February 2, 1906.] 



SCIENCE. 



167 



application of the calculus of probabilities 

 to them, as made by Maxwell, may not lead 

 to an erroneous result. To assert that this 

 is so requires proof. 



An alternative method of dealing with 

 the theory of gases, of which the latest 

 development is due to Jeans, proceeds by 

 treating a gas as a single dynamical system, 

 specified by the positional coordinates and 

 the component velocities of its molecules. 

 "We then, to use Jeans 's words, consider 

 'an infinite number of systems, starting 

 from every conceivable configuration, and 

 moving over every path; and investigate, 

 as far as possible, the motion of this series 

 of systems, in the hope of finding features 

 common to all.' Jeans carries out this in- 

 vestigation by representing each particular 

 phase of the system by a point in a general- 

 ized space. The successive phases through 

 which one of the systems will pass will be 

 represented by the points along a line in 

 the generalized space. Jeans shows that 

 these lines will be 'stream lines' in the 

 space; and that to investigate the infinite 

 number of systems already supposed comes 

 to the same thing as to suppose the gen- 

 eralized space filled with a fluid, moving 

 along stream lines determined by the dy- 

 namical equations of the gas, and to in- 

 vestigate the motion of this fluid. This 

 motion is found to be a 'steady motion.' 

 The advantage of this mode of procedure 

 is that the applications of the calculus of 

 probabilities are made to the elements of 

 the generalized space, and are obviously 

 legitimate. 



By an argument based on this funda- 

 mental mode of representation Jeans shows 

 that all but a negligibly small fraction of 

 the generalized space represents systems in 

 which the density of the gas is uniform; 

 and that within that part of the generalized 

 space which represents states of the system 

 in which the energy is constant, all but an 

 infinitely small fraction represents systems 



in which the velocities are distributed ac- 

 cording to Maxwell's exponential law. 

 Such a state of the system Jeans calls the 

 'normal state,' and his conclusion is that 

 it is infinitely probable that a gas in or- 

 dinary circumstances will be in the normal 

 state. 



We may consider this result as an a pos- 

 teriori proof of the hypothesis of molecular 

 chaos. 



By a treatment that is essentially similar, 

 and without using the hypothesis of molec- 

 ular chaos, it follows that it is infinitely 

 probable that the energy of the gas is, on 

 the average, distributed equally among the 

 degrees of freedom corresponding to mole- 

 cules of different types, and also among the 

 degrees of freedom of each type considered 

 independently. 



When Maxwell attempted to prove the 

 law of equipartition for the time averages 

 of the energy associated with the different 

 degrees of freedom of a single system, he 

 introduced the hypothesis that the system 

 goes through all possible phases, consistent 

 with the conservation of energy, before 

 returning to its initial phase. It is diificult 

 to see how this can be the case in a system 

 self-contained and entirely subject to the 

 laws of dynamics, such as, for example, a 

 gas within the smooth envelope ordinarily 

 postulated in the theory. But if we think 

 of the envelope as itself an assemblage of 

 moving molecules, it may be that the 

 courses of the gas molecules will be changed 

 so irregularly by impacts at the boundaries 

 that the gas may be thought of as ex- 

 periencing so many fortuitoiis disturbances 

 that it will practically fulfil the condition 

 of passing through all possible phases. In 

 this case the law of equipartition can be 

 extended to the time average of any one 

 degree of freedom. 



Lord Kelvin has taken exception to this 

 form of the theorem of equipartition, as- 

 serting that the proof is invalid, and that 



