72 REPORT— 1894. 



an assumption which is convenient but not essential, because ® can only 

 be a function of the generalised co-ordinates, and the investigation applies 

 only to systems in a gwen configuration for which these co-ordinates are 

 therefore constant, Boltzmann supposes with Maxwell that the energy 

 is the same for all the systems, and with these premises he proves that the 

 mean value of each of the terms 



has the same value. He concludes : ' Instead of the law of Maxwell that 

 the mean vis viva has the same value for every co-ordinate, we now obtain 

 the law that the mean value of the vis viva belonging to all momentoids 

 is the same.' 



From this Boltzmann concludes that ' the mean kinetic energies of 

 two given parts of the system are in the ratio of their respective degrees 

 of freedom,' j^fovided that the kinetic energy contains no j^Toducts of a 

 generalised momentum of one of the given 2Mrts into a generalised momentum 

 not belonging to that part. 



22. Now in Appendix A I have shown that for non-colliding rigid bodies 

 laws of permanent distribution exist in which the mean kinetic energies 

 due to the rotations about the three principal axes are unequal. This 

 test case shows, therefore, that Maxwell's result is not always true for et'er?/ 

 possible law of distribution. The following method is shorter than 

 Boltzmann's and Maxwell's, and shows under what circumstances the mean 

 kinetic energies belonging to the momentoids may be unequal. 



Let the law of distribution be given by the formula (9), 



f(E)dpi . . . dp„dq^ . . . dq„, 



and let ^'i, ^'2, . . . k,, be w linear functions of ^j,, 2^2, ■ ■ ■ 2^m such that the 

 kinetic energy is of the form 



T=i(A,2.fA-,2+ . . . ^..2) . . . (11) 

 Then, since these functions are linear, the determinant 



®: 



d (Ai, A'2 . . . k„ 

 d{p^,2H . . . Pn) 



is a function of the co-ordinates g,, . . . q,^ alone, and since we are dealing 

 only with those systems which happen to be in a given configuration at 

 the instant considered, ® is constant, the potential energy V is constant, 

 and so is the nmltiple differential dq^ . . . dq„. The mean value of ^k,.^ 

 for these systems is therefore 



__ \f{h{^i' + h'+ • • ■ k„^) + Y\U,^d2}, . . . dp„ 

 11. 2 J -<= 



2 A,. 



[7{K^,HV+ • . • kn^} + y}dp, t?;.„ 



J —CO 



7{1(^,2+ . . . +^.2) + Y|p_2^^._ . , , ^J,^^ 



J' '60 



/{h{k,'+ . .. +k,,-^) + Y}dk, dk. 



