast number of molecular events are crowded together within 

 a duration that appears very short to us, that small effects 

 have superabundant opportunity of gradually accumulating 

 and becoming conspicuous, within a small fraction (e. g. 

 within the thousandth part) of one second of time. 



In estimating the average energy, the average may be 

 struck either over a great succession of events happening to 

 one molecule, or over what occurs simultaneously to a vast 

 number of molecules f . 



* Certain internal forces also : for example, such as manifest themselves 

 in a change of state or in a chemical reaction. The theorem does not 

 hold over intervals of time extending from an epoch before to an epoch 

 after catastrophes of this kind. 



t In Maxwell's and Boltzmann's investigations the averages have been 

 struck only in the second way, viz., over what occurs simultaneously to 

 a vast number of molecules ; but general dynamical considerations seem 

 to warrant our feeling sure that the average, if struck in the other way, 

 i. e. f over a great succession of events happening to one molecule, would 

 have the same value — on the understanding, of course, that the molecules 

 are alike, and that the condition of the gas, in the respects with which 

 the theorem is concerned, is a persistent condition. 



In fact, it then seems safe to assume that in the course of a sufficiently 

 long experience the coordinates and momenta of any one molecule will 

 have approximately assumed all the combinations of values that the 

 coordinates and momenta of any other molecule shall have had. 



This is the same kind of assumption as the familiar assumption that if 

 n dice (n being a sufficiently large number) were thrown simultaneously, 

 sixes would turn up on a number of these dice which bears to the number 

 of times sixes would turn up on one die thrown n times, a ratio which 

 may be treated as a ratio of equality. 



