Conditions necessary for Equipartition of Energy. 135 



which each system passes do not include all phases consistent 

 with energy E, the law of equipartition is not proved by this 

 method at all. 



We come then to Boltzmann's proof. Here Mr. Jeans 

 begins by saying " we must make Boltzmann's assumption 

 that the gas is in a i molekular-ungeordnet ' state/' I do not 

 see the use of making it unless we can reason from it, and 

 that we cannot do till we know what it means. It is possible 

 to believe what we do not understand. It is not possible to 

 reason about what we do not understand. Now Boltzmann 

 gives us no adequate explanation of " molekular-ungeordnet." 

 Nor does Mr. Jeans. Boltzmann makes no use of the assump- 

 tion in argument. Nor does Mr. Jeans. Unless, indeed, 

 the very definite assumption which they both (as I think) 

 make, and which is stated below, is to be taken as the inter- 

 pretation of " molekular-ungeordnet." Boltzmann assumes, 

 namely, the independence of the molecular motions. If 

 FdTJdVdW be the chance that molecule M shall have velo- 

 cities U . . . JJ-rdTJ &c, and/ dudvdw the chance that molecule 

 m shall have velocities u . . .u + du &c. ; then he assumes that 

 the chance of the two events happening is FfdXJ . . . die. Jn 

 other words, the chances are independent at every instant, 

 and however near M and m be to each other. It is assumed, 

 indeed, only when they are very near to each other, namely, 

 at the point of an encounter. It is right to say that the 

 assumption may be qualified as relating only to molecules 

 approaching encounter. The condition thus assumed I call 

 Condition A. 



I understand Mr. Jeans to make precisely the same assump- 

 tion in his equation (18), using pp' as Boltzmann uses F/\ 



If that assumption be made, the law of equipartition follows 

 from it without any use, or any further use, of 6t molekular 

 ungeordnet." 



If that assumption be, as I maintain, untrue, the product F, 

 cannot be used as Boltzmann uses it, and the reasoning fails 

 notwithstanding the." molekular-ungeordnet " hypothesis. 



I think therefore that the one sufficient and necessary 

 condition for equipartition of energy is that Condition A shall 

 exist at every instant ; neither more nor less than this, nor any 

 different thing. 



I have said that Mr. Jeans's proof rests on the same 

 assumption as Boltzmann's; and must therefore, like Boltz- 

 mann '>. stand or fall with the truth of that assumption. It 

 may nevertheless be that a new proof founded on the same 

 assumption is valuable. 



