Mr. J. 11. Jeans' TlLeory of Gases. 531 



A is untrue, iu^ fact impossible, goes on to i)rov(' that 

 Maxwell's law of distribution, criveii in complete I'orm in 

 art. 3^, is true, that is, that the chance of the system of 

 N molecules having velocities u ..u + dn kc, and coordinates 

 X ....c-\-chv kc. is 



in which ^ i^ the potential of external forces (intermolecular 

 forces are not here considered). But from this result it 

 immediately follows that assumption A, which we have just 

 proved to be false, is true. The above formula (34) is the 

 concise mathematical expression of it. MaxwelFs law and 

 assumption A can no more exist one without the other than 

 head and tail of the same living animal. 



6. The explanation of the contradiction is Jeans' assump- 

 tion, art. 37 {d), '' that at any instant that part of the total 

 energy which is accounted for by the intermolecular forces 

 forms an infinitesimal fraction of the whole." For that 

 implies that the gas considered is an infinitely rare gas. Now 

 3IaxwelF s law is, as I maintain, never exactly true ; but we 

 can make it approximate to the truth as nearly as we please 

 by diminishing the density. If the law be not as Maxwell 

 says A^-^2:m«2^ -^ ^.^^ ^^ Ae-'^Q, in which 



Q = aiUi + hy2UiU2 H- ^2^2" + &c., 



a complete quadratic function of the velocities. But the 

 coefficient h of products of the velocities in this quadratic 

 function diminishes as the density diminishes, and as we 

 approach the infinitely rare gas becomes negligible. There- 

 fore in the only case contemplated in Jeans'* (37) MaxwelFs 

 law, though not accurately, is approximately true. By 

 consequence its satellite the law of equipartition of energy 

 is approximately true ; and so is assumption A. That is the 

 explanation of Jeans' apparent self-contradiction. 



7. Jeans' case (37) is a limiting case. The study of limiting- 

 cases is always instructive, and may be important. It is 

 exceptionally important in the theory of gases, because air 

 under ordinary conditions is supposed to approximate to the 

 state of the infinitely rare gas. Jf, however, we are always 

 to assume the infinite rarity of our gas, we are like the 

 early navigators who never went out of sight of land. They 

 also with their imperfect means obtained imjjortant results, 

 but they did not obtain a complete science of navigation. 



8. The difference between Boltzmann and Jeans is one of 

 method only. Boltzmann uses the ordinary Cartesian coor- 

 dinates, .1". y, z denoting the coordinates of the centre of a 



2M2 



